Multiple-colony Ant Algorithm for Parallel Assembly Line Balancing Problem

Applied Soft Computing 11 (2011) 31863198

Contents lists available at ScienceDirect

Applied Soft Computing

journa l homepage: www.e lsev ier .co

Multip m

Lale Ozb Gora Erciyes Univeb University of rkey

a r t i c l

Article history:Received 16 JuReceived in reAccepted 12 DAvailable onlin

Keywords:Line balancingParallel assemAnt colony opMeta-heuristic

orienBalanly lineg sysParalcapaces onNP-his devrst

isticse ap

ve.

1. Introduction

Assemblconveyor bequantity staperformedrepeatedly.costs for mrequiremencustomer ntake over thighly efcbalanced insome authoby consider[13]. In oin terms oftogether is rtryandacadseparately bprecedenceworkstation

CorresponE-mail add

baykasoglu@g

ancing problem in the literature (SALBP). In addition to the wellknown simple assembly lines, researchers and practitioners devel-

1568-4946/$ doi:10.1016/j.y lines consist of serial workstations connected withlts or other material handling systems to produce highndardized products. Operations of the work pieces areby highly specialized operators on these workstationsThe main purpose of such systems is to reduce the

ass production of standardized products. However, thets of todays manufacturing systems are changing aseeds diversifying. Increasing product variety causes tohe concept of assembly to order. This means that theient and exible assembly lines must be designed andorder to produce different products. As discussed byrs efciency of assembly lines can be further improveding neighboring lines together while balancing themrder to reduce failure sensitivity, increase exibilitycapacity and product variety balancing parallel linesecently attracting the attention of researchers in indus-emia. Traditionally straight assembly linesarebalancedy assigning operations to workstations considering therelations and cycle time to minimize the number ofs. This is themost frequently studied assembly line bal-

ding author. Tel.: +90 3423604383; fax: +90 3423604383.resses: [email protected],mail.com (A. Baykasoglu).

oped several other assembly line designs and balancing techniquessuch as mixed and multi model lines, U shaped lines, assemblyline with parallel workstations or tasks, etc. For an excellent anddetailed explanation and classication of assembly lines with dis-cussions on existing solution procedures refer to Boysen et al. [4,5].A survey is also presented by Lusa [6] in terms of parallel assem-bly lines. This paper is a good survey presenting different typesof parallel assembly lines along with discussions on their mainadvantages and disadvantages. Parallel Assembly Line BalancingProblem (PALBP) has initially been considered by Gokcen et al. [1]in the literature. Scholl and Boysen [2] presented a detailed deni-tion of PALBP. They entitled this problem as Multiproduct ParallelAssembly Line Balancing Problem (MPALBP) and divided it into twoconnected sub-problems; assignment of the products to parallellines and balancing the lines simultaneously.

It has been proven in the literature that meta-heuristicapproaches are quite effective in handling NP-hard problems [7].Ant Colony Optimization (ACO), which is one of the swarm intel-ligence based meta-heuristic algorithms, is successfully applied toseveral other types of assembly line balancing problems in the lit-erature. Bautista and Pereira [8] applied ACO to time and spaceconstrained assembly lines. Baykasoglu and Dereli [9] proposedan algorithm based on ACO for two sided assembly line balancingproblem. McMullen and Tarasewich [10] employed ACO for solvingthe multi-objective assembly line balancing problem. Sabuncuoglu

see front matter 2010 Elsevier B.V. All rights reserved.asoc.2010.12.021le-colony ant algorithm for parallel asse

akira, Adil Baykasoglub,, Beyza Gorkemlia, Latifersity, Faculty of Engineering, Department of Industrial Engineering, Kayseri, TurkeyGaziantep, Faculty of Engineering, Department of Industrial Engineering, Gaziantep, Tu

e i n f o

ne 2009vised form 26 July 2010ecember 2010e 17 December 2010

bly linestimizations

a b s t r a c t

Assembly lines are designed as owdardized products in a serial manner.among the other problems of assembmanufacturing environment increasinbalance and productivity are crucial.these objectives especially when theassembly lines there are a few studiParallel assembly line balancing is anovel multiple-colony ant algorithmproposed algorithm is also one of theswarm intelligence based meta-heurmark problems and performance of ththe proposed approach is very effectim/locate /asoc

bly line balancing problem

kemlia

ted production systems which perform operations on stan-cing of assembly lines is one of the most important problemss like designing and managing. In todays highly competitivetem exibility, reducing failure sensitivity, improving systemlel assembly lines provide some opportunities in improvingity of production system is insufcient. Unlike the traditionalbalancing parallel assembly lines in the present literature.

ard problem similar to other assembly lines. In this paper, aeloped for balancing bi-objective parallel assembly lines. Theattempts in modeling and solving the present problem with. The proposed approach is extensively tested on the bench-proach is compared with existing algorithms. It is shown that

2010 Elsevier B.V. All rights reserved.

L. Ozbakir et al. / Applied Soft Computing 11 (2011) 31863198 3187

et al. [11] developed anACOalgorithm forU-type assembly line bal-ancing problem. Vilarinho and Simaria [12] proposed an ACO basedbalancing procedure for mixed-model assembly lines with parallelworkstations. Simaria and Vilarinho [13] developed an ACO algo-rithm for twthe motivataging resulwe decidedalso extendmultiple couating the pof this pape2. The propthis algorithmance of thpublished r

2. Problem

PALBP coa parallel foto differentthe conceptacademia [1ancing in thGokcen et amatical prowhich contto minimizone assembobjective tamaximizingBenzer et alis based onsingle modprogrammibly lines. Sclinear progrcedure baseScholl and Btion (minimsufcient foinefciencieplace insteconcept as alinked to for(back to baof activemuoriginal objmulti-line ling workersline lengththe parallelwalking dis

There arels/productTraditionallprecedenceto minimizeof lines canaccount thesuch a wayit is possiblethe main pulines simult

2,1- 2,

1,1 1,7 - 1,31,51,4 - 1,2 Line-1

Station-6 Station-7

1,6

Station

Station-2

ig. 1.

ighbo pand 3on oignedilar tstud

denn.s timask tnumucts.atorsator

elop

of tl optthe peta-s wbserin

y deperomed bo maeromcandstag

solutnt alies.eposiyed.

paraterm

ch anta newhe ph

mais suolony is located on different processors and during searchs they exchange information within predetermined periods.e main purpose of the multiple-colony ant algorithms is todifferent regions of search space by the colonies cooperat-nd good solutions by information exchange. In developingsent multiple-colony ant algorithm we motivated from theakdigool and Kachitvichyanukuls [22] study. They devel-multiple-colony ant algorithm which consists of multipleo sided assembly line balancing problems. Based onion of the previous successful applications and encour-ts of our preliminary studies with ACO for PALBP [14]to develop a new and improved ACO algorithm whichs our previous work. The proposed ACO algorithm is alony type and multiple criteria are considered in eval-erformance of the generated solutions. The remainderr is organized as follows: PALBP is explained in Sectionosed multiple colony ant algorithm and components ofm is presented in Section 3. In Section 4, the perfor-e proposed algorithm is tested and compared with theesults. Finally, the conclusions are given in Section 5.

denition

nsists of a number of serial assembly lines arranged inrm. Different products and/or models can be assignedserial lines by considering a joint cycle time. Althoughof parallel assembly lines is known in the industry and5], there are just a few research papers on their bal-e existing literature. The problem is rst introduced byl. [1]. They proposed several procedures and a mathe-gramming model for assembly line balancing problemains parallel lines. The objective of their study wase the number of stations while balancing more thanly line together. Ozcan et al. [16] developed a multi-bu search algorithm for solving PALBP with the aim ofline efciency and minimizing variation of workloads.

. [17] proposed a networkmodel for PALBP. Theirmodelthe shortest route formulationwhichwas developed forel assembly line. Kara et al. [18] proposed a fuzzy goalng model that can be used for balancing parallel assem-holl and Boysen [2] modeled the problem as a binaryamming problem and proposed an exact solution pro-d on the branch and bound algorithm. In a later work,oysen [3] mentioned that the original objective func-izing number of workstations alone) seems not totallyr obtaining a competitive line design as it might lead tos (i.e. longer lines). They advised touse the termwork-ad of workstation and dened the split workplaceplacewhere only directly oppositeworkstations canbemaworkplacewhereworkers can alsowork in parallel

ck). They tried to achieve this by dening the propertylti-line balances and adding a second-order term to theective function which might results in a more compactayout. However, we should mention here that allow-to work in parallel (back to back) may reduce the totalbut this in turn will necessitate more space betweenlines in most of the applications. This also means moretance for an operator who is working on two lines.e two basic sub-problems in PALBP; matching mod-s with lines and balancing adjacent lines together.y, each line is balanced separately considering therelations between the tasks and cycle time in orderthe total number of workstations. But, parallel design

offer a reduced number of workstations by taking intoneighboring lines. If neighboring lines are settled in

that a single worker can operate in both directions thento reduce the number of workstations. In other words,rpose of PALBP is to balance multiple serial assemblyaneously. This is achievedby assigning task(s) fromsev-

F

eral newith twers 1 aoperatiare ass

Simin this

Preceknow

Cycle The t The

prod Oper Oper

3. DevPALBP

Onenatoriatrue ifACO mcolonietrail. Ocessfulnest bthe phincreasare twand phfrom asecondpletedbasic astrategants demplo

InitializeUntil the

For eaBuild

Update t

Theto ACOeach cprocesThus thsearching tothe preUdomsoped aStation-4

2 2,4 Line-2

Station-5Station-3

2,62,5 - 2,3

-1

Schematic representation of a typical parallel assembly line.

oring lines to a multi-skilled operator. Such a designrallel lines is schematically represented in Fig. 1, work-can operate on both neighboring lines (in Fig. 1, 1st

f product one and, 1st and 2nd operations of product twoto worker one).

o previous studies, the following assumptions aremadey for modeling and solving the present PALBP [13]:

ce diagrams of each model produced on each line are

es of each line are known and identical.imes of each product are known.ber of parallel lines is dened as equal to number of

are multi-skilled.transfer times between two adjacent lines are ignored.

ment of multiple-colony ant algorithm for

he most efcient meta-heuristic algorithms on combi-imization problems is known as ACO. This is especiallyroblem has an inherently network structure [19]. The

heuristic mimics the foraging behavior of natural anthich share the central information called pheromonevation of ants in nature shows that ants are very suc-nding shortest path between food sources and theirositing pheromone and choosing their way by usingone concentration. The probability of choosing paths isy having strong pheromone concentration [20]. Therein parts in ant colony algorithms; solution constructionone update. In the rst stage ants select one operationidate list until a complete solution is obtained. In thee ants deposit the pheromone on the way of its com-ion. Ant density, ant quantity and ant cycle are thegorithms which have different pheromone trail updateIn this study ant cycle algorithm which implies thatt pheromone after they have built a complete tour isThe basic ant cycle steps are as follows [21]:

meters and pheromone trailsination criteria is satisedin the colonysolution according to the pheromone trail and heuristic informationeromone trail

n idea behind the multiple-colony ant algorithm is dueitable structure for parallelization. In the present study,

3188 L. Ozbakir et al. / Applied Soft Computing 11 (2011) 31863198

colonies wiing problem

As it is pohave differesearch spacorder to achthemultipltics to sub-csolution byexplained itheir tour,is updated.are rankedcomplete tnumber of baction amothe searchdened andber of succeprocessing t

3.1. Solutio

In each stask-worksCandidate lhave alreadtion is sufFig. 2. Schematic representation of multiple-colo

thin a single processor for solving the job shop schedul-[22].rtrayed in Fig. 2multiple ant colonies aredesignedas tont characteristics to explore the different regions in thee. Different heuristics are assigned to these colonies inieve a more diversied search. In general execution of

e-colony ACO is initialized by assigning ants and heuris-olonies. Each ant of a sub-colony construct a completeusing heuristic information and pheromone trail as

n Section 3.1. After all ants in a sub-colony completepheromone amounts in the local pheromone matrixThis step is repeated for all sub-colonies and all antsaccording to the tness values. After all sub-coloniesheir tours, global pheromone matrix is updated by aest ants. Global pheromone matrix provides the inter-

ng sub-colonies which explore the different regions ofspace. In the present research, three sub-colonies arethree different priority heuristics (Maximum total num-ssor task, Maximum total time of successor tasks, Longestime) are assigned to these sub-colonies.

n construction

tep of the proposed algorithm, ant determines the newtation assignment by using a candidate list strategy.ist consists of tasks in which immediate predecessorsy been assigned and the remaining time of current sta-cient for processing the available tasks. Ant applies a

probabilistia workstatiet al. [23] pin modelinworkstationis depositedepositingand latest ptions 1 andavoid extra

E(p,j) =(p

L(p,j) =K +

where pPE(p,j), L(p,j) a

tively.K de

t(p,j) is the tthe lines. Pof task (p,j)of task (p,j)

Under thtions is perny ant system.

c task selection strategy until all tasks are assigned toon to generate a complete solution for PALBP. Bautistaresented three different trail information usage policiesg of assembly lines with ACO; task to task, task to and task to position. In the present study the trail

d between a task and its assigned workstation. Whilepheromone between task and workstation pair, earliestossible stations for each task are determined by equa-2 as advised by Saltzman and Baybars [24] in order tocomputation.

,j) +

qP(p,j)

(p,q)

(1)

1 (p,j) +

q F

(p,j)

(p,q)

(2)

, j Jp (P: set of products; Jp: set of tasks for product P).re the earliest and latest stations for task (p,j), respec-

notes the maximum number of stations. (p,j) = t(p,j)/c.ask time of task j of product p and c is the cycle time of(p,j) denotes the set of direct and indirect predecessors. F(p,j) denotes the set of direct and indirect successors.ese explanations, assignment of all tasks to worksta-

formed by using the following steps:


START

Initilization

Assign ants tosub-colonies

Compute qualityfunction

Calculate heuristicvalues

Update candidatetask list

Select task-stationpair

Generate candidatetask list

Open new station

Initialize ant

t(p,j)=Station remaining time

Is iterationlimit reached?

All sub-coloniescompleted?

Are all antscompleted tour?

Are alltasks assigned?

Is candidatetask list empty?

Assign heuristics tosub-colonies

Update globalbest solution

Update globalpheremone matrix

Update localpheremone matrix

Return the bestsolution

END

J1: Exploitation

J2: BiasedExploitation

Calculate earliest & latest stationfor each task

Assign Rand(0.1, 0.5) pheromoneamount to each possible

task-station pair

PALBP Data

Yes

Yes

Yes

Yes

No

No

No

No

No

Yes

Fig. 3. Flowchart of the multiple-colony ant algorithm.


Repeat (until all tasks have been assigned)Open a new stationDetermine candidate task list by considering neighboring assembly lines.

Candidate task list contains available tasks.Available task: immediate predecessors must be assigned and its task time must

be less thaSelect task-wAssign the s

Until all tasks

3.1.1. TaskThe task

the pheromtions and thin the candtions 3-4 arexploitation

J1 = arg ma

J2 = p((p, i)

where; r isworkstationtic informaant n afterpheromonevide interacin Eq. (5) derespect to l

k(p,j),z(t) =

Heuristiassigned toues to taskvalues of tatask list is ubetween1 fcandidate lthe importaare close to

The opeowchart iassigned totime informdeterminedis assignedcolony compheromoneall sub-colothe best sol

Assigninassignableassigned antion time. Iremainingtask to beglobal/local(3)(5)). UpBy this way3.2) is compin all sub-cupdated.

Table 1Parameter setting.

Parameters level

Number of aer of i

nt phtancetancemoneportanber of

tness

his seratiizin

he lial. [2minr tobin

ion Mn byve fulutioget vget vval

he inms=1

(

ms=

ize f

erom

enernstal (0.h pathe

the perom(9).

predeonespec

e pre

,z(t, tk

,z(t, t + n) =m

k=1k(p,i),z(t, t + n) (10)

t + n) = (1 )(p,i),z(t) + (p,i),z(t, t + n) (11)r completion of iteration, constant amount of pheromone isted to all arcs to prevent thepremature convergence.Manualn of a small sized sample problem is shown in Appendix An or equal to remaining station time.orkstation pair by using the ant cycle algorithm

elected task to the stationhave been assigned

selection strategyselection from the set of candidate tasks depends onone trail between the task and previously assigned sta-e information provided by the heuristic for each taskidate list. Similar to Vilarinho and Simaria [12] equa-e employed as task selection strategy. Eq. (3) is used forand equation 4 is used to enable biased exploration.

x(p,j)Ak(p,j)

{[(p,j),z][(p,j)]} if r 0.5 (3)

, (p, J2)) =[(p.J2),z]

[(p,J2)]

(p,j)Ak(p,i)

[(p.j),z][(p,j)]

if r > 0.5 (4)

a random number between 0 and 1. z is the set ofs which task (p,j) can be assigned. (p,j) is the heuris-

tion of task (p,j). An(p,i) indicates candidate task list forselection of task (p,i). Ants use both local and globalinformation during the task selection in order to pro-tion among sub-colonies by making use of Eq. (5). wnotes the effect of global pheromone information withocal pheromone information [22].

(1 w)klocal(p,j),z(t) + wkglobal(p,j),z(t) w [0,1] (5)

c information is obtained from the priority ruleseach sub-colony. Instead of assigning static priority val-s, a rank based structure is implemented. The prioritysks are updated when a task is assigned and candidatepdated. Thepriority values of tasks in candidate list varyor the least desirable task and thenumber of tasks in theist for the most desirable task. This structure increasednce of heuristic information when the priority valueseach other.ration of the proposed algorithm is depicted in an Fig. 3. In the owchart rst ants and heuristics aresub-colonies. After obtaining problem data and cycleation, earliest and latest assignable workstations arefor each task. Random (0.1, 0.5) amount of pheromonebetween tasks and workstations. Ants in each sub-plete their tours based on their own heuristic rule andamount. Local pheromone matrices are updated. Afterny tours are completed global pheromone matrix andution is updated.g tasks to workstations: Open a workstation, generatetasks list from the tasks whose predecessors wered their processing times t into remaining worksta-f there is a task with a processing time equal to thestation time, assign this task. Otherwise, select theassigned based on the pheromone information in thepheromone matrices and heuristic knowledge (Eqs.date the assignable tasks list and heuristic values.an ant completes its tour and its tness (see Sectionuted, local pheromone matrix is updated. After all antsolonies nish their tours global pheromone matrix is

NumbQConstaImporImporPherow (ImR num

3.2. Fi

In tconsidmaximsents tLeu etaims toin ordeare comDeviatsolutioobjectiof a soare tarthe tardesiredfrom t

f1 =(

f2 =

Minim

3.3. Ph

In gto a cointervaon eacning oftours,The phby Eq.is thepherom(11), reQ is th

k(p,i)

(p,i)

(p,i),z(

Aftedeposisolutionts 30, 9terations 100, 250

5eromone amount between iterations 1of pheromone intensity () 1of heuristic information () 5evaporation coefcient 0.1ce of global pheromone) 0.7iteration for restart 25

function

tudy, two different performance criteria are taken intoon, minimizing the idle time of workstations andg the line efciency. The function f1 (Eq. (6)) repre-ne efciency. The function f2 (Eq. (7)) as proposed by5] is adopted for minimizing the idle time. This functionimize thenumberofworkstations and station idle timesobtain a balanced solution. These objective functionsed into a single objective function by using Minimumethod (MDM). MDM tries to nd the best compromiseminimizing the sum of the fractional deviations of thenctions [26]. In relation to MDM approach the tnessn is computed by using Eq. (8). f max1 and f

min2 in Eq. (8)

alues. The target value of line efciency is set to 100 andalue of total idle time is set to 0. f 01 and f

02 are the least

ues. The least desired value of f 01 and f02 are obtained

itial ant tour.

(STs)) 100m c) (6)

1((c STs)2)

m+m

s=1(c STs)m

(7)

= fmax1 f1

f max1 f 01+ f

min2 f2

f min2 f 02(8)

one trail update

al, initialization of the pheromone on every arc is setnt initial value. In this study a random number in the1, 0.5) is determined as the initial pheromone valueth in order to enforce the diversication at the begin-algorithm. After the ants in a sub-colony complete theirheromone amount on the explored paths is updated.one amount that ant k deposit to arc ((p,i),z) is denedfk denotes the tness function value of ant k and Qned constant for pheromone update. The amount ofdeposition after iteration is computed by Eqs. (10) andtively. is the pheromone evaporation coefcient anddened constant for pheromone update.

+ n) = Qf

(9)


Table 2Comparative results for PALBP.

Test problems # Tasks (Line1-Line2) Cycle time Theoretical min Independent balance GOK MOD ABS TS mc-ACO

Merten 7 9 6 4+3 7 7 7 76 11 5 3+3 5 5 5 5

13 5 3+2 5 5 5 517 4 2+2 4 4 4 4

Jaeschke 9 9 8 5+4 8 8 8 88 11 7 4+4 7 7 7 7

13 6 3+3 6 6 6 615 5 3+3 5 5 5 517 4 3+2 4 4 4 4

Jackson 11 8 11 7+6 13 13 13 1310 10 9 5+5 9 9 9 9

13 7 4+4 7 7 7 715 6 4+3 6 6 6 619 5 3+3 5 5 5 5

Roszieg 25 14 18 10+10 18 20 18 1824 16 16 8+8 16 16 16 16

17 15 8+8 15 16 15 1522 12 6+6 12 12 12 1230 9 5+5 9 9 9 9

Sawyer 30 25 26 14+13 26 28 26 2628 27 24 13+12 25 26 24 24

30 22 12+11 22 23 22 2236 18 10+9 18 19 18 1841 16 8+8 16 16 16 1654 12 7+6 12 12 12 12

Kilbridge 45 57 20 10+10 20 20 20 20 2043 79 14 7+7 14 15 14 14 14

92 12 6+6 12 12 12 12 12110 10 6+5 10 10 10 10 10138 8 4+4 8 8 8 8 8184 6 3+3 6 6 6 6 6

Hahn 53 2004 14 8+7 14 17 14 14 1451 2338 12 7+7 12 15 12 12 12

2806 10 6+5 10 11 10 10 103507 8 5+4 8 9 8 8 84676 6 4+3 6 7 6 6 6

Tonge 70 160 43 23+22 45 46 44 4466 168 41 22+22 43 45 42 42

207 34 18+17 34 35 34 34234 30 16+15 30 32 30 30270 26 14+13 26 27 26 26293 24 13+12 24 25 24 24

Wee-Mag 75 28 105 63+60 123 123 123 123 12371 29 102 63+60 123 123 123 123 123

31 95 62+60 121 122 121 121 12133 89 61+59 119 120 119 119 11934 87 61+59 119 120 119 119 11941 72 59+57 116 116 116 116 11642 70 55+53 107 108 107 107 10743 69 50+48 98 98 98 98 9849 60 32+31 62 63 62 63 6254 55 31+30 60 61 60 60 60

Arcus1 83 3786 39 21+19 40 42 40 40 4079 3985 37 20+19 38 40 37 38 38

4206 35 19+17 36 38 36 36 364454 33 18+17 34 35 34 34 344732 31 17+16 32 33 32 32 325853 25 14+12 26 26 25 26 256842 22 12+11 22 23 22 22 227571 20 11+10 20 21 20 20 208412 18 10+9 18 19 18 18 18

10816 14 8+7 14 14 14 14 14

Lutz2 89 11 86 49+46 90 97 88 8885 12 79 44+41 82 88 79 80

13 73 40+38 75 81 73 7314 68 37+35 71 79 69 6815 63 34+32 64 68 63 6316 59 31+30 60 66 59 5917 56 29+28 56 60 56 5619 50 26+25 50 53 50 50


Table 2 (Continued)

Test problems # Tasks (Line1-Line2) Cycle time Theoretical min Independent balance GOK MOD ABS TS mc-ACO

20 47 25+24

Lutz3

Mukherje

Arcus2

in order toalgorithm.

4. Comput

In orderalgorithm,set for theline-balanccomparedwmathematicwas solvedand bound[2,3] (ABS f(TS for shortaken into cresults for 5cycle timesmented byIntel Core 2values of thshown in Ta

Numberalgorithm.ants must bperformancto numbertest problemproblems. S13136 tasthe numberproblems dhave 3 sub-as 9 and 30

For para= {1,2,5},Different cofor differen

par.

irectcen euresr ofwr ofimu

(GOKandterw5 GBas s

to-onis n89 75 43 23+2285 79 41 22+22

83 39 21+2187 37 20+1992 35 19+18

94 176 47 25+2490 183 45 24+23

192 43 23+21201 41 22+20211 39 21+19222 37 20+18234 35 19+18248 33 18+17263 31 17+16281 29 16+15301 27 15+14324 26 14+13351 24 13+12

111 5785 52 27+28107 6016 50 26+27

6267 48 25+266540 46 24+256837 44 23+247162 42 22+22

provide a more detailed explanation of the proposed

ational study

to analyze the efciency of the multi-colony ACOtest problems from the literature are solved. The datatest problems can be obtained from www.assembly-ing.de. The performance of the proposed algorithm isith the heuristic of Gokcen et al. [1] (GOK for short); theal programming model of Scholl and Boysen [3] whichby XPressMP solver (MOD for short) and the branchbased exact solution procedure of Scholl and Boysenor short) and tabu search algorithm of Ozcan et al. [16]t). For GOK, MOD and ABS, 95 common cycle times are

similarTable 1

A dInGokprocednumbenumbeon minrithmSchollcompuand 1.500 s wa one-manceonsideration. For TS, since Ozcan et al. [16] reported the5 cycle times, the comparisons are achieved on these

. The proposedmultiple-colony ACO algorithm is imple-C# programming language and runs are performed on aDuo 2.4GHz computer with 2 GB RAM. The parametere proposed algorithm for solving the test problems areble 1.of ants in ACO directly affects the performance of the

Dorigo and Sttzle [27] presented that the number ofe greater than 1. Dorigo et al. [21] suggested that thee of the algorithmgetting better as number of ants closeof cities in travelling salesmen problem. In this study,s are divided into two groups as small and large sized

mall sized problems are the test problems which haveks. Large sized problems have 146217 tasks. Usually,of ants is suggested to be less than 10 for large sized

ue to the excessive computation time [11]. Since wecolonies in this study, the number of ants is determinedfor large and small sized problems, respectively.meters , , Q and w; different combinations of the = {1,2,5}, Q= {1,5,10} w = {0.5,0.7} are evaluated.mbinations of these parameters yield better resultst problems. But the differences are not distinctive, so

generates hsmaller thacomputatioruns of the

The resuby using stametric testcomputes teach case at test procebetween twinTable4. InMean differmc-ACO ansample sizedisplays thvalue is equsignicancewe can concpairs. The nGOK meansthan GOK. Smc-ACO-TSnicant. An48 51 47 48

45 46 44 44 4443 45 41 42 4140 43 39 40 3938 40 37 38 3836 37 35 36 35

48 49 48 48 4746 47 45 46 4544 44 43 44 4342 44 41 42 4140 40 39 40 3938 40 38 38 3736 37 36 36 3534 35 33 34 3332 33 32 32 3230 31 30 30 2928 29 28 28 2826 27 26 26 2624 25 24 24 24

55 54 53 54 5353 52 51 51 5150 50 49 50 4948 50 47 48 4746 48 45 45 4544 44 43 43 43

ameters setting is used for all problems as shown in

comparison of the results is given in Table 2 and Table 3.t al. [1] andOzcan et al. [16] computation times for theirGOK and TS are not given. Since they only present theorkstations,wecompared the solutionswith respect to

workstations. In Table 2, results are presented in termsm number of workstations obtained from each algo-, MOD, ABS, TS and multiple-colony ACO: mc-ACO).Boysen [2,3] were run their algorithms on a personalith an Intel Pentium IVprocessor of 3.0GHz clock speed

yte of RAM. For each MPALBP-instance a time limit ofet for their procedures MOD and ABS. Therefore makinge comparison in terms of computational time perfor-ot possible. However, we can say that our algorithm

igh quality solutions in very short CPU times, muchn 500 cpu sec. In Table 3 we presented details of thenal time performance and results for 10 independentproposed algorithm.lts ofmc-ACOare comparedwithGOK,MOD,ABSandTStistical tests. In this study paired-t test is used as a para-in order to compare the means of two samples. The testhe differences between values of the two samples fornd test whether the average differs from 0. The paired-dure is used to test the hypothesis of no differenceo samples. The results of the paired-t test are presentedTable4 therst columnrepresents the comparedpairs.ence column displays the average difference betweend the compared results. N and df columns represent theanddegree of freedom, respectively. The Sig (p) columne probability of obtaining a t statistics whose absoluteal to or greater than the obtained t statistics. Since thevalue for pair mc-ACO-GOK is less than 0.05 (=0.05)lude that there is a signicant difference between theseegative sign of average difference between mc-ACO-that mc-ACO consists of less number of workstationsimilar conclusions are repeated for mc-ACO-MOD andpairs. The differences between these pairs are also sig-d the negative sign of mean difference values denotes


Table 3Detailed results of multiple-colony ACO.

Test problems Cycle time CPU time (seconds) Fitness function value Number of stations

Average Average Min Max

Merten 9 0.001 2.040 7/1011 0.002 2.000 5/1013 0.002 2.063 5/1017 0.001 2.067 4/10

Jaeschke 9 0.002 2.034 8/1011 0.001 2.000 7/1013 0.002 2.110 6/1015 0.002 2.088 5/1017 0.002 0.000 4/10

Jackson 8 0.004 2.012 13/1010 0.049 1.528 9/1013 0.002 2.000 7/1015 0.002 2.026 6/1019 0.004 2.048 5/10

Roszieg 14 0.027 2.000 18/1016 0.020 2.076 16/1017 0.509 1.672 15/1022 0.016 2.057 12/1030 0.011 2.047 9/10

Sawyer 25 12.943 0.919 26/3 27/727 78.478 0.881 24/5 25/530 0.152 2.010 22/1036 0.217 1.364 18/1041 0.031 2.149 16/1054 0.022 2.034 12/10

Kilbridge 57 0.084 2.125 20/1079 0.055 2.024 14/1092 0.047 2.095 12/10

110 0.038 2.034 10/10138 0.038 2.054 8/10184 0.025 2.094 6/10

Hahn 2004 0.067 2.066 14/102338 0.064 2.049 12/102806 0.038 2.074 10/103507 0.039 2.166 8/104676 0.031 2.033 6/10

Tonge 160 21.047 1.140 44/10168 22.655 1.094 42/10207 10.470 1.687 34/10234 0.297 2.058 30/10270 0.213 2.075 26/10293 0.202 2.063 24/10

Wee-Mag 28 1.806 2.001 123/1029 1.783 2.002 123/1031 6.458 2.002 121/1033 1.720 2.001 119/1034 1.686 2.000 119/1041 1.689 2.000 116/1042 1.664 2.001 107/1043 1.497 2.001 98/1049 1.603 2.016 62/1054 13.636 2.002 60/10

Arcus1 3786 1.053 1.959 40/103985 4.622 1.999 38/104206 0.264 2.018 36/104454 0.394 2.035 34/104732 0.203 2.069 32/105853 212.449 0.634 25/4 26/66842 0.142 2.065 22/107571 0.114 2.064 20/108412 0.109 2.150 18/10

10816 0.081 2.096 14/10

Lutz2 11 335.672 0.618 88/4 89/612 208.594 0.548 80/2 81/813 243.544 0.425 73/6 74/414 272.620 0.499 68/6 69/415 205.817 0.364 63/7 64/316 123.772 0.374 59/5 60/517 2.939 0.961 56/10


Table 3 (Continued)

Test problems Cycle time CPU time (seconds) Fitness function value Number of stations

Average Average Min Max

19 6.239 0.20 1.117 2.

Lutz3 75 0.516 1.79 230.516 0.83 105.656 0.87 0.458 1.92 124.953 0.

Mukherje 176 0.484 1.183 158.534 0.192 42.752 0.201 227.448 0.

0.0.0.0.1.0.1.1.1.

Arcus2 0.0.0.0.0.1.

Table 4Paired-t test re

Pair

mc-ACO-GOmc-ACO-MOmc-ACO-ABmc-ACO-TS

that mc-ACsignicancedifference bnegative sigbetter valuecedure by Snd the samlems. For lafourproblem(however thtion for allproblems).Because thealgorithms.test problemthe same.

Table 5algorithms.which are c

Table 5Summary of re

Algorithms

GOKMODABSTSmc-ACO

f deve of a

#tei=1211 331.078222 2.617234 55.513248 210.097263 0.298281 152.75301 0.247324 0.228351 0.200

5785 26.6036016 200.2316267 126.9256540 20.8586837 13.6817162 0.602

sults of comparisons.

Mean difference N df Sig.(p)

K 0.38947 95 94 9.80641E08

ratio oaverag

ADR :D 1.58947 95 94 2.02711E11S 0.01053 95 94 0.740803

0.30909 55 54 8.65235E06

O generates better results than MOD and TS. Since thevalue for pair mc-ACO-ABS is greater than 0.05, theetween this pair is not statistically signicant. But then of average difference denotes that mc-ACO generatess than ABS. ABS was proposed as an exact solution pro-choll andBoysen [2,3]. Theproposed algorithm is able toe solutionswithABS for all of the smaller size test prob-rger size problems ABS generated better solutions fors andACOgeneratedbetter solutions forveproblems

e difference between them is very small, i.e. 1worksta-cases, moreover ACO seems better for relatively largerA similar comparison is performed on TS and mc-ACO.se two solution approaches are meta-heuristic searchmc-ACO algorithm generates better results for 17 of 55s than TS. The results of remainder test problems are

presents the summary of the results for all comparedThese results denote the average deviation ratios (ADR)alculated by using equation 12. For each test case, the

sults.

#Test cases Average deviationratio (Opt/LB)

95 0.01388895 0.04972295 0.00573455 0.01137395 0.005333

where: WSalgorithmfstations. If tvalue is taksented by Smc-ACOalgBased on ththe proposeapproach fo

5. Conclus

In this pparallel assple colonyof workstaered in evaThe proposapproachesNP-hard prfour othercal analysecompetitiveproblem. ItCPU times.

Acknowled

Prof. Dr.ences (TB561 50/10087 48/10

045 44/10771 41/4 42/6672 39/3 40/7022 38/10437 35/2 36/8

000 47/10580 45/10546 43/10416 41/10401 39/8 40/2863 37/10334 35/10258 33/10109 32/10262 29/2 30/8129 28/10048 26/10129 24/10

868 53/10658 51/10733 49/10894 47/10934 45/10059 43/10

iation from optimum or lower bound is calculated andll test cases is reported.

stcases(WSi Opti)/Opti#testcases

(12)

i represents the number of workstation found by theor test case i,Opti denotes theoptimumnumberofwork-heoptimumisnot proven for this test case, lowerbounden. These optimum and lower bound values were pre-choll and Boysen [2,3]. As it can be seen from Table 5,orithmgenerates theminimumaveragedeviation ratio.e computational results we can safely conclude that

dmc-ACO algorithm is a very promisingmeta-heuristicr solving the multi product PALBP.

ion

aper, an ACO is proposed for balancing multi productembly lines. The proposed ACO algorithm is a multi-type and multiple criteria (minimizing the idle timetions and maximizing the line efciency) are consid-luating the performance of the generated solutions.ed ACO algorithm is also one of the rst meta-heuristicfor solving the present problem which is known as a

oblem. The proposed algorithm is also compared withexisting approaches form the literature. After statisti-s it is found out that the proposed algorithm is veryand eligible solution approach for solving the present

is able to generate high quality solutions in very short

gements

Adil Baykasoglu is grateful to Turkish Academy of Sci-A) for supporting his scientic studies.


1,1

1,2

1,3

1,5

1,45

4

3

1

32,1

2,2

2,3

1

5

3

Fig. A1. Precedence diagrams and task times of the sample problem.

Appendix A.

A.1. Sample problem

A sample problem is used to present how the proposed algo-rithm works. Precedence diagrams and task times of the sampleproblem for two parallel lines are illustrated in Fig. A1. Cycle timefor assembly lines is set to 12.

A.2. Solution

Parameter setting for the sample problem is shown in Table A1.

A.2.1. Initial solution generation Determine the earliest and latest possible stations for each task

by using estation isstation is

Assign Rapair for tglobal ph

Generatetion stringby ants fo

Using theand (7), re

Table A1Parameter sett

Parameters l

Number of aNumber of iQConstant phImportanceImportancePheromonew (ImportanR number of

Update the local bests and the global best, and pheromoneamount of the arcs. The pheromone amount on the arcs is givenin Table A6.

A.2.2. New solution generationAssignment of the rst task (for the rst ant of the rst colony):

Generate candidate task list: {1,1-2,1}. Calculate the heuristic values: Heuristic information of task 1,1

and 2,1 is 2 and 1, respectively. Select a task for assignment. Generate a random number (0,1) r. (r=0.36). Determine the task selection strategy using r: Exploitation strat-

egy is determined. Calculate the k(1,1),1(t) by using Eq. (5).

k(1,1),1(t)

k(2,1),1(t)

Since 8.7

After thants are giv

Pherompheromone

latio

,1),1(

,1),1(

,1(t +

rol th.the c

r upn Tabavio

Finalr thn is g

Table A2Local and glob

Colony

Station/task

1,11,21,31,41,52,12,22,3quations 1 and 2. For task 1,1 earliest station is 1, latest2. And for the other tasks, earliest station is 1, latest3.nd [0.1, 0.5] pheromone amount to each task stationhe initialization of arcs. Table A2 shows the local anderomone matrices.solutions for each colony. Tables A3A5 show the solu-s andtness functionsof solutionswhichare generatedr each colony.initial solutions, determine f 01 and f

02 by using Eqs. (6)

spectively. f 01 = 69.44444, f 02 = 8.32141.

ing for the sample problem.

evel

nts 6terations 50

5eromone amount between iterations 1of pheromone intensity () 1of heuristic information () 5evaporation coefcient 0.1ce of global pheromone) 0.7iteration for restart 25

Calcu

k(1

k(1

(1,1)

Conttinue

Add

Aftegiven i

Beh

A.2.3.Afte

solutio

al pheromone matrices.Localpheromonematrix (colony1) Localpheromonematrix (colony2) L

1 2 3 1 2 3 1

0.5 0.5 0.3 0.3 00.4 0.1 0.1 0.5 0.2 0.4 00.5 0.4 0.5 0.1 0.1 0.2 00.3 0.5 0.2 0.2 0.1 0.3 00.1 0.3 0.4 0.1 0.4 0.4 00.2 0.4 0.2 0.3 0.2 0.5 00.3 0.2 0.2 0.5 0.2 0.1 00.5 0.1 0.1 0.2 0.3 0.4 0= (1 0.7) 8.8 + 0.7 8.7 = 8.7

= (1 0.7) 8.5 + 0.7 8.5 = 8.5

is maximum, select task 1,1 for the assignment.

e assignment of all tasks, obtained solution strings ofen in Tables A7A9.one trail update (for arc 1,1-1 of the rst localmatrix):

ns are given below using Eqs. (9)(11).

t, t + n) = 52

t, t + n) = 7.4

n) = (1 0.1)8.8 + 7.4 = 15.3

e threshold of pheromoneamount: Since0.5


Table A3Solution strings of colony 1.

Ant # Colony 1 solution strings Fitness function

1 1,1-1 2,1-1 2,3-1 1,2-1 1,3-1 1,5-2 1,4-2 2,2-3 2.02 1,1-1 2,1-1 2,2-1 1,3-1 1,2-2 1,4-2 2,3-2 1,5-3 2.07272



1 2,1-1 2,3-1 1,1-1 1,3-1 1,2-1 1,4-2 2,2-2 1,5-3 2.080292 1,1-1 1,3-1 1,2-1 2,1-1 1,5-1 1,4-2 2,2-2 2,3-3 2.08029



1 1,1-1 1,2-1 1,4-1 1,5-2 2,1-2 2,2-2 2,3-2 1,3-3 2.203822 1,1-1 1,2-1 1,4-1 1,5-2 2,1-2 2,2-2 2,3-2 1,3-3 2.20382

Table A6Local and global pheromone matrices.

Colony Localpheromonematrix (colony1) Localpheromonematrix (colony2) Localpheromonematrix (colony3) Global pheromone matrix

Station/task 1 2 3 1 2 3 1 2 3 1 2 3

1,1 8.8 1.5 8.4 1.5 7.9 1.5 - 8.7 1.5 1,2 6.3 3.5 1.5 1.5 1.5 1.5 8.1 1.5 1.5 6.1 3.5 1.51,3 8.8 1.5 1.5 8.3 1.5 1.5 1.5 1.5 8.1 8.6 1.5 1.51,4 1.5 8.8 1.5 1.5 8.3 1.5 8.2 1.5 1.5 1.5 8.8 1.51,5 1.5 6.2 3.7 5.8 1.5 3.7 1.5 7.8 1.5 1.5 6.2 3.82,1 8.5 1.5 1.5 8.4 1.5 1.5 1.5 8.1 1.5 8.5 1.5 1.52,2 3.6 1.5 6.1 1.5 8.3 1.5 1.5 7.8 1.5 3.5 1.5 6.12,3 6.4 3.5 1.5 3.5 1.5 6.1 1.5 8.0 1.5 6.1 3.8 1.5



1 1,1-1 2,1-1 1,2-1 1,3-1 2,3-1 1,4-2 1,5-2 2,2-3 2.02 1,1-1 1,2-1 1,4-1 2,1-2 1,5-2 1,3-2 2,2-2 2,3-3 2.08029



1 1,1-1 1,2-1 1,4-1 2,1-2 2,2-2 1,5-2 2,3-2 1,3-3 2.203822 1,1-1 1,2-1 1,4-1 2,1-2 2,2-2 2,3-2 1,5-2 1,3-3 2.20382



1 1,1-1 1,2-1 1,4-1 1,5-2 1,3-2 2,1-2 2,2-2 2,3-3 2.080292 1,1-1 1,2-1 1,4-1 1,5-2 2,1-2 2,2-2 2,3-2 1,3-3 2.20382

Table A10Local and global pheromone matrices.

Colony Localpheromonematrix (colony1) Localpheromonematrix (colony2) Localpheromonematrix (colony3) Global pheromone matrix

Station/task 1 2 3 1 2 3 1 2 3 1 2 3

1,1 16.3 2.3 15.5 2.3 15.2 2.3 16.2 2.3 1,2 14.1 4.1 2.3 15.7 2.3 2.3 15.4 2.3 2.3 13.9 4.2 2.31,3 13.9 4.7 2.3 10.8 2.3 6.8 2.3 7.1 10.6 13.8 4.7 2.31,4 4.7 13.9 2.3 6.8 10.8 2.3 15.5 2.3 2.3 4.7 13.9 2.31,5 2.3 14.0 4.3 8.7 6.8 4.3 2.3 15.1 2.3 2.3 14.0 4.42,1 13.7 4.7 2.3 11.0 6.8 2.3 2.3 15.4 2.3 13.6 4.7 2.32,2 4.3 4.7 11.5 2.3 15.4 2.3 2.3 15.1 2.3 4.1 4.7 11.52,3 11.8 4.1 4.7 4.2 6.8 8.9 2.3 10.5 7.1 11.5 4.4 4.7

Table A11The best solution after 50 iteration.

Solution string Fitness function

1,1-1 2,1-1 2,3-1 1,2-1 1,3-1 1,5-2 1,4-2 2,2-3 2.0


Fi

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Multiple-colony ant algorithm for parallel assembly line balancing problemIntroductionProblem definitionDevelopment of multiple-colony ant algorithm for PALBPSolution constructionTask selection strategy

Fitness functionPheromone trail update

Computational studyConclusionAcknowledgementsAppendix ASample problemSolutionInitial solution generationNew solution generationFinal solution

References

Multiple-colony Ant Algorithm for Parallel Assembly Line Balancing Problem

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