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Showing papers on "Integer programming published in 1971"


Journal ArticleDOI
TL;DR: Cluster analysis involves the problem of optimal partitioning of a given set of entities into a pre-assigned number of mutually exclusive and exhaustive clusters that lead to different kinds of linear and non-linear integer programming problems.
Abstract: Cluster analysis involves the problem of optimal partitioning of a given set of entities into a pre-assigned number of mutually exclusive and exhaustive clusters. Here the problem is formulated in two different ways with the distance function (a) of minimizing the within groups sums of squares and (b) minimizing the maximum distance within groups. These lead to different kinds of linear and non-linear (0–1) integer programming problems. Computational difficulties are discussed and efficient algorithms are provided for some special cases.

357 citations


Journal ArticleDOI
TL;DR: The paper gives a simple formula for finding the equation of the hyperplane, discusses some ways of strengthening the cut, proposes an algorithm, and gives a finiteness proof.
Abstract: This paper proposes a new class of cutting planes for integer programming. A typical member of the class is generated as follows. Let X be the feasible set, and x the optimal (noninteger) solution...

314 citations


Journal ArticleDOI
TL;DR: The heuristic rules for generating the tree, which are the main features of the method, are presented and numerous parameters allow the user for adjusting the search strategy to a given problem.
Abstract: This paper presents a “branch and bound” method for solving mixed integer linear programming problems. After briefly discussing the bases of the method, new concepts called pseudo-costs and estimations are introduced. Then, the heuristic rules for generating the tree, which are the main features of the method, are presented. Numerous parameters allow the user for adjusting the search strategy to a given problem.

273 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present an exact solution of the fixed-charge transportation problem by decomposing it into a master integer program and a series of transportation subprograms, and a composite algorithm based on Murty's and the author's results is proposed.
Abstract: In the fixed-charge transportation problem, a fixed charge is associated with each route that can be opened, in addition to the variable transportation cost proportional to the amount of goods shipped- This note presents an exact solution of this mixed integer programming problem by decomposing it into a master integer program and a series of transportation subprograms. To reduce the number of vertices that need to be examined, bounds are established on the maximum and minimum values of the total fixed cost, and feasibility conditions for the transportation problem are used extensively. Computational results show the method to be particularly suitable when fixed costs are large compared to variable costs. A composite algorithm based on Murty's and the author's results is proposed.

111 citations


Journal ArticleDOI
TL;DR: In this article, two integer programming methods, Branch-and-Bound (B&B) and Decomposition (DWC), are used to solve vessel scheduling problems, where the branching is performed on one of the essential fractional variables and the bounds are obtained by the decomposition algorithm.
Abstract: In a previous paper, Appelgren (Appelgren, L. 1969. A column generation algorithm for a ship scheduling problem. Trans. Sci. 3 53–68.), a decomposition algorithm for a class of vessel scheduling problems was presented. In some problems, the algorithm gives fractional solutions that cannot be interpreted as feasible schedules. This paper treats two integer programming methods that can be used to resolve these cases. The cutting plane method that was first tested was abandoned because it was not able to solve all the test problems. The second method is a branch-and-bound algorithm, where the branching is performed on one of the “essential” fractional variables and where the bounds are obtained by the decomposition algorithm. All fractional problems that have been found by simulation or in regular use of the algorithm have been solved, mostly with one branching only. There are fundamental difficulties in combining these integer programming methods with the Dantzig-Wolfe decomposition, since the constraints g...

103 citations


Journal ArticleDOI
TL;DR: Two extensions of the successful Beale and Small branch-and-bound mixed-integer algorithm are proposed, utilizing the integer requirements on nonbasic variables to calculate stronger "penalties" when searching down the solution tree and to give a stronger criterion for abandoning unprofitable branches of the tree when backtracking.
Abstract: This note proposes two extensions of the successful Beale and Small branch-and-bound mixed-integer algorithm. The integer requirements on nonbasic variables are utilized to calculate stronger "penalties" when searching down the solution tree and to give a stronger criterion for abandoning unprofitable branches of the tree when backtracking. This stronger criterion is obtained by making use of Gomory cutting-plane constraints. These modifications have produced considerable reductions of the searching effort required for pure integer and predominantly integer problems, and have the further advantage of being very easy to incorporate.

79 citations


Journal ArticleDOI
TL;DR: It is shown that any bounded integer linear programming problem can be trans- formed to an equivalent integerlinear programming problem with a single constraint and the same number of variables.

70 citations


Journal ArticleDOI
TL;DR: The redundancy optimization problem is formulated as an integer programming problem of zero-one type variables and the solution is obtained making use of an algorithm due to Lawler and Bell.
Abstract: The redundancy optimization problem is formulated as an integer programming problem of zero-one type variables. The solution is obtained making use of an algorithm due to Lawler and Bell. Objective function and constraints can be any arbitrary functions. Three different variations of the optimization problem are considered. The formulation is easy and the solution is convenient on a digital computer. The size of the problem that can be solved is not restricted by the number of constraints.

65 citations


Journal ArticleDOI
TL;DR: An improved version of the Bowman-White zero-one integer programming formulation of an assembly line balancing problem is presented and it is demonstrated that certain steps in Geoffrion's 0–1 integer programming algorithm can be simplified or eliminated.
Abstract: In this study we will present an improved version of the Bowman-White zero-one integer programming formulation of an assembly line balancing problem. We will demonstrate that in solving the above problem, certain steps in Geoffrion's 0–1 integer programming algorithm can be simplified or eliminated. Computational results are given which indicate that the proposed algorithm is faster than the present exact methods.

62 citations


Book
21 Feb 1971
TL;DR: This volume contains thirty-three selected general research papers devoted to the theory and application of the mathematics of constrained optimization, including linear programming and its extensions to convex programming, general nonlinear programming, integer programming, and programming under uncertainty.
Abstract: This volume contains thirty-three selected general research papers devoted to the theory and application of the mathematics of constrained optimization, including linear programming and its extensions to convex programming, general nonlinear programming, integer programming, and programming under uncertainty.Originally published in 1971.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

48 citations


Book
01 Jan 1971

Journal ArticleDOI
TL;DR: This paper reviews the principal public literature concerned with expected-profit decision-theoretic closed-competitive-bidding models and makes extensions of previous formulations for static multi-contract bidding situations with and without cost dependencies and resource constraints.
Abstract: This paper reviews the principal public literature concerned with expected-profit decision-theoretic closed-competitive-bidding models. It makes extensions of previous formulations for static multi-contract bidding situations with and without cost dependencies and resource constraints. The same decision model is reformulated to utilize numerical, Lagrange-multiplier, dynamic, and integer linear programming techniques according to the information available to the bidder. A zero-one integer programming formulation is well suited to utilize subjective probabilities of winning estimates. Finally, the paper suggests a format for further development.

Journal ArticleDOI
TL;DR: This paper presents the results of experimentation on the development of an efficient branch-and-bound algorithm for the solution of zero-one linear mixed integer programming problems and a comparison with the computational experience obtained with several other algorithms on a number of problems.
Abstract: This paper presents the results of experimentation on the development of an efficient branch-and-bound algorithm for the solution of zero-one linear mixed integer programming problems. An implicit enumeration is employed using bounds that are obtained from the fractional variables in the associated linear programming problem. The principal mathematical result used in obtaining these bounds is the piecewise linear convexity of the criterion function with respect to changes of a single variable in the interval [0, 1]. A comparison with the computational experience obtained with several other algorithms on a number of problems is included.

Journal ArticleDOI
TL;DR: In this article, it is shown how to relax the non-negativity constraints on a set of basic variables and obtain a knapsack problem, which either gives the solution of the integer problem, or when solved by dynamic programming provides bounds at least as strong as those provided by the group problem.
Abstract: By relaxing the nonnegativity constraints on a set of basic variables, an integer programming problem can be reduced to a shortest route problem over a finite Abelian group. Here it is shown how given a similar relaxation on any set of variables, a structurally simpler problem is obtained, which can be regarded as a shortest route problem over an Abelian (but not necessarily finite) group. In particular if the non-negativity constraints are relaxed on all but one of a set of basic variables, a knapsack (or very close to knapsack) problem is obtained, which either gives the solution of the integer problem, or when solved by dynamic programming provides bounds at least as strong as those provided by the above group problem. The difficulty that arises when the order of groups encountered is very large is also considered in the same framework, and by further relaxation it is shown how the groups can be reduced to manageable size, which can be used to provide simple bounds for the above group problem. Finally ...

Journal ArticleDOI
TL;DR: A theory of equivalent integer programs was developed in this article, where it was shown that every all-integer integer programming problem is equivalent to infinitely many other integer programming problems, such that the solution to any one problem in the equivalence class determines the solution of every other problem.
Abstract: A theory of equivalent integer programs is developed that shows that every all-integer integer programming problem is equivalent to infinitely many other integer programming problems The equivalence is such that the solution to any one problem in the equivalence class determines the solution to every other problem in the class Procedures to construct certain canonical problems in each equivalence class are described The relationship of this theory to computational algorithms is discussed



Journal ArticleDOI
TL;DR: In this article, the problem of minimizing the weight of the subsystems of a life support system subject to several separable nonlinear constraints while maintaining an acceptable level of reliability of the system was considered.
Abstract: The problem considered is to minimize the weight of the subsystems of a life support system subject to several separable nonlinear constraints while maintaining an acceptable level of reliability of the system. Zero-one integer programming is used to solve this problem. The subsystems of the life support system designed by the Space Division of the North American Rockwell Corporation are studied examples. They are the refrigerant circuit, the cryogenic oxygen circuit, the carbonization cell, and the water reclamation unit. The results obtained are compared with the original design proposed by the North American Rockwell Corporation. The comparison indicates that the use of the integer programming for determining the optimization reliability would result in an improved arrangement of the standby units.

Journal ArticleDOI
TL;DR: This paper gives efficient methods for solving four specially structured network problems that arise in connection with certain integer programming methods developed by Cook and Cooper, Hillier, and Glover.
Abstract: This paper gives efficient methods for solving four specially structured network problems that arise in connection with certain integer programming methods developed by Cook and Cooper, Hillier, and Glover. Such problems have also independently been studied in inventory theory by Ignall and Veinott, who have developed extensive qualitative (nonalgorithmic) implications of their structures. In the integer programming context, special subclasses of these network problems are generated and solved as part of a strategy for solving more general integer linear programs. By providing particularly efficient methods for accommodating somewhat broader network structures, our results enable the development of related integer programming solution strategies that generate more complex subproblems. We show that the first of the four network problems can be solved by a procedure that assigns each variable a value exactly once, without subsequent revision. Properties of optimal solutions for the remaining problems are de...

Journal ArticleDOI
TL;DR: The proposed enumerative method allows a greater flexibility in the backtracking step that makes it possible to devise suitable rules of choice to improve the efficiency of the searching process.
Abstract: This paper presents an enumerative method for solving integer programming problems. In the enumerative methods so far available, the variables are ranked in a rigid order by the choices in the forward steps, and backtracking always takes place on the last variable in this order. The proposed method allows a greater flexibility in the backtracking step that makes it possible to devise suitable rules of choice to improve the efficiency of the searching process. A numerical example is devised to illustrate the enumerative method in the direct-search case.

Journal ArticleDOI
TL;DR: A method based on a special case of the knapsack function that replaces each cut or original constraint by a new inequality whose hyperplane passes through as many integer points in 0, 1 space as possible is described.
Abstract: In deriving the well known cuts for cutting-plane methods in 0, 1 integer programming, the integer points outside the 0,1 space can limit the parallel movement of the hyperplane of the cut toward the solution set. Furthermore it is unnecessarily restrictive to limit the movement of this hyperplane to parallel translations. This paper removes these two limitations in order to derive stronger cuts and reduce the total number of cuts required. Thus, it describes a method based on a special case of the knapsack function that replaces each cut or original constraint by a new inequality whose hyperplane passes through as many integer points in 0, 1 space as possible.

Journal ArticleDOI
TL;DR: The group-theoretic ideas are used to provide information about the solutions of the asymptotic mixed integer problem, and it is shown how useful bounds can be obtained from the corresponding all-integer asymPTotic problem.
Abstract: This paper shows how the asymptotic structure of the integer programming problem extends to the mixed integer problem. The group-theoretic ideas are then used to provide information about the solutions of the asymptotic mixed integer problem, and it is also shown how useful bounds can be obtained from the corresponding all-integer asymptotic problem. An example shows the application of some of the results.

Journal ArticleDOI
TL;DR: One of the most frustrating tasks encountered by the OR practitioner is that of attempting to select an existing algorithm for application to an immediate real world problem.
Abstract: One of the most frustrating tasks encountered by the OR practitioner is that of attempting to select an existing algorithm for application to an immediate real world problem. For purposes of illustration only, consider the OR analyst who is faced with having to solve a large zero-one integer programming problem. Being a member of ORSA, TIMS, SIAM, and other organizations, he realizes that a considerable number of publications have been addressed to this topic and thus he might be able to use one of these to solve the problem at hand. This is sometimes his first mistake. Let us assume that he has surveyed the literature and found “n” publications presenting algorithms which deal directly with his particular problem. At this point in his search he will begin to discover that the mere existence of pertinent algorithms in the literature is not nearly sufficient.



Journal ArticleDOI
TL;DR: A procedure is described for investigating optimum running conditions for a thermal power station to find out if it is feasible to extend the life of a coal-fired power station by up to 30 years.
Abstract: A procedure is described for investigating optimum running conditions for a thermal power station.

Journal ArticleDOI
TL;DR: In this paper, a class of two-variable integer programming problems is defined which are indexed by a continuous parameter $t > 0$ and it is shown that, given any nonempty open interval I of positive real numbers and any number N, there can be found a rational value $t_0 \in I$, such that the problem indexed by $t 0 $ requires at least N cuts before convergence to the integer optimal occurs, using the algorithm of Gomory based on the fractional row cut.
Abstract: A class of two-variable integer programming problems is defined which are indexed by a continuous parameter $t > 0$. It is then shown that, given any nonempty open interval I of positive real numbers and any number N, there can be found a rational value $t_0 \in I$, such that the problem indexed by $t_0 $ requires at least N cuts before convergence to the integer optimal occurs, using the algorithm of Gomory based on the fractional row cut.



Journal ArticleDOI
TL;DR: An integer programming algorithm is proposed for optimizing pipe sizes and slopes in sewer design based on principles similar to the ones advanced by Deininger and Holland.
Abstract: In order to cope with the ever increasing problems of the urban environment, new approaches are being sought to their solution. The objective of this paper is to review and evaluate the merits of certain new methods aimed at finding optimal solutions in sewer design. Based on principles similar to the ones advanced by Deininger [1966] and Holland [1968] the authors propose an integer programming algorithm for optimizing pipe sizes and slopes. The new algorithm is applied to an actual situation and compared with a solution arrived at by a traditional design approach.