It has been proposed to use Pythagorean fuzzy sets (PFS), is an extension of fuzzy sets (FSs), to address uncertainty in practical decision– making problems. Then the Shortest Path Problem (SPP) is a well– known network improvement issue with numerous practical applications. Then the shortest Path (SP) and the shortest distance (SD)in an Interval – valued Pythagorean fuzzy graph (I-VPFG) are foundusing a method in the current communication. Nodes and connections are crisp, while the edge weights are I-V triangular Pythagoreanfuzzy numbers (I-VTPFN). Additionally, a numerical example hasbeen used to demonstrate the suggested strategy

M, Asim Basha

M, Mohammed Jabarulla

English

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)

Ratio Mathematica Volume 46, 2023Algorithm approaches for shortest pathproblem in an interval - valued triangularPythagorean fuzzy networkAsim Basha Mubarak*Mohammed Jabarulla Mohamed†AbstractIn this paper, it has been proposed to use Pythagorean fuzzy sets(PFS), is an extensionof fuzzy sets (FSs), to address uncertainty inpractical decision– making problems. Then the Shortest Path Prob-lem (SPP) is a well– known network improvement issue with numer-ous practical applications.Then the shortest Path (SP) and the short-est distance (SD)in an Interval – valued Pythagorean fuzzy graph(I-VPFG) are foundusing a method in the current communication.Nodes and connectionsare crisp, while the edge weights are I-V trian-gular Pythagoreanfuzzy numbers (I-VTPFN). Additionally, a numer-ical example hasbeen used to demonstrate the suggested strategy.Keywords: I - V Pythagorean fuzzy sets (I-VPFS), I – V Pythagoreanfuzzy graph (I-VPFG), I – V triangular Pythagorean fuzzy number (I-VTPFN), ranking function (RF), I – V Pythagorean fuzzy shortestpath problem (I-VPFSPP)2020 AMS subject classifications: 05C72, 12D15, 03D45. 1*PG & Research Depart of mathematics Jamal Mohamed College, Afiliated to BharthidasanUniversity, Tiruchirappalli 620020, India. asimbasha36@gmail.com.†PG & Research Depart of mathematics Jamal Mohamed College, Afiliated to BharthidasanUniversity, Tiruchirappalli 620020, India. m.md.jabarulla@gmail.com.1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,2023. DOI: 10.23755/rm.v46i0.1064. ISSN: 1592-7415. eISSN: 2282-8214. ©M. Asim Bashaet al. This paper is published under the CC-BY licence agreement.117M. Asim Basha and M. Mohammed Jabarulla1 IntroductionThe pair of sets (V, E) that make up a graph are V, which represents the set ofvertices, and E, which represents the set of edges that connect the pair of vertices.Numerous domains, including computer, social and natural research, among oth-ers, use graph theory. Any mathematical, scientific, or engineering problem canbe represented as a graph. Leonard Euler first introduced the idea of graph theoryin the year 1736. In order to address the issue of the seven Konigsberge bridgesconstructed across the Pregel River of Prussia, he produced the first graph. Thekey factor contributing to graph theory’s explosive growth is its application acrossa variety of disciplines. Even though graphs can be used to simulate events thathappen in real-world problems, graph theory became interesting. Graphs are use-ful for studying these issues.In the graphs are a very crucial model for networks. The SPPs are very usefuland are used in various areas, applications for the road network, transportation,communication channel routing, and scheduling issues are a few examples. Theweights of the edges in the basic network issue are taken to be actual numbers.After all, in most constructive applications, the bounds are generally not precise.Therefore, they can be considered fuzzy numbers (FN) in the real world.Zadeh, 1965 extracted the idea of a Fuzzy Set (FS) as a way to identify un-certainty in 1965. As a generalisation of FSs, Zadeh introduced the idea of I – Vfuzzy sets (I-VFS). Intuitionistic fuzzy sets (IFS), which Atanassov [1986] sug-gested in 1986, look more exactly at uncertainty dimension and give the chance toprecisely model the issue based on the available data and consideration I-VFS wasintroduced three years later by Atanassov and Gargov [1989]. Then the definitionof an I-V fuzzy graph (FG) was given by Kumar et al. [2009]. On Intuitionisticfuzzy graph (IFGs), Mohamed and Ali [2018] defined several new applications.Dubois and Prade [1980] were the first to investigate the SPP in an ambiguous andunreliable surroundings. They claim that while the length of the shortest way maybe determined, the network may not contain a path that corresponds to it. Numer-ous academics have looked into the SPP in literature, each in a different way. Linand Chen proposed the Fuzzy SP length in a network created using fuzzy linearprogramming. Furthermore, Yao and Lin [2003] created two distinct fuzzy SP net-work challenges, the first of which makes use of triangular fuzzy integers. Okada[2004] proposed an algorithm to complete the degree of incident for each arc inthe network and introduce the similarity index among the sums of Fuzzy numbers(FN) by taking into consideration interactivity with FNs. Kumar and Kaur [2011]suggested one such approach for locating the SP in a network flow with imprecisearc lengths. A based on model the idea of dynamic scripting was presented byKarunambigai et al. [2007]et al. to locate the SP in IFGs. Additionally, Gani andJabarulla [2010] created a technique for finding the IFSP in a network.118Algorithm approaches for SPP in Pythagorean fuzzy networkTo handle the complex, imprecise, and ambiguity in challenges with effectivedecision-making, Yager [2014] introduced the concept of the PFSs, as a generali-sation of the IFS. The key feature of the PF model is that it relaxes the action thatthe sum of its mem-ship degree and non-mem-ship degree i.e., (0.5+0.4 ≤ 1) withthe squar sum of its mem-ship and non-mem-ship degree i.e., (0.62 + 0.72 ≤ 1).Following Yager [2013] invention of the PFS, Xu and Zhang [2014] granted themathematical form of the PFS as well as the concept of the PFN. Then the ideaof a PFGs was proposed by Naz et al. [2013]. As a generalisation of PFSs, pre-sented the idea of I-VPF sets Xindong and Young [2016]. The idea of an I-VPFGwas suggested by Mohamed and Ali [2018]. In order to accomplish this, we firstoffer a mathematical formulation for SP issues where contradiction costs of arcsare expressed in terms of I-VPF numbers. Then, in order to construct a solutionalgorithm, we present the optimality in I-VPF networks. To compare the costs ofseveral pathways, whose arc are expressed by I-VPF numbers, an improved scorefunction (SF) is employed.In this study, we present a novel method for finding the SP using I-VPFSs.Using the suggested approach, a decision maker can determine the SP and the SDbetween each node and the SN. The following is how the study is structured. Sec2 goes over some fundamental ideas such as I-VTPFNs, arithmetic operations,and RFs. In sec 3, a algorithm for determining the SP and the SD in an I-VPFG isproposed. Sec 4 contains an example of determining the SP and SD between theSN and the DNs. The final section includes some closing remarks.2 PrefaceThis chapter discusses some fundamental ideas, operations in mathematics,and rank order functions for I-VTPFNs are covered.Definition 2.1. (I-VPFS)Xindong and Young [2016]An I-VPFS ÃP̃ , in the global set Z′ is defined byÃP̃ = 〈x, µ̌ÃP̃(x), λ̌ÃP̃(x)〉 : x ∈ Z ′where µ̌ÃP̃= [µ̌ÃP̃L(x), µ̌ÃP̃U(x)] ⊆ [0, 1] and λ̌ÃP̃=[λ̌ÃP̃L(x) , λ̌ÃP̃U(x)]⊆[0, 1] are interval numbers satisfied as 0 ≤[µ̌ÃP̃U(x)]2+[λ̌ÃP̃U(x)]2≤ 1 forseparate element x ∈ Z ′ . For each I-VPFS ÃP̃ and x ∈ X′ ,π̌ÃP̃(x) =[π′ÃP̃L(x) , π′ÃP̃U(x)]=[√1−[µ̌ÃP̃U(x)]2−[λ̌ÃP̃U(x)]2,√1−[µ̌ÃP̃L(x)]2−[λ̌ÃP̃L(x)]2]119M. Asim Basha and M. Mohammed Jabarullais called the hesitancy interval of x to ÃP̃ . For an I-VPFS ÃP̃ is defined thepair〈[µ̌ÃP̃L(x) , µ̌ÃP̃U(x)]〉,〈[λ̌ÃP̃L(x) , λ̌ÃP̃U(x)]〉is called an I-VPFN isdenoted by ÃP̃ =〈[µ̌ÃP̃L, µ̌ÃP̃U]〉,〈[λ̌ÃP̃L, λ̌ÃP̃U]〉.Definition 2.2:- (Interval - valued Pythagorean fuzzy graph)Mohamed and Ali[2018]An I-VPFG with the underneath set v′′ is defined as a pair of G′ =(P̌ , Q̌)where,(i) In the functions µ̌P̃ : v′′ → [0, 1] and λ̌P̃ : v′′ → [0, 1] express thedegree of membership and nonmembership element of the x′ ∈ v′′,respectively, such that 0 ≤ µ̌P̃ (x′) + λ̌P̃ (x′) ≤ 1, ∀ x′ ∈ v′′.(ii) In the functions µ̌Q̃ : E′ ⊆ v′′ × v′′ → [0, 1] , λ̌Q̃ : E ′ ⊆ v′′ × v′′ → [0, 1]is,µ̌Q̃L ((x′, y′)) ≤ min(µ̌P̃L (x′) , µ̌P̃L (y′))andµ̌Q̃L ((x′, y′)) ≥ max(λ̌P̃L (x′) , λ̌P̃L (y′))µ̌Q̃U ((x′, y′)) ≤ min(µ̌P̃U (x′) , µ̌P̃U (y′))andµ̌Q̃U ((x′, y′)) ≥ max(λ̌P̃U (x′) , λ̌P̃U (y′))such that,0 ≤ µ̌2Q̃U((x′, y′)) + λ̌2P̃U((x′, y′)) ≤ 1, ∀ (x′, y′) ∈ E ′.Definition 2.3 Kumar et al. [2015]Let ÃP̃ = (ã1, ã2, ã3)〈[µ̌ÃP̃L, µ̌ÃP̃U]〉,〈[λ̌ÃP̃L, λ̌ÃP̃U]〉andB̃P̃ =(b̃1, b̃2, b̃3)〈[µ̌B̃P̃L, µ̌B̃P̃U]〉,〈[λ̌B̃P̃L, λ̌B̃P̃U]〉be two I-VPTFNs,Then defined as,(i)ÃP̃ ⊕ B̃P̃ = 〈(√ã12 + b̃12,√ã22 + b̃22,√ã32 + b̃32)√(µ̌ÃP̃L)2+(µ̌B̃P̃L)2−(µ̌ÃP̃L)2.(µ̌B̃P̃L)2×√(µ̌ÃP̃U)2+(µ̌B̃P̃U)2−(µ̌ÃP̃U)2.(µ̌B̃P̃U)2×[(λ̌ÃP̃L).(λ̌B̃P̃L),(λ̌ÃP̃U)(λ̌B̃P̃U)]〉 .120Algorithm approaches for SPP in Pythagorean fuzzy networkDefinition 2.4 (Score and accuracy functions) Garg [2016]The SF and the AF of any I-VPTFNÃP̃ =〈(ã, b̃, c̃) [µ̌ÃP̃L, µ̌ÃP̃U].[λ̌ÃP̃L, λ̌ÃP̃U]〉are defined as,S(ÃP̃)= 12[(µ̌ÃP̃L)2+(µ̌ÃP̃U)2−(µ̌ÃP̃L)2−(µ̌ÃP̃U)2], S(ÃP̃)∈[−1, 1] andH(ÃP̃)= 12[(µ̌ÃP̃L)2+(µ̌ÃP̃U)2+(µ̌ÃP̃L)2+(µ̌ÃP̃U)2], H(ÃP̃)∈[0, 1].Remark 2.5Kumar et al. [2015]Let ÃP̃ and B̃P̃ be two I-VTPFVs the collection of real numbers. Then wedefine a method of ranking is,(i) If S(ÃP̃)> S(B̃P̃), then(ÃP̃)>(B̃P̃), that is(ÃP̃)is sup to(B̃P̃),denoted by(ÃP̃)>(B̃P̃).(ii) If S(ÃP̃)= S(B̃P̃)and H(ÃP̃)> H(B̃P̃)then(ÃP̃)>(B̃P̃),that is(ÃP̃)is sup to(B̃P̃)denoted by(ÃP̃)>(B̃P̃).I-VTPF distance 2.6Mohamed and Ali [2018]Let F be any connected I-VPFG. For any path{P̃ : ũ1, ũ2, . . . , ũn}length ofP̃ is described as the weights added together.(W̃i)those arcs in P̃ , in which theweight among two adjacent I-VTriPFNs vertices are J(P̃ ′)=∑ni=1 (W̃′i−1, W̃′i ).any two nodes u, v ∈ F , let P̃ = {P̃i is a u− v path, i = 1, 2, 3, . . .} The defini-tion of the I-VTriPF distance is, IP̃ {u, v} = min{J(P̃ ′i); P̃ , i = 1, 2, . . . , n}.3 Algorithm for SPAn algorithm is suggested in this section for the SP and SD of each node fromthe source node (SN) Okada [2004].The following steps for algorithm are defined byStep – 1: Surmise IP̃i = [0, 0, 0] ; [0, 0] [1, 1] and label the SN is,[0, 0, 0] ; [0, 0] [1, 1] J(G̃)= −−.Step – 2: Calculate IP̃ ′j= min{IP̃ ′i⊕ IP̃ ′j/ i ∈ G̃P̃ (j)}; j = 2, 3, . . . , n.Step – 3: A minimum value appearing from step 2 and matching the uniquevalue of i, the label node next j as IP̃i(x). If the minimum value appears to be121M. Asim Basha and M. Mohammed Jabarullaequal to one of the values of i, then it indicates that there is just one remaining pathbetween the SN as well as the node jbecause the separation between all pathwaysis IP̃j , choose any value for i.Step – 4: The DN is denoted by IP̃nγ, then the SD between the SN and theDN is IP̃n .Step – 5: The DN is now marked IP̃nγ. Check the marking of node γ to findthe SP between the SN and the DN. Let it be IP̃nη, now check the marking of nodeq, and so on. Repeat the process until node 1. The SP can be found by combiningall nodes.4 Numerical explanationConsider an I-VPF – scaled graph, in Fig. 1, where the I-VTPFNs representsthe distance between any two vertices. The problem is find to the SD and the pathenclosed by SN and the DN in network.1245673Figure 1: I-VTPF directed graph..The below (fig 1) network shall be formed in each edges to evaluate the I-VTPFN as follows:ExplanationLet node 7 is the DN is defined as, n = 7.Let us deal with the initial distance is d̃′′1 = [0, 0, 0] ; [0, 0] [1, 1] and the SN[declared nod 1] as[[0, 0, 0] ; [0, 0] [1, 1] , J(G̃)= −−], the values of d̃′′j′ ; j =2, 3, 4, 5, 6, 7 can be assigned the network terminology:Iteration – I Let taken away from fig.1 PN 2 is node 1, so place the valuesfrom i = 1 & j = 2 appropriately in the St – 2 being the above algorithm.122Algorithm approaches for SPP in Pythagorean fuzzy networkEdges I-VTPF distance1− 2 [0.3, 0.5, 0.5] ; [0.4, 0.6] [0.6, 0.8]1− 4 [0.4, 0.5, 0.6] ; [0.5, 0.6] [0.7, 0.8]2− 3 [0.3, 0.5, 0.7] ; [0.6, 0.7] [0.6, 0.7]2− 5 [0.4, 0.5, 0.7] ; [0.4, 0.6] [0.7, 0.8]3− 4 [0.3, 0.5, 0.6] ; [0.6, 0.7] [0.7, 0.7]3− 5 [0.5, 0.5, 0.5] ; [0.5, 0.6] [0.6, 0.8]4− 6 [0.4, 0.5, 0.6] ; [0.6, 0.7] [0.6, 0.8]5− 6 [0.5, 0.6, 0.5] ; [0.5, 0.6] [0.6, 0.8]5− 7 [0.4, 0.5, 0.7] ; [0.4, 0.7] [0.7, 0.7]6− 7 [0.5, 0.6, 0.7] ; [0.5, 0.7] [0.6, 0.7]Table 1: I-VTPFNs edge weightsFrom the path for d̃′′2 is, d̃′′2 = min{d̃′′1 ⊕ d̃′′12}= min {[0, 0, 0] ; [0, 0] [1, 1]⊕ [0.3, 0.5, 0.5] ; [0.4, 0.6] [0.6, 0.8]}d̃′2 = min {[0.3, 0.5, 0.5] ; [0.4, 0.6] [0.6, 0.8]}d̃′2 = {[0.3, 0.5, 0.5] ; [0.4, 0.6] [0.6, 0.8]}From the above calculation, there is a min that corresponds to node 1, and thennode 2 is considered to be{[0.3, 0.5, 0.5] ; [0.4, 0.6] [0.6, 0.8] , J(G̃)= 1}Iteration – II The PN for node 3 is node 2, then the distance between the SNand node 3 according to the above procedure isd̃′′3 = [0.424264, 0.707107, 0.860233] ; [0.68, 0.820731] [0.36, 0.56]and the node 3 is{[0.424264, 0.707107, 0.860233] ; [0.68, 0.820731] [0.36, 0.56] , J(G̃)= 2}Iteration – III The PN nodes for node 4 are nodes 1 and 3, then the value ofd̃′′4 is given,d̃′′4 = min {d̃′′1 ⊕ d̃′′14, d̃′′3 ⊕ d̃′′34}= min[0, 0, 0] ; [0, 0] [1, 1]⊕ [0.4, 0.5, 0.6] ; [0.5, 0.6] [0.7, 0.8] ,[0.424264, 0.707107, 0.860233] ; [0.68, 0.820731] [0.36, 0.56]⊕ [0.3, 0.5, 0.6] ; [0.6, 0.7] [0.7, 0.7]123M. Asim Basha and M. Mohammed Jabarulla= min{[0.5, 0.707107, 0.781025] ; [0.60828, 0.768375] [0.42, 0.64] ,[0.519645, 0.86602, 1.048809] ; [0.809899, 0.912968] [0.252, 0.396]}Applying the ranking method (based on the SF) results in the following valuefor d̃′′4.[0.5, 0.707107, 0.781025] ; [0.60828, 0.768375] [0.42, 0.64]and the term for d̃′′4 is{[0.5, 0.707107, 0.781025] ; [0.60828, 0.768375] [0.42, 0.64] , J(G̃)= 1}.Iteration – IV The PNs for node 5 are nodes 2 & 3, then the value of d̃′′5 isd̃′′5 = min {d̃′′2 ⊕ d̃′′25, d̃′′3 ⊕ d̃′′35}Using the rank order method, the value of d̃′′5 isd̃′′5 = [0.5, 0.707107, 0.860233] ; [0.542586, 0.768375] [0.42, 0.64]d̃′′5 and the name for d̃′′5 is,{[0.5, 0.707107, 0.860233] ; [0.542586, 0.768375] [0.42, 0.64] , J(G̃)= 2}.Iteration – V The PN nodes for node 6 are nodes 4 and 5, then the value ofd̃′′6 isd̃′′6 = min {d̃′′4 ⊕ d̃′′46, d̃′′5 ⊕ d̃′′56}Using the ranking approach, d̃′′6 value isd̃′′6 = [0.707107, 0.92279, 0.994988] ; [0.686148, 0.858986] [0.252, 0.512] andits label is{[0.707107, 0.92279, 0.994988]; [0.686148, 0.858986][0.252, 0.512], J(G̃) =5}.Iteration – VI The PN nodes for node 7 are nodes 5 and 6, then the value ofd̃′′7 isd̃′′6 = min {d̃′′5 ⊕ d̃′′57, d̃′′6 ⊕ d̃′′67}Using the ranking approach, d̃′′7 value isd̃′′7 = [0.640312, 0.86602, 1.109028] ; [0.809919, 0.931328] [0.252, 0.448] andits label is{[0.640312, 0.86602, 1.109028] ; [0.809919, 0.931328] [0.252, 0.448] , J(G̃)= 5}.Path 1→ 2→ 5→ 7 is determined to be the SP from the computations above,and the distance between the SN and DNs is124Algorithm approaches for SPP in Pythagorean fuzzy network[0.640312, 0.86602, 1.109028] ; [0.809919, 0.931328] [0.252, 0.448].The following table lists the SP taken by all nodes from the SN using thesuggested algorithm:Node IP̃j W2 [0.3, 0.5, 0.5] ; [0.4, 0.6] [0.6, 0.8] 1→ 23 [0.424, 0.707, 0.860] ; [0.68, 0.821] [0.36, 0.56] 1→ 2→ 34 [0.3, 0.5, 0.7] ; [0.6, 0.7] [0.6, 0.7] 1→ 45 [0.5, 0.707, 0.860] ; [0.543, 0.768] [0.42, 0.64] 1→ 2→ 56 [0.707, 0.923, 0.995] ; [0.686, 0.859] [0.252, 0.512] 1→ 5→ 67 [0.640, 0.866, 1.109] ; [0.809, 0.931] [0.252, 0.448] 1→ 2→ 5→ 7Table 2: SP and SDwhere W = SP between the SN and the jth node5 ConclusionsIn a SPP, the SD and I-VPFSP provide useful information for making deci-sions. The distance function, which aids in determining the SP, has been definedfor I-VTPFNs in this study. On a network with I-VTPF length, a new approachfor resolving the I-VPFSP issue has been suggested. The ranking of the paths is avery helpful decision-making tool for selecting the best alternative path. The pro-cess of determining the SP has been adequately explained. Furthermore, a used toexplain how the suggested algorithm can be implemented.ReferencesK. Atanassov. Intuitionistic fuzzy sets. Fuzzy sets Systems, 20:87–96, 1986.K. Atanassov and G. Gargov. Interval – valued intuitionistic fuzzy sets. Fuzzysets and Systems, 31:343–349, 1989.D. Dubois and H. . Prade. Fuzzy sets and systems. academic Press, New York,NY, USA, 1980.A. Gani and M. M. Jabarulla. On searching intuitionistic fuzzy shortest path in anetwork,. Applied Mathematical Sciences,, 4:3447–3454, 2010.H. Garg. A novel accuracy function under interval – valued pythagorean fuzzyenvironment for solving multicriteria decision making problems. J Intell FuzzySystems, pages 529–540, 2016.125M. Asim Basha and M. Mohammed JabarullaM. Karunambigai, P. Ranagasmy, K. Atanassov, and N. N. An intuitionistic fuzzygraph method for finding the shortest paths in networks. Advances in soft Com-puting, pages 3–10, 2007.A. Kumar and M. Kaur. A new algorithm for solving network flow with fuzzyarc lenths. An official Journal of Turkish Fuzzy systems Association, pages 1–3,2011.G. Kumar, R. Bajaj, and N. Gandotra. Interval – valued fuzzy sub semi – groupsand subgroups associated by interval – valued fuzzy graphs,. in Proceedingsof the WRI Global congress on Intelligent Systems (GCIS’09), pages 484–487,2009.G. Kumar, R. K. Bajaj, and N. Gandotra. Algorithm for shortest path problem ina network with interval – valued intuitionistic trapezoidal fuzzy number,. Inter-national Conference on Eco-friendly Computing and Communication Systems,,2015.S. Y. Mohamed and A. M. Ali. On strong interval – valued pythagorean fuzzygraph,. Journal of Applied Science and Computations., 5, 2018.S. Naz, S. Ashraf, and M. Akram. A novel approach to decision making withpythagorean fuzzy information,. Mathematics, 5:37–46, 2013.S. Okada. Fuzzy shortest path problems incorporating among paths. Fuzzy setsand Systems, 142:335–357, 2004.P. Xindong and Y. Young. Fundamental properties of interval – valuedpythagorean fuzzy aggregation operators. International Journal of IntelligentSystems, 31:444–487, 2016.Z. Xu and Zhang. Extension of topsis to multiple criteria decision making withpythagorean fuzzy sets. Int. J. Intell. Systems., 29:1061–1078, 2014.R. Yager. Pythagorean fuzzy subsets. In Proceedings of the joint IFSA Worldcongress and NAFIPS Annual Meeting, Edmonton, AB, Canada,, pages 24–28,2013.R. Yager. Pythagorean membership grades in multi-criteria decision making.IEEE Trans. Fuzzy Systems, 22:958–965, 2014.J. Yao and F. Lin. Fuzzy shortest – path network problems with uncertain edgeweights. Journal of Information Science and Engineering, 19:329–351, 2003.L. Zadeh. Fuzzy sets,. Information and Control,, 8:338–353, 1965.126

Algorithm Approaches for Shortest Path Problem in an Interval - Valued Triangular Pythagorean Fuzzy Network

http://eiris.it/ojs/index.php/ratiomathematica/article/download/117-126/pdf

Algorithm Approaches for Shortest Path Problem in an Interval - Valued Triangular Pythagorean Fuzzy Network

Abstract

Similar works

Full text

Available Versions

APAV - Academy of Sciences, Letters, Arts and Technology (E-Journals)