Showing papers on "Integer programming published in 1986"

Theory of Linear and Integer Programming

Abstract: Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Abstract: An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

A survey of exact algorithms for the simple assembly line balancing problem

Abstract: In this survey paper we discuss the development of the simple assembly line balancing problem SALBP; modifications and generalizations over time; present alternate 0-1 programming formulations and a general integer programming formulation of the problem; discuss other well-known problems related to SALBP; describe and comment on a number of exact i.e., optimum-seeking methods; and present a summary of the reported computational experiences. All models discussed here are deterministic i.e., all input parameters are assumed to be known with certainty and all the algorithms discussed are exact. The problem is termed "simple" in the sense that no "mixed-models," "subassembly lines," "zoning restrictions," etc. are considered. Due to the richness of the literature, we exclude from discussion here a the inexact i.e., heuristic/approximate algorithms for SALPB and b the algorithms for the general assembly line balancing problem including the stochastic models.

A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems

Abstract: This paper is concerned with the solution of linearly constrained zero-one quadratic programming problems. Problems of this kind arise in numerous economic, location decision, and strategic planning situations, including capital budgeting, facility location, quadratic assignment, media selection, and dynamic set covering. A new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods. An implicit enumeration algorithm which uses Lagrangian relaxation, Benders' cutting planes, and local explorations is designed to exploit the strength of this linearization. Computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm.

A Branch and Bound Approach for Machine Load Balancing in Flexible Manufacturing Systems

Abstract: A flexible manufacturing system FMS is an integrated system of computer numerically controlled machine tools connected with automated material handling. A set of production planning problems for FMSs has been defined Stecke [Stecke, Kathryn E. 1983. Formulation and solution of nonlinear integer production planning problems for flexible manufacturing systems. Management Sci.29 3, March 273-288.], and this paper considers one called the loading problem. This problem involves assigning to the machine tools, operations and associated cutting tools required for part types that have been selected to be produced simultaneously. The part types will be machined during the upcoming production period say, of one to three weeks duration on average and according to a prespecified part mix. This assignment is constrained by the capacity of each machine's tool magazine as well as by the production capacities of both the system, and each machine type. There are several loading objectives that are applicable in a flexible manufacturing situation. This paper considers the most commonly applied one, that of balancing the workload on all machines. This paper first discusses a nonlinear integer mathematical programming formulation of the loading problem. The problem is formulated in all detail. Then an efficient solution procedure is proposed and illustrated with an example. Computational results are provided to demonstrate the efficiency of the suggested special-purpose procedures.

Sensitivity theorems in integer linear programming

Abstract: We consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right-hand-side vector. Among our results, we show that, for any optimal solution to a linear program max{wx: Ax≤b}, the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixA. Using this, we strengthen several integer programming ‘proximity’ results of Blair and Jeroslow; Graver; and Wolsey. We also show that the Chvatal rank of a polyhedron {x: Ax≤b} can be bounded above by a function of the matrixA, independent of the vectorb, a result which, as Blair observed, is equivalent to Blair and Jeroslow's theorem that ‘each integer programming value function is a Gomory function.’

An exact algorithm for solving a capacitated location-routing problem

Abstract: In location-routing problems, the objective is to locate one or many depots within a set of sites (representing customer locations or cities) and to construct delivery routes from the selected depot or depots to the remaining sites at least system cost. The objective function is the sum of depot operating costs, vehicle acquisition costs and routing costs. This paper considers one such problem in which a weight is assigned to each site and where sites are to be visited by vehicles having a given capacity. The solution must be such that the sum of the weights of sites visited on any given route does not exceed the capacity of the visiting vehicle. The formulation of an integer linear program for this problem involves degree constraints, generalized subtour elimination constraints, and chain barring constraints. An exact algorithm, using initial relaxation of most of the problem constraints, is presented which is capable of solving problems with up to twenty sites within a reasonable number of iterations.

A dynamic programming approach to the complete set partitioning problem

Abstract: The complete set partitioning (CSP) problem is a special case of the set partitioning problem where the coefficient matrix has 2 m −1 columns, each column being a binary representation of a unique integer between 1 and 2 m −1,m⩾1. It has wide applications in the area of corporate tax structuring in operations research. In this paper we propose a dynamic programming approach to solve the CSP problem, which has time complexityO(3 m ), wheren=2 m −1 is the size of the problem space.

Computer and Database Location in Distributed Computer Systems

Abstract: Design of distributed computer systems is a complex task requiring solutions for several difficult problems. Location of computing resources and databases in a wide-area network is one of these problems which has not yet been solved satisfactorily. Solution of this problem involves determining number and size of computer facilities and their locations, configuring databases and allocating these databases among computer facilities. An integer programming formulation of the problem is presented. Heuristic and optimal solution procedures are developed and computational experience with these procedures is reported. Implications of the model for designing distributed systems are discussed.

Heuristics for multilevel lot-sizing with a bottleneck

Abstract: In this paper we present a heuristic method, based on Lagrangian relaxation, for multilevel lot-sizing when there is a single bottleneck facility. A series of Lagrangian relaxations one for each item in the product structure is imbedded in a branch and bound procedure. The objective is to find a production schedule that fits within available capacity at minimum cost. The method has two solution phases, dual and primal. In the dual phase of the procedure, implied costs of setups and production are determined based on a tentative schedule. The primal phase is repeated with these new prices and we iterate to reach a good solution. The solution procedure is first tested on two special cases: uncapacitated multilevel lot-sizing and the capacitated, single-level multi-item lot sizing problem. The results show that the solution procedure can provide better solutions than some heuristics designed especially for those problems. Test results on the bottleneck problem indicate that good feasible solutions are found for problems too difficult to solve with exact methods.

Production planning of style goods with high setup costs and forecast revisions

Abstract: In this paper we study a problem, common to a wide variety of manufacturing companies, of determining the production schedule of style goods, such as clothing and consumer durables, under capacity constraints. Demand for items is stochastic and occurs in the last season of the planning horizon. Demand estimates are revised in each period. We exploit the problem's two-level hierarchical structure, which is characterized by families and items. Production changeover costs from one family to another are high, compared to other costs. However, changeover costs between items in the same family are negligible. We first formulate this problem as a difficult-to-solve stochastic mixed integer programming problem. Then, exploiting the problem's hierarchical structure, we formulate a deterministic, mixed integer programming problem and solve it by means of an algorithm that provides an approximate solution. A lower bound is obtained by applying generalized linear programming to the approximate problem. We illustrate the procedure using the disguised data of a consumer electronics company. The computational results demonstrate the effectiveness of the proposed approach in a practical setting.

Review of distribution system planning models: a model for optimal multistage planning

Abstract: A representative set of electric power distribution system planning models published in the literature has been reviewed. The models have been classified according to their characteristics, from the point of view of stages of the plan and overall time span; the methods of treating distribution feeders and/or substations in terms of cost representation, location and sizing problems; radiality and voltage drop considerations; and the mathematical programming techniques used to solve them. Some of the particular features of models have been discussed in detail. The paper also presents a model to solve the optimal sizing, location and timing of the distribution substations and feeder expansion simultaneously. The model is based on mixed-integer programming and its objective function represents the present value of costs of investment, energy and demand losses of the system which take place throughout the duration of the plan. The objective function is minimised subject to Kirchhoff's current law, power capacity limits, and logical constraints. The model developed allows explicit constraints of radiality and voltage drops to be included in its formulation.

Global minimization of large-scale constrained concave quadratic problems by separable programming

Abstract: The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesx ∈R n , while the linear part is in terms ofy ∈R k. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be ≤ 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteede-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.

Resource-Constrained Assignment Scheduling

Abstract: Many resource-constrained assignment scheduling problems can be modeled as 0-1 assignment problems with side constraints APSC. Unlike the well-known assignment problem of linear programming, APSC is NP-complete. In this paper we define a branch-and-bound algorithm for solving APSC to optimality. The algorithm employs a depth-first, polychotomous branching strategy in conjunction with a bounding procedure that utilizes subgradient optimization. We also define a heuristic procedure for obtaining approximate solutions to APSC. The heuristic uses subgradient optimization to guide the search for a good solution as well as to provide a bound on solution quality. We present computational experience with both procedures, applied to over 400 test problems. The algorithm is demonstrated to be effective across three different classes of resource-constrained assignment scheduling problems. The heuristic generates solutions for these problems that are, on average, within 0.8% of optimality.

Optimal allocation of components in parallel−series and series−parallel systems

Abstract: This paper shows how majorization and Schur-convex functions can be used to solve the problem of optimal allocation of components to parallel-series and series-parallel systems to maximize the reliability of the system. For parallel-series systems the optimal allocation is completely described and depends only on the ordering of component reliabilities. For series-parallel systems, we describe a partial ordering among allocations that can lead to the optimal allocation. Finally, we describe how these problems can be cast as integer linear programming problems and thus the results obtained in this paper show that when some linear integer programming problems are recast in a different way and the techniques of Schur functions are used, complete solutions can be obtained in some instances and better insight in others.

A Mixed-Integer Goal-Programming Formulation of the Standard Flow-Shop Scheduling Problem

Abstract: Until recently, the majority of models used to find an optimal sequence for the standard flow-shop problem were based on a single objective, typically makespan. In many applications, the practitioner may also want to consider other criteria simultaneously, such as mean flow-time or throughput time. As makespan and flow-time are equivalent criteria for optimizing machine idle-time and job idle-time, respectively, these additional criteria could be inherently considered as well. The effect of job idle-time, measuring in-process inventory, could be of particular importance.

Integer programming applied to the map label placement problem

Abstract: This paper describes a point label placement program that uses a mathematical optimization algorithm to determine the best position for each label. The program detects all label overplots, moves labels to new positions to resolve overplot problems, and deletes labels when absolutely necessary. All tasks are performed without human intervention. The program is designed for use in production mapping application in the oil industry where thousands of labels must be placed, and hundreds of label conflicts resolved on a single map in everyday operations. This function is performed accurately and efficiently by this program, independent of the number of labels involved. Based on success in this application, it is reasonable to consider the use of optimization techniques to help solve other problems in automated cartography, including label placement for linear features and the selection of features to be displayed on a map. L'article decrit un programme de placement des ecritures a l'aide d'un algorithme d'opti...

Optimal capacity expansion planning when there are learning effects

Abstract: Production and capacity expansion decisions are difficult to analyze when there is learning. Later production is less costly, and maybe more profitable, but the company must endure high initial production costs. Mixed integer programming models are presented for optimizing coordinated production and capacity expansion plans in the face of such learning effects. An illustrative model is developed, optimized, and the types of strategies it selects are discussed.

An Integer Programming Approach to the Apparel Sizing Problem

Abstract: The paper addresses the problem of determining the measurements of a given number of sizes of apparel so as to maximize expected sales or minimize an index of aggregate discomfort. The problem is formulated as a 'p-median' or facility location problem, and the results are compared with current practice.

Solving nonlinear integer programs with large-scale optimization software

Abstract: This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.