The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.

[1]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[2]  R. Wets,et al.  L-SHAPED LINEAR PROGRAMS WITH APPLICATIONS TO OPTIMAL CONTROL AND STOCHASTIC PROGRAMMING. , 1969 .

[3]  G. Nemhauser,et al.  Integer Programming , 2020 .

[4]  David P. Rutenberg,et al.  Computation in Discrete Stochastic Programs with Recourse , 1973, Oper. Res..

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  M. Garey Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[7]  R. Wets Solving stochastic programs with simple recourse , 1983 .

[8]  Alexander H. G. Rinnooy Kan,et al.  Hierarchical vehicle routing problems , 1984, Networks.

[9]  Stein W. Wallace,et al.  Solving stochastic programs with network recourse , 1986, Networks.

[10]  Stein W. Wallace Investing in arcs in a network to maximize the expected max flow , 1987, Networks.

[11]  Gilbert Laporte,et al.  Models and exact solutions for a class of stochastic location-routing problems , 1987 .

[12]  David K. Smith Theory of Linear and Integer Programming , 1987 .

[13]  Giovanni Andreatta,et al.  Stochastic shortest paths with recourse , 1988, Networks.

[14]  Yuri Ermoliev,et al.  Stochastic programming, an introduction. Numerical techniques for stochastic optimization , 1988 .

[15]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[16]  Yuri Ermoliev,et al.  Numerical techniques for stochastic optimization , 1988 .

[17]  Stein W. Wallace A Two-Stage Stochastic Facility-Location Problem with Time-Dependent Supply , 1988 .

[18]  R. J-B. Wets,et al.  Large Scale Linear Programming Techniques , 1988 .

[19]  R. Rubinstein,et al.  Optimization of static simulation models by the score function method , 1990 .

[20]  Giovanni Rinaldi,et al.  Facet identification for the symmetric traveling salesman polytope , 1990, Math. Program..

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

[22]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[23]  Julia L. Higle,et al.  Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse , 1991, Math. Oper. Res..

[24]  James B. Orlin,et al.  A faster algorithm for finding the minimum cut in a graph , 1992, SODA '92.

[25]  C. Geyer,et al.  Constrained Monte Carlo Maximum Likelihood for Dependent Data , 1992 .

[26]  Gilbert Laporte,et al.  The Vehicle Routing Problem with Stochastic Travel Times , 1992, Transp. Sci..

[27]  G. Infanger,et al.  Planning under uncertainty solving large-scale stochastic linear programs , 1992 .

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

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

[30]  Gilbert Laporte,et al.  A Priori Optimization of the Probabilistic Traveling Salesman Problem , 1994, Oper. Res..

[31]  A. Shapiro Simulation-based optimization—convergence analysis and statistical inference , 1996 .

[32]  Stephen M. Robinson,et al.  Sample-path optimization of convex stochastic performance functions , 1996, Math. Program..

[33]  Julia L. Higle,et al.  Stopping Rules for Stochastic Decomposition , 1996 .

[34]  Andrzej Ruszczynski,et al.  On Optimal Allocation of Indivisibles Under Uncertainty , 1998, Oper. Res..

[35]  A. Shapiro,et al.  On rate of convergence of Monte Carlo approximations of stochastic programs , 1998 .

[36]  Alexander Shapiro,et al.  A simulation-based approach to two-stage stochastic programming with recourse , 1998, Math. Program..

[37]  John M. Wilson,et al.  Introduction to Stochastic Programming , 1998, J. Oper. Res. Soc..

[38]  R. Bixby,et al.  On the Solution of Traveling Salesman Problems , 1998 .

[39]  Georg Ch. Pflug,et al.  A branch and bound method for stochastic global optimization , 1998, Math. Program..

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

[41]  A. Verweij Selected applications of integer programming : a computational study , 2000 .

[42]  Alexander Shapiro,et al.  On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs , 2000, SIAM J. Optim..

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

[44]  David S. Johnson,et al.  8. The traveling salesman problem: a case study , 2003 .