Research Article Multimodal Processes Rescheduling: Cyclic...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 407096 24 pageshttpdxdoiorg1011552013407096

Research ArticleMultimodal Processes Rescheduling Cyclic Steady StatesSpace Approach

Grzegorz Bocewicz1 Robert Woacutejcik2 Zbigniew Antoni Banaszak3 and PaweB Pawlewski4

1 Department of Electronics and Computer Science Koszalin University of Technology Koszalin Poland2 Institute of Computer Engineering Control and Robotics Wrocław University of Technology Wrocław Poland3Department of Business Informatics Warsaw University of Technology Warsaw Poland4 Faculty of Engineering Management Poznan University of Technology Poznan Poland

Correspondence should be addressed to Grzegorz Bocewicz bocewiczietukoszalinpl

Received 29 March 2013 Accepted 22 September 2013

Academic Editor Zhichun Yang

Copyright copy 2013 Grzegorz Bocewicz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The paper concerns cyclic scheduling problems arising in aMultimodal TransportationNetwork (MTN) in which several unimodalnetworks (AGVs hoists lifts etc) interact with each other via common shared workstations as to provide a variety of demand-responsive material handling operations The material handling transport modes provide movement of work-pieces betweenworkstations along their manufacturing routes in the MTN The goal is to provide a declarative framework enabling to state aconstraint satisfaction problem aimed at AGVs fleet match-up scheduling taking into consideration assumed itineraries of concur-rently manufactured product types In that context treating the different product types as a set of cyclic multimodal processes isthe main objective to discuss the conditions sufficient for FMS rescheduling imposed by production orders changes To concludethe conditions sufficient for an FMS rescheduling imposed by changes of production orders treated as cyclic multimodal processesare stated as the paperrsquos main contribution

1 Introduction

The design and operational issues arising in an AutomatedGuided Vehicle (AGV) served Flexible Manufacturing Sys-tem (FMS)when deciding on how to achieve a desired systemperformance are usually hard to determine and evaluateThisis because productivity of an AGV-served flow shop produc-ing a set of different kinds of products repetitively dependson both the job flow sequencing (aimed at maximizingthroughput or minimizing cycle time) and the material han-dling system required to achieve a prespecified throughputthat is AGVs fleet sizing assignment and scheduling [1ndash5]So assuming a deterministic systemwhere demand is knownin advance and the processing times of each job on eachmachine is a known constant in order to improve its pro-ductivity the AGV fleet size and the schedulingdispatchinginfluence on the system throughput are usually examined[1 6ndash9]

In opposite to conventionally applied approach assumingAGVs fleet sizing routing and timetabling while match-ing up prescheduled multiproduct manufacturing flow (ieproviding detailed transportation route and materials loadunload schedule between workstations in FMS) we pro-pose AGVs following a Multimodal Transportation Network(MTN) concept MTN can be seen as composition of severalunimodal networks (AGVs hoists lifts etc) interacting witheach other via common shared workstations as to provide avariety of demand-responsive material handling operationsThematerial handling transportmodes providemovement ofwork-pieces betweenworkstations along theirmanufacturingroutes in the MTN The MTN encompasses a well-knownmultimodal approach to freight transportation supply chainsand passenger transportation systems So similarly to suchsystems having ability to use the appropriate mode of trans-port formovement of goods andpeople the assumedmaterial

2 Mathematical Problems in Engineering

handling transport modes provide movement of work-piecesbetween workstations along their manufacturing routes inthe MTN In that context assuming that the modes can betreated as cyclic local transportation processes and material(eg work-piecestools) flows supported by them can betreated as multimodal processes the problems consideredcan be seen either as aimed at designing of a multimodalnetwork infrastructure guaranteeing assumed materials flowor focused onwork-pieces routing ensuring lag-free worksta-tions service

The problems arising concerning material transportationrouting and scheduling belong to NP-hard ones [2] Since thesteady state of production flows has a cyclic character henceservicing them AGV-served transportation processes (usu-ally executed along loop-like routes) encompasses also cyclicbehavior That means that the periodicity of FMS [10]depends on both the periodicity of production flow cyclicschedule and following this schedule the AGVs periodicityOf course the Multimodal Transportation Network (MTN)throughput is maximized by the minimization of its cycletime

It seems to be obvious that not all the behaviors (includingcyclic ones) are reachable under constraints imposed bythe systemrsquos structure The similar observation concerns thesystemrsquos behavior that can be achieved in systems possessingspecific structural constraints Since system constraints deter-mine its behavior both the system structure and the desiredcyclic schedule have to be considered simultaneously So theproblem solution requires that the system structure mustbe determined for the purpose of processes scheduling yetscheduling must be done to devise the system configurationIn that context our contribution provides a discussionof some solvability issues concerning cyclic processes dis-patching problems especially the conditions guaranteeingsolvability of the cyclic processes scheduling as well as directandor indirect transitions between assumed cyclic steadystatesTheir examinationmay replace exhaustive searches forsolution satisfying required system capabilities

Many models and methods have been considered to date[11] Among them the mathematical programming approach[1 12] max-plus algebra [13] constraint logic programming[14ndash16] evolutionary algorithms Petri nets [17 18] andheuristic frameworks [19] belong to themore frequently usedMost of these are oriented at finding a minimal cycle ormaximal throughputwhile assuming deadlock-free processesflow Note that processesrsquo operations are blocking if theymuststay on a resource (eg the station the machine) afterfinishing when the next resource is occupied by a job fromanother process During this stay the resource is blocked forother processes The approaches trying to estimate the cycletime from cyclic processes structure and the synchroniza-tion mechanism employed (ie mutual exclusion instances)while taking into account deadlock phenomena are quiteunique

In our approach a declarative framework aimed at refine-ment and prototyping of the cyclic steady states for concur-rently executed cyclic processes modelling material handlingsystems is employed These systems are of the type of AGVsfleets in the flexible manufacturing systems (FMS) and are

frequently encountered in industry The following questionsare the main focus of the research Can the assumed materialhandling system for example AGVs meet loadunloaddeadlines imposed by flow of scheduled work-pieces process-ing Does there exist sufficient AGVS enabling to schedulethe AGVs fleet as to ensure lag-free service of scheduled workpieces processing So the main question is Can the assumedAGVs fleet assignment reach its goal subject to constraintsassumed on concurrent multiproduct manufacturing athand

Moreover the declarative framework enables to state aproblem of multimodal processes rescheduling which boilsdown to searching for transient periods allowing the mutualreachability of MTN cyclic behaviors In the case of MTNsdistinguishing many different cyclic steady states (cyclicbehaviors) the following questions play a pivotal role Doesthere exist the smooth (direct) transition between twoassumed cyclic behaviors What conditions guarantee thereachability of a given cyclic behavior from any other onesIs it possible to come back to a given cyclic behavior fromany disturbance-born one for instance caused by operationtime delays damage of devices route modifications and soforth

In other words the paperrsquos objective concerns the MTNinfrastructure assessment from the perspective of possibleFMS oriented requirements imposed by AGVs fleet assign-ment sizing and scheduling (problems of cyclic processesscheduling and rescheduling) Due to the complexity impliedin answering the above questions the combined problemremains unsolved for all practical purposes This is especiallyproblematic as many manufacturing companies would standto reap considerable rewards through better fleet assign-ment and scheduling The presented approach addresses thisthrough solving the combined rather than the separate prob-lems individually

The rest of the paper is organized as follows Section 2introduces the Automated Guided Vehicles System (AGVS)modelled in terms of concurrently flowing cyclic processes(SCCPs) and a cyclic steady state space concept Section 3regarding multimodal processes rescheduling provides aproblem statement concerning AGVs fleet match-up sched-uling with an assumed multiproduct manufacturing flowschedule Sufficient conditions allowing one to search forstates with mutual allocation as well as illustrative exampleof implementing them algorithm usage are discussed Therelated work and concluding remarks are presented in Sec-tions 4 and 5 respectively

2 Multimodal Network Modeling

Considered case of multimodal processes rescheduling prob-lem is presented on the example of Automated Guided Vehi-cle Systems (AGVS) modeled by Systems of ConcurrentlyFlowing Cyclic Processes (SCCPs) To solve this kind of prob-lems a nonempty state space that is containing nonemptyset of possible cyclic behaviors is required For that reasonthe conditions guaranteeing cyclic behavior are presentedprimarily They are specified in terms of a declarative modelformalism distinguishing the structure and behavior as the

Mathematical Problems in Engineering 3

main components of an SCCP Relationship linking thestructure and reachable behaviors is the main subject of thissection

21 Model of AGVS Automated Guided Vehicle Systems areused for material handling within a Flexible ManufacturingSystem (FMS) and provide asynchronous movement palletsof products through a network of guide paths between theworkstations by the AGVs Each workstation is connected tothe guide path network by a pick-updelivery station wherepallets are transferred fromto the AGVs

In AGVS literature most are related to AGVS designissues which include determination of the number of vehiclesrequired flow path design and route planning as well asvehicle dispatching and traffic management Recently anintegrated problem of dispatching and conflict free routingof AGVs that is integrating the simultaneous assignmentscheduling and conflict free routing of the vehicles is receiv-ing increasing attention The above mentioned problems inthe general case belong to the class of NP-hard problemsSince most processes observed in steady state manufacturingare periodic cyclic schedules and following them cyclicschedulingmethods can be considered Since cyclic schedulesencompass repetitive character of manufacturing processesthe cyclic processes modeling approach seems to be a reason-able perspective [13ndash15]

To present some of the design and operational issues thatarise in repetitive manufacturing systems served by AGVs innetwork of loops layout with unidirectional material flow acase example of a simple FMS is shown in Figure 1 The con-current multiproduct flows are depicted by bold green blueand orange color lines while the transportation loop-likenetworks are distinguished by double solid line Both kindsof material (jobs) and transportation (AGVs) flows shown inFigure 1 can be modeled in terms of Systems of ConcurrentlyFlowing Cyclic Processes [14 15] as shown in Figure 2 Inturn the SCCP framework provides a formal model enablingto state and resolve problems of AGV fleet size minimizationas well as steady state cycle timeminimizationThe cycle timeminimization is required to obtain maximum throughputrate

Eight local cyclic processes are considered namely 1198751

1198752 119875

3 119875

4 119875

5 119875

6 119875

7 and 119875

8 The processes follow the

routes are composed of transportation sectors and machines(distinguished in Figure 2 by the set of resources 119877 =

1198771 119877

119888 119877

33 119877

119888mdashthe 119888th resource) Some of the local

cyclic processes are pipeline flow processes that is theycontain streams (representing vehicles from Figure 1) of theprocesses following the same route while occupying differentresources (sectors) For instance processes 119875

4 119875

5 and 119875

6

contain two streams 1198754= 119875

1

4 119875

2

4 119875

5= 119875

1

5 119875

2

5 and 119875

6=

1198751

6 119875

2

6 respectively and 119875

2contains three streams 119875

2=

1198751

2 119875

2

2 119875

3

2 that is the processes (vehicles) moving along the

same route The remaining local processes contain uniquestreams 119875

1= 119875

1

1 119875

3= 119875

1

3 119875

7= 119875

1

7 and 119875

8= 119875

1

8

In other words the streams 1198751

1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5

1198751

6 1198752

6 1198751

7 and 1198751

8represent the 13 vehicles from Figure 1The

119896th stream of the 119894th local process 119875119894is denoted by 119875119896

119894

Apart from local processes we consider threemultimodalprocesses (ie processes executed along the routes consistingof parts of the routes of local processes)119898119875

1119898119875

2 and119898119875

3

For example the production route depicted by the orangeline corresponds to the multimodal process 119898119875

1supported

by AGVs (arbitrarily given) which in turn encompass localtransportation streams 1198751

1 1198751

2 1198753

2 and 1198752

4 This means that

the production route specifying how a multimodal process isexecuted can be considered to be composed of parts of theroutes of local cyclic processes

In the system considered each multimodal process con-sists of four streams 119898119875

119894= 119898119875

1

119894 119898119875

2

119894 119898119875

3

119894 119898119875

4

119894 119894 =

1 2 3 which means that along each production route fourwork- pieces are processed serially (four balls on the oneproduction lines in Figure 1)

Processes can interact with each other through sharedresources that is the transportation sectors The routes ofthe considered local processes (streams) are as follows (seeFigure 2)

1199011

1= (119877

16 119877

3 119877

13 119877

6)

1199011

2= (119877

21 119877

8 119877

18 119877

9 119877

14 119877

6 119877

17 119877

5)

1199012

2= (119877

17 119877

5 119877

21 119877

8 119877

18 119877

9 119877

14 119877

6)

1199013

2= (119877

14 119877

6 119877

17 119877

5 119877

21 119877

8 119877

18 119877

9)

1199011

3= (119877

12 119877

19 119877

9 119877

5)

1199011

4= (119877

29 119877

7 119877

25 119877

8 119877

22 119877

11 119877

26 119877

10)

1199012

4= (119877

26 119877

10 119877

29 119877

7 119877

25 119877

8 119877

22 119877

11)

1199011

5= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

27 119877

1 119877

23)

1199012

5= (119877

27 119877

1 119877

23 119877

2 119877

20 119877

5 119877

24 119877

4)

1199011

6= (119877

32 119877

4 119877

28 119877

7 119877

31)

1199012

6= (119877

31 119877

32 119877

4 119877

28 119877

7)

1199011

7= (119877

33 119877

7 119877

29 119877

10)

1199011

8= (119877

1 119877

27 119877

4 119877

30)

(1)

where 1198771minus119877

2 119877

4minus119877

11 119877

14 119877

17minus119877

18 119877

20minus119877

29 119877

31 and 119877

32

(due to two streams of process1198756 resources119877

31119877

32represent

one sector) are resources shared by local processes that is byat least two streams and 119877

3 119877

12 119877

13 119877

15 119877

16 119877

19 119877

30 and

11987733

are the nonshared resources that is exclusively used byone stream only

In the general case the route 119901119896119894is the sequence of

resources used in order to execute the operations of thestream 119875119896

119894 Note that the streams 1198751

2 1198752

2 and 1198753

2which belong

to 1198752and the streams of processes 119875

4 119875

5 and 119875

6follow the

same route but start from different resources (these streamscorrespond to vehicles moving along the same route)


Input resources Output resources

mP3

mP2

mP1

R1 R4 R7 R10

R11

R12

R2

R3R6 R9

R8R5

P5

P8 P6

P2

P3P1

P7

P4

1

1

2

2

3

3

4

4

1

2

3

4

1

3

1

1

1

1 11

1

2

2

2

2

Pi j - AGV modeled as the jth stream of local process Pi

q

q

- ith machine resource Ri

- Transportation sector - The qth work piece transported by the jth AGVwhile modeled as the jth stream of local processserviced by the qth stream of multimodal process

- Production routesmultimodal processes mPi - Work-piece modeled as the qth stream of

multimodal process mPi (orange process)

Figure 1 Example of an AGVS

Similarly the cyclic multimodal processes1198981198751119898119875

2 and

1198981198753follow the routes (see Figure 2) as follows

119898119901119896

1= (119877

3 119877

13 119877

6 119877

17 119877

5 119877

21 119877

8 119877

22 119877

11)

119896 = 1 4

119898119901119896

2= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

28 119877

7 119877

29 119877

10)

119896 = 1 4

119898119901119896

3= (119877

1 119877

27 119877

4 119877

28 119877

7 119877

25 119877

8 119877

18 119877

9 119877

15 119877

12)

119896 = 1 4

(2)

Let us assume that1198981199011198961119898119901119896

2 and119898119901119896

3can be seen as follows

(see Figure 2)

119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8)

(11987718 119877

9) (119877

15 119877

12)) 119896 = 1 4

(3)

where one has the following (1198773 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8)

and (11987722 119877

11)mdashsubsequences of routes 1199011

1 1199011

2 1199013

2 and 1199012

4

defining the transportation sections of 1198981199011198961 (119877

2 119877

20 119877

5)

(11987724 119877

4) (119877

28 119877

7) and (119877

29 119877

10)mdashsubsequences of routes

1199011

5 1199012

5 1199011

6 and 1199011

7defining the transportation sections of

119898119901119896

2 and (119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9) and

(11987715 119877


8 1199012

6 1199011

4 1199012

2 and 1199011

3

defining the transportation sections of119898119901119896

3

Let us assume also that the considered SCCPs follow theconstraints stated below [15]


Dispatching rules Θ0

12059001 (P25 P

18 P

15 ) 120590012 (P1

3 ) 120590023 (P25 P

15 )

12059002 (P15 P

25 ) 120590013 (P1

1 ) 120590024 (P15 P

25 )

12059003 (P11 ) 120590014 (P3

2 P12 P

22 ) 120590025 (P1

4 P24 )

12059004 (P16 P

18 P

26 P

15 P

25 ) 120590015 (P1

3 ) 120590026 (P24 P

14 )

12059005 (P22 P

32 P

15 P

25 P

12 ) 120590016 (P1

1 ) 120590027 (P25 P

18 P

15 )

12059006 (P32 P

12 P

11 P

22 ) 120590017 (P2

2 P32 P

12 ) 120590028 (P1

6 P26 )

12059007 (P14 P

24 P

16 P

17 P

26 ) 120590018 (P1

2 P22 P

32 ) 120590029 (P1

4 P24 P

17 )

12059008 (P12 P

14 P

22 P

32 P

24 ) 120590019 (P1

3 ) 120590030 (P18 )

12059009 (P12 P

22 P

13 P

32 ) 120590020 (P1

5 P25 ) 120590031 (P2

6 P16 )

120590010 (P24 P

17 P

14 ) 120590021 (P1

2 P22 P

32 ) 120590032 (P1

6 P26 )

120590011 (P14 P

24 ) 120590022 (P1

4 P24 ) 120590033 (P1

7 )

12059011 (mP13 mP2

3 mP33 mP4

3) 120590112 (mP33 mP4

3 mP13 mP2

3)120590123

12059012 (mP12 mP2

2 mP32 mP4

2) 120590113 (mP11 mP2

1 mP31 mP4

1) 120590124 (mP12 mP2

2 mP32 mP4

2)

12059013 (mP11 mP2

1 mP31 mP4

1) 120590114 120590125 (mP33 mP4

3 mP13 mP2

3)

12059014(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)120590115 (mP3

3 mP43 mP1

3 mP23) 120590126

12059015(mP3

1 mP42 mP4

1 mP12

mP11 mP2

2 mP21 mP3

2)120590116 120590127 (mP1

3 mP23 mP3

3 mP43)

12059016 (mP11 mP2

1 mP31 mP4

1) 120590117 (mP11 mP2

1 mP31 mP4

1) 120590128(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)

12059017(mP3

2 mP43 mP4

2 mP13

mP12 mP2

3 mP22 mP3

3)120590118

(mP33 mP4

3 mP13 mP2

3) 120590129 (mP32 mP4

2 mP12 mP2

2)

12059018(mP3

3 mP31 mP4

3 mP41

mP13 mP1

1 mP23 mP2

1)120590119 120590130

12059019(mP3

3 mP43 mP1

3 mP23) 120590120 (mP1

2 mP22 mP3

2 mP42) 120590131

120590110 (mP32 mP4

2 mP12 mP2

2) 120590121 (mP31 mP4

1 mP11 mP2

1) 120590132

120590111 (mP31 mP4

1 mP11 mP2

1) 120590122 (mP31 mP4

1 mP11 mP2

1) 120590133


4 3 2 1

- ith resource with dispatching rules 12059001 12059011

Ri

(1205900i 1205901i )

Ri - ith resource representing transportation sector

Pi - ith local process

- Multimodal processes streams

R6

R1

R12

R10

R11

R9

R7

R8

R4

R5R2

mP1

mP2

mP3

P2

R13 R14 R15

R16

R17 R18

R26R25R24R23

R20 R21 R22

R27 R28 R29

R19

R31

P1 P3

P5

P8

R30 R32

P6

P4

P7

R33

Sector

Machine

4 3 2 1

R3

(12059001 12059011)

(12059002 12059012)

(12059003 12059013)

(12059006 12059016)

(12059009 12059019) (120590012 120590

112)

(120590011 120590111)(12059008 120590

18)(12059005 120590

15)

(12059004 12059014) (12059007 120590

17)

(120590010 120590110)

- Multimodal processes mPi

empty

empty

emptyempty

empty

empty

empty

empty

empty

Figure 2 Example of FMS-SCCP model of an AGVS

(i) a new subsequent operation of the local process maystart on a required resource only if the current oper-ation has been completed and the resource has beenreleased

(ii) each new consecutive operation in the multimodalprocess may start its execution on an assigned re-source only if the current operation has been com-pleted and the resource has been released and an


appropriate local process begins its consecutive oper-ation on this resource

(iii) localmultimodal processes share common resourcesin the mutual exclusion mode the operation of alocalmultimodal process can only be suspended ifthe necessary resource is occupied suspended pro-cesses cannot be released and processes are nonper-ceptible that is a resource may not be taken from aprocess as long as it is used by it

(iv) the SCCP resources can be shared by local and multi-modal processes as well as by both of them

(v) multimodal processes encompassing production flowconveyed by AGVs follow local transportation routes

(vi) different multimodal processes can be executed si-multaneously along the same local process

(vii) local and multimodal processes execute cyclicallywith periods 120572 and119898120572 respectively resources occuruniquely in each transportation route

(viii) in a cyclic steady state each 119894th stream must cover itslocal route the same number of times

A resource conflict (caused by the application of the mutualexclusion protocol) is resolved with the aid of a prioritydispatching rule [14ndash16] which determines the order inwhich streams access shared resources For instance in thecase of the resource 119877

5 the priority dispatching rule 1205900

5=

(1198752

2 119875

3

2 119875

1

5 119875

2

5 119875

1

2) determines the order in which streams of

local processes can access the shared resource 1198775 in the case

considered the stream 1198752

2is allowed to access as first then the

stream 1198753

2as next then streams 1198751

5 1198752

5 and 1198751

2 and then once

again 1198752

2 and so on The stream 119875

119896

119894occurs the same number

of times (in the considered system once) in each dispatchingrule associated with the resources featuring in its route (inthe case considered each stream 119875

119896

119894occurs uniquely) The

SCCP shown in Figure 2 is specified by the following set ofdispatching rules Θ = Θ

0 Θ

1 where Θ0

= 1205900

1 120590

0

33

is set of rules determining the orders of local processes andΘ

1= 120590

1

1 120590

1

33 is set of rules determining the orders of

multimodal processesIn general the following notation is used

(i) A sequence 119901119896119894= (119901

119896

1198941 119901

119896

1198942 119901

119896

119894119895 119901

119896

119894119897119903(119894)) spec-

ifies the route of the stream of a local process 119875119896

119894(the

119896th stream of the 119894th local process 119875119894) Its compo-

nents define the resources used in the execution ofoperations where 119901119896

119894119895isin 119877 (the set of resources

119877 = 1198771 119877

2 119877

119888 119877

119898) denotes the resources

used by the 119896th stream of the 119894th local process inthe 119895th operation in the rest of the paper the 119895thoperation executed on the resource 119901119896

119894119895in the stream

119875119896

119894will be denoted by 119900119896

119894119895 119897119903(119894) is the length of the

cyclic process route (all streams of 119875119894are of the same

length) For example the route 11990111= (119877

16 119877

3 119877

13 119877

6)

of the stream of process 1198751(Figure 2) is the sequence

1199011

1= (119901

1

11 119901

1

12 119901

1

13 119901

1

14) where the first element 1199011

11

is equal to 11987716 whereas the second 1199011

12= 119877

3 and so

on 119901113= 119877

13 1199011

14= 119877

6

(ii) 119909119896119894119895(119897) isin N is the timing of commencement of opera-

tion 119900119896119894119895in the 119897th cycle

(iii) 119905119896119894= (119905

119896

1198941 119905

119896

1198942 119905

119896

119894119895 119905

119896

119894119897119903(119894)) specifies the operation

times of local processes where 119905119896119894119895denotes the time of

execution of operation 119900119896119894119895

(iv) 119898119901119896

119894= (119898119901119903

1199021

1198941

(1198861198941

1198871198941

) 1198981199011199031199022

1198942

(1198861198942

1198871198942

) 119898119901119903119902119910

119894119910

(119886119894119910

119887119894119910

)) specifies the route of the stream 119898119875119896

119894from the

multimodal process 119898119875119894(the 119896th stream of the 119894th

multimodal process119898119875119894) where

119898119901119903119902

119894(119886 119887)

=

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119887) 119886 le 119887

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119897119903(119894) 119901

119902

1198941 119901

119902

119894119887minus1 119901

119902

119894119887) 119886 gt 119887

119886 119887 isin 1 119897119903 (119894)

(4)

is the subsequence of the route 119901119902119894= (119901

119902

1198941 119901

119902

1198942

119901119902

119894119895 119901

119902

119894119897119903(119894)) containing elements from 119901

119902

119894119886to 119901119902

119894119887

In other words the transportation route 119898119901119894is a

sequence of parts of routes of local processes Forinstance the route followed by process 119898119875

1(see

Figure 2) is as follows 1198981199011198961

= ((1198773 119877

13 119877

6)

(11987717 119877

5) (119877

21 119877

8) (119877

22 119877

11)) where 1198981199011199031

1(2 4) =

(1198773 119877

13 119877

6) 1198981199011199031

2(7 8) = (119877

17 119877

5) 1198981199011199033

2(1 2) =

(11987721 119877

8)1198981199011199032

4(5 6) = (119877

22 119877

11)

(v) 119898119909119896119894119895(119897) isin N is the timing of commencement of oper-

ation119898119900119896119894119895in the 119897th cycle

(vi) 119898119905119896119894= (119898119905

119896

1198941 119898119905

119896

1198942 119898119905

119896

119894119895 119898119905

119896

119894119897119898(119894)) specifies

the operation times of multimodal processes where119898119905

119896

119894119895denotes the time of execution of operation119898119900119896

119894119895

and 119897119898(119894) is the length of the cyclic process route119898119901119896119894

(vii) Θ = Θ0 Θ

1 is the set of priority dispatching rules

Θ119894= 120590

119894

1 120590

119894

2 120590

119894

119888 120590

119894

119898 is the set of priority dis-

patching rules for local (119894 = 0)multimodal (119894 = 1)processes where 120590119894

119888= (119904

119894

1198881 119904

119894

119888119889 119904

119894

119888119897119901(119888)) are

sequence components which determine the order inwhich the processes can be executed on the resource119877119888 1199040

119888119889isin 119867 (where 119867 is the set of local streams

eg in the case presented in Figure 2 119867 = 1198751

1 119875

1

2

1198752

2 119875

3

2 119875

1

3 119875

1

4 119875

2

4 119875

1

5 119875

2

5 119875

1

6 119875

2

6 119875

1

7) and 1199041

119888119889isin 119898119867

(where 119898119867 is the set of multimodal streams eg incase from Figure 2 119898119867 = 119898119875

119896

1 119898119875

119896

2 119898119875

119896

3| 119896 =

1 sdot sdot sdot 4)

Using the above notation a SCCP can be defined as a tuple

SC = ((119877 SL) SM) (5)


where one has the following 119877 = 1198771 119877

2 119877

119888 119877

119898mdash

the set of resources 119898mdashthe number of resourcesSL = (119880 119879Θ

0)mdashthe structure of local processes that is

119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)

119899mdashthe set of routes of

local process 119897119904(119894)mdashthe number of streams belongingto the process 119875

119894 119899mdashthe number of local processes

119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)

119899mdashthe set of sequences of

operation times in local processes Θ0= 120590

0

1 120590

0

2 120590

0

119888

1205900

119898mdashthe set of priority dispatching rules for local

processes SM = (119872119898119879Θ1)mdashthe structure of multimodal

processes that is 119872 = 1198981199011

1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)

119908mdashthe set of routes of a multimodal process

119897119904119898(119894)mdashthe number of streams belonging to theprocess 119898119875

119894119908mdashthe number of multimodal processes

119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)

119908mdashthe set of

sequences of operation times in multimodal processes andΘ

1= 120590

1

1 120590

1

2 120590

1

119888 120590

1

119898mdashthe set of priority dispatching

rules for multimodal processesThe SCCP model (5) can be seen as a multilevel model

cf Figure 3 that is a model composed of an ldquo119877 levelrdquo(resources) an ldquoSL levelrdquo (local cyclic processes) and an ldquoSM1

levelrdquo (multimodal cyclic processes) as well as an ldquoSM119902 levelrdquo(the 119902th metamultimodal process) The SL level is definedby the transportation routes structure following the set 119880 oflocal processes and the set of parameters Θ determining therequired system behavior In turn the SM1 level takes intoaccount multimodal processes as well as metamultimodalprocesses (SM2 level) composed of multimodal processesfrom the SM1 level In other words it is assumed that thevariables describing SM119902 are the same as in the case of SMwhereas the routes of the multimodal process of the 119902th levelremain composed of the processes from the (119902 minus 1)th levelThe presented model is an extended version of a simplifiedmodel limited to 119877 and SL levels which is introduced in[15]

Therefore in general the SC = ((119877 SL) SM) model canbe seen as composed of 119897119901 levels as follows

SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)

The SC119897119901 model emphasizes structural characteristics of theSCCP modeled In turn the behavioral characteristics canbe specified in terms of the admissible states reachabilityconcept that is either taking into account the state spaceconcept including both cyclic steady states and leading tothem transient states [16] or the cyclic steady state space only[14 15]

The second way which does not take into account initialstates leading through the transient states to the cyclic steadystates seems to be quite close to the cyclic schedulingmethods widely used in many real-life cases [9 11]

In that context the relevant multilevel cyclic schedule119883119897119901

encompassing the SCCP behavior on each of its processes

level that is local SL and multimodal SM119897119901 processes can bedefined as follows119883

119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)

where one has the following 119883 120572(119898119902119883119898

119902120572)mdashsequence of

commencement of operations and periodicity of localmulti-modal (119894th level) processes executions

It should also be noted that the schedule 119883119897119901 can bedefined as a sequence of ordered pairs describing behaviorsof local (119883 120572) and multimodal (119898119894

119883119898119894120572) processes where

119883 is a sequence of timings 119909119896119894119895

(for 119897 = 0th cycle) of com-mencement of operations 119900119896

119894119895from streams119875119896

119894executed along

local processes 119883 = (1199091

11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))

consists of the timing of commencement of multimodalprocesses operations (from the 119902th level where one has thefollowing 119908(119902)mdashthe number of multimodal processes at the119902th level 119897119904119898(119894 119902)mdashthe number of streams of the 119894th processat the 119902th level and 119897119898(119894 119902)mdashthe number of operation of the119894th process at the 119902th level)

Variables 119909119896119894119895119898

119902119909119896

119894119895isin Z describe the timing of com-

mencement of operations in the 119902th cycle of the SCCP cyclicsteady state behavior 119909119896

119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896

119894119895follow from system structure

hence the cyclic behavior 119883119897119901 of SCCP is determined by itsstructure SM119897119901Moreover themultimodal processes behavior(119898119894119883119898

119894120572) also depends on the local cyclic processes behav-

ior (119883 120572)In the general case besides depending on the structural

characteristics of SCCP the values of the considered variables119909119896

119894119895119898

119902119909119896

119894119895depend on constraints from the mutual exclusion

protocol that is the set of priority dispatching rules Θoperation times and so forth [14 15] as well as on the waythe local and multimodal processes interacting with eachother For example in case of the two levels structure modelthat is including levels SL and SM as shown in Figure 2 theconstraints determining 119909119896

119894119895119898119902

119909119896

119894119895can be expressed by the

following rules [15](i) for local processes the timing of commencement of

operation 119900119896119894119895beginning states for a maximum of the

completion time of operation 119900119896119894119895minus1

preceding 119900119896119894119895and

the release time (with delay Δ119905) of the resource 119901119896119894119895

awaiting for 119900119894119895execution is as follows

moment of operation 119900119896119894119895

beginning

= max (moment of 119901119896119894119895

release + lag time Δ119905)

(moment of operation 119900119896119894119895minus1

completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR

A prospective higher level SM2 level

R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level

e level of local processes

SL = (U T Θ0)

e level ofmultimodal processesSM = (MmT Θ1)

Figure 3 Multilayered model of the SCCP from Figure 2

(ii) for multimodal processes the timing of commence-ment of operation119898119900119896

119894119895beginning is equal to the near-

est admissible value (determined by setX119896

119894119895of values

119898119909119896

119894119895) being a maximum of both the completion time

of operation 119898119900119896119894119895minus1

preceding 119898119900119894119895

and the releasetime (lag time Δ119905119898) of the resource119898119901119896

119894119895awaiting for

119898119900119894119895execution Consider


beginning

= lceilmax (moment of 119898119901119896119894119895


moment of operation 119898119900119896119894119895minus1

completionrceilX119896119894119895

119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)

where one has the followingX119896

119894119895mdashset of values119898119909119896

119894119895

following the set of local processes 119883 (X119896

119894119895mdashthe

set of moments when operation 119898119900119896119894119895of multimodal

process may use required local process) and lceil119886rceil119861mdash

the smallest integer greater than or equal to 119886 in termsof the set 119861 lceil119886rceil

119861= min119896 isin 119861 119896 ge 119886

In other words the timing of commencement of opera-tion 119898119900

119894119895belongs to the set X119896

119894119895and being coincident with

operation processes by relevant local transportation process119875119894The constraints determining (due to introduced rules (8)

and (9)) the timing of commencement of operations for theSCCP from Figure 2 are gathered in Tables 1 and 2

22 Cyclic Steady State Space According to the previoussection a set of cyclic process achieved in the structure SC119897119901

(6) can be represented as a cyclic schedule119883119897119901 (7) that meetsthe constraints (8) and (9) Figure 4 presents an exampleof a cyclic schedule which can be achieved in the systemillustrated in Figure 1 It shows that for a set of productionplan (represented by multimodal processes 119898119875

1 119898119875

2 and

1198981198753) it is possible to organize the work of AGVs responsible

for transportation of elements between workstations (localprocesses 119875

1ndashP

8) in such a way that no deadlocks appear in

the system or there is no process waiting at the resources An


Table 1 Constraints determining the local processes behavior of the SCCP from Figure 2

Constraints of the local processes1199091

11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2

48minus 120572) mdash

1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83

assumption was made that the transportation of work-pieces(streams of multimodal processes) takes one unit of timeand their loading tounloading from AGVs (local processes)takes place in the first and the last unit of each operation (seeFigure 4)

Thework [20] shows that schedules of this kind (ensuringsuch an organization of AGVs that guarantees the accom-plishment of the set of production routes without any stop-pages) can be obtained by means of solving the followingconstraint satisfaction problem (10)

CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)

where one has the following 119883119897119901 119879119897119901mdashdecision variables119883

119897119901mdashcyclic schedule (7) 119879119897119901mdashsequence of operation times119879119897119901= (((119879119898119879

1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash

domains determining admissible value of decision variables

119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862

119872mdashthe set of constraints119862

119871and119862

119872describing SCCP

behavior 119862119871mdashconstraints determining cyclic steady state

of local processes that is their cyclic schedule and 119862119872mdash

constraints determining multimodal processes behaviorThe sequence of operations times 119879119897119901 and the sequence of

their beginningmoments119883119897119901 both of them follow constraints119862

119871119862

119872which are the sufficient conditions for the SCCPcyclic

steady state behavior that is the state following the problem(10) solution In case of two level systems that is SL andSM (see Figure 3) the constraints 119862

119871 119862

119872can be specified in

terms of constraints (8) and their extension (9) [15]Presenting cyclic processes as119883119897119901 schedules is commonly

used both in SCCP [11 12 14 15] and in various problemsof cyclic scheduling Instead of cyclic schedules the so-calledbehavior digraphs [16] that create the state spaceP can alsobe applied


Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898

minus119898120572)(1198981199091 19+1198981199051 19minus119898120572)rceilX1 11

1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime

(units time)mP13 mP2

3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41

- Execution of processrsquos mPji operationmP

ji

- Execution of processrsquos Pji operation

Load time = 1 utUnload time = 1 ut

- Local process P1- Local process P2

- Local process P3

- Local process P4

- Local process P5

- Local process P6


Sa Sb Sc- Transition Sa rarr Sb rarr Sc

- Admissible state SaSa

mPi1

mPi2

mPi3

- Multimodal process mPi1



Figure 4 AGVs fleet cyclic schedule matching the multiproduct manufacturing


The digraph 119866119882

(denoted by 119882 index in case of cyclicprocesses representation) is a digraph whose vertexes repre-sent the admissible states 119878119903 of the system and arcs representtransitions between the states (while depicting a given tran-sition function 120575 [16])

In the considered case the state of SCCP describes thepresent (valid in a given 119905 period) allocation on the resourcesof local (AGVs) and multimodal (technological routes) pro-cesses and describes the current access rights to the resources(determined by dispatching priority rules Θ) Formally inthe state definition three elements are distinguished thatcharacterize processes at the particular behavior levels (seeFigure 3)

In this approach the state of SCCP takes the followingform

119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)

where one has the following (i) 119897119901mdashthe number of levels ofstructure SC119897119901 (6) and (ii) 119878119897119903mdashthe 119903th state of local processesexecuted on the SL level (5) as follows

119878119897119903= (119860

119903 119885

119903 119876

119903) (12)

where one has the following 119860119903= (119886

1

119903 119886

2

119903 119886

119888

119903

119886119898

119903)mdashallocation of processes in the 119903th state 119886

119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896

119894mdashthe 119888th resource 119877

119888is occupied by the local stream

119875119896

119894 and 119886

119888

119903= Δmdashthe 119888th resource 119877

119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898

119903) is the sequence of sema-

phores corresponding to the 119903th state and 119911119888

119903isin 119875 is name of

the stream (specified in the 119888th dispatching rule 120590119888 allocated

to the 119888th resource) which was allowed to occupy the 119888thresource for example 119911

119888

119903= 119875

119896

119894means that stream 119875

119896

119894is

currently allowed to occupy the 119888th resource119876

119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898

119903) is the sequence of sem-

aphore indices corresponding to the 119903th state and 119902119888

119903

determines the position of the semaphore 119911119888

119903 in the prioritydispatching rule120590

119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888

119903isin N For instance 119902

2

119903= 2

and 1199112

119903= 119875

2

1correspond to the semaphore 119911

2

119903= 119875

2

1taking

the 2nd position in the priority dispatching rule 1205902

(iii) 119898119897119878119903

is the 119903th state of multimodal processes executedon the SM119897 level (6) as follows

119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)

where one has the following 119898119897119860

119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)

119897119908(119897))mdashallocation of the process

streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)

where one has the following 119898119897119875ℎ

119894mdashthe ℎth stream

of the 119894th multimodal process from 119897-level of SC119897119901 (6)119898mdashthe number of resources 119877 119898119897

119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ

119894means that the 119888th resource 119877

119888is

occupied by the ℎth stream of multimodal process119898

119897119875119894 and 119898119897

119886119903ℎ


119888which

is released by the ℎth stream of multimodal process119898

119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903

) are the sequences of semaphores andindices defined similarly as 119885119903 and 119876119903

In the context of the introduced notions (allocationsemaphore indexes) the 119883119897119901 schedule presented in Figure 4illustrates only the allocation of processes in time Theparticular allocations are denoted by a frame and symbolldquo ⃝rdquo of the state 119878119903 connected with the given allocationTherefore the cyclic process represented by119883119897119901 schedule canbe described with use of 42 states (S0ndashS41)


= (119881119882 119864

119882) corresponding to 119883119897119901

schedule from Figure 4 was illustrated in Figure 5 The set ofvertexes 119881

119882= 119878

0 119878

41 includes all the states that can

be achieved by the system in the course of implementingprocesses according to 119883119897119901 schedule The set of arcs 119864

119882sube

119881119882times119881

119882defines the transitions between the states Transition

of the system from the state 1198780 to the state 1198781 (denotedby 1198780 rarr 119878

1 and in Figure 5 as ⃝ rarr ⃝) occursaccording to the transition function 120575 whichwas described in[16]

According to this approach the digraph 119866119882= (119881

119882 119864

119882)

depicting the behavior from Figure 4 is a cycle including 42vertexes see Figure 5 Apart from 119878119903 states Figure 5 illustratesalso local states 119878119897119903 (12) which only represent the behaviorof local processes In the discussed case the number of localstates is the same as the number of 119878119903 states Generally it doesnot have to be so as there are situations when one local stateis an element of numerous 119878119903 states [14 15] The local states119878119897

119903 as well as the digraph related to them should be perceivedas a projection of 119878119903 states and 119866

119882digraph onto the level of

local processesIt should be noted that in the discussed example only

one cyclic steady state is reachable that is as a result ofthe solution (10) that is one schedule 119883119897119901 was obtainedalong with one digraph 119866

119882corresponding to it Generally

in the given structure SC119897119901 a lot of cyclic steady states can bereachable (which depends on the initial phases of dispatchingrules) and therefore we can consider the cyclic steady statespace P as a set of digraphs that can be reachable in a givenstructure of 119866

119882digraphs

The cyclic steady state space can be illustrated as a graphbeing the sum of all reachable digraphs P = (119881P 119864P) =

1198661198821

cup 1198661198822

cup sdot sdot sdot cup119866119882119897119892

(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864

119887)) are reachable in a given structureThere are also

situations where in a given structure no cyclic steady statesare reachable then P = (0 0) The main reason for the lackof cyclic behaviors is the possible deadlocks

The form of the space of states P is strictly depen-dent upon the form of SC119897119901 structure In other wordsthe structure determines behaviors reachable in the sys-tem Another example of a cyclic behavior is shown in


Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979

- Transition Se rarr Sf

- Transition Sle rarr Slf

S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)

- The rth local state of SCCP Slr = (Ar Zr Qr)- The rth state of SCCP Sr = (Slr m1S

r)

Figure 5 The digraph 119866119882following Ganttrsquos chart from Figure 4

Figure 6 It is a cyclic schedule 119883119897119901 reachable in SC struc-ture (Figure 2) in which the manner of implementingproduction routes was changed (routes 119898119901119896

2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) There are still seven types

of AGVs used for transport (local processes P1ndashP

7) with

routes remaining unchanged yet different than previouslywhen they performed their operations

This situation can be interpreted as introducing a newproduction order into the system which requires a newmethod of processing elementsThe presence of cyclic sched-ule 119883119897119901 (providing a solution to the problem (10)) and 119866

119882

digraph (Figure 7) related to it mean that such an order canbe implemented

The presence of two different cyclic steady states leadsto the question of their mutual reachability Is it possibleto smoothly change the system behavior from 119866

1198821to 119866

1198822

(Figure 7) In other words is it possible to smoothly (with noneed to stop the system) proceed from implementing the firstproduction order 119866

1198821to the other 119866

1198822

3 Multimodal Processes Rescheduling

31 Problem Formulation The question of the possibilityof smoothly changing the cyclic behaviors of the system isrelated to the problem of mutual reachability of the twobehavior digraphs 119866

1198821 119866

1198822 The digraphs presented so far

(Figure 7) are cycles In general a digraph may take the formof the so-called vortex 119866

119878 that is a digraph including a

cycle representing a cyclic steady states 119866119882

and a tree 119866119879119894

representing transient states leading to cyclic steady statesAn example of two vortexes 119866

1198781 119866

1198782is shown in Figure 8

Formally the digraph of the vortex type is defined as follows

119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866

1198792cup sdot sdot sdot cup 119866

119879119897119905 119897119905 is the number of

trees 119866119879119894

leading to 119866119882cycle

In this approach the problem of mutual reachability ofthe behavior digraphs of mutual space of states is definedas follows there is a given structure SC119897119901 and the cyclicsteady state space P = (119881P 119864P) resulting from it whichincludes two vortexes 119866

1198781 119866

1198782that represent two cyclic

steady processes It is therefore necessary to find answer tothe following question Are the digraphs 119866

1198821 119866

1198822(being

subdigraphs of 1198661198781 119866

1198782) mutually reachable (Figure 8) In

the case they are mutually reachable the next question is howto make a transition between the digraphs

In other words the question of mutual reachability ofcyclic behaviors can be treated as the question of the possibil-ity of direct or indirect transition between the digraphs 119866

1198821

1198661198822

An example of such transitions is shown in Figure 8 Anindirect transition should be understood as changing 119866

1198821

into 1198661198822

resulting from the transition of the state 1198781199091015840 intotransitory path (a path of digraph 119866

119879119894) leading to 119866

1198822 The

direct transition means a transition from the state 119878119909 rightinto the cycle 119866

1198821

32 Searching for States with Mutual Allocation

321 Sufficient Conditions The possibility of transitionsbetween cyclic steady states depends on the state 119878119909(1198781199091015840)which is an element of two digraphs 119866

1198781and 119866

1198782 In other

words it is necessary that in case of direct transition thefollowing transitions between states occur

(i) for all 119878119896 isin 1198811198821

(119878119896rarr sdot sdot sdot rarr 119878119909rarr sdot sdot sdot rarr 119878

119896) forthe digraph 119866

1198821

(ii) for all 119878119897 isin 1198811198822



1198822

In SCCP transitions of this kind are not acceptable Itresults from the property saying that each state has at least


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42

Figure 6 AGVs fleet cyclic schedule matching the multiproduct manufacturing for system with changed multimodal routes

one descendant [16] (a descendant is the consecutive stateafter 119878119909) The presence of state 119878119909 included simultaneouslyinto two digraphs (119866

1198821 119866

1198822) would require the existence of

at least two descendants of the state 119878119909Although there are no mutual states at the level of the

space of states P the different projections of states uponlower levels of behavior as well as their components (spaceof allocation semaphores etc) may be elements belongingto numerous digraphs at the same time

An example of such a situation is shown in Figure 9where the idea of amultilevel model of the cyclic steady statesspaceP has been applied The figure shows the projection of

the space P upon the level of local processes behavior PSL(the second level in Figure 9) and upon the space of allocationA (the lowest level in Figure 9) The elements of the spaceAare the allocations 119860119903 of the local states 119878119897119903 = (119860

119903 119885

119903 119876

119903)

occurring in the space PSL = (119881SL 119864SL) (where 119878119897119903isin 119881SL)

which at the same time is the projection of the space of statesP

It is worth mentioning that some allocations of the spaceA are simultaneously mutual for several states For examplethe allocation 1198602

isin A is at the same time an element ofthe state 1198781198973 = (1198602

1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)

In practice it means that in both states the local processes


Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)

Figure 7 The state space for structures with different multimodal processes routes

- Admissible state- Initial state


GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1

Figure 8 Illustration of the cyclic steady state space composed oftwo vortexes

(representing eg means of transport such as AGVs) areallocated on the same resources However since there arevarious access rights (determined by11988531198763 and11988541198764) theirfurther implantation will be different

The existence of states of common allocation can be usedfor the transitions between various behavior digraphs Thestates 1198781198973 and 1198781198974 differ only with the sequence of semaphoresand indexes Thus if in the state 1198781198973 with allocation 1198602 theform of 11988531198763changes into the form of 11988541198764 we will attainthe state 1198781198974 leading to another cycle

The modification of the state 1198781198973 into 1198781198974 by changingthe form of semaphores and indexes allows for an indirecttransition between cyclic processes of the space P0 Thedirect transition is possible as a result of modifying thesemaphores and indexes of the states 1198781198971 = (119860

1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876

2) mutually sharing the allocation 1198601

Therefore the presence of states of mutual allocation impliesthe possibility of changing the set of cyclic steady states


- Transition Sla rarr Slb

- Cyclic steady state projection of GW1 onto the space 119979SL

- Cyclic steady state represented by cyclic digraph GW1

- The rth state of SCCP Sr = (Slr m1Sr)

- Transition Sa rarr Sb

119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964

- The rth allocation Ar

Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n

- The rth local state of SCCP Slr = (Ar Zr Qr)

(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa

Figure 9 Illustration of cyclic steady state space projectionsPSL andA for behavior digraphs from Figure 8

The considered transitions between 1198781198973 and 1198781198974 as well as119878119897

1 and 1198781198972 refermainly to digraphs occurring only at the locallevel (ie space PSL) In case of digraphs 119866

1198781 119866

1198782of the

spaceP a similar procedure should be followedThese statesare the projections of the corresponding states from the spaceP The state 1198781198971 is the projection of the state 1198781 = (1198781198971 1198981

1198781

)which is an element of the digraph 119866

1198821 and the state 1198782 =

(1198781198972 119898

11198782

) is the projection of the digraph 1198661198822

If among states projecting upon 1198781198971 and 1198781198972 there are

states (eg states 1198781198971 and 1198781198972) with mutual allocation 1198981119860

119909

of themultimodal processes the transition between1198661198821

and119866

1198822(and therefore also between 119866

1198781and 119866

1198782) is possible as

a result of modifications of corresponding semaphores andindexes (from the formof1198981

11988511198981

1198761 into the formof1198981

1198852

1198981119876

2)

The states 1198781 and 1198782 are characterized bymutual allocationof both local andmultimodal processes Bymeans of modify-ing semaphores and indexes in the state 1198781 a direct transitionfrom digraph 119866

1198821to digraph 119866

1198822is possible

In practice the change of semaphores and indexes relatedto themmeans simply the change of the control rules (changeof signaling) of the system the moment the processes areallocated properly (compatible with 1198781) The solution of thiskind neither stops the implementation of the process (workof AGVs technological routes) nor creates the need forchanging their allocation (shift) In case of the modificationof the state 1198786 the transition is immediate and noninvasive itonly requires the change of control

To sum up the above considerations we can say that thetransition between the two digraphs119866

1198821and119866

1198822is possible

if they include states characterized by mutual allocation at


every behavior level This observation leads to the followingtwo properties

Property 1 Digraph 1198661198822

is reachable from the digraph 1198661198821

(which is denoted by 1198661198821

rarr 1198661198822

) if there are states 119878119886 isin1198811198821

and 119878119887 isin 1198811198782

(where1198811198821

is the set of states of the digraph119866

1198821and 119881

1198782is the set of states of the digraph 119866

1198782) sharing

mutual allocation of local and multimodal processes 119860119886=

119860119887119898119897

119860119886

= 119898119897119860

119887

for 119897 = 1 sdot sdot sdot 119897119901

Property 2 Two digraphs 1198661198821

and 1198661198822

are mutually reach-able (which is denoted by 119866

1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821

In this approach the problem of digraphs reachabilitycalls for an answer to the following question Are there twostates 119878119886 isin 119881

1198821and 119878119887 isin 119881

1198782 sharing the same allocations

119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887

for 119897 = 1 sdot sdot sdot 119897119901 among states includedinto 119866

1198781and 119866

1198782 If such states exist the transition between

the states is possible as a result ofmodifications of semaphoresand indexes In this context the direct transition is defined as

sdot sdot sdot 997888rarr 1198781198861 997888rarr 119878

1198862 997888rarr sdot sdot sdot 997888rarr 119878

119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878

119887(119909+1) 997888rarr sdot sdot sdot

(16)

and the indirect transition as

sdot sdot sdot 997888rarr 1198781198861 997888rarr sdot sdot sdot 997888rarr (119878

119886119909 999492999484 119878

119889119909) 997888rarr sdot sdot sdot

997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)

where one has the following 1198781198861 119878119886(119909minus1) 119878119886119909 119878119886119897119886mdashstates of digraph 119866

1198821 1198781198871 119878119887(119909minus1) 119878119887119909 119878119887119897119887mdashstates of

digraph1198661198822

1198781198891 119878119889119909 119878119887(119909+1) 119878119889119897119887mdashstates included intothe transitory digraph leading 119866

119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash

transition between states with SCCP assumption and transi-tory function 120575 [16] 119878119886119909 119878119887119909mdashstates with mutual allocationand 119878119886119909 999492999484 119878

119887119909mdashtransition from the state 119878119886119909 to 119878119887119909 as a

result of states modification consisting of a sequence changeof semaphores and indexes from 119885

119886119909 119876119886

119909 to 119885119887119909 119876119886

119909

322 Searching Algorithm In the situation when digraphs119866

1198821and 119866

1198822and transitory processes leading to them

are known (in other words vortexes 1198661198781

and 1198661198782

(15) areknown) determining the states with mutual allocation is nota complex task Due to a small number of states (in practiceno more than a few hundred states per vortex) all that hasto be done is to successively compare all potential variantsAn algorithm corresponding to such an approach looks asAlgorithm 1

In Algorithm 1 one has the following 1198661198781= (119881

1198781 119864

1198781)

1198661198782= (119881

1198782 119864

1198782)mdashinput data vortexes (15) 119878119886 119878119887mdashstates

(15) included in the digraph 1198661198821

1198661198782 and ACmdashset of pairs

(119878119886 119878

119887) of states with mutual allocation

The result of Algorithm 1 is the set AC including pairsof states (119878119886 119878119887) with mutual allocations The states can beused to determine the direct and indirect transitions between

digraphs 1198661198821

and 1198661198822

Therefore the existence of a non-empty set ACmeans a positive answer to the posed question

To make things simple an assumption can be made thatwe take into consideration that direct transitions and thedigraphs have the same number of states (denoted by 119897119889) thuscomputational complexity of Algorithm 1 is expressed by thefunction 119891(119897119889) = 1198971198892

Algorithm 1 makes it possible to evaluate the mutualreachability of only two digraphs 119866

1198821and 119866

1198822isin DC (set

of all cyclic digraphs existing in the space P) In case ofevaluating the mutual reachability of all digraphs from theset DC it is necessary to make a search of this kind for apair of processes The computational complexity in such caseamounts to

119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)

Owing to polynomial character of the function of com-putational complexity the problem of evaluating the mutualreachability of behavior digraphs is not a difficult one All thecomputational effort is focused on the stage of determiningcyclic digraphs (set DC) that is solving the problem (10) Inorder to do so methods described in [11 12 14ndash16] can beapplied

Among the available methods however there is a deficitof those which can help determine transient states leading tocyclic digraphs (ie digraphs 119866

119879) Their usability is therefore

limited to evaluating direct transitions (where no informationon transient states (processes) is required)

In order to determine the transient states the method ofvortex generation can be used Vortexes determined with useof this method include cyclic processes as well as all transientprocesses leading to them The example below illustratesthe use of this method for evaluating mutual reachability ofbehavior digraphs

33 Numerical Example Consider two AGVs supportingmultiproduct manufacturing flows and their SCCP modelsshown in Figures 2 and 7(b) Let us assume that the two seriesmanufacturing of three different products follow the routesdistinguished by different colored (green orange and blue)bold lines

Each workstation can process only unique work-piecefrom a unique product kind at a given instance Successivework-pieces following the product route of a given productkind while taking into account local material handling routesare treated as subsequent streams ofmanufacturing processes(in a given kind of product manufacturing flow) and aremodeled as streams of multimodal processes For instancesee streams of119898119875

1119898119875

2 and119898119875

3from Figure 2 as follows

119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4


function StatesCoAll (1198661198781= (119881

1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)

and Figure 7(b) as follows

119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)

Both systems use the same fleet of 13 AGVs A fleet sup-ports work-pieces movements between workstations alongthe givenmanufacturing routesTheAGVs flows aremodeledby streams 1198751

1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751

8of local processes P

1ndashP

8 Each workstation (in general

system resource) can be serviced only by one AGV at themomentThatmeans that both kinds of flow that is productsand AGVs flows are synchronized by a mutual exclusionprotocol

Given are the two cyclic schedules of multiproductmanufacturing supported by AVGs (see Figures 4 and 6)and corresponding to them two cyclic digraphs (Figures 7(a)and 7(b)) Due to this schedules in first system four work-pieces 1198981198751

119894 1198981198752

119894 1198981198753

119894 1198981198754

119894of each product (119894 = 1 2 3) are

simultaneously processed in one cycle and one work-pieceof each product is outputted at each 80 ut In second systemfour work-pieces of119898119875119896

1and three work-pieces of119898119875119896

2119898119875119896

3

product are simultaneously processed in one cycle and onework-piece of each product is outputted at each 96 ut

As mentioned before operation times from the schedulesconsidered take into account periods required for work-pieceprocessing as well as material handling operations that isloadunload periods (1 ut for the work-piece load and 1 utfor its unload see Figure 4) Moreover the assumed time lagsΔ119905119898 (see (9) and constraints from Table 2) enable to take

into account the work-pieces transportation between work-stations In the case considered all time lags are the same andequal toΔ119905119898 = 1Moreover it is assumed that the input to119877

1

1198772 and 119877

3and the output of 119877

10 119877

11 and 119877

12workstations

are loaded and unloaded by dedicated conveyorsThe operation times 119898119879 and 119879 of considered systems

are defined in Tables 3 and 4 The timing 119909119896119894119895 119898119909119896

119894119895of

commencement of operations following the constraints (8)(9) are presented in Tables 5 and 6

The values presented in Tables 3ndash6 are the results ofsolving the problem (10) respectively for the system fromFigure 2 and Figure 7(b) In order to achieve this objec-tive OzMozart constraint programming environment wasapplied The problem solving time in both cases did notexceed 3 seconds (Intel Core 2 Duo 3GHz RAM 4GB)

During the analysis of digraphs of the attained sched-ules it turned out that it is possible to smoothly changethe behavior (with use of the state 11987818) from the one presentedin Figure 4 into the behavior from Figure 7 The transitionwas illustrated in Figures 11 and 12 In practice a transitionof this kind means the possibility of changing the currentproduction order into a new one with no need to stop thesystem

Therefore a question may be posed whether an inversesituation is possible when from the behavior represented bydigraph 119866

1198822 behavior 119866

1198821is attained (return to 119866

1198821) In

other words the question is the following Are the consideredbehavior digraphs mutually reachable 119866

1198821harr 119866

1198822

According to Property 2 it is possible if1198661198821

rarr 1198661198822

and119866

1198822rarr 119866

1198821The schedule shown in Figure 10 illustrates the

reachability 1198661198821

rarr 1198661198822

In order to determine whether119866

1198822rarr 119866

1198821is possible it must be stated (Property 1) if

there are states ofmutual allocations in digraphs1198661198822

and1198661198781

(1198661198781mdashdigraph including 119866

1198821and transitory path of states

leading to it)In order to determine these states Algorithm 1 was

applied The use of Algorithm 1 depended upon the knowl-edge of digraph 119866

1198781 It should be emphasized that the solu-

tion of the problem (10) makes it possible to determinethe cyclic schedule and digraph 119866

1198821corresponding to it

It does not allow however determining states belonging totransitory processes In the discussed case digraph 119866

1198781is


Table 3 Times (units) of operations executions of local and multimodal processes from Figure 2

119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896

1mdash mdash 77 mdash 64 77 mdash 20 mdash mdash 20 mdash 1

119898119905119896

2mdash 77 mdash 20 10 mdash 30 mdash mdash 30 mdash mdash 1

119898119905119896

377 mdash mdash 49 mdash mdash 20 50 20 mdash mdash 10 1

1199051

1mdash mdash 76 mdash mdash 2 mdash mdash mdash mdash mdash mdash 1

1199051

2mdash mdash mdash mdash 4 18 mdash 1 1 mdash mdash mdash 1

1199052


1199053


1199051

3mdash mdash mdash mdash mdash mdash mdash mdash 1 mdash mdash 1 1

1199051

4mdash mdash mdash mdash mdash mdash 4 1 mdash 1 1 mdash 1

1199052


1199051

51 61 mdash 1 1 mdash mdash mdash mdash mdash mdash mdash 1

1199052


1199051

6mdash mdash mdash 1 mdash mdash 1 mdash mdash mdash mdash mdash 1

1199052


1199051

7mdash mdash mdash mdash mdash mdash 1 mdash mdash 34 mdash mdash 1

1199051

81 mdash mdash 1 mdash mdash mdash mdash mdash mdash mdash mdash 1

Table 4 Times (units) of operations executions of local and multimodal processes from Figure 7(b)

119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896

1mdash mdash 91 mdash 64 91 mdash 20 20 mdash mdash 20 1

119898119905119896

2mdash 91 mdash 20 24 mdash 68 30 mdash mdash 30 mdash 1

119898119905119896

391 mdash mdash 68 mdash mdash 20 50 mdash mdash mdash mdash 1

1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051


unknown Yet the state of cyclic processes can be determinedfrom digraph 119866

1198821by means of searching for parents of the

states 1198811198821

not belonging to this set Therefore next statefunction 120575 defined in [16] was applied

Based on the determined states a search for mutual allo-cation stateswas carried out An example of such states are thestates 11987819 11987840 A change of cyclic steady states based on thesestates (consisting of changing priority rules semaphores andindexes) was illustrated in Figures 11 and 12

As shown in Figure 12 the considered cyclic steady statesare mutually reachable (119866

1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)

In general case however mutual reachability is not alwaysguaranteed For instance consider SCCP composed of fourcyclic processes shown in Figure 13(a) The cyclic steady

state space P consists of two vortexes 1198661198781

and 1198661198782 see

Figure 13(b)There are five different transient periods linking119866

1198821and 119866

1198822 see orange arcs There is not however any

transient period linking 1198661198782

and 1198661198781 that is there are

not any states shared by 1198661198822

and 1198661198781

while followingProperty 1 Moreover both direct mutual reachability andindirect mutual reachability are not guaranteed in generalcase of a cyclic steady state space

4 Related Work

Many algorithms for the scheduling and routing of AGVshave been proposed However most of the existing resultsare applicable to systems with a small number of AGVs


Table 5 The timing 119909119896119894119895119898119909119896

119894119895of commencement of operations of local and multimodal processes from Figure 2

1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091

1mdash mdash minus10 mdash 146 68 mdash 211 mdash mdash 232 mdash

1198981199091

2mdash 55 mdash 144 133 mdash 165 mdash mdash 190 mdash mdash

1198981199091

39 mdash mdash 87 mdash mdash 137 158 209 mdash mdash 230

1199091

1mdash mdash minus9 mdash mdash 68 mdash mdash mdash mdash mdash mdash

1199091

2mdash mdash mdash mdash 66 47 mdash minus8 minus6 mdash mdash mdash

1199092

2mdash mdash mdash mdash 45 71 mdash 48 49 mdash mdash mdash

1199093

2mdash mdash mdash mdash 47 minus7 mdash 51 70 mdash mdash mdash

1199091

3mdash mdash mdash mdash mdash mdash mdash mdash 68 mdash mdash minus10

1199091

4mdash mdash mdash mdash mdash mdash minus7 minus2 mdash 71 0 mdash

1199092

4mdash mdash mdash mdash mdash mdash minus2 70 mdash minus6 72 mdash

1199091

569 minus9 mdash 57 53 mdash mdash mdash mdash mdash mdash mdash

1199092

5minus9 53 mdash 64 62 mdash mdash mdash mdash mdash mdash mdash

1199091

6mdash mdash mdash 3 mdash mdash 5 mdash mdash mdash mdash mdash

1199092


1199091

7mdash mdash mdash mdash mdash mdash 34 mdash mdash 36 mdash mdash

1199091

85 mdash mdash 7 mdash mdash mdash mdash mdash mdash mdash mdash

Table 6 The timing 119909119896119894119895119898119909119896

119894119895of commencement of operations of local and multimodal processes from Figure 7(b)

1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091

1mdash mdash minus10 mdash 174 82 mdash 239 260 mdash mdash 281

1198981199091

2mdash 55 mdash 172 147 mdash 193 262 mdash mdash 293 mdash

1198981199091

39 mdash mdash 101 mdash mdash 170 mdash mdash 191 mdash mdash

1199091


1199091


1199092


1199093


1199091


1199091


1199092

4mdash mdash mdash mdash mdash mdash 13 74 mdash 11 103 mdash

1199091


1199092


1199091

6mdash mdash mdash minus9 mdash mdash 5 mdash mdash mdash mdash mdash

1199092


1199091


1199091


offering a low degree of concurrency With a drasticallyincreased number of AGVs in recent applications (eg inthe order of a hundred in a container handling system) thecombinatorial nature of the fleet scheduling and assignmentproblems rapidly increases the complexity For instance itappears clearly through constraints which relate to assigninga fleet to a route Suppose that there are 119899 fleet and 119898routes the number of binary decisions variables is119898times 119899 andtheoretically there are 2119899times119898 solutions to be considered Infact according to computational complexity this problem isNP-hard [2] A number of papers are concerned with the fleetassignment problem [1ndash3 8] andmaintenance planning [6 721] However few papers have considered the combinationof fleet assignment and maintenance planning [2 22] Most

publications on fleet assignment have been focused on theassignment of fleets to minimize the fleet operation cost [1 38] Some papers on maintenance planning have been focusedon planningmaintenance tasks to minimize the maintenancecost [6 7 21] Few papers have considered the integrationof fleet assignment and maintenance planning [2 22] Theinventory policy with preventive maintenance considerationismostly studied for production system [6 23 24] Because ofunpredicted situations rescheduling is required to deal withonline decisionmaking Finally scheduling and reschedulingfleet assignment and maintenance planning and inventorypolicy are implemented as a decision support system [22]

Because most of real-life available manufacturing pro-cesses manifest their cyclic steady state the alternative


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji


Load time = 1 utUnload time = 1 utSuspension of process mp

ji

mPji

0 50 100 150 200 250 300 350 400 450

Cyclic steady state 1 Cyclic steady state 2Transient state

(S18 Sd1 ) (Sd34 rarr S2)

Figure 10 The indirect transition between schedules from Figures 4 and 6

approach to AGVs fleet dispatching and scheduling forexample following from the SCCP concept [1 9 25] can beconsidered as well Many models and methods have beenproposed to solve the cyclic scheduling problem so far Mostof these are oriented at finding a minimal cycle or maximalthroughputwhile assuming deadlock-free processes flowTheapproaches trying to estimate the cycle time from cyclicprocesses structure and the synchronization mechanismemployed (ie mutual exclusion instances) are quite unique

In that context our main contribution is to propose a newmodeling framework enabling to state and answer the fol-lowing questions Does there exist a control procedure (ega set of dispatching rules and an initial state) guaranteeingan assumed steady cyclic state subject to SCCPrsquos structureconstraints Does an SCCPrsquos structure exist such that a steadycyclic state can be achieved

Is the cyclic steady sate space empty In the case the spaceis not empty the next question is do the cyclic steady states aremutually reachable

The last question refers to the problem of multimodalprocesses rescheduling Most of so far proposed approachesto the rescheduling of processes executed within multimodaltransportation networks environment [17 26ndash28] are onlylimited to the deadlock-free cases It should be noted thathowever the evolutionary algorithms driven approaches

frequently used [26ndash28] are not effective in cases allowingdeadlocks occurrenceThe approach proposed allows findingdeadlock-free transitions between the reachable cyclic steadystates while avoiding the deadlocks in the course of thetransient state generation

5 Conclusions

Thecomplicated problemwhich integratesAGVsfleet assign-ment routing and scheduling generating a suboptimalcyclic schedule for multiproduct manufacturing is noveltyof this research Such integrated AGVs dispatching problemscan be seen as a special case of the cyclic blocking flow-shop one where the jobs might block either the machine orthe AGV at the processing time Therefore the main class ofproblems of AGVs fleet match-up scheduling with referenceschedule of assumed production flow belong to the NP-hardones

Besides the above mentioned AGVs dispatchingplan-ning issues the research objective regards a quite large classof digital andor logistics networks that share commonproperties even though they have huge intrinsic differencesIn other words besides AGVs that can be treated as a kindof internal transport other emerging trends concerning thelogistics (eg supply chains management) and city traffic


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

Cyclic steady state 1Cyclic steady state 2 Transient state


ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54

Figure 12 The indirect transition between cyclic steady states 1198661198821

rarr 1198661198822

and 1198661198822

rarr 1198661198821

(eg infrastructure of the public transport) issues can bemodeled For instance the relevant multimodal processesof passengers traveling between destination points in anenvironment of local processes encompassing the subwaylines networkmight be considered as well In other words the

passengerrsquos itinerary including different metro lines can beconsidered as a plan of a multimodal process routing withina metro network

The proposed declarative approach aimed at AGVs fleetscheduling stated in terms of constraint satisfaction problem


R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)

Figure 13 Illustration of SCCP composed of four local cyclic processes (a) and cyclic steady state spaceP including transient states depictingmutual reachability 119866

1198821rarr 119866

1198822(b)

representation provides a unified method for performanceevaluation of local as well as supported by them multimodalprocesses

In general case however there is not any guarantee thatthe considered SCCP model of AGVS has a nonempty cyclicsteady states space as well as that in case this space is notempty any direct or indirect mutual reachability can beobserved That means that the decidability of the problem ofthe space P reachability and mutual reachability among itscyclic steady states plays a pivotal role in multimodal pro-cesses rescheduling

References

[1] J Abara ldquoApplying integer linear programming to the fleetassignment problemrdquo Interfaces vol 19 pp 4ndash20 1989

[2] L W Clarke C A Hane E L Johnson and G L NemhauserldquoMaintenance and crew considerations in fleet assignmentrdquoTransportation Science vol 30 no 3 pp 249ndash260 1996

[3] M D D Clarke ldquoIrregular airline operations a review of thestate-of-the-practice in airline operations control centersrdquo Jour-nal of Air Transport Management vol 4 no 2 pp 67ndash76 1998

[4] N G Hall C Sriskandarajah and T Ganesharajah ldquoOpera-tional decisions in AGV-served flowshop loops fleet sizing anddecompositionrdquoAnnals of Operations Research vol 107 no 1ndash4pp 189ndash209 2001

[5] S Hoshino and H Seki ldquoMulti-robot coordination for jams incongested systemsrdquo Robotics and Autonomous Systems vol 61no 8 pp 808ndash820 2013

[6] A Chelbi and D Ait-Kadi ldquoAnalysis of a productioninventorysystemwith randomly failing production unit submitted to reg-ular preventive maintenancerdquo European Journal of OperationalResearch vol 156 no 3 pp 712ndash718 2004

[7] S Deris S Omatu H Ohta L C Shaharudin Kutar and P AbdSamat ldquoShip maintenance scheduling by genetic algorithm andconstraint-based reasoningrdquo European Journal of OperationalResearch vol 112 no 3 pp 489ndash502 1999

[8] C A Hane C Barnhart E L Johnson R E Marsten G LNemhauser and G Sigismondi ldquoThe fleet assignment problemsolving a large-scale integer programrdquo Mathematical Program-ming vol 70 no 2 pp 211ndash232 1995

[9] L Qiu W-J Hsu S-Y Huang and H Wang ldquoScheduling androuting algorithms for AGVs a surveyrdquo International Journal ofProduction Research vol 40 no 3 pp 745ndash760 2002

[10] B Trouillet O Korbaa and J-C Gentina ldquoFormal approach ofFMS cyclic schedulingrdquo IEEE Transactions on SystemsMan andCybernetics C vol 37 no 1 pp 126ndash137 2007

[11] E Levner V Kats D Alcaide Lopez De Pablo and T C ECheng ldquoComplexity of cyclic scheduling problems a state-of-the-art surveyrdquo Computers and Industrial Engineering vol 59no 2 pp 352ndash361 2010

[12] T Von Kampmeyer Cyclic scheduling problems [PhD dis-sertation] Fachbereich MathematikInformatik UniversitatOsnabruck 2006

[13] M Polak P B Z Majdzik and R Wojcik ldquoThe performanceevaluation tool for automated prototyping of concurrent cyclicprocessesrdquo Fundamenta Informaticae vol 60 no 1-4 pp 269ndash289 2004

[14] G Bocewicz and Z Banaszak ldquoDeclarative approach to cyclicscheduling of multimodal processesrdquo in EcoProduction and


Logistics P Golinska Ed pp 203ndash238 Springer HeidelbergGermany 2013

[15] G Bocewicz and Z Banaszak ldquoDeclarative approach to cyclicsteady states space refinement periodic processes schedulingrdquoInternational Journal of Advanced Manufacturing Technologyvol 67 no 1ndash4 pp 137ndash155 2013

[16] G Bocewicz R Wojcik and Z Banaszak ldquoCyclic Steady StateRefinementrdquo in Proceedings of the International Symposium onDistributed Computing andArtificial Intelligence A Abraham JM Corchado S Rodrıguez Gonzalez and J F de Paz SantanaEds vol 91 of Advances in Intelligent and Soft Computing pp191ndash198 Springer Berlin Germany 2011

[17] M Dridi K Mesghouni and P Borne ldquoTraffic control intransportation systemsrdquo Journal of Manufacturing TechnologyManagement vol 16 no 1 pp 53ndash74 2005

[18] J-S Song and T-E Lee ldquoPetri net modeling and schedulingfor cyclic job shops with blockingrdquo Computers and IndustrialEngineering vol 34 no 2-4 pp 281ndash295 1998

[19] A Che and C Chu ldquoOptimal scheduling of material handlingdevices in a PCB production line problem formulation and apolynomial algorithmrdquo Mathematical Problems in Engineeringvol 2008 Article ID 364279 21 pages 2008

[20] G Bocewicz I Nielsen and Z Banaszak ldquoAGVsfleet match-upscheduling with production flow considerationsrdquo EngineeringApplications of Artificial Intelligence In press

[21] N Papakostas P Papachatzakis V Xanthakis D Mourtzisand G Chryssolouris ldquoAn approach to operational aircraftmaintenance planningrdquo Decision Support Systems vol 48 no4 pp 604ndash612 2010

[22] W El Moudani and F Mora-Camino ldquoA dynamic approach foraircraft assignment and maintenance scheduling by airlinesrdquoJournal of Air Transport Management vol 6 no 4 pp 233ndash2372000

[23] N Rezg S Dellagi andA Chelbi ldquoJoint optimal inventory con-trol and preventivemaintenance policyrdquo International Journal ofProduction Research vol 46 no 19 pp 5349ndash5365 2008

[24] T S Vaughan ldquoFailure replacement and preventive mainte-nance spare parts ordering policyrdquo European Journal of Oper-ational Research vol 161 no 1 pp 183ndash190 2005

[25] M Sharma ldquoControl classification of automated guided vehiclesystemsrdquo International Journal of Engineering and AdvancedTechnology vol 2 no 1 pp 191ndash196 2012

[26] B F Chaar and S Hammadi ldquoEvolutionary approach to theregulation of the traffic of a system of multimodal transportrdquoJournal Europeen des Systemes Automatises vol 38 no 7-8 pp901ndash931 2004

[27] A K-S Wong Optimization of container process at multimodalcontainer terminals [PhD thesis] Queensland University ofTechnology Brisbane Australia 2008

[28] S Zidi and S Maouche ldquoAnt colony optimization for therescheduling of multimodal transport networksrdquo in Proceedingsof the IMACS Multiconference on Computational Engineering inSystems Applications vol 1 pp 965ndash9971 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


handling transport modes provide movement of work-piecesbetween workstations along their manufacturing routes inthe MTN In that context assuming that the modes can betreated as cyclic local transportation processes and material(eg work-piecestools) flows supported by them can betreated as multimodal processes the problems consideredcan be seen either as aimed at designing of a multimodalnetwork infrastructure guaranteeing assumed materials flowor focused onwork-pieces routing ensuring lag-free worksta-tions service

The problems arising concerning material transportationrouting and scheduling belong to NP-hard ones [2] Since thesteady state of production flows has a cyclic character henceservicing them AGV-served transportation processes (usu-ally executed along loop-like routes) encompasses also cyclicbehavior That means that the periodicity of FMS [10]depends on both the periodicity of production flow cyclicschedule and following this schedule the AGVs periodicityOf course the Multimodal Transportation Network (MTN)throughput is maximized by the minimization of its cycletime

It seems to be obvious that not all the behaviors (includingcyclic ones) are reachable under constraints imposed bythe systemrsquos structure The similar observation concerns thesystemrsquos behavior that can be achieved in systems possessingspecific structural constraints Since system constraints deter-mine its behavior both the system structure and the desiredcyclic schedule have to be considered simultaneously So theproblem solution requires that the system structure mustbe determined for the purpose of processes scheduling yetscheduling must be done to devise the system configurationIn that context our contribution provides a discussionof some solvability issues concerning cyclic processes dis-patching problems especially the conditions guaranteeingsolvability of the cyclic processes scheduling as well as directandor indirect transitions between assumed cyclic steadystatesTheir examinationmay replace exhaustive searches forsolution satisfying required system capabilities

Many models and methods have been considered to date[11] Among them the mathematical programming approach[1 12] max-plus algebra [13] constraint logic programming[14ndash16] evolutionary algorithms Petri nets [17 18] andheuristic frameworks [19] belong to themore frequently usedMost of these are oriented at finding a minimal cycle ormaximal throughputwhile assuming deadlock-free processesflow Note that processesrsquo operations are blocking if theymuststay on a resource (eg the station the machine) afterfinishing when the next resource is occupied by a job fromanother process During this stay the resource is blocked forother processes The approaches trying to estimate the cycletime from cyclic processes structure and the synchroniza-tion mechanism employed (ie mutual exclusion instances)while taking into account deadlock phenomena are quiteunique

In our approach a declarative framework aimed at refine-ment and prototyping of the cyclic steady states for concur-rently executed cyclic processes modelling material handlingsystems is employed These systems are of the type of AGVsfleets in the flexible manufacturing systems (FMS) and are

frequently encountered in industry The following questionsare the main focus of the research Can the assumed materialhandling system for example AGVs meet loadunloaddeadlines imposed by flow of scheduled work-pieces process-ing Does there exist sufficient AGVS enabling to schedulethe AGVs fleet as to ensure lag-free service of scheduled workpieces processing So the main question is Can the assumedAGVs fleet assignment reach its goal subject to constraintsassumed on concurrent multiproduct manufacturing athand

Moreover the declarative framework enables to state aproblem of multimodal processes rescheduling which boilsdown to searching for transient periods allowing the mutualreachability of MTN cyclic behaviors In the case of MTNsdistinguishing many different cyclic steady states (cyclicbehaviors) the following questions play a pivotal role Doesthere exist the smooth (direct) transition between twoassumed cyclic behaviors What conditions guarantee thereachability of a given cyclic behavior from any other onesIs it possible to come back to a given cyclic behavior fromany disturbance-born one for instance caused by operationtime delays damage of devices route modifications and soforth

In other words the paperrsquos objective concerns the MTNinfrastructure assessment from the perspective of possibleFMS oriented requirements imposed by AGVs fleet assign-ment sizing and scheduling (problems of cyclic processesscheduling and rescheduling) Due to the complexity impliedin answering the above questions the combined problemremains unsolved for all practical purposes This is especiallyproblematic as many manufacturing companies would standto reap considerable rewards through better fleet assign-ment and scheduling The presented approach addresses thisthrough solving the combined rather than the separate prob-lems individually

The rest of the paper is organized as follows Section 2introduces the Automated Guided Vehicles System (AGVS)modelled in terms of concurrently flowing cyclic processes(SCCPs) and a cyclic steady state space concept Section 3regarding multimodal processes rescheduling provides aproblem statement concerning AGVs fleet match-up sched-uling with an assumed multiproduct manufacturing flowschedule Sufficient conditions allowing one to search forstates with mutual allocation as well as illustrative exampleof implementing them algorithm usage are discussed Therelated work and concluding remarks are presented in Sec-tions 4 and 5 respectively

2 Multimodal Network Modeling

Considered case of multimodal processes rescheduling prob-lem is presented on the example of Automated Guided Vehi-cle Systems (AGVS) modeled by Systems of ConcurrentlyFlowing Cyclic Processes (SCCPs) To solve this kind of prob-lems a nonempty state space that is containing nonemptyset of possible cyclic behaviors is required For that reasonthe conditions guaranteeing cyclic behavior are presentedprimarily They are specified in terms of a declarative modelformalism distinguishing the structure and behavior as the







1198752 119875

3 119875

4 119875

5 119875

6 119875

7 and 119875



1198771 119877

119888 119877

33 119877



4 119875

5 and 119875

6


1

4 119875

2

4 119875

5= 119875

1

5 119875

2

5 and 119875

6=

1198751

6 119875

2



2=

1198751

2 119875

2

2 119875

3



1= 119875

1

1 119875

3= 119875

1

3 119875

7= 119875

1

7 and 119875

8= 119875

1

8


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5

1198751

6 1198752

6 1198751

7 and 1198751



119894


1119898119875

2 and119898119875

3


1supported


1 1198751

2 1198753

2 and 1198752

4 This means that



119894= 119898119875

1

119894 119898119875

2

119894 119898119875

3

119894 119898119875

4

119894 119894 =



1199011

1= (119877

16 119877

3 119877

13 119877

6)

1199011

2= (119877

21 119877

8 119877

18 119877

9 119877

14 119877

6 119877

17 119877

5)

1199012

2= (119877

17 119877

5 119877

21 119877

8 119877

18 119877

9 119877

14 119877

6)

1199013

2= (119877

14 119877

6 119877

17 119877

5 119877

21 119877

8 119877

18 119877

9)

1199011

3= (119877

12 119877

19 119877

9 119877

5)

1199011

4= (119877

29 119877

7 119877

25 119877

8 119877

22 119877

11 119877

26 119877

10)

1199012

4= (119877

26 119877

10 119877

29 119877

7 119877

25 119877

8 119877

22 119877

11)

1199011

5= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

27 119877

1 119877

23)

1199012

5= (119877

27 119877

1 119877

23 119877

2 119877

20 119877

5 119877

24 119877

4)

1199011

6= (119877

32 119877

4 119877

28 119877

7 119877

31)

1199012

6= (119877

31 119877

32 119877

4 119877

28 119877

7)

1199011

7= (119877

33 119877

7 119877

29 119877

10)

1199011

8= (119877

1 119877

27 119877

4 119877

30)

(1)


2 119877

4minus119877

11 119877

14 119877

17minus119877

18 119877

20minus119877

29 119877

31 and 119877

32


31119877

32represent


3 119877

12 119877

13 119877

15 119877

16 119877

19 119877

30 and

11987733





2 1198752

2 and 1198753

2which belong


4 119875

5 and 119875

6follow the




mP3

mP2

mP1

R1 R4 R7 R10

R11

R12

R2

R3R6 R9

R8R5

P5

P8 P6

P2

P3P1

P7

P4

1

1

2

2

3

3

4

4

1

2

3

4

1

3

1

1

1

1 11

1

2

2

2

2


q

q







2 and


119898119901119896

1= (119877

3 119877

13 119877

6 119877

17 119877

5 119877

21 119877

8 119877

22 119877

11)

119896 = 1 4

119898119901119896

2= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

28 119877

7 119877

29 119877

10)

119896 = 1 4

119898119901119896

3= (119877

1 119877

27 119877

4 119877

28 119877

7 119877

25 119877

8 119877

18 119877

9 119877

15 119877

12)

119896 = 1 4

(2)


2 and119898119901119896


(see Figure 2)

119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8)

(11987718 119877

9) (119877

15 119877

12)) 119896 = 1 4

(3)


13 119877

6) (119877

17 119877

5) (119877

21 119877

8)

and (11987722 119877


1 1199011

2 1199013

2 and 1199012

4


2 119877

20 119877

5)

(11987724 119877

4) (119877

28 119877

7) and (119877

29 119877


1199011

5 1199012

5 1199011

6 and 1199011


119898119901119896

2 and (119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9) and

(11987715 119877


8 1199012

6 1199011

4 1199012

2 and 1199011

3


3




12059001 (P25 P

18 P

15 ) 120590012 (P1

3 ) 120590023 (P25 P

15 )

12059002 (P15 P

25 ) 120590013 (P1

1 ) 120590024 (P15 P

25 )

12059003 (P11 ) 120590014 (P3

2 P12 P

22 ) 120590025 (P1

4 P24 )

12059004 (P16 P

18 P

26 P

15 P

25 ) 120590015 (P1

3 ) 120590026 (P24 P

14 )

12059005 (P22 P

32 P

15 P

25 P

12 ) 120590016 (P1

1 ) 120590027 (P25 P

18 P

15 )

12059006 (P32 P

12 P

11 P

22 ) 120590017 (P2

2 P32 P

12 ) 120590028 (P1

6 P26 )

12059007 (P14 P

24 P

16 P

17 P

26 ) 120590018 (P1

2 P22 P

32 ) 120590029 (P1

4 P24 P

17 )

12059008 (P12 P

14 P

22 P

32 P

24 ) 120590019 (P1

3 ) 120590030 (P18 )

12059009 (P12 P

22 P

13 P

32 ) 120590020 (P1

5 P25 ) 120590031 (P2

6 P16 )

120590010 (P24 P

17 P

14 ) 120590021 (P1

2 P22 P

32 ) 120590032 (P1

6 P26 )

120590011 (P14 P

24 ) 120590022 (P1

4 P24 ) 120590033 (P1

7 )

12059011 (mP13 mP2

3 mP33 mP4

3) 120590112 (mP33 mP4

3 mP13 mP2

3)120590123

12059012 (mP12 mP2

2 mP32 mP4

2) 120590113 (mP11 mP2

1 mP31 mP4

1) 120590124 (mP12 mP2

2 mP32 mP4

2)

12059013 (mP11 mP2

1 mP31 mP4

1) 120590114 120590125 (mP33 mP4

3 mP13 mP2

3)

12059014(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)120590115 (mP3

3 mP43 mP1

3 mP23) 120590126

12059015(mP3

1 mP42 mP4

1 mP12

mP11 mP2

2 mP21 mP3

2)120590116 120590127 (mP1

3 mP23 mP3

3 mP43)

12059016 (mP11 mP2

1 mP31 mP4

1) 120590117 (mP11 mP2

1 mP31 mP4

1) 120590128(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)

12059017(mP3

2 mP43 mP4

2 mP13

mP12 mP2

3 mP22 mP3

3)120590118

(mP33 mP4

3 mP13 mP2

3) 120590129 (mP32 mP4

2 mP12 mP2

2)

12059018(mP3

3 mP31 mP4

3 mP41

mP13 mP1

1 mP23 mP2

1)120590119 120590130

12059019(mP3

3 mP43 mP1

3 mP23) 120590120 (mP1

2 mP22 mP3

2 mP42) 120590131

120590110 (mP32 mP4

2 mP12 mP2

2) 120590121 (mP31 mP4

1 mP11 mP2

1) 120590132

120590111 (mP31 mP4

1 mP11 mP2

1) 120590122 (mP31 mP4

1 mP11 mP2

1) 120590133


4 3 2 1


Ri

(1205900i 1205901i )




R6

R1

R12

R10

R11

R9

R7

R8

R4

R5R2

mP1

mP2

mP3

P2

R13 R14 R15

R16

R17 R18

R26R25R24R23

R20 R21 R22

R27 R28 R29

R19

R31

P1 P3

P5

P8

R30 R32

P6

P4

P7

R33

Sector

Machine

4 3 2 1

R3

(12059001 12059011)

(12059002 12059012)

(12059003 12059013)

(12059006 12059016)

(12059009 12059019) (120590012 120590

112)

(120590011 120590111)(12059008 120590

18)(12059005 120590

15)

(12059004 12059014) (12059007 120590

17)

(120590010 120590110)


empty

empty

emptyempty

empty

empty

empty

empty

empty














5=

(1198752

2 119875

3

2 119875

1

5 119875

2

5 119875

1





stream 1198753


5 1198752

5 and 1198751

2 and then once

again 1198752


119896



119896



0 Θ

1 where Θ0

= 1205900

1 120590

0

33


1= 120590

1

1 120590

1



(i) A sequence 119901119896119894= (119901

119896

1198941 119901

119896

1198942 119901

119896

119894119895 119901

119896

119894119897119903(119894)) spec-


119894(the




119877 = 1198771 119877

2 119877

119888 119877




119875119896





16 119877

3 119877

13 119877

6)


1199011

1= (119901

1

11 119901

1

12 119901

1

13 119901

1


11


12= 119877

3 and so

on 119901113= 119877

13 1199011

14= 119877

6



(iii) 119905119896119894= (119905

119896

1198941 119905

119896

1198942 119905

119896

119894119895 119905

119896




(iv) 119898119901119896

119894= (119898119901119903

1199021

1198941

(1198861198941

1198871198941

) 1198981199011199031199022

1198942

(1198861198942

1198871198942

) 119898119901119903119902119910

119894119910

(119886119894119910

119887119894119910


119894from the



119898119901119903119902

119894(119886 119887)

=

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119887) 119886 le 119887

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119897119903(119894) 119901

119902

1198941 119901

119902

119894119887minus1 119901

119902

119894119887) 119886 gt 119887

119886 119887 isin 1 119897119903 (119894)

(4)


119902

1198941 119901

119902

1198942

119901119902

119894119895 119901

119902


119902

119894119886to 119901119902

119894119887



1(see


= ((1198773 119877

13 119877

6)

(11987717 119877

5) (119877

21 119877

8) (119877

22 119877

11)) where 1198981199011199031

1(2 4) =

(1198773 119877

13 119877

6) 1198981199011199031

2(7 8) = (119877

17 119877

5) 1198981199011199033

2(1 2) =

(11987721 119877

8)1198981199011199032

4(5 6) = (119877

22 119877

11)



(vi) 119898119905119896119894= (119898119905

119896

1198941 119898119905

119896

1198942 119898119905

119896

119894119895 119898119905

119896

119894119897119898(119894)) specifies


119896


119894119895


(vii) Θ = Θ0 Θ


Θ119894= 120590

119894

1 120590

119894

2 120590

119894

119888 120590

119894



119888= (119904

119894

1198881 119904

119894

119888119889 119904

119894

119888119897119901(119888)) are




1 119875

1

2

1198752

2 119875

3

2 119875

1

3 119875

1

4 119875

2

4 119875

1

5 119875

2

5 119875

1

6 119875

2

6 119875

1

7) and 1199041

119888119889isin 119898119867


119896

1 119898119875

119896

2 119898119875

119896

3| 119896 =

1 sdot sdot sdot 4)


SC = ((119877 SL) SM) (5)



2 119877

119888 119877

119898mdash



119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)




119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)



0

1 120590

0

2 120590

0

119888

1205900




1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)




119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)



1= 120590

1

1 120590

1

2 120590

1

119888 120590

1






SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)






119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)





119883119898119894120572) processes where



119894119895from streams119875119896



11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))


Variables 119909119896119894119895119898

119902119909119896



119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896






119894119895119898

119902119909119896



119894119895119898119902

119909119896





preceding 119900119896119894119895and




beginning

= max (moment of 119901119896119894119895



completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1198752 119875

3 119875

4 119875

5 119875

6 119875

7 and 119875



1198771 119877

119888 119877

33 119877



4 119875

5 and 119875

6


1

4 119875

2

4 119875

5= 119875

1

5 119875

2

5 and 119875

6=

1198751

6 119875

2



2=

1198751

2 119875

2

2 119875

3



1= 119875

1

1 119875

3= 119875

1

3 119875

7= 119875

1

7 and 119875

8= 119875

1

8


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5

1198751

6 1198752

6 1198751

7 and 1198751



119894


1119898119875

2 and119898119875

3


1supported


1 1198751

2 1198753

2 and 1198752

4 This means that



119894= 119898119875

1

119894 119898119875

2

119894 119898119875

3

119894 119898119875

4

119894 119894 =



1199011

1= (119877

16 119877

3 119877

13 119877

6)

1199011

2= (119877

21 119877

8 119877

18 119877

9 119877

14 119877

6 119877

17 119877

5)

1199012

2= (119877

17 119877

5 119877

21 119877

8 119877

18 119877

9 119877

14 119877

6)

1199013

2= (119877

14 119877

6 119877

17 119877

5 119877

21 119877

8 119877

18 119877

9)

1199011

3= (119877

12 119877

19 119877

9 119877

5)

1199011

4= (119877

29 119877

7 119877

25 119877

8 119877

22 119877

11 119877

26 119877

10)

1199012

4= (119877

26 119877

10 119877

29 119877

7 119877

25 119877

8 119877

22 119877

11)

1199011

5= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

27 119877

1 119877

23)

1199012

5= (119877

27 119877

1 119877

23 119877

2 119877

20 119877

5 119877

24 119877

4)

1199011

6= (119877

32 119877

4 119877

28 119877

7 119877

31)

1199012

6= (119877

31 119877

32 119877

4 119877

28 119877

7)

1199011

7= (119877

33 119877

7 119877

29 119877

10)

1199011

8= (119877

1 119877

27 119877

4 119877

30)

(1)


2 119877

4minus119877

11 119877

14 119877

17minus119877

18 119877

20minus119877

29 119877

31 and 119877

32


31119877

32represent


3 119877

12 119877

13 119877

15 119877

16 119877

19 119877

30 and

11987733





2 1198752

2 and 1198753

2which belong


4 119875

5 and 119875

6follow the




mP3

mP2

mP1

R1 R4 R7 R10

R11

R12

R2

R3R6 R9

R8R5

P5

P8 P6

P2

P3P1

P7

P4

1

1

2

2

3

3

4

4

1

2

3

4

1

3

1

1

1

1 11

1

2

2

2

2


q

q







2 and


119898119901119896

1= (119877

3 119877

13 119877

6 119877

17 119877

5 119877

21 119877

8 119877

22 119877

11)

119896 = 1 4

119898119901119896

2= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

28 119877

7 119877

29 119877

10)

119896 = 1 4

119898119901119896

3= (119877

1 119877

27 119877

4 119877

28 119877

7 119877

25 119877

8 119877

18 119877

9 119877

15 119877

12)

119896 = 1 4

(2)


2 and119898119901119896


(see Figure 2)

119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8)

(11987718 119877

9) (119877

15 119877

12)) 119896 = 1 4

(3)


13 119877

6) (119877

17 119877

5) (119877

21 119877

8)

and (11987722 119877


1 1199011

2 1199013

2 and 1199012

4


2 119877

20 119877

5)

(11987724 119877

4) (119877

28 119877

7) and (119877

29 119877


1199011

5 1199012

5 1199011

6 and 1199011


119898119901119896

2 and (119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9) and

(11987715 119877


8 1199012

6 1199011

4 1199012

2 and 1199011

3


3




12059001 (P25 P

18 P

15 ) 120590012 (P1

3 ) 120590023 (P25 P

15 )

12059002 (P15 P

25 ) 120590013 (P1

1 ) 120590024 (P15 P

25 )

12059003 (P11 ) 120590014 (P3

2 P12 P

22 ) 120590025 (P1

4 P24 )

12059004 (P16 P

18 P

26 P

15 P

25 ) 120590015 (P1

3 ) 120590026 (P24 P

14 )

12059005 (P22 P

32 P

15 P

25 P

12 ) 120590016 (P1

1 ) 120590027 (P25 P

18 P

15 )

12059006 (P32 P

12 P

11 P

22 ) 120590017 (P2

2 P32 P

12 ) 120590028 (P1

6 P26 )

12059007 (P14 P

24 P

16 P

17 P

26 ) 120590018 (P1

2 P22 P

32 ) 120590029 (P1

4 P24 P

17 )

12059008 (P12 P

14 P

22 P

32 P

24 ) 120590019 (P1

3 ) 120590030 (P18 )

12059009 (P12 P

22 P

13 P

32 ) 120590020 (P1

5 P25 ) 120590031 (P2

6 P16 )

120590010 (P24 P

17 P

14 ) 120590021 (P1

2 P22 P

32 ) 120590032 (P1

6 P26 )

120590011 (P14 P

24 ) 120590022 (P1

4 P24 ) 120590033 (P1

7 )

12059011 (mP13 mP2

3 mP33 mP4

3) 120590112 (mP33 mP4

3 mP13 mP2

3)120590123

12059012 (mP12 mP2

2 mP32 mP4

2) 120590113 (mP11 mP2

1 mP31 mP4

1) 120590124 (mP12 mP2

2 mP32 mP4

2)

12059013 (mP11 mP2

1 mP31 mP4

1) 120590114 120590125 (mP33 mP4

3 mP13 mP2

3)

12059014(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)120590115 (mP3

3 mP43 mP1

3 mP23) 120590126

12059015(mP3

1 mP42 mP4

1 mP12

mP11 mP2

2 mP21 mP3

2)120590116 120590127 (mP1

3 mP23 mP3

3 mP43)

12059016 (mP11 mP2

1 mP31 mP4

1) 120590117 (mP11 mP2

1 mP31 mP4

1) 120590128(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)

12059017(mP3

2 mP43 mP4

2 mP13

mP12 mP2

3 mP22 mP3

3)120590118

(mP33 mP4

3 mP13 mP2

3) 120590129 (mP32 mP4

2 mP12 mP2

2)

12059018(mP3

3 mP31 mP4

3 mP41

mP13 mP1

1 mP23 mP2

1)120590119 120590130

12059019(mP3

3 mP43 mP1

3 mP23) 120590120 (mP1

2 mP22 mP3

2 mP42) 120590131

120590110 (mP32 mP4

2 mP12 mP2

2) 120590121 (mP31 mP4

1 mP11 mP2

1) 120590132

120590111 (mP31 mP4

1 mP11 mP2

1) 120590122 (mP31 mP4

1 mP11 mP2

1) 120590133


4 3 2 1


Ri

(1205900i 1205901i )




R6

R1

R12

R10

R11

R9

R7

R8

R4

R5R2

mP1

mP2

mP3

P2

R13 R14 R15

R16

R17 R18

R26R25R24R23

R20 R21 R22

R27 R28 R29

R19

R31

P1 P3

P5

P8

R30 R32

P6

P4

P7

R33

Sector

Machine

4 3 2 1

R3

(12059001 12059011)

(12059002 12059012)

(12059003 12059013)

(12059006 12059016)

(12059009 12059019) (120590012 120590

112)

(120590011 120590111)(12059008 120590

18)(12059005 120590

15)

(12059004 12059014) (12059007 120590

17)

(120590010 120590110)


empty

empty

emptyempty

empty

empty

empty

empty

empty














5=

(1198752

2 119875

3

2 119875

1

5 119875

2

5 119875

1





stream 1198753


5 1198752

5 and 1198751

2 and then once

again 1198752


119896



119896



0 Θ

1 where Θ0

= 1205900

1 120590

0

33


1= 120590

1

1 120590

1



(i) A sequence 119901119896119894= (119901

119896

1198941 119901

119896

1198942 119901

119896

119894119895 119901

119896

119894119897119903(119894)) spec-


119894(the




119877 = 1198771 119877

2 119877

119888 119877




119875119896





16 119877

3 119877

13 119877

6)


1199011

1= (119901

1

11 119901

1

12 119901

1

13 119901

1


11


12= 119877

3 and so

on 119901113= 119877

13 1199011

14= 119877

6



(iii) 119905119896119894= (119905

119896

1198941 119905

119896

1198942 119905

119896

119894119895 119905

119896




(iv) 119898119901119896

119894= (119898119901119903

1199021

1198941

(1198861198941

1198871198941

) 1198981199011199031199022

1198942

(1198861198942

1198871198942

) 119898119901119903119902119910

119894119910

(119886119894119910

119887119894119910


119894from the



119898119901119903119902

119894(119886 119887)

=

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119887) 119886 le 119887

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119897119903(119894) 119901

119902

1198941 119901

119902

119894119887minus1 119901

119902

119894119887) 119886 gt 119887

119886 119887 isin 1 119897119903 (119894)

(4)


119902

1198941 119901

119902

1198942

119901119902

119894119895 119901

119902


119902

119894119886to 119901119902

119894119887



1(see


= ((1198773 119877

13 119877

6)

(11987717 119877

5) (119877

21 119877

8) (119877

22 119877

11)) where 1198981199011199031

1(2 4) =

(1198773 119877

13 119877

6) 1198981199011199031

2(7 8) = (119877

17 119877

5) 1198981199011199033

2(1 2) =

(11987721 119877

8)1198981199011199032

4(5 6) = (119877

22 119877

11)



(vi) 119898119905119896119894= (119898119905

119896

1198941 119898119905

119896

1198942 119898119905

119896

119894119895 119898119905

119896

119894119897119898(119894)) specifies


119896


119894119895


(vii) Θ = Θ0 Θ


Θ119894= 120590

119894

1 120590

119894

2 120590

119894

119888 120590

119894



119888= (119904

119894

1198881 119904

119894

119888119889 119904

119894

119888119897119901(119888)) are




1 119875

1

2

1198752

2 119875

3

2 119875

1

3 119875

1

4 119875

2

4 119875

1

5 119875

2

5 119875

1

6 119875

2

6 119875

1

7) and 1199041

119888119889isin 119898119867


119896

1 119898119875

119896

2 119898119875

119896

3| 119896 =

1 sdot sdot sdot 4)


SC = ((119877 SL) SM) (5)



2 119877

119888 119877

119898mdash



119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)




119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)



0

1 120590

0

2 120590

0

119888

1205900




1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)




119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)



1= 120590

1

1 120590

1

2 120590

1

119888 120590

1






SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)






119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)





119883119898119894120572) processes where



119894119895from streams119875119896



11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))


Variables 119909119896119894119895119898

119902119909119896



119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896






119894119895119898

119902119909119896



119894119895119898119902

119909119896





preceding 119900119896119894119895and




beginning

= max (moment of 119901119896119894119895



completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











mP3

mP2

mP1

R1 R4 R7 R10

R11

R12

R2

R3R6 R9

R8R5

P5

P8 P6

P2

P3P1

P7

P4

1

1

2

2

3

3

4

4

1

2

3

4

1

3

1

1

1

1 11

1

2

2

2

2


q

q







2 and


119898119901119896

1= (119877

3 119877

13 119877

6 119877

17 119877

5 119877

21 119877

8 119877

22 119877

11)

119896 = 1 4

119898119901119896

2= (119877

2 119877

20 119877

5 119877

24 119877

4 119877

28 119877

7 119877

29 119877

10)

119896 = 1 4

119898119901119896

3= (119877

1 119877

27 119877

4 119877

28 119877

7 119877

25 119877

8 119877

18 119877

9 119877

15 119877

12)

119896 = 1 4

(2)


2 and119898119901119896


(see Figure 2)

119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8)

(11987718 119877

9) (119877

15 119877

12)) 119896 = 1 4

(3)


13 119877

6) (119877

17 119877

5) (119877

21 119877

8)

and (11987722 119877


1 1199011

2 1199013

2 and 1199012

4


2 119877

20 119877

5)

(11987724 119877

4) (119877

28 119877

7) and (119877

29 119877


1199011

5 1199012

5 1199011

6 and 1199011


119898119901119896

2 and (119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9) and

(11987715 119877


8 1199012

6 1199011

4 1199012

2 and 1199011

3


3




12059001 (P25 P

18 P

15 ) 120590012 (P1

3 ) 120590023 (P25 P

15 )

12059002 (P15 P

25 ) 120590013 (P1

1 ) 120590024 (P15 P

25 )

12059003 (P11 ) 120590014 (P3

2 P12 P

22 ) 120590025 (P1

4 P24 )

12059004 (P16 P

18 P

26 P

15 P

25 ) 120590015 (P1

3 ) 120590026 (P24 P

14 )

12059005 (P22 P

32 P

15 P

25 P

12 ) 120590016 (P1

1 ) 120590027 (P25 P

18 P

15 )

12059006 (P32 P

12 P

11 P

22 ) 120590017 (P2

2 P32 P

12 ) 120590028 (P1

6 P26 )

12059007 (P14 P

24 P

16 P

17 P

26 ) 120590018 (P1

2 P22 P

32 ) 120590029 (P1

4 P24 P

17 )

12059008 (P12 P

14 P

22 P

32 P

24 ) 120590019 (P1

3 ) 120590030 (P18 )

12059009 (P12 P

22 P

13 P

32 ) 120590020 (P1

5 P25 ) 120590031 (P2

6 P16 )

120590010 (P24 P

17 P

14 ) 120590021 (P1

2 P22 P

32 ) 120590032 (P1

6 P26 )

120590011 (P14 P

24 ) 120590022 (P1

4 P24 ) 120590033 (P1

7 )

12059011 (mP13 mP2

3 mP33 mP4

3) 120590112 (mP33 mP4

3 mP13 mP2

3)120590123

12059012 (mP12 mP2

2 mP32 mP4

2) 120590113 (mP11 mP2

1 mP31 mP4

1) 120590124 (mP12 mP2

2 mP32 mP4

2)

12059013 (mP11 mP2

1 mP31 mP4

1) 120590114 120590125 (mP33 mP4

3 mP13 mP2

3)

12059014(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)120590115 (mP3

3 mP43 mP1

3 mP23) 120590126

12059015(mP3

1 mP42 mP4

1 mP12

mP11 mP2

2 mP21 mP3

2)120590116 120590127 (mP1

3 mP23 mP3

3 mP43)

12059016 (mP11 mP2

1 mP31 mP4

1) 120590117 (mP11 mP2

1 mP31 mP4

1) 120590128(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)

12059017(mP3

2 mP43 mP4

2 mP13

mP12 mP2

3 mP22 mP3

3)120590118

(mP33 mP4

3 mP13 mP2

3) 120590129 (mP32 mP4

2 mP12 mP2

2)

12059018(mP3

3 mP31 mP4

3 mP41

mP13 mP1

1 mP23 mP2

1)120590119 120590130

12059019(mP3

3 mP43 mP1

3 mP23) 120590120 (mP1

2 mP22 mP3

2 mP42) 120590131

120590110 (mP32 mP4

2 mP12 mP2

2) 120590121 (mP31 mP4

1 mP11 mP2

1) 120590132

120590111 (mP31 mP4

1 mP11 mP2

1) 120590122 (mP31 mP4

1 mP11 mP2

1) 120590133


4 3 2 1


Ri

(1205900i 1205901i )




R6

R1

R12

R10

R11

R9

R7

R8

R4

R5R2

mP1

mP2

mP3

P2

R13 R14 R15

R16

R17 R18

R26R25R24R23

R20 R21 R22

R27 R28 R29

R19

R31

P1 P3

P5

P8

R30 R32

P6

P4

P7

R33

Sector

Machine

4 3 2 1

R3

(12059001 12059011)

(12059002 12059012)

(12059003 12059013)

(12059006 12059016)

(12059009 12059019) (120590012 120590

112)

(120590011 120590111)(12059008 120590

18)(12059005 120590

15)

(12059004 12059014) (12059007 120590

17)

(120590010 120590110)


empty

empty

emptyempty

empty

empty

empty

empty

empty














5=

(1198752

2 119875

3

2 119875

1

5 119875

2

5 119875

1





stream 1198753


5 1198752

5 and 1198751

2 and then once

again 1198752


119896



119896



0 Θ

1 where Θ0

= 1205900

1 120590

0

33


1= 120590

1

1 120590

1



(i) A sequence 119901119896119894= (119901

119896

1198941 119901

119896

1198942 119901

119896

119894119895 119901

119896

119894119897119903(119894)) spec-


119894(the




119877 = 1198771 119877

2 119877

119888 119877




119875119896





16 119877

3 119877

13 119877

6)


1199011

1= (119901

1

11 119901

1

12 119901

1

13 119901

1


11


12= 119877

3 and so

on 119901113= 119877

13 1199011

14= 119877

6



(iii) 119905119896119894= (119905

119896

1198941 119905

119896

1198942 119905

119896

119894119895 119905

119896




(iv) 119898119901119896

119894= (119898119901119903

1199021

1198941

(1198861198941

1198871198941

) 1198981199011199031199022

1198942

(1198861198942

1198871198942

) 119898119901119903119902119910

119894119910

(119886119894119910

119887119894119910


119894from the



119898119901119903119902

119894(119886 119887)

=

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119887) 119886 le 119887

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119897119903(119894) 119901

119902

1198941 119901

119902

119894119887minus1 119901

119902

119894119887) 119886 gt 119887

119886 119887 isin 1 119897119903 (119894)

(4)


119902

1198941 119901

119902

1198942

119901119902

119894119895 119901

119902


119902

119894119886to 119901119902

119894119887



1(see


= ((1198773 119877

13 119877

6)

(11987717 119877

5) (119877

21 119877

8) (119877

22 119877

11)) where 1198981199011199031

1(2 4) =

(1198773 119877

13 119877

6) 1198981199011199031

2(7 8) = (119877

17 119877

5) 1198981199011199033

2(1 2) =

(11987721 119877

8)1198981199011199032

4(5 6) = (119877

22 119877

11)



(vi) 119898119905119896119894= (119898119905

119896

1198941 119898119905

119896

1198942 119898119905

119896

119894119895 119898119905

119896

119894119897119898(119894)) specifies


119896


119894119895


(vii) Θ = Θ0 Θ


Θ119894= 120590

119894

1 120590

119894

2 120590

119894

119888 120590

119894



119888= (119904

119894

1198881 119904

119894

119888119889 119904

119894

119888119897119901(119888)) are




1 119875

1

2

1198752

2 119875

3

2 119875

1

3 119875

1

4 119875

2

4 119875

1

5 119875

2

5 119875

1

6 119875

2

6 119875

1

7) and 1199041

119888119889isin 119898119867


119896

1 119898119875

119896

2 119898119875

119896

3| 119896 =

1 sdot sdot sdot 4)


SC = ((119877 SL) SM) (5)



2 119877

119888 119877

119898mdash



119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)




119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)



0

1 120590

0

2 120590

0

119888

1205900




1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)




119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)



1= 120590

1

1 120590

1

2 120590

1

119888 120590

1






SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)






119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)





119883119898119894120572) processes where



119894119895from streams119875119896



11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))


Variables 119909119896119894119895119898

119902119909119896



119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896






119894119895119898

119902119909119896



119894119895119898119902

119909119896





preceding 119900119896119894119895and




beginning

= max (moment of 119901119896119894119895



completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











12059001 (P25 P

18 P

15 ) 120590012 (P1

3 ) 120590023 (P25 P

15 )

12059002 (P15 P

25 ) 120590013 (P1

1 ) 120590024 (P15 P

25 )

12059003 (P11 ) 120590014 (P3

2 P12 P

22 ) 120590025 (P1

4 P24 )

12059004 (P16 P

18 P

26 P

15 P

25 ) 120590015 (P1

3 ) 120590026 (P24 P

14 )

12059005 (P22 P

32 P

15 P

25 P

12 ) 120590016 (P1

1 ) 120590027 (P25 P

18 P

15 )

12059006 (P32 P

12 P

11 P

22 ) 120590017 (P2

2 P32 P

12 ) 120590028 (P1

6 P26 )

12059007 (P14 P

24 P

16 P

17 P

26 ) 120590018 (P1

2 P22 P

32 ) 120590029 (P1

4 P24 P

17 )

12059008 (P12 P

14 P

22 P

32 P

24 ) 120590019 (P1

3 ) 120590030 (P18 )

12059009 (P12 P

22 P

13 P

32 ) 120590020 (P1

5 P25 ) 120590031 (P2

6 P16 )

120590010 (P24 P

17 P

14 ) 120590021 (P1

2 P22 P

32 ) 120590032 (P1

6 P26 )

120590011 (P14 P

24 ) 120590022 (P1

4 P24 ) 120590033 (P1

7 )

12059011 (mP13 mP2

3 mP33 mP4

3) 120590112 (mP33 mP4

3 mP13 mP2

3)120590123

12059012 (mP12 mP2

2 mP32 mP4

2) 120590113 (mP11 mP2

1 mP31 mP4

1) 120590124 (mP12 mP2

2 mP32 mP4

2)

12059013 (mP11 mP2

1 mP31 mP4

1) 120590114 120590125 (mP33 mP4

3 mP13 mP2

3)

12059014(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)120590115 (mP3

3 mP43 mP1

3 mP23) 120590126

12059015(mP3

1 mP42 mP4

1 mP12

mP11 mP2

2 mP21 mP3

2)120590116 120590127 (mP1

3 mP23 mP3

3 mP43)

12059016 (mP11 mP2

1 mP31 mP4

1) 120590117 (mP11 mP2

1 mP31 mP4

1) 120590128(mP4

3 mP42 mP1

3 mP12

mP23 mP2

2 mP13 mP1

2)

12059017(mP3

2 mP43 mP4

2 mP13

mP12 mP2

3 mP22 mP3

3)120590118

(mP33 mP4

3 mP13 mP2

3) 120590129 (mP32 mP4

2 mP12 mP2

2)

12059018(mP3

3 mP31 mP4

3 mP41

mP13 mP1

1 mP23 mP2

1)120590119 120590130

12059019(mP3

3 mP43 mP1

3 mP23) 120590120 (mP1

2 mP22 mP3

2 mP42) 120590131

120590110 (mP32 mP4

2 mP12 mP2

2) 120590121 (mP31 mP4

1 mP11 mP2

1) 120590132

120590111 (mP31 mP4

1 mP11 mP2

1) 120590122 (mP31 mP4

1 mP11 mP2

1) 120590133


4 3 2 1


Ri

(1205900i 1205901i )




R6

R1

R12

R10

R11

R9

R7

R8

R4

R5R2

mP1

mP2

mP3

P2

R13 R14 R15

R16

R17 R18

R26R25R24R23

R20 R21 R22

R27 R28 R29

R19

R31

P1 P3

P5

P8

R30 R32

P6

P4

P7

R33

Sector

Machine

4 3 2 1

R3

(12059001 12059011)

(12059002 12059012)

(12059003 12059013)

(12059006 12059016)

(12059009 12059019) (120590012 120590

112)

(120590011 120590111)(12059008 120590

18)(12059005 120590

15)

(12059004 12059014) (12059007 120590

17)

(120590010 120590110)


empty

empty

emptyempty

empty

empty

empty

empty

empty














5=

(1198752

2 119875

3

2 119875

1

5 119875

2

5 119875

1





stream 1198753


5 1198752

5 and 1198751

2 and then once

again 1198752


119896



119896



0 Θ

1 where Θ0

= 1205900

1 120590

0

33


1= 120590

1

1 120590

1



(i) A sequence 119901119896119894= (119901

119896

1198941 119901

119896

1198942 119901

119896

119894119895 119901

119896

119894119897119903(119894)) spec-


119894(the




119877 = 1198771 119877

2 119877

119888 119877




119875119896





16 119877

3 119877

13 119877

6)


1199011

1= (119901

1

11 119901

1

12 119901

1

13 119901

1


11


12= 119877

3 and so

on 119901113= 119877

13 1199011

14= 119877

6



(iii) 119905119896119894= (119905

119896

1198941 119905

119896

1198942 119905

119896

119894119895 119905

119896




(iv) 119898119901119896

119894= (119898119901119903

1199021

1198941

(1198861198941

1198871198941

) 1198981199011199031199022

1198942

(1198861198942

1198871198942

) 119898119901119903119902119910

119894119910

(119886119894119910

119887119894119910


119894from the



119898119901119903119902

119894(119886 119887)

=

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119887) 119886 le 119887

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119897119903(119894) 119901

119902

1198941 119901

119902

119894119887minus1 119901

119902

119894119887) 119886 gt 119887

119886 119887 isin 1 119897119903 (119894)

(4)


119902

1198941 119901

119902

1198942

119901119902

119894119895 119901

119902


119902

119894119886to 119901119902

119894119887



1(see


= ((1198773 119877

13 119877

6)

(11987717 119877

5) (119877

21 119877

8) (119877

22 119877

11)) where 1198981199011199031

1(2 4) =

(1198773 119877

13 119877

6) 1198981199011199031

2(7 8) = (119877

17 119877

5) 1198981199011199033

2(1 2) =

(11987721 119877

8)1198981199011199032

4(5 6) = (119877

22 119877

11)



(vi) 119898119905119896119894= (119898119905

119896

1198941 119898119905

119896

1198942 119898119905

119896

119894119895 119898119905

119896

119894119897119898(119894)) specifies


119896


119894119895


(vii) Θ = Θ0 Θ


Θ119894= 120590

119894

1 120590

119894

2 120590

119894

119888 120590

119894



119888= (119904

119894

1198881 119904

119894

119888119889 119904

119894

119888119897119901(119888)) are




1 119875

1

2

1198752

2 119875

3

2 119875

1

3 119875

1

4 119875

2

4 119875

1

5 119875

2

5 119875

1

6 119875

2

6 119875

1

7) and 1199041

119888119889isin 119898119867


119896

1 119898119875

119896

2 119898119875

119896

3| 119896 =

1 sdot sdot sdot 4)


SC = ((119877 SL) SM) (5)



2 119877

119888 119877

119898mdash



119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)




119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)



0

1 120590

0

2 120590

0

119888

1205900




1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)




119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)



1= 120590

1

1 120590

1

2 120590

1

119888 120590

1






SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)






119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)





119883119898119894120572) processes where



119894119895from streams119875119896



11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))


Variables 119909119896119894119895119898

119902119909119896



119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896






119894119895119898

119902119909119896



119894119895119898119902

119909119896





preceding 119900119896119894119895and




beginning

= max (moment of 119901119896119894119895



completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















5=

(1198752

2 119875

3

2 119875

1

5 119875

2

5 119875

1





stream 1198753


5 1198752

5 and 1198751

2 and then once

again 1198752


119896



119896



0 Θ

1 where Θ0

= 1205900

1 120590

0

33


1= 120590

1

1 120590

1



(i) A sequence 119901119896119894= (119901

119896

1198941 119901

119896

1198942 119901

119896

119894119895 119901

119896

119894119897119903(119894)) spec-


119894(the




119877 = 1198771 119877

2 119877

119888 119877




119875119896





16 119877

3 119877

13 119877

6)


1199011

1= (119901

1

11 119901

1

12 119901

1

13 119901

1


11


12= 119877

3 and so

on 119901113= 119877

13 1199011

14= 119877

6



(iii) 119905119896119894= (119905

119896

1198941 119905

119896

1198942 119905

119896

119894119895 119905

119896




(iv) 119898119901119896

119894= (119898119901119903

1199021

1198941

(1198861198941

1198871198941

) 1198981199011199031199022

1198942

(1198861198942

1198871198942

) 119898119901119903119902119910

119894119910

(119886119894119910

119887119894119910


119894from the



119898119901119903119902

119894(119886 119887)

=

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119887) 119886 le 119887

(119901119902

119894119886 119901

119902

119894119886+1 119901

119902

119894119897119903(119894) 119901

119902

1198941 119901

119902

119894119887minus1 119901

119902

119894119887) 119886 gt 119887

119886 119887 isin 1 119897119903 (119894)

(4)


119902

1198941 119901

119902

1198942

119901119902

119894119895 119901

119902


119902

119894119886to 119901119902

119894119887



1(see


= ((1198773 119877

13 119877

6)

(11987717 119877

5) (119877

21 119877

8) (119877

22 119877

11)) where 1198981199011199031

1(2 4) =

(1198773 119877

13 119877

6) 1198981199011199031

2(7 8) = (119877

17 119877

5) 1198981199011199033

2(1 2) =

(11987721 119877

8)1198981199011199032

4(5 6) = (119877

22 119877

11)



(vi) 119898119905119896119894= (119898119905

119896

1198941 119898119905

119896

1198942 119898119905

119896

119894119895 119898119905

119896

119894119897119898(119894)) specifies


119896


119894119895


(vii) Θ = Θ0 Θ


Θ119894= 120590

119894

1 120590

119894

2 120590

119894

119888 120590

119894



119888= (119904

119894

1198881 119904

119894

119888119889 119904

119894

119888119897119901(119888)) are




1 119875

1

2

1198752

2 119875

3

2 119875

1

3 119875

1

4 119875

2

4 119875

1

5 119875

2

5 119875

1

6 119875

2

6 119875

1

7) and 1199041

119888119889isin 119898119867


119896

1 119898119875

119896

2 119898119875

119896

3| 119896 =

1 sdot sdot sdot 4)


SC = ((119877 SL) SM) (5)



2 119877

119888 119877

119898mdash



119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)




119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)



0

1 120590

0

2 120590

0

119888

1205900




1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)




119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)



1= 120590

1

1 120590

1

2 120590

1

119888 120590

1






SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)






119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)





119883119898119894120572) processes where



119894119895from streams119875119896



11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))


Variables 119909119896119894119895119898

119902119909119896



119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896






119894119895119898

119902119909119896



119894119895119898119902

119909119896





preceding 119900119896119894119895and




beginning

= max (moment of 119901119896119894119895



completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2 119877

119888 119877

119898mdash



119880 = 1199011

1 119901

119897119904(1)

1 119901

1

119899 119901

119897119904(119899)




119879 = 1199051

1 119905

119897119904(1)

1 119905

1

119899 119905

119897119904(119899)



0

1 120590

0

2 120590

0

119888

1205900




1 119898119901

119897119904119898(1)

1 119898119901

1

119908

119898119901119897119904119898(119899)




119898119879 = 1198981199051

1 119898119905

119897119904119898(1)

119908 119898119905

1

119899 119898119905

119897119904119898(119899)



1= 120590

1

1 120590

1

2 120590

1

119888 120590

1






SC119897119901= (((((119877 SL) SM1

) SM2) SM119902

) SM119897119901)

(6)






119897119901= (((((119883 120572) (119898

1119883119898

1120572)) ) (119898

119902119883119898

119902120572))

(119898119897119901119883119898

119897119901120572))

(7)





119883119898119894120572) processes where



119894119895from streams119875119896



11 119909

1

12 119909

119897119904(119899)

119899119897119903(119899)) and by analogy

the sequence 119898119902119883 = (119898

1199021199091

11 119898

1199021199091

12 119898

119902119909119897119904119898(119908119902)

119908(119902)119897119898(119908119902))


Variables 119909119896119894119895119898

119902119909119896



119894119895(119897) = 119909

119896

119894119895+ 119897 sdot 120572119898

119902119909119896

119894119895(119897) = 119898

119902119909119896

119894119895+

119897 sdot 119898119902120572

Since values of 119909119896119894119895119898

119902119909119896






119894119895119898

119902119909119896



119894119895119898119902

119909119896





preceding 119900119896119894119895and




beginning

= max (moment of 119901119896119894119895



completion)

119894 = 1 119899 119895 = 1 119897119903 (119894) 119896 = 1 119897119904 (119894)

(8)


12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12059007

120590031120590032120590030

R32R30R3132R30R31R32R

1205907

R32

000 120590031120590032120590030

12059000

0000 00003030000

000

1205900

1205901205900

R3 R13 R6 R14 R9 R15

R2 R20 R5 R21 R8

R1 R27 R4 R28 R7 R29

e set of resourcesR


R12

R16 R19

R22 R11

R17 R18

R25 R26R24R24

R10

R33

SL level

12059003

120590016

120590013

12059002

120590024

120590020

12059006

12059001

120590017

120590024

120590014

12059005

12059004120590027

120590033

120590021

120590018

12059008

12059009

120590015

120590019

120590012

120590022120590011

120590010120590029

120590026120590025

P1 P3

P7P8 P6

P5 P4

P2

mP1

mP2

mP3

12059011312059016

120590117

12059015 12059012112059018

12059011812059019 120590115 120590112

120590122 120590111

120590125

12059017 120590129 12059011012059012812059014120590124

120590127

120590021

12059013

SM1 level


SL = (U T Θ0)






119894119895of values

119898119909119896



preceding 119898119900119894119895





beginning





119894 = 1 119899 119895 = 1 119897119898 (119894) 119896 = 1 119897119904119898 (119894)

(9)



119894119895






119861= min119896 isin 119861 119896 ge 119886








1 119898119875

2 and



1ndashP






11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11= 119909

1

14+ 119905

1

14minus 120572 119909

2

41= max (1199091

48+ Δ119905 minus 120572) (119909

2

48+ 119905

2


1199091

12= 119909

1

11+ 119905

1

111199092

42= max (1199091

41+ Δ119905) (119909

2

41+ 119905

2

41) 119909

2

51= max (1199091

57+ Δ119905 minus 120572) (119909

2

58+ 119905

2

58minus 120572)

1199091

13= 119909

1

12+ 119905

1

121199092

43= max (1199091

42+ Δ119905) (119909

2

42+ 119905

2

42) 119909

2

52= max (1199091

58+ Δ119905 minus 120572) (119909

2

51+ 119905

2

51)

1199091

14= max (1199091

27+ Δ119905) (119909

1

13+ 119905

1

13) 119909

2

44= max (1199091

43+ Δ119905) (119909

2

43+ 119905

2

43) 119909

2

53= max (1199091

51+ Δ119905) (119909

2

52+ 119905

2

52)

mdash 1199092

45= max (1199091

44+ Δ119905) (119909

2

44+ 119905

2

44) 119909

2

54= max (1199091

52+ Δ119905) (119909

2

53+ 119905

2

53)

1199091

21= max (1199093

26+ Δ119905 minus 120572) (119909

1

28+ 119905

1

28minus 120572) 119909

2

46= max (1199093

27+ Δ119905) (119909

2

45+ 119905

2

45) 119909

2

55= max (1199091

53+ Δ119905) (119909

2

54+ 119905

2

54)

1199091

22= max (1199092

47+ Δ119905 minus 120572) (119909

1

21+ 119905

1

21) 119909

2

47= max (1199091

46+ Δ119905) (119909

2

46+ 119905

2

46) 119909

2

56= max (1199091

54+ Δ119905) (119909

2

55+ 119905

2

55)

1199091

23= max (1199093

28+ Δ119905 minus 120572) (119909

1

22+ 119905

1

22) 119909

2

48= max (1199091

47+ Δ119905) (119909

2

47+ 119905

2

47) 119909

2

57= max (1199091

55+ Δ119905) (119909

2

56+ 119905

2

56)

1199091

24= max (1199093

21+ Δ119905) (119909

1

23+ 119905

1

23) mdash 119909

2

58= max (1199091

56+ Δ119905) (119909

2

57+ 119905

2

57)

1199091

25= max (1199093

22+ Δ119905) (119909

1

24+ 119905

1

24) 119909

1

51= max (1199092

55+ Δ119905 minus 120572) (119909

1

58+ 119905

1


1199091

26= max (1199093

23+ Δ119905) (119909

1

25+ 119905

1

25) 119909

1

52= max (1199092

56+ Δ119905 minus 120572) (119909

1

51+ 119905

1

51) 119909

1

61= max (1199092

63+ Δ119905 minus 120572) (119909

1

65+ 119905

1

65minus 120572)

1199091

27= max (1199093

24+ Δ119905) (119909

1

26+ 119905

1

26) 119909

1

53= max (1199093

25+ Δ119905) (119909

1

52+ 119905

1

52) 119909

1

62= max (1199092

51+ Δ119905) (119909

1

61+ 119905

1

61)

1199091

28= max (1199092

57+ Δ119905) (119909

1

27+ 119905

1

27) 119909

1

54= max (1199092

58+ Δ119905 minus 120572) (119909

1

53+ 119905

1

53) 119909

1

63= max (1199092

65+ Δ119905 minus 120572) (119909

1

62+ 119905

1

62)

mdash 1199091

55= max (1199092

64+ Δ119905) (119909

1

54+ 119905

1

54) 119909

1

64= max (1199092

45+ Δ119905) (119909

1

63+ 119905

1

63)

1199092

21= max (1199091

28+ Δ119905 minus 120572) (119909

2

28+ 119905

2

28minus 120572) 119909

1

56= max (1199091

83+ Δ119905) (119909

1

55+ 119905

1

55) 119909

1

65= max (1199092

62+ Δ119905) (119909

1

64+ 119905

1

64)

1199092

22= max (1199091

21+ Δ119905) (119909

2

21+ 119905

2

21) 119909

1

57= max (1199091

82+ Δ119905) (119909

1

56+ 119905

1

56) mdash

1199092

23= max (1199091

22+ Δ119905) (119909

2

22+ 119905

2

22) 119909

1

58= max (1199092

54+ Δ119905) (119909

1

57+ 119905

1

57) 119909

2

61= max (1199092

63+ Δ119905) (119909

2

65+ 119905

2

65minus 120572)

1199092

24= max (1199091

45+ Δ119905) (119909

2

23+ 119905

2

23) mdash 119909

2

62= max (1199091

62+ Δ119905) (119909

2

61+ 119905

2

61)

1199092

25= max (1199091

24+ Δ119905) (119909

2

24+ 119905

2

24) 119909

1

31= 119909

1

34+ 119905

1

34minus 120572 119909

2

63= max (1199091

84+ Δ119905) (119909

2

62+ 119905

2

62)

1199092

26= max (1199091

25+ Δ119905) (119909

2

25+ 119905

2

25) 119909

1

32= 119909

1

31+ 119905

1

311199092

64= max (1199091

64+ Δ119905) (119909

2

63+ 119905

2

63)

1199092

27= max (1199091

21+ Δ119905) (119909

2

26+ 119905

2

26) 119909

1

33= max (1199092

27+ Δ119905) (119909

1

32+ 119905

1

32) 119909

2

65= max (1199091

73+ Δ119905) (119909

2

64+ 119905

2

64)

1199092

28= max (1199091

11+ Δ119905 minus 120572) (119909

2

27+ 119905

2

27) 119909

1

34= 119909

1

33+ 119905

1

33mdash

mdash mdash 1199091

71= 119909

1

74+ 119905

1

74minus 120572

1199093

21= max (1199092

28+ Δ119905 minus 120572) (119909

3

28+ 119905

3

28minus 120572) 119909

1

41= max (1199091

74+ Δ119905 minus 120572) (119909

1

48+ 119905

1

48minus 120572) 119909

1

72= max (1199091

65+ Δ119905) (119909

1

71+ 119905

1

71)

1199093

22= max (1199092

21+ Δ119905) (119909

3

21+ 119905

3

21) 119909

1

42= max (1199092

61+ Δ119905) (119909

1

41+ 119905

1

41) 119909

1

73= max (1199092

44+ Δ119905) (119909

1

72+ 119905

1

72)

1199093

23= max (1199092

22+ Δ119905) (119909

3

22+ 119905

3

22) 119909

1

43= max (1199092

46+ Δ119905 minus 120572) (119909

1

42+ 119905

1

42) 119909

1

74= max (1199092

43+ Δ119905) (119909

1

73+ 119905

1

73)

1199093

24= max (1199092

23+ Δ119905) (119909

3

23+ 119905

3

23) 119909

1

44= max (1199091

23+ Δ119905) (119909

1

43+ 119905

1

43) mdash

1199093

25= max (1199092

24+ Δ119905) (119909

3

24+ 119905

3

24) 119909

1

45= max (1199092

48+ Δ119905 minus 120572) (119909

1

44+ 119905

1

44) 119909

1

81= max (1199092

53+ Δ119905) (119909

1

84+ 119905

1

84minus 120572)

1199093

26= max (1199092

25+ Δ119905) (119909

3

25+ 119905

3

25) 119909

1

46= max (1199092

41+ Δ119905) (119909

1

45+ 119905

1

45) 119909

1

82= max (1199092

52+ Δ119905) (119909

1

81+ 119905

1

81)

1199093

27= max (1199092

26+ Δ119905) (119909

3

26+ 119905

3

26) 119909

1

47= max (1199092

42+ Δ119905) (119909

1

46+ 119905

1

46) 119909

1

83= max (1199091

63+ Δ119905) (119909

1

82+ 119905

1

82)

1199093

28= max (1199092

27+ Δ119905) (119909

3

27+ 119905

3

27) 119909

1

48= max (1199091

71+ Δ119905 minus 120572) (119909

1

47+ 119905

1

47) 119909

1

84= 119909

1

83+ 119905

1

83



CS119883119879= ((119883

119897119901 119879

119897119901 119863

119883 119863

119879) 119862

119871 119862

119872) (10)



1) 119898119879

119902) 119898119879

119897119901) 119863

119883 119863

119879 119863

120572= Nmdash


119863119883 119898

119902119909119896

119894119895 119909119896

119894119895isin Z 119898119902

120572 120572 isin N 119863119879 119898

119902119905119896

119894119895 119905119896

119894119895isin N

119862119871 119862


119871and119862






119871119862



119871 119862





Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table2Con

straints(9)d

eterminingthem

ultim

odalprocessesb

ehavioro

fthe

SCCP

from

Figure

2Con

straintsfor

stream

1198981198751 1

Con

straintsfor

stream

1198981198751 2

Con

straintsfor

stream

1198981198751 3

1198981199091 11=lceilmax

(1198981199094 12+Δ119905119898


1198981199091 12=lceilmax

(1198981199094 13+Δ119905119898

minus119898120572)(1198981199091 11+1198981199051 11)rceilX1 11

1198981199091 13=lceilmax

(1198981199094 14+Δ119905119898

minus119898120572)(1198981199091 12+1198981199051 12)rceilX1 13

1198981199091 14=lceilmax

(1198981199094 15+Δ119905119898

minus119898120572)(1198981199091 13+1198981199051 13)rceilX1 14

1198981199091 15=lceilmax

(1198981199091 24+Δ119905119898)(1198981199091 14+1198981199051 14)rceilX1 15

1198981199091 16=lceilmax

(1198981199094 17+Δ119905119898)(1198981199091 15+1198981199051 15)rceilX1 16

1198981199091 17=lceilmax

(1198981199091 38+Δ119905119898)(1198981199091 16+1198981199051 16)rceilX1 17

1198981199091 18=lceilmax

(1198981199094 19+Δ119905119898)(1198981199091 17+1198981199051 17)rceilX1 18

1198981199091 19=lceilmax

(1198981199094 19+1198981199054 19+Δ119905119898)(1198981199091 18+1198981199051 18)rceilX1 19

1198981199091 21=lceilmax

(1198981199094 22+Δ119905119898


1198981199091 22=lceilmax

(1198981199094 23+Δ119905119898

minus119898120572)(1198981199091 21+1198981199051 21)rceilX1 21

1198981199091 23=lceilmax

(1198981199094 16+Δ119905119898)(1198981199091 22+1198981199051 22)rceilX1 23

1198981199091 24=lceilmax

(1198981199094 25+Δ119905119898

minus119898120572)(1198981199091 23+1198981199051 23)rceilX1 24

1198981199091 25=lceilmax

(1198981199091 34+Δ119905119898)(1198981199091 24+1198981199051 24)rceilX1 25

1198981199091 26=lceilmax

(1198981199091 35+Δ119905119898)(1198981199091 25+1198981199051 25)rceilX1 26

1198981199091 27=lceilmax

(1198981199091 36+Δ119905119898)(1198981199091 26+1198981199051 26)rceilX1 27

1198981199091 28=lceilmax

(1198981199094 29+Δ119905119898)(1198981199091 27+1198981199051 27)rceilX1 28

1198981199091 29=lceilmax

(1198981199094 29+1198981199054 29+Δ119905119898)(1198981199091 28+1198981199051 28)rceilX1 29

1198981199091 31=lceilmax

(1198981199094 32+Δ119905119898


1198981199091 32=lceilmax

(1198981199094 33+Δ119905119898

minus119898120572)(1198981199091 31+1198981199051 31)rceilX1 31

1198981199091 33=lceilmax

(1198981199094 26+Δ119905119898)(1198981199091 32+1198981199051 32)rceilX1 33

1198981199091 34=lceilmax

(1198981199094 27+Δ119905119898

minus119898120572)(1198981199091 33+1198981199051 33)rceilX1 34

1198981199091 35=lceilmax

(1198981199094 28+Δ119905119898)(1198981199091 34+1198981199051 34)rceilX1 35

1198981199091 36=lceilmax

(1198981199094 37+Δ119905119898)(1198981199091 35+1198981199051 35)rceilX1 36

1198981199091 37=lceilmax

(1198981199094 18+Δ119905119898)(1198981199091 36+1198981199051 36)rceilX1 37

1198981199091 38=lceilmax

(1198981199094 39+Δ119905119898)(1198981199091 37+1198981199051 37)rceilX1 38

1198981199091 39=lceilmax

(1198981199094 310+Δ119905119898)(1198981199091 38+1198981199051 38)rceilX1 39

1198981199091 310=lceilmax

(1198981199094 311+Δ119905119898)(1198981199091 39+1198981199051 39)rceilX1 310

1198981199091 311=lceilmax

(1198981199094 111+1198981199054 111+Δ119905119898)(1198981199091 310+1198981199051 310)rceilX1 311

1198981199091 11isinX1 11=1199091 12(119897 )|1199091 12(119897 )=1199091 12+119897sdot120572119897isinZ

1198981199091 12isinX1 12=1199091 13(119897 )|1199091 13(119897 )=1199091 13+119897sdot120572119897isinZ

1198981199091 13isinX1 13=1199091 14(119897 )|1199091 14(119897 )=1199091 14+119897sdot120572119897isinZ

1198981199091 14isinX1 14=1199091 27(119897 )|1199091 27(119897 )=1199091 27+119897sdot120572119897isinZ

1198981199091 15isinX1 15=1199091 28(119897 )|1199091 28(119897 )=1199091 28+119897sdot120572119897isinZ

1198981199091 16isinX1 16=1199093 25(119897 )|1199093 25(119897 )=1199093 25+119897sdot120572119897isinZ

1198981199091 17isinX1 17=1199093 26(119897 )|1199093 26(119897 )=1199093 26+119897sdot120572119897isinZ

1198981199091 18isinX1 18=1199092 47(119897 )|1199092 47(119897 )=1199092 47+119897sdot120572119897isinZ

1198981199091 19isinX1 19=1199092 48(119897 )|1199092 48(119897 )=1199092 48+119897sdot120572119897isinZ

1198981199091 21isinX1 21=1199091 51(119897 )|1199091 51(119897 )=1199091 51+119897sdot120572119897isinZ

1198981199091 22isinX1 22=1199091 52(119897 )|1199091 52(119897 )=1199091 52+119897sdot120572119897isinZ

1198981199091 23isinX1 23=1199091 53(119897 )|1199091 53(119897 )=1199091 53+119897sdot120572119897isinZ

1198981199091 24isinX1 24=1199092 57(119897 )|1199092 57(119897 )=1199092 57+119897sdot120572119897isinZ

1198981199091 25isinX1 25=1199092 58(119897 )|1199092 58(119897 )=1199092 58+119897sdot120572119897isinZ

1198981199091 26isinX1 26=1199091 63(119897 )|1199091 63(119897 )=1199091 63+119897sdot120572119897isinZ

1198981199091 27isinX1 27=1199091 64(119897 )|1199091 63(119897 )=1199091 64+119897sdot120572119897isinZ

1198981199091 28isinX1 28=1199091 73(119897 )|1199091 73(119897 )=1199091 73+119897sdot120572119897isinZ

1198981199091 29isinX1 29=1199091 74(119897 )|1199091 74(119897 )=1199091 74+119897sdot120572119897isinZ

1198981199091 31isinX1 31=1199091 81(119897 )|1199091 81(119897 )=1199091 81+119897sdot120572119897isinZ

1198981199091 32isinX1 32=1199091 82(119897 )|1199091 82(119897 )=1199091 82+119897sdot120572119897isinZ

1198981199091 33isinX1 33=1199091 83(119897 )|1199091 83(119897 )=1199091 83+119897sdot120572119897isinZ

1198981199091 34isinX1 34=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 35isinX1 35=1199092 64(119897 )|1199092 62(119897 )=1199092 64+119897sdot120572119897isinZ

1198981199091 36isinX1 36=1199091 43(119897 )|1199091 43(119897 )=1199091 43+119897sdot120572119897isinZ

1198981199091 37isinX1 37=1199091 44(119897 )|1199091 43(119897 )=1199091 44+119897sdot120572119897isinZ

1198981199091 38isinX1 38=1199092 25(119897 )|1199092 25(119897 )=1199092 25+119897sdot120572119897isinZ

1198981199091 39isinX1 39=1199092 26(119897 )|1199092 26(119897 )=1199092 26+119897sdot120572119897isinZ

1198981199091 310isinX1 310=1199091 33(119897 )|1199091 33(119897 )=1199091 33+119897sdot120572119897isinZ

1198981199091 311isinX1 311=1199091 34(119897 )|1199091 34(119897 )=1199091 34+119897sdot120572119897isinZ

Con

straintso

fthe

restof

streams119898

1198752 1119898

1198753 1119898

1198754 1119898

1198752 2119898

1198753 2119898

1198754 2119898

1198752 3119898

1198753 3and

1198981198754 3ared

efinedin

thes

amew

ay


R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

400 60 80 100 120 140

Inpu

t res

ourc

esO

utpu

t res

ourc

es

Automatic unloading

Automatic loading

P7

R7 R10

t = 34 t = 36

Load time3534 36

Unload time

Transportationtime


3

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31 mP4

1 mP11

mP21

mP11

mP41

mP13

mP13

mP42

mP43

mP43

mP41

m1P42

mP43

mP31mP3

3

mP42

mP32

mP41

mP31

mP33

S41S18S0S1 S7 S7 S0 S41


ji




- Local process P3

- Local process P4

- Local process P5

- Local process P6




mPi1

mPi2

mPi3










119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119878119903= ((((119878119897

119903 119898

1119878119903

) 119898119897119878119903

) ) 119898119897119901119878119903

) (11)


119878119897119903= (119860

119903 119885

119903 119876

119903) (12)


1

119903 119886

2

119903 119886

119888

119903

119886119898


119888

119903isin 119875 cup Δ

119886119888

119903= 119875

119896



119875119896

119894 and 119886

119888


119888is unoccupied

119885119903= (119911

1

119903 119911

2

119903 119911

119888

119903 119911

119898






119888

119903= 119875

119896


119896

119894is


119903= (119902

1

119903 119902

2

119903 119902

119888

119903 119902

119898



119903



119888 119911

119888

119903= 119904

119888(119902119888

119903) 119902

119888


2

119903= 2

and 1199112

119903= 119875

2


2

119903= 119875

2

1taking


(iii) 119898119897119878119903


119898119897119878119903= (119898

119897119860

119903

119898119897119885

119903

119898119897119876

119903

) (13)


119903

= (119898119897119860

1199031

1

119898119897119860

119903ℎ

119894 119898

119897119860

119903119897119904119898(119897119908(119897)119897)


streams119898119897119875ℎ

119894 that is

119898119897119860

119903ℎ

119894= (119898

119897119886119903ℎ

1198941 119898

119897119886119903ℎ

1198942 119898

119897119886119903ℎ

119894119888 119898

119897119886119903ℎ

119894119898) (14)




119886119903ℎ

119894119888isin 119898

119897119875ℎ

119894 Δ

119898119897119886119903ℎ

119894119888= 119898

119897119875ℎ


119888is


119897119875119894 and 119898119897

119886119903ℎ


119888which


119897119875119894

119898119897119885

119903

=(1198981198971199111

119903

119898119897119911119888

119903

119898119897119911119898

119903

) and119898119897119876

119903

= (1198981198971199021

119903

119898119897119902119896

119903 119898

119897119902119898

119903




= (119881119882 119864



119882= 119878

0 119878



119882sube

119881119882times119881





119882 119864

119882)








119882digraphs


1198661198821

cup 1198661198822


(where 119866119882119886

cup 119866119882119887

= (119881119886cup

119881119887 119864

119886cup119864





Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sl41

Sl30

SSl41

Sl30

Sl0

Sl1

Sl9

Sl9

S30

S0

S1

S41

119979



S20 = (Sl20 m1S20)

Sl20=(A20 Z20 Q20)


r)



2= ((119877

2

11987720 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

25 119877

8 119877

22 119877

11)) and 119898119901119896

3=

((1198771 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877



7) with



119882



1198821to 119866

1198822


1198821to the other 119866

1198822



1198821 119866





and a tree 119866119879119894


1198781 119866



119866119878= (119881

119878 119864

119878) = 119866

119882cup (

119897119905

⋃

119894=1

119866119879119894) (15)

where⋃119897119905

119894=1119866

119879119894= 119866

1198791cup 119866


119879119897119905 119897119905 is the number of

trees 119866119879119894



1198781 119866



1198821 119866

1198822(being

subdigraphs of 1198661198781 119866




1198821

1198661198822


1198821

into 1198661198822


119879119894) leading to 119866

1198822 The


1198821



1198781and 119866

1198782 In other


(i) for all 119878119896 isin 1198811198821



1198821

(ii) for all 119878119897 isin 1198811198822



1198822



R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12

40200 60 80 100 120 140 160 180

Inpu

t res

ourc

esO

utpu

t res

ourc

es


ji

(units time)mP13

mP23

mP12

mP22

mP11

mP43 mP4

2

mP12

mP21

mP31

mP31

mP32

mP32

mP41 mP2

1

mP21

mP11

mP11

mP13

mP13

mP41

mP41

mP41 mP4

2

m1P42

mP43

mP31

mP42

mP42

mP42

mP32

mP32

mP31




- Local process P3

- Local process P4

- Local process P5

- Local process P6


S42



mPi1

mPi2

mPi3




S19 S21S0S1 S15 S15 S0 S42



1198821 119866






119903 119885

119903 119876

119903)





1198853 119876

3) and the state 1198781198974 = (1198602

1198854 119876

4)



Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sl41

Sl30

Sl41

Sl30

Sl0

Sl1

Sl9

S0

S1S20

S41

GW1

119979

SC

S9

42 states

Sl20

11997901199790

(a)

S41

Sl41

Sl30

S41

SSl41

Sl30

S

Sl0

Sl1

Sl9

S0

S1

S42

GW2 S21

SC

119979

43 states

S9

(b)




GW1GW2

GS2

Transient period

Transient period

Sx

Sx998400

GTi

Sx998400

Sx

GS1





1 119885

1 119876

1)

and 1198781198972 = (1198601 119885

2 119876









119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119979SL

119979

Aj

Sl3 =

A1

A2

Sl4 =

119964


Sl2

Slj Sl1

Sj S1

S2

S4

S3GW1

GW1

GW2

GS1

GS2

Eval

uatio

n


(A2 Z4 Q4)

(A2 Z3 Q3)

atio

naaataa




1198781 119866

1198782of the


1198781


1198821 and the state 1198782 =

(1198781198972 119898

11198782




119909


and119866


1198781and 119866



11988511198981


1198852

1198981119876

2)


1198821to digraph 119866

1198822is possible



1198821and119866

1198822is possible







rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














rarr 1198661198822


and 119878119887 isin 1198811198782

(where1198811198821


1198821and 119881


1198782) sharing


119860119887119898119897

119860119886

= 119898119897119860

119887



and 1198661198822


1198821harr 119866

1198822) if 119866

1198821rarr 119866

1198822and

1198661198822

rarr 1198661198821


1198821and 119878119887 isin 119881


119860119886= 119860

119887119898119897119860

119886

= 119898119897119860

119887


1198781and 119866





119886(119909minus1)

997888rarr (119878119886119909 999492999484 119878

119887119909) 997888rarr 119878


(16)



119886119909 999492999484 119878


997888rarr 119878119889119897119889 997888rarr 119878

119887119895 997888rarr 119878

119887(119895+1) sdot sdot sdot

(17)



digraph1198661198822


119879to 119866

1198822 1198781198861 rarr 119878

1198862mdash




119886119909 119876119886

119909 to 119885119887119909 119876119886

119909


1198821and 119866



and 1198661198782



1198781 119864

1198781)

1198661198782= (119881

1198782 119864




(119878119886 119878



digraphs 1198661198821

and 1198661198822




1198821and 119866

1198822isin DC (set


119891 (119897119889 119889119888) =1

2(119889119888

2minus 119889119888) sdot 119897119889

2 where 119889119888 = |DC| (18)








1119898119875

2 and119898119875


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

22 119877

11))

119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7) (119877

29 119877

10))

119896 = 1 4



1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198781 119864

1198781) 119866

1198782= (119881

1198782 119864

1198782))

AC = 0for all 119878119886 isin 119881

1198821

for all 119878119887 isin 1198811198782

if (119860119886= 119860

119887) amp (1198981

119860119886

= 1198981119860

119887

) ampamp (119898119897119901119860

119886

= 119898119897119901119860

119887

) thenAC = AC cup (119878119886 119878119887)

endend

endreturn AC

end

Algorithm 1

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

25 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

(19)


119898119901119896

1= ((119877

3 119877

13 119877

6) (119877

17 119877

5) (119877

21 119877

8) (119877

18 119877

9)

(11987715 119877

12)) 119896 = 1 4

119898119901119896

2= ((119877

2 119877

20 119877

5) (119877

24 119877

4) (119877

28 119877

7)

(11987725 119877

8 119877

22 119877

11)) 119896 = 1 3

119898119901119896

3= ((119877

1 119877

27 119877

4) (119877

28 119877

7) (119877

29 119877

10)) 119896 = 1 3

(20)


1 1198751

2 1198752

2 1198753

2 1198751

3 1198751

4 1198752

4 1198751

5 1198752

5 1198751

6 1198752

6 1198751

7 and

1198751


1ndashP




119894 1198981198752

119894 1198981198753

119894 1198981198754




2119898119875119896

3




1

1198772 and 119877


10 119877

11 and 119877

12workstations



119894119895of





1198822 behavior 119866


1198821) In


1198821harr 119866

1198822


rarr 1198661198822

and119866

1198822rarr 119866



rarr 1198661198822


1198822rarr 119866



and1198661198781









1198781is



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051



119896 = 1 2 3 4 1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

11987713divide 119877

33

119898119905119896


119898119905119896


119898119905119896


1199051


1199051


1199052


1199053


1199051


1199051


1199052


1199051


1199052


1199051


1199052


1199051


1199051




states 1198811198821




1198821rarr 119866

1198822and 119866

1198822rarr 119866

1198821)



and 1198661198782 see


1198821and 119866





and 1198661198781


4 Related Work



Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 5 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091


Table 6 The timing 119909119896119894119895119898119909119896


1198771

1198772

1198773

1198774

1198775

1198776

1198777

1198778

1198779

11987710

11987711

11987712

1198981199091


1198981199091


1198981199091


1199091


1199091


1199092


1199093


1199091


1199091


1199092


1199091


1199092


1199091


1199092


1199091


1199091






R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12mP4

3

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13 mP1

3

mP13

mP13

mP13

mP11

mP11

mP21

mP21

mP31

mP31

mP31

mP31

mP31mP3

1

mP21

mP21 mP2

1

mP21

mP11

mP11 mP1

1

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP42mP3

2

mP32

mP32 mP3

2

mP21 mP3

1

mP31

mP31

mP41

mP41

mP33

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23 mP2

3

mP23

mP32

mP32

mP32

mP42

mP42

mP42

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP11

mP12

mP12 mP1

2

mP12

mP22

mP22

mP22 mP2

2

mP22

mP22


ji



ji

mPji

0 50 100 150 200 250 300 350 400 450









5 Conclusions




R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R11

R12



ji



ji

mPji

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP43

mP13

mP13

mP13

mP13

mP13

mP13

mP13

mP31mP2

1 mP21mP1

1 mP11mP4

1

mP32

mP32

mP32

mP32

mP32

mP33

mP33

mP33

mP33

mP33

mP33

mP23

mP23

mP23

mP23

mP23

mP23

mP23

mP42

mP42

mP42 mP4

2

mP42

mP42

mP42

mP42

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP12

mP22

mP22

mP22

mP22

mP22

mP22

mP22

0 50 100 150 200 250 300 350 400 500450 550

(S19 Sd1 ) (Sd54 rarr S40)

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP31

mP21

mP21

mP21

mP21

mP11

mP11

mP11

mP11

mP11

mP41

mP41

mP41

mP41

mP41

mP41

mP41

mP41


Sl41

Sl301199790

Sl41

Sl30

Sl41

Sl301199790

SSl41

Sl30

Sl0

Sl1

Sl9

Sl20

S9

S30

S0

S1S20

S41

Sl0

Sl1

Sl9

S9

S31

S0

S1

S42

GW1 GW2 S21

119979119979

S18S2

Transient state

Sd1 Sd34

Sd1

Sd54


rarr 1198661198822

and 1198661198822

rarr 1198661198821





R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R3

R2

12059001

R1

12059002R6

R5

R4

R9

12059004

R11

12059007

P11

P12 P1

3 R12

12059005

R10

R8

120590012

R7

120590011

SC

R13

12059005

P14

(a)

119979

GW1

GW2

GS1

GS2

(b)


1198821rarr 119866

1198822(b)



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Multimodal Processes Rescheduling: Cyclic...

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