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Dijkstra’s Algorithm for Solving the Shortest Path Problem on Networks Under Intuitionistic Fuzzy Environment

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Journal of Mathematical Modelling and Algorithms

Abstract

In this paper, a well known problem called the Shortest Path Problem (SPP) has been considered in an uncertain environment. The cost parameters for traveling each arc have been considered as Intuitionistic Fuzzy Numbers (IFNs) which are the more generalized form of fuzzy numbers involving a degree of acceptance and a degree of rejection. A heuristic methodology for solving the SPP has been developed, which aim to exploit tolerance for imprecision, uncertainty and partial truth to achieve tractability, robustness and low cost solution corresponding to the minimum-cost path or the shortest path. The Modified Intuitionistic Fuzzy Dijkstra’s Algorithm (MIFDA) has been proposed in this paper for solving Intuitionistic Fuzzy Shortest Path Problem (IFSPP) using the Intuitionistic Fuzzy Hybrid Geometric (IFHG) operator. A numerical example illustrates the effectiveness of the proposed method.

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References

  1. Atanassov, K.T.: Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84) (1983) (in Bulgarian)

  2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Set Syst. 20, 87–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atanassov, K.T.: More on intuitionistic fuzzy sets. Fuzzy Set Syst. 33(1), 37–46 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atanassov, K.T.: Idea for intuitionistic fuzzy sets equation, in equation and optimization. Notes on IFS 1(1), 17–24 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Atanassov, K.T.: Intuitionistic Fuzzy Sets. Physica, Heidelberg (1999)

    MATH  Google Scholar 

  6. Atanassov, K.T.: Two theorems for Intuitionistic fuzzy sets. Fuzzy Set Syst. 110, 267–269 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Atanassov, K.T., Gargov, G.: Interval-valued intuitionistic fuzzy sets. Fuzzy Set Syst. 31(3), 343–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Atanassov, K.T., Pasi, G., Yager, R.R.: Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision making. Int. J. Syst. Sci. 36, 859–868 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boroujerdi, A., Uhlmann, J.: An efficient algorithm for computing least cost paths with turn constraints. Inf. Process. Lett. 67(6), 317–321 (1998)

    Article  MathSciNet  Google Scholar 

  10. Burillo, P., Bustince, H., Mohedano, V.: Some definition of Intuitionistic Fuzzy numbers, first properties. In: Lakoy, D. (ed.) Proc. of the 1st Workshop on Fuzzy Based Expert Systems, pp. 53–55. Sofia, Bulgaria (1994)

    Google Scholar 

  11. Bustince, H., Burillo, P.: Vague sets are intuitionistic fuzzy sets. Fuzzy Set Syst. 79, 403–405 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, S.M., Tan, J.M.: Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Set Syst. 67, 163–172 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chuang, T.-N., Kung, J.-Y.: The fuzzy shortest path length and the corresponding shortest path in a network. Comput. Oper. Res. 32(6), 1409–1428 (2005)

    Article  MathSciNet  Google Scholar 

  14. Chuang, T.-N., Kung, J.-Y.: A new algorithm for the discrete fuzzy shortest path problem in a network. Appl. Math. Comput. 174(1), 660–668 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959). doi:10.1007/BF01386390

    Article  MathSciNet  MATH  Google Scholar 

  16. Dijkstra, E.W., Misa, T.J. (eds.): An Interview with Edsger W. Dijkstra. Commun. ACM 53(8), 41–47 (2010). doi:10.1145/1787234.1787249

    Article  Google Scholar 

  17. Domenico, C., Simone, F.: Two-levels-greedy: a generalization of Dijkstra’s shortest path algorithm. Electron. Notes Discret. Math. 17, 81–86 (2004)

    Article  Google Scholar 

  18. Galand, L., Perny, P., Spanjaard, O.: Choquet-based optimisation in multiobjective shortest path and spanning tree problems. Eur. J. Oper. Res. 204(2), 303–315 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23(2), 610–614 (1993)

    Article  MATH  Google Scholar 

  20. Guiwu, W.: Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making. Appl. Soft Comput. 10(2), 423–431 (2010)

    Article  Google Scholar 

  21. Gui-Wu, W.: Some generalized aggregating operators with linguistic information and their application to multiple attribute group decision making. Comput. Ind. Eng. 61(1), 32–38 (2011)

    Article  Google Scholar 

  22. Hassanzadeh, R., Mahdavi, I., Mahdavi-Amiri, N., Tajdin, A.: A genetic algorithm for solving fuzzy shortest path problems with mixed fuzzy arc lengths. Math. Comput. Model. (2011, in press). doi:10.1016/j.mcm.2011.03.040

  23. Hernandes, F., Lamata, M.T., Verdegay, J.L., Yamakami, A.: The shortest path problem on networks with fuzzy parameters. Fuzzy Set Syst. 158(14), 1561–1570 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hong, D.H., Choi, C.H.: Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Set Syst. 114, 103–113 (2000)

    Article  MATH  Google Scholar 

  25. Ji, X., Iwamura, K., Shao, Z.: New models for shortest path problem with fuzzy arc lengths. Appl. Math. Model. 31(2), 259–269 (2007)

    Article  MATH  Google Scholar 

  26. Keshavarz, E., Khorram, E.: A fuzzy shortest path with the highest reliability. J. Comput. Appl. Math. 230(1), 204–212 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Klein, C.M.: Fuzzy shortest paths. Fuzzy Set Syst. 39(1), 27–41 (1991)

    Article  MATH  Google Scholar 

  28. Knuth, D.E.: A generalization of Dijkstra’s algorithm. Inf. Process. Lett. 6(1), 1–5 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kung, J.-Y., Chuang, T.-N.: The shortest path problem with discrete fuzzy arc lengths. Comput. Math. Appl. 49(2–3), 263–270 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, D.F.: Multiattribute decision making models and methods using intuitionistic fuzzy sets. J. Comput. Syst. Sci. 70, 73–85 (2005)

    Article  MATH  Google Scholar 

  31. Lin, K.-C., Chern, M.-S.: The fuzzy shortest path problem and its most vital arcs. Fuzzy Set Syst. 58(3), 343–353 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lu, X., Camitz, M.: Finding the shortest paths by node combination. Appl. Math. Comput. 217(13), 6401–6408 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mahdavi, I., Nourifar, R., Heidarzade, A., Mahdavi Amiri, N.: A dynamic programming approach for finding shortest chains in a fuzzy network. Appl. Soft Comput. 9(2), 503–511 (2009)

    Article  Google Scholar 

  34. Misra, J.: A walk over the shortest path: Dijkstra’s algorithm viewed as fixed-point computation. Inf. Process. Lett. 77(2–4), 197–200 (2011)

    Google Scholar 

  35. Mitchell, H.B.: Ranking-intuitionistic fuzzy numbers. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12(3), 377–386 (2004)

    Article  MATH  Google Scholar 

  36. Moazeni, S.: Fuzzy shortest path problem with finite fuzzy quantities. Appl. Math. Comput. 183(1), 160–169 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Noshita, K.: A theorem on the expected complexity of Dijkstra’s shortest path algorithm. J. Algorithm 6(3), 400–408 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Noshita, K., Masuda, E., Machida, H.: On the expected behaviors of the Dijkstra’s shortest path algorithm for complete graphs. Inf. Process. Lett. 7(5), 237–243 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  39. Okada, S., Gen, M.: Fuzzy shortest path problem. Comput. Ind. Eng. 27(1–4), 465–468 (1994)

    Article  Google Scholar 

  40. Okada, S., Soper, T.: A shortest path problem on a network with fuzzy arc lengths. Fuzzy Set Syst. 109(1), 129–140 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Okada, S.: Fuzzy shortest path problems incorporating interactivity among paths. Fuzzy Set Syst. 142(3), 335–357 (2004)

    Article  MATH  Google Scholar 

  42. Orlin, J.B., Madduri K., Subramani, K., Williamson, M.: A faster algorithm for the single source shortest path problem with few distinct positive lengths. J. Discret. Algorithms 8(2), 189–198 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Smith, O.J., Boland, N., Waterer, H.: Solving shortest path problems with a weight constraint and replenishment arcs. Comput. Oper. Res. 39(5), 964–984 (2012)

    Article  MathSciNet  Google Scholar 

  44. Su, Z.-x., Chen, M.-y., Xia, G.-p., Wang, L.: An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making. Expert Syst. Appl. 38(12), 15286–15295 (2011)

    Article  Google Scholar 

  45. Su, Z.-x., Chen, M.-y., Xia, G.-p., Wang, L.: Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making. Expert Syst. Appl. 39(2), 1902–1910 (2012)

    Article  Google Scholar 

  46. Szmidt, E., Kacprzyk, J.: Remarks on some applications of intuitionistic fuzzy sets in decision making. NIFS 2(3), 22–31 (1996)

    MathSciNet  MATH  Google Scholar 

  47. Szmidt, E., Kacrzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Set Syst. 114(3), 505–518 (2000)

    Article  MATH  Google Scholar 

  48. Szmidt, E., Kacprzyk, J.: Using intuitionistic fuzzy sets in group decision making. Control Cybern. 31, 1037–1053 (2002)

    MATH  Google Scholar 

  49. Szmidt, E., Kacprzyk, J.: A consensus-reaching process under intuitionistic fuzzy preference relations. Int. J. Intell. Syst. 18(7), 837–852 (2003)

    Article  MATH  Google Scholar 

  50. Tajdin, A., Mahdavi, I., Mahdavi-Amiri, N., Sadeghpour-Gildeh, B.: Computing a fuzzy shortest path in a network with mixed fuzzy arc lengths using α-cuts. Comput. Math. Appl. 60(4), 989–1002 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Träff, J.L.: An experimental comparison of two distributed single-source shortest path algorithms. Parallel Comput. 21(9), 1505–1532 (1995)

    Article  MathSciNet  Google Scholar 

  52. Xu, M.H., Liu, Y.Q., Huang, Q.L, Zhang, Y.X., Luan, G.F.: An improved Dijkstra’s shortest path algorithm for sparse network. Appl. Math. Comput. 185(1), 247–254 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Xu, Z.S.: An overview of methods for determining OWA weights. Int. J. Intell. Syst. 20, 843–865 (2005)

    Article  MATH  Google Scholar 

  54. Xu, Z.S, Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 35, 417–433 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  55. Xu, Z.S.: Intuitionistic preference relations and their application in group decision making. Inf. Sci. 177, 2363–2379 (2007)

    Article  MATH  Google Scholar 

  56. Xu, Z.S.: Models for multiple attribute decision making with intuitionistic fuzzy information. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 15, 285–297 (2007)

    Article  MATH  Google Scholar 

  57. Xu, Z.S.; Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15, 1179–1187 (2007)

    Article  Google Scholar 

  58. Zadeh, L.A: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhou, L.-G., Chen, H.-Y., Merigó, J.M., Gil-Lafuente, A.M.: Uncertain generalized aggregation operators. Expert Syst. Appl. 39(1), 1105–1117 (2012)

    Article  Google Scholar 

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  1. Bengal College of Engineering and Technology, SS Banerjee Sarani, Bidhannagar, Durgapur, 713212, West Bengal, India

    Sathi Mukherjee

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  1. Sathi Mukherjee
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Mukherjee, S. Dijkstra’s Algorithm for Solving the Shortest Path Problem on Networks Under Intuitionistic Fuzzy Environment. J Math Model Algor 11, 345–359 (2012). https://doi.org/10.1007/s10852-012-9191-7

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  • DOI: https://doi.org/10.1007/s10852-012-9191-7

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