The solution and duality of imprecise network problems

Duality properties have been investigated by many researchers in the recent literature. They are introduced in this paper for a fully fuzzified version of the minimal cost flow problem, which is a basic model in network flow theory. This model illustrates the least cost of the shipment of a commodit...

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Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2008-06, Vol.55 (12), p.2767-2790
Main Authors: Ghatee, Mehdi, Hashemi, S. Mehdi, Hashemi, Behnam, Dehghan, Mehdi
Format: Article
Language:English
Subjects:
Algorithms
Combinatorial analysis
Discrete optimization
Duality theorems
Fuzzy
Fuzzy cost
Fuzzy logic
Fuzzy set theory
Fuzzy supply–demand
Mathematical models
Networks
Optimality conditions
Optimization
Risk taking
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Summary:Duality properties have been investigated by many researchers in the recent literature. They are introduced in this paper for a fully fuzzified version of the minimal cost flow problem, which is a basic model in network flow theory. This model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes. First, we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper. By applying Hukuhara’s difference, approximated and exact multiplication and Wu’s scalar production, we exhibit the flow in network models. Then, we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified MCFP to acquire fuzzy optimal flows. To give duality theorems, we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms. Finally, we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision-making problems.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2007.10.023