Families Of Valid Inequalities Research Articles

Exact methods and a variable neighborhood search for the robust capacitated [formula omitted]-median problem

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.

The multidepot drone general routing problem with duration and capacity constraints

AbstractThis paper studies the multidepot drone general routing problem with duration and capacity constraints (MDdGRP), an extension of the classical general routing problem with several depots. A fleet of drones with limited flight time and payload, each one located in a different depot, must jointly perform the service. In this continuous optimization problem, a set of lines of a network has to be serviced and a set of nodes has to be visited to deliver a given commodity. The aim is to find a set of drone routes performing the complete service with the shortest total duration. Aerial drones can enter a line at any point, service a portion of that line, and exit through another point, without following the lines of the network. While this flexibility allows for potentially better solutions, it also adds complexity to the problem. To deal with this, we digitize the MDdGRP instances by approximating each line with a polygonal chain containing a finite number of points that can be used to enter and exit each line. Thus, an instance of a discrete multidepot general routing problem (MDGRP) is obtained. This paper proposes an integer formulation for the MDGRP, some families of valid inequalities, and a branch‐and‐cut algorithm based on the strengthened formulation. Two matheuristic algorithms are presented, focused on sequentially selecting (and adding) some promising intermediate points to enter and exit a line. Instances involving up to six depots, 22 delivery points, and 88 original lines have been generated to test the performance of the methods proposed.

Theoretical and computational analysis of a new formulation for the Rural Postman Problem and the General Routing Problem

The Rural Postman Problem (RPP) is one of the most well-known problems in arc routing. Given an undirected graph, the RPP consists of finding a closed walk traversing and servicing a given subset of edges with minimum total cost. In the General Routing Problem (GRP), there is also a subset of vertices that must be visited. Both problems were introduced by Orloff and proved to be NP-hard. In this paper, we propose a new formulation for the RPP and the GRP using two sets of binary variables representing the first and second traversal, respectively, of each edge. We present several families of valid inequalities that induce facets of the polyhedron of solutions under mild conditions. Using this formulation and these families of inequalities, we propose a branch-and-cut algorithm, test it on a large set of benchmark instances, and compare its performance against the exact procedure that, as far as we know, produced the best results. The results obtained show that the proposed formulation is useful for solving undirected RPP and GRP instances of very large size.

Maintenance scheduling at high-speed train depots: An optimization approach

Under the preventive maintenance policy, high-speed trains need to conduct periodical inspection/overhaul at dedicated depots, where maintenance facilities and maintenance crews are available. Due to the complicated structure of trains, the maintenance work is rich in a variety of types, characterized by mechanical components repaired, maintenance levels and etc. In practice, these types of maintenance work are integrated into maintenance packets, and different packets may require the specific types of crews. In this paper, a mixed integer linear programming model is proposed for the depot maintenance packet assignment and crew scheduling problem. The objective of the problem is to minimize the overall crew worktime, while the main constraints include the compatibility between maintenance packets and crew types, maintenance packet duration time and execution order, maintenance time window, and crew worktime limit. Besides, two families of valid inequalities are proposed to improve the baseline model. Computational experiments on a set of randomly generated instances show the effectiveness and efficiency of the improved model compared to the baseline one. Finally, a real-world case study from Shanghai South Depot is carried out to further validate the proposed approach. Improvements on both solution time and quality are achieved in contrast with the manual schedule.

Drop-and-pull container drayage with flexible assignment of work break for vehicle drivers

Consideration of driver-related constraints, such as mandatory work break, in vehicle scheduling and routing is of extraordinary importance for safety driving and guaranteeing the interests of vehicle drivers. Drop-and-pull transportation is an efficient mode in container drayage services and creates interdependencies between the routes of vehicle drivers. The situation becomes even more complicated when the drivers must comply with requirements of the mandatory work break. This paper addresses a drop-and-pull container drayage problem with flexible assignment of work break. The time and location of work break for drivers can be flexibly chosen based on scheduling, routing and optimization requirements. The transportation vehicle is comprised of two parts as a tractor and a trailer. The tractor can drag at most one trailer at a time, and is separable from the trailer when executing drayage activities. The problem is formulated as a mixed-integer programming model, which is then strengthened by proposing several families of valid inequalities. To efficiently solve realistic-sized instances, a backtracking adaptive threshold accepting algorithm with a mixed-integer programming model specially coping with the assignment of work break is proposed. Mechanisms of tabu search are incorporated to enhance the searching ability of the algorithm. Experiments indicate that the proposed algorithm can efficiently handle the vehicle scheduling and routing as well as the work break assignment and outperforms the mathematical programming approach. Sensitivity analyses about different policies and varied length of the work break are also conducted with some managerial insights presented.

A location-routing problem for local supply chains

This study addresses a local Supply Chain problem by proposing the Collection and Delivery Location Open Routing Problem (CDLORP), a variant of the classical location-routing problem that considers both collection and distribution routes to decide where to locate facilities. Collection routes are responsible for picking up the required raw material from local suppliers, while distribution routes deliver the final product to local customers. This study is motivated by a public initiative that desires to seize the by-product generated by the agro-industrial local sector to produce and provide healthy and nutritional snacks to students in the public elementary school system in Saltillo, Coahuila, Mexico. A node-based formulation and a flow-based formulation are proposed to solve instances of the problem. The formulations are valid for the symmetric and asymmetric cases of the problem. Four families of valid inequalities from the literature are adapted for the problem and are used to implement two branch-and-cut algorithms based on the proposed formulations. Computational experiments using 42 benchmark instances of different sizes (25 to 260 vertices) are performed to assess the efficacy of the proposed formulations and algorithms. The best results are obtained by the branch-and-cut algorithm using the node-based formulation. Finally, a case study is also addressed and optimally solved with the branch-and-cut algorithm using the node-based formulation. The results obtained for the benchmark instances and the case study show that the proposed approach can solve real-life problems to develop local supply chains for new products.

Strong cutting planes for the capacitated multi-pickup and delivery problem with time windows

In this paper, we propose an improved 2-index mixed integer linear programming formulation for capacitated multi-pickup and delivery problems with time windows. We develop new families of valid inequalities, present some simple separation heuristics and solve the benchmark problems using a cutting plane approach. Specifically, we propose and demonstrate the effectiveness of incompatible request sets based valid inequalities, exploiting thereby the time window and capacity constraints. Further, we introduce two sub-families of these inequalities: node-based and request-based incompatible request set cuts, along with their separation heuristics. The computational results show that the incompatible request set cuts are highly useful as cutting planes for the benchmark instances. Overall, there is a marked improvement in solution performance vis-à-vis the different formulations and solution approaches used in extant literature. The approach in this paper is able to solve 12%, 37% and 43% more instances to optimality, improve the root gap by 18%, 33% and 36%, and final gap by 36%, 45% and 43% vis-à-vis the asymmetric representative formulation, three-index and the two-index formulations respectively.

Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts

The irregular strip-packing problem, also known as nesting or marker making, is defined as the automatic computation of a non-overlapping placement of a set of non-convex polygons onto a rectangular strip of fixed width and unbounded length, such that the strip length is minimized. Nesting methods based on heuristics are a mature technology, and currently, the only practical solution to this problem. However, recent performance gains of the Mixed-Integer Programming (MIP) solvers, together with the known limitations of the heuristics methods, have encouraged the exploration of exact optimization models for nesting during the last decade. Despite the research effort, there is room to improve the efficiency of the current family of exact MIP models for nesting. In order to bridge this gap, this work introduces a new family of continuous MIP models based on a novel formulation of the NoFit-Polygon Covering Model (NFP-CM), called NFP-CM based on Vertical Slices (NFP-CM-VS). Our new family of MIP models is based on a new convex decomposition of the feasible space of relative placements between pieces into vertical slices, together with a new family of valid inequalities, symmetry breakings, and variable eliminations derived from the former convex decomposition. Our experiments show that our new NFP-CM-VS models outperform the current state-of-the-art MIP models. Ten instances are solved up to optimality within one hour for the first time, including one with 27 pieces. Finally, we provide a detailed reproducibility protocol and dataset as supplementary material to allow the exact replication of our models, experiments, and results.

A new MILP formulation for the flying sidekick traveling salesman problem

AbstractNowadays, truck‐and‐drone problems represent one of the most studied classes of vehicle routing problems. The Flying Sidekick Traveling Salesman Problem (FS‐TSP) is the first optimization problem defined in this class. Since its definition, several variants have been proposed differing for the side constraints related to the operating conditions and for the structure of the hybrid truck‐and‐drone delivery system. However, regardless the specific problem under investigation, determining the optimal solution of most of these routing problems is a very challenging task, due to the vehicle synchronization issue. On this basis, this work provides a new arc‐based integer linear programming formulation for the FS‐TSP. The solution of such formulation required the development of a branch‐and‐cut solution approach based on new families of valid inequalities and variable fixing strategies. We tested the proposed approach on different sets of benchmark instances. The experimentation shows that the proposed method is competitive or outperforms the state‐of‐the‐art approaches, providing either the optimal solution or improved bounds for several instances unsolved before.

A branch-and-cut algorithm for the routing and spectrum allocation problem

One of the most promising solutions to deal with huge data traffic demands in large communication networks is given by flexible optical networking, in particular the flexible grid (flexgrid) technology specified in the ITU-T standard G.694.1. In this specification, the frequency spectrum of an optical fiber link is divided into narrow frequency slots. Any sequence of consecutive slots can be used as a simple channel, and such a channel can be switched in the network nodes to create a lightpath. In this kind of networks, the problem of establishing lightpaths for a set of end-to-end demands that compete for spectrum resources is called the routing and spectrum allocation problem (RSA). Due to its relevance, RSA has been intensively studied in the last years. It has been shown to be NP-hard and different solution approaches have been proposed for this problem. In this paper we present several families of valid inequalities, valid equations, and optimality cuts for a natural integer programming formulation of RSA and, based on these results, we develop a branch-and-cut algorithm for this problem. Our computational experiments suggest that such an approach is effective at tackling this problem.

A new formulation and branch-and-cut method for single-allocation hub location problems

A new compact formulation for uncapacitated single-allocation hub location problems with fewer variables than the previous Integer Linear Programming formulations in the literature is introduced. Our formulation works even with costs not based on distances and not satisfying triangle inequality. Moreover, costs can be given in aggregated or disaggregated way. Different families of valid inequalities that strengthen the formulation are developed and a branch-and-cut algorithm based on a relaxed version of the formulation is designed, whose restrictions are inserted in a cut generation procedure together with two sets of valid inequalities. The performance of the proposed methodology is tested on well-known hub location data sets and compared to the most recent and efficient exact algorithms for single-allocation hub location problems. Extensive computational results prove the efficiency of our methodology, that solves large-scale instances in very competitive times.

Complexity, algorithmic, and computational aspects of a dial-a-ride type problem

In dial-a-ride systems involving autonomous vehicles circulating along a circuit, a fleet of vehicles must serve clients who request rides between stations of the circuit such that the total number of pick-up and drop-off operations is minimized. In this paper, we focus on a unitary variant where each client requests a single place in the vehicles and all the clients must be served within a single tour of the circuit. Such unitary variant induces a combinatorial optimization problem for which we introduce a nontrivial special case that is polynomially solvable when the capacity of each vehicle is at most 2 but it is NP-Hard otherwise. We also study the polytope associated with the solutions to this problem. We introduce new families of valid inequalities and give necessary and sufficient conditions under which they are facet-defining. Based on these inequalities, we devise an efficient branch-and-cut algorithm that outperforms the state-of-the-art commercial solvers.

Valid inequalities and extended formulations for lot-sizing and scheduling problem with sequence-dependent setups

In this paper, we propose new valid inequalities and extended formulations for the lot-sizing and scheduling problem with sequence-dependent setups, which are derived by investigating the single-period substructure of the problem. Specifically, we derive two new families of valid inequalities and identify their facet-defining conditions. Additionally, we demonstrate that these inequalities can be separated in polynomial time. After introducing the existing extended formulations for the problem, we provide new extended formulations adapting decision variables representing the time-flow and compare the theoretical strengths of the various formulations and valid inequalities, including the proposed ones. Finally, we conduct computational experiments to demonstrate the effectiveness of the proposed inequalities and formulations. The test results indicate that the proposed inequalities and extended formulations facilitate tightening the linear programming relaxation bounds.

Precedence constrained generalized traveling salesman problem: Polyhedral study, formulations, and branch-and-cut algorithm

The Precedence Constrained Generalized Traveling Salesman Problem (PCGTSP) is an extension of two well-known combinatorial optimization problems — the Generalized Traveling Salesman Problem (GTSP) and the Precedence Constrained Asymmetric Traveling Salesman Problem (PCATSP), whose path version is known as the Sequential Ordering Problem (SOP). Similarly to the classic GTSP, the goal of the PCGTSP, for a given input digraph and partition of its node set into clusters, is to find a minimum cost cyclic route (tour) visiting each cluster in a single node. In addition, as in the PCATSP, feasible tours are restricted to visit the clusters with respect to the given partial order. Unlike the GTSP and SOP, to the best of our knowledge, the PCGTSP still remain to be weakly studied both in terms of polyhedral theory and algorithms. In this paper, for the first time for the PCGTSP, we propose several families of valid inequalities, establish dimension of the PCGTS polytope and prove sufficient conditions ensuring that the extended Balas’ π- and σ-inequalities become facet-inducing. Relying on these theoretical results and evolving the state-of-the-art algorithmic approaches for the PCATSP and SOP, we introduce a family of MILP-models (formulations) and several variants of the branch-and-cut algorithm for the PCGTSP. We prove their high performance in a competitive numerical evaluation against the public benchmark library PCGTSPLIB, a known adaptation of the classic SOPLIB to the problem in question.

Dendrograms, minimum spanning trees and feature selection

Feature selection is a fundamental process to avoid overfitting and to reduce the size of databases without significant loss of information that applies to hierarchical clustering. Dendrograms are graphical representations of hierarchical clustering algorithms that for single linkage clustering can be interpreted as minimum spanning trees in the complete network defined by the database. In this work, we introduce the problem that determines jointly a set of features and a dendrogram, according to the single linkage method. We propose different formulations that include the minimum spanning tree problem constraints as well as the feature selection constraints. Different bounds on the objective function are studied. For one of the models, several families of valid inequalities are proposed and the problem of separating them is studied. For another formulation, a decomposition algorithm is designed. In an extensive computational study, the effectiveness of the different models is discussed, the model with valid inequalities is compared with the decomposition algorithm. The computational results also illustrate that the integration of feature selection to the optimization model allows to keep a satisfactory percentage of information.

Formulations and valid inequalities for the capacitated dispersion problem

AbstractThis work focuses on the capacitated dispersion problem for which we study several mathematical formulations in different spaces using variables associated with nodes, edges, and costs. The relationships among the presented formulations are investigated by comparing the projections of the feasible sets of the LP relaxations onto the subspace of natural variables. These formulations are then strengthened with families of valid inequalities and variable‐fixing procedures. The separation problems associated with the valid inequalities that are exponential in number are shown to be polynomially solvable by reducing them to longest path problems in acyclic graphs. The dual bounds obtained from stronger but larger formulations are used to improve the strength of weaker but smaller formulations. Several sets of computational experiments are conducted to illustrate the usefulness of the findings, as well as the aptness of the formulations for different types of instances.

Exact approaches for the Minimum Subgraph Diameter Problem

This work addresses the Minimum Subgraph Diameter Problem (MSDP), an NP-hard problem with applications to network design. Given an undirected graph with lengths and costs on the edges, the MSDP’s goal is to find a spanning subgraph with total cost limited by a given budget, such that the subgraph’s diameter is minimum. We propose a Mixed-Integer Linear Programming (MILP) model for the MSDP, along with two families of valid inequalities, which form the basis of the first exact approaches for the MSDP. We present an approach to apply lazy constraints on the MILP and a heuristic to generate upper bounds from fractional solutions. We propose a comprehensive benchmark for the MSDP and conduct a thorough computational study. The results show the effectiveness of each contribution towards reducing optimality gaps and computational times.

New variants of the simple plant location problem and applications

This work revisits novel variants of the Simple Plant Location Problem (SPLP), which were originally presented in the Ph.D. thesis Set packing, location and related problems (2019). They consist in considering different requirements on clients allocation to facilities, namely incompatibilities or co-location for some specific pairs of clients. These correspond to “cannot-link” and “must-link” constraints previously studied in the related topic of clustering. Set packing formulations of these new location problems are presented, together with families of valid inequalities and facets. Computational experience supporting the effectiveness of the resulting cuts into a branch and cut procedure is exposed. The comprehensive revision on these new variants of the SPLP is completed by illustrative applications, including the design of telecommunication networks and interesting connections in the field of Genetics, and also by the analysis of open questions and future research directions.

Multi-Trip Multi-Trailer Drop-and-Pull Container Drayage Problem

This paper addresses a novel and interesting container drayage problem in which a tractor can pull multiple trailers and can perform multiple trips during a working day. A state-based method and an IF-THEN method are proposed to model the multi-trip feature. The problem is mathematically formulated as two mixed-integer programming models. Three families of valid inequalities are introduced to strengthen the models. An adaptive large neighborhood search (ALNS) algorithm with an embedded decoding mathematical model is proposed to solve realistic-sized instances of the problem. Numerical experiments based on a large number of benchmark instances indicate that the inequalities are valid and that the model based on IF-THEN constraints shows excellent performance. The ALNS algorithm outperforms the strengthened mathematical model considering large-sized instances. The multi-trailer mode is more suitable if customers are clustered in the area far away from the depot when compared to the single-trailer mode.

Social distancing and revenue management—A post-pandemic adaptation for railways

The SARS-CoV-2 pandemic has had a significant impact on rail operations worldwide. Adopting control measures such as a 50% occupancy rate can contribute to a safer travel environment, though at the expense of operational efficiency. This paper addresses the issues of social distancing and revenue maximization for a train operating company in a post-pandemic world. Although the two objectives appear to be highly contradictory, we believe that judicious planning can optimize both to a great extent. Existing research on social distancing on public transport has only considered the risk of virus transmission during travel. This is the first attempt to recognize the risk of virus spread in different cities along with transmission risk as part of developing a social distancing plan. We study the problem of assigning seats to passenger groups on long-distance trains while ensuring social distancing within coaches. A novel seating assignment policy is proposed that takes into account several factors that govern the spread of virus. In an effort to reduce the spread of the virus and improve revenue simultaneously, a mixed-integer programming (MIP) model is proposed to assign seats to passengers. Several families of valid inequalities and preprocessing steps are proposed to strengthen the MIP formulation, which represents a substantial contribution to the literature on group seat assignment problem. The validity of the model and the effectiveness of the valid inequalities have been evaluated using real-life data from Indian Railways. The computational results demonstrate a significant reduction in the risk of contagion and an increase in seat utilization compared to the current approach employed by operators.