An efficient scaling procedure for gain networks

Truemper, Klaus

doi:10.1002/net.3230060207

Cited by 24 publications

(2 citation statements)

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“…where j ϭ q and n ϭ g. In either case, we have g pq g fg ϭ g pg g fq . Truemper [22] gives a series of equivalent conditions such that a generalized network is transformable to a pure network. One such condition is that, for every cycle, the product of weights on the forward arcs equals the product of weights on the backward arcs.…”

Section: Ratio Conditionsmentioning

confidence: 99%

Solving generalized transportation problems via pure transportation problems

Gottlieb

2002

Naval Research Logistics

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This paper investigates certain issues of coefficient sensitivity in generalized network problems when such problems have small gains or losses. In these instances, it might be computationally advantageous to temporarily ignore these gains or losses and solve the resultant "pure" network problem. Subsequently, the optimal solution to the pure problem could be used to derive the optimal solution to the original generalized network problem. In this paper we focus on generalized transportation problems and consider the following question: Given an optimal solution to the pure transportation problem, under what conditions will the optimal solution to the original generalized transportation problem have the same basic variables? We study special cases of the generalized transportation problem in terms of convexity with respect to a basis. For the special case when all gains or losses are identical, we show that convexity holds. We use this result to determine conditions on the magnitude of the gains or losses such that the optimal solutions to both the generalized transportation problem and the associated pure transportation problem have the same basic variables. For more general cases, we establish sufficient conditions for convexity and feasibility.

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Section: Ratio Conditionsmentioning

confidence: 99%

Solving generalized transportation problems via pure transportation problems

Gottlieb

2002

Naval Research Logistics

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“…For example, in 'networks with gains', we imagine that each arc has a multiplier which converts the incoming flow xtj to an output flow ; th variety of interesting new applications (Jewell 1962), but requires complicated labelling schemes, since the solutions are not integral, and conservation in-the-large is not satisfied. M aurras (1972) reports on recent computational experience; Glover & Klingm an (1973) show th at some networks with multipliers can, in fact, be reduced by scaling to ordinary networks, and Truemper (1976) discusses scaling in general.…”

Section: N E T W O R K Fl O W Modelsmentioning

confidence: 99%

The analytic methods of operations research

1977

Phil. Trans. R. Soc. Lond. A

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Originally, the techniques used by operational research teams were borrowed from other scientific fields. However, the scope of problems addressed by this new discipline soon led to the development of special analytic methods, including such now familiar terms as linear programming, game theory, dynamic programming, queueing theory, and so forth. This growth was accelerated by the parallel development of the high-speed digital computer and the modern concepts of solution algorithms and simulation models. When computational storage and speed were limited, emphasis was on the exploitation of a problem’s special structure; as capabilities have expanded and costs have diminished, emphasis has shifted to the problems of data management for largerscale problems. Efficient data-structure methods have led to new methods for ‘unsolvable’ combinatorial problems. Concurrently, there has been an explosion in the theoretical literature. Specialty journals flourish in the various methodological areas, and conference offerings have grown to an unwieldy size. Most major universities now offer courses and degree programmes in O.R., based on topics which were unknown twenty years ago, and which use a wide variety of available textbooks. Furthermore, it is increasingly difficult to draw a firm line between O.R. and other disciplines, as successful techniques are routinely taught and used in their fields of application, including a variety of new disciplines such as transportation and urban planning, waste management, energy analysis, environmental engineering, health care systems, etc. O.R. methods have also had a large influence on theoretical fields, such as mathematics, statistics and economics. Recent developments in selected methodological areas are surveyed to indicate the variety and sophistication of O.R. techniques, and current research trends. Selected bibliographic references provide an introduction to the techniques, or to important new developments. In conclusion, the current crises which have appeared within the profession are discussed, and the prospects of this now-mature field are analysed.

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Bibliography

2009

Linear Programming and Network Flows

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An efficient scaling procedure for gain networks

Abstract: A simple and efficient procedure is presented which scales a network with gains to an equivalent one with positive or unit gains, provided certain necessary and sufficient conditions are satisfied.

Cited by 24 publications

References 11 publications

Solving generalized transportation problems via pure transportation problems

Solving generalized transportation problems via pure transportation problems

The analytic methods of operations research

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