Solving multi-item lot-sizing problems with an MIP solver using classification and reformulation

Management Science

Volumn 48, Issue 12, 2002, Pages 1587-1602

  Wolsey, Laurence A ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Classification; Convex hull; Extended formulation; Lot sizing; Mixed integer programming; Production planning

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; CONSTRAINT THEORY; INTEGER PROGRAMMING; OPTIMIZATION; PLANNING; PROBLEM SOLVING; PRODUCTION;

EXTENDED FORMULATION; LOT SIZING; MIXED-INTEGER PROGRAMMING;

MANAGEMENT SCIENCE;

EID: 0036968625     PISSN: 00251909     EISSN: None     Source Type: Journal    
DOI: 10.1287/mnsc.48.12.1587.442     Document Type: Article

References (61)

1. Aggarwal, A., J. Park. 1993. Improved algorithms for economic lot-size problems. Oper. Res. 41 549-571.
2. Agra, A., M. Constantino. 1999. Lotsizing with backlogging and start-ups: The case of Wagner-Whitin costs. Oper. Res. Lett. 25 81-88.
3. ____, ____. 2001. Polyhedral description of basic knapsack problems with a continuous variable. Department of Mathematics, University of Lisbon, Lisbon, Portugal.
4. Barany, I., J. Edmonds, L. A. Wolsey. 1986. Packing and covering a tree by subtrees. Combinatorica 6 245-257.
5. ____, T. J. Van Roy, L. A. Wolsey. 1984. Uncapacitated lot sizing: The convex hull of solutions. Math. Programming Stud. 22 32-43.
6. Belvaux, G., L. A. Wolsey. 2000. BC-PROD: A specialized branch-and-cut system for lot-sizing problems. Management Sci. 46 724-738.
7. ____, ____. 2001. Modelling practical lot-sizing problems as mixed integer programs. Management Sci. 47 993-1007.
8. Bitran, G. R., H. H. Yanasse. 1982. Computational complexity of the capacitated lot size problem. Management Sci. 28 1174-1186.
9. The CHES Problems. 1989. Chesapeake Decision Sciences Inc., New Providence, NJ.
10. Clark, A. J., H. Scarf. 1960. Optimal policies for multi-echelon inventory problems. Management Sci. 6 475-490.
11. Constantino, M. 1995. A polyhedral approach to production planning models: Start-up costs and times, upper and lower bounds on production. Ph.D. thesis, Faculté des Sciences Appliquées, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
12. ____. 1996. A cutting plane approach to capacitated lot-sizing with start-up costs. Math. Programming 75 353-376.
13. ____. 1998. Lower bounds in lot-sizing models: A polyhedral study. Math. Oper. Res. 23 101-118.
14. Cordier, C., H. Marchand, R. Laundry, L. A. Wolsey. 1999. bc-opt: A branch-and-cut code for mixed integer programs. Math. Programming 86 335-354.
15. ILOG Cplex 7.0 User's Manual. 2000. Gentilly, France.
16. Eppen, G. D., R. K. Martin. 1987. Solving multi-item lot-sizing problems using variable redefinition. Oper. Res. 35 832-848.
17. Federgruen, A., M. Tsur. 1991. A simple forward algorithm to solve general dynamic lot-size models with n periods in O(n log n) or O(n) time. Management Sci. 37 909-925.
18. Fleischmann, B. 1990. The discrete lot-sizing and scheduling problem. Eur. J. Oper. Res. 44 337-348.
19. ____ 1994. The discrete lotsizing and scheduling problem with sequence-dependent setup costs. Eur. J. Oper. Res. 75 395-404.
20. Florian, M., M. Klein. 1971. Deterministic production planning with concave costs and capacity constraints. Management Sci. 18 12-20.
21. Günlük, O., Y. Pochet. 2001. Mixing mixed integer inequalities. Math. Programming 90 429-457.
22. Karmarkar, U. S., L. S. Schrage. 1985. The deterministic dynamic product cycling problem. Oper. Res. 33 326-345.
23. Krarup, J., O. Bilde. 1977. Plant location, set covering and economic lot sizes: An O(mn) algorithm for structured problems. L. Collatz et al. eds. Optimierung bei Graphentheoretischen und Ganzzahligen Probleme. Birkhauser Verlag, Basel, Switzerland, 155-180.
24. Kuik, R., M. Salomon, L. N. van Wassenhove. 1994. Batching decisions: Structure and models. Eur. J. Oper. Res. 75 243-263.
25. Leung, J. M. Y., T. L. Magnanti, R. Vachani. 1989. Facets and algorithms for capacitated lot-sizing. Math. Programming 45 331-359.
26. Loparic, M. 2001. Stronger mixed 01 models for lot-sizing problems. Ph.D. thesis, Faculty of Engineering, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
27. ____, H. Marchand, L.A. Wolsey. 2002. Dynamic knapsack sets and capacitated lot-sizing. Math. Programming B. Forthcoming.
28. ____, Y. Pochet, L. A. Wolsey. 2001. The uncapacitated lot-sizing problem with sales and safety stocks. Math. Programming 89 487-504.
29. Love, S. F. 1973. Bounded production and inventory models with piecewise concave costs. Management Sci. 20 313-318.
30. Marchand. H. 1998. A polyhedral study of the mixed knapsack set and its use to solve mixed integer programs. Ph.D. thesis, Faculty of Engineering, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
31. Miller, A., L. A. Wolsey. 2001a. Tight MIP formulations for multi-item discrete lot-sizing problems. Universite Catholique de Louvain, Louvain-la-Neuve, Belgium.
32. ____, ____. 2001b. Tight formulations for some simple MIPs and convex objective IPs. CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
33. ____, G. L. Nemhauser, M. W. P. Savelsbergh. 2000a. On the polyhedral structure of a multi-item capacitated lot-sizing problems with set-up times by branch-and-cut. CORE DP 2000/52, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
34. ____, ____, ____. 2000b. Solving a multi-item production planning model with set-up times. CORD DP 2000/39. Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
35. Padberg, M. W., T. J. Van Roy, L. A. Wolsey. 1985. Valid inequalities for fixed charge problems. Oper. Res. 33 842-861.
36. Pochet, Y. 1988. Valid inequalities and separation for capacitated economic lot-sizing. Oper. Res. Lett. 7 109-116
37. ____. 2001. Mathematical programming models and formulations for deterministic production planning problems. M. Jünger, D. Naddef, eds. Computational Combinatorial Optimization. Lecture Notes in Computer Science LNCS 2241, Springer, Berlin-Heidelberg, Germany, 57-111.
38. ____, L. A. Wolsey. 1988. Lot-size models with backlogging: Strong formulations and cutting planes. Math. Programming 40 317-335.
39. ____, ____. 1993. Lot-sizing with constant batches: Formulation and valid inequalities. Math. Oper. Res. 18 767-785.
40. ____, ____. 1994. Polyhedra for lot-sizing with Wagner-Whitin costs. Math. Programming 67 297-324.
41. ____, ____. 1995. Algorithms and reformulations for lot-sizing problems. W. Cook, L. Lovasz, P. Seymour, eds. Combinatorial Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 20. American Mathematical Society, Providence, RI.
42. Rardin, R., L. A. Wolsey. 1993. Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems. Eur. J. Oper. Res. 71 95-109.
43. Simpson, N. C., S. S. Erenguc. 1998. Production planning in multiple stage manufacturing environments with joint costs, limited resources and set-up times. Technical Report, Department of Management Science and Systems, University of Buffalo, Buffalo, NY.
44. Vanderbeck, F. 1998. Lot-sizing with start-up times. Management Sci. 44 1409-1425.
45. Van Eijl, C. A. 1996. A polyhedral approach to the discrete lot-sizing and scheduling problem. Ph.D. thesis, Technical University of Eindhoven, Eindhoven, The Netherlands.
46. ____, C.P.M. van Hoesel. 1997. On the discrete lot-sizing and scheduling problem with Wagner-Whitin costs. Oper. Res. Lett. 20 7-13.
47. van Hoesel, C. P. M. 1991. Models and algorithms for single-item lot-sizing problems. Ph.D. thesis, Erasmus University, Rotterdam, The Netherlands.
48. ____, A.W.J. Kolen. 1993. A class of strong valid inequalities for the discrete lot-sizing and scheduling problem. Eindhoven University of Technology, Eindhoven, The Netherlands.
49. ____, R. Kuik, M. Salomon, L. N. van Wassenhove. 1994. The single item discrete lot-sizing and scheduling problem: Optimization by linear and dynamic programming. Discrete Appl. Math. 48 289-303.
50. ____, ____. 1994. A linear description of the discrete lot-sizing and scheduling problem. Eur. J. Oper. Res. 75(2) 342-353.
51. 3) algorithm for the economic lot-sizing problem with constant capacities. Management Sci. 42 142-150.
52. ____, ____. 2001. Fully polynomial approximation schemes for single-item capacitated economic lot-sizing problems. Math. Oper. Res. 26 339-357.
53. ____, ____. L. A. Wolsey. 1994. Polyhedral characterization of the economic lot-sizing problem with start-up costs. SIAM J. Discrete Math. 7 141-151.
54. 3) for the single item lot-sizing problem with constant capacities and backlogging. Internal Report, CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
55. ____. 2002. Algorithms for constant capacity lot-sizing problems. Internal Report, CORE, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, Forthcoming.
56. Wagelmans, A. P. M., C. P. M. van Hoesel, A. W. J. Kolen. 1992. Economic lot-sizing: An O(n log n) algorithm that runs in linear time in the Wagner-Whitin case. Oper. Res. 40(Supplement 1) 145-156.
57. Wagner, H. M., T. M. Whitin. 1958. Dynamic version of the economic lot size model. Management Sci. 5 89-96.
58. Wolsey, L. A. 1989. Uncapacitated lot-sizing problems with start-up costs. Oper. Res. 37 741-747.
59. ____. 1999. Integer programming for production planning and scheduling. Presented at Chorin Winter School, Chorin, Germany.
60. XPRESS-MP Optimiser, Release 12.5. 2001. Dash Associates, Northants, U.K.
61. Zangwill, W. I. 1969. A backlogging model and a multi-echelon model of a dynamic economic lot size production system-A network approach. Management Sci. 15, 506-526.