Given a set [Formula: see text] of size n, a nonnegative, integer-valued distance matrix D of dimensions [Formula: see text], an integer [Formula: see text] and an integer-valued weight vector [Formula: see text], the discrete ordered median problem (DOMP) consists of selecting a subset [Formula: see text] of exactly p points from [Formula: see text] (also referred to as the centers) so as to: 1) assign each point in [Formula: see text] to its closest center in [Formula: see text]; 2) rank the resulting distances (between every point and its center) from smallest to largest in a sorted vector that we denote [Formula: see text]; 3) minimize the scalar product [Formula: see text]. The DOMP generalizes several classical location problems such as the p-center, the p-median and the obnoxious median problem. We introduce an exact branch-and-bound algorithm to solve the DOMP. This branch-and-bound decouples the ranking attribute of the problem to form a series of simpler subproblems which are solved using innovative binary search methods. We consider several acceleration techniques such as warm-starts, primal heuristics, variable fixing, and symmetry breaking. We perform a thorough computational analysis and show that the proposed method is competitive against several MIP models from the scientific literature. We also comment on the limitations of our method and propose avenues of future research. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants 2017-06106, 2020-06311, and 2021-03327]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0059 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0059 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In this article we propose an iterative algorithm to address the optimization problem of distributing a set of multiple highly perishable commodities in a healthcare network. In the biomedical sample transportation problem, numerous commodities with short lifespans presume multiple transportation requests at the same facility in a day and restrict the maximum time to reach their destination. These two characteristics create an interdependency between the routing and the pickup decisions in time that is highly complex. To address these timing issues, we model this problem as a service network design problem over a time-expanded network. Our solution method aggregates the network at two levels. First, the commodities are aggregated and artificially consolidated, reducing the symmetry arising when multiple transportation requests are solicited within a short period of time. Second, the space-time nodes in the network are constructed dynamically, thus reducing the size of the mathematical model to be solved at each iteration. Moreover, the method creates auxiliary networks to calculate good-quality primal bounds to the problem. Our algorithm proves to be efficient to solve a set of real-life instances from the Quebec laboratory network under the management of the Ministère de la Santé et des Services sociaux (Ministry of Health and Social Services) with a detailed network of up to 2,377 periods and 277 transportation requests. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grants 2018-04609, 2020-06311]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0061 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0061 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

This study addresses large-scale personnel scheduling problems in the service industry by combining mathematical programming with data mining techniques to enhance efficiency. The studied problem aims at efficiently scheduling skilled employees over a one-week planning horizon, minimizing costs while meeting diverse job demands. In service industries, shift planning is intricately tied to customer presence, leading to a multitude of potential shifts and a difficult optimization problem that cannot be easily solved using a commercial mixed-integer programming solver. Nevertheless, these problems are categorized as recurrent problems, where distinct instances share common characteristics and solution structures that differ only in a few parameters over time. We propose to use a data mining technique, namely, the \(k\) -nearest neighbors algorithm, to expedite the solution process while upholding solution quality. We suggest using schedules of past solutions to reduce the problem size. Thus, for an upcoming instance, we identify similar historical instances and streamline the enumeration of shifts to align with the comparable historical instances’ schedules. This approach allows us to solve the problem using a commercial solver within a reasonable timeframe while preserving solution quality. Moreover, our methodology offers decision-makers the flexibility to determine the extent to which they wish to scale down the problem. Our experiments conducted on instances generated from real historical data with up to 12 jobs and 252 employees, yield an average removal of up to 85.5% of decision variables. This resulted in an average speedup factor of up to 15.5, with a marginal average cost increase of approximately 1.2%.

Given a ground-set of elements and a family of subsets, the set covering problem consists in choosing a minimum number of elements such that each subset contains at least one of the chosen elements. This research focuses on the set covering polytope , which is the convex hull of integer solutions to the set covering problem. We investigate the connection between the study of the facets of the set covering polytope and tilting theory. This theory studies how inequalities can be rotated around their contact points with a polyhedron in order to obtain inequalities inducing higher dimensional faces . To study this connection, we introduce the concept of tilting vectors which characterize the degrees of freedom of rotation of an inequality. These vectors characterize facet-defining inequalities and can be used to tilt inequalities with a similar procedure to the one used for arbitrary polyhedra. Additionally, we demonstrate that the computational effort needed to tilt an inequality can be reduced when the inequality has many null coefficients. Finally, we use the tilting vectors to extend several necessary and/or sufficient conditions for facets of the set covering polytope presented by several previous works of the literature.

In this short paper, we argue that the standard approach adopted by modern mixed-integer linear programming solvers of using very little cutting plane generation in the branch-and-bound tree can be too conservative and lead to the loss of significant opportunities. Our observation is motivated by some relatively simple computational investigation on a couple of instances in the MIPlib 2010 collection for which the benefit of generating globally valid cuts in the tree is significant. History: This “Challenge” paper was invited by the Editor-in-Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida.

We present a progressive approximation algorithm for the exact solution of several classes of interdiction games in which two noncooperative players (namely an attacker and a follower) interact sequentially. The follower must solve an optimization problem that has been previously perturbed by means of a series of attacking actions led by the attacker. These attacking actions aim at augmenting the cost of the decision variables of the follower’s optimization problem. The objective, from the attacker’s viewpoint, is that of choosing an attacking strategy that reduces as much as possible the quality of the optimal solution attainable by the follower. The progressive approximation mechanism consists of the iterative solution of an interdiction problem in which the attacker actions are restricted to a subset of the whole solution space and a pricing subproblem invoked with the objective of proving the optimality of the attacking strategy. This scheme is especially useful when the optimal solutions to the follower’s subproblem intersect with the decision space of the attacker only in a small number of decision variables. In such cases, the progressive approximation method can solve interdiction games otherwise intractable for classical methods. We illustrate the efficiency of our approach on the shortest path, 0-1 knapsack and facility location interdiction games. Summary of Contribution: In this article, we present a progressive approximation algorithm for the exact solution of several classes of interdiction games in which two noncooperative players (namely an attacker and a follower) interact sequentially. We exploit the discrete nature of this interdiction game to design an effective algorithmic framework that improves the performance of general-purpose solvers. Our algorithm combines elements from mathematical programming and computer science, including a metaheuristic algorithm, a binary search procedure, a cutting-planes algorithm, and supervalid inequalities. Although we illustrate our results on three specific problems (shortest path, 0-1 knapsack, and facility location), our algorithmic framework can be extended to a broader class of interdiction problems.

Column generation (CG) algorithms are well known to suffer from convergence issues due, mainly, to the degenerate structure of their master problem and the instability associated with the dual variables involved in the process. In the literature, several strategies have been proposed to overcome this issue. These techniques rely either on the modification of the standard CG algorithm or on some prior information about the set of dual optimal solutions. In this paper, we propose a new stabilization framework, which relies on the dynamic generation of aggregated rows from the CG master problem. To evaluate the performance of our method and its flexibility, we consider instances of three different problems, namely, vehicle routing with time windows (VRPTW), bin packing with conflicts (BPPC), and multiperson pose estimation (MPPEP). When solving the VRPTW, the proposed stabilized CG method yields significant improvements in terms of CPU time and number of iterations with respect to a standard CG algorithm. Huge reductions in CPU time are also achieved when solving the BPPC and the MPPEP. For the latter, our method has shown to be competitive when compared with a tailored method. Summary of Contribution: Column generation (CG) algorithms are among the most important and studied solution methods in operations research. CG algorithms are suitable to cope with large-scale problems arising from several real-life applications. The present paper proposes a generic stabilization framework to address two of the main issues found in a CG method: degeneracy in the master problem and massive instability of the dual variables. The newly devised method, called dynamic separation of aggregated rows (dyn-SAR), relies on an extended master problem that contains redundant constraints obtained by aggregating constraints from the original master problem formulation. This new formulation is solved in a column/row generation fashion. The efficacy of the proposed method is tested through an extensive experimental campaign, where we solve three different problems that differ considerably in terms of their constraints and objective function. Despite being a generic framework, dyn-SAR requires the embedded CG algorithm to be tailored to the application at hand.

The vehicle routing problem with time windows (VRPTW) consists of finding least-cost vehicle routes to satisfy the demands of customers that can be visited within specific time windows. We introduce two enhancements for the exact solution of the VRPTW by branch-price-and-cut (BPC). First, we develop a sharper form of the limited-memory subset-row inequalities by representing the memory as an arc subset rather than a node subset. Second, from the elementary inequalities introduced by Balas in 1977, we derive a family of inequalities that dominate them. These enhancements are embedded into an exact BPC algorithm that includes state-of-the-art features such as bidirectional labeling, decremental state-space relaxation, completion bounds, variable fixing, and route enumeration. Computational results show that these enhancements are particularly effective for the most difficult instances and that our BPC algorithm can solve all 56 Solomon instances with 100 customers and 51 of 60 Gehring and Homberger instances with 200 customers.

In this article we introduce the quadratic capacitated vehicle routing problem (QCVRP) motivated by two applications in engineering and logistics: the capacitated vehicle routing problem with angle penalties (angle-CVRP) and the capacitated vehicle routing problem with reload costs (CVRP-RC). We introduce a three-index vehicle-flow formulation of the problem, which is strengthened with valid inequalities, and we derive a branch-and-cut algorithm capable of providing tight lower bounds and solving small- to medium-size instances in short to moderate computing times. Furthermore, we present a hybrid metaheuristic capable of providing high quality solutions in short computing times. The two algorithms are tested on several instances from the CVRP literature modified to mimic the two problems that motivate our study.

In this article, we present a comparative study of several strategies that can be applied to achieve the so‐called elementary lower bound in vehicle routing problems, that is, the bound obtained when all positive‐valued variables in an optimal solution of the linear relaxation of the set‐partitioning formulation correspond to vehicle routes without cycles. This bound can be achieved by solving the resource‐constrained elementary shortest path problem—an ‐hard problem—as the pricing problem in a column generation algorithm, but several other strategies can be used to ultimately produce the same lower bound in less computational effort. State‐of‐the‐art algorithms for vehicle routing problems rely on the quality of this lower bound to either bound the size of the search tree in a branch‐and‐price algorithm or the complexity of an enumeration procedure used to limit the number of variables in the set‐partitioning model. We consider several strategies for imposing elementarity that involve ng ‐paths, strong degree constraints, and decremental state‐space relaxation. We compare the performance of these strategies on some selected instances of the vehicle routing problem with time windows. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(1), 88–99. 2015

In this paper we present an exact algorithm for the capacitated location-routing problem (CLRP) based on cut-and-column generation. The CLRP is formulated as a set-partitioning problem that also inherits all of the known valid inequalities for the flow formulations of the CLRP. We introduce five new families of inequalities that are shown to dominate some of the cuts from the two-index formulation. The problem is solved by column generation, where the subproblem consists in finding a shortest path of minimum reduced cost under capacity constraints. We first use the two-index formulation for enumerating all of the possible subsets of depot locations that could lead to an optimal solution of cost less than or equal to a given upper bound. For each of these subsets, the corresponding multiple depot vehicle routing problem is then solved by means of column generation. The results show that we can improve the bounds found in the literature, solve to optimality some previously open instances, and improve the upper bounds on some other instances.