scispace - formally typeset
Search or ask a question

Showing papers on "Integer programming published in 1973"


Journal ArticleDOI
TL;DR: Rules are given that enable the transformation of a0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints.
Abstract: This paper gives rules that enable the transformation of a 0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints. Rules are also given that provide reduced numbers of variables when the true variables of interest are not individual cross-product terms, but sums of such terms or polynomials of the form ∑xjp.

118 citations


Journal ArticleDOI
TL;DR: It is shown that no computing device can be programmed to compute the optimum criterion value for all problems in this class of integer programming problems in which squares of variables may occur in the constraints.
Abstract: This paper studies a class of integer programming problems in which squares of variables may occur in the constraints, and shows that no computing device can be programmed to compute the optimum criterion value for all problems in this class.

114 citations


Journal ArticleDOI
TL;DR: The flexibility of this technique is examined through experiments with different branching and subproblem selection strategies, and the efficacy of these various heuristics is assessed.
Abstract: The branch and bound method of solving the mixed integer linear programming problem is summarized. The flexibility of this technique is examined through experiments with different branching and subproblem selection strategies, and the efficacy of these various heuristics is assessed.

92 citations


Journal ArticleDOI
TL;DR: In this article, an efficient heuristic procedure for a special class of mixed integer programming problems called the uncapacitated warehouse (plant) location problem was introduced. But this procedure is derived from the branching decision rules proposed for the branch and bound algorithm by the author in earlier paper.
Abstract: This paper introduces an efficient heuristic procedure for a special class of mixed integer programming problems called the uncapacitated warehouse (plant) location problem. This procedure is derived from the branching decision rules proposed for the branch and bound algorithm by the author in an earlier paper. It can be viewed as tracing a single path of the branch and bound tree (from the initial node to the terminal node), the path being determined by the particular branching decision rule used. Unlike branch and bound the computational efficiency of this procedure is substantially less than linearly related to the number of potential warehouse locations (integer variables) in the problem. Its computational efficiency is tested on problems found in the literature.

84 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that transitive majority decisions can be characterized as basic solutions of a set of linear inequalities, and that constrained majority rule is equivalent to an integer programming problem.
Abstract: In this paper we are concerned with imposing constraints directly on the admissible majority decisions so as to insure transitivity without restricting individual preference orderings. We demonstrate that this corresponds to requiring that majority decisions be confined to the extreme points of a convex polyhedron. Thus, transitive majority decisions can be characterized as basic solutions of a set of linear inequalities. Through the use of a majority decision function (which is not restricted to be linear) it is shown that constrained majority rule is equivalent to an integer programming problem. Some special forms of majority decision functions are studied including the generalized lp norm and an indicator function. Implications of an integer programming solution, including alternate optima and post optimality analysis, are also discussed.

62 citations


Journal ArticleDOI
TL;DR: Pierce and Pierce as mentioned in this paper used combinatorial programming to solve a class of integer programming problems in which all elements are zero or one, by representing the problem elements in a binary computer as bits in a word and employing logical "and" and "or" operations in the problem-solving process.
Abstract: In an earlier paper [Pierce, J. F. 1968. Application of combinatorial programming to a class of all-zero-one integer programming problems. Management Sci.15 3, November 191--209.] combinatorial programming procedures were presented for solving a class of integer programming problems in which all elements are zero or one. By representing the problem elements in a binary computer as bits in a word and employing logical “and” and “or” operations in the problem-solving process, a number of problems involving several hundred integer variables were solved in a matter of seconds. In the present paper a number of improvements in these earlier algorithms are presented, including a new search strategy, methods for reducing the original problem, and mechanisms for feasibility filtering in multi-word problems. With these improvements problem-solving efficiency has been increased in many instances by an order of magnitude. In addition, the present paper contains computational experience obtained in solving problems for the k-best solutions.

56 citations


Journal ArticleDOI
TL;DR: The innovative subroutines are shown to be efficient to compute and effective in finding good integer programming solutions and providing strong lower bounds for the branch and bound search.
Abstract: This paper gives specific computational details and experience with a group theoretic integer programming algorithm. Included among the subroutines are a matrix reduction scheme for obtaining group representations, network algorithms for solving group optimization problems, and a branch and bound search for finding optimal integer programming solutions. The innovative subroutines are shown to be efficient to compute and effective in finding good integer programming solutions and providing strong lower bounds for the branch and bound search.

42 citations


Journal ArticleDOI
01 Feb 1973-Infor
TL;DR: In this article, an improved method for calculating shortest paths is presented, and an integer programming application is presented for integer programming with the shortest path algorithm, which is based on integer programming.
Abstract: An improved method for calculating feth shortest paths is presented. We point out an integer programming application.

31 citations


Journal ArticleDOI
TL;DR: An equivalence is established between solving 0-1 integer programs with quadratic or linear objective functions and solving a cut problem on a related graph.
Abstract: This paper is concerned with the relation between 0-1 integer programs and graphs. An equivalence is established between solving 0-1 integer programs with quadratic or linear objective functions and solving a cut problem on a related graph.

31 citations


01 Jul 1973
TL;DR: In this article, a generalized polar set for nonconvex linear constrained quadratic programs is proposed, where the convexity assumption on the maximand is introduced and a second generalized polar is defined.
Abstract: : In the paper the authors discuss a new method for solving linearly constrained, nonconvex quadratic programs in which the maximand is convex (or the minimand is concave). The approach is based on the adaptation and generalization of the concept of outer polars, first developed in the context of integer programming; but the paper is intended to be self-contained. The authors first construct a generalized polar set for arbitrary (linearly constrained) quadratic programs. The convexity assumption on the maximand is introduced and a second generalized polar is defined. The properties of these generalized polars are used to generate a cutting plane (the polar cut), akin to the intersection cuts used in integer programming. The polar cut is then compared to other cuts proposed in the literature (Hoang Tuy, Ritter, Konno). Next, the authors describe a constraint-activating algorithm, which does not use cutting planes, but constructs a polytope containing the feasible set and contained in its generalized polar, and such that a global maximum, if one exists, is attained at a vertex of this polytope. Finally, the constraint-activating algorithm is illustrated on a numerical example. (Modified author abstract)

28 citations


Journal ArticleDOI
TL;DR: A finite algorithm for solving a special class of 0-1 integer programs is developed from parallels between the work of Hoang Tui and R. D. Young and general "intersection" or "convexity" cut approaches.
Abstract: This paper connects some of the recent developments in concave and integer programming. In particular, it points out parallels between the work of Hoang Tui and R. D. Young. From these methods, a finite algorithm for solving a special class of 0-1 integer programs is developed. Our approach contrasts with an earlier extension of Tui's method due to M. Raghavachari and general "intersection" or "convexity" cut approaches.

Journal ArticleDOI
TL;DR: In this article, a general integer programming algorithm consisting of problem relaxation, solution of the relaxed problem parametrically by dynamic programming, and generation of kth best solutions until a feasible solution is found is presented.
Abstract: When regarded as a shortest route problem, an integer program can be seen to have a particularly simple structure. This allows the development of an algorithm for finding thekth best solution to an integer programming problem with max{O(kmn), O(k logk)} operations. Apart from its value in the parametric study of an optimal solution, the approach leads to a general integer programming algorithm consisting of (1) problem relaxation, (2) solution of the relaxed problem parametrically by dynamic programming, and (3) generation ofkth best solutions until a feasible solution is found. Elementary methods based on duality for reducingk for a given problem relaxation are then outlined, and some examples and computational aspects are discussed.

Journal ArticleDOI
TL;DR: The optimal allocation of personnel in this situation is formulated as an integer linear programming problem, and the special structure of the formulation is exploited in order to develop an efficient solution algorithm which requires only hand calculations.
Abstract: Certain types of organizations operate continuously and must allocate personnel over time to meet fluctuating, but cyclical, staff requirements. The optimal allocation of personnel in this situation is formulated as an integer linear programming problem, and the special structure of the formulation is exploited in order to develop an efficient solution algorithm which requires only hand calculations. The relationship between optimal solutions and flexibility in the original data is also considered.

Journal ArticleDOI
TL;DR: In this paper, a general algorithm for solving the hyperbolic integer program which reduces to solving a sequence of linear integer problems is proposed, which is similar to the Isbell-Marlow procedure.
Abstract: The hyperbolic integer program is treated as a special case of a hyperbolic program with a finite number of feasible points. The continuous hyperbolic program also belongs to this class since its solution can be obtained by considering only the extreme points of the feasible set. A general algorithm for solving the hyperbolic integer program which reduces to solving a sequence of linear integer problems is proposed. When the integer restriction is removed, this algorithm is similar to the Isbell-Marlow procedure. The geometrical aspects of the hyperbolic problem are also discussed and several cutting plane algorithms are given.

Journal ArticleDOI
TL;DR: An eight phase algorithm is described that exploits special structure of the working regulations for full-time and part-time shifts and enables variables to be grouped together and the integer constraints successively introduced.
Abstract: The problem of rostering toll collectors for a toll facility such as a bridge, tunnel, or expressway is formulated as an integer program. The working regulations for full-time and part-time shifts give special structure to the program. An eight phase algorithm is described that exploits this special structure and enables variables to be grouped together and the integer constraints successively introduced. The algorithm has been tested for a wide variety of practical schedule problems and has proved successful in obtaining acceptable schedules in comparatively short computer run times.

Book ChapterDOI
01 Jan 1973
TL;DR: In this article, the authors discuss integer programming by emphasizing computation and relations among models and present a general enumeration algorithm for the mathematical-programming problem of integer linear program (ILP).
Abstract: Publisher Summary This chapter discusses integer programming by emphasizing computation and relations among models. Integer programming deals with the class of mathematical programming problems in which some or all of the variables are required to be integers. The chapter discusses the case in which both the objective function and constraints are linear. It describes integer linear program (ILP) models and some relationships among integer programming models and presents the classification of them according to computational complexity. A difficulty in reporting computational results is the number of different computers used. There is a problem in reporting results on approximate methods. There is virtually no point in reporting times to find a suboptimal solution without also giving its deviation from optimality. Unfortunately, for the large problems for which approximate methods are designed, optimal solutions are generally unknown. The chapter describes the most widely-used algorithms and presents a general enumeration algorithm for the mathematical-programming problem.

01 Jan 1973
TL;DR: In this article, the authors presented a study in association with the Electric Power Systems Engineering Laboratory and Dept. of Civil Engineering, M.I.T., which was conducted at the University of Sheffield.
Abstract: Prepared in association with Electric Power Systems Engineering Laboratory and Dept. of Civil Engineering, M.I.T.

Journal ArticleDOI
TL;DR: This paper develops an algorithm for pure integer programming problems that first transforms the integer programming problem to an algebraically equivalent Hermite canonical problem, and then employs the Fourier-Motzkin elimination to calculate an optimal solution.
Abstract: This paper develops an algorithm for pure integer programming problems. It first transforms the integer programming problem to an algebraically equivalent Hermite canonical problem, and then employs the Fourier-Motzkin elimination. These algebraic operations transform the problem into a form that leads to an efficient implicit enumeration scheme to calculate an optimal solution. The algorithm constructs, in a finite number of operations, an optimal solution to an integer program with n variables and n or n + 1 inequality constraints. If the original problem has more than n + 1 constraints, then the integer program with only the constraints that are binding at an optimal linear programming solution is solved in place of the original problem. Computational results are presented.

Journal ArticleDOI
TL;DR: The purpose of this paper is to show that the group problem associated with the fixed-charge transportation problem can be viewed as a multiparametric integer programming problem having a totally unimodular constraint matrix.
Abstract: The multiparametric integer programming problem for the right-hand side is to minimize c′t subject to At = b(y), t ≧ 0, t = 0 (mod 1), where b(y) can be expressed in the form b(y) = b + F(y), where F is a matrix of constant coefficients, and y is an integer vector parameter. The group problem associated with any integer programming problem may be viewed as a multiparametric integer programming problem. The purpose of this paper is to show that the group problem associated with the fixed-charge transportation problem can be viewed as a multiparametric integer programming problem having a totally unimodular constraint matrix.

Proceedings ArticleDOI
27 Aug 1973
TL;DR: The lock box problem involves the location of post office boxes within a company's distribution area to locate the boxes in a way that will minimize processing cost and the opportunity costs associated with the remittances while in transit.
Abstract: The lock box problem involves the location of post office boxes within a company's distribution area. Customer remittances are mailed to these boxes and the checks are processed by a local bank. The problem is to locate the boxes in a way that will minimize processing cost and the opportunity costs associated with the remittances while in transit (float costs). For m potential lock box locations and n customer groups, the problem can be formulated as a zero-one integer programming problem with mn + n variables and m + n constraints. The problem, however, can be partitioned in a way that results in a zero-one integer programming problem with only m variables. Once values have been established for these m variables, values for the remaining mn zero-one variables can be determined by a trivial process. Thus the problem reduces to determining values for the m zero-one variables. This is accomplished by an implicit enumeration procedure. Computational results are reported for problems involving up to 5050 variables.

Journal ArticleDOI
TL;DR: This paper presents a generalization, called polaroid, of the concept of polar sets, indicating that the new concept will be fruitfully used in an area of non‐convex programming as well as in integer programming, by means of polaroid cuts.
Abstract: : The paper presents a generalization, called polaroid, of the concept of polar sets. A list of properties satisfied by polaroids is established indicating that the new concept may be fruitfully used in an area of non-convex (called here polar) programming as well as in integer programming, by means of polaroid cuts; this class of new cuts contains the ones defined by Tuy for concave programming (a special case of polar programming) and by Balas for integer programming; it furthermore provides for new degrees of freedom in the construction of algorithms in the above-mentioned areas of mathematical programming.


Book ChapterDOI
Harlan Crowder1, Ellis L. Johnson1
01 Jan 1973
TL;DR: The use of cyclic groups, cutting planes, and shortest paths to provide bounds is very much along the lines of Tomlin's use of Gomory's mixed integer cut as discussed by the authors.
Abstract: Publisher Summary The use of cyclic groups, cutting planes, and shortest paths to provide bounds is very much along the lines of Tomlin's use of Gomory's mixed integer cut. That cut is what the cutting plane algorithm gives after one iteration. Initially, the linear program obtained by dropping all of the integer restrictions is solved as a linear program. The algorithm has two general steps: (1) choose some linear program and solve it as a linear program and (2) choose some variable that is required to be integer-valued but whose value is not integer. Then, create two new linear programs. There are two choices that must be made: the linear program to solve in step 1 and the variable to branch on in step 2. Once a variable in step 2 is chosen creating two new linear programs, one always chooses the next linear program in step 1 to be one of these two linear programs. Whenever one returns to step 1 from one of the three possible cases in step 1, one have not just created two new linear programs and the choice of linear program is completely open. Each such choice begins a new major cycle. The first major cycle consists of solving the linear program, branching on some variable, solving one of these two linear programs, and continuing until the linear program chosen is either infeasible or has an integer optimum solution.

Journal ArticleDOI
TL;DR: An heuristic method for determining low-cost solutions of one-machine scheduling problems with delay costs, which involves starting with a logically determined initial sequence and then successively improving the sequence by reordering, using insertion, exchange, and displacement operations.
Abstract: An heuristic method is presented for determining low-cost solutions of one-machine scheduling problems with delay costs. The algorithm involves starting with a logically determined initial sequence and then successively improving the sequence by reordering, using insertion, exchange, and displacement operations. Computational results on a set of 197 problems, with up to 40 jobs, are compared to branch and bound, and integer programming computational experience. The reordering algorithm produced optimal solutions for all problems in this set.

01 Nov 1973
TL;DR: Evidence is presented, in terms of the number of variables which have to be fixed to locate the 1% and 5% optima, which strongly supports the view that this order-of-magnitude speed-up will be independent of the precise branch-and-bound algorithm that is used.
Abstract: : The paper reports an empirical discovery in integer programming. A version of the branch-and-bound approach is used as a control, and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree. It is shown that the imbedded linear search locates solutions within 1%, and solutions within 5%, of the theoretical optimum, which in fact can be seen to have this proximity to the theoretical optimum at the time of termination of computation, over (10) times faster than the control. Evidence is presented, in terms of the number of variables which have to be fixed to locate the 1% and 5% optima, which strongly supports the view that this order-of-magnitude speed-up will be independent of the precise branch-and-bound algorithm that is used. (Author)

Journal ArticleDOI
TL;DR: In this article, a mixed integer programming approach to selection of municipal water source-facility combinations overcomes many of the limitations of linear programming is presented. But the mixed integer concept allows separation of capital investment and operating costs for detailed analysis of the least cost of insuring desired level of service.
Abstract: The mixed integer programming approach to selection of municipal water source-facility combinations overcomes many of the limitations of linear programming. The mixed integer concept allows separation of capital investment and operating costs for detailed analysis of the least cost of insuring desired level of service. Standard of pipe and equipment require discrete, not continuous variables. Chance constrained programming allows uncertainty in both supply and demand to be considered in the model. An optimizing algorithm based upon the Branch Bound concept of mixed integer programming selects the optimum combination of source-facilities from all possible alternatives. The procedure is demonstrated by developing a model for an example problem and the optimum solution is given.

Journal ArticleDOI
TL;DR: In this article, the authors illustrate some computational experience using existing integer algorithms to solve a set of capital budgeting problems and to begin to catalog the performance of integer codes on financial problems.
Abstract: Solving capital budgeting problems with linear and integer programming has been part of the finance literature for some time [21, 22, 23, 7, 14, and 18]. Capital budgeting problems have unique properties that distinguish them from other integer linear problems discussed in the mathematical programming literature. Capital budgeting problems generally have the following characteristics: (1) the matrix tends to be rectangular with more variables than constraints; (2) they are all maximization problems with ≤ constraints and nonnegativity conditions in the general form 0≤xi≤1 in the case of linear programming and xi = 0, 1 in the case of integer problems; and (3) there are often mutually exclusive projects among the variables. The purposes of this note are to illustrate some computational experience using existing integer algorithms to solve a set of capital budgeting problems and to begin to catalog the performance of integer codes on financial problems.

Journal ArticleDOI
TL;DR: While the group-theoretic approach is in many ways useful for 0-1 problems too, in this latter case the sufficient condition given by Gomory for a solution to the group problem to solve the initial integer program is never satisfied.
Abstract: While the group-theoretic approach is in many ways useful for 0-1 problems too, in this latter case the sufficient condition given by Gomory for a solution to the group problem to solve the initial integer program is never satisfied. This underscores the need to use information about the constraints that are slack at the linear programming optimum.

Journal ArticleDOI
01 May 1973
TL;DR: An algorithm is presented for solving all-integer linear programming problems which successively searches hyperplanes parallel to the optimal hyperplane containing the solution to the associated linear programming problem with the integer restrictions suppressed.
Abstract: An algorithm is presented for solving all-integer linear programming problems which successively searches hyperplanes parallel to the optimal hyperplane containing the solution to the associated linear programming problem with the integer restrictions suppressed. These hyperplanes are searched for integer solution points using a dynamic programming formulation. Numerical examples of this technique and some computational results are also presented.

Journal ArticleDOI
TL;DR: In this paper, a method of reliability optimization of a system subject to multiple constraints with integer coefficients is described, where the coefficients are generally integers; in case they are not, they can be converted into integer form by simple manipulation.