Coordinated Capacitated Lot-Sizing Problem with Dynamic Demand: A Lagrangian Heuristic

Coordinated replenishment problems are common in manufacturing and distribution when a family of items shares a common production line, supplier, or a mode of transportation. In these situations the coordination of shared, and often limited, resources across items is economically attractive. This paper describes a mixed-integer programming formulation and Lagrangian relaxation solution procedure for the single-family coordinated capacitated lot-sizing problem with dynamic demand. The problem extends both the multi-item capacitated dynamic demand lot-sizing problem and the uncapacitated coordinated dynamic demand lot-sizing problem. We provide the results of computational experiments investigating the mathematical properties of the formulation and the performance of the Lagrangian procedures. The results indicate the superiority of the dual-based heuristic over linear programming-based approaches to the problem. The quality of the Lagrangian heuristic solution improved in most instances with increases in problem size. Heuristic solutions averaged 2.52% above optimal. The procedures were applied to an industry test problem yielding a 22.5% reduction in total costs.

[1]  Edward P. C. Kao,et al.  A Multi-Product Dynamic Lot-Size Model with Individual and Joint Set-up Costs , 1979, Oper. Res..

[2]  Chien-Hua M. Lin,et al.  An effective algorithm for the capacitated single item lot size problem , 1994 .

[3]  Y. Roll,et al.  A Heuristic Algorithm for the Multi-Item Lot-Sizing Problem with Capacity Constraints , 1982 .

[4]  Linet Özdamar,et al.  The capacitated lot sizing problem with overtime decisions and setup times , 2000 .

[5]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[6]  William W. Trigeiro,et al.  A Dual-Cost Heuristic For The Capacitated Lot Sizing Problem , 1987 .

[7]  L Van Wassenhove,et al.  Lagrangean Relaxation for the Multi-Item Capacitated Lot-Sizing Problem , 1985 .

[8]  Kenneth C. Gilbert,et al.  A Heuristic for a Class of Production Planning and Scheduling Problems , 1991 .

[9]  Ömer Kirca,et al.  A new heuristic approach for the multi-item dynamic lot sizing problem , 1994 .

[10]  M. Florian,et al.  DETERMINISTIC PRODUCTION PLANNING WITH CONCAVE COSTS AND CAPACITY CONSTRAINTS. , 1971 .

[11]  Vincent A. Mabert,et al.  Cyclical schedules for capacitated lot sizing with dynamic demands , 1991 .

[12]  Chien-Hua M. Lin,et al.  An OT2 Algorithm for the NI/G/NI/ND Capacitated Lot Size Problem , 1988 .

[13]  Harish C. Bahl,et al.  A Lagrangean Relaxation Approach for Very-Large-Scale Capacitated Lot-Sizing , 1992 .

[14]  James R. Evans.  An efficient implementation of the Wagner-Whitin algorithm for dynamic lot-sizing , 1985 .

[15]  J. K. Lenstra,et al.  Deterministic Production Planning: Algorithms and Complexity , 1980 .

[16]  Leon S. Lasdon,et al.  An Efficient Algorithm for Multi-Item Scheduling , 1971, Oper. Res..

[17]  G. D. Eppen,et al.  Solving Multi-Item Capacitated Lot-Sizing Problems Using Variable Redefinition , 1987, Oper. Res..

[18]  Tony J. Van Roy,et al.  Cross decomposition for mixed integer programming , 1983, Math. Program..

[19]  Ömer Kirca,et al.  An optimal procedure for the coordinated replenishment dynamic lot‐sizing problem with quantity discounts , 2000 .

[20]  Khalil S. Hindi,et al.  Solving the CLSP by a Tabu Search Heuristic , 1996 .

[21]  Gabriel R. Bitran,et al.  The Multi-Item Capacitated Lot Size Problem: Error Bounds of Manne's Formulations , 1986 .

[22]  Ömer Kirca.  An efficient algorithm for the capacitated single item dynamic lot size problem , 1990 .

[23]  Dev Joneja,et al.  The Joint Replenishment Problem: New Heuristics and Worst Case Performance Bounds , 1990, Oper. Res..

[24]  S. Lippman.  Optimal inventory policy with multiple set-up costs , 1968 .

[25]  S. S. Erenguc,et al.  Multi‐Item Inventory Models with Co‐ordinated Replenishments: A Survey , 1988 .

[26]  R. Kipp Martin,et al.  Generating Alternative Mixed-Integer Programming Models Using Variable Redefinition , 1987, Oper. Res..

[27]  James R. Evans.  Network-based optimization algorithms for the capacitated multi-item lot sizing problem , 1985 .

[28]  S. Selcuk Erenguc,et al.  Multiproduct dynamic lot-sizing model with coordinated replenishments , 1988 .

[29]  Alan S. Manne,et al.  Programming of Economic Lot Sizes , 1958 .

[30]  Vahid Lotfi,et al.  An Algorithm for the Single-item Capacitated Lot-Sizing Problem with Concave Production and Holding Costs , 1994 .

[31]  William W. Trigeiro,et al.  Capacitated lot sizing with setup times , 1989 .

[32]  Laurence A. Wolsey,et al.  Uncapacitated lot-sizing: The convex hull of solutions , 1984 .

[33]  Jatinder N. D. Gupta,et al.  OR Practice - Determining Lot Sizes and Resource Requirements: A Review , 1987, Oper. Res..

[34]  Dong X. Shaw,et al.  An Algorithm for Single-Item Capacitated Economic Lot Sizing with Piecewise Linear Production Costs and General Holding Costs , 1998 .

[35]  Sebastián Lozano,et al.  Primal-dual approach to the single level capacitated lot-sizing problem , 1991 .

[36]  M. Lambrecht,et al.  Heuristic Procedures for the Single Operation, Multi-Item Loading Problem , 1979 .

[37]  A. Federgruen,et al.  A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n Periods in 0n log n or 0n Time , 1991 .

[38]  P. R. Kleindorfer,et al.  A Lower Bounding Structure for Lot-Size Scheduling Problems , 1975, Oper. Res..

[39]  S. Selçuk Erengüç,et al.  A branch and bound algorithm for a single item nonconvex dynamic lot sizing problem with capacity constraints , 1990, Comput. Oper. Res..

[40]  Jatinder N. D. Gupta,et al.  Determining lot sizes and resource requirements: A review , 1987 .

[41]  G. Bitran,et al.  Computational Complexity of the Capacitated Lot Size Problem , 1982 .

[42]  Jean-Michel Thizy,et al.  Analysis Of Lagrangian Decomposition For The Multi-Item Capacitated Lot-Sizing Problem , 1991 .

[43]  P. S. Eisenhut.  A Dynamic Lot Sizing Algorithm with Capacity Constraints , 1975 .

[44]  Edward A Silver,et al.  Coordinated replenishments of items under time‐varying demand: Dynamic programming formulation , 1979 .

[45]  Willard I. Zangwill,et al.  A Deterministic Multiproduct, Multi-Facility Production and Inventory Model , 1966, Oper. Res..

[46]  Harvey M. Wagner,et al.  Dynamic Version of the Economic Lot Size Model , 2004, Manag. Sci..

[47]  Laurence A. Wolsey,et al.  Strong Formulations for Multi-Item Capacitated Lot Sizing , 1984 .

[48]  Hanif D. Sherali,et al.  Convex envelope results and strong formulations for a class of mixed‐integer programs , 1996 .

[49]  E. Newson.  Multi-Item Lot Size Scheduling by Heuristic Part I: With Fixed Resources , 1975 .

[50]  Albert P. M. Wagelmans,et al.  Economic Lot Sizing: An O(n log n) Algorithm That Runs in Linear Time in the Wagner-Whitin Case , 1992, Oper. Res..

[51]  E. Newson.  Multi-Item Lot Size Scheduling by Heuristic Part II: With Variable Resources , 1975 .

[52]  Tony J. Van Roy,et al.  A Cross Decomposition Algorithm for Capacitated Facility Location , 1986, Oper. Res..

[53]  F. Toledo,et al.  A network flow model for the capacitated lot-sizing problem , 1999 .

[54]  Albert P. M. Wagelmans,et al.  Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems , 2001, Math. Oper. Res..

[55]  E. Pratsini,et al.  The capacitated dynamic lot size problem with variable technology , 2000 .

[56]  Khalil S. Hindi,et al.  Algorithms for Capacitated, Multi-Item Lot-Sizing without Set-Ups , 1995 .

[57]  Ke Ding,et al.  A Tabu-Search Heuristic for the Capacitated Lot-Sizing Problem with Set-up Carryover , 2001, Manag. Sci..

[58]  Thomas L. Magnanti,et al.  Facets and algorithms for capacitated lot sizing , 1989, Math. Program..

[59]  Stephen F. Love.  Bounded Production and Inventory Models with Piecewise Concave Costs , 1973 .

[60]  K. R. Baker,et al.  An Algorithm for the Dynamic Lot-Size Problem with Time-Varying Production Capacity Constraints , 1978 .

[61]  Ömer Kirca,et al.  The coordinated replenishment dynamic lot-sizing problem with quantity discounts , 1996 .

[62]  S. Graves.  A multiechelon inventory model with fixed replenishment intervals , 1996 .

[63]  Ömer Kirca,et al.  A primal‐dual algorithm for the dynamic lotsizing problem with joint set‐up costs , 1995 .

[64]  S. Selcuk Erenguc,et al.  A multifamily dynamic lot‐sizing model with coordinated replenishments , 1990 .

[65]  Harvey H. Millar,et al.  Lagrangian heuristics for the capacitated multi-item lot-sizing problem with backordering , 1994 .

[66]  John O. McClain,et al.  Mathematical Programming Approaches to Capacity-Constrained MRP Systems: Review, Formulation and Problem Reduction , 1983 .

[67]  N. Adam,et al.  The Dynamic Lot-Sizing Problem for Multiple Items Under Limited Capacity , 1981 .

[68]  W. Zangwill.  A Deterministic Multi-Period Production Scheduling Model with Backlogging , 1966 .

[69]  C. R. Sox,et al.  The capacitated lot sizing problem with setup carry-over , 1999 .

[70]  Arthur F. Veinott,et al.  Minimum Concave-Cost Solution of Leontief Substitution Models of Multi-Facility Inventory Systems , 1969, Oper. Res..

[71]  Marshall L. Fisher,et al.  The Lagrangian Relaxation Method for Solving Integer Programming Problems , 2004, Manag. Sci..

[72]  B. P. Dzielinski,et al.  Optimal Programming of Lot Sizes, Inventory and Labor Allocations , 1965 .

[73]  Hanan Luss,et al.  Production planning with time-dependent capacity bounds , 1982 .

[74]  J Jaap Wessels,et al.  Multi item lot size determination and scheduling under capacity constraints , 1976 .

[75]  Alf Kimms,et al.  Lot sizing and scheduling -- Survey and extensions , 1997 .

[76]  Fokko ter Haseborg.  On the optimality of joint ordering policies in a multi-product dynamic lot size model with individual and joint set-up costs , 1982 .

[77]  K. Hindi.  Solving the single-item, capacitated dynamic lot-sizing problem with startup and reservation costs by tabu search , 1995 .

[78]  Li-Lian Gao,et al.  A Dual Ascent Procedure for Multiproduct Dynamic Demand Coordinated Replenishment with Backlogging , 1996 .

[79]  E. Arkin,et al.  Computational complexity of uncapacitated multi-echelon production planning problems , 1989 .

[80]  Harish C. Bahl,et al.  Capacitated lot-sizing and scheduling by Lagrangean relaxation , 1992 .

[81]  Luk N. Van Wassenhove,et al.  Multi-Item Single-Level Capacitated Dynamic Lot-Sizing Heuristics: A General Review , 1988 .

[82]  Alok Aggarwal,et al.  Improved Algorithms for Economic Lot Size Problems , 1993, Oper. Res..

[83]  Peter A. Pinto,et al.  Integrative Cycle Scheduling Approach for a Capacitated Flexible Assembly System , 1993 .

[84]  Donald Erlenkotter,et al.  A Dual-Based Procedure for Uncapacitated Facility Location , 1978, Oper. Res..

[85]  K. Hindi.  Computationally efficient solution of the multi-item, capacitated lot-sizing problem , 1995 .

[86]  Harvey H. Millar,et al.  An application of Lagrangean decomposition to the capacitated multi-item lot sizing proble , 1993, Comput. Oper. Res..

[87]  E. Silver,et al.  A heuristic solution procedure for the multi-item, single-level, limited capacity, lot-sizing problem , 1981 .