Simplex Method Of Linear Programming Research Articles

A simplified molp algorithm: The MOLP-S procedure

A number of MOLP-algorithms have been developed to establish the set of non-dominated solutions, using a number of different approaches and theorems that may be non-trivial to the non-expert user. This article presents a simplified MOLP-algorithm (MOLP-S), based on a straightforward extension of the simplex-method of linear programming, to trace out the set of non-dominated solutions. The proposed methodology exhibits computational characteristics that may render the method more efficient as compared to other algorithms currently in use. The proposed method is tested on a number of problems from the literature which exhibit varying degrees of complexity.

Implementing the recursive APL code for dynamic programming

Despite the importance and wide applicability of dynamic programming technique in optimization, the complexity and uniqueness of various dynamic programming models often make the comprehension of their solution procedures difficult, particularly for a novice. Although it can be argued that, unlike the Simplex method in linear programming, no general dynamic programming algorithm exists, with the recursive capability of APL, it is feasible to develop a “systematic” solution approach for deterministic dynamic programming. In this paper, we introduce a set of such APL code developed be solve various types of dynamic programming problems with minor modifications on some of its key functions to accommodate the nature of each type of dynamic programming problems.

A “build-down” scheme for linear programming

We propose a “build-down” scheme for Karmarkar's algorithm and the simplex method for linear programming. The scheme starts with an optimal basis “candidate” setΞ including all columns of the constraint matrix, then constructs a dual ellipsoid containing all optimal dual solutions. A pricing rule is developed for checking whether or not a dual hyperplane corresponding to a column intersects the containing ellipsoid. If the dual hyperplane has no intersection with the ellipsoid, its corresponding column will not appear in any of the optimal bases, and can be eliminated fromΞ. As these methods iterate,Ξ is eventually built-down to a set that contains only the optimal basic columns.

An exponential example for Terlaky's pivoting rule for the criss-cross simplex method

Recently T. Terlaky has proposed a new pivoting rule for the criss-cross simplex method for linear programming and he proved that his rule is convergent. In this note we show that the required number of iterations may be exponential in the number of variables and constraints of the problem.

Eliminating columns in the simplex method for linear programming

We propose a column-eliminating technique for the simplex method of linear programming. A pricing criterion is developed for checking whether a dual hyperplane corresponding to a column intersects a simplex containing all of the optimal dual feasible solutions. If the dual hyperplane has no intersection with this simplex, we can eliminate the corresponding column from further computation during the course of the simplex method.

Management of pavement rehabilitation

The development of a decision support system (DSS) for pavement management is described. The system contains forecasting and optimization models. The forecasting model is used to predict the condition of stretches of road. The optimization model has the form of a linear programming problem with additional logical constraints. The solution method proposed for this model is based on the simplex method for linear programming, with a restricted basis entry procedure. Results of a sensitivity analysis on the optimization model are reported, and the implementation of the DSS on a microcomputer is discussed.

A GENERAL ALGORITHM FOR THE GENERATION OF QUASILATTICES BY MEANS OF THE CUT AND PROJECTION METHOD USING THE SIMPLEX METHOD OF LINEAR PROGRAMMING AND THE MOORE-PENROSE GENERALIZED INVERSE

A computational algorithm for the generation of quasiperiodic tiles based on the cut and projection method is presented. The algorithm is capable of projecting any type of lattice embedded in any euclidean space onto any subspace making it possible to generate quasiperiodic tiles with any desired symmetry. The simplex method of linear programming and the Moore-Penrose generalized inverse are used to construct the cut (strip) in the higher dimensional space which is to be projected.

A parallelization of the simplex method

This paper presents a parallelization of the simplex method for linear programming. Current implementations of the simplex method on sequential computers are based on a triangular factorization of the inverse of the current basis. An alternative decomposition designed for parallel computation, called the quadrant interlocking factorization, has previously been proposed for solving linear systems of equations. This research presents the theoretical justification and algorithms required to implement this new factorization in a simplex-based linear programming system.

Chebyshev approximation for linear phase nonrecursive digital filters

A procedure has been developed for the design of nonrecursive digital filters with prescribed passband and stopband amplitude characteristics. The proposed procedure is based on an efficient algorithm utilizing the simplex method of linear programming. The design algorithm yields equiripple approximation and it is an alternative to the one which is based on the Remz exchange algorithm. The design procedure allows exact specifications for arbitrary passband and stopband edges. Furthermore, no prior knowledge of the degree of the filter is required. To demonstrate the potential of the design algorithm, several examples with different requirements are worked out and a sample is presented. The obtained results show that the design procedure performed very well where the various parameters of the filter were taken into consideration.

Degeneracy in the presence of roundoff errors

A technique is described for resolving degeneracy in the simplex method for linear programming. It is shown that this technique enables a guarantee of termination to be given, not only for exact arithmetic but also for inexact arithmetic when roundoff errors are present. The method is recursive, and each level of recursion is obtained by localizing and dualizing the problem at a lower level. The existence of a cost function at any level is exploited to provide the guarantee. A number of interesting theoretical properties of the method are given. Data structures are described which enable the method to be implemented with very little overhead beyond what is normally required for the simplex method. There is no difficulty in using any of the currently popular techniques for representing and updating matrix factors. The technique is extended to handle degeneracy in l1 linear programming problems, including linear l1 approximation, and it is shown that all the above properties can be preserved. Some comparative numerical experiments are described.

A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. Part I of this paper has developed a general and direct simplex algorithm for piecewise-linear programming, under convenient assumptions that guarantee a finite number of basic solutions, existence of basic feasible solutions, and nondegeneracy of all such solutions. Part II now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise-linear simplex algorithm. The theory of piecewise-linear programming is thereby extended, and numerous features of linear programming are generalized or are seen in a new light. An analysis of the algorithm's computational requirements and a survey of applications will be presented in Part III.

Maximum-weight markings in marked graphs: Algorithms and interpretations based on the simplex method

The problem of determining a maximum-weight marking in a marked graph is mathematically dual to the transshipment problem of operations research. The special structure of the transshipment problem facilitates efficient implementation of the simplex method of linear programming, for solving such problems. In this paper, we first show that the maximum-weight marking problem possesses as much structure as its dual, and then present an implementation of simplex for this problem in terms of marked graph concepts and operations. The pivoting operation in the simplex method is shown to correspond to the subgraph firing operation in marked graphs. A diakoptic reachability theorem is also proved. The formulations presented in this paper cover both live- and nonlive-marked graphs with or without capacity constraints.

MINIMUM PHASE SELECTIVE BANDPASS DIGITAL FILTERS WITH CONTROLLED PASS-BAND LOSS AND DELAY

A simple numerical procedure is described for the construction of minimum phase selective band-pass digital filters satisfying simultaneous pass-band amplitude and phase constraints for any specified stop-band loss. It is shown that, if the pass-band loss of the filter is expressed as a linear combination of cosine functions, then the filter's group delay can be linearly expressed in terms of these functions together with any stop-band loss constraints. The simplex method of linear programming can then be invoked to minimize the pass-band delay deviation from constancy for any specified peak pass-band loss. Tlie_resulting transfer functions are shown to be less sensitive with reference to coefficients trancations. Illustrative examples are also given.

A Family of Simplex Variants Solving an m × d Linear Program in Expected Number of Pivot Steps Depending on d Only

We present a family of variants of the Simplex method, which are based on a Constraint-By-Constraint procedure: the solution to a linear program is obtained by solving a sequence of subproblems with an increasing number of constraints. We discuss several probabilistic models for generating linear programs. In all of them the underlying distribution is assumed to be invariant under changing the signs of rows or columns in the problem data. A weak regularity condition is also assumed. Under these models, for linear programs with d variables and m + d inequality constraints, the expected number of pivots required by these algorithms is bounded by a function of min(m, d) only. In particular this means that, for a fixed number of variables, the expected number of pivots is bounded by a constant when the number of constraints tends to infinity. Since Smale's original model [Smale, S. 1983. On the average speed of the simplex method of linear programming. Math. Programming 27.] satisfies our probabilistic assumptions, the same results apply to his model, although not to the particular algorithm he analyzes. We also present some results for models generating only feasible linear programs, and for Bland's pivoting rule. We conclude with a discussion of our probabilistic models, and show why they are inadequate for obtaining meaningful results unless d and m are of the same order of magnitude.

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.

Imposed Lack of Fit as a Means of Enhancing Space Truss Design

Space trusses with a high degree of static redundancy, such as double-layer grids, are highly sensitive to imperfections. Substantial reductions in load-bearing capacity are caused primarily by random lack of fit (LOF) of members, associated with brittle-type compressive member buckling. A substantial increase in load-bearing capacity and/or reduction in weight can be achieved by imposing lack of fit on selected members in a controlled manner. Such prestressing by imposed lack of fit utilizes the compressive capacity of nominally tensile members and can be used to compensate for the deleterious effect of random lack of fit. A simple algorithm is presented for obtaining LOF patterns that minimize weight or maximize load-bearing capacity for a class of trusses in which members' cross-section properties are proportionally fixed. The algorithm is based on the flexibility method of structural analysis and on the simplex method of linear programming. Some illustrative examples suggest the possibility of doubling load-bearing capacity or halving weight in some design cases.

LOCAL ORDER IN THE DISTRIBUTION OF Si AND Al IN TECTOALUMINOSILICATES

Abstract Local order of Si,Al distribution in tectosilicates is examined under the restriction of both Loewenstein and Dempsey rules. The distribution of various cluster types permissible for a given Si/Al ratio is predicted by the application of the simplex method of linear programming. There is a good agreement between the predicted and observed values.

On the average number of steps of the simplex method of linear programming

The goal is to give some theoretical explanation for the efficiency of the simplex method of George Dantzig. Fixing the number of constraints and using Dantzig's self-dual parametric algorithm, we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average.

Devising high-strength weldable steel by the methods of mathematical planning of experiments

1. The use of the method of Latin squares, and specifying or excluding those alloying elements whose optimum content is known and is largely independent of the concentrations of the principal alloying elements makes it possible to reduce the number of investigated experimental melts in working out the composition of steel with the required properties. 2. We obtained multiple regression equations type "chemical composition-properties" for weldable low carbon alloy steel, heat-treated to the same strength (σ0 · 2 ⩾ 750 MPa). 3. The simplex method of linear programming as used to optimize the composition of steel with σ0 · 2 ⩾ 750 MPa according to the criteria of ductility, impact toughness, and crack resistance, with constraints on the chemical composition for obtaining satisfactory weldability. The devised weldable steel in sections up to 250 mm after hardening in water and temnering has good ductility, impact toughness, and crack resistance.

A Primal Algorithm for Finding Minimum-Cost Flows in Capacitated Networks With Applications

Algorithms for finding a minimum-cost, single-commodity flow in a capacitated network are based on variants of the simplex method of linear programming. We describe an implementation of a primal algorithm which is fast and can solve large problems. The major ideas incorporated are (i) the sparsity of the network is used to reduce the time and computer storage space requirements; (ii) basic solutions are stored compactly as spanning trees of the network; (iii) a candidate stack is used to allow flexible strategies in choosing an arc to enter the basis tree; (iv) the predecessor and thread data structures are used to efficiently traverse the tree and to update the solution at each iteration; (v) rules are implemented to avoid cycling or stalling caused by degeneracy; and (vi) piecewise-linear, convex arc costs are handled implicitly. The Primal Network Flow Convex (PNFC) code implements this algorithm and three examples, from communication networks, that can be solved with PNFC are discussed: (i) solving the area transfer problem; (ii) scheduling the collection of traffic data records; and (iii) planning the placement of pair-gain systems.