State-Variable Modeling for a Class of Two-Stage Stochastic Optimization Problems

Abstract

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a standard stochastic programming formulation of these problems involves an exponential number of scenarios, therefore an exponential number of variables and constraints. We propose a new mixed-integer programming modeling approach where the number of variables and constraints is independent of the number of scenarios and scales at most pseudopolynomially with the problem size. The proposed modeling approach relies on state variables that track the system’s state as the uncertainty realizes sequentially. We demonstrate the advantages of the proposed approach in two applications arising in project scheduling and operating room allocation. Summary of Contribution: This paper proposes a new modeling approach for a class of two-stage stochastic optimization problems that is computationally more efficient than the traditional scenario-based stochastic integer programming models. The proposed modeling approach relies on state variables that track the system's state as the uncertainty realizes sequentially. We demonstrated the efficiency of the proposed approach by computational results on two applications in project scheduling and operating room allocation.