Design and planning under uncertainty: issues on problem formulation and solution

Abstract

Firms operate today in a rapidly changing and risky environment, where such factors as market and technology are inevitably shrouded in uncertainties. They must make design and operating decisions to satisfy several conflicting goals such as maximizing expected profit, minimizing risk, and sustaining long-term viability and competitiveness. Proper formulation is both essential and critical for finding appropriate solutions to such problem. We show how one can formulate this problem as a Markov decision process with recourse that considers decision making throughout the process life cycle and at different hierarchical levels. This formulation incorporates multiple kinds of uncertainties such as market conditions and technology evolution. It allows decision-makers to provide multiple criteria—such as expected profit, expected downside risk, and process lifetime—that reflect various conflicting or incommensurable goals. The formulation integrates design decisions and future planning by constructing a multi-period decision process in which one makes decisions sequentially at each period. The decision process explicitly incorporates both the upper-level investment decisions and lower-level production decisions as a two-stage optimization problem. This problem formulation leads to a multi-objective Markov decision problem, searching for Pareto optimal design strategies that prescribe design decisions for each state the environment and process could occupy. We can often recast this class of problem in order to exploit a rigorous multi-objective stochastic dynamic programming algorithm. This approach decomposes the problem into a sequence of single-period subproblems, each of which is a two-stage stochastic program with recourse. We show how one can solve these subproblems to obtain and propagate the Pareto optimal solutions set recursively backward in time. A small illustrative example appears throughout the paper to demonstrate the formulation and solution issues. The scalability of the rigorous algorithm is limited due to the “curse of dimensionality”, suggesting the need for approximating approaches to solve realistic problems.