Showing papers on "Linear programming published in 1974"

Validation of subgradient optimization

Abstract: The "relaxation" procedure introduced by Held and Karp for approximately solving a large linear programming problem related to the traveling-salesman problem is refined and studied experimentally on several classes of specially structured large-scale linear programming problems, and results on the use of the procedure for obtaining exact solutions are given It is concluded that the method shows promise for large-scale linear programming

Multicommodity Distribution System Design by Benders Decomposition

Abstract: A commonly occurring problem in distribution system design is the optimal location of intermediate distribution facilities between plants and customers. A multicommodity capacitated single-period version of this problem is formulated as a mixed integer linear program. A solution technique based on Benders Decomposition is developed, implemented, and successfully applied to a real problem for a major food firm with 17 commodity classes, 14 plants, 45 possible distribution center sites, and 121 customer zones. An essentially optimal solution was found and proven with a surprisingly small number of Benders cuts. Some discussion is given concerning why this problem class appears to be so amenable to solution by Benders’ method, and also concerning what we feel to be the proper professional use of the present computational technique.

Optimal Facility Location with Concave Costs

Abstract: The following problem is considered: select plant sites from a given set of sites and choose their production and distribution levels to meet known demand at discrete points at minimum cost. The construction and operating cost of each plant is assumed to be a concave function of the total production at that plant, and the distribution cost between each plant and demand point is assumed to be a concave function of the amount shipped. There may be capacity restrictions on the plants. A branch-and-bound algorithm for identifying an optimal solution is described; it is equivalent to the solution of a finite sequence of transportation problems. The algorithm is developed as a particular case of a simplified algorithm for minimizing separable concave functions over linear polyhedra. Computational results are cited for a computer code implementing the algorithm.

Implementation and computational comparisons of primal, dual and primal-dual computer codes for minimum cost network flow problems

Abstract: : The paper presents extensive computational experience with a special purpose primal simplex algorithm. The performance is compared to that of several 'state of the art' out-of-kilter computer codes. The computational characteristics of several different primal feasible start procedures and pivot selection strategies are also examined. The study discloses the advantages, in both computation time and memory requirements, of the primal approach over the out-of-kilter method. The test environment has the following distinguishing properties: (1) all of the codes are tested on the same machine and the same problems, (2) the test set includes capacitated and uncapacitated transhipment networks, transportation problems, and assignment problems, and (3) problem sizes ranging from 100 to 8,000 nodes with up to 35,000 arcs are examined. (Author)

Transmission Expansion by Branch-and-Bound Integer Programming with Optimal Cost - Capacity Curves

Abstract: A new method is proposed in this paper to solve a single-stage expansion problem for a transmission network, given future generation and load patterns, and alternative types of lines available, subject to overload, reliability and right-of-way constraints. The problem is formulated as a series of zero-one integer programs which are solved by an efficient branch-and-bound algorithm. Complexity is reduced by the concepts of optimal cost-capacity curves and screening algorithms. A sample study is shown and the method is implemented in a computer program.

Optimal planning of power networks using mixed-integer programming. Part 1: Static and time-phased network synthesis

Abstract: A mixed-integer linear programming approach to the planning of electrical-power networks is described. The method is based on an interpretation of fixed-cost transportation-type models, and includes both network security and costs of network losses. Both single-period and multitime-period planning problems are considered. A large general-purpose mathematical programming system is used to obtain solutions using branch-and-bound algorithms, and the practical aspects of organising the problems and controlling the branch-and-bound tree search within a modern m.p.s. package are discussed. Three case studies are presented for illustration: the optimal design of a 132 kV subtransmission system, the optimal time-phased design of the same system over. eight years, and the optimal layout and cable selection for a new housing-estate low-voltage network. The paper concludes by indicating potential further applications of the method for scheduling, for example, new plant at substations.

Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems

Abstract: Two minimum cardinality set covering problems of similar structure are presented as difficult test problems for evaluating the computational efficiency of integer programming and set covering algorithms. The smaller problem has 117 constraints and 27 variables, and the larger one, constructed by H.J. Ryser, has 330 constraints and 45 variables. The constraint matrices of the two set covering problems are incidence matrices of Steiner triple systems. An optimal solution to the problem that we were able to solve (the smaller one) gives some new information on the 1-widths of members of this class of (0,1)-matrices.

An Algorithm for Large Set Partitioning Problems

Abstract: An algorithm is presented for the special integer linear program known as the set partitioning problem. This problem has a binary coefficient matrix, binary variables, and unit resources. Furthermore, all of its constraints are equations. In spite of its very special form, the set partitioning problem has many practical interpretations. The algorithm is of the branch and bound type. A special class of finite mappings is enumerated rather than the customary set of binary solution vectors. Linear programming is used to obtain bounds on the minimal costs of the subproblems that arise. Computational results are reported for several large problems.

Linear programming design of IIR digital filters with arbitrary magnitude function

Abstract: This paper discusses the use of linear programming techniques for the design of infinite impulse response (IIR) digital filters. In particular, it is shown that, in theory, a weighted equiripple approximation to an arbitrary magnitude function can be obtained in a predictable number of applications of the simplex algorithm of linear programming. When one implements the design algorithm, certain practical difficulties (e.g., coefficient sensitivity) limit the range of filters which can be designed using this technique. However, a fairly large number of IIR filters have been successfully designed and several examples will be presented to illustrate the range of problems for which we found this technique to be useful.

An Interactive Procedure for Sizing and Timing Distribution Substations Using Optimization Techniques

Abstract: This paper defines a mathematical model which simulates the growth of a power system and determines the least cost expansion plan for a system of distribution substations. A new approach employing linear and integer programming is used to optimize the system's substation capacities subject to the constraints of cost, load, voltage, and reserve requirements. The model has been successfully applied to a 1600 square mile urban area served by 70 distribution substations.

On fourier’s analysis of linear inequality systems

Abstract: Fourier treated a system of linear inequalities by a method of elimination of variables. This method can be used to derive the duality theory of linear programming. Perhaps this furnishes the quickest proof both for finite and infinite linear programs. For numerical evaluation of a linear program, Fourier’s procedure is very cumbersome because a variable is eliminated by adding each pair of inequalities having coefficients of opposite sign. This introduces many redundant inequalities. However, modifications are possible which reduce the number of redundant inequalities generated. With these modifications the method of Fourier becomes a practical computational algorithm for a class of parametric linear programs.

Duality, Indifference and Sensitivity Analysis inr Multiple Objective Linear Programming

Abstract: Many decision situations can be described as multiple objective linear programmes. In this paper, the duality of such programmes is investigated, and the duality theorem is used to illustrate aspects of sensitivity analysis with multiple objectives. Both optimizing and satisficing situations are considered.

Ground-Water Hydraulics in Aquifer Management

Abstract: The method first replaces the differential equations of ground-water flow by finite-difference approximations that include unknown sink/source terms. The resulting system of algebraic linear equations has a rectangular matrix of coefficients. This system, together with linear inequalities relating sink/source terms, heads or both, and together with an objective function, forms a linear programming (LP) model. The method is applied to small-scale models of confined and unconfined saturated flow for steady-state and transient cases. The steady-state LP models are solved using available computer codes. For the transient confined model, the Crank-Nicolson scheme is used, and a single LP problem is solved covering all of the time steps. For the transient unconfined model, a predictor technique is used, and a LP problem is solved at each corrector step. The optimal solutions are consistent with the results of traditional analyses.

The Cutting Stock Problem in the Flat Glass Industry

Abstract: The cutting stock problem occurs where large rectangles of some material require cutting into smaller rectangles, in the most appropriate way, to satisfy an order book. A linear programming approach to the problem has been suggested by P. C. Gilmore and R. E. Gomory. An application of this approach in the glass industry is described which is shown to be inadequate since it only satisfies a wastage criterion. In practice, multiple criteria must be satisfied and two alternative approaches using linear programming and heuristic scheduling are proposed.

Application of Programs with Maximin Objective Functions to Problems of Optimal Resource Allocation

Abstract: A mathematical program with a maximin objective function is defined as an optimization problem of the following type: Maxz = mini cixi, subject to AX = b, X ≧ 0. Although the ci can be in the interval (− ∞, ∞), the paper discusses the more common practical case where all ci ≧ 0. It shows that problems of this type arise in a variety of applications where it is required to maximize a production function of the “fixed proportion” type subject to a set of linear constraints. Although it is well known that the solution to this type of problem can be found by linear programming, this paper shows that, if the existence of a certain condition can be demonstrated, then a simplified method can be used to determine the optimum solution. Many problems of practical interest can be solved by this simplified method; an example involving the readiness of a ship is presented.

A Transshipment Model for Cash Management Decisions

Abstract: The cash management problem is concerned with optimally financing net cash outflows and investing net inflows of a firm while simultaneously determining payment schedules for incurred liabilities. The problem is formulated as a transshipment model to minimize the total cost of allocating sources of funds to different uses while retaining the possibility of transferring cash between sources. A numerical example is formulated using this model and its optimal solution is shown to be essentially the same as that of a linear programming formulation proposed by Orgler. Extensions of the methodology for examining the effects of different “minimum cash balance” requirements and for incorporating other institutional constraints are also outlined. The transshipment formulation is useful in organizing data for financial control. It is intuitively appealing and computationally more efficient by a factor of about 30 to 1 than the linear programming formulation of the problem.

Postoptimality analysis in integer programming by implicit enumeration: The mixed integer case

Abstract: This paper develops a method for doing postoptimality analysis on the mixed integer programming problem. The proposed procedures form a natural adjunct to enumerative I.P. algorithms that are linear programming based, and they are designed, in effect, to capitalize on insights generated as the problem is initially solved to do subsequent analysis upon it. In particular, limited ranging analysis is possible on selected parameters, as is the efficient resolving of the problem following parameter changes.

Capital Cost Minimization of Drainage Networks

Abstract: This paper presents and compares three mathematical programming models for the optimization of drainage networks. The three models are based on two extensions of linear programming: separable-convex and mixed- integer programming. The first produces a continuous range of diameters while the second limits the solution to discrete commercially available sizes. It is shown that while the first can be formulated with pipes flowing full, the second must allow for partial flow. A solution which combines the use of both of these techniques can produce a minimum cost system with partially-full flows and commercially available diameters. This solution requires less computer time than those based strictly on mixed-integer programming. An example seven-link drainage network is designed by the three proposed methods and the results are reported in the paper.

A mixed integer programming model for scheduling orders in a steel mill

Abstract: The problem of scheduling orders at each facility of a large integrated steel mill is considered. Orders are received randomly, and delivery dates are established immediately. Each order is filled by converting raw materials into a finished saleable steel product by a fixed sequence of processes. The application of a deterministic mixed integer linear programming model to the order scheduling problem is given. One important criterion permitted by the model is to process the orders in a sequence which minimizes the total tardiness from promised delivery for all orders; alternative criteria are also possible. Most practical constraints which arise in steelmaking can be considered within the formulation. In particular, sequencing and resource availability constraints are handled easily. The order scheduling model given here contains many variables and constraints, resulting in computational difficulties.

A note on the nucleolus

Abstract: It was shown byKohlberg [1972] that the nucleolus can be obtained by solving a linear program of extremely large size (2 n ! constraints). We show here how this program can be reduced to a more tractable size (4 n constraints).

Minimum Weight Design of Stress Limited Trusses

Abstract: A procedure based on the force method of analysis and using linear programming is presented for a minimum weight optimum design of stress limited indeterminate trusses. The problem is initially cast as a linear optimization problem considering only equilibrium conditions and stress limits. Thereafter, additional approximate linear conditions are annexed to the initial formulation and the solution is repeated yielding a more nearly compatible design. The process is then repeated refining these additional conditions in each iteration until a compatible design is obtained. The simplex method is used to solve the linear programming problem in each iteration cycle. The relatively simple mathematical programming formulation is shown to be efficient as well as having the ability to modify the configuration of the structure by vanishing unnecessary members. A computer program based on this method and the results of some examples are presented.

Optimal control of multiunit interbasin water resource systems

Abstract: A method for finding optimal operating policies for a multiunit water resource system that extends over two river basins and serves multiple demands is presented. The method was developed and tested for one of several systems that have been proposed for further development of the water resource in the urbanizing Piedmont Triad region of North Carolina. Monthly operating decisions are given by solutions of a piecewise linear programing problem, the objective function for which consists of two parts, immediate economic losses within the month and the expected present value of future losses as a function of end-of-month storage levels in the reservoirs. The latter function is estimated by imbedding the linear programing problem in a stochastic dynamic programing problem. An approximate solution technique for the larger problem is described, and computational experience is reported.

An Extremal Principle for Accounting Balance of a Resource Value-Transfer Economy: Existence, Uniqueness and Computation

Abstract: : An extremal principle for accounting balance of a resource value- transfer economy of W. P. Drews is developed. Through it, existence and uniqueness are completely characterized in terms of an associated linear programming problem of pure distribution (transportation) type. An existence theorem of Dantzig's (established by the Brouwer fixed point theorem under economically unrealistic conditions) is an immediate special corollary. The principle is equivalent to unconstrained minimization of a simple strictly convex function and can be computed thereby. Alternately, it is an extended geometric programming problem whose dual is minimization of a strictly convex function subject to the same distribution constraints mentioned above.