A Relax-and-Cut algorithm for the set partitioning problem

Relax-and-Cut algorithms offer an alternative to strengthen Lagrangian relaxation bounds. The main idea behind these algorithms is to dynamically select and dualize inequalities (cuts) within a Lagrangian relaxation framework. This paper proposes a Relax-and-Cut algorithm for the Set Partitioning Problem. Computational tests are reported for benchmark instances from the literature. For Set Partitioning instances with integrality gaps, a variant of the classical Lagrangian relaxation is often used in the literature. It introduces a knapsack constraint to the standard formulation of the problem. Our results indicate that the proposed Relax-and-Cut algorithm outperforms the latter approach in terms of lower bound quality. Furthermore, it turns out to be very competitive in terms of CPU time. Decisive in achieving that performance was the implementation of dominance rules to manage inequalities in the cut pool. The Relax-and-Cut framework proposed here could also be used as a preprocessing tool for Linear Integer Programming solvers. Computational experiments demonstrated that the combined use of our framework and XPRESS improved the performance of that Linear Integer Programming solver for the test sets used in this study.     

Lagrangian and branch-and-cut approaches for upgrading spanning tree problems

Problems aiming at finding budget constrained optimal upgrading schemes to improve network performance have received attention over the last two decades. In their general setting, these problems consist of designing a network and, simultaneously, allocating (limited) upgrading resources in order to enhance the performance of the designed network.In this paper we address two particular optimal upgrading network design problems; in both cases, the sought-after layout corresponds to a spanning tree of the input network and upgrading decisions are to be taken on nodes. We design Mixed Integer Programming-based algorithmic schemes to solve the considered problems: Lagrangian relaxation approaches and branch-and-cut algorithms. Along with the designed algorithms, different enhancements, including valid inequalities, primal heuristic and variable fixing procedures, are proposed.Using two set of instances, we experimentally compare the designed algorithms and explore the benefits of the devised enhancements. The results show that the proposed approaches are effective for solving to optimality most of the instances in the testbed, or manage to obtain solutions and bounds giving very small optimality gaps. Besides, the proposed enhancements turn out to be beneficial for improving the performance of the algorithms.     

A set partitioning reformulation of a school bus scheduling problem

We present an integer programming model for the integrated optimization of bus schedules and school starting times, which is a single-depot vehicle scheduling problem with additional coupling constraints among the time windows. For instances with wide time windows the linear relaxation is weak and feasible solutions found by an ILP solver are of poor quality. We apply a set partitioning relaxation to compute better lower bounds and, in combination with a primal construction heuristic, also better primal feasible solutions. Integer programs with at most two non-zero coefficient per constraint play a prominent role in our approach. Computational results for several random and a real-world instance are given and compared with results from a standard branch-and-cut approach.     

A Benders decomposition based framework for solving cable trench problems

In this work, we present an algorithmic framework based on Benders decomposition for the Capacitated p-Cable Trench Problem with Covering. We show that our approach can be applied to most variants of the Cable Trench Problem (CTP) that have been considered in the literature. The proposed algorithm is augmented with a stabilization procedure to accelerate the convergence of the cut loop and with a primal heuristic to derive high-quality primal solutions. Three different variants of the CTP are considered in a computational study which compares the Benders approach with two compact integer linear programming formulations that are solved with CPLEX. The obtained results show that the proposed algorithm significantly outperforms the two compact models and that it can be used to tackle significantly larger instances than previously considered algorithms based on Lagrangean relaxation.     

Capacitated closed-loop supply chain network design under uncertainty

This study optimizes the design of a closed-loop supply chain network, which contains forward and reverse directions and is subject to uncertainty in demands for new & returned products. To address uncertainty in decision-making, we formulate a two-stage stochastic mixed-integer non-linear programming model to determine the distribution center locations and their corresponding capacity, and new & returned product flows in the supply chain network to minimize total design and expected operating costs. We convert our model to a conic quadratic programming model given the complexity of our problem. Then, the conic model is added with certain valid inequalities, such as polymatroid inequalities, and extended with respect to its cover cuts so as to improve computational efficiency. Furthermore, a tabu search algorithm is developed for large-scale problem instances. We also study the impact of inventory weight, transportation weight, and marginal value of time of returned products by the sensitivity analysis. Several computational experiments are conducted to validate the effectiveness of the proposed model and valid inequalities.     

A Branch-and-Cut method for the Capacitated Location-Routing Problem

Recent researches in the design of logistic networks have shown that the overall distribution cost may be excessive if routing decisions are ignored when locating depots. The Location-Routing Problem (LRP) overcomes this drawback by simultaneously tackling location and routing decisions. The aim of this paper is to propose an exact approach based on a Branch-and-Cut algorithm for solving the LRP with capacity constraints on depots and vehicles. The proposed method is based on a zero-one linear model strengthened by new families of valid inequalities. The computational evaluation on three sets of instances (34 instances in total), with 5–10 potential depots and 20–88 customers, shows that 26 instances with five depots are solved to optimality, including all instances with up to 40 customers and three with 50 customers.     

A heuristic algorithm for optimal location of flow‐refueling capacitated stations

Constructing refueling stations in the transportation network is one of the most important steps toward the promotion of alternative‐fuel vehicles. The capacity of these stations is usually limited. In this paper, a new capacitated refueling station location model and a solution algorithm are proposed. The algorithm is divided into two main steps. At first step, a restricted capacitated problem on core sets is constructed. Then, a modified Lagrangean iterative method is used for obtaining solutions. The Lagrangean method decomposes the restricted problem into two subproblems that are easy to solve. Information from subproblems is used to generate valid inequalities for tightening the upper and lower bounds. The approach is evaluated by considering a set of networks and randomly generated instances. The obtained results indicate that the large problems are efficiently tractable by the proposed algorithm in a reasonable time.     

Discrete time/cost trade-off problem: A decomposition-based solution algorithm for the budget version

This paper investigates the budget variant of the discrete time/cost trade-off problem (DTCTP). This multi-mode project scheduling problem requires assigning modes to the activities of a project so that the total completion time is minimized and the budget and the precedence constraints are satisfied. This problem is often encountered in practice as timely completion of the projects without exceeding the budget is crucial. The contribution of this paper to the literatures is to describe an effective Benders Decomposition-based exact algorithm to solve the DTCTP instances of realistic sizes. Although Benders Decomposition often exhibits a very slow convergence, we have included several algorithmic features to enhance the performance of the proposed tailored approach. Computational results attest to the efficacy of the proposed algorithm, which can solve large-scale instances to optimality.     

A simulated annealing algorithm for the job shop cell scheduling problem with intercellular moves and reentrant parts

This paper addresses the problem of scheduling parts in job shop cellular manufacturing systems by considering exceptional parts that need to visit machines in different cells and reentrant parts which need to visit some machines more than once in non-consecutive manner. Initially, an integer linear programming (ILP) model is presented for the problem to minimize the makespan, which considers intercellular moves and non-consecutive multiple processing of parts on a machine. Due to the complexity of the model, a simulated annealing (SA) based solution approach is developed to solve the problem. To increase the efficiency of the search algorithm, a neighborhood structure based on the concept of blocks is applied. Subsequently, the efficiency of the ILP model and the performance of the proposed SA are assessed over a set of problem instances taken from the literature. The proposed ILP model was coded in Lingo 8.0 and the solution obtained by the proposed SA was compared to the optimal values. The computational results demonstrate that the proposed ILP model and SA algorithm are effective and efficient for this problem.     

The fixed set search applied to the power dominating set problem

In this article, we focus on solving the power dominating set problem and its connected version. These problems are frequently used for finding optimal placements of phasor measurement units in power systems. We present an improved integer linear program (ILP) for both problems. In addition, a greedy constructive algorithm and a local search are developed. A greedy randomised adaptive search procedure (GRASP) algorithm is created to find near optimal solutions for large scale problem instances. The performance of the GRASP is further enhanced by extending it to the novel fixed set search (FSS) metaheuristic. Our computational results show that the proposed ILP has a significantly lower computational cost than existing ILPs for both versions of the problem. The proposed FSS algorithm manages to find all the optimal solutions that have been acquired using the ILP. In the last group of tests, it is shown that the FSS can significantly outperform the GRASP in both solution quality and computational cost.     

Maximizing lifetime in wireless sensor networks with multiple sensor families

Wireless sensor networks are generally composed of a large number of hardware devices of the same type, deployed over a region of interest in order to perform a monitoring activity on a set of target points. Nowadays, several different types of sensor devices exist, which are able to monitor different aspects of the region of interest (including sound, vibrations, proximity, chemical contaminants, among others) and may be deployed together in a heterogeneous network. In this work, we face the problem of maximizing the amount of time during which such a network can remain operational, while maintaining at all times a minimum coverage guarantee for all the different sensor types. Some global regularity conditions in order to guarantee a fair level of coverage for each sensor type to each target are also taken into account in a second variant of the proposed problem. For both problem variants we developed an exact approach, which is based on a column generation algorithm whose subproblem is either solved heuristically by means of a genetic algorithm or optimally by an appropriate ILP formulation. In our computational tests the proposed genetic algorithm is shown to be able to dramatically speed up the procedure, enabling the resolution of large-scale instances within reasonable computational times.     

Models and algorithms for packing rectangles into the smallest square

We consider the problem of determining the smallest square into which a given set of rectangular items can be packed without overlapping. We present an ILP model, an exact approach based on the iterated execution of a two-dimensional packing algorithm, and a randomized metaheuristic. Such approaches are valid both for the case where the rectangles have fixed orientation and the case where they can be rotated by 90°. We computationally evaluate the performance and the limits of the proposed approaches on a large set of instances, including a number of classical benchmarks from the literature, for both cases above, and for the special case where the items are squares.     

Undirected postman problems with zigzagging option: A cutting-plane approach

The paper devises a new model and associated cutting-plane and branch-and-cut approaches for a variant of the undirected Chinese and rural postman problem where some of the edges offer the flexibility of either being serviced twice by two separate traversals or by a single zigzag traversal. The kernel of the proposed cutting-plane algorithm is a separation procedure for generalized blossom inequalities. We show that the currently best known separation procedure of Letchford, Reinelt and Theis [A faster exact separation algorithm for blossom inequalities. In: Nemhauser G, Bienstock D, editors, Integer programming and combinatorial optimization, vol. 3064. Berlin: Springer; 2004. (chapter 10)] is applicable and leads to a highly efficient solution approach which can handle large-scale problem instances.     

Existence of binary invariant sets in feedback neural networks withapplication to synthesis

The design of fully connected discrete-time neural networks, with threshold units, can be performed by solving a set of linear inequalities. Using this formulation, the design problem can be handled by several powerful algorithms. This approach is extended to the design of continuous-time neural networks with sigmoidal units, which represent a more realistic model of analog circuit implementations. The proposed method is based on some theoretical results proved in this paper.     

A discrete time exact solution approach for a complex hybrid flow-shop scheduling problem with limited-wait constraints

We study a real-world complex hybrid flow-shop scheduling problem arising from a bio-process industry. There are a variety of constraints to be taken into account, in particular zero intermediate capacity and limited waiting time between processing stages. We propose an exact solution approach for this optimization problem, based on a discrete time representation and a mixed-integer linear programming formulation. The proposed solution algorithm makes use of a new family of valid inequalities exploiting the fact that a limited waiting time is imposed on jobs between two successive production stages. The results of our computational experiments confirm that the proposed method produces good feasible schedules for industrial instances.     

An exact approach for the 0–1 knapsack problem with setups

We consider the 0–1 Knapsack Problem with Setups. We propose an exact approach which handles the structure of the ILP formulation of the problem. It relies on partitioning the variables set into two levels and exploiting this partitioning. The proposed approach favorably compares to the algorithms in literature and to solver CPLEX 12.5 applied to the ILP formulation. It turns out to be very effective and capable of solving to optimality, within limited CPU time, all instances with up to 100, 000 variables.     

Time constrained maximal covering salesman problem with weighted demands and partial coverage

In a routing framework, it may not be viable to visit every single customer separately due to resource limitations or efficiency concerns. In such cases, utilizing the notion of coverage; i.e., satisfying the demand of multiple customers by visiting a single customer location, may be advantageous. With this motivation, we study the time constrained maximal covering salesman problem (TCMCSP) in which the aim is to find a tour visiting a subset of customers so that the amount of demand covered within a limited time is maximized. We provide flow and cut formulations and derive valid inequalities. Since the connectivity constraints and the proposed valid inequalities are exponential in the size of the problem, we devise different branch-and-cut schemes. Computational experiments performed on a set of problem instances demonstrate the effectiveness of the proposed valid inequalities in terms of strengthening the linear relaxation bounds as well as speeding up the solution procedure. Moreover, the results indicate the superiority of using a branch-and-cut methodology over a flow-based formulation. Finally, we discuss the relation between the problem parameters and the structure of optimal solutions based on the results of our experiments.     

一种多属性目标监测的无线感知网络覆盖算法

本文研究了一种无线感知网络应用中多属性目标的覆盖问题。这种覆盖问题与单一类型数据的目标不同,其待测区域中的每个目标同时包含多种类型的现场数据。如果布置一个无线感知网络去担任监测任务,其节点需要配置多种不同类型的传感器单元。针对这种需要采集多种类型的数据才能对目标进行监测的无线感知网络的应用,节能而有效的的覆盖目标更是一个突出的问题。本文首先用ILP模型将问题进行了形式化,然后通过设计一种分布式算法求得问题了模拟仿真。仿真结果表明,这种分布式算法比直接求解ILP求出的网络寿命很接近。由于直接求解ILP问题必须依靠中心节点完成,对于节点较多并且电量受限的无线感知网络,这种分布式算法更适合。     

Introducing possibilistic logic in ILP for dealing with exceptions

In this paper we propose a new formalization of the inductive logic programming (ILP) problem for a better handling of exceptions. It is now encoded in first-order possibilistic logic. This allows us to handle exceptions by means of prioritized rules, thus taking lessons from non-monotonic reasoning. Indeed, in classical first-order logic, the exceptions of the rules that constitute a hypothesis accumulate and classifying an example in two different classes, even if one is the right one, is not correct. The possibilistic formalization provides a sound encoding of non-monotonic reasoning that copes with rules with exceptions and prevents an example to be classified in more than one class. The benefits of our approach with respect to the use of first-order decision lists are pointed out. The possibilistic logic view of ILP problem leads to an optimization problem at the algorithmic level. An algorithm based on simulated annealing that in one turn computes the set of rules together with their priority levels is proposed. The reported experiments show that the algorithm is competitive to standard ILP approaches on benchmark examples.     

Solving school bus routing using the multiple vehicle traveling purchaser problem: A branch-and-cut approach

School bus routing problems, combining bus stop selection and bus route generation, look simultaneously for a set of bus stops to pick up students from among a group of potential locations, and for bus routes to visit the selected stops and carry the students to their school. These problems, classified as Location-Routing problems, are of interest in densely populated urban areas.This article introduces a generalization of the vehicle routing problem called the multi-vehicle traveling purchaser problem, modeling a family of routing problems combining stop selection and bus route generation. It discusses a Mixed Integer Programming formulation extending previous studies on the classical single vehicle traveling purchaser problem. The proposed model is based on a single commodity flow formulation combining continuous variables with binary variables by means of coupling constraints. Additional valid inequalities are proposed with the purpose of strengthening its Linear Programming relaxation. These valid inequalities are obtained by projecting out the flow variables.We develop a branch-and-cut algorithm that makes use of the proposed model and valid inequalities. This cutting plane algorithm is implemented and tested on a large family of symmetric and asymmetric instances derived from randomly generated problems, showing the usefulness of the proposed valid inequalities.