Showing papers on "Linear programming published in 1969"
TL;DR: A zero-one 0-1 linear programming formulation of multiproject and job-shop scheduling problems is presented that is more general and computationally tractable than other known formulations.
Abstract: A zero-one 0-1 linear programming formulation of multiproject and job-shop scheduling problems is presented that is more general and computationally tractable than other known formulations. It can accommodate a wide range of real-world situations including multiple resource constraints, due dates, job splitting, resource, substitutability, and concurrency and nonconcurrency of job performance requirements. Three possible objective functions are discussed; minimizing total throughput time for all projects: minimizing the time by which all projects are completed i.e., minimizing makespan; and minimizing total lateness or lateness penalty for all projects.
747 citations
IBM1
TL;DR: In this article, a theory and algorithms for asymptotic integer programs are described and a class of polyhedra is introduced, which are cross sections of more symmetric higher dimensional polyhedras whose properties are then studied.
407 citations
TL;DR: In this paper, a linear decision rule is proposed to specify the release during any period of reservoir operation as the difference between the storage at the beginning of the period and a decision parameter for the period.
Abstract: With the aid of a linear decision rule, reservoir management and design problems often can be formulated as easily solved linear programing problems. The linear decision rule specifies the release during any period of reservoir operation as the difference between the storage at the beginning of the period and a decision parameter for the period. The decision parameters for the entire study horizon are determined by solving the linear programing problem. Problems may be formulated in either the deterministic or the stochastic environment.
290 citations
TL;DR: In this article, the LU decomposition is computed with row interchanges of the basic matrix of Dantzig's simplex method, which is based on the LU matrix decomposition.
Abstract: Standard computer implementations of Dantzig's simplex method for linear programming are based upon forming the inverse of the basic matrix and updating the inverse after every step of the method. These implementations have bad round-off error properties. This paper gives the theoretical background for an implementation which is based upon the LU decomposition, computed with row interchanges, of the basic matrix. The implementation is slow, but has good round-off error behavior. The implementation appears as CACM Algorithm 350.
232 citations
TL;DR: In this article, a simple implicit enumeration algorithm fitted with optional imbedded linear programming machinery was implemented and tested extensively on an IBM 7044 and shown to reduce solution times by a factor of about 100.
Abstract: This paper synthesizes the Balasian implicit enumeration approach to integer linear programming with the approach typified by Land and Doig and by Roy, Bertier, and Nghiem. The synthesis results from the use of an imbedded linear program to compute surrogate constraints that are as "strong" as possible in a sense slightly different from that originally used by Glover. A simple implicit enumeration algorithm fitted with optional imbedded linear programming machinery was implemented and tested extensively on an IBM 7044. Use of the imbedded linear program greatly reduced solution times in virtually every case, and seemed to render the tested algorithm superior to the five other implicit enumeration algorithms for which comparable published experience was available. The crucial issue of the sensitivity of solution time to the number of integer variables was given special attention. Sequences were run of set-covering, optimal-routing, and knapsack problems with multiple constraints of varying sizes up to 90 variables. The results suggest that use of the imbedded linear program in the prescribed way may mitigate solution-time dependence on the number of variables from an exponential to a low-order polynomial increase. The dependence appeared to be approximately linear for the first two problem classes, with 90-variable problems typically being solved in about 15 seconds; and approximately cubic for the third class, with 80-variable problems typically solved in less than 2 minutes. In the 35-variable range for all three classes, use of the imbedded linear program reduced solution times by a factor of about 100.
214 citations
TL;DR: This paper describes an algorithm for a ship scheduling problem, obtained from a Swedish shipowning company, that uses the Dantzig-Wolfe decomposition method for linear programming and some integer programming experiments have been made.
Abstract: This paper describes an algorithm for a ship scheduling problem, obtained from a Swedish shipowning company The algorithm uses the Dantzig-Wolfe decomposition method for linear programming The subprograms are simple network flow problems that are solved by dynamic programming The master program in the decomposition algorithm is an LP problem with only zero-one elements in the matrix and the right-hand side Integer solutions are not guaranteed, but generation and solution of a large number of problems indicates that the frequency of fractional solutions is as small as 1–2 per cent Problems with about 40 ships and 50 cargoes are solved in about 25 minutes on an IBM 7090 In order to resolve the fractional cases, some integer programming experiments have been made The results will be reported in a forthcoming paper
166 citations
TL;DR: This paper considers the solution of the corresponding nonlinear minimax approximation problems, by solving a sequence of linear discrete T-problems.
Abstract: are now well understood. In particular the equivalence of (i) with a linear programming problem permits its solution under very general conditions (see, for example, Kelley (1959), Stiefel (1960), Rice (1964), Osborne and Watson (1967, 1968)). In this paper, we consider the solution of the corresponding nonlinear minimax approximation problems, by solving a sequence of linear discrete T-problems. Certain properties of the solution of the linear problem are required, in particular, properties relevant to its solution as a linear programming problem, and these we now summarise. If we write r=f-Aa, (1.1)
109 citations
TL;DR: In this paper, conditions which are necessary and sufficient for convergence of a nonlinear programming algorithm are discussed, and it is shown that the convergence conditions can be easily applied to most programming algorithms.
Abstract: Conditions which are necessary and sufficient for convergence of a nonlinear programming algorithm are stated. It is also shown that the convergence conditions can be easily applied to most programming algorithms. As examples, algorithms by Arrow, Hurwicz and Uzawa; Cauchy; Frank and Wolfe; and Newton-Raphson are proven to converge by direct application of the convergence conditions. Also the Topkis-Veinott convergence conditions for feasible direction algorithms are shown to be a special case of the conditions stated in this paper.
96 citations
TL;DR: The Criss-Cross Method of solving linear programming problems, a primal-dual scheme, normally begins with a problem solution that is neither primal nor dual feasible, and generates an optimal feasible solution in a finite number of iterations.
Abstract: This paper1 describes the Criss-Cross Method of solving linear programming problems.
The method, a primal-dual scheme, normally begins with a problem solution that is neither primal nor dual feasible, and generates an optimal feasible solution in a finite number of iterations. Convergence of the method is proved and flow charts of the method are presented. The method has been programmed in FORTRAN and has been run on a number of computers including the IBM 1620, the IBM 7044, the CDC G-20, and the CDC 6400. A number of problems have been solved using the Criss-Cross method, and some comparisons between the Criss-Cross method and the Simplex method have been made. The results, though scanty, are favorable for the Criss-Cross method.
A means of using the product form of the inverse with the Criss-Cross method is also discussed.
85 citations
01 Jun 1969
TL;DR: In this paper, duality for linear and nonlinear programs with arbitrary variables is discussed and algorithms based on duality constructions based on them are discussed. But the most important class of such problems is that of mixed-integer (linear and non-linear) programs.
Abstract: : The paper discusses duality for linear and nonlinear programs in which some of the variables are arbitrarily constrained. The most important class of such problems is that of mixed-integer (linear and nonlinear) programs. The paper introduces the duality constructions and discusses algorithms based on them.
65 citations
TL;DR: In this article, the problem of ranking n objects from 'best' to 'worst', using the results of a paired comparison experiment, is formulated as a linear programming problem with the logarithm of the likelihood as the objective function.
Abstract: SUMMARY The problem of ranking n objects from 'best' to 'worst', using the results of a paired comparison experiment, is formulated as a linear programming problem with the logarithm of the likelihood as the objective function. The constraints are determined by the transitivity relationships implied in a ranking. The linear program will determine maximum likelihood rankings for the k fold replicated paired comparison experiment with or without the possibility of ties. The method is demonstrated with two examples from recent literature.
TL;DR: In this paper, the assignment model of linear programming is extended to allow for vector optimizations and dynamic interactions between assigned personnel and positions in each of which a variety of possible measures and approaches are explored.
Abstract: The assignment model of linear programming is here extended to allow for vector optimizations and dynamic interactions between assigned personnel and positions in each of which a variety of possible measures and approaches are explored. Formulations involving people-to-people as well as people-to-position matchings are also examined from the standpoint of organizations in which jobs may be fitted to people or vice versa as well as in weighted combinations. Possible uses of such models for dealing with the problems of placing disadvantsged or handicapped persons are noted, but the analysis stops short of the still further possibilities offered by new types of machine-technology and information systems designs.
TL;DR: In this article, the reliability optimization problems with N stages or subsystems in series, utilizing parallel or series redundant units, can be formulated and solved as integer programming problems and expressions for the probability of failure in each case are developed.
Abstract: Reliability optimization problems with N stages or subsystems in series, utilizing parallel or series redundant units, can be formulated and solved as integer programming problems. The systems considered have subsystems with components which can fail in several modes and are subject to linear and nonlinear constraints. Two situations are considered in which components within the subsystem 1) all fail in the same mode, or 2) all may fail in different modes. Expressions are developed for the probability of failure in each case. Two examples are solved.
TL;DR: In this paper, the authors discuss some technical aspects of the linear programming approach, including the formulation of the problem, some alternative constraint possibilities, branch and bound techniques to avoid splitting neighborhoods, and improvement in execution speed.
Abstract: The recent Supreme Court decision barring “freedom of choice” pupil assignment plans has forced a large number of school districts to devise alternative approaches to school desegregation. One possibility is establishing attendance boundaries to mix races and linear programming is one method of planning such boundary lines. This paper discusses some technical aspects of the linear programming approach, including the formulation of the problem, some alternative constraint possibilities, branch and bound techniques to avoid splitting neighborhoods, and improvement in execution speed.
TL;DR: In this article, a reformulation of the Simplex method in which all tableaux are in standard form, so that use can be made of the symmetry properties in every tableau is presented.
Abstract: : For the solution of convex quadratic programming problem, a number of efficient methods have been developed. The most well-known methods are the Simplex method for quadratic programming, discovered by Dantzig and, together with the closely related dual method, further developed by van de Panne and Whinston, and methods developed by Beale, Houthakker and Wolfe. The authors have shown that the methods by Beale and Houthakker can be considered as variants of the Simplex method for quadratic programming or are closely related to it. Compared with the Simplex tableaux used in linear programming, quadratic programming tableaux have a larger size. A tableau for a linear programming problem with n variables and m constraints had (m + l) (n + l) nontrivial elements, while a Simplex tableau for a quadratic programming problem with the same number of variables and constraints has (m + n + l) elements. In the Simplex method for quadratic programming, a considerable number of tableaux will be in standard form, which means that the tableau can be divided in symmetric and skew-symmetric parts, so that the number of elements to be computed and stored is reduced by nearly one half. However, nonstandard tableaux do not have these symmetry properties, so that all elements of these tableaux must be computed. This paper gives a reformulation of the Simplex method in which all tableaux are in standard form, so that use can be made of the symmetry properties in every tableau. The actual number of nontrivial elements in a quadratic Simplex tableau is therefore decreased by a factor of 2. This symmetric formulation has other advantages as well. (Author)
TL;DR: In this article, the authors considered the infinite horizon stationary case and showed that two curves in the plane determine an optimal policy with slopes between minus one and zero, differentiable, and bounded by two straight lines with a slope of minus one.
Abstract: In an earlier paper [5], we generalized and extended Beckmann'a results [1] for a production and inventory problem with proportional smoothing costs and demands being random variables. Our previous results concerned the finite horizon nonstationary case. Here we consider the infinite horizon stationary case. Two curves in the plane determine an optimal policy. They are shown to have slopes between minus one and zero, to be differentiable, and to be bounded by two straight lines with a slope of minus one. These results are used (a) to accelerate each iteration of a successive approximations algorithm and (b) to formulate a linear programming problem from whose solution an optimal policy can be determined.
TL;DR: In this paper, a mathematical model of regional water quality management using bypass piping is presented, which is developed within the framework of linear programming and a large-scale problem is solved using semi-realistic data from the Delaware Estuary.
Abstract: This paper presents a mathematical model of regional water quality management using by-pass piping. The model is developed within the framework of linear programming and a large-scale problem is solved using semi-realistic data from the Delaware Estuary. The technique of generation of elements is used in conjunction with the truncated tableau to provide efficient solutions. A possible method of taxation is indicated based on the values of the dual variables.
TL;DR: This article discusses the optimization of a water supply system by linear programming and the various formulas used are discussed and a mathematical model is developed.
Abstract: This article discusses the optimization of a water supply system by linear programming. The various formulas used are discussed and a mathematical model is developed. A problem is solved to illustrate the use of the mathematical formulation developed earlier. Various limitations of the mathematical model are also discussed.
TL;DR: This survey examines eight mathematical programming models of pattern classification and finds that six of the models produce decision rules that maximize a function the authors call the quality of a decision rule.
Abstract: This survey examines eight mathematical programming models of pattern classification. The paper determines the range of applicability and computational merits of each model. The flexibility and large sample properties of the models are also discussed. We find that six of the models produce decision rules that maximize a function we call the quality of a decision rule. The remaining two models minimize a weighted sum of errors.
TL;DR: A new algorithm for solving the pure-integer linear programming problem with general integer variables is presented and evaluated and encouraging computational experience is reported that suggests that this algorithm should compare favorably in efficiency with existing algorithms.
Abstract: A new algorithm for solving the pure-integer linear programming problem with general integer variables is presented and evaluated. Roughly speaking, this algorithm proceeds by obtaining tight bounds or conditional bounds on the relevant values of the respective variables, and then identifying a sequence of constantly improving feasible solutions by scanning the relevant solutions. Encouraging computational experience is reported that suggests that this algorithm should compare favorably in efficiency with existing algorithms. Plans for investigating ways of further increasing the efficiency of the algorithm and of extending it to more general problems also are outlined.
TL;DR: The analysis attempts to define relatively distribution-free tolerance levels and the incidence of nonnormality in chance-constrained linear programming.
Abstract: The approach of chance-constrained linear programming is analyzed here in the context of safety-first principles based on Tchebycheff-type inequalities. The analysis attempts to define relatively distribution-free tolerance levels and the incidence of nonnormality in chance-constrained linear programming.
TL;DR: In this paper, the authors demonstrate the use of a linear programming model in open pit mine evaluation, where the primary conditions to be incorporated into the model are the course and the inclination of the ore vein, and during the whole life of the mine, safe slope angles to be maintained.
Abstract: The purpose of this paper is to demonstrate the use of a Linear Programming model in open pit mine evaluation. The primary conditions to be incorporated into the model are the course and the inclination of the ore vein, and, during the whole life of the mine, safe slope angles to be maintained. Subject to this the objective of running a mine is assumed to be maximizing the direct operating profits which consist of proceeds from the sale of ore minus direct operating cost of extraction, transportation, and upgrading. First the Linear Programming model is developed, and subsequently some computational aspects are treated.
TL;DR: In this article, a network of road links is constructed and the traffic demand is modeled as a relation between two nodes in the network, and the overall system is decomposed into a NET OF NETWORK PROBLEMS, where each node will represent the road system in one time period.
TL;DR: This article describes an analytic approach to flight scheduling within an airlift system that consists of a monthly planning model that produces an initial schedule and a daily model for making periodic changes in the schedule, formulated as two-stage stochastic linear programs.
Abstract: This article describes an analytic approach to flight scheduling within an airlift system. The model takes explicit account of the uncertainty present in cargo requirements or demands. For computational feasibility, the approach consists of two related models: (1) a monthly planning model that produces an initial schedule, and (2) a daily model for making periodic changes in the schedule. Both are formulated as two-stage stochastic linear programs. A detailed mathematical description of each model and its physical interpretation is given.
The monthly model determines the number of flights each type of aircraft in the fleet. Excess demands on certain routes are assumed to be met, at least in part, by spot procurement of commerical lift from outside the system. The flight assignment is determined by minimizing the expected total system cost, which consists of operating costs, costs of reallocating aircraft to different routes, spot commercial procurement costs, and other penalty costs of excess demand. The model accounts for limitations on the number of flying hours and the carrying capacities of various aircraft in satisfying demands.
In the daily model the number of aircraft of each type to switch from one route to another and the number of commercial flights on spot contract to add on the current day are the principal decision variables. These are determined by balancing operating, procurement, and redistribution costs against the expected costs of additional cargo delay. The current state of the system — the amount of unmoved cargo on various routes andathe position of aircraft throughout the system — plays a role in determining these decisions.
A description of two variants of an algorithm recently developed for this class of problems is presented. Both versions, which use ideas from convex programming, make extensive use of linear programming codes for the brunt of the calculations. The models may thus be solved by augmenting existing linear programming routines.
Book•
31 Dec 1969
TL;DR: Linear Equations and Straight Lines Matrices Linear Programming, A Geometric Approach The Simplex Method Sets and Counting Probability Probability and Statistics Markov Processes The Theory of Games The Mathematics of Finance Difference Equations, Mathematical Models Logic Graphs as mentioned in this paper
Abstract: Linear Equations and Straight Lines Matrices Linear Programming, A Geometric Approach The Simplex Method Sets and Counting Probability Probability and Statistics Markov Processes The Theory of Games The Mathematics of Finance Difference Equations and Mathematical Models Logic Graphs
TL;DR: In this article, it is shown how to obtain the dual variables of a linear program if the problem is solved by using the Dantzig-Wolfe decomposition principle, which is a generalization of the simplex method.
Abstract: It is well known that the dual variables of a linear program may be obtained easily if the simplex method is vised to solve the problem. This note shows how to obtain these dual variables if the problem is solved by using the Dantzig-Wolfe decomposition principle.
01 Jan 1969
TL;DR: Ten pivot selection rules for representing the inverse of a sparse basis in triangularized product form are compared and one of the rules yield inverses that were only slightly less sparse than the original basis.
Abstract: : The authors empirically compared ten pivot selection rules for representing the inverse of a sparse basis in triangularized product form. On examples drawn from actual applications, one of the rules yield inverses that were only slightly less sparse than the original basis. The rule was used in the M5 mathematical programming system and has resulted in substantial reduction in running time.