The discrete facility location problem with balanced allocation of customers

We consider a discrete facility location problem where the difference between the maximum and minimum number of customers allocated to every plant has to be balanced. Two different Integer Programming formulations are built, and several families of valid inequalities for these formulations are developed. Preprocessing techniques which allow to reduce the size of the largest formulation, based on the upper bound obtained by means of an ad hoc heuristic solution, are also incorporated. Since the number of available valid inequalities for this formulation is exponential, a branch-and-cut algorithm is designed where the most violated inequalities are separated at every node of the branching tree. Both formulations, with and without the improvements, are tested in a computational framework in order to discriminate the most promising solution methods. Difficult instances with up to 50 potential plants and 100 customers, and largest easy instances, can be solved in one CPU hour.     

An adaptive memory algorithm for the split delivery vehicle routing problem

The split delivery vehicle routing problem (SDVRP) relaxes routing restrictions forcing unique deliveries to customers and allows multiple vehicles to satisfy customer demand. Split deliveries are used to reduce total fleet cost to meet those customer demands. We provide a detailed survey of the SDVRP literature and define a new constructive algorithm for the SDVRP based on a novel concept called the route angle control measure. We extend this constructive approach to an iterative approach using adaptive memory concepts, and then add a variable neighborhood descent process. These three new approaches are compared to exact and heuristic approaches by solving the available SDVRP benchmark problem sets. Our approaches are found to compare favorably with existing approaches and we find 16 new best solutions for a recent 21 problem benchmark set.     

A unified exact method for solving different classes of vehicle routing problems

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.      

A branch-and-cut algorithm for the plant-cycle location problem

The Plant-Cycle Location Problem (PCLP) is defined on a graph G=(I∪J, E), where I is the set of customers and J is the set of plants. Each customer must be served by one plant, and the plant must be opened to serve customers. The number of customers that a plant can serve is limited. There is a cost of opening a plant, and of serving a customer from an open plant. All customers served by a plant are in a cycle containing the plant, and there is a routing cost associated to each edge of the cycle. The PCLP consists in determining which plants to open, the assignment of customers to plants, and the cycles containing each open plant and its customers, minimizing the total cost. It is an NP-hard optimization problem arising in routing and telecommunications. In this article, the PCLP is formulated as an integer linear program, a branch-and-cut algorithm is developed, and computational results on real-world data and randomly generated instances are presented. The proposed approach is able to find optimal solutions of random instances with up to 100 customers and 100 potential plants, and of instances on real-world data with up to 120 customers and 16 potential plants.     

The undirected m-Capacitated Peripatetic Salesman Problem

In the m-Capacitated Peripatetic Salesman Problem (m-CPSP) the aim is to determine m Hamiltonian cycles of minimal total cost on a graph, such that all the edges are traversed less than the value of their capacity. This article introduces three formulations for the m-CPSP. Two branch-and-cut algorithms and one branch-and-price algorithm are developed. Tests performed on randomly generated and on TSPLIB Euclidean instances indicate that the branch-and-price algorithm can solve instances with more than twice the size of what is achievable with the branch-and-cut algorithms.     

Exact solution of the soft-clustered vehicle-routing problem

The soft-clustered vehicle-routing problem (SoftCluVRP) extends the classical capacitated vehicle-routing problem by one additional constraint: The customers are partitioned into clusters and feasible routes must respect the soft-cluster constraint, that is, all customers of the same cluster must be served by the same vehicle. In this article, we design and analyze different branch-and-price algorithms for the exact solution of the SoftCluVRP. The algorithms differ in the way the column-generation subproblem, a variant of the shortest-path problem with resource constraints (SPPRC), is solved. The standard approach for SPPRCs is based on dynamic-programming labeling algorithms. We show that even with all the recent acceleration techniques (e.g., partial pricing, bidirectional labeling, decremental state space relaxation) available for SPPRC labeling algorithms, the solution of the subproblem remains extremely difficult. The main contribution is the modeling and solution of the subproblem using a branch-and-cut algorithm. The conducted computational experiments prove that branch-and-price equipped with this integer programming-based approach outperforms sophisticated labeling-based algorithms by one order of magnitude. The largest SoftCluVRP instances solved to optimality have more than 400 customers or more than 50 clusters.     

Decomposition methods for the two-stage stochastic Steiner tree problem

A new algorithmic approach for solving the stochastic Steiner tree problem based on three procedures for computing lower bounds (dual ascent, Lagrangian relaxation, Benders decomposition) is introduced. Our method is derived from a new integer linear programming formulation, which is shown to be strongest among all known formulations. The resulting method, which relies on an interplay of the dual information retrieved from the respective dual procedures, computes upper and lower bounds and combines them with several rules for fixing variables in order to decrease the size of problem instances. The effectiveness of our method is compared in an extensive computational study with the state-of-the-art exact approach, which employs a Benders decomposition based on two-stage branch-and-cut, and a genetic algorithm introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms existing ones, both on benchmark instances from literature, as well as on large-scale telecommunication networks.     

Branch-and-cut with lazy separation for the vehicle routing problem with simultaneous pickup and delivery

We propose a branch-and-cut algorithm for the VRPSPD where the constraints that ensure that the capacities are not exceeded in the middle of a route are applied in a lazy fashion. The algorithm was tested in 87 instances with 50–200 customers, finding improved lower bounds and several new optimal solutions.     

Order and static stability into the strip packing problem

This paper investigates the two-dimensional strip packing problem considering the case in which items should be arranged to form a physically stable packing satisfying a predefined item unloading order from the top of the strip. The packing stability analysis is based on conditions for the static equilibrium of rigid bodies, differing from others strategies which are based on area and percentage of support. We consider an integer linear programming model for the strip packing problem with the order constraint, and a cutting plane algorithm to handle stability, leading to a branch-and-cut approach. We also present two heuristics: the first is based on a stack building algorithm; and, the last is a slight modification of the branch-and-cut approach. The computational experiments show that the branch-and-cut model can handle small and medium-sized instances, whereas the heuristics found almost optimal solutions quickly for several instances. With the combination of heuristics and the branch-and-cut algorithm, many instances are solved to near optimality in a few seconds.     

The Clustered Orienteering Problem

In this paper we study a generalization of the Orienteering Problem (OP) which we call the Clustered Orienteering Problem (COP). The OP, also known as the Selective Traveling Salesman Problem, is a problem where a set of potential customers is given and a profit is associated with the service of each customer. A single vehicle is available to serve the customers. The objective is to find the vehicle route that maximizes the total collected profit in such a way that the duration of the route does not exceed a given threshold. In the COP, customers are grouped in clusters. A profit is associated with each cluster and is gained only if all customers belonging to the cluster are served. We propose two solution approaches for the COP: an exact and a heuristic one. The exact approach is a branch-and-cut while the heuristic approach is a tabu search. Computational results on a set of randomly generated instances are provided to show the efficiency and effectiveness of both approaches.     

Branch-and-Cut-and-Price for Capacitated Connected Facility Location

We consider a generalization of the Connected Facility Location Problem (ConFL), suitable to model real world network extension scenarios such as fiber-to-the-curb. In addition to choosing a set of facilities and connecting them by a Steiner tree as in ConFL, we aim to maximize the resulting profit by potentially supplying only a subset of all customers. Furthermore, capacity constraints on potential facilities need to be considered. We present two mixed integer programming based approaches which are solved using branch-and-cut and branch-and-cut-and-price, respectively. By studying the corresponding polyhedra we analyze both approaches theoretically and show their advantages over previously presented models. Furthermore, using a computational study we are able to additionally show significant advantages of our models over previously presented ones from a practical point of view.     

A node-based layered graph approach for the Steiner tree problem with revenues,budget and hop-constraints

The Steiner tree problem with revenues, budget and hop-constraints (STPRBH) is a variant of the classical Steiner tree problem. The goal is to find a tree maximizing the collected revenue, which is associated with nodes, subject to a given budget for the edge cost of the tree and a hop-limit for the distance between the given root node and any other node in that tree. In this work, we introduce a novel generic way to model hop-constrained tree problems as integer linear programs and apply it to the STPRBH. Our approach is based on the concept of layered graphs that gained widespread attention in the recent years, due to their computational advantage when compared to previous formulations for modeling hop-constraints. Contrary to previous MIP formulations based on layered graphs (that are arc-based models), our model is node-based. Thus it contains much less variables and allows to tackle large-scale instances and/or instances with large hop-limits, for which the size of arc-based layered graph models may become prohibitive. The aim of our model is to provide a good compromise between quality of root relaxation bounds and the size of the underlying MIP formulation. We implemented a branch-and-cut algorithm for the STPRBH based on our new model. Most of the instances available for the DIMACS challenge, including 78 (out of 86) previously unsolved ones, can be solved to proven optimality within a time limit of 1000 s, most of them being solved within a few seconds only. These instances contain up to 500 nodes and 12,500 edges, with hop-limit up to 25.     

Arc-based integer programming formulations for three variants of proportional symbol maps

Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data associated with specific locations. The symbols we consider are opaque disks, which may be partially covered by other overlapping disks. We address the problem of creating a suitable drawing of the disks that maximizes one of two quality metrics: the total and the minimum visible length of disk boundaries. We study three variants of this problem, two of which are known to be NP-hard and another whose complexity is open. We propose novel integer programming formulations for each problem variant and test them on real-world instances with a branch-and-cut algorithm. When compared with state-of-the-art models from the literature, our models significantly reduce computation times for most instances.     

A GRASP + ILP-based metaheuristic for the capacitated location-routing problem

In this paper we present a three-phase heuristic for the Capacitated Location-Routing Problem. In the first stage, we apply a GRASP followed by local search procedures to construct a bundle of solutions. In the second stage, an integer-linear program (ILP) is solved taking as input the different routes belonging to the solutions of the bundle, with the objective of constructing a new solution as a combination of these routes. In the third and final stage, the same ILP is iteratively solved by column generation to improve the solutions found during the first two stages. The last two stages are based on a new model, the location-reallocation model, which generalizes the capacitated facility location problem and the reallocation model by simultaneously locating facilities and reallocating customers to routes assigned to these facilities. Extensive computational experiments show that our method is competitive with the other heuristics found in the literature, yielding the tightest average gaps on several sets of instances and being able to improve the best known feasible solutions for some of them.     

需求可分的车辆路径问题模型与算法

需求可分的车辆路径问题(SDVRP)无论是从运输距离还是派车数量上,都可进一步优化传统的车辆路径问题。为了降低SDVRP的求解难度,本文在分析最优解性质的基础上,加强模型的约束条件,将原模型转变为等价的改进SDVRP,并在使用蚂蚁算法求解改进SDVRP模型的过程中,采用开发新路径和2-opt相结合的方法,以避免出现迭代停滞的现象。实验表明,算法计算结果稳定,最差解与最好解的偏差仅为1.80%。     

A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery

This paper addresses a location-routing problem with simultaneous pickup and delivery (LRPSPD) which is a general case of the location-routing problem. The LRPSPD is defined as finding locations of the depots and designing vehicle routes in such a way that pickup and delivery demands of each customer must be performed with same vehicle and the overall cost is minimized. We propose an effective branch-and-cut algorithm for solving the LRPSPD. The proposed algorithm implements several valid inequalities adapted from the literature for the problem and a local search based on simulated annealing algorithm to obtain upper bounds. Computational results, for a large number of instances derived from the literature, show that some instances with up to 88 customers and 8 potential depots can be solved in a reasonable computation time.     

The two-echelon production-routing problem

This paper introduces the Two-Echelon Production-Routing Problem. This problem is motivated from the petrochemical industry, enlarging the supply chain integration by taking into account production, inventory, and routing decisions in a two-echelon vendor-managed inventory system. We describe, model, and design a branch-and-cut (B&C) to solve the problem under different inventory policies. We also propose a novel exact algorithm, by employing parallel computing techniques, in order to combine local search procedures within a traditional B&C scheme. We evaluate the performance of our methods through extensive computational experiments, both by comparing the algorithms, the effectiveness of the different inventory policies, and the impact of these policies on the partial costs. We derive many managerial insights based on the results. We also validate our new exact algorithm by solving similar problems from the literature, such as the two-echelon multi-depot inventory-routing (2E-MDIRP) and the classical multi-vehicle production-routing problem (MV-PRP). Computational experiments show that our method is very competitive. Based on 512 experiments for the 2E-MDIRP, our algorithm was able to find 111 new best known solutions (BKS), besides proving 412 optimal solutions, against 298 from the literature. For 336 experiments over small and medium size MV-PRP instances, we proved 242 optimal solutions, 11 more than the exact methods from the literature, besides providing 95 new BKS. Moreover, we were the first to tackle large MV-PRP instances exactly, and in this case, our algorithm provides all BKS for instances up to 50 customers, 20 periods and 5 vehicles, outperforming all meta/matheuristics procedures from the literature.     

Multiple depot ring star problem: a polyhedral study and an exact algorithm

The multiple depot ring-star problem (MDRSP) is an important combinatorial optimization problem that arises in optical fiber network design and in applications that collect data using stationary sensing devices and autonomous vehicles. Given the locations of a set of customers and a set of depots, the goal is to (i) find a set of simple cycles such that each cycle (ring) passes through a subset of customers and exactly one depot, (ii) assign each non-visited customer to a visited customer or a depot, and (iii) minimize the sum of the routing costs, i.e., the cost of the cycles and the assignment costs. We present a mixed integer linear programming formulation for the MDRSP and propose valid inequalities to strengthen the linear programming relaxation. Furthermore, we present a polyhedral analysis and derive facet-inducing results for the MDRSP. All these results are then used to develop a branch-and-cut algorithm to obtain optimal solutions to the MDRSP. The performance of the branch-and-cut algorithm is evaluated through extensive computational experiments on several classes of test instances.