On Solution of Interval Valued Intuitionistic Fuzzy Assignment Problem Using Similarity Measure and Score Function

Authors: Gaurav Kumar, Rakesh Kumar Bajaj

Abstract:

The primary objective of the paper is to propose a new method for solving assignment problem under uncertain situation. In the classical assignment problem (AP), z_pqdenotes the cost for assigning the qth job to the pth person which is deterministic in nature. Here in some uncertain situation, we have assigned a cost in the form of composite relative degree F_pq instead of  and this replaced cost is in the maximization form. In this paper, it has been solved and validated by the two proposed algorithms, a new mathematical formulation of IVIF assignment problem has been presented where the cost has been considered to be an IVIFN and the membership of elements in the set can be explained by positive and negative evidences. To determine the composite relative degree of similarity of IVIFS the concept of similarity measure and the score function is used for validating the solution which is obtained by Composite relative similarity degree method. Further, hypothetical numeric illusion is conducted to clarify the method’s effectiveness and feasibility developed in the study. Finally, conclusion and suggestion for future work are also proposed.

Keywords: Assignment problem, Interval-valued Intuitionistic Fuzzy Sets, Similarity Measures, score function.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1093594

References:

[1] M. L .Bali ski, A Competitive (dual) simplex method for the assignment problem, Math. Program, 34(2) (1986), 125-141.
[2] R.S. Barr, F. Glover, D. Kingman, The alternating basis algorithm for assignment problems, Math. Program, 13(1) (1977), 1-13.
[3] M.S. Hung, W.O. Rom, Solving the assignment problem by relaxation, Oper. Res., 28(4) (1980), 969-982.
[4] L.F. McGinnis, Implementation and testing of a primal-dual algorithm for the assignment problem, Oper. Res., 31(2) (1983), 277-291.
[5] H.W. Kuhn, the Hungarian method for the assignment problem, Naval Research Logistics Quartely, 2 (1955), 83-97.
[6] S.P. Eberhardt, T. Duad, A. Kerns, T.X. Brown,A.P. Thakoor, Competitive neural architecture for hardware solution to the assignment problem, Neural Networks, 4(4) (1991), 431-442.
[7] D. Avis, L.Devroye, An analysis of a decomposition heuristic for the assignment problem, Oper. Res. Lett., 3(6) (1985), 279-283.
[8] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338–353.
[9] Lin Chi-Jen, Wen Ue-Pyng, A labeling algorithm for the fuzzy assignment problem, Fuzzy Sets and Systems 142 (2004), 373-391.
[10] Atanassov, K.T.: Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, 20, 69–78 (1986).
[11] Atanassov, K.T.: Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 64, 159–174 (1994).
[12] Atanassov, K.T. and Gargov, G.: Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31, 343–349 (1989).
[13] Hwang, C. L. and Yoon, K.: Multiple attributes decision making methods and applications. Springer-Verlag, Berlin, 1981.
[14] Yoon, K.: The propagation of errors in multiple attribute decision analysis: a practical approach. Journal of the Operational Research Society, 40, 681–686 (1984).
[15] Edwards, W.: How to use multi-attribute utility measurement for social decision making. IEEE Trans. on Systems, Man and Cybernetics, 7, 326–340 (1977).
[16] Saaty T. L.: A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15, 234–281 (1977).
[17] Yang, J. B. and Xu, D. L.: Nonlinear information aggregation via evidential reasoning in multi-attribute decision analysis under uncertainty. I.E.E.E.Trans. on Systems, Man and Cybernetics-Part A, 32, 376–393 (2002).
[18] Bustine H. and Burillo P.: Vague sets are intuitionistic fuzzy sets, Fuzzy sets and Systems, 79, 403–405 (1996).
[19] Park, J.H., Lim, K.M. and Park, J.S.: Distances between interval-valued intuitionistic fuzzy sets,2007 International Symposium on Nonlinear Dynamics, Journal of Physics: Conference Series, 96, 1–8 (2008).
[20] Xu, Z.S., Chen, and J.and Wu J.J.: Clustering algorithm for intuitionistic fuzzy sets, Information Sciences, 178, 3775–3790 (2008).
[21] Xu, Z. and Ronald R Y.: Intuitionistic and interval valued intuitionistic fuzzy preference relations and their measure of similarity for the evaluation of agreement within a group, Fuzzy Optimization Decision Making, 8, 123–139 (2009).
[22] Xu, Z.S.: "On similarity measures of interval-valued intuitionistic fuzzy sets and their application to pattern recognitions” Journal of Southeast University (English Edition), 23, 139–143 (2007).
[23] Xu, Z.S.: "Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making,” Control and Decision, 22, 215–219 (2007).
[24] Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied Soft Computing, 10 423–431 (2010).
[25] Xu, Z.S.: A method based on distance measure for interval-valued intuitionistic fuzzy group decision making, Information Sciences, 180 (2010), 181-190.
[26] Xu, Z.S.: Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences, 180, 726–736 (2010).
[27] Jahan, A., Ismail, M.Y., Mustapha F. and Sapuan, S.M.: Material selection based on ordinal data, Materials and Design, 31 3180–3187 (2010).
[28] Mukherjee, S. and Basu, K.: Solving Intuitionistic Fuzzy Assignment Problem by Using Similarity Measures and Score Functions, 2, 1–18 (2011).