An efficient trajectory method for probabilistic production–inventory–distribution problems

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.