Discrete optimization via simulation with stochastic constraints

Abstract

In this thesis, we first develop a new method called penalty function with memory (PFM). PFM consists of a penalty parameter and a measure of constraint violation and it converts a discrete optimization via simulation (DOvS) problem with stochastic constraints into a series of DOvS problems without stochastic constraints. PFM determines a penalty of a visited solution based on past results of feasibility checks on the solution. Specifically, assuming a minimization problem, a penalty parameter of PFM, namely the penalty sequence, diverges to infinity for an infeasible solution but converges to zero almost surely for any strictly feasible solution under certain conditions. For a feasible solution located on the boundary of feasible and infeasible regions, the sequence converges to zero either with high probability or almost surely. As a result, a DOvS algorithm combined with PFM performs well even when optimal solutions are tight or nearly tight. Second, we design an optimal water quality monitoring network for river systems. The problem is to find the optimal location of a finite number of monitoring devices, minimizing the expected detection time of a contaminant spill event while guaranteeing good detection reliability. When uncertainties in spill and rain events are considered, both the expected detection time and detection reliability need to be estimated by stochastic simulation. This problem is formulated as a stochastic DOvS problem with the objective of minimizing expected detection time and with a stochastic constraint on the detection reliability; and it is solved by a DOvS algorithm combined with PFM. Finally, we improve PFM by combining it with an approximate budget allocation procedure. We revise an existing optimal budget allocation procedure so that it can handle active constraints and satisfy necessary conditions for the convergence of PFM.