Showing papers on "Linear programming published in 1978"
TL;DR: A nonlinear (nonconvex) programming model provides a new definition of efficiency for use in evaluating activities of not-for-profit entities participating in public programs and methods for objectively determining weights by reference to the observational data for the multiple outputs and multiple inputs that characterize such programs.
25,433 citations
TL;DR: Inner approximation algorithms have had two major roles in the mathematical programming literature: their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle and recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs.
Abstract: Inner approximation algorithms have had two major roles in the mathematical programming literature. Their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle. However, recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs. Avriel and Williams' Avriel, M., A. C. Williams. 1970. Complementary geometric programming. SIAM J. Appl. Math.19 125-141. complementary geometric programming algorithm, Duffin and Peterson's Duffin, R. J., E. L. Peterson. 1972. Reversed geometric programs treated by harmonic means. Indiana Univ. Math. J.22 531-550. reversed geometric programming algorithms, Reklaitis and Wilde's Reklaitis, G. V., D. J. Wilde. 1974. Geometric programming via a primal auxiliary problem. AIIE Trans.6 308-317. primal reversed geometric programming algorithm, and Bitran and Novaes' Bitran, G. R., A. G. Novaes. 1973. Linear programming with a fractional objective function. Opns. Res.21 22-29. linear fractional programming algorithm are all examples of this class of inner approximation algorithms. A sequence of approximating convex programs are solved in each of these algorithms. Rosen's Rosen, J. B. 1966. Iterative solution of nonlinear optimal control problems. SIAM J. Control4 223-244. inner approximation algorithm is a special case of the general inner approximation algorithm presented in this note.
957 citations
TL;DR: This approach has obtained and verified optimal solutions to all the Kuehn-Hamburger location problems in well under 0.1 seconds each on an IBM 360/91 computer, with no branching required.
Abstract: We develop and test a method for the uncapacitated facility location problem that is based on a linear programming dual formation. A simple ascent and adjustment procedure frequently produces optimal dual solutions, which in turn often correspond directly to optimal integer primal solutions. If not, a branch-and-bound procedure completes the solution process. This approach has obtained and verified optimal solutions to all the Kuehn-Hamburger location problems in well under 0.1 seconds each on an IBM 360/91 computer, with no branching required. Computational tests on problems with as many as 100 potential facility locations provide evidence that this approach is superior to several other methods.
914 citations
TL;DR: An algorithm for solving large-scale nonlinear programs with linear constraints is presented, which combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities.
Abstract: An algorithm for solving large-scale nonlinear programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.
510 citations
TL;DR: A method for determining the unit commitment schedule for hydro-thermal systems using extensions and modifications of the Branch and Bound method for Inteler Programming has been developed and significant features include its computational practicability for realistic systems and proper representation of reserves associated with different risk levels.
Abstract: A method for determining the unit commitment schedule for hydro-thermal systems using extensions and modifications of the Branch and Bound method for Inteler Programming has been developed. Significant features of the method include its computational practicability for realistic systems and proper representation of reserves associated with different risk levels. Contracts relating to power interchange have also been adequately modelled for such an approach.
372 citations
TL;DR: It is shown that no more than O(m 2) preemptions are necessary, in order to schedule n jobs on m unrelated processors so as to minimize makespan.
Abstract: It IS shown that certain problems of optimal preemptive scheduling of unrelated parallel processors can be formulated and solved as hnear programming problems As a by-product of the linear programming formulaUons of these problems, upper bounds are obtained on the number of preempuons required for optimal schedules In particular it is shown that no more than O(m 2) preemptions are necessary, m order to schedule n jobs on m unrelated processors so as to minimize makespan
285 citations
TL;DR: In this article, the first paper in a two-part report on linear programming (LP) applied to power system security control calculations is presented, where both corrective and preventive security control strategies are dealt with.
Abstract: This is the first paper in a two-part report on linear programming (LP) applied to power system security control calculations. The advantages of an LP approach are complete computational reliabilit and very high speed, making it suitable for real-time or study-mode purposes. Both corrective and preventive security control strategies are dealt with. Active- power control is exercised over generators, phase shifters and hvdc links. Load shedding and emergency start-up of plant are included. Part I presents the basic LP formulation and algorithm, and gives the computational details necessary to obtain very fast solutions. Specific applications to practical power systems are described in Part II1.
248 citations
01 Sep 1978
TL;DR: In this paper, the problem of estimating the state variables from measurements in an electric-power system is considered, and a linear program is proposed for real-time power system monitoring and control where process variables have unknown statistics.
Abstract: The problem of estimating the state variables from measurements in an electric-power system is considered. The conventional linearised least-squares solution is shown to be ineffective in the presence of gross measurement errors. Reformulating the problem as a linear program leads to a state estimator that combines the advantages of noise filtering and bad-data elimination, and may be implemented straightforwardly by application of the simplex method. The solution of various examples based on three test networks confirms the advantages of the method especially where the data are corrupted by a number of gross errors. Depending on the degree of redundancy in the measurement set, the computational requirements of the method are comparable with conventional least-squares solution. For real-time power-system monitoring and control where process variables have unknown statistics, the linear-programming method is believed to be more efficient than conventional algorithms.
167 citations
TL;DR: Simple characterizations of the efficiency of an edge incident to a nondegenerate or a degenerate efficient vertex are given and form the basis of an algorithm for enumerating all efficient vertices.
Abstract: In this paper we develop a method for finding all efficient extreme points for multiple objective linear programs. Simple characterizations of the efficiency of an edge incident to a nondegenerate or a degenerate efficient vertex are given. These characterizations form the basis of an algorithm for enumerating all efficient vertices. The algorithm appears to have definite computational advantages over other methods. Some illustrative examples are included.
125 citations
TL;DR: An interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem.
Abstract: In many situations it is under legislative mandate to manage publicly owned forest resources for multiple uses e.g., timber production, hunting, grazing. The major obstacle that has been encountered in applying previously developed mathematical programming procedures to multiple-use forest management has been the difficulty in assessing the appropriate criterion weights required. To avoid the criterion weight estimation problem, an interactive multiple-objective linear programming approach, which does not require criterion weights of any kind, was developed in response to the needs of the multiple-use forest management problem. The procedure uses a combination of linear programming and vector-maximum techniques. At each iteration the cone generated by the gradients of the multiple objectives is contracted. On the last two iterations the most acceptable efficient extreme point is identified with the aid of a filtering device. As illustrated, the method has been applied to prepare preliminary management plans for a 10,000-acre sub-unit of a national forest.
110 citations
TL;DR: The numerical results reported here, combined with the fact that in the absence of constraints the present algorithm reduces to the earlier unconstrained $l_1$ algorithm, indicate that this algorithm is very efficient.
Abstract: We describe an algorithm, based on the simplex method of linear programming, for solving the discrete $l_1$ approximation problem with any type of linear constraints. The numerical results reported here, combined with the fact that in the absence of constraints the present algorithm reduces to our earlier unconstrained $l_1$ algorithm, indicate that this algorithm is very efficient.
01 Dec 1978
TL;DR: Existence andDuality properties for multiple objective linear programs are developed which contain the fundamental existence and duality results of linear programming as special cases.
Abstract: This paper is a contribution to the theory of multiple objective linear programming. Existence and duality properties for multiple objective linear programs are developed which contain the fundamental existence and duality results of linear programming as special cases. Several implications of the duality results will be indicated.
01 Jan 1978
TL;DR: In this paper, it was shown that the linear complementarity problem of finding an n-by-1 vector x such that Mx+q≧0, x≧ 0, and x T(Mx +q)=0, where T is a linear program.
Abstract: It is shown that the linear complementarity problem of finding an n-by-1 vector x such that Mx+q≧0, x≧0, and x T(Mx+q)=0, where M is a given n-by-n real matrix and q is a given n-by-1 vector, is solvable if and only if the linear program: minimize p T x subject to Mx+q≧0, x≧0, is solvable, where p is an n-by-1 vector which satisfies certain conditions. Furthermore each solution of the linear program, solves the linear complementarity problem. For a number of special cases including those when M has nonpositive off-diagonal elements, or when M is strictly or irreducibly diagonally dominant, or when M is a positive matrix with a dominant diagonal columnwise, p is very easily determined and the linear complementarity problem can be solved as an ordinary linear program. Examples of linear complementarity problems are given which can be solved as linear programs, but not by Lemke's method or the principal pivoting method.
TL;DR: Computational results are reported to show that linear programming often produces integer solutions to uncapacitated problems as required, and it is suggested that this represents a practical solution approach.
Abstract: This paper considers a class of feasible set fixed-charge depot location problems which have been formulated as mixed-integer programmes. Computational results are reported to show that linear programming often produces integer solutions to uncapa- citated problems as required. It is suggested that this represents a practical solution approach. Computational evidence suggests this convenient property does not extend to capacitated problems. Discussion of reducing infinite set problems to such feasible set problems is included. This paper considers depot location-demand allocation problems where loca- tions are to be chosen from a finite set of candidate sites. This corresponds to the "feasible set approach", discussed by Rand.' The objective is to locate depots so that all customer demand is allocated among the depots while minimizing the sum of variable and fixed depot costs associated with satisfy- ing that demand. Demand is assumed known in each of a number of customer zones. The modelling intent is to answer the four fundamental questions listed by Rand as: How many depots should there be? Where should they be? Which customers should they serve? How big should they be? A mixed integer programming model-sometimes called the fixed-charge plant loca- tion model is used for which efficient special purpose algorithms exist (see Elshafei2 and Geoffrion and Graves3). However, it is a nontrivial task to develop the necessary computer programs. The purpose of this paper is to provide computational results which indicate that ordinary linear program- ming typically produces integer solutions to uncapacitated problems. It is suggested that this represents a practical solution approach if the problem size is not too large, but computational results show that this approach seems inappropriate for capacitated problems. Discussion of reducing certain infinite set problems to feasible set problems is presented in an Appendix.
TL;DR: In this paper, a modified random walk production-inventory heuristic for the aggregate planning problem is proposed. But, the proposed approach is contrasted with linear programming, parametric programming, and linear decision rule optimal and near optimal solutions for several well known production situations.
Abstract: A number of approaches to the aggregate planning problem have been proposed in the literature, yet experience suggests that industrial concerns seldom use these models in actual planning situations. This paper describes a modified random walk production-inventory heuristic for the problem which should appeal to managers on the basis of simplicity as well as efficiency. The proposed approach is contrasted with linear programming, parametric programming, and linear decision rule optimal and near optimal solutions for several well known production situations. The simple production switching heuristic produces schedules which exceed optimal schedules by only 1 to 2 percent of total production costs in all cases.
Journal Article•
TL;DR: In this paper, an integrated form of yield line method incorporating finite elements and linear programming is developed and applied to some illustrative problems, which is simple to operate and provides an upper bound to the collapse load of the continuous slab.
Abstract: An integrated form of yield line method incorporating finite elements and linear programming is developed and applied to some illustrative problems. The method is simple to operate and provides an upper bound to the collapse load of the continuous slab. The method is a further example of a significant trend towards the solution of problems of engineering plasticity by mathematical programming.(a) /TRRL/
TL;DR: In this paper an algorithm is developed and computational experience provided for solving zero-one integer programs with many variables and few constraints.
01 Jan 1978
TL;DR: The paper relates three duality concepts in multiple objective linear programming — the concepts of Gale-Kuhn-Tucker, Isermann and Kornbluth — to each other and indicates some decision-oriented implications of duality.
Abstract: The paper relates three duality concepts in multiple objective linear programming — the concepts of Gale-Kuhn-Tucker, Isermann and Kornbluth — to each other and indicates some decision-oriented implications of duality.
TL;DR: This paper describes the experimental results of testing a “large-scale” program for solving minimum-cost network flow problems, a variant of the primal revised simplex method that substantially improves computer processing time and core storage, especially for relatively large network problems.
Abstract: This paper describes the experimental results of testing a “large-scale” program for solving minimum-cost network flow problems. With this program, general structure transshipment problems with over ten thousand nodes and thirty thousand arcs have been easily solved without resorting to auxiliary storage. The algorithm is a variant of the primal revised simplex method; the computer code is called LPNET illustrating the close connection between linear programming and network graphs. This approach substantially improves computer processing timeand core storage, especially for relatively large network problems. The results of these experiments are provided. It is emphasized that an organized experimental design and a detailed series of empirical tests are crucial for an efficient implementation.
TL;DR: A primal simplex procedure to solve transshipment problems with an arbitrary additional constraint is developed, which incorporates efficient methods for pricing-out the basis, determining certain key vector representations, and implementing the change of basis.
Abstract: This paper develops a primal simplex procedure to solve transshipment problems with an arbitrary additional constraint. The procedure incorporates efficient methods for pricing-out the basis, determining certain key vector representations, and implementing the change of basis. These methods exploit the near triangularity of the basis in a manner that takes advantage of computational schemes and list structures used to solve the pure transshipment problem. We have implemented these results in a computer code, I/OPNETS-I. Computational results necessarily limited confirm that this code is significantly faster than APEX-III on some large problems. We have also developed a fast method for determining near optimal integer solutions. Computational results show that the near optimum integer solution value is usually within 0.5% of the value of the optimum continuous solution value.
TL;DR: The present work describes a direct method which treats constraints as limiting surfaces or subspaces and the solution is autanatically restricted within the limiting subspaced.
Abstract: The optimal scheduling of hydrothermal power system, either on long term or on short term basis, is basically non-linear programming problam with non-linear objective function and a mixture of linear and non-linear constraints. The existing solution techniques are based on orienting the problem so that methods of unconstrained minimization are applicable with the constraints being taken care of indirectly. The present work describes a direct method which treats constraints as limiting surfaces or subspaces and the solution is autanatically restricted within the limiting subspaces.
TL;DR: In this paper, a branch-and-bound-based method for finding the exact optimal solution of reliability allocation problems is presented, which is generalized to handle nonlinear constraints and nonseparable problems.
Abstract: The paper presents an efficient method for finding the exact optimal solutions of reliability allocation problems that are formulated as an integer nonlinear programming problem generalized to handle nonlinear constraints and nonseparable problems. The method is based on branch-and-bound and developed by considering separation and relaxation techniques.
01 Jan 1978
TL;DR: In this paper, the authors discuss techniques for implementing Lemke's algorithm for the linear complementarity problem in a numerically robust way as well as a method for recovering from loss of feasibility or singularity of the basis.
Abstract: This note discusses techniques for implementing Lemke's algorithm for the linear complementarity problem in a numerically robust way as well as a method for recovering from loss of feasibility or singularity of the basis. This recovery method is valid for both positive semi-definite M matrices and those with positive principal minors. It also allows a user to start from an advanced basis for such problems.
TL;DR: It is pointed out how the preemptive goal-programming approach is incompatible with utility preferences and the tendency of optimal solutions for standard linear goal programs to occur at extreme points is observed.
Abstract: After first formulating the problem of the Marine Environmental Protection program of the Coast Guard as a multiple-objective linear program, we investigate the applicability and limitations of goal programming. We point out how the preemptive goal-programming approach is incompatible with utility preferences. Then we observe the tendency of optimal solutions for standard linear goal programs to occur at extreme points. We also note problems of more general approaches, such as dealing with additively separable approximations to preferences.
TL;DR: A class of methods is presented for solving standard linear programming problems that move from one feasible solution to another at each iteration, improving the objective function as they go, like the simplex method.
Abstract: A class of methods is presented for solving standard linear programming problems. Like the simplex method, these methods move from one feasible solution to another at each iteration, improving the objective function as they go. Each such feasible solution is also associated with a basis. However, this feasible solution need not be an extreme point and the basic solution corresponding to the associated basis need not be feasible. Nevertheless, an optimal solution, if one exists, is found in a finite number of iterations (under nondegeneracy). An important example of a method in the class is the reduced gradient method with a slight modification regarding selection of the entering variable.
01 Nov 1978
TL;DR: A simple algorithm is described which determines the satisfiability over the reals of a conjunction of linear inequalities, none of which contains more than two variables, which is particularly suited to applications in mechanical program verification.
Abstract: A simple algorithm is described which determines the satisfiability over the reals of a conjunction of linear inequalities, none of which contains more than two variables. In the worst case the algorithm requires time O(${mn}^{\lceil \log^2 n \rceil + 3}$), where n is the number of variables and m the number of inequalities. Several considerations suggest that the algorithm may be useful in practice: it is simple to implement, it is fast for some important special cases, and if the inequalities are satisfiable it provides valuable information about their so1ution set. The algorithm is particularly suited to applications in mechanical program verification.
TL;DR: In this paper, a sequential explicitly stochastic linear programing model (SESLP) is proposed to determine both a design and a management policy for a multi-reservoir system.
Abstract: A sequential explicitly stochastic linear programing model (SESLP) which consists of a nonlinear program and an algorithm for obtaining an approximate solution is presented. The SESLP model can be used either to determine for a multipurpose multiple-reservoir system both a design and a management policy or to determine only a management policy. A discrete lag-one Markov process is explicitly included in the model as the streamflow description. The approximate solution to the nonlinear program is obtained by sequentially solving two linear programs which are subsets of the nonlinear program. A method of significantly reducing the computational burden and data requirements of the SESLP model is also presented. The method (system coordinated performance-individual operation (Scorpio)) is effective because within the SESLP/Scorpio model, although the operating rules for each reservoir are dependent on events occurring only at that reservoir site, system-wide performance levels are measured and the operation of each reservoir is coordinated with all other reservoirs.
TL;DR: In this article, the problem of locating a single new facility in the plane relative to several existing facilities is treated, and simultaneous consideration is given to minisum and minimax criteria.
Abstract: The problem of locating a single new facility in the plane relative to several existing facilities is treated. Simultaneous consideration is given to minisum and minimax criteria. Rectilinear distances are assumed. In addition to linear programming formulations, search procedures are developed based on the special structure of the location problems examined.