Compact Integer Linear Programming Research Articles

Optimizing modular equipment in the design of multi-product reconfigurable production lines

This work addresses the problem of the design and balancing of a modular multi-product line in a reconfigurable environment. The study emphasizes the importance of modularity in the reconfigurable manufacturing systems (RMS), which allows fast, efficient, and cost-effective adaptation to changes by rearranging, moving, or removing modules. The line consists of a fixed number of modular machines arranged linearly and connected by a material handling system. Each machine has a limited number of slots for modules. The line can handle several product types per batch, and each batch requires a specific line configuration to be produced. The objective consists in identifying the minimum number of modules needed and their arrangement in different configurations to produce each product type. To solve this problem, a compact integer linear programming (ILP) formulation is initially investigated. Then an ILP reformulation, based on a decomposition approach, as well as a heuristic that uses this ILP reformulation are developed. The results demonstrate the effectiveness of the proposed methods.

Virtual network function reconfiguration in 5G networks: An optimization perspective

AbstractOne of the major challenges in managing 5G networks is the reconfiguration of network slices. The task covers in particular reconfiguration and relocation of virtual network functions (VNFs) so as to match the service requirements of the slices and the availability of resources of the data centers. In this article, we study in deep the problem of optimal VNFs reconfiguration analyzing a number of its variants. We define two exact, compact integer linear programming formulations of the VNFs reconfiguration problem, and derive two approximate solution algorithms, one of them based on the column generation method. Numerical tests and comparisons illustrate the performance of the algorithms and the quality of the provided solutions.

Variable and constraint reduction techniques for the temporal bin packing problem with fire-ups

The aim of this letter is to design and computationally test several improvements for the compact integer linear programming (ILP) formulations of the temporal bin packing problem with fire-ups (TBPP-FU). This problem is a challenging generalization of the classical bin packing problem in which the items, interpreted as jobs of given weight, are active only during an associated time window. The TBPP-FU objective function asks for the minimization of the weighted sum of the number of bins, viewed as servers of given capacity, to execute all the jobs and the total number of fire-ups. The fire-ups count the number of times the servers are activated due to the presence of assigned active jobs. Our contributions are effective procedures to reduce the number of variables and constraints of the ILP formulations proposed in the literature as well as the introduction of new valid inequalities. By extensive computational tests we show that substantial improvements can be achieved and several instances from the literature can be solved to proven optimality for the first time.

Exact and heuristic algorithms for the maximum weighted submatrix coverage problem

The maximum weighted submatrix coverage problem is a recently introduced problem with applications in data mining. It is concerned with selecting K submatrices of a given numerical matrix such that the sum of the matrix-entries, which occur in at least one of the selected submatrices, is maximized. In the paper introducing the problem, a problem-specific constraint programming approach was developed and embedded in a large neighborhood-search to obtain a heuristic. A compact integer linear programming formulation was also presented, but deemed inefficient due to its size.In this paper, we introduce new integer linear programming formulations for the problem, one of them is based on Benders decomposition. The obtained Benders decomposition-based formulation has a nice combinatorial structure, i.e., there is no need to solve linear programs to separate Benders cuts. We present preprocessing procedures and valid inequalities for all formulations. We also develop a greedy randomized adaptive search procedure for the problem, which is enhanced with a local search.A computational study using the instances from literature is done to evaluate the effectiveness of our new approaches. Our algorithms manage to find improved primal solutions for ten out of 17 real-world instances, and optimality is proven for two real-world instances. Moreover, for over 700 of 1617 large-scale synthetic instances, our algorithms find improved primal solutions compared to the heuristics from the literature.

Algorithms for hierarchical and semi-partitioned parallel scheduling

We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming (ILP) formulation for the job assignment subproblem, and show how to turn arbitrary ILP solutions into valid schedules. We also derive a polynomial-time 2-approximation algorithm for the problem. Extensions that incorporate memory capacity constraints are also discussed.

Compact integer linear programming formulations for the temporal bin packing problem with fire-ups

In this article we examine a specific version of the temporal bin packing problem (TBPP) that occurs in job-to-server scheduling. The TBPP represents a generalization of the well-known bin packing problem (BPP) with respect to an additional time dimension, and it requires to find the minimum number of bins (servers) to accommodate a given list of items (jobs) at any instant of time. In addition to this goal, recent publications suggest to also incorporate the number of fire-ups of the respective servers as an important factor in sustainable and energy-efficient overall operation. Addressing both these objectives by a weighted sum method typically increases the modeling complexity, thus leading to challenging ILP formulations, two of which have already been described in the literature. It has been shown that the parameter used to weight the two objectives strongly influences the applicability of heuristics and, therefore, the difficulty of the overall problem, so that only favorable choices can be handled so far. Even for these tailored scenarios, the current approaches already fail to compute an exact solution of many moderately-sized instances in reasonable times. For this reason, we propose various new preprocessing techniques to strengthen the LP bound and/or to reduce the numbers of variables or constraints appearing in the existing ILP formulations as well as one alternative modeling approach for the problem under consideration. Based on numerical tests with differently characterized sets of benchmark instances, the new and improved formulations are shown to lead (on average) to better performances (than compact state-of-the-art models) in terms of instances solved to optimality and computation times.

A column-and-row generation approach for the flying sidekick travelling salesman problem

The flying sidekick traveling salesman problem (FS-TSP) is a parcel delivery problem arising in the last-mile logistics, where the distribution plan of a driver-operated truck assisted by a drone (unmanned aerial vehicle, UAV) has to be defined. The FS-TSP is a variant of the TSP where routing decisions are integrated with customer-to-drone and customer-to-truck assignment decisions and truck-and-drone synchronization constraints. The objective is the minimization of the time required to serve all the customers, taking into account drone payload capacity and battery power constraints. In this work we provide a new representation of the FS-TSP based on the definition of an extended graph. This representation allows to model the problem by a new and compact integer linear programming formulation, where the synchronization issue is tackled in a column generation fashion, thus avoiding the usage of big-M constraints, representing one of the main drawbacks of the models present in literature. The proposed formulation has been solved by an exact approach which combines a Branch-and-Cut algorithm and a column generation procedure, strengthened by variable fixing strategies and new valid inequalities specifically defined for the problem. The proposed method has been experienced on a large set of benchmark instances. Computational results show that the proposed approach either is competitive or outperforms the best exact approach present in literature for the FS-TSP. Indeed, it is able to provide the optimal solution for all small size instances with 10 customers and for several medium size instances with 20 customers, some of them never solved before.

An Exact Approach for a Variant of the FS-TSP

In this work we focus on the flying sidekick traveling salesman problem (FS-TSP). The FS-TSP arises in the last-mile distribution context and it is a variant of the TSP aimed at determining the distribution plan of a driver-operated truck assisted by a drone (unmanned aerial vehicle, UAV), where the synchronization between the two vehicles allows to parallelize the delivery operations, so providing a reduction of the overall completion time. Several variants of the FS-TSP have been considered in literature, most of them sharing the assumption that launching and rendezvous position of a drone sortie must be different. In this work, we relax this assumption allowing the truck to wait for the drone at the launching position (FS-TSP*). This introduces an important flexibility element to deal with deliveries in rural or in urban areas with no-fly zones. We propose an original and compact integer linear programming formulation which allows to exactly solve the FS-TSP* and consequently also the FS-TSP. The results on several test instances show that our method can effectively solve to optimality instances with up to 20 customers and quantify the advantages of the FS-TSP* with respect to FS-TSP in terms of overall completion time.

Alignment of biological networks by integer linear programming: virus-host protein-protein interaction networks

BackgroundThe alignment of protein-protein interaction networks was recently formulated as an integer quadratic programming problem, along with a linearization that can be solved by integer linear programming software tools. However, the resulting integer linear program has a huge number of variables and constraints, rendering it of no practical use.ResultsWe present a compact integer linear programming reformulation of the protein-protein interaction network alignment problem, which can be solved using state-of-the-art mathematical modeling and integer linear programming software tools, along with empirical results showing that small biological networks, such as virus-host protein-protein interaction networks, can be aligned in a reasonable amount of time on a personal computer and the resulting alignments are structurally coherent and biologically meaningful.ConclusionsThe implementation of the integer linear programming reformulation using current mathematical modeling and integer linear programming software tools provided biologically meaningful alignments of virus-host protein-protein interaction networks.

A branch-and-price method for the vehicle allocation problem

The Vehicle Allocation Problem (VAP) consists of allocating a fleet of vehicles to attend to the expected demand for freight transportation between terminals along a finite multiperiod planning horizon. The objective is to maximize the profits generated for the completed services. The previous deterministic and stochastic approaches used heuristic procedures and approximations for solving large-scale instances of this problem. This paper proposes a Branch-and-Price method which is the first tailored exact solution approach for the VAP. This method provides proven optimal solutions within reasonable computational times, even for large-scale problem instances, and it is based on reformulating a compact Integer Linear Programming model of the VAP through the Dantzig–Wolfe decomposition and using efficient procedures for solving each component of the reformulation. The Primal–Dual Column Generation Method (PDCGM) is used to solve the master problem, while the subproblem is modeled as a Maximum Cost Flow Problem and is solved using the aggregation of the optimal longest paths on Directed Acyclic Graphs (DAG). Finally, we use three branching procedures (branching on a set of arcs, on the original variables and on the demand constraints) to obtain the optimal integer solution of the VAP. Computational experiments with 30 instances from a case study and 200 random realistic-sized instances are presented and analyzed, which show that the method has superior performance with respect to other exact approaches in solving large-scale VAP instances.

The constrained two-dimensional guillotine cutting problem with defects: an ILP formulation, a Benders decomposition and a CP-based algorithm

This paper addresses a variant of two-dimensional cutting problems in which rectangular small pieces are obtained by cutting a rectangular object through guillotine cuts. The characteristics of this variant are (i) the object contains some defects, and the items cut must be defective-free; (ii) there is an upper bound on the number of times an item type may appear in the cutting pattern; (iii) the number of guillotine stages is not restricted. This problem commonly arises in industrial settings that deal with defective materials, e.g. either by intrinsic characteristics of the object as in the cutting of wooden boards with knotholes in the wood industry, or by the manufacturing process as in the production of flat glass in the glass industry. We propose a compact integer linear programming (ILP) model for this problem based on the discretisation of the defective object. As solution methods for the problem, we develop a Benders decomposition algorithm and a constraint-programming (CP) based algorithm. We evaluate these approaches through computational experiments, using benchmark instances from the literature. The results show that the methods are effective on different types of instances and can find optimal solutions even for instances with dimensions close to real-size.

The Steiner Traveling Salesman Problem and its extensions

This paper considers the Steiner Traveling Salesman Problem, an extension of the classical Traveling Salesman Problem on an incomplete graph where not all vertices have demand. Some extensions including several depots or location decisions are introduced, modeled and solved. A compact integer linear programming formulation is proposed for each problem, where the routes are represented with two-index decision variables, and parity conditions are modeled using cocircuit inequalities. Exact branch-and-cut algorithms are developed for all formulations. Computational results obtained confirm the good performance of the algorithms. Instances with up to 500 vertices are solved optimally.

Column generation for the discrete UC problem with min-stop ramping constraints

Abstract The discrete unit commitment problem with min-stop ramping constraints optimizes the daily production of thermal power plants. For this problem, compact Integer Linear Programming (ILP) formulations have been designed to solve exactly small instances and heuristically real-size instances. This paper investigates whether Dantzig-Wolfe reformulation allows to improve the previous exact method and matheuristics. The extended ILP formulation is presented with the column generation algorithm to solve its linear relaxation. The experimental results show that the Dantzig-Wolfe reformulation does not improve the quality of the linear relaxation of the tightest compact ILP formulations. Computational experiments suggest also a conjecture which would explain such result: the compact ILP formulation of min-stop ramping constraints would be tight. Such results validate the quality of the exact methods and matheuristics based on compact ILP formulations previously designed.

A note on computational aspects of the Steiner traveling salesman problem

AbstractThe Steiner traveling salesman problem (StTSP) is a variant of the classical traveling salesman problem (TSP). In the StTSP, we are given a graph with edge distances, and a set of terminal nodes, which are a subset of all nodes. The goal is to find a minimum distance closed walk to visit each terminal node at least once. Two recent articles proposed solution approaches to the problem, namely, on the exact side, Letchford et al. proposed compact integer linear programming models, and on the heuristic side, Interian and Ribeiro proposed greedy randomized adaptive search procedure, enhanced with a local search. Using the exact approach, instances with up to 250 nodes could be solved to optimality, with runtimes up to 1400 seconds, and the heuristic approach was used to tackle instances with up to 3353 nodes, with runtimes up to 8500 seconds. In this note, we show that by transforming the problem to the classical TSP, and using a state‐of‐the‐art TSP solver, all instances from the literature can be solved to optimality within 20 seconds, most of them within a second. We provide optimal solution values for 14 instances, where the optimal solution was not known.

A multi-criteria repair/recovery framework for the tail assignment problem in airlines

Given a list of flights to be operated, the tail assignment problem aims at assigning each flight to a single airplane. When dealing with tail assignment problems, airline companies are willing to define assignments which minimize not only operation costs, but also robust solutions which are able to “resist” perturbations. Even with robust solutions, unexpected events can occur requiring to reconsider the plan under execution. This paper presents a general methodology for repair/recovery in tail assignment problems. When considering repair/recovery solutions for the initial plan under implementation, the decision maker may want to minimize operating costs, but also limit the changes with respect to the initial plan, minimize flights cancellations etc. Hence, we formulate the repair/recovery problem as a multiobjective integer linear programming problem minimizing specified functions for various repair criteria. The choice among the efficient solutions of this multiobjective program is supported by a multi-criteria model based on an additive value function elicited indirectly from past repair/recovery instances. The proposed repair framework applies to tail assignment, and is sufficiently generic to apply to any operations management problem formulated as a compact integer linear program.

Complexity of Energy Efficient Localization With the Aid of a Mobile Beacon

Localization is an essential service in wireless sensor networks. Trilateration is a commonly used solution to range-based localization for providing such services. It might be, however, impossible to localize the entire network at once using trilateration due to low connectivity on sparse deployments. In such scenarios, a mobile beacon with a known position is used to move among and locate the nodes with low connectivity to aid trilateration. Given a network graph, finding a minimum energy route traveled by the mobile beacon is a key problem in many real-world applications. We prove in this letter that this problem called mobile assisted trilateration based energy optimum localization is NP-hard. To the best of our knowledge, this is the first such result in an attempt to computationally classify this important problem. We also provide a compact integer linear programming formulation for the problem.

A Branch-and-Cut Algorithm for the Multidepot Rural Postman Problem

This paper considers the Multidepot Rural Postman Problem, an extension of the classical Rural Postman Problem in which there are several depots instead of only one. The aim is to construct a minimum cost set of routes traversing each required edge of the graph, where each route starts and ends at the same depot. The paper makes the following scientific contributions: (i) It presents optimality conditions and a worst case analysis for the problem; (ii) It proposes a compact integer linear programming formulation containing only binary variables, as well as a polyhedral analysis; (iii) It develops a branch-and-cut algorithm that includes several new exact and heuristic separation procedures. Instances involving up to four depots, 744 vertices, and 1,315 edges are solved to optimality. These instances contain up to 140 required components and 1,000 required edges.

A reactive container rerouting model for container flow recovery in a hub-and-spoke liner shipping network

ABSTRACTThis paper focuses on the container rerouting due to a disruption, aims at making the optimal container flow recovery plan for the affected liner shipping company. First, we make the initial effort to bring up with a basic framework of disruption management for liner shipping. Second, we present a compact integer linear programming model for addressing the container rerouting problem under the proposed framework in a hub-and-spoke liner shipping network, based on a given recovery vessel schedule that determines to omit a port of call. Other shipping companies’ services and other modes (roadway, railway, and airline) as candidate alternative means to transport the miss-connected containers are also incorporated in the proposed model. The container flow recovery plan would select the optimal alternative paths for the miss-connected containers balancing the trade-off between container transport costs and delivery delay penalty costs. Finally, a case study from a global liner shipping company is investigated and the computational results indicate the model can be solved effectively and efficiently for the real-scale problem. Thus, the proposed approach in this paper can supply real-time decision support tool for the liner shipping operators on handling the process of container flow recovery.

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.

On Computing Breakpoint Distances for Genomes with Duplicate Genes.

A fundamental problem in comparative genomics is to compute the distance between two genomes in terms of its higher level organization (given by genes or syntenic blocks). For two genomes without duplicate genes, we can easily define (and almost always efficiently compute) a variety of distance measures, but the problem is NP-hard under most models when genomes contain duplicate genes. To tackle duplicate genes, three formulations (exemplar, maximum matching, and any matching) have been proposed, all of which aim to build a matching between homologous genes so as to minimize some distance measure. Of the many distance measures, the breakpoint distance (the number of nonconserved adjacencies) was the first one to be studied and remains of significant interest because of its simplicity and model-free property. The three breakpoint distance problems corresponding to the three formulations have been widely studied. Although we provided last year a solution for the exemplar problem that runs very fast on full genomes, computing optimal solutions for the other two problems has remained challenging. In this article, we describe very fast, exact algorithms for these two problems. Our algorithms rely on a compact integer-linear program that we further simplify by developing an algorithm to remove variables, based on new results on the structure of adjacencies and matchings. Through extensive experiments using both simulations and biological data sets, we show that our algorithms run very fast (in seconds) on mammalian genomes and scale well beyond. We also apply these algorithms (as well as the classic orthology tool MSOAR) to create orthology assignment, then compare their quality in terms of both accuracy and coverage. We find that our algorithm for the "any matching" formulation significantly outperforms other methods in terms of accuracy while achieving nearly maximum coverage.