Shortest path problem in fuzzy, intuitionistic fuzzy and neutrosophic environment: an overview

Complex & Intelligent Systems https://doi.org/10.1007/s40747-019-0098-z ORIGINAL ARTICLE Shortest path problem in fuzzy, intuitionistic fuzzy and neutrosophic environment: an overview Said Broumi1 · Mohamed Talea1 · Assia Bakali2 · Florentin Smarandache3 · Deivanayagampillai Nagarajan4 · Malayalan Lathamaheswari4 · Mani Parimala5 Received: 1 December 2018 / Accepted: 9 March 2019 © The Author(s) 2019 Abstract In the last decade, concealed by uncertain atmosphere, many algorithms have been studied deeply to workout the shortest path problem. In this paper, we compared the shortest path problem with various existing algorithms. Finally, we concluded the best algorithm for certain environment. Keywords Fuzzy sets · Intuitionistic fuzzy sets · Vague sets · Neutrosophic sets · Shortest path problem Introduction SPP is a cardinal issue among familiar connectional problems which occur in different areas of engineering and science, such as application in highway networks, portage and conquer in intelligence channels and problem of scheduling. The SPP focuses on recommending the path which has minimum length enclosed by two vertices. The length of the arc/ edge produces the quantities of the real life, namely cost, time, etc. In the case of conventional method of measuring SP, the length of each bend is assumed as a crisp numbers. If there is uncertainty on the parameters in the network, then the length can be represented by fuzzy number. In the current preceding, many of the SPPs with various types of input data have been examined in junction with fuzzy, intuitionistic, vague, interval fuzzy, interval-valued intuitionistic fuzzy and neutrosophic sets [2, 3, 8, 9, 11, 13, 14, 17–20, 23, 30, 39, 46–52, 83–92]. Up until now plenty of new algorithms have been designed. The paper is arranged as: section “Preliminaries” comprehends the primary definitions and overviewed SPP under different sets in sections, “SPP in vague environment”, “SPP in fuzzy environment”, “SPP in intuitionistic fuzzy environment” and “SPP in neutrosophic environment”, respectively. Lastly, conclusion has been presented for the objective of the paper. 1 Mohamed Talea Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Sidi Othman, B.P 7955, Casablanca, Morocco 2 Assia Bakali Ecole Royale Navale, Boulevard Sour Jdid, B.P 16303 Casablanca, Morocco 3 Florentin Smarandache Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA 4 Deivanayagampillai Nagarajan Department of Mathematics, Hindustan Institute of Technology and Science, Chennai 603 103, India 5 Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, Tamil Nadu 638401, India * Said Broumi Malayalan Lathamaheswari Mani Parimala 13 Vol.:(0123456789) Complex & Intelligent Systems Preliminaries Here, we principally recollected some of the concepts connected to neutrosophic sets (NSs), single-valued neutrosophic sets (SVNSs) related to the present work. See especially [10, 12] for further details and background. Definition 2.1. Let X be a nonempty set. A fuzzy set A drawn from X is defined as, { } A = x, 𝜇A (x)|x ∈ X , (1) where 𝜇A ∶ X → [0, 1], is called the membership function of A and defined over a universe of discourse X. Definition 2.2. A type-2 fuzzy set, denoted by A is characterized by a type-2 membership function 𝜇A (x, u), where x ∈ X,u ∈ J x ⊆ [0, 1], i.e., A= ) {( (x, u), 𝜇A (x, u) |x ∈ X, } ∀u ∈ Jx ⊆ [0, 1] . (2) Definition 2.3. An interval-valued fuzzy set is a special case of type-2 fuzzy sets [ ] by representing the membership function 𝜇A = 𝜇A , 𝜇A , where 𝜇A is a lower membership func- tion and 𝜇A is an upper membership function. The area between these lower and upper membership functions is called a footprint of uncertainty (FOU), which represents the level of uncertainty of the set. Definition 2.4. Let X be a nonempty set. An intuitionistic fuzzy set (IFS) A in X is an object having the form � � A = ⟨x, 𝜇A (x), 𝜈A (x)⟩�x ∈ X , (3) where the functions 𝜇A (x), 𝜈A (x) ∶ X → [0, 1] define the degree of membership and nonmembership, respectively, of the element x ∈ X to A, for the entire element x ∈ X 0 ≤ 𝜇A (x) + 𝜈A (x) ≤ 1.Also,𝜋A (x) = 1 − 𝜇A (x) − 𝜈A (x) is called the index of IFS, and is the degree of indeterminacy of x ∈ X to the IFS A, which expresses the lack of knowledge of whether x belongs to IFS or not. Also 𝜋A (x) ∈ [0, 1], i.e., 𝜋A (x) ∶ X → [0, 1] and 0 ≤ 𝜋A (x) ≤ 1, ∀x ∈ X. Definition 2.5. An interval-valued intuitionistic fuzzy set (IVIFS) A in X is defined as an object of the form � � A = ⟨x, PA (x), QA (x)⟩�x ∈ X , (4) where the functions PA (x) ∶ X → [0, 1], QA (x) ∶ X → [0, 1] denote the degree of membership and[ non-membership ] U L P P = P of A , respectively. Also, and (x) (x), (x) A A A [ L ] U U U QA (x) = QA (x), QA (x) ,0 ≤ PA (x) + QA (x) ≤ 1, ∀x ∈ X { } Definition 2.6. Let U be the universe, U = x1 , x2 , … , xn , with a generic element of U denoted by xi , i = 1, 2, … , n. 13 A vague set {⟨ ( )is defined ( )⟩ as an } object of the form A = xi , TA xi , FA xi |xi ∈ X in U is characterized by a truth membership function TA and a false membership function ( ) FA , i.e., TA ∶ U → [0, 1], FA ∶ U → [0, 1], where TA xi ( is)the lower bound on the grade of membership of xi , FA xi is the lower bound on the (negation ) (of )xi , derived from the evidence against xi and TA xi + FA xi ≤ 1. The grade of membership [ ( of) xi in the( vague )] set A is bounded T x , 1 − F x to the subinterval A i A i [ ( ) ( )] of the interval [0, 1]. F The vague value TA xi ,(1 − ) A xi indicates that the exact grade of membership But it is 𝜇 x A ( i )of xi may be ( ) ( unknown. ) bounded by TA xi ≤ 𝜇A xi ≤ 1 − FA xi . Definition 2.7. An interval-valued vague set A over a universe of discourse as {⟨ [ X is ]defined [ ]⟩ an object } of the A = xi , TAL , TAU , FAL , FAU |xi ∈ X , w h e r e form 0 ≤ TAL ≤ TAU ≤ 1 and 0 (≤ )TAU ≤ TAL ≤(1.)For each ( )intervalvalued vague set A, 𝜋A xi = 1 − TAL xi − FAL xi and are called degree of hesitancy of xi . Definition 2.8 Consider the space X consists of universal elements characterized by x.{(The NS A is a phenomenon ) } which has the structure A = TA (x), IA (x), FA (x) ∕x ∈ X , where the three grades of memberships are from X to ] −0, 1+ [ of the element x ∈ X to the set A, with the criterion: − 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3+ . (5) The functions, and are the truth, indeterminate and falsity grades which lie in real standard/non-standard subsets of ]−0, 1+[. Since there is a complication of applying NSs to realistic issues, Samarandache and Wang wt al. [11, 12] proposed the notion of SVNS, which is a specimen of NS and it is useful for realistic applications of all the fields. Definition 2.9. Let X be the space of objects which contains global elements. A SVNS is represented by degrees of membership grades mentioned in Definition 2.1. For all x in X, TA (...truncated)