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Multi-level multi-item capacitated lot sizing problem

서울대학교 산업공학과 제조 통합 자동화 연구실 2002/02/21. Multi-level multi-item capacitated lot sizing problem. 발표자 : 정 성 원. Contents. Main Topic

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Multi-level multi-item capacitated lot sizing problem

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  1. 서울대학교 산업공학과 제조 통합 자동화 연구실 2002/02/21 Multi-level multi-item capacitated lot sizing problem 발표자 : 정 성 원

  2. Contents • Main Topic • An integrated Lagrangean relaxation-simulated annealing approach to the multi-level multi-item capacitated lot sizing problem, L. Ozdamar, International journal of production economics 68(2000) • Sub Topic • An application oriented guide to Lagrangean relaxation , M.L. Fisher, Interfaces 15(1985) • Discussion Topic • Production Planning & Distribution Planning (G7 IMS Project 2002) 1/24

  3. An Application Oriented Guide to Lagrangean Relaxation Marshall L. Fisher Department of Decision Sciences The Wharton School University of Pennsylvania, Philadelphia, Pennsylvania Interfaces 15, March-April 1985

  4. Introduction • Combinatorial Optimization Problem • Polynomial time에 풀리는 쉬운 문제 • Exponential time을 요구하는 어려운 문제 • Lagrangean Method • Complicating constraint are replaced with a penalty term in the objective function involving the amount of violation of the constraints and their dual variables Lagrangean Relaxation LP Relaxation Original Problem Easy problem to solve Easy problem to solve Hard problem to solve 3/24

  5. 물품 i는 오직 하나의 배낭 j에 할당된다. 풀이) 물품 I는 값이 가장 작은 배낭 j에 할당 배낭 j에 할당된 물품의 총량은 용량 bi이하이다. Example • The generalized assignment problem (1) (2) 4/24

  6. Proof) Original Problem Lagrangean Relaxation X*를 원문제의 최적해라고 가정 X*는 Lagrangean Relaxation서 가능해 중 하나 Lagrangean Relaxation을 이용해서 나온 최적 해는 원문제의 최적 해의 상한 값(최대화 문제) /하한 값(최소화 문제)을 제공할 수 있다. Question I • 원 문제를 Lagrangean Relaxation을 이용하여 변형하여 풀어서 나온 값은 어떻게 이용하나? • Lagrangean Relaxation을 이용해서 나온 해는 원문제의 가능해 인가? • 값의 설정에 따라 가능해가 나올 수도, 비가해가 나올 수도 있다. • Lagrangean Relaxation을 이용해서 나온 해는 원문제의 상한 값이다.(최대화문제에서) • Lagrangean Relaxation을 이용해서 나온 해는 값에 관계없이 원문제의 상한 값이다. 5/24

  7. Question II • Which constraints should be relaxed? • The relaxation should make the problem significantly easier to solve • How to compute good multiplier? • What is the good multiplier?  provide a tight bound • There is a general purpose procedure called the subgradient method • How to deduce a good feasible solution to the original problem? • Problem specific (Lagrangean heuristic) 6/24

  8. Generic Lagrangean relaxation algorithm • Generic Lagrangean relaxation algorithm (Maximize Problem) Construction of Branch And Bound Tree A Node of the tree Z*  best known feasible value U0 initial multiplier value K0 Upper bound and possibly feasible solution Adjustment of Multipliers B uk xk = Min ZLR(u) Tight upper bound를 구하기 위한 과정 Subgradient Method를 많이 사용 Solution of Lagrangean Problem C B, C 과정 반복 7/24

  9. An integrated Lagrangean relaxation-simulated annealing approach to the multi-level multi-item capacitated lot sizing problem Linet Ozdamar*, Gulay Barbarosoglu** *Istanbul kultur University, Computer Engineering Department, Istanbul, Turkey **Bogazici University, Industrial Engineering Department, Istanbul Turkey International Journal of Production Economics 68(2000, December)

  10. Introduction • 모델 Sub Item Final Item Ex) Resource1 : 480 min/day Car A : 2 min/unit (Capacity), 30min for setup Inventory Cost : 10$/unit , Production cost : 5000$/unit Setup Cost : 100$ for setup a Resource1 A Resource3 b B Resource1 Resource3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 T D c 0 100 0 0 150 0 0 0 200 0 0 150 0 0 C Resource2 Resource4 d NP-Hard Problem !! Multi Item, Multi-level, Capacitated Resource4 9/24

  11. 모든 기간의 아이템에 대한 (생산비용+ 재고유지비용+ 준비비용) 최소화 재고 및 수요 조건 생산 용량 조건 생산여부변수 = 0/1 Formulation of the basic model • 목적식 • 제약식 10/24

  12. Lagrangean relaxation problem • Lagrangean relaxation (2) (1) Uncapacitated multi-level lot sizing problem  not to easy to solve optimally (2) (1), (2) Uncapacitated single-level lot sizing problem (WW)  not to easy to find feasible solution 11/24

  13. Generic Lagrangean relaxation algorithm (1/2) • Lagrangean relaxation algorithm (Z) Original Problem K=0 • 원 문제의 제약식을 목적식에 올려 풀기 쉬운 Lagrangean Problem로 변환한다. • Lagrangean Problem의 최적해 Xk* 를 구한다. • 최적해 Xk* 서 Heuristic 기법을 적용하여 원문제의 가능해 Xfk를 구한다. • Lagrange multiplier 값을 변경한다. • Z*의 상한과 하한 값의 차가 상한 값의 일정 비율 이하일 경우 알고리즘을 종료한다. Lagrangean Problem (ZD) Xk* Lagrangean Heuristic Xfk Adjustment of Multipliers K=k+1 uk Lagrangean Heuristic  Simulated Annealing , Phase-1 procedure 12/24

  14. Generic Lagrangean relaxation algorithm (2/2) • The step of this algorithms • Step 1 : Solve the subproblems resulting from relaxation and update the lower bound for the original problem • Step 2 : Using the solution obtained in Step 1, determine the sub-gradients for the relaxed constraints and carry out sub-gradient optimization to update the value of the Lagrange multiplier • Step 3 : Execute a Lagrangean heuristic on the solution obtained in Step 1 to get a feasible solution and update the upper bound. Repeat Steps 1 and 2 with the new values of the Lagrange multipliers obtained in Step 2 until a termination criterion is satisfied LP-Relaxation : 1) HL(u), 2) FL(u) Lagrangean heuristic : 1) SA(Simulated Annealing) , 2) SAI(Simulated Annealing with only improving moves) 3) Phase-1 procedure 13/24

  15. Phase-1 procedure • Phase-1 procedure • An iterative heuristic which carries out a search around a given solution either until a capacity feasible solution is obtained or a given number of iterations is reached without obtaining a feasible solution • Main step • Determine randomly an overloaded resource k and time period t. • Select randomly an item i requiring resource k in period t • Carry out backward shifting and multi-item shifting if necessary • Carry out forward single-item shifting without violating requirements of the successors • Continue with (ii) if there still remain unconsidered items requiring resource k and resource k is still capacity infeasible Forward shifting Backward shifting ttt (tt=t+1, t+2, … ) ttt (tt=t -1, t -2, … ) 14/24

  16. Simulated Annealing • Main step • Decide randomly upon the shift type of whether to increase or decrease a lot • Select randomly an item i and the time period t in which the first decision is to be implemented • Determine the resource k required by item i in period t • Determine the quantity to be shifted • Main purpose • All-feasible solution • Improve the current objective function value without violating capacity and inventory feasibility • All-infeasible solution • Generate a feasible solution with the best objective function value out of a solution which is infeasible • Only-inventory-feasible • Generate a feasible solution out of a solution which is only feasible with inventory constraint 15/24

  17. The number of Lagrangean relaxation is 100 The global search procedures are executed with 5000 iteration during the Lagrangean relaxation Phase-1 procedure is execute at most 10 times to find an “all-feasible solution” Computational Result • Test data • Test data from H. Tempelmeir, M. Derstroff [Management Science 42 , 1996] • Approach • 1) HL+SAI, 2) HL+SA, 3) HL+P1+SAI, 4) HL+P1+SA, 5) FL+SAI, 6) FL+SA • Result 16/24

  18. Conclusion • Contribution • Develops a heuristic approach for solving the MLCLSP • The first attempts integrating Lagrangean relaxation and Simulated Annealing • Show that the integration of Lagrangean relaxation with meta-heuristic is mutually beneficial for both approaches • Discussion • Is it a good approach to use the SA (or SAI) to make infeasible solution be feasible solution ? 17/24

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