The "Best" Algorithm for solving Stochastic Mixed Integer Programs

We present a new algorithm for solving two-stage stochastic mixed-integer programs (SMIPs) having discrete first-stage variables, and continuous or discrete second-stage variables. For a minimizing SMIP, the BEST algorithm (1) computes an upper Bound on the optimal objective value (typically a probabilistic bound), and identifies a deterministic lower-bounding function, (2) uses the bounds to Enumerate a set of first-stage solutions that contains an optimal solution with pre-specified confidence, (3) for each first-stage solution, Simulates second-stage operations by repeatedly sampling random parameters and solving the resulting model instances, and (4) applies statistical Tests (e.g., "screening procedures") to the simulated outcomes to identify a near-optimal first-stage solution with pre-specified confidence. We demonstrate the algorithm's performance on a stochastic facility-location problem

[1]  R. Rockafellar,et al.  Stochastic Convex Programming: Relatively Complete Recourse and Induced Feasibility , 1976 .

[2]  Eitan Israeli,et al.  System Interdiction and Defense. , 1999 .

[3]  Barry L. Nelson,et al.  A fully sequential procedure for indifference-zone selection in simulation , 2001, TOMC.

[4]  Matthew Sta.  A Product-Mix Capacity Planning Model , 1997 .

[5]  David A. Kendrick,et al.  GAMS : a user's guide, Release 2.25 , 1992 .

[6]  A. Ben-Tal,et al.  Bounds on the Expectation of a Convex Function of a Random Variable: With Applications to Stochastic Programming , 1977, Oper. Res..

[7]  Alexander Shapiro,et al.  The Sample Average Approximation Method for Stochastic Discrete Optimization , 2002, SIAM J. Optim..

[8]  Gilbert Laporte,et al.  Exact Solution to a Location Problem with Stochastic Demands , 1994, Transp. Sci..

[9]  C. HuangC.,et al.  Bounds on the Expectation of a Convex Function of a Random Variable , 1977 .

[10]  Julie L. Swann,et al.  Simple Procedures for Selecting the Best Simulated System When the Number of Alternatives is Large , 2001, Oper. Res..

[11]  Linos F. Frantzeskakis,et al.  A Successive Linear Approximation Procedure for Stochastic, Dynamic Vehicle Allocation Problems , 1990, Transp. Sci..

[12]  G. Laporte,et al.  EXACT SOLUTION OF A STOCHASTIC LOCATION PROBLEM BY AN INTEGER L-SHAPED ALGORITHM , 1990 .

[13]  David P. Morton,et al.  Stochastic Vehicle Routing with Random Travel Times , 2003, Transp. Sci..

[14]  W. K. Klein Haneveld,et al.  Stochastic integer programming: state of the art , 1998 .

[15]  A. Madansky.  Bounds on the Expectation of a Convex Function of a Multivariate Random Variable , 1959 .

[16]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[17]  David P. Morton,et al.  Monte Carlo bounding techniques for determining solution quality in stochastic programs , 1999, Oper. Res. Lett..

[18]  David P. Morton,et al.  Assessing solution quality in stochastic programs , 2006, Algorithms for Optimization with Incomplete Information.

[19]  David P. Morton,et al.  Restricted-Recourse Bounds for Stochastic Linear Programming , 1999, Oper. Res..

[20]  David P. Morton,et al.  Stochastic Network Interdiction , 1998, Oper. Res..

[21]  B. Nelson,et al.  Using common random numbers for indifference-zone selection and multiple comparisons in simulation , 1995 .

[22]  Peter Kall,et al.  Approximation Techniques in Stochastic Programming , 1988 .

[23]  A. Tamhane.  Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons , 1995 .

[24]  Sanjay Mehrotra,et al.  On the Value of Binary Expansions for General Mixed-Integer Linear Programs , 2002, Oper. Res..