Browsing by Author "Wolsey, L. A."
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Item Open Access Continuous knapsack sets with divisible capacities(Springer, 2016) Wolsey, L. A.; Yaman, H.We study two continuous knapsack sets (Formula presented.) and (Formula presented.) with (Formula presented.) integer, one unbounded continuous and (Formula presented.) bounded continuous variables in either (Formula presented.) or (Formula presented.) form. When the coefficients of the integer variables are integer and divisible, we show in both cases that the convex hull is the intersection of the bound constraints and (Formula presented.) polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable. The latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the (Formula presented.) case. We also provide an extended formulation for the (Formula presented.) case. It follows that, given a specific objective function, optimization over both (Formula presented.) and (Formula presented.) can be carried out by solving (Formula presented.) polynomial size linear programs. A further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 (after scaling) in any non-trivial facet-defining inequality of the convex hull of such sets.Item Open Access Convex hull results for the warehouse problem(Elsevier B.V., 2018) Wolsey, L. A.; Yaman, HandeGiven an initial stock and a capacitated warehouse, the warehouse problem aims to decide when to sell and purchase to maximize profit. This problem is common in revenue management and energy storage. We extend this problem by incorporating fixed costs and provide convex hull descriptions as well as tight compact extended formulations for several variants. For this purpose, we first derive unit flow formulations based on characterizations of extreme points and then project out the additional variables using Fourier-Motzkin elimination. It turns out that the nontrivial inequalities are flow cover inequalities for some single node flow set relaxations.Item Open Access Relaxations for two-level multi-item lot-sizing problems(Springer, 2014-08) Vyve, M. V.; Wolsey, L. A.; Yaman, H.We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.