Yves Crama

A class of valid inequalities for multilinear 0-1 optimization problems

Abstract This paper investigates the polytope associated with the classical standard linearization technique for the unconstrained optimization of multilinear polynomials in 0–1 variables. A new class of valid inequalities, called 2-links, is introduced to strengthen the LP relaxation of the standard linearization. The addition of the 2-links to the standard linearization inequalities provides a complete description of the convex hull of integer solutions for the case of functions consisting of at most two nonlinear monomials. For the general case, various computational experiments show that the 2-links improve both the standard linearization bound and the computational performance of exact branch & cut methods. The improvements are especially significant for a class of instances inspired from the image restoration problem in computer vision. The magnitude of this effect is rather surprising in that the 2-links are in relatively small number (quadratic in the number of terms of the objective function).

Large neighborhood search for multi-trip vehicle routing

We consider the multi-trip vehicle routing problem, in which each vehicle can perform several routes during the same working shift to serve a set of customers. The problem arises when customers are close to each other or when their demands are large. A common approach consists of solving this problem by combining vehicle routing heuristics with bin packing routines in order to assign routes to vehicles. We compare this approach with a heuristic that makes use of specific operators designed to tackle the routing and the assignment aspects of the problem simultaneously. Two large neighborhood search heuristics are proposed to perform the comparison. We provide insights into the configuration of the proposed algorithms by analyzing the behavior of several of their components. In particular, we question the impact of the roulette wheel mechanism. We also observe that guiding the search with an objective function designed for the multi-trip case is crucial even when exploring the solution space of the vehicle routing problem. We provide several best known solutions for benchmark instances.

Quadratic reformulations of nonlinear binary optimization problems

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables by introducing additional auxiliary variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all “minimal” quadratizations of negative monomials.

Revealed preference theory: an algorithmic outlook

Revealed preference theory is a domain within economics that studies rationalizability of behavior by (certain types of) utility functions. Given observed behavior in the form of choice data, testing whether certain conditions are satisfied gives rise to a variety of computational problems that can be analyzed using operations research techniques. In this survey, we provide an overview of these problems, their theoretical complexity, and available algorithms for tackling them. We focus on consumer choice settings, in particular individual choice, collective choice and stochastic choice settings.

Berge-acyclic multilinear 0-1 optimization problems

Abstract The problem of optimizing a multilinear polynomial f inxa00–1 variables arises in applications from many different areas. We are interested in resolution methods based on reformulating the polynomial problem into an equivalent linear one, an approach that attempts to draw benefit from the extensive literature in integer linear programming. More precisely, we characterize instances for which the classical standard linearization procedure guarantees integer optimal solutions. We show that the standard linearization polytope PH is integer if and only if the hypergraph H defined by the higher-degree monomials of f is Berge-acyclic, or equivalently, when the matrix defining PH is balanced. This characterization follows from more general conditions that guarantee integral optimal vertices for a relaxed formulation depending on the sign pattern of the monomials of f.

Stochastic Inventory Routing for Perishable Products

Different solution methods are developed to solve an inventory routing problem for a perishable product with stochastic demands. The solution methods are empirically compared in terms of average profit, service level, and actual freshness. The benefits of explicitly considering demand uncertainty are quantified. The computational study highlights that in certain situations although a simple ordering policy can achieve very good performance, statistically and economically significant improvements are achieved when using more advanced solution methods. Managerial insights concerning the impact of shelf life and store capacity on profit are also obtained.

Special issue: Twelfth Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2015)

This special issue of the Journal of Scheduling contains ten papers presented at the Twelfth Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2015), held from June 8 to June 12, 2015, in La Roche-enArdenne, Belgium. MAPSP is a biennial workshop dedicated to all theoretical and practical aspects of scheduling, planning, and timetabling. Previous MAPSP meetings have been held in Menaggio, Italy (1993), Wernigerode, Germany (1995), Cambridge, UK (1997), Renesse, Netherlands (1999), Aussois, France (2001), Aussois, France (2003), Siena, Italy (2005), Istanbul, Turkey (2007), Kerkrade, Netherlands (2009), Nymburk, Czech Republic (2011), and Pont àMousson, France (2013). The twelfth edition of the workshop featured invited talks by Onno Boxma (TU Eindhoven), Michel Goemans (MIT), Rolf Niedermeier (TU Berlin), Willem-Jan Van Hoeve (Carnegie-Mellon University), and Stephan West-