Section 2 presents some basic concepts, definitions, a fuzzy time-varying, a fuzzy time-windows and the fuzzy shortest dynamic f-augmenting path with a fuzzy dynamic residual network. The reminder of this paper consists of six sections including organized Introduction. The new version of the problem becomes so-called the Fuzzy Minimum Cost Flow Problem with Fuzzy Time-Windows (FMCFPFTW).
For each arc there is a fuzzy transit time For each node there is a fuzzy time-windows subject to the condition that the schedule remains optimal for any fuzzy service time By the shortest dynamic f-augmenting path with a fuzzy dynamic residual network. A more realistic model is to consider the fuzzy time needed to traverse an arc. All attributes of the fuzzy network, including a fuzzy cost to send a fuzzy flow on the arc, the fuzzy capacity of the arc, are a fuzzy time-invariant.
Traditionally, this problem is considered as a static one, where it is assumed that it takes a zero time to traverse the arc. The problem is to determine how given the amount of the fuzzy flow can be sent from one node s to another node The fuzzy minimum cost on a fuzzy shortest dynamic f-augmenting path of the fuzzy dynamic residual network, subject to the fuzzy capaci ty limits on the arcs, see 10, 11, 15, 17. The fuzzy flow must arrive at node before a fuzzy time if it arrives before it just a fuzzy waiting time before treating the fuzzy flow at the node (a fuzzy waiting time see 4, 5, 6, 7, 18. For each node there is an associated of a three integer parameters, a waiting fuzzy cost a node fuzzy capacity and a fuzzy time-windows where, is a non-negative fuzzy time service, see, 3, 16, 19, 20, 24.įor each arc there is an associated of a three integer parameters, a positive fuzzy transit time a fuzzy transit cost and a positive fuzzy capacity limit All these parameters are functions of the fuzzy time service The source node s and a sink node has a fuzzy time-windows respectively, see 13, 14, 21, 22. We consider be a fuzzy network without parallel arcs and loops, where N is a set of nodes and A is a set of arcs. The MCFP has been studied extensively, see, 1, 2, 8, 9, 18, 23, 24. Over the past 40 years, the MCFP has been an area of research that has attracted many researchers. The Minimum Cost Flow Problem (MCFP) is a basic problem in the network flow theory, which is one of the classical combinatorial optimizations and an NP-hard problem with several applications.
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