1 Best Permutations for the Dynamic Plant Layout Problem Jose M. Rodriguez †, F. Chris MacPhee ‡, David J. Bonham †, Joseph D. Horton ‡, Virendrakumar

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Best Permutations for the Dynamic Plant Layout Problem

Jose M. Rodriguez†, F. Chris MacPhee‡, David J. Bonham†, Joseph D. Horton‡,

Virendrakumar C. Bhavsar‡

†Department of Mechanical Engineering‡Faculty of Computer Science

University of New Brunswick Fredericton, N.B., Canada, E3B 5A3

[email protected]

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Outline

• Introduction

• Problem Statement

• Description of the Algorithm

• Experimental Results

• Conclusions

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IntroductionPlant layout: an engineering design problem

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Facilities Planning Process

S

T

O

Strategic Planning stages:

1. Planning (strategic level)

2. Design (tactical level)

3. Implementation (operational level).

S Objectives are determined

T Other plant layout requirements are determined

O Plant layout is selected and maintained.

Plant layout objectives are determined according to:

• Selection of a strategy to manufacture the products (i.e., Manufacturing strategy)

• Definition of the products (i.e., Product design)

• Specification of the process plan (i.e., Process design)

• Definition of the production plan (i.e., Schedule design).

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Facilities Design Process (In the solution of “real life” problems the layout decision usually results from

combining the use of heuristics with a “satisfactory” design) Phase I Phase II Phase III Optimization — GATS Discrete Event Simulation Decision Making

Layout Set #1 (FLP/DPLP)

Performance Measure

Layout Set #2 (FLP/DPLP)

Performance Measure

Layout Set #s (FLP/DPLP)

Performance Measure

Set of Agile Plant Layouts (i.e., one per period over the planning horizon, P )

Competitive Priorities Low-cost manufacturing Product mix/volume flexibility

Schematic diagram of the integrated planning system (IPS), where manufacturing process simulation is based on layouts generated by GATS that are used to measure system performance (e.g., throughput rate, machine utilization, and cycle time).

… …

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Problem Statement

The Dynamic PlantLayout Problem (DPLP)

as an optimization problem

[J. Balakrishnan and C. H. Cheng, “Dynamic Layout Algorithms: A State-of-the-art Survey,” International Journal of Management Science, vol. 26:4, pp. 507-521, 1998.]

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DPLP Formulation

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Ytijl - 0,1 dependant variable for including cost of shifting facility i from location j to location l in period t

Atijl - fixed cost of shifting facility i from location j to location l in period t

Ctijkl - cost of material flow between facility i located at j and facility k located at l in period t

djl - distance from location j to location l

ftik - flow of material between facility i and facility k in period t

P - number of periods

n - number of facilities and locations

t - a given period of the planning horizon

i,k - facilities in the layout

j,l - locations in the layout

DPLP Formulation

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Genetic and Tabu Search Algorithm (GATS)

• Overview of the GATS code

• The GATS search space

• The triangular evolutionary technique used by GATS

• Research questions

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Overview of the GATS Code

BEGIN

Initialize Variables

QAP DPLP

Parameter Optimizer

Engine

END

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The GATS Search Space

Location of a DPLP instance cost in the GATS search space is defined by (P, CF, N), where:

• Population (P) refers to the number of chromosomes or layouts in the parent pool

• Convergence factor (CF) is the evolution threshold; a better solution must be found every CF generations or the tabu search parameters are modified

• Mutation (N) refers to the number of mutations to be performed between the crossover and tabu search routines

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The Triangular Evolutionary Technique Used by GATS

Engine (CF) M TL

M=1 TL=0 M=2 TL=0 M=3 TL=0

M=2 TL=1 M=3 TL=1

M=3 TL=2

• • • • • • • • • • • •

P, N

P+1, N

• • •

Synergetic Evolution (i.e., improving population quality at each generation)

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Research Questions

• How many optimal layout sets are there and what are they?

• How many times is the layout changed during the five periods (i.e., re-layouts) and at what cost?

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Flow/Distance matrices & shifting costs for the Rosenblatt (1986) Instance

Material Flow and Shifting Costs

To To From 1 2 3 4 5 6 From 1 2 3 4 5 6 Period 1 Period 4 1 0 63 605 551 116 136 1 0 112 15 199 665 649 2 63 0 635 941 50 191 2 153 0 116 173 912 671 3 104 71 0 569 136 55 3 10 28 0 182 855 542 4 65 193 622 0 77 90 4 29 69 15 0 552 751 5 162 174 607 591 0 179 5 198 71 42 24 0 758 6 156 13 667 611 175 0 6 62 109 170 90 973 0 Period 2 Period 5 1 0 175 804 904 56 176 1 0 663 23 128 119 50 2 63 0 743 936 45 177 2 820 0 5 98 141 66 3 168 85 0 918 138 134 3 822 650 0 137 78 91 4 51 94 962 0 173 39 4 826 570 149 0 93 151 5 97 104 730 634 0 144 5 915 515 53 35 0 177 6 95 115 983 597 24 0 6 614 729 178 10 99 0 Period 3 1 0 90 77 553 769 139 2 168 0 114 653 525 185 Shifting cost for facilities 3 32 35 0 664 898 87 887 964 213 367 289 477 4 27 166 42 0 960 179 5 185 56 44 926 0 104 6 72 128 173 634 687 0

Distance (Between Locations) Matrix

1 2 3 4 5 6 1 0 1 2 1 2 3 2 1 0 1 2 1 2 3 2 1 0 3 2 1 4 1 2 3 0 1 2 5 2 1 2 1 0 1 6 3 2 1 2 1 0

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GATS Early Convergence

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GATS Final Convergence

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GATS End of Evolution

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An Optimal Layout Set

## Optimal --> Cost: 71187 , NUM_MOVE: 4 , TABU_LEN: 1 , G: 138 , Real G: 38

The Layout is: P= 11, cost= 71187No.1 Period: 6 4 2 5 3 1 No.2 Period: 6 4 2 5 3 1 No.3 Period: 6 4 2 3 5 1 No.4 Period: 4 6 2 3 5 1 No.5 Period: 4 1 2 3 5 6

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All Layout Sets Found by GATS

Cost P/N Permutation Set 71187 3/0 No. 0 Period: 5 3 1 6 4 2

No. 1 Period: 5 3 1 6 4 2 No. 2 Period: 3 5 1 6 4 2 No. 3 Period: 3 5 1 4 6 2 No. 4 Period: 3 5 6 4 1 2

11/0 No. 0 Period: 6 4 2 5 3 1 No. 1 Period: 6 4 2 5 3 1 No. 2 Period: 6 4 2 3 5 1 No. 3 Period: 4 6 2 3 5 1 No. 4 Period: 4 1 2 3 5 6



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Experimental Results

The GATS site:http://acrl.cs.unb.ca/research/gats/

Experiments were performed on infrastructure managed by the Advanced Computational Research Laboratory at the University of New Brunswick

http://acrl.cs.unb.ca/research/gats/

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ACRL Infrastructure

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ACRL Usage

UNB (Chem)

UNB (CS)

UNB (Geo)

UNB (ME:GATS)

UNB (ME:Other)

UNB (Phys)

UNB (Other)

MTA (Chem)

External (Other)

Non-Research

From April 2003 - March 2004, GATS utilized over 11 CPU years of compute

time on the ACRL chorus cluster.

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DPLP AlgorithmsGenetic Algorithms

CVGA - Conway, D.G. and Venkataramanan, M.A.

NLGA - Balakrishnan, J. and Cheng, C.H.

GADP - Balakrishnan, J., Cheng, C.H., Conway, D.G., and Lau, C.M.

CCGA - Chang, M., Sugiyama, M., Ohkura, K., and Ueda, K.

SymEA - Chang, M., Ohkura, K., Ueda, K., and Sugiyama, M.

Simulated Annealing Algorithms

SA - Baykasoglu, A. and Gindy, N.N.Z. SA, GA, DP

Dynamic Programming, Genetic, and Simulated Annealing Algorithms

DP-GA-SA - Erel, E., Ghosh, J.B., Simon, J.T.

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DPLP ResultsTotal cost of 6 department / 5 period instances

Prob 1 Prob 2 Prob 3 Prob 4 Prob 5 Prob 6 Prob 7 Prob 8

Optimal 106419 104834 104320 106399 105628 103985 106439 103771

GATS 106419 104834 104320 106399 105628 103985 106439 103771

CVGA 108976 105170 104520 106719 105628 105606 106439 104485

NLGA 106419 104834 104320 106515 105628 104053 106978 103771

GADP 106419 104834 104529 106583 105628 104315 106447 103771

SA 107249 105170 104800 106515 106282 103985 106447 103771

DP-GA-SA 106419 104834 104320 106399 105628 103985 106439 103771

CCGA 108574 105717 105081 107718 106164 105351 108063 106561

SymEA 106419 104834 104320 106509 105628 103985 106439 103771

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Optimal 214313 212134 207987 212530 210906 209932 214252 212588

GATS 214313 212134 207987 212530 210906 209932 214252 212588

CVGA 218407 215623 211028 217493 215363 215564 220529 216291

NLGA 214397 212138 208453 212953 211575 210801 215685 214657

GADP 214313 212134 207987 212741 210944 210000 215452 212588

SA 215200 214713 208351 213331 213812 211213 215630 214513

DP-GA-SA 214313 212134 207987 212741 211022 209932 214252 212588

CCGA 218531 215063 209782 216776 213900 212560 217919 217072

SymEA 214921 212138 207987 212741 211072 209932 214252 212588


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Optimal 480453 484761 488748 484446 487753 486493 486268 490551

GATS 504759 514718 516063 508532 515599 509384 512508 514839

CVGA 511854 507694 518461 514242 512834 513763 512722 521116

NLGA 493707 494476 506684 500826 502409 497382 494316 500779

GADP(R) 484090 485352 489898 484625 489885 488640 489378 500779

GADP(U) 501447 506236 512886 504956 509636 508215 508848 512320

SA 481738 485167 491310 485862 489304 488452 487576 493030

DP-GA-SA 481738 485167 491310 485862 489304 488452 487576 493030

CCGA 492622 496628 502320 495660 499054 497993 497706 501835

SymEA 485489 490791 494219 490945 493573 490735 492301 496457


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GATS 979472 977338 981547 972569 977197 967654 979866 983837

CVGA 1055536 1061940 1073603 1060034 1064692 1066370 1066617 1068216

NLGA 1047596 1037580 1056185 1026789 1033591 1028606 1043823 1048853

GADP(R) 1004806 1006790 1012482 1001795 1005988 1002871 1019645 1010772

GADP(U) 987887 980638 985886 976025 982778 973912 982872 987789

SA 1017741 1016567 1021075 1007713 1010822 1007210 1013315 1019092

DP-GA-SA 984013 982714 988465 976456 982191 973199 982790 988304

CCGA 1005038 1003340 1010564 997930 1003228 994717 1001368 1006655

SymEA 990646 990023 993861 986403 988881 977384 991393 993981


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GATS 575371 569023 573493 566072 557277 565388 567217 572992

CVGA 632737 647585 642295 634626 639693 637620 640482 635776

NLGA 611794 611873 611664 611766 604564 606010 607134 620183

GADP(R) 603339 589834 592475 586064 580624 587797 588347 590451

GADPU 578689 572232 578527 572057 559777 566792 567873 575720

SA 604408 604370 603867 596901 591988 599862 600670 610474

DP-GA-SA 579741 570906 577402 569596 561078 567154 568196 575273

CCGA 607957 602878 602973 595257 588956 597866 602682 603391

SymEA 590200 581043 584195 577629 571133 578039 581913 587653


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GATS 1162191 1161594 1156750 1147112 1129559 1143721 1151021 1168560

CVGA 1362513 1379640 1365024 1367130 1356860 1372513 1382799 1383610

NLGA 1228411 1231978 1231829 1227413 1215256 1221356 1212273 1245423

GADPR 1194084 1199001 1197253 1184422 1179673 1178091 1186145 1208436

GADPU 1169474 1168878 1166366 1154192 1133561 1145000 1145927 1168657

SA 1223124 1231151 1230520 1200613 1210892 1221356 1212273 1231408

DP-GA-SA 1171178 1169138 1165525 1152684 1128136 1143824 1142494 1167163

CCGA 1229547 1229469 1222068 1208759 1195233 1213141 1223283 1224250

SymEA 1199376 1200300 1191673 1177912 1163035 1178097 1185496 1193189


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Conclusions: Results• GATS has been developed to solve QAP and

DPLP instances. We have challenged the well-known QAPLIB, a selected DPLP dataset, and other difficult instances.

• Optimum or best-known permutations have been generated for over 82% of 210 available QAP and DPLP instances.

• Better solutions than those known to date for the DPLP have been found. Of the attempted 51 DPLP instances, 29 now have new best-known solution found by GATS.

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Conclusions: Benefits

• Multiple global optima provide many benefits– equally optimal layouts by different qualitative criteria – solutions can be chosen that may have fewer layout

changes, requiring fewer interruptions in the production system.

• Most results published in the literature include only the layout cost (no permutation). This has lead to false best known solutions and retractions. Published results should always include both costs and permutations for verification purposes.

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Conclusions: Future Work

• A provisional patent regarding GATS has been filed with the United States Patent and Trademark Office . Next steps…

• Although the design of a factory is a planning problem, response time is as important as solution quality. The concept of ‘iterative design’ is implicit in GATS and can only be realized with a high performance computing (HPC) infrastructure.

Documents

1 Best Permutations for the Dynamic Plant Layout Problem Jose M. Rodriguez †, F. Chris MacPhee ‡, David J. Bonham †, Joseph D. Horton ‡, Virendrakumar