Fuzzy Optimization and Decision Making

, Volume 4, Issue 4, pp 293–312 | Cite as

Shortest Path Problem on a Network with Imprecise Edge Weight

  • SK. MD. Abu. Nayeem
  • Madhumangal Pal


A network with its arc lengths as imprecise number, instead of a real number, namely, interval number and triangular fuzzy number is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. To obtain a fuzzy shortest path from a source vertex to all other vertices, a common algorithm is developed which works well on both types of imprecise numbers under consideration. In the proposed algorithm, a decision-maker is to negotiate with the obtained fuzzy shortest paths according to his/her view only when the means are same but the widths are different of the obtained paths. Otherwise, a fuzzy optimal path is obtained to which the decision-maker always satisfies with different grades of satisfaction. All pairs fuzzy shortest paths can be found by repeated use of the proposed algorithm.

Key words

fuzzy shortest path interval numbers triangular fuzzy numbers network 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Blue, M., Bush, B., Puckett, J. 2002“Unified approach to Fuzzy Graph Problems”Fuzzy Sets and Systems125355368CrossRefMathSciNetGoogle Scholar
  2. Bortolan, G., Degani, R. 1985“A review of some methods for ranking fuzzy subsets”Fuzzy Sets and Systems15119CrossRefGoogle Scholar
  3. Chen, S.-H. 1985“Ranking Fuzzy Numbers with Maximizing Set and Minimizing Set”Fuzzy Sets and Systems17113129CrossRefGoogle Scholar
  4. Cheng, C.-H. 1998“A New Approach for Ranking Fuzzy Numbers by Distance Method”Fuzzy Sets and Systems95307317CrossRefMathSciNetGoogle Scholar
  5. Delgado, M., Verdegay, J.L., Vila, M.A. 1988“A Procedure for Ranking Fuzzy Numbers using Fuzzy Relations”Fuzzy Sets and Systems264962CrossRefGoogle Scholar
  6. Dijkstra, E.W. 1959“A Note on Two Problems in Connection with Graphs”Numerische Mathematik1269271CrossRefGoogle Scholar
  7. Dubois, D., Prade, H. 1980Fuzzy Sets and Systems: Theory and ApplicationsAcademic PressNew YorkGoogle Scholar
  8. Dubois, D., Prade, H. 1983“Ranking Fuzy Numbers in the Setting of Possiblity Theory”Information Sciences30183224CrossRefGoogle Scholar
  9. Ishibuchi, H., Tanaka, H. 1990“Multiobjective Programming in Optimization of The Interval Objective Function”.European Journal of Operational Research48219225CrossRefGoogle Scholar
  10. Klein, C.M. 1991“Fuzzy Shortest Paths”Fuzzy Sets and Systems392741CrossRefGoogle Scholar
  11. Koczy, L.T. 1992“Fuzzy Graphs in the Evaluation and Optimization of Networks”Fuzzy Sets and Systems46307319CrossRefGoogle Scholar
  12. Li, Y., Gen, M., Ida, K. 1996“Solving Fuzzy Shortest Path Problem by Neural Networks”Comput. Ind. Eng31861865CrossRefGoogle Scholar
  13. Lin, K., Chen, M. 1994“The Fuzzy Shortest Path Problem and its Most Vital Arcs”Fuzzy Sets and Systems58343353CrossRefMathSciNetGoogle Scholar
  14. Moore, R.E. 1997Method and Application of Interval AnalysisSIAMPhiladelphiaGoogle Scholar
  15. Okada, S., Gen, M. 1993“Order Relation Between Intervals and its Application to Shortest Path Problem”Comput. Ind. Eng25147150CrossRefGoogle Scholar
  16. Okada, S., Gen, M. 1994“Fuzzy Shortest Path Problem”Comput. Ind. Eng27465468CrossRefGoogle Scholar
  17. Okada, S., Soper, T. 2000“A Shortest Path Problem on a Network with Fuzzy Arc Lengths”Fuzzy Sets and Systems109129140CrossRefMathSciNetGoogle Scholar
  18. Sengupta, A., Pal, T.K. 2000“Theory and Methodology on Comparing Interval Numbers”European Journal of Operational Research1272843CrossRefMathSciNetGoogle Scholar
  19. Sengupta, J.K. 1981Optimal Decision under UncertaintySpringerNew YorkGoogle Scholar
  20. Shaocheng, T. 1994“Interval Number and Fuzzy Number Linear Programmings”Fuzzy Sets and Systems66301306CrossRefGoogle Scholar
  21. SPLIB ( Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Applied Mathematics with Oceanology and Computer ProgrammingVidyasagar UniversityMidnaporeIndia
  2. 2.Department of Applied Mathematics with Oceanology and Computer ProgrammingVidyasagar UniversityMidnaporeIndia

Personalised recommendations