A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems

In this paper, we present a branch-and-price method to solve special structured multistage stochastic integer programming problems. We validate our method on two different versions of a multistage stochastic batch-sizing problem (SBSP). One version adopts a recourse formulation, and the other is based on probabilistic constraints. Our algorithmic approach is applicable to both formulations. Our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch-and-price tree. The success of our approach calls for extensions in methodology as well as applications.

[1]  Marc Salomon,et al.  Batching decisions: structure and models , 1994 .

[2]  Julia L. Higle,et al.  The C3 Theorem and a D2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification , 2005, Math. Program..

[3]  George L. Nemhauser,et al.  Airline Crew Scheduling: A New Formulation and Decomposition Algorithm , 1997, Oper. Res..

[4]  Rüdiger Schultz,et al.  Dual decomposition in stochastic integer programming , 1999, Oper. Res. Lett..

[5]  Martin W. P. Savelsbergh,et al.  Branch-and-Price: Column Generation for Solving Huge Integer Programs , 1998, Oper. Res..

[6]  Laurence A. Wolsey,et al.  An exact algorithm for IP column generation , 1994, Oper. Res. Lett..

[7]  François Vanderbeck,et al.  On Dantzig-Wolfe Decomposition in Integer Programming and ways to Perform Branching in a Branch-and-Price Algorithm , 2000, Oper. Res..

[8]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[9]  David L. Woodruff,et al.  Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming , 1996, J. Heuristics.

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

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

[12]  Hanif D. Sherali,et al.  A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer recourse , 2002, J. Glob. Optim..

[13]  David L. Woodruff,et al.  Progressive hedging as a meta-heuristic applied to stochastic lot-sizing , 2001, Eur. J. Oper. Res..

[14]  Martin W. P. Savelsbergh,et al.  A Branch-and-Price Algorithm for the Generalized Assignment Problem , 1997, Oper. Res..

[15]  Gilbert Laporte,et al.  The integer L-shaped method for stochastic integer programs with complete recourse , 1993, Oper. Res. Lett..

[16]  Martin W. P. Savelsbergh,et al.  Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition , 2000, INFORMS J. Comput..

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

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

[19]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[20]  Nikolaos V. Sahinidis,et al.  A finite branch-and-bound algorithm for two-stage stochastic integer programs , 2004, Math. Program..

[21]  Leen Stougie,et al.  An algorithm for the construction of convex hulls in simple integer recourse programming , 1996, Ann. Oper. Res..