A finite branch-and-bound algorithm for two-stage stochastic integer programs

Abstract

This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.

Dynamic Capacity Acquisition and Assignment under Uncertainty

“…A proof of the above result in the context of general two-stage stochastic programs with pure integer recourse can be found in [1]. Note that the set C(k) in Theorem 4.1 is a hyper-rectangle since it is the Cartesian product of intervals.…”

“…The parameter d jt is the processing requirement of task j in period t, and accordingly, the second constraint in problem (1) reflects that the processing requirement of all tasks assigned to a resource in any period cannot exceed the installed capacity in that period. The third constraint enforces that each task needs to be assigned to exactly once resource in each of the periods.…”

“…Thus, explicit enumeration of all lattice points is, in general, computationally prohibitive. Recently, Ahmed et al [1] developed an efficient decomposition based branch and bound algorithm for general two-stage stochastic integer programs with mixed-integer first-stage variables and pure integer second-stage variables. Unlike existing algorithms, this method avoids complete enumeration and is guaranteed to terminate finitely.…”

“…In the next section, we describe a decomposition based branch and bound algorithm for the stochastic dynamic capacity acquisition and assignment problem based upon the scheme of [1].…”

“…In general, it is not obvious how one can partition the search space by subdividing along the discontinuities within a branch and bound framework. Ahmed et al [1] proposed a transformation of the general two-stage stochastic integer program with mixed-integer first-stage variables, and pure integer second-stage variables that causes the discontinuities to be orthogonal to the variable axes. Thus, a rectangular partitioning strategy can potentially isolate the discontinuous pieces of the value function, thereby allowing upper and lower bounds to collapse finitely.…”