Analysis of a chance-constrained new product risk model with multiple customer classes

• Published in: European journal of operational research Vol. 272; no. 3; pp. 999 - 1016
• Main Authors: Fontem, Belleh, Smith, Jeremiah
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2019
• Subjects: Chance-constrained optimization; Multiplicative uncertainty; Risk management; Stochastic programming; Risk management; Chance-constrained optimization; Multiplicative uncertainty; Stochastic programming
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2018.07.042

•We consider a cost-minimization nonlinear program featuring multiplicative noise.•We derive a closed-form expression for the optimal solution in a special case.•For a related special case, we reduce the problem to a sequence of linear programs.•For other cases, we propose a heuristic and compare it to sample average approximation.•Numerical experiments illustrate how various parameters influence the optimal cost. We consider the non-convex problem of minimizing a linear deterministic cost objective subject to a probabilistic requirement on a nonlinear multivariate stochastic expression attaining, or exceeding a given threshold. The stochastic expression represents the output of a noisy system featuring the product of mutually-independent, uniform random parameters each raised to a linear function of one of the decision vector’s constituent variables. We prove a connection to (i) the probability measure on the superposition of a finite collection of uncorrelated exponential random variables, and (ii) an entropy-like affine function. Then, we determine special cases for which the optimal solution exists in closed-form, or is accessible via sequential linear programming. These special cases inspire the design of a gradient-based heuristic procedure that guarantees a feasible solution for instances failing to meet any of the special case conditions. The application motivating our study is a consumer goods firm seeking to cost-effectively manage a certain aspect of its new product risk. We test our heuristic on a real problem and compare its overall performance to that of an asymptotically optimal Monte-Carlo-based method called sample average approximation. Numerical experimentation on synthetic problem instances sheds light on the interplay between the optimal cost and various parameters including the probabilistic requirement and the required threshold.