P. Féniès, P. Lacomme, N.Tchernev LIMOS UMR CNRS 6158 CRCGM 5 décembre 2006 June, 15 MONTREAL –...

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P. Féniès, P. Lacomme, N.TchernevLIMOS UMR CNRS 6158CRCGM

5 décembre 2006

June, 15MONTREAL – IESM09

A framework for financial supply chain optimization

Pierre Fénièsa,b, Philippe Lacommea, Nikolay Tcherneva

aLIMOS UMR CNRS 6158bCRCGM

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1. Problem description 2. A genetic algorithm based framework

3. Numerical experiments

4. Conclusion

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1. Problem description

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Problem description

A supply chain is a coalition of autonomous entities coordonated by the same logistic process...

An opened set crossed by flows…;

A system with physical entities and autonomous organization…

An activities set which could be modelled as a value chain …

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Information flow

Goods and Services flows

Financial Flows

Planning budgeting

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Entity

Entity

Entity

Entity

Entity Entity

Entity

Entity

Entity

Coordonate physical and financial flows in decision tools and models for supply chain mangement

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Inclusion of cash flow in scheduling problem:

-) the Resource Investment Problem (RIP) (Najafi, 2006)

-) the Payment Scheduling Problem (PSP) (Ulusoy, 2000).

based on cash flows in networks structure, defined by (Russell, 1970, 1986).

Depending on the objective:

-)Net present value (NPV) (Elmaghraby and Herroelen, 1990);

-)NPV and extra restrictions as bonus-penalty structure (Russell, 1986) (Zhengwen and Xu, 2007)

-)Discounted cash-flows (Najafi, 2006) (Icmeli, 1996).

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Financial Flows Optimization and Supply Chain Management

Decisional level

Objective function

Operational level Maximization of cash position (Badell et al., 2005) (Bertel et al., 2008) ; Few links with physical flows; financial papers focus on payment term and interest

rate, not on the impact of physical flow in financial flows.

Tactical level Net present value maximization under cash position constraints (Russel., 1970) (Comelli et al., 2008).

Links are proposed for a single company, not for a supply chain.

Strategic level Net present value maximization is a classical approach in network design in supply chain management (Vidal et al., 2001)

Few works at operational level; Supply chain is always modeled as a flow-shop or an hybrid flow shop.

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Physical flows optimization at operational level

financial flows optimization at operational level

Makespan Cash Position (Comelli et al., 2008)

Cash Flows (Bertel et al., 2008)

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reveals the cash which is available at the end of a specific period

reveals the cash generation during a specific period

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We propose to model Supply Chain as a « Job shop » and to take into account cash management constraints:

- allows to extend financial constraints on physical flows- allows to take into account phenomena such as reverse logistics - allows different routing in Supply Chain

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A machine represents a Supply Chain entity (factory, warehouse…)

A job represents a manufacturing batch.

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Step 1 Step 2 Step 3LAST STEP

Step 4

Supplier FIRM C

ManufacturerFIRM A

Manufacturer FIRM C

RetailerFIRM A

Customer

Customer

RetailerFIRM A

Distribution CenterFIRM C

WarehouseFIRM A

SupplierFIRM C

Customer

Manufacturer FIRM B

RetailerFIRM B

RetailerFIRM B

WarehouseFIRM C

SupplierFirm C

RetailerFIRM B

Manufacturer FIRM C

ManufacturerFIRM A

Manufacturer FIRM B

Routing 1

€$£€$£€$£€$£

Routing 2

Routing 1

Routing 3

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• J1 : M1 (10), M2 (20), M3 (10)

• J2 : M2 (5), M1 (20), M3(10)

• J3 : M3 (10), M1 (10), M2 (5)

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Mach. Processing time

Price Time payment delay

Op1 Job 1 M1 10 12 2

Op2 Job 1 M2 20 24 3

Op3 Job 1 M3 10 40* 8

Op1 Job 2 M2 5 1 8

Op2 Job 2 M1 20 40 2

Op3 Job 2 M3 10 66* 12

Op1 Job 3 M3 10 2 3

Op2 Job 3 M1 10 15 5

Op3 Job 3 M2 5 20* 3

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R1 R2

Q Price Q Price

Op1 Job 1 3 2 4 1

Op2 Job 1 1 1 1 1

Op3 Job 1 2 3 4 1

Op1 Job 2 4 1 1 1

Op2 Job 2 8 2 5 3

Op3 Job 2 2 3 4 1

Op1 Job 2 2 3 4 2

Op2 Job 2 3 2 4 5

Op3 Job 2 1 1 1 1

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Delay R1 Delay R2

Machine 1 2 1

Machine 2 1 1

Machine 3 5 1

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CashP0 CashMin

Machine 1 24 -40

Machine 2 20 -40

Machine 3 24 -40

Supply Chain68 0

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• J1 : M1 (10), M2 (20), M3 (10)• J2 : M2 (5), M1 (20), M3(10)• J3 : M3 (10), M1 (10), M2 (5)

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5 10 15 20 25 30 35 40 45 50

J1

J2

J3

J1

J2 J3

J3

J1J2

M1

M2

M3

5 10 15 20 25 30 35 40 45 50

J1

J2

J3

J1

J2 J3

J3

J1J2

M1

M2

M3

Job Shop semi active solution

Job shop non semi active solution

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Job Shop semi active solution

Cash position evaluationCash

Time

Job shop non semi active solution

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Problem formulation:

• Min Maxespan with CashP>Cashmin

Cash position evaluation

Cash

Time

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A LINEAR MODEL FOR THE JOB SHOP SCHEDULING PROBLEM WITH CASH FLOW (JSPCF)

-extends the classical mathematical model of the job-shop scheduling problem;

- is written by extension of both classical linear formulations of the job-shop and of the AON-flow formulation of the RCPSP (Artigues et al., 2003).

-Financial constraints are added: CashPosition ≥ Cash Min

-A flow network model is therefore defined and takes into account financial flows constraints.

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2. A genetic algorithm based framework

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Makespan and critical path

Critical path analysis

Non Oriented Disjunctive graph

G

Oriented Disjunctive Graph (Job-Shop only)

Longest Path Computation

Flow resolution

Oriented Disjunctive Graph(Job-Shop constraint and

Disjunctive Cash Flow Arcs)

G

,G

isi,

Generation of a permutation job list

Oriented disjunctive graph

(flow graph based on )

FNG

FNG

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G



Flow resolution



G

,G

isi,




FNG

FNG

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O11 O12 O13

O21 O22 O23

O31 O32 O33

s t

10 20

10

5 20 10

10 10

5

0

0

0

O11 O12 O13

O21 O22 O23

O31 O32 O33

s t

10 20

10

5 20 10

10 10

5

0

0

0

1020

10

5

10

0 10

30 35

30

40

20

55 65

55 70

Non oriented disjunctive graph G defining a job-shop problem

Oriented disjunctive graph λG

representing a solution of makespan 70

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J1

J1

J1

J2

J2 J3

J2J3

J3

Overdraft: unacceptable solution, in a financial point of view

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G



Flow resolution



G

,G

isi,




FNG

FNG

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June, 15MONTREAL – IESM09F C

t

)1(*D )1(*F

s

)(* iD )(* i

F

)(* kiD )(* ki

F

)(* nD )(* n

F

Example of fully oriented disjunctive graph

)(FNG

)(FNG

Finding a flow in a graph is not straightforward and

could be time consuming…

Our proposals

- computing a flow - in the graph

complies with the Bierwith’ sequence since arcs are introduced from :

any node

to any node

Financial flows constraints

)(* iF

)(* kiD

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Heuristic resolution of the flow problem (HRFP)

Financial flows constraints

Procedure name: HRFP Input parameters

: Bierwith’ sequence Output parameters

: the flow Local variables

q : array 1..n of integer

d : array 1..n of integer res : outflow of the current operation i

Begin

)(* ii cqi ; )(* ii rdi ;j=1

While ( nj and 0)0( q ) loop

];min[ )(,*

)(*jsDs

qdj

; sd = sd -)(*

,j

Ds

k=j+1; While ( 0)(* jd ) and ( nk ) loop

];min[ )(,*)(*

)(*)(*kd

DFqj

kj

)(* j

d = )(* j

d - )(*)(*

,kj

DF

; )(*)(*

**,)()(

kjDFkk qq

End loop

)(*

)(*, j

j

dtF

; j:=j+1

end loop end

Such a flow could be denoted

and could be computed by any max flow algorithm

(Dinic, 1970),

(Edmonds and Karp, 1972),

(Cheriyan et al., 1999) (Goldberg and Tarjan, 1988)…

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G



Flow resolution



G

,G

isi,




FNG

FNG

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The flow fully :

- defined the extra arcs denoted DFA (Disjunctive Financial Arcs)

- permits to define the fully oriented graph which encompasses both job-shop constraints (including job precedence constraints and machines precedence constraints) and financial constraints.

,G

,G

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The graph ,G : a solution of the problem

O12 O13

O22 O23

O31 O32 O33

s t

10 20

10

5

20

10

10 10

5

0

0

0

10 2010

5

10

0 10

30 48

40

20

90 100

68 105

22

O11

O21

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The graph ,G : a solution of the problem

O12 O13

O22 O23

O31 O32 O33

s t

10 20

10

5

20

10

10 10

5

0

0

0

10 2010

5

10

0 10

30 48

40

20

90 100

68 105

1818

18

22

22

O11

O21

21

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G



Flow resolution



G

,G

isi,




FNG

FNG

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The memetic algorithm based framework is quite conventional including well known refinement for optimization:

• chromosome representation and evaluation

• A local search procedure which takes advantage of the critical path analysis.

Details are given in the paper

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Note the critical path is composed either of disjunctive arcs from machines (Job-Shop constraints) or of disjunctive cash flow arcs

21

O12 O13

O22 O23

O31 O32 O33

s t

10 20

10

520

10

10 10

5

0

0

0

10 20 10

5

10

0 10

30 48

30

40

20

90 100

68 10518 18

18

22

22

O11

O21

$£€

$£€

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III. Numerical experiments

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31 2

Implementations and benchmarks

- All procedures are implemented under Delphi 6.0 package

- Experiments were carried out on a 1.8 GHz computer under Windows XP with 1 GO of memory.

-The benchmark is concerned with instances based on the OR-library which instances concern classical shop problems (job-shop, flow-shop).

-The instances with financial consideration can be downloaded at: http://www.isima.fr/lacomme/Job_Shop_Financial.html

-The framework performance is studied over experiments including both flow-shop and job-shop instances and 30 instances with financial consideration.

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-the objective is to underline, the capabilities of the framework to provide new solutions for job-shop instances with both inflow and outflow.

-The results presented below push us into accepting that the framework encompasses a wide range of problems:

Experiments objectives

-) with some merits in both job-shop and flow-shop instances;

-) with new results in job-shop instances with financial consideration.

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nj : number of jobsnm : number of machinesn : total number of operations to scheduleOPT : denotes the optimal solutionLB : denotes a lower boundBKS : Best Known Solution (asterisk denotes

optimal solution) S* : the best solutionDev.% : deviation in percentage from S* to OPT or BKSAvg. : average I* : iteration number where S* has been foundT* : computational time (in seconds) to found S*TT : total computational time (in seconds)

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Framework (Rego and Duarte, 2008)

Instances Jn mn n Opt. S* Gap.% Time S* Gap Time

LA01 10 5 50 666 666 0.00 1 666 0.00 1 LA02 10 5 50 655 655 0.00 4 655 0.00 2 LA03 10 5 50 597 597 0.00 4 597 0.00 1 LA04 10 5 50 590 593 0.51 5 590 0.00 1 LA05 10 5 50 593 593 0.00 0 593 0.00 1 LA06 15 5 75 926 926 0.00 2 926 0.00 1 LA07 15 5 75 890 890 0.00 6 890 0.00 1 LA08 15 5 75 863 863 0.00 3 863 0.00 1 LA09 15 5 75 951 951 0.00 2 951 0.00 1 LA10 15 5 75 958 958 0.00 2 958 0.00 1 LA11 20 5 100 1222 1222 0.00 4 1222 0.00 1 LA12 20 5 100 1039 1039 0.00 5 1039 0.00 1 LA13 20 5 100 1150 1150 0.00 4 1150 0.00 1 LA14 20 5 100 1292 1292 0.00 4 1292 0.00 1 LA15 20 5 100 1207 1207 0.00 12 1207 0.00 1 LA16 10 10 100 945 956 1.16 92 947 0.21 1 LA17 10 10 100 784 793 1.15 12 784 0.00 2 LA18 10 10 100 848 848 0.00 51 848 0.00 1 LA19 10 10 100 842 842 0.00 91 846 0.48 1 LA20 10 10 100 902 912 1.11 105 917 1.66 3

Avg. 0.20 20.45 0.12 1.20

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Framework

Instances Jn mn n 0r 1nr Opt. Job-Shop S* Gap.% Time

LA01_Financial 10 5 50 600 805 666 666 0.00 19 LA02_Financial 10 5 50 200 524 655 737 12.52 340 LA03_Financial 10 5 50 250 272 597 922 54.44 434 LA04_Financial 10 5 50 200 590 590 635 7.63 111 LA05_Financial 10 5 50 400 182 593 760 28.16 416 LA06_Financial 15 5 75 355 1225 926 926 0.00 16 LA07_Financial 15 5 75 150 802 890 906 1.80 413 LA08_Financial 15 5 75 450 551 863 863 0.00 131 LA09_Financial 15 5 75 450 335 951 1025 7.78 610 LA10_Financial 15 5 75 450 157 958 1004 4.80 208 LA11_Financial 20 5 100 150 228 1222 1278 4.58 429 LA12_Financial 20 5 100 150 330 1039 1129 8.66 109 LA13_Financial 20 5 100 500 1169 1150 1150 0.00 34 LA14_Financial 20 5 100 350 1061 1292 1292 0.00 28 LA15_Financial 20 5 100 500 1217 1207 1209 0.17 327 LA16_Financial 10 10 100 800 788 945 1237 30.90 789 LA17_Financial 10 10 100 400 925 784 903 15.18 324 LA18_Financial 10 10 100 350 769 848 1039 22.52 328 LA19_Financial 10 10 100 150 652 842 1383 64.25 873 LA20_Financial 10 10 100 300 577 902 1188 31.71 131

Avg. 14.76 303.5

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Typical cash flow profile solving job-shop (La01 instance)

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Typical cash flow profile solving JSPCF (La01 instance)

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Job shop makespan

JSPCF makespan

Typical cash flow profile solving JSPCF and Job shop (La03 instance)

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4. CONCLUSION

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IV. Conclusion

431 2

This work is a step forward definition of wide-ranging methods for shop problem, supply chain management and cash management.

The key features of this current study are to define the JSPCF for simultaneously addressing during optimization:

• physical metrics (makespan)

• financial metrics (cash position, cash flow)

Our proposal is relevant for a company supply chain

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• A framework based on modeling the problem as a disjunctive graph with flow consideration is introduced;

• A memetic algorithm based approach is proposed;

• The memetic algorithm encompasses features including min cost max flow resolution for financial consideration, local search based on analysis of the critical path;

• The framework permits to address a wide range of job-shop problems including the “classical” one;

• The numerical experiment proves that our framework obtain almost optimal solutions in a rather short computational time for classical shop problem in terms of quality of results.

• The proposed framework is more time consuming than dedicated methods, this is not surprising since the framework has a wide range class of application.

431 2

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Perspectives• Simultaneous financial consideration of all financial machines; (each

machine has a cash position)

• Stochastic delays in payment allowing to determine robust solutions from the financial point of view (Hinderer et Waldmann, 2001);

• Splitting in machine operation depending on the financial resource units;

• Exchange and interest rates in cash flow.

431 2

Future works will be relevant for a global supply chain, and will give supply chain manager the possibility to share value (cash flows) between supply chain entities

P. Féniès, P. Lacomme, N.Tchernev LIMOS UMR CNRS 6158 CRCGM 5 décembre 2006 June, 15 MONTREAL –...

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