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1
NC-TODIM Based MAGDM under Neutrosophic
2
Cubic Set Environment
3
Surapati Pramanik 1, Shyamal Dalapati 2,*, Shariful Alam 2 and Tapan Kumar Roy 2
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Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District –North 24 Parganas, Pin code-743126, West Bengal, India; sura_pati@yahoo.co.in 2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah-711103, West Bengal, India; salam50in@yahoo.co.in (S.A.); roy_t_k@yahoo.co.in (T.K.R.) * Correspondence: shyamal.rs2015@math.iiests.ac.in; Tel.: +91-9804234197 1
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Abstract: Neutrosophic cubic set is the hybridization of the concept of neutrosophic set and interval neutrosophic set. Neutrosophic cubic set has the capacity to express the hybrid information of both the interval neutrosophic set and the single valued neutrosophic set simultaneously. As newly defined, little research on the operations and applications of neutrosophic cubic sets appear in the current literature. In the present paper we propose the score, accuracy functions for neutrosophic cubic sets and prove their basic properties. We firstly develop TODIM method in neutrosophic cubic set environment, which we call NC-TODIM. We establish a new NC-TODIM method in neutrosophic cubic set environment for solving MAGDM in neutrosophic cubic set environment problems. We illustrate the proposed NC-TODIM method for solving a MAGDM problem to show applicability and effectiveness of the developed method. We also conduct sensitivity analysis to show the impact of ranking order of the alternatives for different values of attenuation factor of losses for multi-attribute group decision making problem. Keywords: neutrosophic cubic set; single valued neutrosophic set; interval neutrosophic set; multi attribute group decision making; TODIM method; NC-TODIM
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1. Introduction
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While modelling multi attribute decision making (MADM) and multi attribute group decision making (MAGDM), it is often observed that the parameters of the problem are not precisely known. The parameters often involve uncertainty. To deal uncertainty, Zadeh [1] left an important mark to represent and compute with imperfect information by introducing fuzzy set. Fuzzy set fostered a broad research community, and their impact has also been clearly felt at the application level in MADM [2-4] and MAGDM [5-9]. Atanassov [10] incorporated non membership function as independent component and defined intuitionistic fuzzy set (IFS) at first to express uncertainty in more meaningful way. IFSs have been applied in many MADM problems [11-13]. Smarandache [14] proposed the notion of neutrosophic set (NS) by introducing indeterminacy as independent component. Wang et al. [15] grounded the concept of single valued neutrosophic set (SVNS), an instance of neutrosophic set to deal with incomplete, inconsistent and indeterminate information in realistic way. Wang et al. [16] proposed the interval neutrosophic sets (INS) as a subclass of neutrosophic sets in which the values of truth, indeterminacy and falsity membership degrees are interval numbers. Applications of SVNSs and INSs are found in [17-20] and [21-23] for MADM and MAGDM respectively.
41
information. By fusing neutrosophic set and INS, Ali et al. [24] proposed neutrosophic cubic set and
42
defined external and internal neutrosophic cubic sets and established some of their properties. Jun
43
et al. [25] also defined neutrosophic cubic set by combining neutrosophic set and INS. Neutrosophic
44
cubic set is more capable to express the hybrid information of both the INS and the SVNS
45
simultaneously. However, there are only few studies in the literature to deal with MADM and
Neutrosophic sets and INS are both capable of handling uncertainty and incomplete
Š 2017 by the author(s). Distributed under a Creative Commons CC BY license.
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MAGDM in neutrosophic cubic set environment. Banerjee et al. [26] developed grey relational
47
analysis [27-28] based new MADM method in neutrosophic cubic set environment.
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Similarity measure is an important mathematical tool in decision-making problems.
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Pramanik et al. [29] at first defined similarity measure for neutrosophic cubic sets and proved its
50
basic properties. In the same study, Pramanik et al. [29] developed a new MAGDM method in
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neutrosophic cubic set environment. Lu and Ye [30] proposed cosine measures between
52
neutrosophic cubic sets and proved their basic properties. In the same study, Lu and Ye [30]
53
proposed a new cosine measures-based MADM method under a neutrosophic cubic environment.
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Due to little research on operations and application of neutrosophic cubic sets, Pramanik et al.
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[31] proposed several operational rules on neutrosophic cubic sets and defined Euclidean distance
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and arithmetic average operator in neutrosophic cubic sets environment. Pramanik et al. [31] also
57
employed information entropy scheme to calculate unknown weights of the attributes and
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developed a new extended TOPSIS method for MADM under neutrosophic cubic set environment.
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Zhan et al. [32] developed a new algorithm for multi-criteria decision making (MCDM) in
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neutrosophic cubic set environment based on weighted average operator and weighted geometric
61
operator. Ye [33] established the concept of a linguistic neutrosophic cubic number (LNCN). In the
62
same study, Ye [33] developed a new MADM method based on LNCN weighted arithmetic
63
averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA)
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operator under a linguistic neutrosophic cubic environment.
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In the literature there are only five methods [26-33] for MADM and MAGDM in neutrosophic
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cubic set environment. However, we say that none of them is generally superior to all others. So,
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new methods for MADM and MAGDM should be explored under neutrosophic cubic set
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environment.
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TODIM (an acronym in Portuguese for Interactive Multi-criteria Decision Making) is an important MADM method, since it considers decision makers’ bounded rationality. Firstly, Gomes and Lima [34] introduced TODIM method based on prospect theory [35]. Krohling and Souza [36] defined fuzzy TODIM method to solve MCDM problems. Several researchers applied fuzzy TODIM method in various fuzzy MADM or MAGDM problems [37-39]. Fan et al [40] introduced extended TODIM method to deal with the hybrid MADM problems. Krohling et al. [41] extended TODIM method from fuzzy environment to intuitionistic fuzzy environment by extending TODIM method to process the intuitionistic fuzzy information. Wang [42] introduced TODIM method to neutrosophic environment. Zhang et al. [43] proposed TODIM method for MAGDM problems under neutrosophic environment. Ji et al [44] proposed TODIM method under multi valued neutrosophic environment and applied it to personal selection. In 2017, Xu et al. [45] develop TODIM in single valued neutrosophic setting. In neutrosophic cubic set environment TODIM is yet to appear. To fill the gap, we initiate the study of TODIM in neutrosophic cubic set environment which we call as NC-TODIM.
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In this paper we develop a TODIM method (for short, NC-TODIM method) for MAGDM in neutrosophic cubic set environment. We solve an illustrative numerical example of MAGDM problem in neutrosophic cubic set environment to show the applicability and effectiveness of the proposed NC-TODIM method.
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Remainder of the paper is divided into five sections that are organized as follows: Section 2 presents some basic definition of neutrosophic sets, interval-valued neutrosophic sets, neutrosophic cubic sets. Section 3 is devoted to present the proposed NC-TODIM method. Section 4 presents an illustrative numerical example. Section 5 is devoted to analyse the ranking order with
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different values of attenuation factor of losses. Finally, Section 6 presents conclusion and future scope of research.
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2. Preliminaries
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In this section, we review some basic definitions which are important to develop the paper.
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Definition 1. [14] Neutrosophic set (NS)
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Let U be a space of points (objects) with a generic element in U denoted by u i.e. u ∈ U. A neutrosophic set R in U is characterized by truth-membership function t R , indeterminacymembership function iR and falsity-membership function f R , where t R , i R , f R are the functions from − − + + U to ] 0, 1 [ i.e. tR , iR , f R :U → ] 0, 1 [ that means t R (u), i R (u), f R (u) are the real standard − + or non-standard subset of ] 0, 1 [. Neutrosophic set can be expressed as R = {<u; ( tR (u), iR (u), f R − + (u))>: ∀ u ∈ U}. Since t R (u), i R (u), f R (u) are the subset of ] 0, 1 [ , there the sum ( tR (u) + iR − + − + (u) + f R (u)) lies between 0 and 3 , where 0 = 0 - ε and 3 = 3 + ε , ε >0. Example 1. Suppose that U = { u1, u 2 , u3 , ... } be the universal set. Let R 1 be any neutrosophic set in U. Then R 1 expressed as R 1 = {< u 1 ; (.6, .3, .4)>: u 1 ∈ U}.
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Definition 2. [16] Interval neutrosophic set (INS)
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Let G be a non-empty set. An interval neutrosophic set G in G is characterized by truth-membership function t G~ , the indeterminacy membership function iG~ and falsity ~ membership function f G~ . For each g ∈ G, t G~ (g), iG~ (g), f G~ (g) ⊆ [0, 1] and G defined as ~ − + − + + + G = {<g; [ t G~ (g), t G~ (g)], [ iG−~ (g), iG~ (g)], [ f G−~ (g), f G~ (g)]: ∀ g∈ G}. Here, t G~ (g), t G~ (g), iG−~ (g), + + − + − iG~ (g), f G~ (g), f G~ (g) : G → ] 0, 1 [ and − 0 ≤ sup t +~ (g) + supi +~ (g) + sup f +~ (g) ≤ 3+ , G G G In real problems it is difficult to express the truth-memberships function, − + indeterminacy-membership function and falsity-membership function in the form of t G~ (g), t G~ + + − + − − (g), iG~ (g), iG~ (g), f G~ (g), f G~ (g) : G → ] 0, 1 [ . − + + + Here, t G~ (g), t G~ (g), iG−~ (g), iG~ (g), f G−~ (g), f G~ (g): G → [0, 1].
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Example 2.
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~ be any interval neutrosophic set. Then Suppose that G = { g1 , g 2 , g3 , ... , gn} be a non-empty set. Let G 1 ~ expressed as ~ = {< g ; [.39, .47], [.17, .43], [.18, .36]: g ∈ G}. G1 G1 1 1
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Definition 3. [24] Neutrosophic cubic set (NCS)
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~ A neutrosophic cubic set in a non-empty set G is defined as © = {<g; G (g), R (g)>: ∀ g ∈ G}, where ~ G and R are the interval neutrosophic set and neutrosophic set in G respectively. Neutrosophic ~ cubic set can be presented as an order pair © = < G , R >, then we call it as neutrosophic cubic number (NC-number).
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Example 3.
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Suppose that G = { g1 , g 2 , g3 , ... , gn} be a non-empty set. Let © 1 be any NC-number. Then © 1 can be express as © 1 = {< g1 ; [.39, .47], [.17, .43], [.18, .36], (.6, .3, .4)>: g1 ∈ G}
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~
Some operations of NC-numbers: i. Union of any two NC-numbers ~ ~ Let ©1 = < G 1 , R 1 > and © 2 = < G 2 , R 2 > be any two NC-numbers in a non-empty set G. Then the union of © 1 and © 2 denoted by ©1 ∪ © 2 and defined as ~ ~ © ~1 ∪ © 2 ~= < G 1 (g ) ∪ G 1 (g ), R−~1 (g ) ∪−R 2 (g ) ∀g ∈ G+ >, where G1 (g) ∪ G1 (g) = {<g, [max { t G1 (g), t G~ 2 (g)},max { t G~1 (g), t G+~ 2 (g)}], [max { iG−~ 1 (g), iG−~ 2 (g)}, max { iG+~ 1 (g), + + − − iG+~ 2 (g)}], [min { f G~1 (g), f G~2 (g)}, min { f G~1 (g), f G~ 2 (g)}]>: g ∈ G} and R 1 (g ) ∪ R 2 (g ) = {<g, max { t R1 (g), t R 2 (g)}, max { iR1 (g), iR 2 (g)}, min { f R1 (g), f R 2 (g)}>: ∀ g∈ U}.
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Example 4. Let ©1 and © 2 be two NC-numbers in G presented as follows: © 1 = < [.39, .47], [.17, .43], [.18, .36], (.6, .3, .4)> and © 2 = < [.56, .70], [.27, .42], [.15, .26], (.7, .3, .6)>. Then ©1 ∪ © 2 = < [.56, .7], [.27, .43], [.15, .26], (.7, .3, .4)>.
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ii.
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Intersection of any two NC-numbers Intersection of two NC-numbers denoted and defined as follows: ~ ~ ~ ~ − − ©1 ∩ © 2 = < G 1 (g ) ∩ G 1 (g ), R 1 ( g ) ∩ R 2 (g ) ∀g ∈G >, where G1 (g) ∩ G1 (g) = {<g, [min { t G~1 (g), t G~ 2
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(g)},min { t G+~ 1 (g), t G+~ 2 (g)}], [min { iG−~ 1 (g), iG−~ 2 (g)}, min { iG+~ 1 (g), iG+~ 2 (g)}], [max { f G−~1 (g), f G−~ 2 (g)}, max
144
{ f G~1 (g),
+
+ f G~ 2 (g)}]>: g ∈ G} and R 1 (g) ∩ R 2 (g) = {<g, min { t R1 (g), t R 2 (g)}, min { iR1 (g), iR 2 (g)},
148 149
max { f R1 (g), f R 2 (g)}>: ∀ g∈ U}. Example 5. Let ©1 and © 2 be any two NC-numbers in G presented as follows: © 1 = < [.45, .57], [.27, .33], [.18, .46], (.7, .3, .5)> and © 2 = < [.67, .75], [.22, .44], [.17, .21], (.8, .4, .4)>. Then ©1 ∩ © 2 = < [.45, .57], [.22, .33], [.18, .46], (.7, .3, .4)>.
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iii. Compliment of a NC-number ~ ~ Let ©1 = < G 1 , R 1 > be any neutrosophic cubic set in G. Then compliment of ©1 = < G 1 , R 1 > ~c c c denoted by ©1 = {<g, G 1 (g), R 1 (g)>: ∀ g ∈ G}. ~ c Here, G 1 = {<g, [ t G−~ c (g), t G+~ c (g)], [ iG−~ c (g), iG+~ c (g)], [ f G−~ c (g), f G+~ c (g)]>: ∀ g∈ G}, where, t G−~ c (g) = 1 1 1 1 1 − 1 − − +1 + + f G~1 (g), t G+~ 1c (g) = f G~1 (g), iG−~1c (g) = {1} - iG~1 (g), iG+~1c (g) = {1} - iG~1 (g), f G−~1c (g) = tG~1 (g), f G+~1c (g) = f G~1 + c (g) and t R1c (g) = f R 1 (g) , iR?1 (g) = { 1 } - iR1 (g), f R1c (g) = t R 1 (g). Example 6. Assume that © 1 be any NC-number in G in the form: c ©1 = < [.45, .57], [.27, .33], [.18, .46], (.7, .3, .5)>. Then compliment of ©1 is obtained as ©1 = < [.18, .46], [.73, .67], [.45, .57], (.5, .7, .7) >.
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Definition 4. Score function
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Let ©1 be a NC-number in a non-empty set G. Then, a score function of ©1 , Sc (© 1) is defined as:
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1 2 + a1 + a 2 − 2b1 − 2b 2 − c1 − c 2 1 + a − 2b − c (2.1) Sc (© 1) = [( )+( )] 2 4 2 where, ©1 = < [a1 , a2], [b1, b2], [c1, c2], (a, b, c) > and Sc (© 1) ∈ [-1, 1]. Proposition 1. Score function of two NC-numbers lies between -1 to 1. Proof. Using the definition of interval neutrosophic set and neutrosophic set, we have all a1 , a2, b1, b2,c1, c2, a, b, and c ∈[0,1] . Since, 0 ≤ a1 ≤1 , 0 ≤ a 2 ≤1
171
0 ≤ a1 + a 2 ≤ 2 ,
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2 ≤ 2 + a1 + a 2 ≤ 4
(2.2)
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0 ≤ b1 ≤1 0 ≤ 2b1 ≤ 2 , and 0 ≤ b2 ≤1 0 ≤ 2b2 ≤ 2
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− 2 ≤ − 2b1 ≤ 0
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− 2 ≤ − 2b2 ≤ 0
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− 4 ≤ − 2b1 − 2b2 ≤ 0
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0 ≤ c1 ≤1 −1 ≤ − c1 ≤ 0
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0 ≤ c2 ≤1
(2.3)
−1 ≤ − c2 ≤ 0
179
− 2 ≤ − c1 − c2 ≤ 0
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Adding (2.2), (2.3) and (2.4), we obtain
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− 4 ≤ 2 + a1 + a 2 − 2b1 − 2b2 − c1 − c2 ≤ 4 ,
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−1 ≤
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Again,
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0 ≤ a ≤1 1≤1 + a ≤ 2 ,
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0 ≤ b ≤ 1 0 ≤ 2b ≤ 2 ,
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0 ≤ c ≤1 ,
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0 ≤ 2b + c ≤ 3 ,
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− 3 ≤ − 2b − c ≤ 0
(2.4)
2 + a1 + a 2 − 2b1 − 2b 2 − c1 − c2 4
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Adding (2.6) and (2.7), we obtain
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− 2 ≤ 1 + a − 2b − c ≤ 2 ,
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−1 ≤
1+ a − 2b − c ≤1 2
≤1
(2.5)
(2.6)
(2.7)
(2.8)
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Adding (2.5) and (2.8) and dividing by 2, we obtain
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1 2 + a1 + a 2 − 2b1 − 2b 2 − c1 − c 2 1 + a − 2b − c −1 ≤ [( )+( )] ≤1 2 4 2
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Sc (© 1) ∈ [-1, 1],
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Hence complete the proof.
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Example 7.
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Let ©1 and © 2 be two NC-numbers in G presented as follows:
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© 1 = < [.39, .47], [.17, .43], [.18, .36], (.6, .3, .4)> and © 2 = < [.56, .70], [.27, .42], [.15, .26], (.7, .3, .6)>.
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Then, by applying Definition 4, we obtain Sc ( © 1) = -.01 and Sc (© 2) = .07, In this case, we can say
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that ©2 > ©1 .
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Definition 5. Accuracy function
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Let ©1 be a NC-number in a non-empty set G, an accuracy function of ©1 is defined as:
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Ac (© 1) =
1 1 [ (a1 + a 2 − b 2 (1 − a 2 ) − b1 (1 − a1 ) − c 2 (1 − b1 ) − c1 (1 − b 2 )) + a − b(1 − a ) − c(1 − b)] 2 2
(2.9)
Here, Ac (© 1) ∈ [-1,
1]. When the value of Ac (© 1) increases, we say that the degree of accuracy of the NC-number ©1 increases. Proposition 2. Accuracy function of two NC-numbers lies between -1 to 1. Proof. The values of accuracy function depend upon 1 { (a1 + a 2 − b 2 (1 − a 2 ) − b1 (1 − a1 ) − c 2 (1 − b1 ) − c1 (1 − b 2 )) and { a − b(1 − a ) − c(1 − b)} The values of 2
1 { (a1 + a 2 − b 2 (1 − a 2 ) − b1 (1 − a1 ) − c2 (1 − b1 ) − c1 (1 − b 2 ))} and { a − b (1 − a ) − c(1 − b )} lies between -1 2 to 1 from [18]. Thus, − 1 ≤ Ac (© 1) ≤1 . Hence complete the proof. Example 8. Let ©1 and © 2 be two NC-numbers in G presented as follows: © 1 = < [.41, .52], [.10, .18], [.06, .17], (.48, .11, .11) > and
221
©2 =
222
and Ac (© 2) = .30. In this case, we can say that alternative © 2 is better than © 1 .
223 224
With respect to the score function Sc and the accuracy function Ac , a method for comparing NC-numbers can be defined as follows:
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Comparison procedure of two NC-numbers
226 227
Let ©1 and ©2 be any two NC-numbers. Then we define comparison method as follows: i. If Sc ( © 1) > Sc (© 2 ) , then ©1 > ©2. (2.10)
Then, by applying Definition 5, we obtain Ac (© 1) = .14
< [.40, .51], [.10, .20], [.10, .19], (.50, .11, .11) .
228
ii.
If Sc ( © 1) = Sc (© 2 ) and Ac (© 1) > Ac (© 2) , then ©1 > ©2.
(2.11)
229
iii.
If Sc ( © 1) = Sc (© 2 ) and Ac (© 1) = Ac (© 2) , then ©1 = ©2.
(2.12)
230
Example 9.
231
Let ©1 and © 2 be two NC-numbers in G presented as follows:
232
©1 =
233
©2 =
< [.25, .31], [.35, .44], [.35, .44], (.28, .28, .28) > .
234
Sc (© 2 )
= .13. Applying Definition 5, we obtain Ac (© 1) = -.20 and Ac (© 2) = -.18. In this case, we say
235
that alternative © 2 > © 1 . (Score values and Accuracy values taking correct up to two decimal
236
places)
237
Definition 6.
238
Let ©1 and ©2 be any two NC-numbers, then distance between them is defined by
239
∂ (©1, ©2) =
240
where, ©1 = < [a1 , a2], [b1, b2], [c1, c2], (a, b, c) > and ©2 = < [d1 , d2], [e1, e2], [f1, f2], (d, e, f) >.
241 242
Example 10. Let © 1 and © 2 be two NC-numbers in G presented as follows:
< [.23, .29], [.37, .46], [.34, .42], (.26, .26, .26) >
and Then, applying Definition 4, we obtain Sc ( © 1) = .13 and
1 [ a1 − d1 + a 2 − d 2 + b1 − e1 + b 2 − e2 + c1 − f1 + c2 − f 2 + a − d + b − e + c − f ] 9
(2.13)
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© 1 = < [.66, .75], [.25, .32], [.17, .34], (.53, .17, .22)> and © 2 = < [.35, .55], [.12, .25], [.12, .20], (.60, .23, .43)>. Then, applying Definition 6, we obtain ∂ (©1, ©2) = .12.
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Definition 7.
246
Let ©ij = {<[tij− , tij+ ],[iij− , iij+ ],[f ij− , f ij+ ],(t, i, f ) >} be any neutrosophic cubic value. ©ij used to
247
evaluate i-th alternative with respect to j-th criterion. The normalized form of ©ij is defined
248
as follows: −
+
tij
©ij⊗ ={< [
249
m
( ( t i =1
− 2 ij )
+ (t
1 + 2 2 ) ) ij
m
( ( t i =1
−
250
m
( (f i =1
251
− 2 ij )
+ (f
1 + 2 2 ) ) ij
t ij
[ m
2
2
( ( t ij) + (iij) + (f i =1
252 253 254
− 2 ij )
+ (t
1 + 2 2 ) ) ij
+
iij
], [
m
− 2 ij )
( (i i =1
1 + 2 2 ) ) ij
+ (i
iij
, m
− 2 ij )
( (i i =1
1 + 2 2 ) ) ij
],
+ (i
+
f ij
[
−
tij
,
1 2 2 ) ) ij
f ij
, m
( (f i =1
− 2 ij )
+ (f
1 + 2 2 ) ) ij
]
iij
, m
2
2
( ( t ij) + (iij) + (f i =1
1 2 2 ) ) ij
f ij
, m
2
2
( ( t ij) + (iij) + (f i =1
1 2 2 ) ) ij
] >}.
(2.14)
2.1. A conceptual model of evolution of neutrosophic cubic set is shown in Figure 1.
Classical set
255 256 257
Fuzzy set
258 259 260
Intuitionistic fuzzy set
261 262 263 264 265 266 267 268 269 270 271 272 273
Neutrosophic set Neutrosophic cubic set
Interval-valued neutrosophic set Figure.1. Evolution of neutrosophic cubic set
3. NC-TODIM method for solving MAGDM problem under neutrosophic cubic set environment
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Classical TODIM is not enough to deal neutrosophic MAGDM problems due to
275
presence of indeterminacy and complexity of decision environment. However, NC-numbers
276
can express the indeterminate information. In this study we extend the TODIM method to
277
NC-TODIM to solve the MAGDM problems under neutrosophic cubic set environment.
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3.1. Description about MAGDM problems
279
Assume that A = {A1, A2, …, Am} (m ≥ 2), C = {C1, C2, …, Cn} (n ≥ 2 ) be the discrete set of
280
alternatives and attributes respectively. W = {W1, W2, …, Wn} is the weight vector of attribute
281
Cj (j = 1, 2, …, n), where Wj > 0 and W j =1 . Let E = {E1, E2, …, Er} be the set of decision
282
makers and γ = {γ1 , γ 2 , ..., γ r} be the weight vector of decision makers, where γk > 0 and γ k =1
283
n
j =1
r
k =1
3.2. NC-TODIM method
284 285 286
Now, we describe the procedure of NC-TODIM method to solve the MAGDM problems with NC-numbers. The method consists of following steps: Step1. Formulate the decision matrix k
Assume that M =
287
(© ) k ij
m×n
~k k be the decision matrix, where ©ijk = < Gij , R ij > is the rating
288
value provided by the decision maker Ek for alternative Ai with respect to attribute Cj. The
289
matrix form of M is presented below
k
k . .. © © ©1n k k k © 21 © 22 © 2n . ... . k k k ©m1 © m2 ... ©mnj
290
A1 k M = A2 . A m
291
Step 2. Normalize the decision matrix
C1 C 2 ... .C n k 11
k 12
(3.1)
292
MAGDM problem generally consists of cost criteria and benefit criteria. So, the
293
decision matrix needs to be normalized. For cost criterion Cj we use the Equation (7) to
294
normalize the decision matrix (Equation (3.1)) provided by the decision makers. For benefit
295
criterion Cj we don’t need to normalize the decision matrix. When Cj is a cost criterion, the
296
normalized form of decision matrix (see Equation (3.1)) is presented below.
298
C1 C 2 ... .C n ⊗k ⊗k ⊗k A1 ©11 ©12 ... ©1n ⊗k M = A 2 ©⊗21k ©⊗22k ©⊗2nk . . ... . ⊗ k ⊗ k ⊗ k A ©m1 ©m2 ... ©mnj m ⊗k Here © ij is the normalized form of NC-number.
299
Step 3. Determine the relative weight of each criterion
300
Relative weight Wch of each criterion is obtained by the following equation.
301
W ch =
302
where, Wh = max {W1, W2, …, Wn}.
303
Step 4. Calculate score values
297
WC Wh
(3.3)
(3.2)
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304
Using Equation (2.1), calculate score value Sc (©ij⊗ k ) (i = 1, 2, …, m; j = 1, 2, …, n) of ©ij⊗ k if Cj is a
305
cost criterion. Using Equation (2.1), calculate score value Sc (? ij) (i = 1, 2, …, m; j = 1, 2, …, n) of © ijk
306
if Cj is a benefit criterion.
307
Step 5: Calculate accuracy values
308
Using Equation (2.9), calculate accuracy value Ac (©ij⊗ k ) (i= 1, 2, …, m; j= 1, 2, …, n) of ©ij⊗ k if Cj is a
309
cost criterion. Using Equation (2.9), calculate accuracy value Ac (©ijk ) (i= 1, 2, …, m; j= 1, 2, …, n) of
310
k
k © ij if Cj is a benefit criterion.
311
Step 6. Formulate the dominance matrix
312
Calculate the dominance of each alternative Ai over each alternative Aj with respect to the criteria C
313
(C1, C2, …, Cn), of the k-th decision maker Ek by the following Equation (3.4) and Equation (3.5).
314
(For cost criteria)
c =1 ⊗k ⊗k =0 , if ©ic = © jc n W 1 c =1 ch ⊗k ⊗k ⊗k ⊗k =− ∂ (©ic , © jc ) , if ©ic < © jc ( α WCh
WCh ∂ ( ⊗k , ⊗k ) , if ⊗k > ⊗k k ©ic © jc ©ic © jc Ψc (Ai , A j) = ( n Wch 315
316
(For benefit criteria)
Wch c =1 k k =0 , if ©ic = © jc n Wch 1 c =1 k k k k =− ∂(©ic , , © jc ) , if ©ic < © jc ( α WCh
k Ψc (Ai , A j) = (
317
(3.4)
WCh
n
∂ (©ick , ©kjc ) , if ©ick > ©kjc
(3.5)
318
Where, parameter ‘ α ’ represents the attenuation factor of losses and α must be positive.
319
Step 7. Formulate the individual total dominance matrix
320
Using Equation (3.6), calculate the individual total dominance matrix of each alternative Ai over each
321
alternative Aj.
322
k k λ = (A i , A j) = Ψ c (A i , A j)
323
Step 8. Aggregate the dominance matrix
324
Using Equation (3.7), calculate the collective overall dominance of alternative Ai over each
325
alternative Aj.
n
c =1
(3.6)
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m
326
λ (A i , A j ) = γ k λ k (A i , A j)
327
Step 9. Calculate global values
328
Using the Equation (3.8), we calculate global value of each alternative
(3.7)
k =1
n
n
λ (Ai , A j ) − min ( λ (Ai , A j ))
329
Ωi =
1≤ i ≤ m j =1
j =1
n
(3.8)
n
max ( λ (Ai , A j )) − min ( λ (Ai , A j ))
1≤ i ≤ m j = 1
1≤ i ≤ m j =1
330
Step 10. Rank the priority
331
Sorting the values of Ωi provides the rank of each alternative. A set of alternatives can be preference
332
ranked according to the descending order of Ωi . Highest global value corresponds to the best
333
alternative.
334
3.3. A conceptual model of the proposed approach is shown in Figure 2.
335 336
Evaluation steps Decision makers
Step 1: Formulate the decision matrix
337 338 339 340
For cost type criteria For benefit type criteria
341 342 343 344 345 346
Step 3: Determine the relative weight of each criterion Step 4: Calculate score values
Step 2: Normalize the decision matrix Step 5: Calculate accuracy values
347 348 349 350 351
Step 7: Formulate the individual total dominance matrix Step 8: Aggregate the dominance matrix
Step 6: Formulate the dominance matrix
Step 9: Calculate global values
352 353 354 355 356 357 358 359
Stop
Figure 2: A flow chart of the proposed method.
Step 10: Rank the priority
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360
4. Illustrative example
361
In this section, a MAGDM problem is adapted from the study [18] under neutrosophic cubic set
362
environment. An investment company wants to select a best alternative among the set of feasible
363
alternatives. The feasible alternatives are
364
1.
365
2.
Food company (A2)
366
3.
Computer company (A3)
367
4.
368
The best alternative is selected based on the following criteria:
369
1.
Risk analysis (C1)
370
2.
Growth analysis (C2)
371
3.
Environmental impact analysis (C3)
372
An investment company forms a panel of three decision makers {E1, E2, E3} who evaluate four
373
alternatives in decision making process. The weight vector of attributes and decision makers are
374
considered as W = (.4, .35, .25)T γ =(.32, .33, .35)T respectively.
375
The proposed method is presented using the following steps:
376
Step 1. Formulate the decision matrix
Car company (A1)
Arms company (A4)
377
Formulate the decision matrices Mk (k = 1, 2, 3) using the rating values of alternatives with respect
378
to three criteria provided by the three decision makers in terms of neutrosophic cubic numbers.
379
Assume that the NC-numbers
380
maker Ek for alternative Ai with respect to attribute Cj. Using these rating values ©ijk ( k = 1, 2, 3; i = 1,
381
k 2, 3, 4; ; j = 1, 2, 3) , three decision matrices M = (©ijk)4×3 ( k = 1, 2, 3) are constructed (see Equations
382
(4.1), (4.2) and (4.3)).
383 384
385 386
387
388
389
~k k ©ijk = < Gij , R ij > presents rating value provided by the decision
Decision matrix for E1 C1 C3 C2 A < [.41,.52],[.10,.18],[.06,.17],(.48,.11,.11)> < [.40,.51],[.10,.20],[.10,.19],(.50,.11,.11)> < [.22,.27],[.41,.52],[.41,.52],(.31,.31,.31)> 1 A2 < [.35,.46],[.18,.27],[.17,.34],(.43,.16,.21)> < [.22,.28],[.40,.50],[.39,.48],(.28,.28,.28) > < [.38,.49],[.10,.21],[.10,.21],(.57,.12,.12) > A3 < [.23,.29],[.36,.45],[.34,.42],(.26,.26,.26)> < [.34,.45],[.20,.30],[.19,.39],(.44,.16,.22)> < [.22,.27],[.41,.52],[.41,.52],(.31,.31,.31)> < [.17,.23],[.45,.55],[.42,.59],(.21,.32,.37)> < [.22,.28],[.40,.50],[.39,.48],(.28,.28,.28) > < [.38,.49],[.10,.21],[.10,.21],(.57,.12,.12)> A4
M1 =
(4.1)
Decision matrix for E2 M2 =
A1 A2 A3 A 4
C1 C2 C3 < [.17,.23],[.46,.55],[.42,.59],(.21,.32,.37)> < [.25,.31],[.35,.44],[.35,.44],(.28,.28,.28)> < [.34,.43],[.13,.27],[.13,.27],(.49,.11,.11)> < [.23,.29],[.37,.46],[.34,.42],(.26,.26,.26)> < [.25,.31],[.35,.44],[.35,.44],(.28,.28,.28)><[.34,.43],[.13,.27],[.13,.27],(.49,.11,.11)> < [.41,.52],[.10,.18],[.10,.17],(.48,.11,.11)> < [.44,.57],[.10,.17],[.10,.17],(.51,.11,.11)><[.19,.24],[.53,.67],[.53,.67],(.27,.27,.27)> < [.35,.46],[.20,.28],[.17,.34],(.42,.16,.21)> < [.25,.31],[.35,.44],[.35,.44],(.28,.28,.28)><[.34,.43],[.13,.27],[.13,.27],(.49,.11,.11)>
(4.2)
Decision matrix for E3
M3=
C1 C2 C3 A1 < [.22,.27],[.42,.52],[.42,.52],(.28,.28,.28) > < [.22,.28],[.40,.50],[.39,.48],(.28,.28,.28) >< [.41,.52],[.10,.18],[.10,.17],(.48,.11,.11)> A 2 < [.22,.27],[.42,.52],[.42,.52],(.28,.28,.28)> < [.40,.51],[.10,.20],[.10,.19],(.50,.11,.11)> < [.23,.29],[.36,.45],[.34,.42],(.26,.26,.26)> A3 < [.38,.49],[.10,.21],[.10,.21],(.50,.11,.11)> < [.34,.45],[.20,.30],[.19,.39],(.44,.16,.22) > < [.38,.49],[.10,.21],[.10,.21],(.50,.11,.11)> A 4 < [.38,.49],[.10,.21],[.10,.21],(.50,.11,.11)> < [.22,.28],[.40,.50],[.39,.48],(.28,.28,.28) >< [.17,.23],[.45,.54],[.42,.59],(.21,.32,.37)>
(4.3)
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390
Step 2. Normalize the decision matrix
391
Since all the criteria are benefit type, we do not need to normalize the decision matrix.
392
Step 3. Determine the relative weight of each criterion
393
Using Equation (3.3), we obtain the relative weight of criteria Wch as follows:
394
Wch = (1, .875, .625)T.
395
Step 4. Calculate score values
396
The score values of each alternative relative to each criterion obtained by Equation (2.1) are presented in the
397
Tables 1, 2 and 3.
398 399
Table 1. Score values for M1. C1
C2
C3
A1
.56
.54
.06
A2
.40
.09
.54
A3
.50
.38
.06
A4
-.03
.09
.54
400
Table 2. Score values for M2 C1
C2
C3
A1
-.03
.13
.49
A2
.13
.13
.49
A3
.56
.60
-.04
A4
.39
.13
.49
401
Table 3. Score values for M3 C1
C2
C3
A1
.07
.09
.56
A2
.07
.52
.13
A3
.51
.37
.39
A4
.51
.09
-.03
402
Step 5. Calculate accuracy values
403
The accuracy values of each alternative relative to each criterion obtained by Equation (2.9). are presented in
404
Tables 4, 5 and 6.
405 406
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407
Table 4. Accuracy values for M1. C1
C3
C2
A1
.14
.30
-.24
A2
.12
-.23
.32
A3
-.20
.09
-.24
A4
-.38
-.23
.32
408
Table 5. Accuracy values for M2 C1
C3
C2
A1
-.38
-.18
.21
A2
-.20
-.18
.21
A3
.14
.36
-.21
A4
.12
-.18
.21
409
Table 6. Accuracy values for M3 C1
C3
C2
A1
-.24
-.23
.41
A2
-.24
.30
-.20
A3
.26
.09
.12
A4
.26
-.23
-.38
410
Step 6. Formulate the dominance matrix
411
Using Equation (3.5), we construct dominance matrix for α = 1 The dominance matrixes are
412
represented in matrix form (See Equations (4.4), (4.5), (4.6), (4.7), (4.8), (4.9), (4.10), (4.11), and (4.12)).
413
414
The dominance matrix
1 Ψ1 =
The dominance matrix
1
Ψ3
A1 = A2 A3 A 4
A1
A2
0
-1
.25
0
0
−1
.25
0
A4 -1 .26 0 0 −1 .26 0
Ψ12
The dominance matrix
A1 A2 A3 A4 A1 0 .18 .30 .35 A2 − .46 0 −.58 .30 0 .19 A3 - .74 .23 A − .88 − .74 − .47 0 4
415
416
Ψ11
(4.4)
Ψ 13
A1 A1 0 = A2 −.82 A3 - .51 A − .81 4
1
Ψ2
A3
The dominance matrix
A3 0
(4.6)
A4 .29 .18 .28 0 −.69 0 .24 0 .29 0 −.65 0 A2
2
Ψ1
A1 = A2 A3 A 4
(4.5)
Ψ12
A2 A3 A4 - .46 − .88 − .74 .18 0 −.75 − .58 .35 .09 0 .04 .30 .23 .19 0 A1 0
(4.7)
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417
The dominance matrix Ψ 22 A1 2 = Ψ 2 A2 A3 A 4
418
419
A2
0
0 − .84
0
0 −.84
.29 .29 0
A3
0
0 − .84
A4 0 0 .29 0
(4.8)
Ψ 13
The dominance matrix A1 Ψ 13 = A2 A3 A 4
420
421
422
A1
A1 0 0 .31 .31
A3 A4 0 −.78 − .78 0 − .78 − .78 .31 0 0 .31 0 0
Ψ 33
=
A1 2 = Ψ 3 A2 A3 A 4
A2 A3 A4 − .94 −.59 − 1.1 .23 0 − .73 .15 - .59 .18 0 .23 − 1.1 − .58 − .94 0
(4.10)
A1
A2
A3
0
0
.26
0
0
.26
−1
−1
0
0
0
.26
Ψ32
=
Ψ 32
A4 0 0 −1 0
The dominance matrix
A2
The dominance matrix A1 A2 A3 A 4
The dominance matrix
(4.9)
Ψ 32
A1 A2 A3 A4 A1 0 − .83 − .65 0 0 .18 .29 A2 .29 0 .23 A3 .23 − .51 A 0 − .83 − .65 0 4
(4.11)
Ψ 33
A1 0
(4.12)
423
Step 7. Formulate the individual overall dominance matrix
424
The individual overall dominance matrix is calculated by the Equation (3.6) and The dominance
425
matrixes are represented in matrix form (see Equations (4.13), (4.14), and (4.15)).
426
First decision maker’s overall dominance matrix λ1
427
A1 A2 A3 A 4 A1 0 − .53 .47 − .37 1 0 − 1 .30 λ = A2 − 1 0 − .52 A3 - 1.3 − .53 A − 1.5 − .74 − .86 0 4
428
429
430
(4.13)
Second decision maker’s overall dominance matrix λ2 A1 A2 A3 A4 A1 0 − .46 − 1.5 − .74 2 0 − 1.3 − .58 λ = A2 .18 0 − .67 A3 - .36 − .62 A .30 .23 − .39 0 4
Third decision maker’s overall dominance matrix A4 − 1.9 − .34 .46 0
(4.14)
λ3
431
A1 A2 A3 A1 0 − 1.8 − 2 λ3 = A2 .52 0 − 1.3 0 A3 - .05 − .02 A − .79 . − 1.1 − 1.6 4
432
Step 8. Aggregate the dominance matrix
433
Using Equation (3.7), the aggregate dominance matrix is constructed (see Equation 4.16).
434
Aggregate the dominance matrix λ
(4.15)
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A1 A2 A3 A4 A1 0 − .94 − 1.1 − .53 0 − 1.23 − .22 A2 −.10 0 − .23 A3 - .54 − .38 A − .64 . − .55 − .96 0 4
435
λ =
436
Step 9. Calculate global values
437
Using Equation (3.8 ) we calculate the values of Ω i (i = 1, 2, 3, 4) and represented in Table 7.
438
(4.16)
Table 7. Global values of alternatives Ai
A1 .49
Ωi
A2 .61
A3 1
A4 0
439
Step 10. Rank the priority
440 441 442 443
Since Ω3 > Ω2 > Ω1 > Ω4 , alternatives are then preference ranked as follows: A3> A2> A1 >A4. Hence A3 is the best alternative. From the illustrative example, we see that the proposed NC-TODIM method is more suitable for real
444
scientific and engineering applications because it can handle hybrid information consisting of INS
445
and SVNS information simultaneously to cope indeterminate and inconsistent information. Thus,
446
NC-TODIM extends the existing decision-making methods and provides a sophisticated
447
mathematical tool for decision makers.
448
5. Rank of alternatives with different values of α
449 450 451
Table 8 shows that the ranking order of alternatives depends on values of attenuation factor, which reflects the importance of attenuation factor in NC-TODIM method.
452
Table 8. Global values and ranking of alternatives for different values of α
Values of α 0.5
Global values of alternative ( Ω i )
Rank order of Ai
Ω1 = 0, Ω2 =.89, Ω3 = 1, Ω4 = .46
A3> A2> A4 >A1
Ω3 > Ω2 > Ω4 > Ω1 1
Ω1 = .49, Ω2 =.61, Ω3 = 1, Ω4 = 0
A3> A2> A1 >A4
Ω3 > Ω2 > Ω1 > Ω4 1.5
Ω1 = 0, Ω2 =.72, Ω3 = 1, Ω4 = .44
A3> A2> A4 >A1
Ω3 > Ω2 > Ω4 > Ω1 2
Ω1 = 0, Ω2 =1, Ω3 = .81, Ω4 = .38 Ω2 > Ω3 > Ω4 > Ω1
A2> A3> A4 >A1
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3
Ω1 = 0, Ω2 =.56, Ω3 = 1, Ω4 = .45
A3> A2> A4 >A1
Ω3 > Ω2 > Ω4 > Ω1
453
5.1. Analysis on influence of the parameter α to ranking order
454 455 456 457 458 459 460 461 462 463
The impact of parameter α on ranking order is examined by comparing the ranking orders taken with varying the different values of α . When α = .5, 1, 1.5, 2, 3, ranking order are presented in Table 8. We draw Figure 3 and Figure 4 to compare the ranking order for different values of α . When α =.5, α = 1.5 and α =3 the ranking order is unchanged and A3 is the best alternative, A1 is the worst alternative. When α = 1, the ranking order is changed and A3 is the best alternative and A4 is the worst alternative. For α = 2, the ranking order is changed and A2 is the best alternative and A1 is the worst alternative. From Table 8 we see that A3 is the best alternative in four cases and A1 is the worst. We can say that ranking order depends on parameter α and A3 is the best alternative and A1 is the worst alternative.
464 465 466 467
Figure.3. Global values of the alternatives for different values of attenuation factor α = .5, 1, 1.5, 2, 3.
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468 469
Figure.4. Ranking of the alternatives for α = .5, 1, 1.5, 2, 3.
470
6. Conclusion
471
In many real world decision-making problems, decision makers encounter uncertain decision
472
parameters that are incomplete, indeterminate and inconsistent in nature. As a result, the decision
473
makers cannot easily reflect their judgments on the alternatives with exact and crisp values. To
474
tackle the situation, we propose the NC-TODIM for MAGDM problems under neutrosophic cubic
475
information, where the preference values of alternatives over the attributes and the importance of
476
attributes are expressed in terms of neutrosophic cubic numbers. In this study, we propose score
477
function, accuracy functions and established some of their properties. We develop NC-TODIM
478
method, which is capable to tackle MAGDM problems affected by uncertainty and indeterminacy
479
represented by neutrosophic cubic numbers. The standard TODIM, in its original formulation, is
480
only applicable to a crisp environment. Existing neutrosophic TODIM methods deal with single
481
valued neutrosophic information only. Therefore, NC-TODIM provides more flexibility to deal with
482
real world problems. We solve a numerical example to show the applicability and effectiveness of
483
the proposed NC-TODIM. We investigate the influence of attenuation factor of losses α on ranking
484
order of alternatives. The proposed NC-TODIM method can be applied to other MAGDM problems
485
characterized by neutrosophic hybrid environments.
486 487
Acknowledgments: The authors would like to acknowledge the constructive comments and suggestions of the anonymous referees.
488 489 490
Author Contributions: “Surapati Pramanik conceived and designed the problem; Shyamal Dalapati
491 492
Conflicts of Interest: The authors declare that there is no conflict of interest for publication of the
493
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