Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 367 https://internationalpubls.com Fuzzy Traveling Salesman Problem of Fuzzy Hexagonal Numbers using Ranking Method and Dijkstra's Algorithms S.Revathy 𝟏, M.Pandikani 𝟐,V.Nirmala 𝟑∗,P.Megala 𝟒,S.Yamini 𝟓 1 Trinity College for Women Coimbatore, Periyar University, Namakkal 2 Sri Sairam College of Engineering, Visvesvaraya Technological University,Karnataka -562106 3 Department of Mathematics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore 4 Sri Eshwar College of Engineering, Anna University, Coimbatore. 5 Karpagam Institute of Technology, Anna University, Coimbatore. *Corresponding author, e-mail: aaradhananirmala@gmail.com (V.Nirmala) Article History: Received: 01-08-2024 Revised: 11-09-2024 Accepted: 20-09-2024 Abstract The aim of this article is to find the fuzzy shortest possible distance for a fuzzy traveling salesman problem of hexagonal fuzzy numbers. We used to formulate the numerical example using dijkstra's algorithm. Introduction The traveling salesman problem (TSM) involves finding the shortest possible route to multiple destination and returning to the starting point. —_ there we several applications of TSP, such as vehicle routing, scheduling. The TSP in a serious challenge for the logistics and supply chain industry because of involves optimizing the delivery router for multiple destinations while considering various constraints such as, traffic, delivery windows and customer request with multiple vehicle, more cities and multiple sales professionals, TSP become a more challenging to solve. In this article we proposed shorter possible distance route from multiple destination and returning to the starting point using ranking method of hexagonal fuzzy numbers. Conclusions: The ranking approach is used in this paper to transform hexagonal fuzzy numbers into expected time, or standard time, for each activity. Thus, using the DIJKSTRA algorithm, the fuzzy shortest path is found. It assists decision makers in selecting the optimal, shortest path in a fuzzy environment by applying the ranking algorithm. Keywords: fuzzy number, Hexagonal fuzzy number, dijkstra's algorithm Ranking algorithm, sales man. 1. Introduction The traveling salesman problem (TSM) involves finding the shortest possible route to multiple destination and returning to the starting point. —_ there we several applications of TSP, such as vehicle routing, scheduling. The TSP in a serious challenge for the logistics and supply chain industry because of involves optimizing the delivery router for multiple destinations while considering various constraints such as, traffic, delivery windows and customer request.With multiple vehicle, more cities and multiple sales professionals, TSP become a more challenging to solve. In this article we proposed shorter possible distance route from multiple destination and returning to the starting point using ranking method of hexagonal fuzzy numbers. mailto:aaradhananirmala@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 368 https://internationalpubls.com 2 Preliminaries 2.1 Definition Let the universe be 𝑈. The definition of the fuzzy setI on 𝑈 is as follows, I = {𝑥, 𝜒A(𝑥), 𝑥 ∈ U} In which 𝜒A(𝑥): U → [0,1] is called membership 2.2 Definition If there is an element 𝑥 ∈ U whose membership value is one, that is 𝜒I(𝑥) = 1. Then a fuzzy set I is normal. 2.3 Definition Fuzzy number is defined as fuzzy set I of real line R with membership function 𝜒I(𝑥): R → [0,1] if (i) I is normal and convexity (ii) I must be bounded (iii) 𝛼I must be closed interval for every 𝛼 ∈ [0,1] 2.4 Definition A fuzzy number I𝐻 is an HFN and is represented as I𝐻 = (𝑛1, 𝑛2, 𝑛3, 𝑛4, 𝑛5, 𝑛6). In which real numbers ( 𝑛1, 𝑛2, 𝑛3, 𝑛4, 𝑛5, and 𝑛6 ) are found. The function its membership is listed below f(x) = { 0.5 [ 𝑥 − 𝑛1 𝑛3 − 𝑛1 ] for 𝑛1 ≤ 𝑥 ≤ 𝑛2 0.5 + 0.5 [ 𝑥 − 𝑛2 𝑛3 − 𝑛2 ] for 𝑛2 ≤ 𝑥 ≤ 𝑛3 1 for 𝑛3 ≤ 𝑥 ≤ 𝑛4 1 − 0.5 [ 𝑥 − 𝑛4 𝑛5 − 𝑛4 ] for 𝑛4 ≤ 𝑥 ≤ 𝑛5 0.5 [ 𝑛6 − 𝑥 𝑛6 − 𝑛5 ] for 𝑛5 ≤ 𝑥 ≤ 𝑛6 3 Operation of hexagonal fuzzy number Assuming that S𝐻 = (𝑔1, 𝑔2, 𝑔3, 𝑔4, 𝑔5, 𝑔6) and T𝐻 = (ℎ1, ℎ2, ℎ3, ℎ4, ℎ5, ℎ6). are two hexagonal fuzzy numbers, the three operations that can be carried out on them are as follows, 1. Addition: S𝛼 + T𝛼 = (𝑔1 + ℎ1, 𝑔2 + ℎ2, 𝑔3 + ℎ3, 𝑔4 + ℎ4, 𝑔5 + ℎ5, 𝑔6 + ℎ6) 2. subtraction: S𝛼 − T𝛼 = (𝑔1 − ℎ1, 𝑔2 − ℎ2, 𝑔3 − ℎ3, 𝑔4 − ℎ4, 𝑔5 − ℎ5, 𝑔6 − ℎ6) 3. multiplication : S𝛼 ∗ T𝛼 = (𝑔1 ∗ ℎ1, 𝑔2 ∗ ℎ2, 𝑔3 ∗ ℎ3, 𝑔4 ∗ ℎ4, 𝑔5 ∗ ℎ5, 𝑔6 ∗ ℎ6) 3.1𝛼-cut of Hexagonal Fuzzy Number The 𝛼-cut of normal hexagonal fuzzy number S𝐻 = (𝑔1, 𝑔2, 𝑔3, 𝑔4, 𝑔5, 𝑔6) given by the definition (2.2). therefore 𝜒I(𝑥) = 1 for all 𝛼 ∈ [0,1] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 369 https://internationalpubls.com S𝐻 = { [2𝛼(𝑔3 − 𝑔1) + 𝑔1, −2𝛼(𝑔6 − 𝑔5) + 𝑔6] for 𝛼 ∈ [0,0.5) [2𝛼(𝑔3 − 𝑔2) − 𝑔3 + 2𝑔2, −2𝛼(𝑔5 − 𝑔4) + 2𝑔5 − 𝑔4] for 𝛼 ∈ [0.5,1] 3.2 Preposition 1. Let us consider two HFN𝐻 = (𝑔1, 𝑔2, 𝑔3, 𝑔4, 𝑔5, 𝑔6) and T𝐻 = (ℎ1, ℎ2, ℎ3, ℎ4, ℎ5, ℎ6). Then the addition of two hexagonal fuzzy number. Proof Let us add the alpha cut S𝛼 and T𝛼 of S𝐻 and T𝐻 using interval arithmetic. S𝛼 + T𝛼 = { [2𝛼(𝑔3 − 𝑔1) + 𝑔1, −2𝛼(𝑔6 − 𝑔5) + 𝑔6] + [2𝛼(ℎ3 − ℎ1) + ℎ1, −2𝛼(ℎ6 − ℎ5) + ℎ6], for 𝛼 ∈ [0,0.5) [2𝛼(𝑔3 − 𝑔2) − 𝑔3 + 2𝑔2, −2𝛼(𝑔5 − 𝑔4) + 2𝑔5 − 𝑔4] + [2𝛼(ℎ3 − ℎ2) − ℎ3 + 2ℎ2, −2𝛼(ℎ5 − ℎ4) + 2ℎ5 − ℎ4], for 𝛼 ∈ [0.5,1] S𝛼 = [2𝛼 + 𝑔1, −2𝛼 + 𝑔6] for 𝛼 ∈ [0,0.5) T𝛼 = [4𝛼 + ℎ1, −4𝛼 + ℎ6] for 𝛼 ∈ [0.5,1] So that, S𝛼 + T𝛼 = [6𝛼 + (𝑔1 + ℎ1),−6𝛼 + (𝑔6 + ℎ6)] Consider the example of HFN S𝐻 = (2,5,6,3,7,3) and T𝐻 = (9,7,5,1,6,4) S𝛼 = [2𝛼 + 2,−2𝛼 + 3] for 𝛼 ∈ [0,0.5) T𝛼 = [4𝛼 + 9, −4𝛼 + 4] for 𝛼 ∈ [0.5,1] Since for both 𝛼 ∈ [0,0.5) and 𝛼 ∈ [0.5,1] arithmetic intervals are same. Therefore S𝛼 + T𝛼 = [6𝛼 + 11,−6𝛼 + 7] for all 𝛼 ∈ [0,1] 𝛼 = 0 ⟹ S0 + T0 = [11,7] 𝛼 = 0.5 ⟹ S0.5 + T0.5 = [13,4] 𝛼 = 1 ⟹ S1 + T1 = [16,1] Hence, S𝛼 + T𝛼 = [11,13,16,1,4,7], Every point corresponds to part of the two HFN. Consequently, the interval is covered by the addition of two 𝛼-cut. 2 If we assume two hexagonal fuzzy numbers S = (𝑔1, 𝑔2, 𝑔3, 𝑔4, 𝑔5, 𝑔6) and H = (𝑡1, 𝑡2, 𝑡3, 𝑡4, 𝑡5, 𝑡6). Then the subtraction of two numbers. Proof Let us add the alpha cut S𝛼 and T𝛼 of S𝐻 and T𝐻 using interval arithmetic. S𝛼 + T𝛼 = { [2𝛼(𝑔3 − 𝑔1) + 𝑔1, −2𝛼(𝑔6 − 𝑔5) + 𝑔6] − [2𝛼(ℎ3 − ℎ1) + ℎ1, −2𝛼(ℎ6 − ℎ5) + ℎ6], for 𝛼 ∈ [0,0.5) [2𝛼(𝑔3 − 𝑔2) − 𝑔3 + 2𝑔2, −2𝛼(𝑔5 − 𝑔4) + 2𝑔5 − 𝑔4] − [2𝛼(ℎ3 − ℎ2) − ℎ3 + 2ℎ2, −2𝛼(ℎ5 − ℎ4) + 2ℎ5 − ℎ4], for 𝛼 ∈ [0.5,1] . S𝛼 = [2𝛼 + 𝑔1, −2𝛼 + 𝑔6] for 𝛼 ∈ [0,0.5) T𝛼 = [4𝛼 + ℎ1, −4𝛼 + ℎ6] for 𝛼 ∈ [0.5,1] since, S𝛼 − T𝛼 = [−2𝛼 + (𝑔1 − ℎ1), 2𝛼 + (𝑔6 − ℎ6)] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 370 https://internationalpubls.com Consider the example of HFN S𝐻 = (1,3,5,7,9,2) and T𝐻 = (2,4,6,8,1,10) S𝛼 = [2𝛼 + 1,−2𝛼 + 2] for 𝛼 ∈ [0,0.5) T𝛼 = [4𝛼 + 2, −4𝛼 + 10] for 𝛼 ∈ [0.5,1] Since for both 𝛼 ∈ [0,0.5) and 𝛼 ∈ [0.5,1] arithmetic intervals are same. Therefore S𝛼 + T𝛼 = [−2𝛼 − 1,2𝛼 − 8] for all 𝛼 ∈ [0,1] 𝛼 = 0 ⟹ S0 + T0 = [−1,−8] 𝛼 = 0.5 ⟹ S0.5 + T0.5 = [−2, −7] 𝛼 = 1 ⟹ S1 + T1 = [−3,−6] Hence, S𝛼 − T𝛼 = [−1,−2,−3,−6,−7,−8], Every point corresponds to part of the two HFN. Consequently, the interval is covered by the addition of two 𝛼-cut. 3 If we assume two hexagonal fuzzy numbers S = (𝑔1, 𝑔2, 𝑔3, 𝑔4, 𝑔5, 𝑔6) and H = (𝑡1, 𝑡2, 𝑡3, 𝑡4, 𝑡5, 𝑡6). Then the multiplication of two numbers. proof Let us add the alpha cut S𝛼 and T𝛼 of S𝐻 and T𝐻 using interval arithmetic. S𝛼 + T𝛼 = { [2𝛼(𝑔3 − 𝑔1) + 𝑔1, −2𝛼(𝑔6 − 𝑔5) + 𝑔6] ∗ [2𝛼(ℎ3 − ℎ1) + ℎ1, −2𝛼(ℎ6 − ℎ5) + ℎ6], for 𝛼 ∈ [0,0.5) [2𝛼(𝑔3 − 𝑔2) − 𝑔3 + 2𝑔2, −2𝛼(𝑔5 − 𝑔4) + 2𝑔5 − 𝑔4] ∗ [2𝛼(ℎ3 − ℎ2) − ℎ3 + 2ℎ2, −2𝛼(ℎ5 − ℎ4) + 2ℎ5 − ℎ4], for 𝛼 ∈ [0.5,1] . S𝛼 = [2𝛼 + 𝑔1, −2𝛼 + 𝑔6] for 𝛼 ∈ [0,0.5) T𝛼 = [4𝛼 + ℎ1, −4𝛼 + ℎ6] for 𝛼 ∈ [0.5,1] since, S𝛼 ∗ T𝛼 = [(2𝛼 + 𝑔1)(4𝛼 + ℎ1), (−2𝛼 + 𝑔6)(−4𝛼 + ℎ6)] 4 If we assume two hexagonal fuzzy numbers S = (𝑔1, 𝑔2, 𝑔3, 𝑔4, 𝑔5, 𝑔6) and H = (𝑡1, 𝑡2, 𝑡3, 𝑡4, 𝑡5, 𝑡6). Then the division of two numbers. Proof Let us add the alpha cut S𝛼 and T𝛼 of S𝐻 and T𝐻 using interval arithmetic. S𝛼/T𝛼 = { [2𝛼(𝑔3 − 𝑔1) + 𝑔1, −2𝛼(𝑔6 − 𝑔5) + 𝑔6]/ [2𝛼(ℎ3 − ℎ1) + ℎ6, −2𝛼(ℎ6 − ℎ5) + ℎ6], [2𝛼(𝑔3 − 𝑔2) − 𝑔3 + 2𝑔2, −2𝛼(𝑔5 − 𝑔4) + 2𝑔5 − 𝑔4]/ [2𝛼(ℎ3 − ℎ2) − ℎ3 + 2ℎ2, −2𝛼(ℎ5 − ℎ4) + 2ℎ5 − ℎ4], for 𝛼 ∈ [0,0.5) S𝛼 = [2𝛼 + 𝑔1, −2𝛼 + 𝑔6] for 𝛼 ∈ [0.5,1] T𝛼 = [2𝛼 + ℎ1, −4𝛼 + ℎ6] for 𝛼 ∈ [0.5,1] since, S𝛼/T𝛼 = [(2𝛼 + 𝑔1)/(4𝛼 + ℎ6), (−2𝛼 + 𝑔6)/(−4𝛼 + ℎ6)] 4 New Ranking Function The classical set I𝛼 called alpha cut set is the set of elements whose degree of membership is the set of elements whose degree of membership in I𝐻 = (𝑛1, 𝑛2, 𝑛3, 𝑛4, 𝑛5, 𝑛6) is no less than, 𝛼 it is defined as I = {𝑥 ∈ U/𝜒I𝐻(𝑥) ≥ 𝛼} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 371 https://internationalpubls.com = { [P1(𝛼), P2(𝛼)], for 𝛼 ∈ [0,0.5) [Q1(𝛼), Q2(𝛼)], for 𝛼 ∈ [0.5,1] = { [P1(𝛼), Q1(𝛼)], for 𝛼 ∈ [0,0.5) [P2(𝛼), Q2(𝛼)], for 𝛼 ∈ [0.5,1] = [𝑛1 + 𝛼(𝑛3 − 𝑛1) + 𝑛6 + 𝛼(𝑛4 − 𝑛6)] for 𝛼 ∈ [0,1] It provides results, I is a fuzzy number then the ranking is defined by R(I𝐻) = ∫ 0 1  2(0.5)(Iℎ𝛼 𝐿 , Iℎ𝛼 𝑈)𝐷𝛼 where (Iℎ𝛼 𝐿 , Iℎ𝛼 𝑈) is the 𝛼 level cut of the fuzzy number I𝐻 R(I𝐻) = ∫ 0 1  2(0.5)[𝑛1 + 𝛼(𝑛3 − 𝑛1) + 𝑛6 + 𝛼(𝑛4 − 𝑛6)]𝐷𝛼 5 Description of the Model Using the Ranking algorithm R[I𝐻], hexagonal fuzzy numbers are transformed into anticipated time (Normal time) for every activity. These numbers are interpreted as the typical travel time between the nodes and the provided algorithm is used to find the path, or least distance. 5.1 Algorithm step 1: Create the network diagram in accordance with the task assigned. step 2: Find the (normal) Expected time using the Ranking algorithm given the HFN. step 3: Find out how many paths, or possible routes, there are from the starting node to the finishing nodes. step 4: Finally, DIJKSTRA's algorithm will be used to find the shortest path or minimal traveling time. 6 Numerical example-1 Think of a project that has activities and nodes. HFN is a representation of the separation between them. Step 1: Create the network diagram in accordance with the task assigned. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 372 https://internationalpubls.com Activity HFN 1 − 2 (4,3,2,7,6,5) 1 − 3 (6,5,1,14,3,9) 3 − 4 (9,1,3,12,10,6) 2 − 4 (1,3,10,9,7,18) 3 − 6 (2,4,1,9,6,8) 2 − 6 (8,4,3,14,1,9) 5 − 7 (2,9,8,11,10,17) 4 − 7 (4,8,10,12,13,18) 6 − 7 (16,11,12,17,10,13) Table 1: Activities and hexagonal fuzzy number step 2: Find the (normal) Expected time using the Ranking algorithm given the HFN. Step 3 : Find out how many paths, or possible routes, there are from the starting node to the finishing nodes. 1 − 2 − 4 − 7 45 1 − 2 − 5 − 7 45 1 − 3 − 4 − 7 52 1 − 3 − 6 − 7 54 Table 2: path and time using shortest traveling path method Step 4: Finally, DIJKSTRA's algorithm will be used to find the shortest path or minimal traveling time. Fuzzy shortest traveling path is 1 − 2 − 4 − 7 = 45 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 373 https://internationalpubls.com 7 Numerical example-2 one more numerical illustration of finding the shortest path. With a few activities and nodes in mind, the HFN represents the distance between them. Step 1:Create the network diagram in accordance with the task assigned. Table 3: Activities and hexagonal fuzzy number Step 2:Find the (normal) Expected time using the Ranking algorithm given the HFN. Step 3: Find out how many paths, or possible routes, there are from the starting node to the finishing nodes. 1 − 2 − 3 − 4 55 1 − 6 − 5 − 4 35 1 − 6 − 2 − 3 − 4 68 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 374 https://internationalpubls.com 1 − 6 − 5 − 3 − 4 64 1 − 2 − 3 − 6 − 5 − 4 77 Table 4: path and time using shortest traveling path method Step 4: Finally, "DIJKSTRA's algorithm" will be used to find the shortest path or minimal traveling time. Fuzzy shortest traveling path is 1 − 6 − 5 − 4 = 35 8 Conclusion The ranking approach is used in this paper to transform hexagonal fuzzy numbers into expected time, or standard time, for each activity. Thus, using the DIJKSTRA algorithm, the fuzzy shortest path is found. It assists decision makers in selecting the optimal, shortest path in a fuzzy environment by applying the ranking algorithm. 9 Declaration 9.1 Funding of interests: No funding was received to assist with the preparation of this manuscript. 9.2 Conflicts of interests: The authors have no compelling interests to declare that are relevant to the content of this article. Refrences [1] Abbasi.F and Allahviranloo.T (2021) The Fuzzy Arithmetic Operations of Transmission Average on Pseudo- Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis, Fuzzy Information and Engineering ,Volume 13,58-78 [2] Ching-Hsue Cheng (1998) A new approach for ranking fuzzy numbers by distance method, A new approach for ranking fuzzy numbers by distance method,Volume 95, 307-317. [3] Deng Feng, Jiang Xia Nan,Mao Jun Zhang, Venkataraman. 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