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
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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
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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 = 푓 a ()x 2 1 2 푎̃1 푎̃2, 푎̃1 푎̃2 and 푎̃1 푎̃2 we x c () d x i a c x d, have 푑(푎̃1, 푎̃1) = 0. Therefore 푑(푎̃1, 푎̃2) ≥ 0. dc ii. The proof of straightforward. 0 otherwise iii. The normalized Hamming distance between 푎̃1 and 푎̃3 and its falsity membership function is is defined as follows: b x () x a f 푑(푎̃ , 푎̃ ) a a x b, 1 3 ba |푎 (2 + 푡 − 푖 − 푓 ) − 푎 (2 + 푡 − 푖 − 푓 )| 1 푎̃1 푎̃1 푎̃1 3 푎̃3 푎̃3 푎̃3 f b x c a , 1 +|푏1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푏3(2 + 푡푎̃3 − 푖푎̃3 − 푓푎̃3)| a ()x = x c () d x f 12 +|푐1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푐3(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ )| a c x d, 1 1 1 3 3 3 dc (+|푑1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푑2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ )|) 1 1 1 3 3 3 0 otherwise 푎1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푎2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) where 0 ≤ 푇푎̃ (푥) ≤ 1, 0 ≤ 퐼푎̃(푥) ≤ 1, 0 ≤ 퐹푎̃(푥) ≤ 1 and | 1 1 1 2 2 2 | +푎 (2 + 푡 − 푖 − 푓 ) − 푎 (2 + 푡 − 푖 − 푓 ) 0 ≤ 푇푎̃(푥) + 퐼푎̃(푥) + 퐹푎̃(푥) ≤ 3; 푎, 푏, 푐, 푑 ∈ 푅. Then 푎̃ = 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3
([푎, 푏, 푐, 푑]; 푡 푖 푓 ) is called a neutrosophic trapezoidal 푏1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푏2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) 푎̃ , 푎̃ , 푎̃ , + | 1 1 1 2 2 2 | number. 1 +푏2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2) − 푏3(2 + 푡푎̃3 − 푖푎̃3 − 푓푎̃3) = 12 푐1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푐2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) Definition 5. [50,51] Let 푎̃ = ([푎 , 푏 , 푐 , 푑 ]; 푡 푖 푓 ) + | 1 1 1 2 2 2 | 1 1 1 1 1 푎̃1 , 푎̃1, 푎̃1, +푐 (2 + 푡 − 푖 − 푓 ) − 푐 (2 + 푡 − 푖 − 푓 ) [ ] 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3 and 푎̃2 = ( 푎2, 푏2, 푐2, 푑2 ; 푡푎̃2 ,푖푎̃2,푓푎̃2,) be two neutrosophic 푑1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푑2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) trapezoidal fuzzy numbers and 휆 ≥ 0, then + | 1 1 1 2 2 2 | +푑 (2 + 푡 − 푖 − 푓 ) − 푑 (2 + 푡 − 푖 − 푓 ) [ ] ( 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3 ) 1. 푎̃1 ⊕ 푎̃2 = ( 푎1 + 푎2, 푏1 + 푏2, 푐1 + 푐2, 푑1 + 푑2 ; 푡푎̃1 +
푡푎̃2 − 푡푎̃1푡푎̃2, 푖푎̃1 푖푎̃2, 푓푎̃1푓푎̃2); |푎1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푎2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)| [ ] 2. 푎̃1 ⊗ 푎̃2 = ( 푎1푎2, 푏1푏2, 푐1푐2, 푑1푑2 ; 푡푎̃1푡푎̃2, 푖푎̃1 + +|푎2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2) − 푎3(2 + 푡푎̃3 − 푖푎̃3 − 푓푎̃3)| 푖 − 푖 푖 , 푓 + 푓 − 푓 푓 ) 푎̃2 푎̃1 푎̃2 푎̃1 푎̃2 푎̃1 푎̃2 ; |푏 (2 + 푡 − 푖 − 푓 ) − 푏 (2 + 푡 − 푖 − 푓 )| 1 푎̃1 푎̃1 푎̃1 2 푎̃2 푎̃2 푎̃2 3. 휆 푎̃1 = ([휆 푎1 , 휆 푏1, 휆 푐1, 휆 푑1]; 1 − (1 − 1 +|푏2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2) − 푏3(2 + 푡푎̃3 − 푖푎̃3 − 푓푎̃3)| 휆 휆 휆 = 푡 ) , (푖 ) , (푓 ) ; 12 |푐 (2 + 푡 − 푖 − 푓 ) − 푐 (2 + 푡 − 푖 − 푓 )| 푎̃1 푎̃1 푎̃1 1 푎̃1 푎̃1 푎̃1 2 푎̃2 푎̃2 푎̃2 휆 휆 휆 휆 휆 휆 휆 +|+푐 (2 + 푡 − 푖 − 푓 ) − 푐 (2 + 푡 − 푖 − 푓 )| 4. (푎̃) = ([푎 , 푏 , 푐 , 푑 ]; (푡 ) , 1 − (1 − 푖 ) , 1 − 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3 1 1 1 1 푎̃1 푎̃1 휆 |푑1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푑2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)| (1 − 푓푎̃1) ) ( +|푑2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2) − 푑3(2 + 푡푎̃3 − 푖푎̃3 − 푓푎̃3)| ) [ ] Definition 6. Let 푎̃1 = ( 푎1, 푏1, 푐1, 푑1 ; 푡푎̃1 ,푖푎̃1,푓푎̃1,) and |푎1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푎2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)| 푎̃2 = ([푎2, 푏2, 푐2, 푑2]; 푡푎̃ ,푖푎̃ ,푓푎̃ ,) be two neutrosophic 2 2 2 1 +|푏1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푏2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ )| ≤ 1 1 1 2 2 2 trapezoidal fuzzy numbers, then the normalized Hamming 12 +|푐1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푐2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)| distance between 푎̃1 and 푎̃2 is defined as follows: (+|푑1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푑2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)|) 푑(푎̃1, 푎̃2) = |푎 (2 + 푡 − 푖 − 푓 ) − 푎 (2 + 푡 − 푖 − 푓 )| |푎1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푎2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)| 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3
1 +|푏1(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푏2(2 + 푡푎̃ − 푖푎̃ − 푓푎̃ )| 1 +|푏2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2) − 푏3(2 + 푡푎̃3 − 푖푎̃3 − 푓푎̃3)| 1 1 1 2 2 2 (4) + 12 12 +|푐 (2 + 푡 − 푖 − 푓 ) − 푐 (2 + 푡 − 푖 − 푓 )| +|푐1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푐2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)| 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3 +|푑 (2 + 푡 − 푖 − 푓 ) − 푑 (2 + 푡 − 푖 − 푓 )| (+|푑1(2 + 푡푎̃1 − 푖푎̃1 − 푓푎̃1) − 푑2(2 + 푡푎̃2 − 푖푎̃2 − 푓푎̃2)|) ( 2 푎̃2 푎̃2 푎̃2 3 푎̃3 푎̃3 푎̃3 )
Property 1 The normalized Hamming distance measure ≤ 푑(푎̃1, 푎̃2) + 푑(푎̃2, 푎̃3) . □ 푑(. ) of 푎̃1 and 푎̃2 satisfies the following properties: i. 푑(푎̃1, 푎̃2) ≥ 0, 2.1 TOPSIS Strategy for MADM ii. 푑(푎̃ , 푎̃ ) = 푑(푎̃ , 푎̃ ), 1 2 2 1 The idea behind the TOPSIS strategy [5] is to find out iii. 푑(푎̃ , 푎̃ ) ≤ 푑(푎̃ , 푎̃ ) + 푑(푎̃ , 푎̃ ), where 1 3 1 2 2 3 the optimal alternative that has the shortest distance from 푎̃ = ([푎 , 푏 , 푐 , 푑 ]; 푡 푖 푓 ) is a SVTrNN. 3 3 3 3 3 푎̃3 , 푎̃3, 푎̃3, the positive ideal solution and the farthest distance from the
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers 32 Neutrosophic Sets and Systems, Vol. 19, 2018
negative ideal solution, simultaneously. The schematic 푖푎̃푖푗 ≤ 1 , 0 ≤ 푓푎̃푖푗 ≤ 1 and 0 ≤ 푡푎̃푖푗 + 푖푎̃푖푗 + 푓푎̃푖푗 ≤ 3; structure of classical TOPSIS strategy is presented in the 푎, 푏, 푐, 푑 ∈ 푅. following figure (see Fig. 1) Here, 푡푎̃푖푗 denotes the truth membership degree, 푖푎̃푖푗 denotes
the indeterminate membership degree, and 푓푎̃푖푗 denotes the falsity membership degree to consider the trapezoidal Construct a decision matrix number [푎푖푗, 푏푖푗, 푐푖푗, 푑푖푗] as the rating values of 퐴푖 over the attribute 퐶푗. An MADM problem can be expressed by a decision matrix in which the entries represent the evaluation information of all alternatives with respect to the attributes. Normalize the decision matrix Then we construct the following neutrosophic decision matrix, whose elements are SVNTrNs:
푎̃11 푎̃12 … 푎̃1푛 푎̃21 푎̃22 … 푎̃2푛 Calculate the weighted normalized 퐷 = (푎̃푖푗) = ( ) (5) decision matrix 푚×푛 ⋮ ⋮ ⋱ ⋮ 푎̃푚1 푎̃푚2 … 푎̃푚푛 Due to different attribute weights, we assume that the weight vector of all attributes is given by 푤 = (푤1, 푇 Determine the positive and negative 푤2, … , 푤푛) , where 0 ≤ 푤푗 ≤ 1, 푗 = 1,2, … , 푛 , and 푤푗 is ideal solutions the weight of each attribute. The information about attribute weights is usually incomplete in decision making problems under uncertain environment. For convenience, we assume be a set of the known weight information [56-59], where Calculate the distance measure of each can be constructed by the following forms, for 푖 ≠ 푗: alternative from ideal solution Form 1. A weak ranking: {푤푖 ≥ 푤푗}; Form 2. A strict ranking: {푤푖 − 푤푗 ≥ 훼푗}(훼푗 > 0); Form 3. A ranking of difference: {푤 − 푤 ≥ 푤 − 푤 }, for Calculate relative closeness co-efficients 푖 푗 푘 푙 푗 ≠ 푘 ≠ 푙; of the alternatives Form 4. A ranking with multiples: {푤푖 ≥ 훼푗푤푗} (0 ≤ 훼푗 ≤ 1);
Form 5. An interval form: {훼푖 ≤ 푤푖 ≤ 훼푖 + 휖푖}(0 ≤ 훼푗 ≤ Rank the alternatives 훼푖 + 휖푖 ≤ 1). Now we develop a strategy for solving the MADM problems, in which the information about attribute weights Figure 1. A schematic structure of TOPSIS strategy is completely unknown or partially known and the attribute values are expressed by SVTrNNs. 3 TOPSIS strategy for multi-attribute decision mak- The following steps are considered to develop the model. ing with neutrosophic trapezoidal number In this section, we put forward a framework for determining 3.1 Standardize the decision matrix the attribute weights and the ranking orders for all the Let Da ij be a neutrosophic decision matrix, where alternatives with incomplete weight information under mn neutrosophic environment. a a1,,,;,, a 2 a 3 a 4 t i f the SVTrNNs ij ij ij ij ij aij a ij a ij is the rating Consider a MADM problem, where 퐴 = {퐴 , 퐴 , … , 퐴 } is 1 2 푚 values of alternative Ai with respect to attribute C j . Now to a set of 푚 alternatives and 퐶 = {퐶 , 퐶 , … , 퐶 } is a set of 푛 1 2 푛 eliminate the effect from different physical dimensions into attributes. The attribute value of alternative 퐴 (푖 = 푖 decision making process, we should standardize the 1,2, … , 푚) over the attribute 퐶푗(푗 = 1,2, … , 푛) assumes the decision matrix aij based on two common types of form of neutrosophic trapezoidal number 푎̃푖푗 = mn attributes such as benefit type attribute and cost type ([푎 , 푏 , 푐 , 푑 ]; 푡 푖 푓 ), where 0 ≤ 푡 ≤ 1, 0 ≤ 푖푗 푖푗 푖푗 푖푗 푎̃푖푗 , 푎̃푖푗, 푎̃푖푗, 푎̃푖푗 attribute. We consider the following technique to obtain the
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers Neutrosophic Sets and Systems, Vol. 19, 2018 33
Rr denotes the neutrosophic Hamming distance between two standardized decision matrix ij mn , in which the SVTrNNs 푎̃푖푗 and 푎̃푘푗 . r k r r1,,,;,, r 2 r 3 r 4 t i f component ij of the entry ij ij ij ij ij rij r ij r ij in The deviation value of all the alternatives to other R alternatives for the attribute 퐶푗 can be obtained as follows: the matrix are considered as: 푚 푚 푚 1. For benefit type attribute: 퐷푗(푤) = ∑ 푑푖푗(푤) = ∑ ∑ 푑(푎̃푖푗, 푎̃푘푗) 푤푗 a1 a 2 a 3 a 4 푖=1 푖=1 푘=1 ij ij ij ij 푚 푚 1 4 푝 rij ,,,;,,tr i r f r (6) = ∑ ∑ ( ∑ |푎 (2 + 푡 − 푖 − uuuu ij ij ij 푖=1 푘=1 12 푝=1 푖푗 푎̃푖푗 푎̃푖푗 jjjj 푝 푓̃ ) − 푎 (2 + 푡 ̃ − 푖 ̃ − 푓̃ )|) 푤 . (10) 2. For cost type attribute: 푎푖푗 푘푗 푎푘푗 푎푘푗 푎푘푗 푗 Similarly, the deviation value of all the alternatives to other alternatives for all the criteria can be taken as: uuuujjjj 푛 푛 푚 rij ,,,;,,tr i r f r , (7) a4 a 3 a 2 a 1 ij ij ij ( ) ( ) ( ) ij ij ij ij 퐷 푤 = ∑ 퐷푗 푤 = ∑ ∑ 푑푖푗 푤 4 푗=1 푗=1 푖=1 where uj max{ aij | i 1,2,... m }and 푛 푚 푚 푚 = ∑푗=1 ∑푖=1 ∑푖=1 ∑푘=1 푑(푎̃푖푗, 푎̃푘푗) 푤푗 1 umin{ a | i 1,2,... m } for jn 푝 (2 + 푡 − 푖 − 푓 ) j ij 1,2,... . 1 푎푖푗 푎̃푖푗 푎̃푖푗 푎̃푖푗 =∑푛 ∑푚 ∑푚 ( ∑4 | |) 푤 Then we obtain the following standardized decision matrix: 푗=1 푖=1 푘=1 12 푝=1 푝 푗 −푎푘푗 (2 + 푡푎̃푘푗 − 푖푎̃푘푗 − 푓푎̃푘푗 ) r11 r 12 r1n If the information about the attribute weights is partially r r r 21 22 2n known or completely unknown, then we propose two Rrij (8) mn models to obtain the attribute weights. rm12 r m rmn 3.2.1 Information about the weights of attributes is 3.2 Determine the attribute weight partially known. To determine the attribute weights, we use maximum deviation strategy, which was proposed by Wang [60]. In order to obtain the weight vector, we construct a non-lin- According to Wang [60], ear programming model that maximizes all deviation values i. The attribute that has the larger deviation value among of attributes. The model can be presented as follows: p n m m 4 a(2 t i f ) alternatives should be assigned larger weight. 1 ij aij a ij a ij max D w w ii. The attribute having deviation value among alternatives p j 12 j1 i 1 k 1 p 1 a(2 t i f ) should be assigned smaller weight. (M 2) kj akj a kj a kj iii. The attribute having no deviation among alternatives n subject tow , wjj 1, w 0, for j 1,2,.., n . should be assigned zero weight. j1 (11) Following the idea of maximum deviation method, we By solving the model (M-1), we obtain the optimal solution construct an optimization model to determine the optimal to be used as the weight vector. weights of attributes with SVTrNNs. The deviation of the alternative 퐴푖 to all the other alternatives for the attribute 3.2.2 Information about the weights of attributes is un- C j can be defined as follows: known. 푚 푑푖푗(푤) = ∑푘=1 푑( 푎̃푖푗 , 푎̃푘푗)푤푗 , 푖 = 1,2, … , 푚; 푗 = 1,2, … , 푛 If the information about attribute weight is completely where unknown, then we can establish the following programming 푑(푎̃푖푗, 푎̃푘푗) = model: p |푎푖푗1 (2 + 푡푎̃푖푗 − 푖푎̃푖푗 − 푓푎̃푖푗) − 푎푘푗1 (2 + 푡푎̃푘푗 − 푖푎̃푘푗 − 푓푎̃푘푗)| n m m 4 a(2 t i f ) 1 ij aij a ij a ij D w w max p j + |푎푖푗2 (2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) − 푎푘푗2 (2 + 푡푎̃ − 푖푎̃ − 푓푎̃ )| 1 푖푗 푖푗 푖푗 푘푗 푘푗 푘푗 12 j1 i 1 k 1 p 1 akj(2 t a i a f a ) M kj kj kj 12 (9) ( 2) + |푎푖푗3 (2 + 푡푎̃푖푗 − 푖푎̃푖푗 − 푓푎̃푖푗) − 푎푘푗3 (2 + 푡푎̃푘푗 − 푖푎̃푘푗 − 푓푎̃푘푗)| n subject tow , w2 1, w 0, for j 1,2,.., n . + |푎 (2 + 푡 − 푖 − 푓 ) − 푎 (2 + 푡 − 푖 − 푓 )| jj ( 푖푗4 푎̃푖푗 푎̃푖푗 푎̃푖푗 푘푗4 푎̃푘푗 푎̃푘푗 푎̃푘푗 ) j1
푎푖푗푝 (2 + 푡푎̃ − 푖푎̃ − 푓푎̃ ) (12) 1 4 푖푗 푖푗 푖푗 = ∑푝=1 | | To solve the model (M-2), we develop the Lagrange 12 −푎 (2 + 푡 − 푖 − 푓 ) 푘푗푝 푎̃푘푗 푎̃푘푗 푎̃푘푗 function:
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers 34 Neutrosophic Sets and Systems, Vol. 19, 2018
p w w w w n m m 4 a(2 t i f ) 12, ,...,n . 1 ij aij a ij a ij L w w , p j 12 j1 i 1 k 1 p 1 a(2 t i f ) 3.3 Determine the ideal solutions kj akj a kj a kj (13) n 2 In the normalized decision matrix Rr ij , the wj 1 mn 24 j1 neutrosophic trapezoidal local positive ideal solution where is a real number and denoting the Lagrange (NTrPIS) and the neutrosophic trapezoidal local negative multiplier variable. Then the partial derivative of L with ideal solution (NTrNIS) are defined as follows respect to wj ( j 1,2,..., n ) and are obtained as: r r12, r ,..., rn and r r12, r ,..., rn (21)
p mm4 a(2 t i f ) where, L ij aij a ij a ij ww 0 (14) 1 2 3 4 p jj r r,,,,,, r r r t i f wj i1 k 1 p 1 a(2 t i f ) j j j j j j j j kj akj a kj a kj n 1 2 3 4 L 2 maxrij ,max rij ,max rij ,max rij ; i i i i wj 10 (15) (22) j1 maxtij ,min iij ,min fij It follows from Eq. (14) that i ii p r r1 ,,,,,, r 2 r 3 r 4 t i f mm4 aij(2 t a i a f a ) j j j j j j j j ij ij ij p 1 2 3 4 i1 k 1 p 1 a(2 t i f ) kj akj a kj a kj minrij ,min rij ,min rij ,min rij ; i i i i wj for jn1,2,..., . (23) mintij ,max iij ,max fij (16) i ii Moreover, the trapezoidal neutrosophic global positive ideal Putting the values of w j in Eq.(15), we obtain 2 solution and the trapezoidal neutrosophic global trapezoidal p n m m 4 aij(2 t a i a f a ) global negative ideal solution can be directly considered as 2 ij ij ij (17) p j1 i 1 k 1 p 1 a(2 t i f ) rj 1,1,1,1 ,1,0,0 and rj 0,0,0,0 ,0,1,1 (24) kj akj a kj a kj
2 p 3.4 Determine the separation measures from ideal n m m 4 a(2 t i f ) ij aij a ij a ij solutions to each alternative for 0. p j1 i 1 k 1 p 1 a(2 t i f ) kj akj a kj a kj The separation measures di and di of each alternative (18) from the ideal solutions can be determined by Eq.(9), Then combining Eq.(16) and Eq.(18), we obtain the Eq.(20) and Eq.(21), respectively, as follows: following formula for determining the weight of attribute C( j 1,2,..., n ) n j : di wj d r ij, r j p mm4 a(2 t i f ) j1 ij aij a ij a ij p p n 4 r(2 t i f ) i1 k 1 p 1 akj(2 t a i a f a ) 1 ij rij r ij r ij kj kj kj w im wj . (19) j p for 1,2,..., (25) 2 r t i f p 12 jp11j (2 r r r ) n m m 4 a(2 t i f ) j j j ij aij a ij a ij n p j1 i 1 k 1 p 1 akj(2 t a i a f a ) d w d r r kj kj kj i j ij, j j1
We make their sum into a unit by normalizing p n 4 r(2 t i f ) 1 ij rij r ij r ij and get the optimal weight of attribute w j p for (26) 12 jp11rj (2 t i f ) : rj r j r j p mm4 a(2 t i f ) ij aij a ij a ij 3.5 Determine the relative closeness co-efficient p w i1 k 1 p 1 akj(2 t a i a f a ) w j kj kj kj j n p (20) The relative closeness co-efficient of an alternative Ai with w n m m 4 a(2 t i f ) j j ij aij a ij a ij 1 respect to ideal alternative A is defined as the following p j1 i 1 k 1 p 1 a(2 t i f ) kj akj a kj a kj formula: Then we get the normalized weight vector of attributes:
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers Neutrosophic Sets and Systems, Vol. 19, 2018 35
d i 2. Past experience()C2 , RC() Ai (27) ddii 3. General aptitude ()C 3 and where 0RC ( Ai ) 1 for im1,2,... . According to the 4. Self- confidence ()C 4 . closeness co-efficient RC() Ai , the ranking orders of all alternatives and the best alternative can be selected. The schematic diagram of the proposed TOPSIS is presented in The decision maker evaluates the ratings of alternatives Figure-2. Ai ( i 1,2,..., m ) with respect to the attributes Ci ( i 1,2,..., n )
with the decision matrix Da ()ij 44 (see Table 1). Table 1. Rating values of alternatives
C1 C2 Construct a decision [7,8,9,10]; 5,6,7,8 ; matix Problem A1 0.90,0.10,0.05 0.65,0.35,0.30 formulation 6,7,8,9 ; Standardize the A2 decision matrix 0.80,0.20,0.15 4,5,6,7 ; A3 0.50,0.50,0.45 The maximum Determine the attribute A4 deviation weights strategy C3 C4 7,8,9,10 ; A1 0.90,0.10,0.05 Determine the ideal solutions A2
Determine the separation A3 measures TOPSIS strategy A4 with SVTrNNs Calculate the relative closeness co-efficients The information of the attributes is incompletely known and the weight information is given as follows:
0.20ww12 0.30,0.05 0.20, Select the best 4 (28) 0.20w 0.35,0.15 w 0.35; w 1 alternative 34 j j1 Figure 2. The schematic diagram of the proposed startegy To determine the best alternative, we use the proposed strategy involving the following steps: 4 An illustrative example Step 1. Standardize the decision matrix In this section, we consider an illustrative example of med- Since the selective attributes are benefit type attributes, then ical representative selection problem to demonstrate and ap- using Eq. (6), we have the following standardized decision plicability of the proposed. matrix: Rr ()ij 44 (see Table 2.) Consider a MADM problem, where a pharmacy com- pany wants to recruit a medical representative. After initial Table 2. Standardized rating values of alternatives scrutiny four candidates Aii ( 1,2,3,4) have been consid- [0.7,0.8,0.9,1.0]; 0.5,0.6,0.7,0.8 ; ered for further evaluation with respect to the four attributes A1 0.90,0.10,0.05 0.65,0.35,0.30 C (j 1,2,3,4)namely, j 0.6,0.7,0.8,0.9 ; A2 1. Oral communication skill ()C1 ; 0.80,0.20,0.15
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers 36 Neutrosophic Sets and Systems, Vol. 19, 2018
0.4,0.5,0.6,0.7 ; 0.5,0.6,0.7,0.8 ; Step 4. Determine the separation measures from ideal A3 0.50,0.50,0.45 0.65,0.35,0.30 solutions to each alternative. Case 1. Employing Eq.(25), we obtain the separation A4 measures di of each alternative Aii ( 1,2,3,4) from A : C3 C4 d A1, A 0.0673 , d A2 , A 0.1538 , d A3, A 0.4792 , 0.6,0.7,0.8,0.9 ; 0.7,0.8,0.9,1.0 ; d A, A 0.3807. A1 4 0.80,0.20,0.15 0.90,0.10,0.05 Similarly, using Eq.(26), we obtain the separation measures d of each alternative from A : A2 i d A1, A 0.4119 , d A2 , A 0.3254 , d A3,0 A ,
A3 d A4 , A 0.0985.
Case 2. A4 The separation measures of each alternative from : d A1, A 0.0721 , d A2 , A 0.1494 , d A3, A 0.4615 ,
Step 2. Determine the attribute weight d A4 , A 0.3708. Case 1. Weight information is incompletely known. Similarly, the separation measures of each alternative Using the model (M-1), we construct the following single- from : objective programming problem: d A1, A 0.3894 , d A2 , A 0.3120 , , maxD w 3.2133 w1 1.1401 w2 3.4250 w3 2.9700 w4 4 (29) d A4 , A 0.0907. subject tow , wjj 1, w 0, for j 1,2,..,4. j1 Step 5. Calculate the relative closeness coefficient. Solving this model with optimization software LINGO 13, we get the optimal weight vector as Using Eq.(27), we calculate the relative closeness
w 0.30,0.05,0.35,0.30 . coefficient RC() Ai of alternative for Case 1 Case 2. Weight information is completely unknown. and Case 2, respectively. We put the result in Table 3. Following Eq.(20), we obtain the following optimal weight Table 3. Rating values of alternatives vector: RC(Ai) Case 1 Case 2 w 0.2990,0.1061,0.3186,0.2763 . RC(A1) 0.8596 0.8438 RC(A ) 0.6790 0.6824 Step 3. Determine the ideal solutions 2 RC(A3) 0 0 Since the chosen attributes are benefit type attribute, then RC(A4) 0.2056 0.1965 following Eq.(22) we determine the neutrosophic trapezoidal positive ideal solution as Following Table 3, we rank the alternatives [0.7,0.8,0.9,1.0];0.90,0.10,0.05 , according to the values of relative closeness coefficient for both cases: AAAA . Therefore A is the [0.6,0.7,0.8,0.9];0.80,0.20,0.15 , 1 2 4 3 1 A (30) [0.7,0.8,0.9,1.0];0.90,0.10,0.05 , best alternative. [0.7,0.8,0.9,1.0];0.90,0.10,0.05 Similarly, using Eq.(23), we determine the neutrosophic trapezoidal negative ideal solution 5 Conclusions 0.4,0.5,0.6,0.7 ;0.50,0.50,0.45 , 0.5,0.6,0.7,0.8 ;0.65,0.35,0.30 , TOPSIS strategy is a useful strategy for solving MADM A (31) problem under different environment. In this paper, we have 0.4,0.5,0.6,0.7 ;0.50,0.50,0.45 , investigated MADM problems with SVTrNNs. The weight 0.4,0.5,0.6,0.7 ;0.50,0.50,0.45
Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers Neutrosophic Sets and Systems, Vol. 19, 2018 37
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Pranab Biswas, Surapati Pramanik, and Bibhas C. Giri: TOPSIS strategy for MADM with trapezoidal neutrosophic numbers