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We present and study a mixed integer programming model that arises as a sub-structure in many industrial applications. This model provides a relaxation of various capacitated production planning problems, more general fixed charge network flow problems, and other structured mixed integer programs. After distinguishing the general case, which is N P–hard,(More)
The allocation of surgeries to operating rooms (ORs) is a challenging combinatorial optimization problem. There is also significant uncertainty in the duration of surgical procedures, which further complicates assignment decisions. In this article, we present stochastic optimization models for the assignment of surgeries to ORs on a given day of surgery.(More)
Instances of the multi–item capacitated lot–sizing problem with setup times (MCL) often appear in practice, either in standard form or with additional constraints , but they have generally been difficult to solve to optimality. In MCL demand for multiple items must be met over a time horizon, items compete for a shared capacity, and each setup uses up some(More)
We study a special case of a structured mixed integer programming model that arises in production planning. For the most general case of the model, called PI, we have earlier identified three families of facet–defining valid inequalities: (l, S) inequalities (introduced for the uncapacitated lot–sizing problem by Barany, Van Roy, and Wolsey), cover(More)
Multi-level production planning problems in which multiple items compete for the same resources frequently occur in practice, yet remain daunting in their difficulty to solve. In this paper we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot-sizing problems. In addition, unlike many other(More)
We consider the single item capacitated lot–sizing problem, a well-known production planning model that often arises in practical applications, and derive new classes of valid inequalities for it. We first derive new, easily computable valid inequalities for the continuous 0–1 knapsack problem, which has been introduced recently and has been shown to(More)
We study the polyhedral structure of simple mixed integer sets that generalize the two variable set {(s, z) ∈ IR 1 + × Z Z 1 : s ≥ b − z}. These sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs. For four such sets we give a complete description by valid inequalities and/or an integral(More)
This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (, S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (, S) inequalities to a general class of valid inequalities,(More)
We study mixed integer programming formulations of variants of the discrete lot–sizing problem. Our approach is to identify simple mixed integer sets within these models and to apply tight formulations for these sets. This allows us to define integral linear programming formulations for the discrete lot–sizing problem in which backlogging and/or safety(More)