scispace - formally typeset
Search or ask a question

Showing papers on "Integer programming published in 2010"


Book ChapterDOI
01 Jan 2010
TL;DR: It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8].
Abstract: It is a pleasure to write this commentary because it offers an opportunity to express my gratitude to several people who helped me in ways that turned out to be essential to the birth of [8]. They also had a good deal to do with shaping my early career and, consequently, much of what followed.

1,101 citations


Journal ArticleDOI
TL;DR: In this article, the optimal operation of a wind turbine, a solar unit, a fuel cell and a storage battery is searched by a mixed-integer linear programming implemented in General Algebraic Modeling Systems (GAMS).

497 citations


Journal ArticleDOI
TL;DR: A bi-objective possibilistic mixed integer programming model to deal with closed-loop supply chain network design problems and an interactive fuzzy solution approach is developed by combining a number of efficient solution approaches from the recent literature.

450 citations


Journal ArticleDOI
TL;DR: A compact mixed integer program (MIP) formulation and a continuum approximation (CA) model are proposed to study the reliable uncapacitated fixed charge location problem (RUFL) which seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios.
Abstract: Reliable facility location models consider unexpected failures with site-dependent probabilities, as well as possible customer reassignment. This paper proposes a compact mixed integer program (MIP) formulation and a continuum approximation (CA) model to study the reliable uncapacitated fixed charge location problem (RUFL), which seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios. The MIP determines the optimal facility locations as well as the optimal customer assignments and is solved using a custom-designed Lagrangian relaxation (LR) algorithm. The CA model predicts the total system cost without details about facility locations and customer assignments, and it provides a fast heuristic to find near-optimum solutions. Our computational results show that the LR algorithm is efficient for mid-sized RUFL problems and that the CA solutions are close to optimal in most of the test instances. For large-scale problems, the CA method is a good alternative to the LR algorithm that avoids prohibitively long running times.

431 citations


Journal ArticleDOI
TL;DR: This paper proposes a model for integrated logistics network design to avoid the sub-optimality caused by a separate, sequential design of forward and reverse logistics networks, and develops a bi-objective mixed integer programming formulation to minimize the total costs and maximize the responsiveness of a logistics network.

429 citations


Book
01 Jan 2010
TL;DR: In this paper, the Hungarian method for the assignment problem was used to solve the traveling salesman problem and a group-theoretic approach in mixed integer linear programming was proposed for solving the problem.
Abstract: I The Early Years.- Solution of a Large-Scale Traveling-Salesman Problem.- The Hungarian Method for the Assignment Problem.- Integral Boundary Points of Convex Polyhedra.- Outline of an Algorithm for Integer Solutions to Linear Programs An Algorithm for the Mixed Integer Problem.- An Automatic Method for Solving Discrete Programming Problems.- Integer Programming: Methods, Uses, Computation.- Matroid Partition.- Reducibility Among Combinatorial Problems.- Lagrangian Relaxation for Integer Programming.- Disjunctive Programming.- II From the Beginnings to the State-of-the-Art.- Polyhedral Approaches to Mixed Integer Linear Programming.- Fifty-Plus Years of Combinatorial Integer Programming.- Reformulation and Decomposition of Integer Programs.- III Current Topics.- Integer Programming and Algorithmic Geometry of Numbers.- Nonlinear Integer Programming.- Mixed Integer Programming Computation.- Symmetry in Integer Linear Programming.- Semidefinite Relaxations for Integer Programming.- The Group-Theoretic Approach in Mixed Integer Programming.

412 citations


Journal ArticleDOI
TL;DR: The DPFSP is characterized and six different alternative mixed integer linear programming (MILP) models that are carefully and statistically analyzed for performance are proposed.

353 citations


Book
12 Jan 2010
TL;DR: A review of Linear Programming Fundamentals and Network Optimization Problems and Solutions, with a focus on modeling Combinatorial Optimization problems I.
Abstract: PREFACE. PART I MODELING. 1 Introduction. 1.1 Integer Programming. 1.2 Standard Versus Nonstandard Forms. 1.3 Combinatorial Optimization Problems. 1.4 Successful Integer Programming Applications. 1.5 Text Organization and Chapter Preview. 1.6 Notes. 1.7 Exercises. 2 Modeling and Models. 2.1 Assumptions on Mixed Integer Programs. 2.2 Modeling Process. 2.3 Project Selection Problems. 2.4 Production Planning Problems. 2.5 Workforce/Staff Scheduling Problems. 2.6 Fixed-Charge Transportation and Distribution Problems. 2.7 Multicommodity Network Flow Problem. 2.8 Network Optimization Problems with Side Constraints. 2.9 Supply Chain Planning Problems. 2.10 Notes. 2.11 Exercises. 3 Transformation Using 0 1 Variables. 3.1 Transform Logical (Boolean) Expressions. 3.2 Transform Nonbinary to 0 1 Variable. 3.3 Transform Piecewise Linear Functions. 3.4 Transform 0 1 Polynomial Functions. 3.5 Transform Functions with Products of Binary and Continuous Variables: Bundle Pricing Problem. 3.6 Transform Nonsimultaneous Constraints. 3.7 Notes. 3.8 Exercises. 4 Better Formulation by Preprocessing. 4.1 Better Formulation. 4.2 Automatic Problem Preprocessing. 4.3 Tightening Bounds on Variables. 4.4 Preprocessing Pure 0 1 Integer Programs. 4.5 Decomposing a Problem into Independent Subproblems. 4.6 Scaling the Coefficient Matrix. 4.7 Notes. 4.8 Exercises. 5 Modeling Combinatorial Optimization Problems I. 5.1 Introduction. 5.2 Set Covering and Set Partitioning. 5.3 Matching Problem. 5.4 Cutting Stock Problem. 5.5 Comparisons for Above Problems. 5.6 Computational Complexity of COP. 5.7 Notes. 5.8 Exercises. 6 Modeling Combinatorial Optimization Problems II. 6.1 Importance of Traveling Salesman Problem. 6.2 Transformations to Traveling Salesman Problem. 6.3 Applications of TSP. 6.4 Formulating Asymmetric TSP. 6.5 Formulating Symmetric TSP. 6.6 Notes. 6.7 Exercises. PART II REVIEW OF LINEAR PROGRAMMING AND NETWORK FLOWS. 7 Linear Programming Fundamentals. 7.1 Review of Basic Linear Algebra. 7.2 Uses of Elementary Row Operations. 7.3 The Dual Linear Program. 7.4 Relationships Between Primal and Dual Solutions. 7.5 Notes. 7.6 Exercises. 8 Linear Programming: Geometric Concepts. 8.1 Geometric Solution. 8.2 Convex Sets. 8.3 Describing a Bounded Polyhedron. 8.4 Describing Unbounded Polyhedron. 8.5 Faces, Facets, and Dimension of a Polyhedron. 8.6 Describing a Polyhedron by Facets. 8.7 Correspondence Between Algebraic and Geometric Terms. 8.8 Notes. 8.9 Exercises. 9 Linear Programming: Solution Methods. 9.1 Linear Programs in Canonical Form. 9.2 Basic Feasible Solutions and Reduced Costs. 9.3 The Simplex Method. 9.4 Interpreting the Simplex Tableau. 9.5 Geometric Interpretation of the Simplex Method. 9.6 The Simplex Method for Upper Bounded Variables. 9.7 The Dual Simplex Method. 9.8 The Revised Simplex Method. 9.9 Notes. 9.10 Exercises. 10 Network Optimization Problems and Solutions. 10.1 Network Fundamentals. 10.2 A Class of Easy Network Problems. 10.3 Totally Unimodular Matrices. 10.4 The Network Simplex Method. 10.5 Solution via LINGO. 10.6 Notes. 10.7 Exercises. PART III SOLUTIONS. 11 Classical Solution Approaches. 11.1 Branch-and-Bound Approach. 11.2 Cutting Plane Approach. 11.3 Group Theoretic Approach. 11.4 Geometric Concepts. 11.5 Notes. 11.6 Exercises. 12 Branch-and-Cut Approach. 12.1 Introduction. 12.2 Valid Inequalities. 12.3 Cut Generating Techniques. 12.4 Cuts Generated from Sets Involving Pure Integer Variables. 12.5 Cuts Generated from Sets Involving Mixed Integer Variables. 12.6 Cuts Generated from 0 1 Knapsack Sets. 12.7 Cuts Generated from Sets Containing 0 1 Coefficients and 0 1 Variables. 12.8 Cuts Generated from Sets with Special Structures. 12.9 Notes. 12.10 Exercises. 13 Branch-and-Price Approach. 13.1 Concepts of Branch-and-Price. 13.2 Dantzig Wolfe Decomposition. 13.3 Generalized Assignment Problem. 13.4 GAP Example. 13.5 Other Application Areas. 13.6 Notes. 13.7 Exercises. 14 Solution via Heuristics, Relaxations, and Partitioning. 14.1 Introduction. 14.2 Overall Solution Strategy. 14.3 Primal Solution via Heuristics. 14.4 Dual Solution via Relaxation. 14.5 Lagrangian Dual. 14.6 Primal Dual Solution via Benders Partitioning. 14.7 Notes. 14.8 Exercises. 15 Solutions with Commercial Software. 15.1 Introduction. 15.2 Typical IP Software Components. 15.3 The AMPL Modeling Language. 15.4 LINGO Modeling Language. 15.5 MPL Modeling Language. REFERENCES. APPENDIX: ANSWERS TO SELECTED EXERCISES. INDEX.

292 citations


Book ChapterDOI
01 Dec 2010
TL;DR: This chapter is a study of a simple version of general nonlinear integer problems, where all constraints are still linear, and focuses on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure.
Abstract: Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

278 citations


Journal ArticleDOI
TL;DR: A generalized closed-loop model for the logistics planning was proposed by formulating a cyclic logistics network problem into an integer linear programming model and a Genetic Algorithm, which is based on spanning tree structure was developed.

274 citations


Book ChapterDOI
01 Jan 2010
TL;DR: This paper reviews techniques developed to take advantage of the symmetry in an ILP during its solution, and surveys related topics, such as symmetry detection, polyhedral studies of symmetric ILPs, and enumeration of all non isomorphic optimal solutions.
Abstract: An integer linear program (ILP) is symmetric if its variables can be permuted without changing the structure of the problem. Areas where symmetric ILPs arise range from applied settings (scheduling on identical machines), to combinatorics (code construction), and to statistics (statistical designs construction). Relatively small symmetric ILPs are extremely difficult to solve using branch-and-cut codes oblivious to the symmetry in the problem. This paper reviews techniques developed to take advantage of the symmetry in an ILP during its solution. It also surveys related topics, such as symmetry detection, polyhedral studies of symmetric ILPs, and enumeration of all non isomorphic optimal solutions.

Journal ArticleDOI
TL;DR: In this paper, a mixed-integer linear programming model (MILP-1) is developed for FJSPs and compared to an alternative model in the literature (Model F) in terms of computational efficiency.

Proceedings ArticleDOI
25 Jul 2010
TL;DR: In this paper, the authors analyze the N-1 reliable DC optimal dispatch with transmission switching and demonstrate that these networks can be operated to satisfy N 1 standards while cutting costs by incorporating transmission switching into the dispatch.
Abstract: In this paper, we analyze the N-1 reliable DC optimal dispatch with transmission switching. The model is a mixed integer program (MIP) with binary variables representing the state of the transmission element (line or transformer) and the model can be used for planning and/or operations. We then attempt to find solutions to this problem using the IEEE 118-bus and the RTS 96 system test cases. The IEEE 118-bus test case is analyzed at varying load levels. Using simple heuristics, we demonstrate that these networks can be operated to satisfy N-1 standards while cutting costs by incorporating transmission switching into the dispatch. In some cases, the percent savings from transmission switching was higher with an N-1 DCOPF formulation than with a DCOPF formulation.

Journal ArticleDOI
TL;DR: In this paper, a new discrete firefly meta-heuristic was proposed to minimize the makespan for the permutation flow shop scheduling problem, and the results of implementation of the proposed method are compared with other existing ant colony optimization technique.
Abstract: Article history: Received 23 January 2010 Received in revised form 23 April 2010 Accepted 26 April 2010 Available online 26 April 2010 During the past two decades, there have been increasing interests on permutation flow shop with different types of objective functions such as minimizing the makespan, the weighted mean flow-time etc. The permutation flow shop is formulated as a mixed integer programming and it is classified as NP-Hard problem. Therefore, a direct solution is not available and metaheuristic approaches need to be used to find the near-optimal solutions. In this paper, we present a new discrete firefly meta-heuristic to minimize the makespan for the permutation flow shop scheduling problem. The results of implementation of the proposed method are compared with other existing ant colony optimization technique. The preliminary results indicate that the new proposed method performs better than the ant colony for some well known benchmark problems. © 2010 Growing Science Ltd. All rights reserved.

Journal ArticleDOI
TL;DR: The paper shows the application of the proposed approach to a medium-voltage 120 buses network with five wind plants, one photovoltaic field, ten dispatchable generators, and two transformers equipped with on-load tap changers.
Abstract: Among the innovative contributions to electric distribution systems, one of the most promising and qualified is the possibility to manage and control distributed generation. Therefore, the latest distribution management systems tend to incorporate optimization functions for the short-term scheduling of the various energy and control resources available in the network (e.g., embedded generators, reactive power compensators and transformers equipped with on-load tap changers). The short-term scheduling procedure adopted in the paper is composed by two stages: a day-ahead scheduler for the optimization of distributed resources production during the following day, an intra-day scheduler that every 15 min adjusts the scheduling in order to take into account the operation requirements and constraints of the distribution network. The intra-day scheduler solves a non-linear multi-objective optimization problem by iteratively applying a mixed-integer linear programming (MILP) algorithm. The linearization of the optimization function and the constraints is achieved by the use of sensitivity coefficients obtained from the results of a three-phase power flow calculation. The paper shows the application of the proposed approach to a medium-voltage 120 buses network with five wind plants, one photovoltaic field, ten dispatchable generators, and two transformers equipped with on-load tap changers.

Journal ArticleDOI
TL;DR: A strict formulation of a generalization of the classical pickup and delivery problem is presented, and it is concluded that there exist some configurations in which a scheme allowing transfers results in better quality optimal solutions.

Journal ArticleDOI
TL;DR: A new approach based on the interval analysis is introduced to solve the DOCRs coordination problem considering uncertainty in the network topology, and the application of the proposed method to the IEEE 14- and 30-bus test systems proves the ability of the interval method in modeling topology uncertainty inThe large-scale coordination problem.
Abstract: In real power systems, the network topology is subjected to uncertainty due to single-line outage contingencies, maintenance activities, and network reconfigurations. These changes in the network topology may lead to miscoordination of directional overcurrent relays (DOCRs). To overcome this drawback, corresponding to each primary/backup relay pair, a set of inequality coordination constraints which is related to different network topologies should be satisfied. In this paper, a new approach based on the interval analysis is introduced to solve the DOCRs coordination problem considering uncertainty in the network topology. The basic idea is to convert the set of inequality constraints corresponding to each relay pair to an interval constraint. In this situation, the DOCR coordination problem is formulated as an interval linear programming (ILP) problem. Using well-known mathematical theorems, the obtained ILP problem, which has no equality constraints, can be converted to standard linear programming (LP). As a result, the number of coordination constraints is significantly reduced in the proposed methods. The application of the proposed method to the IEEE 14- and 30-bus test systems proves the ability of the interval method in modeling topology uncertainty in the large-scale coordination problem.

BookDOI
01 Jan 2010
TL;DR: This book discusses Integer Programming and Algorithmic Geometry of Numbers, specifically the group-Theoretic approach in Mixed Integer Programming.

Journal ArticleDOI
TL;DR: The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run times than do previously described approaches.
Abstract: We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different integer linear programming (ILP)-based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP relaxation and the original problem and then introduce a new core concept for the multidimensional knapsack problem (MKP), which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run times are also performed to compare the solutions of our approaches to the best-known solutions of another so-far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run times than do previously described approaches.

Journal ArticleDOI
TL;DR: In this article, the authors formulated the road network design problem as a single-level optimization problem with equilibrium constraints, and then they transformed the equilibrium constraints into a set of mixed-integer constraints and linearized the travel time function.
Abstract: The road network design problem, typically formulated as a bi-level program or a mathematical program with equilibrium constraints, is generally non-convex. The non-convexity stems from both the traffic assignment equilibrium conditions and the non-linear travel time function. In this study, we formulate the network design problem as a single-level optimization problem with equilibrium constraints, and then we transform the equilibrium constraints into a set of mixed-integer constraints and linearize the travel time function. The final result is that we cast the network design problem with equilibrium flows into a mixed-integer linear program, whose solution possesses the desirable property of global optimality, subject to the resolution of the linearization scheme adopted.

Journal ArticleDOI
TL;DR: This article studies the facility reliability problem: how to design a reliable supply chain network in the presence of random facility disruptions with the option of hardening selected facilities, and develops a Lagrangian Relaxation‐based solution algorithm.
Abstract: Having a robustly designed supply chain network is one of the most effective ways to hedge against network disruptions because contingency plans in the event of a disruption are often significantly limited. In this article, we study the facility reliability problem: how to design a reliable supply chain network in the presence of random facility disruptions with the option of hardening selected facilities. We consider a facility location problem incorporating two types of facilities, one that is unreliable and another that is reliable (which is not subject to disruption, but is more expensive). We formulate this as a mixed integer programming model and develop a Lagrangian Relaxation-based solution algorithm. We derive structural properties of the problem and show that for some values of the disruption probability, the problem reduces to the classical uncapacitated fixed charge location problem. In addition, we show that the proposed solution algorithm is not only capable of solving large-scale problems, but is also computationally effective. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010

Proceedings Article
21 Jun 2010
TL;DR: Comprehensive experimental results show that the proposed method can obtain better or competitive performance compared with existing SVM-based feature selection methods in term of sparsity and generalization performance, and can effectively handle large-scale and extremely high dimensional problems.
Abstract: A sparse representation of Support Vector Machines (SVMs) with respect to input features is desirable for many applications. In this paper, by introducing a 0-1 control variable to each input feature, l0-norm Sparse SVM (SSVM) is converted to a mixed integer programming (MIP) problem. Rather than directly solving this MIP, we propose an efficient cutting plane algorithm combining with multiple kernel learning to solve its convex relaxation. A global convergence proof for our method is also presented. Comprehensive experimental results on one synthetic and 10 real world datasets show that our proposed method can obtain better or competitive performance compared with existing SVM-based feature selection methods in term of sparsity and generalization performance. Moreover, our proposed method can effectively handle large-scale and extremely high dimensional problems.

Journal ArticleDOI
TL;DR: A mixed integer programming formulation is proposed for hub- and-spoke network design in a competitive environment that addresses the competition between a newcomer liner service provider and an existing dominating operator, both operating on hub-and-spokes networks.
Abstract: A mixed integer programming formulation is proposed for hub-and-spoke network design in a competitive environment. It addresses the competition between a newcomer liner service provider and an existing dominating operator, both operating on hub-and-spoke networks. The newcomer company maximizes its market share—which depends on the service time and transportation cost—by locating a predefined number of hubs at candidate ports and designing its network. While general-purpose solvers do not solve instances of even small size, an accelerated Lagrangian method combined with a primal heuristic obtains promising bounds. Our computational experiments on real instances of practical size indicate superiority of our approach.

Journal ArticleDOI
TL;DR: A heuristic method-based on the GRASP and path relinking methodologies-for finding approximate solutions to the max-min diversity problem and results indicate that the proposed hybrid implementations compare favorably to previous metaheuristics, such as tabu search and simulated annealing.

Book ChapterDOI
14 Jun 2010
TL;DR: This work studies the application of an automated algorithm configuration procedure to different MIP solvers, instance types and optimization objectives, and shows that this fully-automated process yields substantial improvements to the performance of three MIPsolvers: Cplex, Gurobi, and lpsolve.
Abstract: State-of-the-art solvers for mixed integer programming (MIP) problems are highly parameterized, and finding parameter settings that achieve high performance for specific types of MIP instances is challenging. We study the application of an automated algorithm configuration procedure to different MIP solvers, instance types and optimization objectives. We show that this fully-automated process yields substantial improvements to the performance of three MIP solvers: Cplex, Gurobi, and lpsolve. Although our method can be used “out of the box” without any domain knowledge specific to MIP, we show that it outperforms the Cplex special-purpose automated tuning tool.

Journal ArticleDOI
TL;DR: The problem is formulated as a mixed integer programming model that gives the departure times of vehicles in lines so that passengers can transfer between lines at transfer stations with minimum waiting times and a genetic algorithm approach is represented to more easily solve larger networks.
Abstract: This paper studies the transit network scheduling problem and aims to minimize the waiting time at transfer stations. First, the problem is formulated as a mixed integer programming model that gives the departure times of vehicles in lines so that passengers can transfer between lines at transfer stations with minimum waiting times. Then, the model is expanded to a second model by considering the extra stopping time of vehicles at transfer stations as a new variable set. By calculating the optimal values for these variables, transfers can be better performed. The sizes of the models, compared with the existing models, are small enough that the models can be solved for small- and medium-sized networks using regular MIP solvers, such as CPLEX. Moreover, a genetic algorithm approach is represented to more easily solve larger networks. A simple network is used to describe the models, and a medium-sized, real-life network is used to compare the proposed models with another existing model in the literature. The results demonstrate significant improvement. Finally, a large-scale, real-life network is used as a case study to evaluate the proposed models and the genetic algorithm approach.

Journal ArticleDOI
TL;DR: This paper presents a hybrid multi-objective model that combines integer programming (IP) and variable neighbourhood search (VNS) to deal with highly-constrained nurse rostering problems in modern hospital environments and believes it can be applied to other resource allocation problems with a large number of constraints.

Journal ArticleDOI
TL;DR: Two easy-to-implement methods for the piecewise linear approximation of functions of two variables and a detailed description of how the methods can be embedded in a MILP model are considered.

Book ChapterDOI
01 Jan 2010
TL;DR: The first 50 years of Integer and Mixed-Integer Programming have taken us to a very stable paradigm for solving problems in a reliable and effective way, but a lot of work must still be done for improving the solvers and extending their modeling capability.
Abstract: The first 50 years of Integer and Mixed-Integer Programming have taken us to a very stable paradigm for solving problems in a reliable and effective way. We run over these 50 exciting years by showing some crucial milestones and we highlight the building blocks that are making nowadays solvers effective from both a performance and an application viewpoint. Finally, we show that a lot of work must still be done for improving the solvers and extending their modeling capability.

Journal ArticleDOI
TL;DR: This paper presents a mathematical formulation in order to solve a Stackelberg game for a network-constrained energy market using integer programming and reformulated as mixed-integer linear program (MILP) by using disjunctive constraints and linearization.