Showing papers on "Integer programming published in 1977"
TL;DR: In this paper, a class of surrogate constraint heuristics are proposed for integer programming problems. But they are based on a simple framework that illuminates the character of several earlier heuristic proposals and provides a variety of new alternatives.
Abstract: This paper proposes a class of surrogate constraint heuristics for obtaining approximate, near optimal solutions to integer programming problems. These heuristics are based on a simple framework that illuminates the character of several earlier heuristic proposals and provides a variety of new alternatives. The paper also proposes additional heuristics that can be used either to supplement the surrogate constraint procedures or to provide independent solution strategies. Preliminary computational results are reported for applying one of these alternatives to a class of nonlinear generalized set covering problems involving approximately 100 constraints and 300–500 integer variables. The solutions obtained by the tested procedure had objective function values twice as good as values obtained by standard approaches (e.g., reducing the best objective function values of other methods from 85 to 40 on the average. Total solution time for the tested procedure ranged from ten to twenty seconds on the CDC 6600.
1,326 citations
TL;DR: This paper presents a new approach to the solution of multi-target tracking problems that is approached as an unsupervised pattern recognition problem and has the computational structure of the set packing and set partitioning problems of 0-1 integer programming.
Abstract: This paper presents a new approach to the solution of multi-target tracking problems. 0-1 integer programming methods are used to alleviate the combinatorial computing difficulties that accompany any but the smallest of such problems. Multitarget tracking is approached as an unsupervised pattern recognition problem. A multiple-hypothesis test is performed to determine which particular combination of the many feasible tracks is most likely to represent actual targets. This multiple hypothesis test is shown to have the computational structure of the set packing and set partitioning problems of 0-1 integer programming. Multitarget tracking problems that are translated into this form can be rapidly solved, using well-known discrete optimization techniques such as implicit enumeration.
311 citations
TL;DR: A branch-and-bound algorithm is proposed using a Lagrangian method to decompose the problem into single generator problems and a sub gradient method is used to select the Lagrange multipliers that maximize the lower bound produced by the relaxation.
Abstract: Two major decisions are made when scheduling the operations of a fossil-fuel power-generating system over a short time horizon. The “unit commitment” decision indicates what generating units are to be in use at each point in time. The “economic dispatch” decision is the allocation of system demand among the generating units in operation at any point in time. Both these decisions must be considered to achieve a least-cost schedule over the short time horizon. In this paper we present a mixed integer programming model for the short time horizon power-scheduling problem. The objective of the model is to minimize the sum of the unit commitment and economic dispatch costs subject to demand, reserve, and generator capacity and generator schedule constraints. A branch-and-bound algorithm is proposed using a Lagrangian method to decompose the problem into single generator problems. A sub gradient method is used to select the Lagrange multipliers that maximize the lower bound produced by the relaxation. We present...
268 citations
TL;DR: This paper is a state-of-art review of the literature related to optimal system reliability with redundancy and describes the current state of the art in this area.
Abstract: This paper is a state-of-art review of the literature related to optimal system reliability with redundancy. The literature is classified as follows. Optimal system reliability models with redundancy Series Parallel Series-parallel Parallel-series Standby Complex (nonseries, nonparallel) Optimization techniques for obtaining optimal system configuration Integer programming Dynamic programming Maximum principle Linear programming Geometric programming Sequential unconstrained minimization technique (SUMT) Modified sequential simplex pattern search Lagrange multipliers and Kuhn-Tucker conditions Generalized Lagrangian function Generalized reduced gradient (GRG) Heuristic approaches Parametric approaches Pseudo-Boolean programming Miscellaneous
254 citations
TL;DR: In this paper, a modified version of Benders' original mixed-integer programming algorithm is proposed, in which the solution of a sequence of integer programs is replaced by the solution for a sequence linear programs plus some hopefully few integer programs.
Abstract: As applied to mixed-integer programming, Benders' original work made two primary contributions: 1 development of a “pure integer” problem Problem P that is equivalent to the original mixed-integer problem, and 2 a relaxation algorithm for solving Problem P that works iteratively on an LP problem and a “pure integer” problem. In this paper a modified algorithm for solving Problem P is proposed, in which the solution of a sequence of integer programs is replaced by the solution of a sequence of linear programs plus some hopefully few integer programs. The modified algorithm will still allow for taking advantage of any special structures e.g., an LP subproblem that is a “network problem” just as in Benders' original algorithm. The modified Benders' algorithm is explained and limited computational results are given.
197 citations
TL;DR: The usual constrained reliability optimization problem is extended to include determining the optimal level of component reliability and the number of redundancies in each stage and the heuristic approach by Aggarwal et al, is used to solve the problem.
Abstract: The usual constrained reliability optimization problem is extended to include determining the optimal level of component reliability and the number of redundancies in each stage. With cost, weight, and volume constraints, the problem is one in which the component reliability is a variable, and the optimal trade-off between adding components and improving individual component reliability is determined. This is a mixed integer nonlinear programming problem in which the system reliability is to be maximized as a function of component reliability level and the number of components used at each stage. The model is illustrated with three general non linear constraints imposed on the system. The Hooke and Jeeves pattern search technique in combination with the heuristic approach by Aggarwal et al, is used to solve the problem. The Hooke and Jeeves pattern search technique is a sequential search routine for maximizing the system reliability, RS (R, X). The argument in the Hooke and Jeeves pattern search is the component reliability, R, which is varied according to exploratory moves and pattern moves until the maximum of RS (R, X) is obtained. The heuristic approach is applied to each value of the component reliability, R, to obtain the optimal number of redundancies, X, which maximizes RS (R, X) for the stated R.
168 citations
TL;DR: In this article, a zero-one integer programming formulation of the project scheduling problem is presented which maximizes the discounted value of cash flows in a project when progress payments and cash outflows are made upon the completion of certain activities.
Abstract: A zero-one integer programming formulation of the project scheduling problem is reported which maximizes the discounted value of cash flows in a project when progress payments and cash outflows are made upon the completion of certain activities. While this problem has been treated previously, it has not been addressed in a manner which allows for the type of constraints treated in this paper. And while cash can be treated as any other resource in the noncash flow, resource-constrained version of this problem, the effects of cash inflows and outflows and the time sequencing alternatives for them allow for a different treatment of cash resources than has been previously reported.
158 citations
IBM1
TL;DR: An attempt to provide a powerful mathematical programming language, allowing an easy programming of specific studies on medium-size models such as the recursive use of LP or the build-up of algorithms based on the simplex method is described.
Abstract: First, this paper presents the results of experiments with algorithmic techniques for efficiently solving medium and large scale linear and mixed integer programming problems. The techniques presented here are either original or recent.
142 citations
TL;DR: In this article, the authors take stock of what is known and suggest some conceptual foundations for future progress in the areas of postoptimality analysis and parametric optimization techniques for integer programming.
Abstract: The purpose of this paper is to take stock of what is known and to suggest some conceptual foundations for future progress in the areas of postoptimality analysis and parametric optimization techniques for integer programming.
131 citations
TL;DR: It is proved that the gap in optimal value, between a mixed-integer program in rationals and its corresponding linear programming relaxation, is bounded as the right-hand-side is varied.
116 citations
TL;DR: An optimization program that is used to help electric utilities plan investments for power generation offers special economies which make it attractive to power system planners.
Abstract: This paper describes the development and application of an optimization program that is used to help electric utilities plan investments for power generation. For each year over a planning horizon the program determines what types and sizes of generating plants should be constructed, so as to minimize total discounted cost while meeting reliably the system's forecasted demands for electricity. The problem is formulated as a large-scale, chance constrained, mixed integer program. The solution algorithm employs Benders' Partitioning Principle, a mixed integer linear programming code, and a successive linearization procedure. Computation costs are low and, in the important area of sensitivity analysis, the program offers special economies which make it attractive to power system planners. Computational results are presented for a full sized generation planning problem for the six New England states where the algorithm is currently being used for planning generating facilities.
TL;DR: A subset of the constraints of an integer programming problem is said to be binding if, when the remaining constraints are eliminated, the smaller problem has the same optimal solution as the original problem as mentioned in this paper.
Abstract: A subset of the constraints of an integer programming problem is said to be binding if, when the remaining constraints are eliminated, the smaller problem has the same optimal solution as the original problem. It is shown that an integer programming problem with n variables has a set of binding constraints of cardinality less than or equal to 2n-1. The bound is sharp.
TL;DR: Theoretical results are developed for zero–one linear multiple objective programs for the main problem, having as a feasible set the vertices of the unit hypercube.
Abstract: Theoretical results are developed for zero–one linear multiple objective programs. Initially a simpler program, having as a feasible set the vertices of the unit hypercube, is studied. For the main problem an algorithm, computational experience, parametric analysis and indifference sets are presented. The mixed integer version of the main problem is briefly discussed.
TL;DR: In this paper, it was shown that if an integer linear program in n variables has more than 2n linear inequality constraints, then either some of the constraints are unnecessary or there is at least one feasible integer point.
Abstract: This note shows that if an integer linear program in n variables has more than 2n linear inequality constraints, then either some of the constraints are unnecessary or there is at least one feasible integer point.
TL;DR: The m-center problem is to locate a given number of emergency facilities anywhere along a road network so as to minimize the maximum distance between these facilities and fixed demand locations assigned to them as mentioned in this paper.
Abstract: The m-Center Problem is to locate a given number of emergency facilities anywhere along a road network so as to minimize the maximum distance between these facilities and fixed demand locations assigned to them. Fundamental properties of the m-Center Problem are examined. The problem is modeled using integer programming, and is successfully attacked using a binary search technique and a combination of exact tests and heuristics. Computational results are given.
IBM1
TL;DR: This paper discusses heuristic “branch and bound” methods for solving mixed integer linear programming problems and introduces new heuristic rules for generating a tree which make use of pseudo-costs and estimations.
Abstract: This paper discusses heuristic "branch and bound" methods for solving mixed integer linear programming problems. The research presented on here is the follow on to that recorded in [3].
After a resume of the concept of pseudo-costs and estimations, new heuristic rules for generating a tree which make use of pseudo-costs and estimations are presented. Experiments have shown that models having a low percentage of integer variables behave in a radically different way from models with a high percentage of integer variables. The new heuristic rules seem to apply generally to the first type of model.
Later, other heuristic rules are presented that are used with models having a high percentage of integer variables and with models having a special structure (models including special ordered sets.)
The rules introduced here have been implemented in the IBM Mathematical Programming System Extended/370. They are used to solve large mixed integer linear programming models.
Numerical results that permit comparisons to be made among the different rules are provided and discussed.
TL;DR: In this article, a method for solving multiple criteria integer linear programming problems is presented, and two types of algorithms for extending the Zionts-Wallenius algorithm are briefly presented.
Abstract: Although it may seem counterintuitive, a method for solving multiple criteria integer linear programming problems is not an obvious extension of methods that solve multiple criteria linear programming problems The main difficulty is illustrated by means of an example Then a way of extending the Zionts-Wallenius algorithm [6] for solving integer problems is given, and two types of algorithms for extending it are briefly presented An example is presented for one of the two types Computational considerations are also discussed
TL;DR: In this article, an algorithm for solving optimally, the mixed-model sequencing problem when assembly line stations are balanced for each model is described. But the problem is not addressed in this paper.
Abstract: This paper describes an algorithm for solving optimally, the mixed-model sequencing problem when assembly line stations are balanced for each model. An optimal sequence is obtained with the minimization of the overall assembly line length for zero station idle time. The algorithm incorporates two basic steps. The first involves a search procedure that generates all cycle sequences; i.e. sequences having identical ‘start’ and ‘finish’ positions and whose work content can be executed within a defined station length. The second step uses integer programming (IP) to determine the number and combination of the various cycle sequences, such that the production demand is satisfied.
TL;DR: In this article, the authors consider a class of nonlinear programs in which the imposition of integrality constraints on the variables makes it possible to solve the problem by a single, easily-constructed linear program.
Abstract: Although the addition of integrality constraints to the existing constraints of an optimization problem will, in general, make the determination of an optimal solution more difficult, we consider here a class of nonlinear programs in which the imposition of integrality constraints on the variables makes it possible to solve the problem by a single, easily-constructed linear program. The class of problems addressed has a separable convex objective function and a totally unimodular constraint matrix. Such problems arise in logistic and personnel assignment applications.
TL;DR: A modification of the Dinkelbach's algorithm is proposed to exploit the fact that good feasible solutions are easily obtained for both the fractional knapsack problem and the ordinary knapsacks problem, and an upper bound of the number of iterations is derived.
Abstract: The fractional knapsack problem to obtain an integer solution that maximizes a linear fractional objective function under the constraint of one linear inequality is considered. A modification of the Dinkelbach's algorithm [3] is proposed to exploit the fact that good feasible solutions are easily obtained for both the fractional knapsack problem and the ordinary knapsack problem. An upper bound of the number of iterations is derived. In particular it is clarified how optimal solutions depend on the right hand side of the constraint; a fractional knapsack problem reduces to an ordinary knapsack problem if the right hand side exceeds a certain bound.
IBM1
TL;DR: In this paper, a method for solving pure integer programs by a subadditive method was presented, which uses some elements of both enumeration and cutting plane theory in a unified setting.
Abstract: A method is presented for solving pure integer programs by a subadditive method. This work extends to the integer linear problem a method for solving the group problem. It uses some elements of both enumeration and cutting plane theory in a unified setting. The method generates a subadditive function and solves the original integer linear program.
TL;DR: The last dual problem in the sequence yields an optimal solution to the integer programming problem and it is shown that this dual problem approximates the convex hull of the feasible integer solutions in a neighborhood of the optimal solution it finds.
Abstract: We present a constructive procedure for generating a finite sequence of increasingly stronger dual problems to a given integer programming problem. The last dual problem in the sequence yields an optimal solution to the integer programming problem. We show that this dual problem approximates the convex hull of the feasible integer solutions in a neighborhood of the optimal solution it finds. The theory is applicable to any bounded integer programming problem with rational data.
TL;DR: In this paper, an analysis of a grain subterminal location problem within northwestern Indiana is presented, where the problem is solved as a mixed integer program with Benders decomposition, and discussion is presented on the technique.
Abstract: An analysis of a grain subterminal location problem within northwestern Indiana is presented. The problem is solved as a mixed integer program with Benders Decomposition, and discussion is presented on the technique. The empirical analysis concentrates upon the optimum number of subterminals within the region under different assumptions. Upon the concentration of grain in the export market, the results show that initially subterminals reduce costs somewhat regardless of location but as the optimum number is approached location becomes more critical.
TL;DR: A method is given for constructing fully symmetric integration rules over fully asymmetric two- and three-dimensional regions of any degree d with a minimal number of evaluation points based on the solution of integer programming problems.
Abstract: A method is given for constructing fully symmetric integration rules over fully symmetric two- and three-dimensional regions of any degree d with a minimal number of evaluation points. This method is based on the solution of integer programming problems in which the constraints are the conditions for linear consistency of the system of nonlinear algebraic equations ensuring exactness of the rule for all polynomials of degree$ \leqq d$. The solution of the integer programming problem determines the number of generators of each type in the system and indicates how to solve it. In the Appendix, almost all known real fully symmetric integration rules for the regions $C_n $, $S_n $, $E_n^{r^2 } $ and $E_n^r $, $n = 2,3$ are classified in accordance. with the results of the integer programs developed in the paper. These results are used to compute new ninth-degree integration rules with a minimal number of points for the above regions with $n = 3$.
TL;DR: In this paper, the authors identify classes of problems that are efficiently solvable because of special patterns of 0's in the 0-1 constraint matrices, and identify several interesting idle-time scheduling problems are shown to be solvable as finite and bounded series of network flow or matching problems.
Abstract: Integer linear programs with 0-1 constraint matrices arise frequently in scheduling and staffing models For many such problems it is natural to model processor availability by columns of the constraint matrix, where 1's indicate processor availability for production and 0's indicate idle periods Efficiently solvable problems have generally been identified in terms of special patterns of 1's within the matrix Exploiting the fact that idle time is the complement of production time, we identify classes of problems that are efficiently solvable because of special patterns of 0's In particular, several interesting idle-time scheduling problems are shown to be solvable as finite and bounded series of network flow or matching problems
TL;DR: In this article, a primal algorithm for solving (IFP) problems was presented, which can be applied to solve the mixed integer fractional programming problem in finitely many iterations.
Abstract: We construct in this paper new cutting plane algorithms for solving the Integer Fractional Programming (IFP) and the Mixed Integer Fractional Programming (MIFP) problems. By using Charnes and Cooper's approach for solving continuous fractional programs we develop two types of cutting planes, which can be systematically generated and applied while solving (IFP) problems. Similar results are obtained for the (MIFP) problem. By employing Martos' approach for solving continuous fractional programs together with Young's primal algorithm for solving Integer Programming problems, we are able to construct a primal algorithm for solving (IFP) problems in finitely many iterations.
TL;DR: In this paper, the problem of designing forest harvesting units and assigning logging equipment to those units can be formulated as a facilities location problem which exhibits a unique structure that has not previously appeared in the literature.
Abstract: The problem of designing forest harvesting units and assigning logging equipment to those units can be formulated as a facilities location problem which exhibits a unique structure that has not previously appeared in the literature. This structure is referred to here as a “cascading fixed charge” structure; for small problems, exact solutions can be obtained by means of 0–1 integer programming systems. Problems of practical size for use in forest planning, however, would involve thousands of 0–1 variables and constraints. Such problems are intractable for presently available integer programming codes. This paper presents an approximation algorithm which has been developed to solve facilities location problems exhibiting the special cascading fixed charge structure. Experience with an application of the algorithm to an actual forest planning area is cited.
TL;DR: In this paper, a duality theorem for disjunctive programming has been proved for this class of problems, which generalizes to the class of linear programming problems and generalizes the corresponding result for linear programming.
Abstract: We state a duality theorem for disjunctive programming, which generalizes to this class of problems the corresponding result for linear programming.
TL;DR: This note presents a refinement to earlier work that reduces the size of the coefficients in the equivalent constraint and points out advantages as well as computational considerations for solving problems by this method.
Abstract: Integer programming problems with bounded variables can be solved by combining the constraints into an equivalent single constraint. This note presents a refinement to earlier work that reduces the size of the coefficients in the equivalent constraint and points out advantages as well as computational considerations for solving problems by this method.