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New models for shortest path problem with fuzzy arc lengths. (English) Zbl 1152.90674
Summary: This paper considers the shortest path problem with fuzzy arc lengths. According to different decision criteria, the concepts of expected shortest path, $$\alpha$$-shortest path and the most shortest path in fuzzy environment are originally proposed, and three types of models are formulated. In order to solve these models, a hybrid intelligent algorithm integrating simulation and genetic algorithm is provided and some examples are given to illustrate its effectiveness.

##### MSC:
 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
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##### References:
 [1] Bellman, E., On a routing problem, Quart. appl. math., 16, 87-90, (1958) · Zbl 0081.14403 [2] Dijkstra, E.W., A note on two problems in connection with graphs, Numer. math., 1, 269-271, (1959) · Zbl 0092.16002 [3] Dreyfus, S., An appraisal of some shortest path algorithms, Oper. res., 17, 395-412, (1969) · Zbl 0172.44202 [4] Frank, H., Shortest paths in probability graphs, Oper. res., 17, 583-599, (1969) · Zbl 0176.48301 [5] Fu, L.; Rilett, L.R., Expected shortest paths in dynamic and stochastic traffic networks, Transport. res., 32, 7, 499-516, (1998) [6] Hall, R., The fastest path through a network with random time-dependent travel time, Transport. sci., 20, 3, 182-188, (1986) [7] Loui, P., Optimal paths in graphs with stochastic or multidimensional weights, Comm. ACM, 26, 670-676, (1983) · Zbl 0526.90085 [8] Mirchandani, B.P., Shortest distance and reliability of probabilistic networks, Comput. oper. res., 3, 347-676, (1976) [9] Murthy, I.; Sarkar, S., A relaxation-based pruning technique for a class of stochastic shortest path problems, Transport. sci., 30, 3, 220-236, (1996) · Zbl 0879.90093 [10] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049 [11] Klein, C.M., Fuzzy shortest paths, Fuzzy set. syst., 39, 27-41, (1991) · Zbl 0728.90090 [12] Yager, R., Paths of least resistance on possibilistic production systems, Fuzzy set. syst., 19, 121-132, (1986) · Zbl 0601.90018 [13] Lin, K.; Chen, M., The fuzzy shortest path problem and its most vital arcs, Fuzzy set. syst., 58, 343-353, (1994) · Zbl 0804.90138 [14] Okada, S.; Soper, T., A shortest path problem on a network with fuzzy arc lengths, Fuzzy set. syst., 109, 1, 129-140, (2000) · Zbl 0956.90070 [15] Okada, S., Fuzzy shortest path problems incorporating interactivity among paths, Fuzzy set. syst., 142, 3, 335-357, (2004) · Zbl 1045.90092 [16] Liu, B., Theory and practice of uncertain programming, (2002), Physica-Verlag Heidelberg · Zbl 1029.90084 [17] Liu, B., Uncertainty theory: an introduction to its axiomatic foundations, (2004), Springer-Verlag Berlin · Zbl 1072.28012 [18] Liu, B.; Liu, Y.K., Expected value of fuzzy variable and fuzzy expected value model, IEEE trans. fuzzy syst., 10, 445-450, (2002) [19] Ahuja, R.K.; Magnanti, T.L.; Orlin, J.B., Network flows, (1993), Prentice-Hall Englewood CliRs, NJ · Zbl 1201.90001 [20] Murty, K.G., Network programming, (1992), Prentice-Hall Englewood CliRs, NJ [21] Liu, B., Toward fuzzy optimization without mathematical ambiguity, Fuzzy optim. decis. making, 1, 1, 43-63, (2002) · Zbl 1068.90618 [22] Liu, B., Minimax chance constrained programming models for fuzzy decision systems, Inform. sci., 112, 1-4, 25-38, (1998) · Zbl 0965.90058 [23] Liu, B.; Iwamura, K., Chance constrained programming with fuzzy parameters, Fuzzy set. syst., 94, 2, 227-237, (1998) · Zbl 0923.90141 [24] Liu, B.; Iwamura, K., A note on chance constrained programming with fuzzy coefficients, Fuzzy set. syst., 100, 1-3, 229-233, (1998) · Zbl 0948.90156 [25] Liu, B., Dependent-chance programming with fuzzy decisions, IEEE trans. fuzzy syst., 7, 3, 354-360, (1999) [26] Liu, B., Dependent-chance programming in fuzzy environments, Fuzzy set. syst., 109, 1, 97-106, (2000) · Zbl 0955.90153 [27] Holland, J., Adaptatin in natural and artificial system, (1975), University of Michigan Press Ann Arbor, MI [28] Gen, M.; Cheng, R., Genetic algorithms and engineer optimization, (2000), John Wiley and Sons, Inc. New York [29] Cherkassky, B.V.; Goldberg, A.V.; Radzik, T., Shortest paths algorithms: theory and experimental evaluation, Math. prog., 73, 129-174, (1996) · Zbl 0853.90111
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