Showing papers on "Integer programming published in 1980"

Pivot and Complement–A Heuristic for 0-1 Programming

Abstract: Pivot and Complement is a heuristic for finding approximate solutions to 0-1 programming problems. It uses the fact that a 0-1 program is equivalent to the associated linear program with the added requirement that all slack variables, other than those in the upper bounding constraints, be basic. The procedure starts by solving the linear program; then performs a sequence of pivots aimed at putting all slacks into the basis at a minimal cost; finally, it attempts to improve the 0-1 solution obtained in this way by a local search based on complementing certain sets of 0-1 variables. The computational effort involved in the procedure is bounded by a polynomial in the number of constraints and variables. For the 92 test problems with 5 to 200 constraints and 20 to 900 variables on which the procedure was run, the time used to solve the linear program on the average considerably exceeded the time used for everything else. As to the quality of the solutions obtained, in 45 cases, or 49% of the total, the proced...

An integer linear programming approach to the steiner problem in graphs

Abstract: Consider a connected undirected graph G[N; E] with N = S ∪ P, the set of nodes, where P is designated as the set of Steiner points. A weight is associated with each edge ei of the set E. The problem of obtaining a minimal weighted tree which spans the set S of nodes has been termed in literature as the Steiner problem in graphs. A specialized integer programming (set covering) formulation is presented for the problem. The number of constraints in this formulation grows exponentially with the size of the problem. A method called the row generation scheme is developed to solve the above problem. The method requires knowing the constraints only implicitly. Several other problems which can be put in a similar framework can also be handled by the above scheme. The generality of the scheme and its efficiency is discussed. Finally the computational result is demonstrated.

A Cutting Planes Algorithm for the m-Salesmen Problem

Abstract: The existence of reliable and flexible FORTRAN programs for integer linear programming has recently enabled the development of very efficient algorithms for the travelling salesman problem. The main characteristic of these algorithms is the relaxation of most of the constraints of the problem during its solution. The same approach can be used for the solution of the m-salesmen problem in which m salesmen starting from the same city must visit only once n cities at minimum cost. The number of salesmen can be fixed in advance or allowed to vary, upper and lower bounds set on the number of salesmen and even fixed costs associated with the salesmen. The results obtained so far are very encouraging. Problems of up to 100 cities have been solved optimally for the m-travelling salesmen case and other more complex problems are currently under study.

An Implicit Enumeration Algorithm for Quadratic Integer Programming

Abstract: We present an implicit enumeration algorithm for a nonseparable quadratic integer programming problem. We utilize fathoming criteria derived from Lemke's complementary pivot algorithm, and compare the use of pseudo-costs versus generalized penalties as a guide to branching. Computational experience is provided.

Integer Programming Models for Sales Resource Allocation

Abstract: A practical conceptual framework for sales resource allocation modeling is presented in this paper. A literature review of sales resource allocation models is described in terms of this framework. The conceptual framework also lends itself to several integer programming models which may be used to address the variety of sales resource allocation decisions faced by every sales organization. A general model for sales resource allocation is developed which incorporates multiple sales resources, multiple time periods and carryover effects, non-separability, and risk. Several actual model implementations are discussed which illustrate the practical application of the integer programming models. The model implementations utilize recent advances in integer programming theory which enables sales managers and sales representatives to quickly develop and evaluate alternative sales resource allocation strategies.

A Dual-Bounded Algorithm for the p-Median Problem

Abstract: The p-median problem consists of locating p facilities on a network, so that the sum of shortest distances from each of the nodes of the network to its nearest facility is minimized. A dual bound, based on the dual of the LP relaxation of the integer programming formulation of the problem, is developed and tested in a branch-and-bound algorithm. Computational results show that the resulting solution procedure has some advantages over existing exact methods for this problem.

Two stage linear programming under uncertainty with 0–1 integer first stage variables

Abstract: Stochastic programs with continuous variables are often solved using a cutting plane method similar to Benders' partitioning algorithm. However, mixed 0–1 integer programs are also solved using a similar procedure along with enumeration. This similarity is exploited in this paper to solve two stage linear programs under uncertainty where the first stage variables are 0–1. Such problems often arise in capital investment. A network investment application is given which includes as a special case a coal transportation problem.

Optimal Design of Distributed Information Systems

Abstract: In this paper a model is developed for the optimization of distributed information systems. Compared with the previous work in this area, the model is more complete, since it considers simultaneously the distribution of processing power, the allocation of programs and databases, and the assignment of communication line capacities. It also considers the return flow of information, as well as the dependencies between programs and databases. In addition, an algorithm, based on the "bounded branch and bound" integer programming technique, has been developed to obtain the optimal solution of the model. The algorithm is more efficient than several existing general nonlinear integer programming algorithms. Also, it avoids some of the disadvantages of heuristic and decomposition algorithms which are used widely in the optimization of computer networks and distributed databases. The algorithm has been implemented in Fortran, and the computation times of the algorithm for several test problems have been found very reasonable.

Topological optimization of networks: A nonlinear mixed integer model employing generalized Benders decomposition

Abstract: A class of network topological optimization problems is formulated as a nonlinear mixed integer programming model, which can be used to design transportation and computer communication networks subject to a budget constraint. The approach proposed for selecting an optimal network consists in separating the continuous part of the model from the discrete part by Generalized Benders' Decomposition. One then solves a sequence of master and subproblems. The subproblems of the minimal convex cost multicommodity flow type are used to generate cutting planes for choosing potential topologies by means of the master problems. Computational techniques suited to solving the master and subproblems are suggested, and very encouraging experimental results are reported.

A Polynomial Algorithm for the Two-Variable Integer Programming Problem

Abstract: A polynomial time algorithm is presented for solving the following two-variable integer programming problem maximize ClXl + c2x2 subject to a, lxl + a,2x2 = O, integers, where a,j, cj, and b, are assumed to be nonnegattve integers This generahzes a result of Htrschberg and Wong, who developed a polynomial algorithm for the same problem with only one constraint (l e, where n = 1) However, the techniques used here are quite different KEY WORDS AND PHRASES integer programming, knapsack problem, polynomial algorithm, coefficient size, feasible region decomposition CR CATEGORIES 3 15, 5 25, 5 30, 5 40 Introduction We consider the following integer programming problem: (1) maximize clxa + czx2 subject to aaxl + a,2xz ~ b,, a= 1,2 .... n, and xl, x2 => 0, integers, where au, G, and b, are assumed to be nonnegative integers. We first show that the solution to (1) can be obtained easily from the solutions to at most n problems, each of which is of the form (2) maximize

The Linear Multiple Choice Knapsack Problem

Abstract: A fast algorithm is presented for the linear programming relaxation of the Multiple Choice Knapsack Problem. If N is the total number of variables and J and Jmax denote the total number of multiple choice variables and the cardinality of the largest multiple choice set, respectively, the running time of the algorithm is then bounded by 0J log Jmax + 0N. Under certain conditions it is possible to reduce this bound to 0N steps on the average. Possible further improvements are also discussed.

A Lagrangean Relaxation Algorithm for the Two Duty Period Scheduling Problem

Abstract: The two duty period scheduling problem is an integer programming problem with 0-1 constraint coefficients. It is recognized that the problem can be reformulated as a one duty period problem with side constraints. Since the one duty period problem can be solved as a minimal cost network flow problem, we dualize with respect to the side constraints, forming a Lagrangean relaxation which is easily solved. Subgradient optimization is used to maximize the Lagrangean. Computational results are reported for several large problems.

Fractional vertices, cuts and facets of the simple plant location problem

Abstract: This paper investigates the structure of the Integer Programming Polytope of an uncapacitated (simple) plant location problem. One can describe families of fractional vertices and derive from them valid inequalities for the integer problem. Some of these will actually be shown to be facets of the integer polytope. Also some families of SPLP with large duality gaps will be described, together with facets which bridge these gaps. Much of the motivation stems from algorithmic work in which the exploitation of “good” cutting planes within a direct dual algorithm have been shown to be of crucial importance.

A Mixed-Integer Programming Approach to Air Cargo Fleet Planning

Abstract: This paper deals with the mathematical programming aspects of a long range planning study done for the Flying Tiger Line, an all-cargo airline. The study addressed two strategic problems: the design of the service network and the selection and deployment of the aircraft fleet. We show how the concept of a spider graph provides a natural building block for network design and present a mixed-integer programming model that enables the planner to evaluate any network constructed from spider graphs by determining the most profitable selection of aircraft and routing of cargo.

Branch and bound experiments in nonlinear integer programming

Abstract: The branch and bound principle has been established as an effective computational tool for solving linear integer programming problems. In the present dissertation the feasibility of branch and bound procedures in solving nonlinear integer programming problems is investigated. Since the implementation of the branch and bound requires solving of a number of nonlinear continuous problems, the algorithms available for solving nonlinear continuous problems and their corresponding computer codes are first investigated and an efficient code (OPT--based on generalized reduced gradient method) is selected. The efficiency of the branch and bound technique generally depends on the various selection parameters such as selection of branching variables, and branching nodes. Among others, the concepts of 'pseudo-costs' and 'estimations' are implemented in selecting the branching variables and the branching nodes. The efficiency of the algorithm also depends on the availibility of a quick upper bound on the objective minimum. Therefore, heuristics are developed which would first locate an integer feasible solution and thus provide an upper bound on the objective. The different rules for selecting the branching variables, nodes and heuristics form a total of 27 branch and bound strategies. A computer code capable of invoking each of these 27 strategies is developed. Since the generation of subproblems requires storing of a number of nodes, an effective computer storage scheme is developed for the Intermediate storage purpose. Each of the 27 strategies is tested on a set of 22 test problems. The numerical results are provided, and a statistical analysis is performed to evaluate and compare the overall effectiveness of these strategies. Additional experiments are also carried out to evaluate the effect due to the problem parameters such as number of integer variables and constraints.

Integer programming : facets, subadditivity, and duality for group and semi-group problems

Abstract: Integer Programming Cuts, Knapsacks, and a Cyclic Group Problem Finite Abelian Groups Gomory's Corner Polyhedra Blocking Polyhedra and Master Group Problems Araoz's Semigroup Problem Blockers and Polars for Master Semigroup Problems Subadditive and Minimal Valid Inequalities Subadditive Characterizations Duality.

Technical Note—Equivalence of the 0-1 Integer Programming Problem to Discrete Generalized and Pure Networks

Abstract: We show how to formulate a 0-1 integer programming problem as a “mixed integer” generalized network and as a discrete “0-U” pure network problem. Special integer programming structures allow convenient simplifications. The usefulness of these formulations is in providing new relaxations for integer programming that can take advantage of recent advances in the development of efficient computer programs for network problems. We cite three practical applications in which these ideas have led to marked improvement in solution efficiency.

An Interactive Branch and Bound Procedure for Multicriterion Integer Linear Programming

Abstract: An interactive Branch and Bound Procedure for solving multicriteria integer linear programming problems is suggested. This scheme is a natural extension of branch and bound methodology to the multicriteria framework, and is based on a LIPO branch and bound strategy. Initial results of its computational performance are offered.

Finding All Solutions for a Class of Parametric Quadratic Integer Programming Problems

Abstract: We describe a practical procedure for finding all solutions to a parametric family of nonseparable quadratic integer programs that differ in their resource availabilities. We outline a new method for optimizing these quadratic integer programs, and demonstrate how to solve a sequence of such problems parametrized against the right-hand-side of a single constraint. Several methods for accelerating the basic procedure are presented, and computational experience is provided.

The use of dynamic programming methodology for the solution of a class of nonlinear programming problems

Abstract: This paper presents an application of a method for finding the global solution to a problem in integers with a separable objective function of a very general form. This report shows that there is a relationship between an integer problem with a separable nonlinear objective function and many constraints and a series of nonlinear problems with only a single constraint, each of which can be solved sequentially using dynamic programming. The first solution to any of the individual smaller problems that satisfies the original constraints in addition, will be the optimal solution to the multiply-constrained problem.

Parametric Integer Programming Analysis: A Contraction Approach

Abstract: An algorithm is presented for solving families of integer linear programming problems in which the problems are "related" by having identical objective coefficients and constraint matrix coefficients. The righthand-side constants have the form b + θd where b and d are conformable vectors and θ varies from zero to one.

Automatic Identification of Generalized Upper Bounds in Large-Scale Optimization Models

Abstract: To solve contemporary large-scale linear, integer and mixed integer programming problems, it is often necessary to exploit intrinsic special structure in the model at hand. One commonly used technique is to identify and then to exploit in a basis factorization algorithm a generalized upper bound GUB structure. This report compares several existing methods for identifying GUB structure. Computer programs have been written to permit comparison of computational efficiency. The GUB programs have been incorporated in an existing optimization system of advanced design and have been tested on a variety of large-scale real-life optimization problems. The identification of GUB sets of maximum size is shown to be among the class of NP-complete problems; these problems are widely conjectured to be intractable in that no polynomial-time algorithm has been demonstrated for solving them. All the methods discussed in this report are polynomial-time heuristic algorithms that attempt to find, but do not guarantee, GUB sets of maximum size. Bounds for the maximum size of GUB sets are developed in order to evaluate the effectiveness of the heuristic algorithms.

Rational Mixed-Integer and Polyhedral Union Minimization Models

Abstract: The minimization model concept is defined, and its applications to nonlinear optimization are described. Necessary conditions and sufficient conditions are established for functions to have minimization models of certain types. These necessary conditions may also be thought of as properties of the optimal value functions of certain optimization problems subject to linear RHS perturbations.

Unit commitment of thermal generation

Abstract: The problem of determining a feasible generation schedule for a thermal power system that minimises the cost subject to operating and electrical constraints is formulated as a mixed integer programming problem. The model caters for linear and nonlinear fuel cost relations, exponential start up costs, nonlinear loading rates and limits on generators outputs. The method of solution is based on a branch-and-bound capacitated transshipment approach that allows the exploitation of the engineering characteristics of the problem, resulting in an efficient solution procedure.

An Investment Staging Model for a Bridge Replacement Problem

Abstract: This paper discusses the formulation and solution of a mathematical model of the selection and staging of bridge replacements in rural road networks. Both relaxation and decomposition strategies are used. First, a relaxation of the problem into a decomposable shortest path problem allows the identification of the bridge replacement alternatives which are most promising with respect to user cost. Next, the planning horizon is divided into time periods which are then ordered according to their relative importance. Finally, another decomposition in the form of a lexicographic optimization is used to select and schedule the bridge replacements. Within this lexicographic optimization a sequence of computationally feasible integer programming problems is solved.