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Showing papers on "Integer programming published in 1976"


Journal ArticleDOI
TL;DR: For P- complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete.
Abstract: For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete. In contrast with these results, a linear time approximation algorithm for the clustering problem is presented.

1,718 citations


Journal ArticleDOI
TL;DR: In this paper, a man-machine interactive mathematical programming method is presented for solving the multiple criteria problem involving a single decision maker, where all decision-relevant criteria or objective functions are concave functions to be maximized, and the constraint set is convex.
Abstract: In this paper a man-machine interactive mathematical programming method is presented for solving the multiple criteria problem involving a single decision maker. It is assumed that all decision-relevant criteria or objective functions are concave functions to be maximized, and that the constraint set is convex. The overall utility function is assumed to be unknown explicitly to the decision maker, but is assumed to be implicitly a linear function, and more generally a concave function of the objective functions. To solve a problem involving multiple objectives the decision maker is requested to provide answers to yes and no questions regarding certain trade offs that he likes or dislikes. Convergence of the method is proved; a numerical example is presented. Tests of the method as well as an extension of the method for solving integer linear programming problems are also described.

732 citations


Journal ArticleDOI
TL;DR: Two methods for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval are described.
Abstract: The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick [4] points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland [2] or Soland [6], or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin [1], have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin [3] show how they have benefitted from the vast improvements in branch and bound integer programming capabilities over the last few years, as a result of being incorporated in a general mathematical programming system. Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis. This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two by-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval. The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.

235 citations


Journal ArticleDOI
TL;DR: This paper describes an integer programming formulation of the vehicle scheduling problem and illustrates how such a formulation can be extended to incorporate restrictions on work load, coverage and service that occur in real world vehicle scheduling problems.
Abstract: This paper describes an integer programming formulation of the vehicle scheduling problem and illustrates how such a formulation can be extended to incorporate restrictions on work load, coverage and service that occur in real world vehicle scheduling problems. The integer programme is solved using the Revised Simplex method, additional constraints being introduced to retain integrality during convergence. The feasible region of this integer programme is initially restricted so that only routes constructed through sets of radially contiguous locations are considered. The effect of relaxing these over-constraints is explored. The method is demonstrated on fifteen problems ranging in size from 21 to 100 locations and the results generally show an improvement on previously published results. This is particularly true of the larger problems. This method compares favourably with other methods in computational efficiency.

231 citations


ReportDOI
01 Jan 1976
TL;DR: In the context of integer linear programming, postoptimality analysis and parametric optimization techniques are fully developed aspects of linear programming as discussed by the authors, however, these aspects have barely begun to be developed.
Abstract: : Postoptimality analysis and parametric optimization techniques are fully developed aspects of linear programming. In the context of integer linear programming, however, these aspects have barely begun to be developed. The purpose of this paper is to take stock of what is known about this topic and to lay the foundation for future progress. Our conceptual starting point is the notion that, in practical applications, typically one is faced not with a single numerical integer linear program to solve but rather with an entire family of numerical problems of interest. The members of the family may all have the same structure but differ as to the values of one or more coefficients, or they may even have different (but related) structures. The scope of this paper is limited to be first mentioned case.

125 citations


Journal ArticleDOI
01 Jun 1976-Infor
TL;DR: In this article, a binary mixed integer programming (BILP) formulation is presented to solve the problem of minimizing total transportation costs given by the sum of all distance-flow products.
Abstract: This paper considers the location of n departments on one line. These departments are of different lengths and the material ilow between each pair of departments is known. The objective is to minimize total transportation costs given by the sum of all distance-flow products. The distance between two departments is the separation between their centroids. A binary mixed integer programming formulation is presented to solve this problem. The formulation involves ½n(n – 1) binary variables. Computational results are presented.

111 citations


Journal ArticleDOI
01 Aug 1976
TL;DR: An experimental method of scheduling the necessary maintenance activities on generator units in a power system based on the branch-and-bound technique, which results in a practically implementable solution, if a feasible solution exists.
Abstract: An experimental method of scheduling the necessary maintenance activities on generator units in a power system is developed. The problem is identified as an integer programming problem, and a method based on the branch-and-bound technique is developed. The maintenance scheduling problem is characterized by a large number of complex constraints. The method presented is capable of taking into account all these constraints and hence, results in a practically implementable solution, if a feasible solution exists. Other features of the method are employment of a number of different objective functions and discovery of a feasible solution if one exists. Furthermore, unlike most present methods, it actually finds the optimal solution. The operation of the method is exemplified by application to a realistic system.

109 citations


Journal ArticleDOI
TL;DR: In this article, an improved model for solving the long-run multiple warehouse location problem was proposed, which provides a synthesis of a mixed integer programming formulation for the single-period warehouse location model with a dynamic programming procedure for finding the optimal sequence of configurations over multiple periods.
Abstract: This paper proposes an improved model for solving the long-run multiple warehouse location problem. The approach used provides a synthesis of a mixed integer programming formulation for the single-period warehouse location model with a dynamic programming procedure for finding the optimal sequence of configurations over multiple periods. We show that only the Rt, best rank order solutions in any single period need be considered as candidates for inclusion in the optimal multi-period solution. Thus the computational feasibility of the dynamic programming procedure is enhanced by restricting the state space to these Rt best solutions. Computational results on the ranking procedure are presented, and a problem involving two plants, five warehouses, 15 customer zones, and five periods is solved to illustrate the application of the method.

103 citations


Journal ArticleDOI
P. Miliotis1
TL;DR: The generality of the method and the modest solution times achieved leads the author to believe that such an LP approach to other combinatorial problems deserves further consideration.
Abstract: The availability of an LP routine where we can add constraints and reoptimize, makes it possible to adopt an integer programming approach to the travelling-salesman problem. Starting with some of the constraints that define the problem we use either a branching process or a cutting planes routine to eliminate fractional solutions. We then test the resulting integer solution against feasibility and if necessary we generate the violated constraints and reoptimize until a "genuine" feasible solution is achieved. Usually only a small number of the omitted constraints is generated. The generality of the method and the modest solution times achieved leads us to believe that such an LP approach to other combinatorial problems deserves further consideration.

102 citations


Journal ArticleDOI
TL;DR: The Fourier-Motzkin Elimination Method, which can be used for solving Linear Programming Problems, can be extended to deal with Integer Programming Problems through a known decision procedure for the formal theory of a fragment of arithmetic which excludes multiplication.

90 citations


Journal ArticleDOI
TL;DR: An algorithm which recursively generates the complete family of undominated feasible solutions to separable nonlinear multidimensional knapsack problems is developed by exploiting discontinuity preserving properties of the maximal convolution.
Abstract: An algorithm which recursively generates the complete family of undominated feasible solutions to separable nonlinear multidimensional knapsack problems is developed by exploiting discontinuity preserving properties of the maximal convolution. The “curse of dimensionality,” which is usually associated with dynamic programming algorithms, is successfully mitigated by reducing an M-dimensional dynamic program to a 1-dimensional dynamic program through the use of the imbedded state space approach. Computational experience with the algorithm on problems with as many as 10 state variables is also reported and several interesting extensions are discussed.

Journal ArticleDOI
TL;DR: This paper investigates the problem of assigning faculty to courses at a university by developing a program which is both efficient in that integer programming is not required, and effective in that it facilitates interaction by administration in determining the optimal solution.
Abstract: This paper investigates the problem of assigning faculty to courses at a university. A program is developed which is both efficient in that integer programming is not required, and effective, in that it facilitates interaction by administration in determining the optimal solution. The results of some empirical tests are also reported.

Proceedings ArticleDOI
29 Mar 1976
TL;DR: An efficient method to solve the file allocation problem for medium-scale networks is proposed, and a near-optimal heuristic is presented, along with computational results.
Abstract: The problem of allocating files in a computer network is a complex combinatorial problem due to the number of integer design parameters involved. These parameters include system cost, number of copies of each file to be stored, and sites at which the copies should be stored. The tradeoffs between these parameters are discussed. The design problem is formulated as an integer programming problem. A branch and bound algorithm is proposed to solve the problem. A linear programming formulation which ignores integer restrictions (and allows a fraction of a file to reside at a site) is shown to yield integer solutions in most cases. In other words integer restrictions are satisfied automatically. A near-optimal heuristic is presented, along with computational results. An efficient method to solve the file allocation problem for medium-scale networks is proposed.

Journal ArticleDOI
TL;DR: It is shown that a planned subdivision of the physical data base can yield noticeable operating cost improvements over conventional data base designs.

Journal ArticleDOI
TL;DR: In this article, the problem of minimizing the sum of transportation costs is formulated as a binary mixed integer program, where the number of integer variables involved equals the total number of facilities squared.
Abstract: The problem involves the assignment of n facilities to n specified locations. Each facility has a given nonnegative flow from each of the other facilities. The objective is to minimize the sum of transportation costs. Assume these n locations are given as points on a two-dimensional plane and transportation costs are proportional to weighted rectangular distances. Then the problem is formulated as a binary mixed integer program. The number of integer variables (all binary) involved equals the number of facilities squared. Without increasing the number of integer variables, the formulation is extended to include "site costs." Computational results of the formulation are presented.

Journal ArticleDOI
TL;DR: In this article, a linear integer programming model and heuristic solution procedure to realign sales territories is described. But this model is not suitable for the real-time optimization of sales territories.
Abstract: A recent article described a mathematical programming model and heuristic solution procedure to realign sales territories. This report presents two linear integer programming models for sales terri...

Journal ArticleDOI
TL;DR: In this article, a stream water quality simulation model is coupled with a treatment cost minimization model to deal with the problem of determining treatment facilities needed to maintain stream quality standards, where the optimization model is structured as an integer programming problem in which the integer decision variables are wastewater treatment levels to be determined for each discharger.
Abstract: One of several considerations in river basin water quality management is determining treatment facilities needed to maintain stream quality standards. To deal with this problem, a stream water quality simulation model is coupled with a treatment cost minimization model. The optimization model is structured as an integer programming problem in which the integer decision variables are wastewater treatment levels to be determined for each discharger. The stream water quality simulation model, which relates pollution loading to stream quality response, generates the coefficients for the constraint equations in the optimization model. The model is applied to an example problem consisting of four pollution discharge points, at which seven possible treatment levels are available to remove four pollutant constituents. Water quality standards are imposed at five control points along the stream. Results of the optimal solution indicate the treatment level to be applied at each discharge point, the minimum total cost, and the stream quality conditions.

Journal ArticleDOI
TL;DR: According to the computational results, for the majority of the functions the first type of minimal networks is identical to the second type, and for no function were networks of the third type found to exist.
Abstract: Based on the intuitive observation that smaller numbers of gates and connections would usually lead to a more compact network on an integrated circuit (IC), a monotonically increasing function of gate count and connection count is concluded to be a reasonable cost function to be minimized in the logical design of a network implemented in IC. Then it is shown that all minimal solutions of such a cost function always can be found among the following: minimal networks with a minimal number of gates as the first objective and a minimal number of connections as the second objective; minimal networks with a minimal number of connections as the first objective and a minimal number of gates as the second objective; and minimal networks which are associated with the above two types of minimal networks. All three of these types of minimal networks of NOR gates, as an example, are calculated by logical design programs based on integer programming, for all functions of 3 or less variables and also some functions of 4 variables which require 5 or less NOR gates. According to the computational results, for the majority of the functions the first type of minimal networks is identical to the second type, and for no function were networks of the third type found to exist.


Journal ArticleDOI
TL;DR: In this article, a group-theory based algorithm for exact solution of fixed-charge network problems is presented, which exploits the special structures of network problems and results are reported for problems with as many as 100 fixed charge arcs.
Abstract: Many well-known transportation, communication, and facilities location problems in operations research can be formulated as fixed charge network problems, i.e. as minimum cost flow problems on a capacitated network in one commodity where some arcs have both fixed and variable costs. One approach to solving such problems is to use group theoretic concepts from the theory of integer programming to provide bounds for a branch-and-bound procedure. This paper presents such a group-theory based algorithm for exact solution of fixed charge network problems which exploits the special structures of network problems. Computational results are reported for problems with as many as 100 fixed charge arcs.

Proceedings ArticleDOI
01 Dec 1976
TL;DR: An optimal control approach to a problem in national settlement system planning is presented and it is shown how the special structure of the model and the singular nature of the control can be used to reduce the solution of a nonlinear programming problem to the Solution of sets of linear equations.
Abstract: In this paper, an optimal control approach to a problem in national settlement system planning is presented. The problem description is the same as considered by MacKinnon [6] and by Evtushenko and Mackinnon [4]. It is shown how the special structure of the model and the singular nature of the control can be used to reduce the solution of a nonlinear programming problem to the solution of sets of linear equations. A branch and bound integer programming algorithm is used to handle inequality constraints on the control variables. The organization of the paper is as follows. Section I considers problem formulation and an optimal control solution is discussed in Section II. A branch and bound technique for determining active constraints is presented in Section III. A more general problem is considered in Section IV and conclusions are stated in Section V.

Journal ArticleDOI
TL;DR: A new optimal solution of the numerical example in which the objective function to be minimized is smaller than and the system reliability is higher than that of the integer programming method, respectively.
Abstract: In this paper we propose an effective method for solving the optimization problem of the redundant allocation and unit selection in system reliability with several failure-modes by using the implicit enumeration algorithm. The quantitative evaluation for the proposed method is indicated clearly. This shows that the number of constraints and variables in the proposed one are few than those of the integer programming method, respectively. Recently, McLeavey points out an example in which an algorithm reported by Ghare and Taylor for determining optimum redundancy in a series system dose not produce an optimal solution. We also report a new optimal solution of the numerical example in which the objective function to be minimized is smaller than and the system reliability is higher than that of the integer programming method, respectively. Consequently, the computer CPU time is shorter than that of the integer programming method with the same computer and implicit enumeration as a solution algorithm.

Journal ArticleDOI
TL;DR: In this article, a primal cutting plane algorithm is proposed for integer fractional programming, where n n maximize (cO + L c.x.)/(do + L d.x).
Abstract: A primal cutting plane algorithm is proposed for the integer fractional programming problem: n n maximize (cO + L c.x.)/(do + L d.x.J j=l ] ] j=l ] ] n subject to L a .. x. ~ b. , i=1,2,···,m j=l ~J ] ~ x. > 0 , integer, j=1,2,··· ,n ] The algorithm is obtained by slightly modifying Young's simplified primal algorithm developed for the ordinary integer programming problem, and is based on the parametric programming approach to the fractional problem given by Jagannathan and Dinkelbach.

Journal ArticleDOI
R.J. Aust1
TL;DR: An alternative relaxation is proposed in which the integer conditions are maintained but the feasibility conditions are relaxed in a special way, which is applicable to both linear and non-linear pure integer problems.

Book
01 Jan 1976
TL;DR: In this paper, an algorithm for combining Quadratic and Multi-Objective Programming (MOP) is proposed for multi-criteria decision-making in the public sector.
Abstract: From Optimisation to Multi-Criteria Decision Aid: Three Main Operational Attitudes.- An Algorithm for Combined Quadratic and Multiobjective Programming.- Quasi-Kernels of Outranking Relations.- Existence and Duality in Multiple Objective Linear Programming.- On the Relationship of the Tchebycheff Norm and the Efficient Frontier of Multiple-Criteria Objectives.- Large Group Decision Making with Multiple Criteria.- Multi-Person Multi-Criteria Decision-Making: A Sample Approach.- Theoretical Analysis and Empirical Application of Goal Programming with Preemptive Priority Structures.- R&D Project Selection Behavior: Study Designs and some Pilot Results.- An Interactive Objective Function Generator for Goal Programmes.- A Five Phase Procedure for Implementing a Vector-Maximum Algorithm for Multiple Objective Linear Programming Problems.- Multiple Criteria Public Investment Decision Making by Mixed Integer Programming.- A Comparative Study of Four Multiple-Criteria Methods.- Multiple-Criteria Decision Making with a Special Application on Defense Problems.- Multi-Level Planning in the Public Sector.- Rational Solution Principles and Information Requirements as Elements of a Theory of Multiple Criteria Decision Making.- Pareto Optimality with Nondifferentiable Cost Functions.- The Notion of Characteristic Set and Its Implication for the Analysis and Development of Multicriterion Methods.- Two-Level Planning with Conflicting Goals.- Possibilities to Consider Multiple Criteria in Decision Situations.- Quantifying Corporate Preferences for Policy Analysis.- A Series-Parallel Multiple-Criteria Model for a Scheduling Problem in the Dress-Making Industry.- Some Tests of an Interactive Programming Method for Multicriterion Optimization and an Attempt at Implementation.- A New Method for Interactive Multiobjective Optimization: A Boundary Point Ranking Method.- A New Approach to Multiple Criteria Decision-Making.- Selecting a Strategy for Joint Ventures in Fisheries: A First Approximation.- La Promotion de l'electricite dans l'industrie et l'utilisation de methodes multicriteres.- Some Behavioural Aspects of Information Use in Decision Making: A Study of Clinical Judgements.- A Behavioural Model of Company Development.- Program.- List of Participants.

Journal ArticleDOI
TL;DR: In this article, the classical problem of sequencing jobs on a single processor is given a new formulation, which results in a zero-one mixed-integer linear programming problem and includes sequence-dependent setup costs and times, inventory costs for completed jobs, and penalty costs for completion delays.
Abstract: The classical problem of sequencing jobs on a single processor is given a new formulation. Included in the formulation are sequence-dependent setup costs and times, inventory costs for completed jobs, and penalty costs for completion delays. The formulation results in a zero-one mixed integer linear programming problem.

Journal ArticleDOI
TL;DR: This paper presents five simplified algorithms and compares their computational results with the group-theoretic algorithms developed by Gomory, Hu, and Shapiro.
Abstract: Gomory has shown that the group-theoretic problem associated with an integer programming problem can be treated as a shortest-route problem Thus one may solve it by a standard shortest-route algorithm However, because of the special properties of the constructed problem, one can simplify and modify the algorithm This paper presents five such simplified algorithms and compares their computational results with the group-theoretic algorithms developed by Gomory, Hu, and Shapiro



Book
01 Jan 1976
TL;DR: In this paper, a branch-and-bound algorithm for the decomposition of block angular integer programs is presented. But it is not shown how to use the LP-optimal dual multipliers and any slacks which appear in the optimal integer solutions to the subproblems to guide the search.
Abstract: : Linear programming models in which the constraint matrix has a block angular structure arise frequently in many applications. While much work has been devoted to exploiting this special structure when the problem variables are assumed to be continuous, little consideration has been given to models of this type in which the variables are required to take on only integer values. In this report, an algorithm for the decomposition of block angular integer programs is presented. The block angular integer program consists of several subproblems which would operate independently except that they are tied together by a set of linking constraints. Conceptually, these linking constraints are viewed as representing common resources which the subproblems must share. The problem thus becomes that of determining an optimal allocation of these resources among the subproblems. Toward this end, a branch-and-bound search routine is developed. It is shown how the LP-optimal dual multipliers and any slacks which appear in the optimal integer solutions to the subproblems can be used to guide the search, as well as for bounding and fathoming purposes. Special structures which arise when there is only a single linking constraint are discussed in detail. Since the problem decomposes completely once an allocation of the linking resources is specified, only the subproblems ever need be solved explicitly. Computational results obtained with the decomposition algorithm are reported. (Author)