Abstract
Path finding models attempt to provide efficient approaches for finding shortest paths in networks. A well-known shortest path algorithm is the Dijkstra algorithm. This paper redesigns it in order to tackle situations in which the parameters of the networks may be uncertain. To be precise, we allow that the parameters take the form of special picture fuzzy numbers. We use this concept so that it can flexibly fit the vague character of subjective decisions. The main contributions of this article are fourfold: \(\mathrm{(i)}\) The trapezoidal picture fuzzy number along with its graphical representation and operational laws is defined. \(\mathrm{(ii)}\) The comparison of trapezoidal picture fuzzy numbers on the basis of their expected values is proposed in terms of their score and accuracy functions. \(\mathrm{(iii)}\) Based on these elements, we put forward an adapted form of the Dijkstra algorithm that works out a picture fuzzy shortest path problem, where the costs associated with the arcs are captured by trapezoidal picture fuzzy numbers. Also, a pseudocode for the application of our solution is provided. \(\mathrm{(iv)}\) The proposed algorithm is numerically evaluated on a transmission network to prove its practicality and efficiency. Finally, a comparative analysis of our proposed method with the fuzzy Dijkstra algorithm is presented to support its cogency.
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References
Akram M, Arshad M (2019) A novel trapezoidal bipolar fuzzy TOPSIS method for group decision-making. Group Decis Negot 28(3):565–584
Akram M, Zafar F (2020) Hybrid soft computing models applied to graph theory. Stud Fuzz Soft Comput Springer 380:1–434
Alcantud JCR, Giarlotta A (2019) Necessary and possible hesitant fuzzy sets: a novel model for group decision making. Inf Fusion 46:63–76
Alcantud JCR, Laruelle A (2014) Dis&approval voting: a characterization. Soc Choice Welfare 43(1):1–10
Atanassov KT (1986) Intuitionistic fuzzy sets: theory and applications. Fuzzy Sets Syst 20:87–96
Bellman R (1958) On a routing problem. Q Appl Math 16(1):87–90
Burillo, P., Bustince, H., & Mohedano, V. (1994, September). Some definitions of intuitionistic fuzzy number. First properties. In: Proceedings of the 1st workshop on fuzzy based expert systems, pp 53–55. Bulgaria: Sofia
Chanas S, Nowakowski M (1988) Single value simulation of fuzzy variable. Fuzzy Sets Syst 25(1):43–57
Chen SJ, Hwang CL (1992). Fuzzy multiple attribute decision making methods. In: Fuzzy multiple attribute decision making, Springer, Berlin, pp 289–486
Chen SM, Tan JM (1994) Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 67(2):163–172
Chou CC (2003) The canonical representation of multiplication operation on triangular fuzzy numbers. Comput Math Appl 45(10–11):1601–1610
Cuong BC (2013, May). Picture fuzzy sets–first results, Part 1. In: Seminar neuro-fuzzy systems with applications. Preprint 03/2013. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi-Vietnam.
Cuong BC (2013). Picture fuzzy sets–first results, Part 2. In: Seminar neuro-fuzzy systems with applications. Preprint 04/2013. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi-Vietnam
Cuong BC, Kreinovich V (2013, December). Picture Fuzzy Sets-a new concept for computational intelligence problems. In: 2013 third world congress on information and communication technologies (WICT 2013), pp 1-6. IEEE
Deng Y, Chen Y, Zhang Y, Mahadevan S (2012) Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment. Appl Soft Comput 12(3):1231–1237
Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer Math 1(1):269–271
Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30(3):183–224
Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626
Dubois D, Prade H (1980). Fuzzy sets and systems: theory and applications, Vol. 144. Academic Press, Amsterdam
Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24(3):279–300
Floyd RW (1962) Algorithm 97: shortest path. Commun ACM 5(6):345
Geetharamani, G., & Jayagowri P (2012, February). Using similarity degree approach for shortest path in intuitionistic fuzzy network. In: 2012 international conference on computing, communication and applications, IEEE, pp 1–6
Grzegorzewski P (1998) Metrics and orders in space of fuzzy numbers. Fuzzy Sets Syst 97(1):83–94
Heilpern S (1992) The expected value of a fuzzy number. Fuzzy Sets Syst 47(1):81–86
Hong DH, Choi CH (2000) Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114(1):103–113
Ji X, Iwamura K, Shao Z (2007) New models for shortest path problem with fuzzy arc lengths. Appl Math Model 31(2):259–269
Jian-qiang W (2008) Overview on fuzzy multi-criteria decision-making approach. Control Decis 23(6):601–606
Jianqiang W, Zhong Z (2009) Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems. J Syst Eng Electron 20(2):321–326
Klein CM (1991) Fuzzy shortest paths. Fuzzy Sets Syst 39(1):27–41
Lee-Kwang H, Song YS, Lee KM (1994) Similarity measure between fuzzy sets and between elements. Fuzzy Sets Syst 62(3):291–293
Liu ST, Kao C (2004) Network flow problems with fuzzy arc lengths. IEEE Trans Syst Man Cybernet, Part B Cybernet 34(1):765–769
Mizumoto M, Tanakp K (1976) The four operations of arithmetic on fuzzy numbers. Syst Comput Controls 7(5):73–81
Mou J, Gao L, Guo Q, Xu R, Li X (2019) Hybrid optimization algorithms by various structures for a real-world inverse scheduling problem with uncertain due-dates under single-machine shop systems. Neural Comput Appl 31(9):4595–4612
Mp M, Friedman M, Kandel A (1999) A new fuzzy arithmetic. Fuzzy Sets Syst 108(1):83–90
Mukherjee S (2012) Dijkstra’s algorithm for solving the shortest path problem on networks under intuitionistic fuzzy environment. J Math Modell Algorithms 11(4):345–359
Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1(2):97–110
Nenavath H, Jatoth RK (2019) Hybrid SCA-TLBO: a novel optimization algorithm for global optimization and visual tracking. Neural Comput Appl 31(9):5497–5526
Okada S (2004) Fuzzy shortest path problems incorporating interactivity among paths. Fuzzy Sets Syst 142(3):335–357
Peyer S, Rautenbach D, Vygen J (2009) A generalization of Dijkstra’s shortest path algorithm with applications to VLSI routing. J Discret Algorithms 7(4):377–390
Porchelvi RS, Banumathy A (2018) Generalized Dijkstra’s algorithm for shortest path problem in intuitionistic fuzzy environment. Int J Adv Res Sci Eng 7(6):82–88
Rangasamy P, Akram M, Thilagavathi S (2013) Intuitionistic fuzzy shortest hyperpath in a network. Inf Process Lett 113(17):599–603
Shu-Xi W (2012) The improved Dijkstra’s shortest path algorithm and its application. Proced Eng 29:1186–1190
Shu MH, Cheng CH, Chang JR (2006) Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly. Microelectron Reliab 46(12):2139–2148
Singh P (2015) Correlation coefficients for picture fuzzy sets. J Intell Fuzzy Syst 28(2):591–604
Thong PH (2017) Some novel hybrid forecast methods based on picture fuzzy clustering for weather nowcasting from satellite image sequences. Appl Intell 46(1):1–15
Thong PH (2015). A new approach to multi-variable fuzzy forecasting using picture fuzzy clustering and picture fuzzy rule interpolation method. In: Knowledge and systems engineering, Springer, Cham, pp 679–690
Wang JQ, Zhang Z (2008) Programming method of multi-criteria decision-making based on intuitionistic fuzzy number with incomplete certain information. Control Decis 23(10):1145–1148
Warshall S (1962) A theorem on Boolean matrices. J ACM 9:11–12
Xu Z, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35(4):417–433
Yager RR (1981) A procedure for ordering fuzzy subsets of the unit interval. Inf Sci 24(2):143–161
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Sciences Information (1975) The concept of a linguistic variable and its application to approximate reasoning I. Learn Syst Intell Robot 8(3):199–249
Zhang X, Liu P (2010) Method for aggregating triangular fuzzy intuitionistic fuzzy information and its application to decision making. Technol Econ Dev Econ 16(2):280–290
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Akram, M., Habib, A. & Alcantud, J.C.R. An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers. Neural Comput & Applic 33, 1329–1342 (2021). https://doi.org/10.1007/s00521-020-05034-y
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DOI: https://doi.org/10.1007/s00521-020-05034-y