This exposition presents two algorithms for linear programs which allow a value change in more than one nonbasic variable at each iteration. The computational formulae are developed and errors which have appeared in t...
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This exposition presents two algorithms for linear programs which allow a value change in more than one nonbasic variable at each iteration. The computational formulae are developed and errors which have appeared in the literature are noted. One algorithm is a multiple basis exchange procedure while the second is a feasible direction method. There remain many computational challenges in the area of linearprogramming and we hope that this investigation will encourage additional work in the directions indicated in this exposition.
This paper presents a microcomputer system, MicroOR/LP, that addresses the major optimization problem classes with solution methods which are based on linearprogramming. This system is predicted upon an algebraic mod...
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This paper presents a microcomputer system, MicroOR/LP, that addresses the major optimization problem classes with solution methods which are based on linearprogramming. This system is predicted upon an algebraic modeling language that makes it possible to formulate linear programs, quadratic programs, separable programs and integer (mixed as well as binary) programs within the MicroOR/LP system in a form which is very close to their standard mathematical representations. The system provides a text editor working environment with pulldown menus for command selection. All modeling activities are performed from within the text editor, no other software is required.
Many optimization problems occur in both theory and practice when one has to optimize an objective function while an infinite number of constraints must be satisfied. The aim of this paper in to describe methods of ha...
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Many optimization problems occur in both theory and practice when one has to optimize an objective function while an infinite number of constraints must be satisfied. The aim of this paper in to describe methods of handling such problems numerically in an effective manner. We also indicate a number of applications.
The objective function of any solvable linear program can be perturbed by a differentiable, convex or Lipschitz continuous function in such a way that (a) a solution of the original linear program is also a Karush–K...
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The objective function of any solvable linear program can be perturbed by a differentiable, convex or Lipschitz continuous function in such a way that (a) a solution of the original linear program is also a Karush–Kuhn–Tucker point, local or global solution of the perturbed program, or (b) each global solution of the perturbed problem is also a solution of the linear program.
This paper shows how to generalize the Dantzig-Wolfe decomposition principle to integer programming. It does this in a unified way, regardless of the choice between the two main solution methods: ‘Branch and bound’ ...
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This paper shows how to generalize the Dantzig-Wolfe decomposition principle to integer programming. It does this in a unified way, regardless of the choice between the two main solution methods: ‘Branch and bound’ or ‘cutting plane’. In both instances the authority at the central level issues price directives in the form of a polyhedral, concave price function, where the purpose is to charge the sublevels for the use of central resources including a penalty for any attempts to violate the integrality of the result. The sublevels then respond with their optimal activities given this price function. Finite convergence of the procedure is established.
An extension of the classical fixed charge transportation problem is developed that allows a wide variety of practical production, distribution, and inventory planning models to be addressed. Computational results are...
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An extension of the classical fixed charge transportation problem is developed that allows a wide variety of practical production, distribution, and inventory planning models to be addressed. Computational results are presented for problems with up to one thousand network constraints and five thousand network variables.
Since Karmarkar (1984) introduced the projective algorithm for linear programs, there has been much interest in the issues of implementation and computational efficiency. Many of these efforts have involved attempts ...
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Since Karmarkar (1984) introduced the projective algorithm for linear programs, there has been much interest in the issues of implementation and computational efficiency. Many of these efforts have involved attempts to duplicate the original claim by Karmarkar that the algorithm can solve large linearprogramming problems more rapidly than the simplex method. The main ideas of the affine scaling algorithm are reviewed. While the basic algorithm performed poorly when compared to the simplex method, the enhanced version, which uses a row partitioning scheme, proved to be 2-3 times as fast as the IMSL FORTRAN subroutine for solving medium-sized linear programs.
We present a real-world problem that illustrates how different linearprogramming formulations allow different interpretations of solution results and sensitivity analyses. The problem involves the allocation of looms...
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We present a real-world problem that illustrates how different linearprogramming formulations allow different interpretations of solution results and sensitivity analyses. The problem involves the allocation of looms to weave various fabrics in a textile mill and the determination of which fabrics should be purchased outside of the mill in order to meet demand. The case has been used as a teaching tool for emphasizing often over-looked model formulation issues.
This paper discusses the experiences gained from the implementation of a large linearprogramming system on a personal computer for a client in the meat industry. The paper focuses on the use of a database for prepari...
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This paper discusses the experiences gained from the implementation of a large linearprogramming system on a personal computer for a client in the meat industry. The paper focuses on the use of a database for preparing and editing large amounts of data for input into a linearprogramming optimizer. Some useful pointers are given for others implementing similar systems. Finally, some thoughts are offered about the ideal linearprogramming system for large problems on a personal computer.
New methods are presented that are designed to improve a given set of vehicle routes that serve to deliver a commodity to customers over a weekly time period. Developing an efficient delivery system for a set of cust...
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New methods are presented that are designed to improve a given set of vehicle routes that serve to deliver a commodity to customers over a weekly time period. Developing an efficient delivery system for a set of customers over a long time period is a difficult problem, particularly when the customers are permitted to stock out. Analysis focuses on a comparison among 3 procedures which, when applied to a feasible set of weekly vehicle routes, reoptimize the entire set of routes, each creating a new and more efficient set of weekly vehicle routes. The 3 heuristic route improvement schemes are based on the concept of node interchange between different routes. The 3 algorithms are presented together with their computational performance when applied to the inventory routing problem for 12 consecutive weekly periods. It is found that all 3 procedures improve the initial solution constructed by a modified Clark and Wright algorithm by over 50%, as measured by the change in the value of the objective function.
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