Dijkstra algorithm for shortest path problem under interval-valued Pythagorean fuzzy environment

Pythagorean fuzzy set as an extension of fuzzy set has been presented to handle the uncertainty in real-world decision-making problems. In this work, we formulate a shortest path (SP) problem in an interval-valued Pythagorean fuzzy environment. Here, the costs related to arcs are taken in the form of interval-valued Pythagorean fuzzy numbers (IVPFNs). The main contributions of this paper are fourfold: (1) the interval-valued Pythagorean fuzzy optimality conditions in directed networks are described to design of solution algorithm. (2) To do this, an improved score function is used to compare the costs between different paths with their arc costs represented by IVPFNs. (3) Based on these optimality conditions and the improved score function, the traditional Dijkstra algorithm is extended to find the cost of interval-valued Pythagorean fuzzy SP (IVPFSP) and corresponding IVPFSP. (4) Finally, a small sized telecommunication network is provided to illustrate the potential application of the proposed method.

[1]  Madjid Tavana,et al.  A novel artificial bee colony algorithm for shortest path problems with fuzzy arc weights , 2016 .

[2]  Sankaran Mahadevan,et al.  Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment , 2012, Appl. Soft Comput..

[3]  Zeshui Xu,et al.  Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets , 2014, Int. J. Intell. Syst..

[4]  Harish Garg,et al.  Generalised Pythagorean fuzzy geometric interactive aggregation operators using Einstein operations and their application to decision making , 2018, J. Exp. Theor. Artif. Intell..

[5]  Timothy Soper,et al.  A shortest path problem on a network with fuzzy arc lengths , 2000, Fuzzy Sets Syst..

[6]  Harish Garg,et al.  Generalized Pythagorean Fuzzy Geometric Aggregation Operators Using Einstein t‐Norm and t‐Conorm for Multicriteria Decision‐Making Process , 2017, Int. J. Intell. Syst..

[7]  Harish Garg,et al.  A Linear Programming Method Based on an Improved Score Function for Interval-Valued Pythagorean Fuzzy Numbers and Its Application to Decision-Making , 2018, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[8]  Harish Garg,et al.  Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision‐making process , 2018, Int. J. Intell. Syst..

[9]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .

[10]  Harish Garg,et al.  A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem , 2016, J. Intell. Fuzzy Syst..

[11]  Harish Garg,et al.  A Novel Improved Accuracy Function for Interval Valued Pythagorean Fuzzy Sets and Its Applications in the Decision‐Making Process , 2017, Int. J. Intell. Syst..

[12]  Harish Garg,et al.  Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process , 2017, Computational and Mathematical Organization Theory.

[13]  Sathi Mukherjee,et al.  Dijkstra’s Algorithm for Solving the Shortest Path Problem on Networks Under Intuitionistic Fuzzy Environment , 2012, J. Math. Model. Algorithms.

[14]  Esmaile Khorram,et al.  A fuzzy shortest path with the highest reliability , 2009 .

[15]  Harish Garg,et al.  New exponential operational laws and their aggregation operators for interval‐valued Pythagorean fuzzy multicriteria decision‐making , 2018, Int. J. Intell. Syst..

[16]  Ronald R. Yager,et al.  Pythagorean Membership Grades in Multicriteria Decision Making , 2014, IEEE Transactions on Fuzzy Systems.

[17]  Yaling Dou,et al.  Solving the fuzzy shortest path problem using multi-criteria decision method based on vague similarity measure , 2012, Appl. Soft Comput..

[18]  Harish Garg,et al.  A Novel Correlation Coefficients between Pythagorean Fuzzy Sets and Its Applications to Decision‐Making Processes , 2016, Int. J. Intell. Syst..

[19]  Gaurav Kumar,et al.  Algorithm for Shortest Path Problem in a Network with Interval-valued Intuitionistic Trapezoidal Fuzzy Number , 2015 .

[20]  Harish Garg,et al.  HESITANT PYTHAGOREAN FUZZY SETS AND THEIR AGGREGATION OPERATORS IN MULTIPLE ATTRIBUTE DECISION-MAKING , 2018 .

[21]  Harish Garg,et al.  A New Generalized Pythagorean Fuzzy Information Aggregation Using Einstein Operations and Its Application to Decision Making , 2016, Int. J. Intell. Syst..

[22]  G. Geetharamani,et al.  Using similarity degree approach for shortest path in Intuitionistic fuzzy network , 2012, 2012 International Conference on Computing, Communication and Applications.

[23]  Bashir Alam,et al.  An Algorithm for Extracting Intuitionistic Fuzzy Shortest Path in a Graph , 2013, Appl. Comput. Intell. Soft Comput..

[24]  Xiaolu Zhang,et al.  Multicriteria Pythagorean fuzzy decision analysis: A hierarchical QUALIFLEX approach with the closeness index-based ranking methods , 2016, Inf. Sci..

[25]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[26]  Homayun Motameni,et al.  Constraint Shortest Path Problem in a Network with Intuitionistic Fuzzy Arc Weights , 2018, IPMU.

[27]  Harish Garg,et al.  Some methods for strategic decision‐making problems with immediate probabilities in Pythagorean fuzzy environment , 2018, Int. J. Intell. Syst..

[28]  Ali Ebrahimnejad,et al.  Particle swarm optimisation algorithm for solving shortest path problems with mixed fuzzy arc weights , 2015, Int. J. Appl. Decis. Sci..