Stochastic lot-sizing problem with inventory-bounds and constant order-capacities

In this paper, we study the stochastic version of lot-sizing problems with inventory bounds and order capacities. Customer demands, inventory bounds, and costs are subject to uncertainty and dependent with each other throughout the finite planning horizon. Two models in stochastic programming are developed: the first one has inventory-bound constraints, and the second one has both inventory-bound and order-capacity constraints. We explore structural properties of the two models and develop and dynamic programming algorithms for them, respectively. Our model also generalizes the deterministic lot-sizing problem with inventory bounds. For some cases, when applied to the deterministic versions, our algorithms outperform existing deterministic algorithms.

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

[2]  Alper Atamtürk,et al.  An O(n2) algorithm for lot sizing with inventory bounds and fixed costs , 2008, Oper. Res. Lett..

[3]  Huseyin Topaloglu,et al.  Using Stochastic Approximation Methods to Compute Optimal Base-Stock Levels in Inventory Control Problems , 2008, Oper. Res..

[4]  W. Zangwill.  Minimum Concave Cost Flows in Certain Networks , 1968 .

[5]  H. Edwin Romeijn,et al.  Lot-sizing with non-stationary cumulative capacities , 2007, Oper. Res. Lett..

[6]  K. Richter,et al.  The reverse Wagner/Whitin model with variable manufacturing and remanufacturing cost , 2001 .

[7]  Albert P. M. Wagelmans,et al.  An $O(T^3)$ algorithm for the economic lot-sizing problem with constant capacities , 1993 .

[8]  Joaquin Sicilia,et al.  A polynomial algorithm for the production/ordering planning problem with limited storage , 2007, Comput. Oper. Res..

[9]  Yuyue Song,et al.  A dynamic analysis of the single-item periodic stochastic inventory system with order capacity , 2003, Eur. J. Oper. Res..

[10]  Dorit S. Hochbaum,et al.  Technical Note - Solving Linear Cost Dynamic Lot-Sizing Problems in O(n log n) Time , 2008, Oper. Res..

[11]  Joaquin Sicilia,et al.  A new characterization for the dynamic lot size problem with bounded inventory , 2003, Comput. Oper. Res..

[12]  Chung-Yee Lee,et al.  A Dynamic Lot-Sizing Model with Demand Time Windows , 2001 .

[13]  Yongpei Guan,et al.  A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem , 2006, Math. Program..

[14]  Jinxing Xie,et al.  Storage-Space Capacitated Inventory System with (r, Q) Policies , 2007, Oper. Res..

[15]  Kai Huang,et al.  The Value of Multistage Stochastic Programming in Capacity Planning Under Uncertainty , 2009, Oper. Res..

[16]  Harvey M. Wagner.  A POSTSCRIPT TO DYNAMIC PROBLEMS IN THE THEORY OF THE FIRM , 1960 .

[17]  Awi Federgruen,et al.  An Efficient Algorithm for Computing an Optimal (r, Q) Policy in Continuous Review Stochastic Inventory Systems , 1992, Oper. Res..

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

[19]  Knut Richter,et al.  Remanufacturing planning for the reverse Wagner/Whitin models , 2000, Eur. J. Oper. Res..

[20]  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 .

[21]  Eugeniusz Toczyłowski.  An O(T/sup 2/) algorithm for the lot-sizing problem with limited inventory levels , 1995, Proceedings 1995 INRIA/IEEE Symposium on Emerging Technologies and Factory Automation. ETFA'95.

[22]  Shabbir Ahmed,et al.  A Multi-Stage Stochastic Integer Programming Approach for Capacity Expansion under Uncertainty , 2003, J. Glob. Optim..

[23]  Yu-Sheng Zheng,et al.  A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems , 1991, Journal of Applied Probability.

[24]  Marc Lambrecht,et al.  X-Y Band and Modified (s, S) Policy , 1996, Oper. Res..

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

[26]  Yongpei Guan,et al.  Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems , 2008, Oper. Res..

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

[28]  Laurence A. Wolsey,et al.  Lot-sizing with production and delivery time windows , 2005, Math. Program..

[29]  Peter Kall,et al.  Stochastic Programming , 1995 .

[30]  Patrick Suppes,et al.  Mathematical methods in the social sciences, 1959 : proceedings of the first Stanford Symposium , 1962 .

[31]  Diego Klabjan,et al.  A Fully Polynomial Time Approximation Scheme for Single-item Stochastic Inventory Control with Discrete Demand , 2022 .

[32]  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..

[33]  Tieming Liu.  Economic lot sizing problem with inventory bounds , 2008, Eur. J. Oper. Res..

[34]  A. F. Veinott,et al.  Computing Optimal (s, S) Inventory Policies , 1965 .

[35]  Albert P. M. Wagelmans,et al.  Four equivalent lot-sizing models , 2008, Oper. Res. Lett..

[36]  David Simchi-Levi,et al.  A Fully Polynomial Time Approximation Scheme for Single-Item Stochastic Lot-Sizing Problems with Discrete Demand , 2006 .

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

[38]  Awi Federgruen,et al.  Finding Optimal (s, S) Policies Is About As Simple As Evaluating a Single Policy , 1991, Oper. Res..

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

[40]  Yongpei Guan,et al.  On formulations of the stochastic uncapacitated lot-sizing problem , 2006, Oper. Res. Lett..

[41]  Kai Huang,et al.  On stochastic lot-sizing problems with random lead times , 2008, Oper. Res. Lett..

[42]  D. Iglehart.  Optimality of (s, S) Policies in the Infinite Horizon Dynamic Inventory Problem , 1963 .

[43]  Laurence A. Wolsey,et al.  Lot-Sizing on a Tree , 2006, Oper. Res. Lett..

[44]  K. H. Daniel,et al.  Multistage inventory models and techniques , 1965 .