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TOPSIS Strategy for Multi-Attribute Decision Making with Trapezoidal Neutrosophic Numbers

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TOPSIS Strategy for Multi-Attribute Decision Making with Trapezoidal Neutrosophic Numbers Neutrosophic Sets and Systems Volume 19 Article 6 1-1-2018 TOPSIS Strategy for Multi-Attribute Decision Making with Trapezoidal Neutrosophic Numbers Pranab Biswas Surapati Pramanik Bibhas C. Giri Follow this and additional works at: https://digitalrepository.unm.edu/nss_journal Recommended Citation Biswas, Pranab; Surapati Pramanik; and Bibhas C. Giri. "TOPSIS Strategy for Multi-Attribute Decision Making with Trapezoidal Neutrosophic Numbers." Neutrosophic Sets and Systems 19, 1 (2019). https://digitalrepository.unm.edu/nss_journal/vol19/iss1/6 This Article is brought to you for free and open access by UNM Digital Repository. It has been accepted for inclusion in Neutrosophic Sets and Systems by an authorized editor of UNM Digital Repository. For more information, please contact [email protected]. Neutrosophic Sets and Systems, Vol. 19, 2018 29 University of New Mexico TOPSIS Strategy for Multi-Attribute Decision Making with Trapezoidal Neutrosophic Numbers Pranab Biswas1, Surapati Pramanik2, and Bibhas C. Giri3 1Department of Mathematics, Jadavpur University, Kolkata-700032, India. E-mail: [email protected] 2* Department of Mathematics, Nandalal B.T. College, Panpur, Narayanpur-743126, India. E-mail: [email protected] 3 Department of Mathematics, Jadavpur University, Kolkata-700032, India. E-mail: [email protected] *Corresponding author: [email protected] Abstract. Technique for Order Preference by Similarity to the maximum deviation strategy, we develop an optimiza- Ideal Solution (TOPSIS) is a popular strategy for Multi- tion model to obtain the weight of the attributes. Then we Attribute Decision Making (MADM). In this paper, we develop an extended TOPSIS strategy to deal with extend the TOPSIS strategy of MADM problems in trape- MADM with single-valued trapezoidal neutrosophic num- zoidal neutrosophic number environment. The attribute bers. To illustrate and validate the proposed TOPSIS strat- values are expressed in terms of single-valued trapezoidal egy, we provide a numerical example of MADM problem. neutrosophic numbers. The weight information of attrib- ute is incompletely known or completely unknown. Using Keywords: Single-valued trapezoidal neutrosophic number, multi-attribute decision making, TOPSIS. 1 Introduction extended the TOPSIS strategy for MCDM problem in intu- itionistic fuzzy environment. Zhao [25] also studied TOP- Multi-attribute decision making (MADM) plays an im- SIS strategy for MADM under interval intuitionistic fuzzy portant role in decision making sciences. MADM is a pro- environment and utilized the strategy in teaching quality cess of finding the best alternative that has the highest de- evaluation. Xu [19] proposed TOPSIS strategy for hesitant gree of satisfaction over the predefined conflicting attributes. fuzzy multi-attribute decision making with incomplete The preference values of alternatives are generally assessed weight information. quantitatively and qualitatively according to the nature of attributes. When the preference values are imprecise, inde- However fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy terminate or incomplete, the decision maker feels comfort to sets have some limitations to express indeterminate and in- evaluate the alternatives in MADM in terms of fuzzy sets complete information in decision making process. Recently, [1], intuitionistic fuzzy sets [2], hesitant fuzzy sets [3], neu- single valued neutrosophic set (SVNS) [26] has been suc- trosophic sets [4], etc., rather than crisp sets. A large number cessfully applied in MADM or multi-attribute group deci- of strategies has been developed for MADM problems such sion [27-37]. SVNS [26] and interval neutrosophic set (INS) as technique for order preference by similarity to ideal solu- [38], and other hybrid neutrosophic sets have caught atten- tion (TOPSIS) [5], PROMETHEE [6], VIKOR [7], ELEC- tion of the researchers for developing TOPSIS strategy. TRE [7, 8], AHP [9], etc. MADM problem has been studied Biswas et al. [39] developed TOPSIS strategy for multi-at- extensively in fuzzy environment [10-14], intuitionistic tribute group decision making (MAGDM) for single valued fuzzy environment [15-22]. neutrosophic environment. Sahin et al. [40] proposed an- other TOPSIS strategy for supplier selection in neutrosophic TOPSIS [5] is one of the sophisticated strategy for solving environment. Chi and Liu developed TOPSIS strategy to MADM. The main idea of TOPSIS is that the best alterna- deal with interval neutrosophic sets in MADM problems. tive should have the shortest distance from the positive ideal Zhang and Wu [41] proposed TOPSIS strategies for MCDM solution (PIS) and the farthest distance from the negative in single valued neutrosophic environment and interval neu- ideal solution (NIS), simultaneously. Since its proposition, trosophic set environment where the information about cri- researchers have extended the TOPSIS strategy to deal with terion weights are incompletely known or completely un- different environment. Chen [23] extended the TOPSIS known. Ye [42] put forward TOPSIS strategy for MAGDM strategy for solving multi-criteria decision making with single-valued neutrosophic linguistic numbers. Peng et (MCDM) problems in fuzzy environment. Boran et al. [24] al. [43] presented multi-attributive border approximation area comparison (MBAC), TOPSIS, and similarity measure Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers 30 Neutrosophic Sets and Systems, Vol. 19; 2018 approaches for neutrosophic MADM. Pramanik et al. [44] x a) w a x b, extended TOPSIS strategy for MADM in neutrosophic soft ba expert set environment. Different TOPSIS strategies [45-49] w b x c, have been studied in different hybrid neutrosophic set envi- A ()x ()d x w ronment. c x d, Single valued trapezoidal neutrosophic number (SVTrNN ) dc [50, 51] is another extension of single-valued neutrosophic 0 otherwise sets. SVTrNN presents the situation, in which each element is characterized by trapezoidal number that has truth mem- where 푎, 푏, 푐, 푑 are real number satisfying 푎 ≤ 푏 ≤ 푐 ≤ 푑 bership degree, indeterminate membership degree, and fal- and 푤 is the membership degree. sity membership degree. Recently, Deli and Şubaş [52] pro- Definition 3.[4] Let 푋 be a universe of discourse. An posed a ranking strategy of single valued neutrosophic num- neutrosophic sets 퐴 over 푋 is defined by ber and utilized this strategy in MADM problems. Biswas et A{, x T (),(),()| x I x F x x X } al. [53] also proposed value and ambiguity based ranking AAA (2) strategy of single valued trapezoidal neutrosophic number whereTx(), Ix()and Fx() are real standard or non- and applied it to MADM. A A A However, TOPSIS strategy of MADM has not been studied standard subsets of ] 0,1 [ that is TA (x) : X ] 0,1 [ , earlier with trapezoidal neutrosophic numbers, although I A (x) : X ] 0,1 [ and FA (x) : X ] 0,1 [. The these numbers effectively deal with uncertain information in membership functions satisfy the following properties: MADM model. In this study, our objective is to develop an TIF MADM model, where the attribute values assume the form 0AAA (x) (x) (x) 3 . of SVTrNNs and the weight information of attribute is in- Definition 4. [26] Let 푋 be a universe of discourse. A completely known or completely unknown. The existing single-valued neutrosophic set 퐴̃ in 푋 is given by TOPSIS strategy of MADM cannot handle with such situa- tions. Therefore, we need to extend the TOPSIS strategy in (3) SVTrNN environment. where TA( x ): X [0,1] , IA( x ): X [0,1] and To develop the model, we consider the following sections: F( x ): X [0,1] with the condition Section 2 presents a preliminaries of fuzzy sets, neutro- A sophic sets, single-valued neutrosophic sets, and single-val- 0TAAA ( x ) I ( x ) F ( x ) 3 for all xX . ued trapezoidal neutrosophic number IFS, SVNS. Section 3 The functions TxA , IxA and FxA represent, contains the extended TOPSIS strategy for MADM with respectively, the truth membership function, the SVTrNNs. Section 4 presents an illustrative example. Fi- indeterminacy membership function and the falsity nally, Section 5 presents conclusion and future direction re- membership function of the element to the set A. search. Definition 5. [50, 51] Let a is a single valued trapezoidal 2 Preliminaries neutrosophic trapezoidal number (SVNTrN). Then its truth In this section, we review some basic definitions of fuzzy membership function is sets, neutrosophic sets, single-valued neutrosophic sets, and x a) t a a x b, single-valued trapezoidal neutrosophic number. ba t b x c Definition 1. X a , [1] Let be a universe of discourse, then a Txa () 퐴 ()d x ta fuzzy set is defined by c x d, A{ x , ( x ) | x X } (1) dc A which is characterized by a membership function 0 otherwise A :X [0,1], where A()x is the degree of membership of the element x to the set A . Its indeterminacy membership function is Definition 2. [54,55] A generalized trapezoidal fuzzy number 퐴 denoted by 퐴 = (푎, 푏, 푐, 푑; 푤) is described as a fuzzy subset of the real number ℝ with membership function 휇퐴 which is defined by Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers Neutrosophic Sets and Systems, Vol. 19, 2018 31 Proof: b x ( x a ia a x b, ( ) ba i. The distance measure 푑 푎̃1, 푎̃2 is obviously non-neg- ative. If 푎̃ ≈ 푎̃ that is for 푎 = 푎 , 푏 = 푏 , 푐 = i b x c, 1 2 1 2 1 2 1 a 푐 , 푑 = 푑 , 푡 = 푡 푖 = 푖 , 푓
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