Mixed integer lifted cover inequalities for knapsack problems with a single continuous variable

The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility. In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope. We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable. To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.

Encoding transformation-based differential evolution algorithm for solving knapsack problem with single continuous variable

We consider the weight-constrained minimum spanning tree problem which has important applications in telecommunication networks design.We discuss and compare several formulations. In order to strengthen these formulations, new classes of valid inequalities are introduced. They adapt the well-known cover, extended cover and lifted cover inequalities. They incorporate information from the two subsets: the set of spanning trees and the knapsack set. We report computational experiments where the best performance of a standard optimization package was obtained when using a formulation based on the well-known Miller-Tucker-Zemlin variables combined with separation of cut-set inequalities.

Background: By observing the high prevalence of failures in the surgical treatment of myocardial revascularization (MR), with the use of the Left Internal Thoracic Artery (LITA) as a graft, evidenced by the international literature, it was sought to demonstrate the prevalence of lesions that would not allow the use of LITA as a graft in myocardial revascularization surgery, with possible alteration in the surgical management performed by the cardiac surgeon, and reduction of the morbimortality of these patients. Objectives: To evaluate the prevalence of atherosclerotic lesions of the LITA, through selective preoperative angiography, in patients submitted to coronary angiography and indicated for myocardial revascularization. We also analyzed other lesions that made the use of LITA unfeasible as a main graft in cases of myocardial revascularization surgery (MRS). Methods: This was a cross-sectional, prevalence study that evaluated, through selective angiography, the LITA of 39 patients with a median age of 63 years, submitted to coronary angiography, with indication of Coronary Artery Bypass Graft (CABG). Categorical variables were compared by chi-square test and Fisher's exact test. The single continuous variable, age, was tested for normality by the Kolmogorov-Smirnov test, described in median (P25; P75) and the groups compared with the Mann-Whitney test. The level of statistical significance adopted was p < 0.05. The analyzes were performed in SPSS® software version 20. Results: It was identified the presence of 7.7% of disorders in the LITA that made it unfeasible to be used. In all of the patients there was no specific symptomatology evidencing the lesion. No variable was shown as a predictor for the occurrence of the outcomes. Conclusion: The prevalence of the lesions found in the study was significant, indicating that a preoperative evaluation of LITA could bring future benefits to the patients submitted to CABG.

A single‐product maritime inventory routing problem (MIRP) is studied in which the production and consumption rates vary over the planning horizon. The problem involves a heterogeneous fleet and multiple production and consumption ports with limited storage capacity. Two discrete time formulations are developed: an original model and a reformulated model that is a pure fixed charge network flow (FCNF) model with side constraints. Mixed integer sets arising from the decomposition of the formulations are identified. In particular, several lot‐sizing relaxations are derived for the formulations and used to establish valid inequalities to strengthen the proposed formulations. Until now, the derivation of models and valid inequalities for MIRPs has mainly been inspired by the developments in the routing community. Here, we have developed a new model leading to new valid inequalities for MIRPs obtained by generalizing valid inequalities from the recent lot‐sizing literature. Considering a set of instances based on real data, a computational study is conducted to test the formulations and the effectiveness of the valid inequalities. The FCNF formulation is generally much stronger than the original formulation. The developed valid inequalities reduce the integrality gap significantly for both formulations. By using a branch‐and‐bound scheme based on the strengthened FCNF formulation, most of our test instances are solved to optimality. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(4), 297–314 2013

We study the polyhedral structure of simple mixed integer sets that generalize the two variable set {(s,z)R 1 + ×ℤ1:s≥z−b}. 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 extended formulation, and we also indicate what constraints can be added without destroying integrality. We apply these results to provide tight formulations for certain piecewise–linear convex objective integer programs, and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot-sizing problems.

A cutting plane algorithm for the capacitated arc routing problem

A latent variable representation of count data models to accommodate spatial and temporal dependence: Application to predicting crash frequency at intersections

From a practical perspective, mixed integer optimization represents a very powerful modeling paradigm. Its modeling power, however, comes with a price. The presence of both integer and continuous variables results in a significant increase in complexity over the pure integer case with respect to geometric, algebraic, combinatorial and algorithmic properties. Specifically, the theory of cutting planes for mixed integer linear optimization is not yet at a similar level of development as in the pure integer case. The goal of this paper is to discuss four research directions that are expected to contribute to the development of this field of optimization. In particular, we examine a new geometric approach based on lattice point free polyhedra and use it for developing a cutting plane theory for mixed integer sets. We expect that these novel developments will shed some light on the additional complexity that goes along with mixing discrete and continuous variables.

Collaborative routing aided by drones in last-mile delivery (LMD) has been an extensively studied topic in recent years. In this article, a new optimization problem for collaborative routing with a truck and a fleet of drones in LMD is introduced. The problem proposes a new strategy to coordinate a truck and a fleet of drones to serve a given set of customers. The plan is based on identifying the locations where the truck can park and where the drones fly to serve the customers. A procedure to locate these parking lots conveniently is given. For the transportation network, which includes these points, a mixed linear integer programming formulation is provided, which aims to minimize the makespan of serving all the customers. Computational experiments on a set of problem instances were conducted to analyze the model solutions' characteristics and find their computational limits. The experiments showed that the proposed model could significantly improve the results obtained when only the truck delivers. Additionally, we propose a GRASP metaheuristic to solve instances of greater size. Its computational performance was studied when applied to a set of instances with different characteristics. The paper discusses some insights obtained from the computational experimentation and presents future research directions.

The capacitated p-median problem (CPMP) involves placing p identical facilities in a network and assigning customer nodes to these facilities to satisfy all customer demands with minimal transportation costs. In practical applications, demand and distance parameters are often uncertain during the planning process, leading to infeasible or excessively costly solutions if these uncertainties are disregarded. This paper addresses the robust CPMP (RCPMP), which incorporates demand uncertainty into the problem using the robust optimization paradigm. We propose a general framework to model and solve the RCPMP, considering different polyhedral uncertainty sets, namely the cardinality-constrained and the knapsack sets. We develop exact approaches encompassing compact models, different families of valid inequalities, and branch-and-cut and branch-and-price algorithms, exploring both implemented uncertainty sets and problem structure. Furthermore, we implement an efficient Variable Neighborhood Search (VNS) heuristic to solve these robust variants, which incorporates state-of-the-art algorithms, parallelization techniques, and optimized data structures. Computational experiments using adapted benchmark instances with up to 400 nodes indicate the effectiveness of the proposed approaches. The results show that using parallelization and hash tables within the VNS heuristic promotes significant performance improvements and yields near-optimal solutions for most instances, as well as outperforming the exact approaches in several instances where the optimal solution was not found. Moreover, these results highlight the benefits of using robust solutions in practical settings, especially when considering different uncertainty sets to generate solutions with advantageous trade-offs between cost and risk.

In this paper, we study the efficiency (both theoretically and computationally) of a class of valid inequalities for the minimum weighted elementary directed cycle problem (MWEDCP) in planar digraphs with negative weight elementary directed cycles. These valid inequalities are called cycle valid inequalities and are parametrized by an integer called inequality’s order. From a theoretical point of view, we prove that separating cycle valid inequalities of order 1 in planar digraph can be done in polynomial time. From a computational point of view, we present a cutting plane algorithm featuring the efficiency of a lifted form of the cycle valid inequalities of order 1. In addition to these lifted valid inequalities, our algorithm is also based on a mixed integer linear formulation of the MWEDCP. The computational results are carried out on randomly generated planar digraph instances of the MWEDCP. For all 29 instances considered, we obtain in average 26.47% gap improvement. Moreover, for some of our instances the strengthening process directly displays the optimal integer elementary directed cycle.

Ignoring information in binary choice with continuous variables: When is less “more”?

Improved solutions for inventory-routing problems through valid inequalities and input ordering

We explore one method for finding the convex hull of certain mixed integer sets. The approach is to break up the original set into a small number of subsets, find a compact polyhedral description of the convex hull of each subset, and then take the convex hull of the union of these polyhedra. The resulting extended formulation is then compact, its projection is the convex hull of the original set, and optimization over the mixed integer set is reduced to solving a linear program over the extended formulation. The approach is demonstrated on three different sets: a continuous mixing set with an upper bound and a mixing set with two divisible capacities both arising in lot-sizing, and a single node flow model with divisible capacities that arises as a subproblem in network design.