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M ulti- L evel C apacitated L ot- S izing Problem ( 비용 및 생산 제약 조건을 고려한 자재 소요 계획 ). 2004 / 8 / 21. 발표자 : 정성원. Contents. Main Topic A memetic algorithm for a multistage capacitated lot-sizing problem Berretta et al. International Journal of Production Economics, Vol 87, 2004 Sub Topic

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2004 / 8 / 21

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  1. Multi-Level Capacitated Lot-Sizing Problem (비용 및 생산 제약 조건을 고려한 자재 소요 계획) 2004 / 8 / 21 발표자 : 정성원

  2. Contents • Main Topic • A memetic algorithm for a multistage capacitated lot-sizing problem • Berretta et al. International Journal of Production Economics, Vol 87, 2004 • Sub Topic • A brief Introduction to genetic algorithm & memetic algorithm • Handbook of Metaheuristics, Kluwer Academic Publishers, 2003 • A heuristic method for lot-sizing in multi-stage systems • França et al. Computer and Operations Research, Vol. 24, No. 9, 1997 • Discussion – MLLS Problem 1/30

  3. Introduction I • What is the weakness of the MRP ? 1) It does not take into account the limitations of the production resources 2) It does not provide the optimal production plan • Research topics relating with the above problems • Multi-Level Lot-Sizing Problem (MLLS) : MRP + Cost • Multi-Level Capacitated Lot-Sizing Problem (MLCLS) : MRP + Cost + Capacity 2/30

  4. Lagrangean Relaxation, Shifting Procedure MS1) 42,5 (1996) Tempelmeier C&OR2) 24,9 (1997) França W/W algorithm, Shifting Procedure C&MA4) 44 (2002) Xie IJPE3) 87 (2004) Berretta Genetic Algorithm Memetic algorithm Lagrangean Relaxation, Simulated Annealing IJPE3) 68 (2000) Őzdamar Introduction II • A Brief Literature Review - MLCLS Publication Method Author 1) MS : Management Science, 2) C&OR : Computers & Operations Research 3) IJPE : International Journal of Production Economics 4) C&MA : Computers & mathematics with applications 3/30

  5. A heuristic method for lot-sizing in multi-stage system Paulo M. França, Vinícius A. Armentano, Regina E. Berretta and Alistair R. Clark Computers and Operations Research, Vol. 24, No. 9. 1997

  6. Problem Formulation I2 I1 R1 • 목적식 • 제약조건 I5 I4 R2 I3 ① ② ③ I7 I6 R3 ④ ⑤ ⑥ ① : 생산비용 ②: 재고비용 ③: 생산준비비용 ④: 재고-생산-수요 관계 제약조건 ⑤: 생산용량 제약 조건 ⑥: 생산 여부 파악을 위한 제약조건 Iit : inventory stock of item i at the end of period t Xit : lot size of item i in period t Yit : a binary variable which assumes value 1 if item i is produces in period t and 0 otherwise. 4/30

  7. Heuristic Method • The procedure of the proposed heuristic method • P1 : Procedure to obtain an initial solution • Wagner Whitin Algorithm (one item, no capacity constraints) P1 – Initial Solution feasible infeasible P2 – Smoothing P3 – Improvement P4 – Merging 5/30

  8. Heuristic Method • The procedure of the proposed heuristic method • P2 : Smoothing procedure • Try to find a feasible solution by moving production between periods • P3 : Improvement • Try to find another feasible solution by moving production between periods P1 – Initial Solution feasible P2 – Smoothing P3 – Improvement feasible infeasible P4 – Merging 6/30

  9. Heuristic Method • The procedure of the proposed heuristic method • P4 : Merging procedure • Try to prevent a large overuse of resources P1 – Initial Solution P2 – Smoothing P3 – Improvement feasible infeasible P4 – Merging 7/30

  10. Heuristic Method • The pseudo-code of the proposed heuristic method r : interation counter S[r] : solution obtained at iteration r F(S[r]) : cost value of S[r] S* = incumbent solution ITMAX = maximum number of interations of the heuristic Notation Code 8/30

  11. A brief introduction to genetic algorithm & memetic algorithm Handbook of Metaheuristics KLUWER ACADEMIC PUBLISHERS (2003)

  12. Genetic Algorithm • History • The term genetic algorithm was first used by John Hollandin 1975 whose book “Adaptation in Natural and Artificial Systems” • Ken Dejongin 1975first provide a thorough treatment of the GA’s capabilities in optimization • David Goldberg produced first an award-winning doctoral thesis on his application to gas pipeline optimization in 1985 and then ,in 1989, an influential book – “Genetic Algorithms in Search, Optimization, and Machine Learning” • Is it a good algorithm? • D.H. Wolpert and W.G. Macready in 1997 presented “No-Free-Lunch-Theorem (NFLT)” 9/30

  13. Genetic Algorithm • 개념 소개 • 아래의 문제에 있어서 GA를 적용해 보자 • 이러한 문제를 X1=0~64, X2=0~64까지 임의의 수를 발생하여 최적의 해를 찾아보자. • 이런 경우 최적 해를 찾을 확률은 1/4096 이다. • 운이 좋으면 몇 번 만에 최적 해를 찾을 수 있고 그렇지 못하면 4096번 실행해도 최적 해를 찾지 못할 수 있다. • 어떻게 하면 최적 해를 빠른 시간 내에 찾을 수 있을까? 해답은 ? X1=14, X2=24인 경우 13이 최적의 해이다. 10/30

  14. 최적 해는 어떤 패턴을 가지고 있는가? Genetic Algorithm • 개념 소개 • 최적 해의 특징 최적해는 이진법으로 볼 때 X1=001110, X2=011000 인 경우이다. 001110011000 Chromosome X2 X1 Gene 001110011000 11/30

  15. 001110011000 X2 X1 Genetic Algorithm • 개념 소개 • 스키마 : 특정 염색체(Chromosome)가 가질 수 있은 특징 예) 001 : 이 Chromosome이 가지고 있는 패턴? • 최적 해의 스키마  최적 해를 결정하는 중요한 요소 존재 가능한 스키마 수 : 23개 12/30

  16. Genetic Algorithm • 개념 소개 다음의 임의의 해들을 생각해보자. 15 x1=15 , x2=25 001111011001 001110011000 77 x1=14 , x2=16 001110010000 X2 X1 743 x1=15 , x2=51 001111110011 351 x1=31 , x2=31 011111011111 우수한 해는 최적해의 스키마를 가지고 있다. 71 x1=11 , x2=31 001011011111 1127 x1=47 , x2=29 101111011101 001***01**** 743 x1=15 , x2=51 001111110011 1183 x1=47 , x2=15 101111001111 1310 x1=15 , x2=60 001111111100 최적해의 특징을 잘 나타내는 위의 스키마를 편의상 특징 스키마로 명명하자. 15 x1=15 , x2=23 001111010111 2063 x1=45 , x2=57 101101111001 13/30

  17. Genetic Algorithm • 유전 알고리즘에서의 주요 연산자 • 교배 (Crossover) • 임의의 두 해를 교배하여 새로운 해를 만드는 것 • 변이 (Mutation) • 단순히 교배 연산만을 통하여 세대를 진행시킬 때 국부최적화에 빠질 위험을 방지하게 하기 위하여 새로운 유전자 형질(allele)를 나타내게 한다. 14 x1=15 , x2=24 15 001111011000 x1=15 , x2=25 001111011001 62 77 x1=14 ,x2=17 001110010001 x1=14,x2=16 001110010000 Crossover 스키마는 보존하면서 우수한 해 생성 14/30

  18. Genetic Algorithm • 유전 알고리즘 적용 과정 Initial Population Choose an initial population of chromosomes : while termination condition not satisfied do repeat if crossover condition satisfied then {select parent chromosomes; choose crossover parameters; perform crossover}; if mutation condition satisfied then {choose mutation points; perform mutation}; evaluate fitness of offspring Until sufficient offspring created; Select new population endwhile Select Two Parents Perform Crossover Choose Mutation Points Perform Mutation Sufficient Offspring ? No Yes Select New Population Terminate ? No Yes END 15/30

  19. Memetic Algorithm • Main idea • The generic denomination of ‘Memetic Algorithms’ (MAs) is used to encompass a broad class of metaheuristic. • Unlike traditional Evolutionary Computation methods, MAs are intrinsically concerned with exploiting all available knowledge about the problem under study. • This functioning philosophy is perfectly illustrated by the term “memetic”, in which the word ‘meme’ denotes an analogous to the gene in the context of cultural evolution - P.Moscato Examples of memes are tunes, ideas, catch-phrase, clothes fashions, ways of making pots or of building arches. Justas genes propagate themselves in the gene pool by leaping from body to body via sperms or eggs, so memes propagate themselves in the meme pool by leaping from brain to brain via a process which can be called imitation - R. Dawkins ‘Selfish Gene’ 16/30

  20. Memetic Algorithm • Designing a Memetic Algorithm Initial Population Begin Initialize pop using GenerateInitialPopulation() repeat newpop  GenerateNewPopulation(pop); pop  UpdatePopulation (pop,newpop); if pop has converged then pop  RestartPopulation(pop); endif untilTerminationCriterion(); End Generate New Population Select New Population Converge ? No Yes Restart Population Terminate ? No Yes END 17/30

  21. Memetic Algorithm • Comparison with the genetic algorithm Initial Population Select Two Parents Perform Crossover Generate New Population MA Perform Crossover Select New Population Local Optimization Converge ? No Choose Mutation Points GA Yes Perform Mutation Restart Population Sufficient Offspring ? No Terminate ? Yes No Yes END 18/30

  22. A memetic algorithm for a multistage capacitated lot-sizing problem Regina E. Berretta and Luiz Fernando Rodirigues International Journal of Production Economics, 87, 2004

  23. Problem Formulation I2 I1 R1 • 목적식 • 제약조건 I5 I4 R2 I3 ① ② ③ I7 I6 R3 ④ ⑤ ⑥ ① : 생산비용 ②: 재고비용 ③: 생산준비비용 ④: 재고-생산-수요 관계 제약조건 ⑤: 생산용량 제약 조건 ⑥: 생산 여부 파악을 위한 제약조건 Iit : inventory stock of item i at the end of period t Xit : lot size of item i in period t Yit : a binary variable which assumes value 1 if item i is produces in period t and 0 otherwise. 19/30

  24. The proposed memetic algorithm Initialize Population • Pseudo code Restart? No Yes Restart Population UPDATE_POCKET_CURRENT() TREE_ORDERNATION() CROSS_OVER() SMOOTING() IMPROVEMENT() MUTATION() No Terminate? Yes END 20/30

  25. The proposed memetic algorithm Initialize Population • Solution Representation Represented by a matrix of size 2N*T (Inventory, Lot-size) • Population size and structure Restart? No Yes Restart Population UPDATE_POCKET_CURRENT() TREE_ORDERNATION() CROSSOVER() SMOOTING() IMPROVEMENT() MUTATION() No Terminate? Yes END 21/30

  26. The proposed memetic algorithm Initialize Population • INITIALIZE_POPULATION() • Generate 13 Agents : Pocket, Current • WW algorithm • Cause perturbations in the values of the setup cost • UPDATE_POCKET_CURRENT() • TREE_ORDERNATION() Restart? No Yes Restart Population UPDATE_POCKET_CURRENT() TREE_ORDERNATION() CROSSOVER() SMOOTING() IMPROVEMENT() MUTATION() No Terminate? Yes END CURRENT POCKET 22/30

  27. The proposed memetic algorithm Initialize Population • Crossover() • In each subpopulation, the Pocket of a leader agent will recombine with all Pockets of the supporter’s agents. • Each new solution replace the current of the supporter. Restart? No Yes Restart Population UPDATE_POCKET_CURRENT() TREE_ORDERNATION() CROSSOVER() SMOOTING() crossover IMPROVEMENT() MUTATION() No Terminate? Yes END CURRENT POCKET 23/30

  28. The proposed memetic algorithm Initialize Population • Smoothing() • Try to find a feasible solution by moving production between periods • Improvement() • Try to find another feasible solution by moving production between periods • Mutation() • Try to prevent a large overuse of resources Restart? No Yes Restart Population UPDATE_POCKET_CURRENT() TREE_ORDERNATION() CROSSOVER() SMOOTING() IMPROVEMENT() P1 – Initial Solution MUTATION() P2 – Smoothing P3 – Improvement No Terminate? Yes P4 – Merging END 24/30

  29. 1차 납품업체 a 고객 공장a 물류창고 a 고객 1차 납품업체 b 물류창고 b 공장b 고객 고객 1차 납품업체 c 전체 공급 사슬에서의 발생 비용을 최소화 시키는 생산 및 분배 계획 작성 목적 Discussion – Future Research • Multi-Level Lot-Sizing Problem • MRP + Cost • Multi-Level Capacitated Lot-Sizing Problem • MRP + Cost + Capacity • Multi-Facility Capacitated Lot-Sizing Problem in Supply Chain 25/30

  30. 1 3 2 Discussion - MLLS • Chromosome Representation 아이템 1을 생산하기 위해서는 아이템 2, 3이 필요하고 아이템 1의 기간별 수요는 다음과 같다. 例) 1 1 (1) (2) X Chromosome = O Chromosome = N. Dellaert , J. Jeunet, N.Jonard International journal of Production Economics 68 (2000) 26/30

  31. 1 3 2 Discussion - MLLS • Chromosome Representation 아이템 1을 생산하기 위해서는 아이템 2, 3이 필요하고 아이템 1의 기간별 수요는 다음과 같다. 例) 1 1 (1) (2) Epistasis!! 27/30

  32. 1 3 2 Discussion - MLLS 1 1 (1) (2) • Crossover - Original • Mutation - Original T1 T2 T3 T4 T5 T6 T1 T2 T3 T4 T5 T6 Item 1 Item 1 + Item 2 Item 2 Item 3 Item 3 Child 1 Child 2 T1 T2 T3 T4 T5 T6 T1 T2 T3 T4 T5 T6 Item 1 Item 1 Item 2 Item 2 Item 3 Item 3 28/30

  33. Discussion - MLLS • Crossover – Proposed (우성 vs 열성 비교) 20 40 10 + 120 30 20 Child 29/30

  34. Discussion - MLLS Select Two Parents • Mutation – Proposed (Mutation Rate의 변화) • 우수한 Chromosome의 경우 Mutation Rate를 낮추고 • 열등한 Chromosome의 경우 Mutation Rate를 높임 • Local Optimization (Best Solution과의 비교) Perform Crossover Perform Crossover Local Optimization Choose Mutation Points Perform Mutation Sufficient Offspring ? No Current Yes Best 30/30

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