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An Efficient Trajectory Method for Probabilistic Production-Inventory-Distribution Problems

Miguel A. Lejeune, Andrzej Ruszczyski

Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213
Department of Management Science and Information Systems, Rutgers University, 94 Rockefeller Road, Piscataway, New Jersey 08854
mlejeune{at}andrew.cmu.edu
rusz{at}business.rutgers.edu

We consider a supply chain operating in an uncertain environment: The customers’ demand is characterized by a discrete probability distribution. A probabilistic programming approach is adopted for constructing an inventory-production-distribution plan over a multiperiod planning horizon. The plan does not allow the backlogging of the unsatisfied demand, and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level. It is a strategic plan that hedges against undesirable outcomes, and that can be adjusted to account for possible favorable realizations of uncertain quantities. A modular, integrated, and computationally tractable method is proposed for the solution of the associated stochastic mixed-integer optimization problems containing joint probabilistic constraints with dependent right-hand side variables. The concept of p-efficiency is used to construct a finite number of demand trajectories, which in turn are employed to solve problems with joint probabilistic constraints. We complement this idea by designing a preordered set-based preprocessing algorithm that selects a subset of promising p-efficient demand trajectories. Finally, to solve the resulting disjunctive mixed-integer programming problem, we implement a special column-generation algorithm that limits the risk of congestion in the resources of the supply chain. The methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient: it finds a solution with a minimal integrality gap and provides substantial cost savings.

Subject classifications: inventory/production; multistage supply chain; programming; integer; stochastic; transportation; scheduling.
History: Received February 2005; revision received December 2005; accepted March 2006.