Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment

NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment information Article NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment 1 , 2 2 2 ID ID Surapati Pramanik * , Shyamal Dalapati , Shariful Alam and Tapan Kumar Roy Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District–North 24 Parganas, West Bengal 743126, India Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah, West Bengal 711103, India; [email protected] (S.D.); [email protected] (S.A.); [email protected] (T.K.R.) * Correspondence: [email protected]; Tel.: +91-947-703-5544 Received: 19 October 2017; Accepted: 14 November 2017; Published: 18 November 2017 Abstract: A neutrosophic cubic set is the hybridization of the concept of a neutrosophic set and an interval neutrosophic set. A 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 has been reported in the current literature. In the present paper, we propose the score and accuracy functions for neutrosophic cubic sets and prove their basic properties. We also develop a strategy for ranking of neutrosophic cubic numbers based on the score and accuracy functions. We firstly develop a TODIM (Tomada de decisao interativa e multicritévio) in the neutrosophic cubic set (NC) environment, which we call the NC-TODIM. We establish a new NC-TODIM strategy for solving multi attribute group decision making (MAGDM) in neutrosophic cubic set environment. We illustrate the proposed NC-TODIM strategy for solving a multi attribute group decision making problem to show the applicability and effectiveness of the developed strategy. We also conduct sensitivity analysis to show the impact of ranking order of the alternatives for different values of the attenuation factor of losses for multi-attribute group decision making strategies. Keywords: neutrosophic cubic set; single valued neutrosophic set; interval neutrosophic set; multi attribute group decision making; TODIM strategy; NC-TODIM 1. Introduction 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 with uncertainty, Zadeh [1] left an important mark to represent and compute with imperfect information by introducing the fuzzy set. The fuzzy set fostered a broad research community, and its impact has also been clearly felt at the application level in MADM [2–4] and MAGDM [5–9]. Atanassov [10] incorporated the non-membership function as an independent component and defined the intuitionistic fuzzy set (IFS) at first to express uncertainty in a more meaningful way. IFSs have been applied in many MADM problems [11–13]. Smarandache [14] proposed the notion of the neutrosophic set (NS) by introducing indeterminacy as an independent component. Wang et al. [15] grounded the concept of the single valued neutrosophic set (SVNS), an instance of the neutrosophic set, to deal with incomplete, inconsistent, and indeterminate information in a realistic way. Wang et al. [16] proposed the interval neutrosophic set (INS) as a subclass of neutrosophic sets in which the values of truth, indeterminacy, and falsity membership degrees are interval numbers. Theoretical development Information 2017, 8, 149; doi:10.3390/info8040149 www.mdpi.com/journal/information Information 2017, 8, 149 2 of 21 and applications of SVNSs and INSs are found in [17–37] for MADM or MAGDM. Some studies on MADM in single valued neutrosophic hesitant fuzzy set environments are found in [38–41]. NS and INS are both capable of handling uncertainty and incomplete information. By fusing NS and INS, Ali et al. [42] proposed the neutrosophic cubic set (NCS) and defined external and internal neutrosophic cubic sets, and established some of their properties. In the same study, Ali et al. [42] proposed an adjustable strategy to NCS-based decision making. Jun et al. [43] also defined NCS by combining NS and INS. In decision making process, the advantage of NCSs is that the decision makers can employ the hybrid information comprising of INSs and SVNSs for evaluating and rating of the alternatives with respect to their predefined attributes. However, there are only a few studies in the literature to deal with MADM and MAGDM in the NCS environment. Banerjee et al. [44] established grey relational analysis (GRA) [45–47] based on the new MADM strategy in the NCS environment. In the same study, Banerjee et al. [44] proposed the Hamming distances for weighted grey relational coefficients and ideal grey relational coefficients, and offered the concept of relative closeness coefficients for presenting the ranking order of the alternatives based on the descending order of their relative closeness coefficients. Similarity measure is an important mathematical tool in decision-making problems. Pramanik et al. [48] at first defined similarity measure for NCSs and proved its basic properties. In the same study, Pramanik et al. [48] developed a new MAGDM strategy in the NCS environment. Lu and Ye [49] proposed cosine measures between NCSs and established their basic properties. In the same study, Lu and Ye [49] proposed three new cosine measures-based MADM strategies under a NCS environment. Due to little research on the operations and application of NCSs, Pramanik et al. [50] proposed several operational rules on NCSs, and defined Euclidean distances and arithmetic average operators of NCSs. In the same study, Pramanik et al. [50] also employed the information entropy scheme to calculate the unknown weights of the attributes, and developed a new extended TOPSIS strategy for MADM under the NCS environment. Zhan et al. [51] proposed a new algorithm for multi-criteria decision making (MCDM) in an NCS environment based on a weighted average operator and a weighted geometric operator. Ye [52] established the concept of a linguistic neutrosophic cubic number (LNCN). In the same study, Ye [52] developed a new MADM strategy based on a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator under a linguistic NCS environment. In the literature, there are only six strategies [44,48–52] for MADM and MAGDM in NCS environment. However, we say that none of them is generally superior to all others. So, new strategies for MADM and MAGDM should be explored under the NCS environment for the development of neutrosophic studies. TODIM (an acronym in Portuguese for interactive multi-criteria decision making strategy named Tomada de decisao interativa e multicritévio) is an important MADM strategy, since it considers the decision makers’ bounded rationality. Firstly, Gomes and Lima [53] introduced the TODIM strategy based on prospect theory [54]. Krohling and Souza [55] defined the fuzzy TODIM strategy to solve MCDM problems. Several researchers applied the TODIM strategy in various fuzzy MADM or MAGDM problems [56–58]. Fan et al. [59] introduced the extended TODIM strategy to deal with the hybrid MADM problems. Krohling et al. [60] extended the TODIM strategy from fuzzy environment to intuitionistic fuzzy environment to process the intuitionistic fuzzy information. Wang [61] introduced TODIM strategy into multi-valued neutrosophic set environment. Zhang et al. [62] proposed the TODIM strategy for MAGDM problems under a neutrosophic number environment. Ji et al. [63] proposed the TODIM strategy under a multi-valued neutrosophic environment and employed it to solve personal selection problems. In 2017, Xu et al. [64] developed the TODIM strategy in a single valued neutrosophic setting and extended it into interval neutrosophic setting. Neutrosophic TODIM [64] is capable of dealing with only single-valued neutrosophic information or interval neutrosophic information. Information 2017, 8, 149 3 of 21 NCS can be used to express the interval neutrosophic information and neutrosophic information in the process of MADGM. It seems that TODIM in NCSs has an enormous chance of success to deal with group decision making problems. In the NCS environment, the TODIM strategy is yet to appear. Motivated by these, we initiated the study of TODIM in the NCS environment, which we call NC-TODIM. However, NCSs comprise of hybrid information of INSs and SVNSs simultaneously, which are more flexible and elegant for expressing neutrosophic cubic information. To apply NCSs to MADGM problems, we introduce some basic operations of neutrosophic cubic (NC) numbers and the score, and accuracy functions of NC numbers, and the ranking strategy of NC numbers. In this paper we develop a TODIM strategy (for short, NC-TODIM strategy) for MAGDM in the NCS environment. The proposed NC-TODIM strategy was proven to be capable of successfully dealing with MAGDM problems by solving an illustrative example. What is more, a comparative analysis ensured the feasibility of the proposed NC-TODIM strategy. The remainder of the paper is divided into seven sections that are organized as follows: Section 2 presents some basic definitions of NS, SVNS, INS, and NCS. Section 3 presents comparison strategy of two NC-numbers. Section 4 is devoted to present the proposed NC-TODIM strategy. Section 5 presents an illustrative numerical example of MAGDM in the NCS environment. Section 6 is devoted to analyzing the ranking order with different values of attenuation factors of losses. Section 7 presents a comparative analysis between the developed strategy and other existing strategies in the NCS environment. Section 8 presents the conclusion and the future scope of research. 2. Preliminaries In this section, we review some basic definitions which are important to develop the paper. Definition 1. [14] NS. Let U be a space of points (objects) with a generic element in U denoted by u, i.e., u 2 U. A neutrosophic set R in U is characterized by truth-membership function t , indeterminacy-membership function i , and falsity-membership function f , where t , i , f are the functions from U to ] 0, 1 [ i.e., t , R R R R R R i , f : U!] 0, 1 [ that means t (u), i (u), f (u) are the real standard or non-standard subset of ] 0, R R R R R 1 [. The neutrosophic set can be expressed as R = {<u; (t (u), i (u), f (u))>: 8u2U}. Since t (u), i (u), R R R R R + + f (u) are the subset of ] 0, 1 [, there the sum (t (u) + i (u) + f (u)) lies between 0 and 3 , where 0 = 0 " R R R R and 3 = 3 + ", " > 0. Example 1. Suppose that U = {u , u , u , . . .} is the universal set. Let R be any neutrosophic set in U. 1 2 3 1 Then R expressed as R = {<u ; (0.6, 0.3, 0.4)>: u 2U}. 1 1 1 1 Definition 2. [15] SVNS. Let U be a space of points (objects) with a generic element in U denoted by u. A single valued neutrosophic set H in U is expressed by H = {<u, (t (u), i (u), f (u))>; u2U}, where t (u), i (u), H H H H H f (u)2[0, 1]. Therefore for each u2U, t (u), i (u), f (u)2[0, 1] and 0  t (u) + i (u) + f (u)  3. H H H H H H H Definition 3. [16] INS. Let G be a non-empty set. An interval neutrosophic set G in G is characterized by truth-membership function t , the indeterminacy membership function i and falsity membership function f . e e e G G G For each g2G, t (g), i (g), f (g)  [0, 1] and G defined as e e e G G G + + + G = f< g; [t (g), t (g)], [i (g), i (g), ], [f (g), f (g)] : 8g 2 Gg. e e e e e e G G G G G G Here, + + + + t (g), t (g), i (g), i (g), f (g), f (g) : G !] 0, 1 [, e e e e e e G G G G G G and + + + + 0  sup t (g) + sup i (g) + sup f (g)  3 . e e G G G Information 2017, 8, 149 4 of 21 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), t (g), i (g), i (g), f (g), f (g) : G !] 0, 1 [. e e e e e e G G G G G G Here, + + + t (g), t (g), i (g), i (g), f (g), f (g) : G ! [0, 1]. e e e e e e G G G G G G e e Example 2. Suppose that G = {g , g , g , . . ., g } is a non-empty set. Let G be an INS. Then G is n 1 1 1 2 3 expressed as G = f< g ; [0.39, 0.47], [0.17, 0.43], [0.18, 0.36] : g 2 Gg. 1 1 e e Definition 4. [42,43] NCS. A NCS in a non-empty set G is defined as © = {<g; G(g), R(g)>: 8g2G}, where G and R are the INS and NS in G respectively. NCS can be presented as an order pair © = <G, R>, then we call it as a neutrosophic cubic (NC) number. Example 3. Suppose that G = {g , g , g , . . ., g } is a non-empty set. Let © be any NC-number. Then © n 1 1 1 2 3 can be expressed as © = {<g ; [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4)>: g 2G}. 1 1 Some operations of NC-numbers: i. Union of any two NC-numbers e e Let © =< G , R > and © =< G , R > be any two NC-numbers in a non-empty set G. 1 1 1 2 2 2 Then the union of © and © denoted by © [ © and defined as 1 2 1 2 e e © [ © =< G (g) [ G (g), R (g) [ R (g)8g 2 G >, 1 2 1 1 1 2 + + + e e where G (g) [ G (g) = {<g, [max {t (g), t (g)},max {t (g), t (g)}], [max {i (g), i (g)}, max {i (g), 1 1 e e e e e e e G G G1 G G G G 1 2 2 1 2 1 + + + i (g)}], [min {f (g), f (g)}, min {f (g), f (g)}]>: g2G} and R (g) [ R (g) = {<g, max {t (g), t (g)}, 1 2 R R e e e e e 1 G G G G G 2 1 2 1 2 max {i (g), i (g)}, min {f (g), f (g)}>:8g2U}. R R R R 2 2 1 1 Example 4. Let © and © be two NC-numbers in G presented as follows: © =< [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4) > and © =< [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6) > . Then © [ © =< [0.56, 0.70], [0.27, 0.43], [0.15, 0.26], (0.7, 0.3, 0.4) > . 1 2 ii. Intersection of any two NC-numbers Intersection of two NC-numbers denoted and defined as follows: e e © \ © =<G (g) \ G (g), R (g) \ R (g)8g 2 G >, 1 2 1 1 1 2 + + + e e where G (g) \ G (g) = {<g, [min {t (g), t (g)},min {t (g), t (g)}], [min {i (g), i (g)}, min {i (g), 1 1 e e e e e e e G G G1 G G G G 1 2 2 1 2 1 + + + i (g)}], [max {f (g), f (g)}, max {f (g), f (g)}]>: g2G} and R (g) \ R (g)= {<g, min {t (g), t (g)}, 1 2 R R e e e e e 1 2 G G G G G 2 1 2 1 2 min {i (g),i (g)}, max {f (g), f (g)}>:8g2U}. R R R R 1 2 1 2 Information 2017, 8, 149 5 of 21 Example 5. Let © and © be any two NC-numbers in G presented as follows: 1 2 © =< [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5) > and © =< [0.67, 0.75], [0.22, 0.44], [0.17, 0.21], (0.8, 0.4, 0.4) > . Then © \ © =< [0.45, 0.57], [0.22, 0.33], [0.18, 0.46], (0.7, 0.3, 0.4) > . 1 2 iii. Compliment of a NC-number e e Let © =< G , R > be a NCS in G. Then, the compliment of © =< G , R > denoted by 1 1 1 1 1 1 c c © = {<g, G (g), R (g)>: 8g2G}. 1 1 c + + + Here, G = {<g, [t (g), t (g)], [i (g), i (g)], [f (g), f (g)]>: 8g2G}, where, c c c c e c e c e e e e G G G G G G 1 1 1 1 1 1 + + + + t (g) = f (g), t (g) = f (g), i (g) = {1} i (g), i (g) = {1} i (g), f (g) = t (g), e c e e c e e c e e c e e c e G G G G G G G G G G 1 1 1 1 1 1 1 1 1 1 + + c + c c f (g) = f (g) and t (g) = f (g), i (g) = {1 } i (g), f (g) = t (g). R R´ R R R 1 R e c e 1 1 1 G G 1 1 1 1 Example 6. Assume that © be any NC-number in G in the form: © =< [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5) > Then compliment of © is obtained as © =< [0.18, 0.46], [0.73, 0.67], [0.45, 0.57], (0.5, 0.7, 0.7) > . Definition 5. Score function. Let © be a NC-number in a non-empty set G. Then, a score function of © , 1 1 denoted by Sc (© ) is defined as: 1 2 + a + a 2b 2b c c 1 + a 2b c 1 2 1 2 1 2 Sc (© ) = [( ) + ( )] (1) 2 4 2 where, © = <[a , a ], [b , b ], [c , c ], (a, b, c) > and Sc (© )2[–1, 1]. 1 1 2 1 2 1 2 Proposition 1. Score function of two NC-numbers lies between 1 to 1. Proof. Using the definition of INS and NS, we have all a , a , b , b , c , c , a, b, and c [0, 1]. 1 2 1 2 1 2 Since, 0  a  1, 0  a  1 1 2 0  a + a  2, (2) 1 2 ) 2  2 + a + a  4 1 2 0  b  1 ) 0  2b  2 and 0  b  1 ) 0  2b  2 2 2 1 1 ) 2  2b  0 (3) ) 2  2b  0 ) 4  2b 2b  0 1 2 0  c  1 ) 1  c  0 1 1 0  c  1 ) 1  c  0 (4) 2 2 ) 2  c c  0 1 2 Information 2017, 8, 149 6 of 21 Adding Equations (2)–(4), we obtain ) 4  2 + a + a 2b 2b c c  4, 1 2 1 2 1 2 (5) 2 + a + a 2b 2b c c 1 2 1 2 1 2 ) 1   1 Again, 0  a  1) 1  1 + a  2 , (6) 0  b  1 ) 0  2b  2, 0  c  1, (7) ) 0  2b + c  3, ) 3  2b c  0 Adding (6) and (7), we obtain 2  1 + a 2b c  2, (8) 1 + a 2b c ) 1   1 Adding (5) and (8) and dividing by 2, we obtain 1 2 + a + a 2b 2b c c 1 + a 2b c 1 2 1 2 1 2 1  [( ) + ( )]  1 2 4 2 Sc (© ) 2 [1, 1]. Hence the proof is complete. Example 7. Let © and © be two NC-numbers in G, presented as follows: 1 2 © =< [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4) > and © =< [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6) > . Then, by applying Definition 5, we obtain Sc (© ) = 0.01 and Sc (© ) = 0.07, In this case, we can say 1 2 that © > © . Definition 6. Accuracy function. Let © be a NC-number in a non-empty set G, an accuracy function of © is 1 1 defined as: 1 1 Ac(© ) = /2 [ /2(a + a b (1 a ) b (1 a ) c (1 b ) c (1 b ) + a b(1 a) c(1 b)] (9) 1 1 2 2 2 1 1 2 1 1 2 Here, Ac (© )2[–1, 1]. When the value of Ac (© ) increases, we say that the degree of accuracy of the NC-number © increases. 1 1 Proposition 2. Accuracy function of two NC-numbers lies between 1 to 1. Proof. The values of accuracy function depend upon f (a + a b (1 a ) b (1 a ) c (1 b ) c (1 b )) andf a b(1 a) c(1 b)g 1 2 2 2 1 1 2 1 1 2 The values of f (a + a b (1 a ) b (1 a ) c (1 b ) c (1 b ))g 1 2 2 2 1 1 2 1 1 2 2 Information 2017, 8, 149 7 of 21 and f a b(1 a) c(1 b)g lie between 1 to 1 from [37]. Thus, 1  Ac (© )  1. Hence the proof is completed. Example 8. Let © and © be two NC-numbers in G presented as follows: © =< [0.41, 0.52], [0.10, 0.18], [0.06, 0.17], (0.48, 0.11, 0.11) > and © =< [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > . Then, by applying Definition 6, we obtain Ac (© ) = 0.14 and Ac (© ) = 0.30. In this case, we can say 1 2 that alternative © is better than © . 2 1 With respect to the score function Sc and the accuracy function Ac, a strategy for comparing NC-numbers can be defined as follows: 3. Comparison Strategy of Two NC-Numbers Let © and © be any two NC-numbers. Then we define comparison strategy as follows: 1 2 i. If Sc(© ) > Sc(© ), then © > © . (10) 1 2 1 2 ii. If Sc(© ) = Sc(© ) and Ac(© ) > Ac(© ), then © > © . (11) 1 2 1 2 1 2 iii. If Sc(© ) = Sc(© ) and Ac(© ) = Ac(© ), then © = © . (12) 1 2 1 2 1 2 Example 9. Let © and © be two NC-numbers in G, presented as follows: 1 2 © =< [0.23, 0.29], [0.37, 0.46], [0.34, 0.42], (0.26, 0.26, 0.26) > and © =< [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > . Then, applying Definition 5, we obtain Sc (© ) = 0.13 and Sc (© ) = 0.13. Applying Definition 6, 1 2 we obtain Ac(© ) =0.20 and Ac (© ) =0.18. In this case, we say that alternative © > © . (Score values 1 2 1 and Accuracy values taking correct up to two decimal places). Definition 7. Let © and © be any two NC-numbers, then the distance between them is defined by 1 2 ¶(© , © ) = [ a d + a d + b e + b e + c f + c f + a d + b e + c f ] (13) j j j j j j j j j j j j j j j j j j 1 2 1 1 2 2 1 1 2 2 1 1 2 2 where, © = <[a , a ], [b , b ], [c , c ], (a, b, c)> and © = <[d , d ], [e , e ], [f , f ], (d, e, f)>. 1 1 2 1 2 1 2 2 1 2 1 2 1 2 Example 10. Let © and © be two NC-numbers in G presented as follows: 1 2 © =< [0.66, 0.75], [0.25, 0.32], [0.17, 0.34], (0.53, 0.17, 0.22) > 1 Information 2017, 8, 149 8 of 22 ∂ (©1, ©2) = [ a − d + a − d + b − e + b − e + c − f + c − f + a − d + b − e + c − f ] 1 1 2 2 1 1 2 2 1 1 2 2 (13) where, ©1 = <[a1, a2], [b1, b2], [c1, c2], (a, b, c)> and ©2 = <[d1, d2], [e1, e2], [f1, f2], (d, e, f)>. Example 10. Let ©1 and ©2 be two NC-numbers in G presented as follows: = <[0.66, 0.75], [0.25, 0.32], [0.17, 0.34], (0.53, 0.17, 0.22)> and = < [0.35, 0.55], [0.12, 0.25], [0.12, 0.20], (0.60, 0.23, 0.43)> © 2 Then, applying Definition 7, we obtain ∂ (©1, ©2) = 0.12. Information 2017, 8, 149 8 of 21 − + − + − + {<[ , ],[ , ],[ , ],(t,i,f ) >} Definition 8. Let ©ij = be any neutrosophic cubic value. ©ij used to evaluate t t i i f f ij ij ij ij ij ij and i-th alternative with respect to j-th criterion. The normalized form of ©ij is defined as follows: © =< [0.35, 0.55], [0.12, 0.25], [0.12, 0.20], (0.60, 0.23, 0.43) > − + t t ⊗ ij ij ©ij ={< [ , ] , Then, applying Definition 7, we obtain ¶ (© , © ) 1= 0.12. 1 1 2 m m − 2 + 2 − 2 + 2 2 2 ( ( ) + ( ) ) ( ( ) + ( ) )  t t  t t ij ij ij ij i=1 i=1 + + Definition 8. Let © = f< [t , t ], [i , i ], [f , f ], (t, i, f) >g be any neutrosophic cubic value. ij ij ij ij ij ij ij © used to evaluate i-th alternative with respect to j-th criterion. The normalized form of © is defined as follows: − + ij ij i i ij ij [ , ] , 1 1 m m − 2 + t 2 − 2 + 2 t 2 2 ij ij ( ( ) + ( ) ) ( ( ) + ( ) )   i i i i ij ij ij ij © = f< [ , ], m m 1 1 ij i=1 2 i2 =1 2 2 + + 2 2 ( S (t ) +(t ) ) ( S (t ) +(t ) ) ij ij ij ij i=1 i=1 (14) i i − + ij ij f f [ ij , ij ], m m 1 1 2 2 2 2 [ + , + ] 2 2 ( S (i ) +(i ) ) ( S (i ) +(i ) ) 1 1 ij ij ij ij m m i=1 i=1 − 2 + 2 − 2 + 2 2 2 (14) ( ( ) + ( ) ) ( ( ) + ( ) )  f f  f + f ij ij ij ij f f ij ij i=1 i=1 [ , ] m 1 m 1 2 2 2 2 + + 2 2 ( S (f ) +(f ) ) ( S (f ) +(f ) ) ij ij ij ij i=1 i=1 ij iij f ij t i f ij ij ij [ , , ] >}. [ , , ] >g. 1 1 1 m m m 1 1 1 m m m 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 ( S (t ) +(i ) +(f ) ) 2 ( S (t ) +(i ) +(f ) ) 2 ( S (t ) +(i ) +(f ) ) 2 ij ij ij ij ij ij ij ij ij ( ( ) + ( ) + ( ) ) ( ( ) + ( ) + ( ) ) ( ( ) + ( ) + ( ) )  t i f  t i f  t i f i=1 ij ij ij i=1 ij ij ij i=1 ij ij ij i=1 i=1 i=1 A conceptual model of the evolution of the neutrosophic cubic set is shown in Figure 1. A conceptual model of the evolution of the neutrosophic cubic set is shown in Figure 1. Figure 1. Evolution of the neutrosophic cubic set. Figure 1. Evolution of the neutrosophic cubic set. 4. NC-TODIM Based MAGDM under a NCS Environment 4. NC-TODIM Based MAGDM under a NCS Environment Assume that A = {A , A , . . . , A } (m  2) and C = {C , C , . . . , C } (n  2) are the discrete set m n 1 2 1 2 of alternatives and attributes respectively. W = {W , W , . . . , W } is the weight vector of attributes 1 2 n C (j = 1, 2, . . . , n), where W > 0 and S W = 1. Let E = {E , E , . . . , E } be the set of decision makers j j j 1 2 r j=1 and g = fg , g , . . . , g g be the weight vector of decision makers, where g > 0 and S g = 1. 1 2 r k k k=1 NC-TODIM Strategy Now, we describe the NC-TODIM strategy to solve the MAGDM problems with NC-numbers. The NC-TODIM strategy consists of the following steps: Information 2017, 8, 149 9 of 21 Step 1. Formulate the decision matrix k k k k Assume that M = (© ) be the decision matrix, where © = <G , R > is the rating value ij ij ij ij m n provided by the k-th (E decision maker for alternative A , with respect to attribute C . The matrix i j k) form of M is presented as: 0 1 C C . . . C 1 2 n k k k B C A © © . . . © B 11 12 1n C B C k k k k M = A © © © (15) B C 21 22 2n B C @ A . . . . . . k k k A © © . . . © m1 m2 mnj Step 2. Normalize the decision matrix The MAGDM problem generally consists of cost criteria and benefit criteria. So, the decision matrix needs to be normalized. For cost criterion C , we use the Definition 8 to normalize the decision matrix (Equation (15)) provided by the decision makers. For benefit criterion C we don’t need to normalize the decision matrix. When C is a cost criterion, the normalized form of decision matrix (see Equation (15)) is presented below. 0 1 C C . . . C 1 2 n B C A © © . . . © B C 11 12 1n B C M = A © © © (16) B C 21 22 2n B C @ A . . . . . . A © © . . . © m1 m2 mnj Here © is the normalized form of the NC-number. ij Step 3. Determine the relative weight of each criterion The relative weight W of each criterion is obtained by the following equation. ch W = (17) ch where, W = max {W , W , . . . , W }. h 1 2 Step 4. Calculate score values Using Equation (1), calculate the score value Sc (© ) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) of © if C ij ij is a cost criterion. Using Equation (1), calculate the score value Sc ((c) ) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) ij of © if C is a benefit criterion. ij Step 5. Calculate accuracy values Using Equation (9), calculate the accuracy value Ac (© ) (I = 1, 2, . . . , m; j= 1, 2, . . . , n) of ij k k © if C is a cost criterion. Using Equation (9), calculate the accuracy value Ac (© ) (I = 1, 2, . . . , m; j ij ij j = 1, 2, . . . , n) of © if C is a benefit criterion. ij Step 6. Formulate the dominance matrix Calculate the dominance of each alternative A over each alternative A with respect to the criteria i j C (C C , . . . , C ), of the k-th decision maker E by the following Equations (18) and (19). 1, 2 n k Information 2017, 8, 149 10 of 21 (For cost criteria) Ch Y (A , A ) = ( ¶(© , © ) , if © > © i j n c ic jc ic jc S W ch > c=1 = 0, if © = © (18) ic jc S W > ch 1 c=1 k ; = ( ¶(© , © ) , if © < © ic jc ic jc Ch (For benefit criteria) W k k k k k Ch Y (A , A ) = ( ¶(© , © ) , if © > © > i j n c ic jc ic jc S W ch c=1 k k = 0, if © = © (19) ic jc S W > ch 1 c=1 k k k k = ( ¶(© , , © ) , if © < © W ic jc ic jc Ch where, parameter represents the attenuation factor of losses and must be positive. Step 7. Formulate the individual overall dominance matrix Using Equation (20), calculate the individual total dominance matrix of each alternative A over each alternative A under the criterion C . j j k k = (A , A ) = S Y (A , A ) (20) i j c i j c=1 Step 8. Aggregate the dominance matrix Using Equation (21), calculate the collective overall dominance of alternative A over each alternative A . (A , A ) = S g  (A , A ) (21) i j k i j k=1 Step 9. Calculate global values We present the global value of each alternative as follows: n n S  (A , A ) min ( S  (A , A )) i j i j j=1 1 i m j=1 W = (22) n n max ( S  (A , A )) min ( S  (A , A )) i j i j 1 i m j=1 1 i m j=1 Step 10. Rank the priority Sorting the values of W provides the rank of each alternative. A set of alternatives can be preference-ranked according to the descending order of W . The highest global value corresponds to the best alternative. A conceptual model of the NC-TODIM strategy is shown in Figure 2. Information 2017, 8, 149 11 of 22 Step 10. Rank the priority Sorting the values of Ω provides the rank of each alternative. A set of alternatives can be preference-ranked according to the descending order of Ω . The highest global value corresponds Information 2017, 8, 149 11 of 21 to the best alternative. A conceptual model of the NC-TODIM strategy is shown in Figure 2. Figure 2. A flow chart of the proposed neutrosophic cubic set (NC)-TODIM strategy. Figure 2. A flow chart of the proposed neutrosophic cubic set (NC)-TODIM strategy. 5. Illustrative Example 5. Illustrative Example In this section, a MAGDM problem is adapted from the study [18] under the NCS environment. In this section, An investme a MAGDM nt company pr wants to se oblem is lect the best al adapted tfr ernat om ive the amostudy ng the set [18 of ]fea under sible altthe ernatNCS ives. environment. The feasible alternatives are An investment company wants to select the best alternative among the set of feasible alternatives. 1. Car company (A1) The feasible alternatives are 2. Food company (A2) 3. Computer company (A3) 1. Car company (A ) 4. Arms company (A4). 2. Food company (A ) The best alternative is selected based on the following criteria: 3. Computer company (A ) 1. Risk analysis (C1) 2. Growth analysis (C2) 4. Arms company (A ). 3. Environmental impact analysis (C3). The best alternative is selected based on the following criteria: An investment company forms a panel of three decision makers {E1, E2, E3} who evaluate four alternatives in decision making process. The weight vector of attributes and decision makers are 1. Risk analysis (C ) 1 T T considered as W = (0.4, 0.35, 0.25) γ = (0.32, 0.33, 0.35) respectively. 2. Growth analysis (C ) The proposed strategy is presented using the following steps: 3. Environmental impact analysis (C ). An investment company forms a panel of three decision makers {E , E , E } who evaluate four 1 2 3 alternatives in decision making process. The weight vector of attributes and decision makers are T T considered as W = (0.4, 0.35, 0.25) g = (0.32, 0.33, 0.35) respectively. The proposed strategy is presented using the following steps: Step 1. Formulate the decision matrix Formulate the decision matrices M (k = 1, 2, 3) using the rating values of alternatives with respect to three criteria provided by the three decision makers in terms of NC-numbers. Assume k k that the NC-numbers © = <G , R > present the rating value provided by the decision maker E for ij ij k ij alternative A with respect to attribute C . Using these rating values © (k = 1, 2, 3; i = 1, 2, 3, 4; j = 1, i j ij k k 2, 3), three decision matrices M = (© ) (k = 1, 2, 3) are constructed (see Equations (23)–(25)). ij 43 Decision matrix for E 0 1 C C C 1 2 3 B C A < [0.41, 0.52], [0.10, 0.18], [0.06, 0.17], (0.48, 0.11, 0.11) > < [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > < [0.22, 0.27], [0.41, 0.52], [0.41, 0.52], (0.31, 0.31, 0.31) > B 1 C B C M = BA < [0.35, 0.46], [0.18, 0.27], [0.17, 0.34], (0.43, 0.16, 0.21) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.57, 0.12, 0.12) >C (23) B C @ A A < [0.23, 0.29], [0.36, 0.45], [0.34, 0.42], (0.26, 0.26, 0.26) > < [0.34, 0.45], [0.20, 0.30], [0.19, 0.39], (0.44, 0.16, 0.22) > < [0.22, 0.27], [0.41, 0.52], [0.41, 0.52], (0.31, 0.31, 0.31) > A < [0.17, 0.23], [0.45, 0.55], [0.42, 0.59], (0.21, 0.32, 0.37) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.57, 0.12, 0.12) > Decision matrix for E 0 1 C C C 1 2 3 B C A < [0.17, 0.23], [0.46, 0.55], [0.42, 0.59], (0.21, 0.32, 0.37) > < [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > < [0.34, 0.43], [0.13, 0.27], [0.13, 0.27], (0.49, 0.11, 0.11) > B 1 C B C M = BA < [0.23, 0.29], [0.37, 0.46], [0.34, 0.42], (0.26, 0.26, 0.26) > < [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > < [0.34, 0.43], [0.13, 0.27], [0.13, 0.27], (0.49, 0.11, 0.11) >C (24) B C @ A A3 < [0.41, 0.52], [0.10, 0.18], [0.10, 0.17], (0.48, 0.11, 0.11) > < [0.44, 0.57], [0.10, 0.17], [0.10, 0.17], (0.51, 0.11, 0.11) > < [0.19, 0.24], [0.53, 0.67], [0.53, 0.67], (0.27, 0.27, 0.27) > A < [0.35, 0.46], [0.20, 0.28], [0.17, 0.34], (0.42, 0.16, 0.21) > < [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > < [0.34, 0.43], [0.13, 0.27], [0.13, 0.27], (0.49, 0.11, 0.11) > 4 Information 2017, 8, 149 12 of 21 Decision matrix for E 0 1 C C C 1 2 3 B C A < [0.22, 0.27], [0.42, 0.52], [0.42, 0.52], (0.28, 0.28, 0.28) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.41, 0.52], [0.10, 0.18], [0.10, 0.17], (0.48, 0.11, 0.11) > B 1 C B C M = BA < [0.22, 0.27], [0.42, 0.52], [0.42, 0.52], (0.28, 0.28, 0.28) > < [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > < [0.23, 0.29], [0.36, 0.45], [0.34, 0.42], (0.26, 0.26, 0.26) >C (25) B C @ A A < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.50, 0.11, 0.11) > < [0.34, 0.45], [0.20, 0.30], [0.19, 0.39], (0.44, 0.16, 0.22) > < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.50, 0.11, 0.11) > A < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.50, 0.11, 0.11) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.17, 0.23], [0.45, 0.54], [0.42, 0.59], (0.21, 0.32, 0.37) > Step 2. Normalize the decision matrix Since all the criteria are benefit type, we do not need to normalize the decision matrix. Step 3. Determine the relative weight of each criterion Using Equation (17), we obtain the relative weight vector W of criteria as follows: ch W = (1, 0.875, 0.625) . ch Step 4. Calculate score values The score values of each alternative relative to each criterion obtained by Equation (1) are presented in the Tables 1–3. Table 1. Score values for M . C C C 1 2 2 A 0.56 0.54 0.06 A 0.40 0.09 0.54 A 0.50 0.38 0.06 A 0.03 0.09 0.54 Table 2. Score values for M . C C C 1 2 2 A 0.03 0.13 0.49 A 0.13 0.13 0.49 A 0.56 0.60 0.04 A 0.39 0.13 0.49 Table 3. Score values for M . C C C 1 2 2 A 0.07 0.09 0.56 A 0.07 0.52 0.13 A 0.51 0.37 0.39 A 0.51 0.09 0.03 Step 5. Calculate accuracy values The accuracy values of each alternative relative to each criterion obtained by Equation (9) are presented in Tables 4–6. Table 4. Accuracy values for M . C C C 1 2 2 A 0.14 0.30 0.24 A 0.12 0.23 0.32 A 0.20 0.09 0.24 A 0.38 0.23 0.32 4 Information 2017, 8, 149 13 of 21 Table 5. Accuracy values for M . C C C 1 2 2 A 0.38 0.18 0.21 A 0.20 0.18 0.21 A 0.14 0.36 0.21 A 0.12 0.18 0.21 Table 6. Accuracy values for M . C C C 1 2 2 A 0.24 0.23 0.41 A 0.24 0.30 0.20 A 0.26 0.09 0.12 A 0.26 0.23 0.38 Step 6. Formulate the dominance matrix Using Equation (19), we construct dominance matrix for = 1. The dominance matrices are represented in matrix form (See Equations (26)–(34)). 1 1 The dominance matrix Y , the dominance matrix Y 1 2 0 1 A A A A 1 2 3 4 B C A 0 0.18 0.30 0.35 B 1 C B C Y = (26) BA 0.46 0 0.58 0.30C B C @ A A 0.74 0.23 0 0.19 A 0.88 0.74 0.47 0 0 1 A A A A 1 2 3 4 B C A 0 0.29 0.18 0.28 B 1 C B C Y = A 0.82 0 0.69 0 (27) B C B C @ A A 0.51 0.24 0 0.29 A 0.81 0 0.65 0 1 2 The dominance matrix Y , the dominance matrix Y 3 1 0 1 A A A A 1 2 3 4 B C A 0 1 0 1 B 1 C B C Y = A 0.25 0 0.26 0 (28) B C B C @ A A 0 1 0 1 A 0.25 0 0.26 0 0 1 A A A A 1 2 3 4 B C A 0 0.46 0.88 0.74 B 1 C B C Y =BA 0.18 0 0.75 0.58C (29) B C @ A A 0.35 0.09 0 0.04 A 0.30 0.23 0.19 0 4 Information 2017, 8, 149 14 of 21 2 2 The dominance matrix Y , the dominance matrix Y 2 3 0 1 A A A A 1 2 3 4 B C A 0 0 0.84 0 B 1 C B C Y =BA 0 0 0.84 0 C (30) B C @ A A 0.29 0.29 0 0.29 A 0 0 0.84 0 0 1 A A A A 1 2 3 4 B C A 0 0 0.26 0 B 1 C B C Y =BA 0 0 0.26 0 C (31) 3 2 B C @A 1 1 0 1A A 0 0 0.26 0 3 3 The dominance matrix Y , the dominance matrix Y 1 2 0 1 A A A A 2 3 1 4 B C A 0 0 0.78 0.78 B 1 C B C Y =BA 0 0 0.78 0.78C (32) B C @ A A 0.31 0.31 0 0 A 0.31 0.31 0 0 0 1 A A A A 1 2 3 4 B C A 0 0.83 0.65 0 B 1 C B C Y = (33) B A 0.29 0 0.18 0.29C B C @ A A 0.23 0.51 0 0.23 A 0 0.83 0.65 0 The dominance matrix Y 0 1 A A A A 1 2 3 4 B C A 0 0.94 0.59 1.1 B 1 C B C Y = A 0.23 0 0.73 0.15 (34) B C B C @ A A 0.59 0.18 0 0.23 A 1.1 0.58 0.94 0 Step 7. Formulate the individual overall dominance matrix The individual overall dominance matrix is calculated by the Equation (20) and the dominance matrices are represented in matrix form (see Equations (35)–(37)). First decision maker ’s overall dominance matrix 0 1 A A A A 1 2 3 4 B C A 0 0.53 0.47 0.37 B 1 C B C = BA 1 0 1 0.30 C (35) B C @A 1.3 0.53 0 0.52 A A 1.5 0.74 0.86 0 4 Information 2017, 8, 149 15 of 21 Second decision maker ’s overall dominance matrix 0 1 A A A A 1 2 3 4 B C A 0 0.46 1.5 0.74 B 1 C B C = BA 0.18 0 1.3 0.58C (36) B C @ A A 0.36 0.62 0 0.67 A 0.30 0.23 0.39 0 Third decision maker ’s overall dominance matrix 0 1 A A A A 1 2 3 4 B C A 0 1.8 2 1.9 B 1 C B C =BA 0.52 0 1.3 0.34C (37) B C @ A A 0.05 0.02 0 0.46 A 0.79 1.1 1.6 0 Step 8. Aggregate the dominance matrix Using Equation (21), the aggregate dominance matrix  is constructed (see Equation (38)) as follows: 0 1 A A A A 1 2 3 4 B C A 0 0.94 1.1 0.53 B 1 C B C = A 0.10 0 1.23 0.22 (38) B C B C @ A A 0.54 0.38 0 0.23 A 0.64 0.55 0.96 0 Step 9. Calculate global values Using Equation (22), we calculate the values of W (i = 1, 2, 3, 4) and represented in Table 7. Table 7. Global values of alternatives. A A A A A i 1 2 3 4 W 0.49 0.61 1 0 Step 10. Rank the priority Since W > W > W > W , alternatives are then preference ranked as follows: A > A > A > A . 3 2 1 4 3 2 1 4 Hence A is the best alternative. From the illustrative example, we see that the proposed NC-TODIM strategy is more suitable for real scientific and engineering applications because it can handle hybrid information consisting of INS and SVNS information simultaneously to cope with indeterminate and inconsistent information. Thus, NC-TODIM extends the existing decision-making strategies and provides a sophisticated mathematical tool for decision makers. 6. Rank of Alternatives with Different Values of Table 8 shows that the ranking order of alternatives depends on the values of the attenuation factor, which reflects the importance of the attenuation factor in the NC-TODIM strategy. Information 2017, 8, 149 16 of 21 Table 8. Global values and ranking of alternatives for different values of . Values of Global Values of Alternative (W ) Rank Order of A i i Information 2017, 8, 149 17 of 22 W = 0, W = 0.89, W = 1, W = 0.46 1 2 3 4 0.5 A > A > A > A 3 2 4 1 W > W > W > W 3 2 4 1 Ω Ω Ω = 0, = 0.72, Ω = 1, = 0.44 W = 0.49, 1 W = 0.61, 2 W = 1, W3 = 0 4 1 2 3 4 1 A > A > A > A 1.5 3 2 1A3 > A 4 2 > A4 > A1 W > W > W > W 3 2 1 4 Ω Ω Ω Ω > > > 3 2 4 1 W = 0, W = 0.72, W = 1, W = 0.44 1 2 3 4 1.5 A > A > A > A 3 2 4 1 Ω Ω Ω = 0, = 1, Ω = 0.81, = 0.38 W > W > W > W 1 2 3 4 3 2 4 1 2 A2 > A3 > A4 > A1 W = 0, W = 1, W = 0.81, W = 0.38 1 2 3 4 Ω Ω Ω > Ω > > 2 A > A > A > A 2 3 4 1 2 3 4 1 W > W >W > W 2 3 4 1 Ω Ω Ω W = 0, W = =0, 0.56, W = 0. = 5 1,6W , Ω = 0.45 = 1, = 0.45 1 1 2 2 3 4 3 4 3 A > A > A > A 3 2 4 1 3 A3 > A2 > A4 > A1 W > W > W > W 3 2 4 1 Ω Ω Ω Ω > > > 3 2 4 1 Analysis on Influence of the Parameter to Ranking Order Analysis on Influence of the Parameter α to Ranking Order The impact of parameter on ranking order is examined by comparing the ranking orders taken The impact of parameter on ranking order is examined by comparing the ranking orders with varying the different values of . When = 0.5, 1, 1.5, 2, 3, ranking order are presented in taken with varying the different values of α . When α = 0.5, 1, 1.5, 2, 3, ranking order are presented Table 8. We draw Figures 3 and 4 to compare the ranking order for different values of . When = 0.5, in Table 8. We draw Figures 3 and 4 to compare the ranking order for different values of . When = 1.5 and = 3, the ranking order is unchanged and A is the best alternative, while A is the worst 3 1 α = 0.5, α = 1.5 and α = 3, the ranking order is unchanged and A3 is the best alternative, while alternative. When = 1, the ranking order is changed and A is the best alternative and A is the worst 3 4 A1 is the worst alternative. When α = 1, the ranking order is changed and A3 is the best alternative alternative. For = 2, the ranking order is changed and A is the best alternative and A is the worst 2 1 and A4 is the worst alternative. For α = 2, the ranking order is changed and A2 is the best alternative. From Table 8, we see that A is the best alternative in four cases and A is the worst for 3 1 alternative and A1 is the worst alternative. From Table 8, we see that A3 is the best alternative in four four cases. We can say that ranking order depends on parameter . cases and A1 is the worst for four cases. We can say that ranking order depends on parameter α . Figure 3. Global values of the alternatives for different values of attenuation factor = 0.5, 1, 1.5, 2, 3. Figure 3. Global values of the alternatives for different values of attenuation factor = 0.5, 1, 1.5, 2, 3. Information 2017, 8, 149 17 of 21 Information 2017, 8, 149 18 of 22 Figure 4. Ranking of the alternatives for = 0.5, 1, 1.5, 2, 3. Figure 4. Ranking of the alternatives for α = 0.5, 1, 1.5, 2, 3. 7. Comparative Analysis and Discussion 7. Comparative Analysis and Discussion On comparing with the existing neutrosophic decision making strategies [26–29,33–35,64–69], On comparing with the existing neutrosophic decision making strategies [26–29,33–35,64–69], we see that the decision information used in the proposed NC-TODIM strategy is NC numbers, we see that the decision information used in the proposed NC-TODIM strategy is NC numbers, which comprises of interval neutrosophic information and single-valued neutrosophic information which comprises of interval neutrosophic information and single-valued neutrosophic information simultaneously; whereas the decision information in the existing literature is either SVNSs or INSs. simultaneously; whereas the decision information in the existing literature is either SVNSs or INSs. Since NC numbers comprises of much more information, the NC numbers based on the TODIM Since NC numbers comprises of much more information, the NC numbers based on the TODIM strategy proposed in this paper is more elegant, typical and more general in applications, while the strategy proposed in this paper is more elegant, typical and more general in applications, while the existing neutrosophic decision-making strategies cannot deal with the NC number decision-making existing neutrosophic decision-making strategies cannot deal with the NC number decision-making problem developed in this paper. problem developed in this paper. The first decision making paper in NCS environment was studied by Banerjee et al. [44]. The first decision making paper in NCS environment was studied by Banerjee et al. [44]. On On comparison with existing GRA-based NCS decision making strategies [44], we observe that comparison with existing GRA-based NCS decision making strategies [44], we observe that the the proposed NC-TODIM strategy uses the score, and accuracy functions, while the decision proposed NC-TODIM strategy uses the score, and accuracy functions, while the decision making-strategy in [44] uses Hamming distances for weighted grey relational coefficients and standard making-strategy in [44] uses Hamming distances for weighted grey relational coefficients and (ideal) grey relational coefficients, and ranks the alternatives based on the relative closeness coefficients. standard (ideal) grey relational coefficients, and ranks the alternatives based on the relative Hence, the proposed NC-TODIM strategy is relatively simple in the decision making process. closeness coefficients. Hence, the proposed NC-TODIM strategy is relatively simple in the decision On comparing with cosine measures of NCSs [49], we observe that the proposed NC-TODIM making process. involves multiple decision makers, while in [49] only a single decision maker is involved. This shows On comparing with cosine measures of NCSs [49], we observe that the proposed NC-TODIM that [49] cannot deal with group decision making, while the proposed NC-TODIM strategy is more involves multiple decision makers, while in [49] only a single decision maker is involved. This sophisticated as it can deal with single as well as group decision making in the NCS environment. shows that [49] cannot deal with group decision making, while the proposed NC-TODIM strategy On comparison with extended TOPSIS [50] with neutrosophic cubic information, we observe is more sophisticated as it can deal with single as well as group decision making in the NCS that nine components are present in NCSs. Therefore, by calculation of a weighted decision matrix, environment. a neutrosophic cubic positive ideal solution (NCPIS), and a neutrosophic cubic negative ideal solution, On comparison with extended TOPSIS [50] with neutrosophic cubic information, we observe the distance measures of alternatives from NCPIS and NCNIS (NCNIS,) and entropy weight, and use of that nine components are present in NCSs. Therefore, by calculation of a weighted decision matrix, a an aggregation operator are lengthy, time consuming, and hence expensive. The proposed NC-TODIM neutrosophic cubic positive ideal solution (NCPIS), and a neutrosophic cubic negative ideal solution, strategy is free from different kinds of typical aggregation operators. The calculations required for the distance measures of alternatives from NCPIS and NCNIS (NCNIS,) and entropy weight, and the proposed strategy are relatively straightforward and time-saving. Therefore, the final ranking use of an aggregation operator are lengthy, time consuming, and hence expensive. The proposed obtained by the proposed strategy is more conclusive than those produced by the other strategies, NC-TODIM strategy is free from different kinds of typical aggregation operators. The calculations and it is evident that the proposed strategy is accurate and reliable. required for the proposed strategy are relatively straightforward and time-saving. Therefore, the On comparison with the strategy proposed by Zhan et al. [51], we see that they employ score, final ranking obtained by the proposed strategy is more conclusive than those produced by the accuracy, and certainty functions, and a weighted average operator and weighted geometric operator other strategies, and it is evident that the proposed strategy is accurate and reliable. of NCSs for decision making problem involving only a single decision maker. This reflects that the On comparison with the strategy proposed by Zhan et al. [51], we see that they employ score, strategy introduced by Zhan et al. [51] is only applicable for decision making problems involving accuracy, and certainty functions, and a weighted average operator and weighted geometric single decision maker. However, our proposed NC-TODIM strategy is more general as it is capable of operator of NCSs for decision making problem involving only a single decision maker. This reflects dealing with group decision-making problems. that the strategy introduced by Zhan et al. [51] is only applicable for decision making problems involving single decision maker. However, our proposed NC-TODIM strategy is more general as it is capable of dealing with group decision-making problems. Information 2017, 8, 149 18 of 21 A comparative study is conducted with the existing strategy [48] for group decision making under a NCS environment (See Table 9). Since the philosophy of two strategies are different, the obtained results (ranking order) are different. At a glance, it cannot be said which strategy is superior to the other. However, on comparison with similarity measure-based strategies studied in [48], we observed that ideal solutions are needed for ranking of alternatives but in a real world ideal solution, this is an imaginary case, which means that an indeterminacy arises automatically, whereas in our proposed NC-TODIM strategy we can calculate the rank of the alternatives based on global values of alternatives. So, the proposed NC-TODIM strategy is relatively easy to implement and apply for solving MAGDM problems. Table 9. Ranking order of alternatives using three different decision making strategies in the neutrosophic cubic set (NCS) environment. Proposed NC-TODIM Strategy Similarity Measure [48] W = 0, W = 0.89, W = 1, W = 0.46  = 0.20,  = 0.80,  = 0.22,  = 0.19 1 2 3 4 1 2 3 4 Ranking order: A > A > A > A Ranking order: A > A > A > A 3 2 4 1 2 3 1 4 8. Conclusions NCSs can better describe hybrid information comprising of INSs and NSs. In this study, we proposed a score function and an accuracy function, and established their properties. We developed a NC-TODIM strategy, which is capable for tackling MAGDM problems affected by uncertainty and indeterminacy represented by NC numbers. The standard TODIM, in its original formulation, is only applicable to a crisp environment. Existing neutrosophic TODIM strategies deal with single valued neutrosophic information or interval neutrosophic information. Therefore, proposed NC-TODIM strategy demonstrates the advantages of presenting and manipulating MAGDM problems with NCSs comprising of the hybrid information of INSs and NSs. Furthermore, NC-TODIM strategy that considers the risk preferences of decision makers, is significant to solve MAGDM problems. The proposed NC-TODIM strategy was verified to be applicable, feasible, and effective by solving an illustrative example regarding the selection problem of investment alternatives. In addition, we investigated the influence of attenuation factor of losses on ranking the order of alternatives. The contribution of this study can be concluded as follows. First, this study utilized NCSs to present the interval neutrosophic information and neutrosophic information in the MAGDM process. Second, the NC-TODIM strategy established in this paper is simpler and easier than the existing strategy proposed by Pramanik et al. [48] for group decision making with neutrosophic cubic information based on similarity measure and demonstrates the main advantage of its simple and easy group decision making process. Third, TODIM strategy was extended to the NCS environment. Fourth, we defined the NC number. Fifth, we defined the score and accuracy functions and proved their basic properties. Sixth, we developed the ranking of NC numbers using score and accuracy functions. Therefore, two functions namely, score function, accuracy function, and proofs of their basic properties, ranking of NC numbers, and NC-TODIM strategy for MAGDM are the main contributions of the paper. Several directions for future research are generated from this study. First, this study employs the NC-TODIM strategy to deal with MAGDM. In addition to MAGDM, MAGDM problems in a variety of other fields can be solved using the NC-TODIM strategy, including logistics center selection, personnel selection, teacher selection, renewable energy selection, medical diagnosis, image processing, fault diagnosis, etc. Second, this study considers the risk preferences of decision makers i.e., the essence of TODIM, while the interrelationship between criteria are ignored. In future research, the NC-TODIM strategy will be improved to address this deficiency. Third, the proposed strategy can only deal with crisp weights of attributes and decision makers, rather than NCS, which reflects its main limitation. This limitation will be effectively addressed in our future research. Fourth, in our illustrative Information 2017, 8, 149 19 of 21 example, three criteria are considered as an example. However, in real world group decision making problems, many other criteria should be included. A comprehensive framework for MAGDM problem comprising of all relevant criteria should be designed based on prior studies and the proposed NC-TODIM strategy in future research. Finally, we conclude that the developed NC-TODIM strategy offers a novel and effective strategy for decision makers under the NCS environment, and will open up a new avenue of research into the neutrosophic hybrid environment. Acknowledgments: The authors would like to acknowledge the constructive comments and suggestions of the anonymous referees. Author Contributions: Surapati Pramanik conceived and designed the problem; Shyamal Dalapati solved the problem; Surapati Pramanik, Shariful Alam and Tapan Kumar Roy analyzed the results; Surapati Pramanik and Shyamal Dalapati wrote the paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–356. [CrossRef] 2. Chen, S.J.; Hwang, C.L. Fuzzy Multiple Attribute Decision-Making, Methods and Applications; Lecture Notes in Economics and Mathematical Systems; Springer-Verlag: Berlin/Heidelberg, Germany, 1992; Volume 375. 3. Chang, T.H.; Wang, T.C. Using the fuzzy multi-criteria decision making approach for measuring the possibility of successful knowledge management. Inf. Sci. 2009, 179, 355–370. [CrossRef] 4. Pramanik, S.; Mondal, K. Weighted fuzzy similarity measure based on tangent function and its application to medical diagnosis. Int. J. Innovat. Res. Sci. Engg. Tech. 2015, 4, 158–164. [CrossRef] 5. Chen, C.T. Extension of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst. 2000, 114, 1–9. [CrossRef] 6. Zhang, G.; Lu, J. An integrated group decision-making method dealing with fuzzy preferences for alternatives and individual judgments for selection criteria. Group Decis. Negot. 2003, 12, 501–515. [CrossRef] 7. Krohling, R.A.; Campanharo, V.C. Fuzzy TOPSIS for group decision making: A case study for accidents with oil spill in the sea. Expert Syst. Appl. 2011, 38, 4190–4197. [CrossRef] 8. Xia, M.; Xu, Z. A novel method for fuzzy multi-criteria decision making. Int. J. Inf. Technol. Decis. Mak. 2014, 13, 497–519. [CrossRef] 9. Mehlawat, M.K.; Guptal, P. A new fuzzy group multi-criteria decision making method with an application to the critical path selection. Int. J. Adv. Manuf. Technol. 2016, 83, 1281–1296. [CrossRef] 10. Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [CrossRef] 11. Xu, Z. Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optim. Decis. Mak. 2007, 6, 109–121. [CrossRef] 12. Mondal, K.; Pramanik, S. Intuitionistic fuzzy multi criteria group decision making approach to quality-brick selection problem. J. Appl. Quant. Methods 2014, 9, 35–50. 13. Dey, P.P.; Pramanik, S.; Giri, B.C. Multi-criteria group decision making in intuitionistic fuzzy environment based on grey relational analysis for weaver selection in Khadi institution. J. Appl. Quant. Methods 2015, 10, 1–14. 14. Smarandache, F. A unifying field in logics. In Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, DE, USA, 1999. 15. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct. 2010, 4, 410–413. 16. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. 17. Kharal, A. A neutrosophic multi-criteria decision making method. New Math. Nat. Comput. 2014, 10, 143–162. [CrossRef] 18. Ye, J. Similarity measures between interval neutrosophic sets and their multi criteria decision-making method. J. Intell. Fuzzy Syst. 2014, 26, 165–172. 19. Ye, J. Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single valued neutrosophic environment. J. Intell. Fuzzy Syst. 2014, 27, 2927–2935. 20. Mondal, K.; Pramanik, S. Multi-criteria group decision making approach for teacher recruitment in higher education under simplified neutrosophic environment. Neutrosophic Sets Syst. 2014, 6, 28–34. Information 2017, 8, 149 20 of 21 21. Biswas, P.; Pramanik, S.; Giri, B.C. Entropy based grey relational analysis method for multi-attribute decision making under single valued neutrosophic assessments. Neutrosophic Sets Syst. 2014, 2, 102–110. 22. Biswas, P.; Pramanik, S.; Giri, B.C. A new methodology for neutrosophic multi-attribute decision-making with unknown weight information. Neutrosophic Sets Syst. 2014, 3, 44–54. 23. Biswas, P.; Pramanik, S.; Giri, B.C. Cosine similarity measure based multi-attribute decision-making with trapezoidal fuzzy neutrosophic numbers. Neutrosophic Sets Syst. 2014, 8, 46–56. 24. Mondal, K.; Pramanik, S. Neutrosophic decision making model for clay-brick selection in construction field based on grey relational analysis. Neutrosophic Sets Syst. 2015, 9, 64–71. 25. Mondal, K.; Pramanik, S. Neutrosophic tangent similarity measure and its application to multiple attribute decision making. Neutrosophic Sets Syst. 2015, 9, 85–92. 26. Pramanik, S.; Biswas, P.; Giri, B.C. Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment. Neural Comput. Appl. 2017, 28, 1163–1176. [CrossRef] 27. Sahin, R.; Küçük, A. Subsethood measure for single valued neutrosophic sets. J. Intell. Fuzzy Syst. 2015, 29, 525–530. [CrossRef] 28. Ye, J. An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers. J. Intell. Fuzzy Syst. 2015, 28, 247–255. 29. Biswas, P.; Pramanik, S.; Giri, B.C. TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 2016, 27, 727–737. [CrossRef] 30. Biswas, P.; Pramanik, S.; Giri, B.C. Value and ambiguity index based ranking method of single-valued trapezoidal neutrosophic numbers and its application to multi-attribute decision making. Neutrosophic Sets Syst. 2016, 12, 127–138. 31. Biswas, P.; Pramanik, S.; Giri, B.C. Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making. Neutrosophic Sets Syst. 2016, 12, 20–40. 32. Smarandache, F.; Pramanik, S. (Eds.) New Trends in Neutrosophic Theory and Applications; Pons Editions: Brussels, Belgium, 2016; pp. 15–161, ISBN 978-1-59973-498-9. 33. Sahin, R.; Liu, P. Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput. Appl. 2016, 27, 2017–2029. [CrossRef] 34. Sahin, R. Cross-entropy measure on interval neutrosophic sets and its applications in multi criteria decision making. Neural Comput. Appl. 2017, 28, 1177–1187. [CrossRef] 35. Sahin, R.; Liu, P. Possibility-induced simplified neutrosophic aggregation operators and their application to multi-criteria group decision-making. J. Exp. Theor. Artif. Intell. 2017, 29, 769–785. [CrossRef] 36. Biswas, P.; Pramanik, S.; Giri, B.C. Multi-attribute group decision making based on expected value of neutrosophic trapezoidal numbers. New Math. Natural Comput. 2015. [CrossRef] 37. Sahin, ¸ R. Multi-Criteria Neutrosophic Decision Making Method Based on Score and Accuracy Functions under Neutrosophic Environment. arXiv, 2014. [CrossRef] 38. Biswas, P.; Pramanik, S.; Giri, B.C. GRA method of multiple attribute decision making with single valued neutrosophic hesitant fuzzy set information. In New Trends in Neutrosophic Theory and Applications; Smarandache, F., Pramanik, S., Eds.; Pons Editions: Brussels, Belgium, 2016; Volume 1, pp. 55–63, ISBN 978-1-59973-498-9. 39. Biswas, P.; Pramanik, S.; Giri, B.C. Some distance measures of single valued neutrosophic hesitant fuzzy sets and their applications to multiple attribute decision making. In New Trends in Neutrosophic Theory and Applications; Smarandache, F., Pramanik, S., Eds.; Pons Editions: Brussels, Belgium, 2016; Volume 1, pp. 27–34, ISBN 978-1-59973-498-9. 40. Sahin, R.; Liu, P. Distance and similarity measure for multiple attribute with single–valued neutrosophic hesitant fuzzy information. In New Trends in Neutrosophic Theory and Applications; Smarandache, F., Pramanik, S., Eds.; Pons Editions: Brussels, Belgium, 2016; Volume 1, pp. 35–54, ISBN 978-1-59973-498-9. 41. Sahin, R.; Liu, P. Correlation coefficient of single-valued neutrosophic hesitant fuzzy sets and its applications in decision making. Neural Comput. Appl. 2017, 28, 1387–1395. [CrossRef] 42. Ali, M.; Deli, I.; Smarandache, F. The theory of neutrosophic cubic sets and their applications in pattern recognition. J. Intell. Fuzzy Syst. 2016, 30, 1957–1963. [CrossRef] 43. Jun, Y.B.; Smarandache, F.; Kim, C.S. Neutrosophic Cubic sets. New Math. Natural Comput. 2017, 13, 41–54. [CrossRef] 44. Banerjee, D.; Giri, B.C.; Pramanik, S.; Smarandache, F. GRA for multi attribute decision making in neutrosophic cubic set environment. Neutrosophic Sets Syst. 2017, 15, 60–69. [CrossRef] Information 2017, 8, 149 21 of 21 45. Deng, J.L. Introduction to grey system theory. J. Grey Syst. 1989, 1, 1–24. 46. Pramanik, S.; Mukhopadhyaya, D. Grey relational analysis based intuitionistic fuzzy multi criteria group decision-making approach for teacher selection in higher education. Int. J. Comput. Appl. 2011, 34, 21–29. 47. Chang, K.H.; Chang, Y.C.; Tasi, I.T. Enhancing FMEA assessment by integrating grey relational analysis and the decision making trial and evaluation laboratory approach. Eng. Fail. Anal. 2013, 31, 211–224. [CrossRef] 48. Pramanik, S.; Dalapati, S.; Alam, S.; Roy, T.K.; Smarandache, F. neutrosophic cubic MCGDM method based on similarity measure. Neutrosophic Sets Syst. 2017, 16, 44–56. [CrossRef] 49. Lu, Z.; Ye, J. Cosine measures of neutrosophic cubic sets for multiple attribute decision-making. Symmetry 2017, 9, 121. [CrossRef] 50. Pramanik, S.; Dey, P.P.; Giri, B.C.; Smarandache, F. An extended TOPSIS for multi-attribute decision making problems with neutrosophic cubic information. Neutrosophic Sets Syst. 2017, 17, 20–28. 51. Zhan, J.; Khan, M.; Gulistan, M. Applications of neutrosophic cubic sets in multi-criteria decision-making. Int. J. Uncertain. Quantif. 2017, 7, 377–394. [CrossRef] 52. Ye, J. Linguistic neutrosophic cubic numbers and their multiple attribute decision-making method. Information 2017, 8, 110. [CrossRef] 53. Gomes, L.; Lima, M. TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Found. Comput. Decis. Sci. 1991, 16, 113–127. 54. Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk. Econ. J. Econ. Soc. 1979, 47, 263–291. [CrossRef] 55. Krohling, R.A.; De Souza, T.T.M. Combining prospect theory and fuzzy numbers to multi-criteria decision making. Expert Syst. Appl. 2012, 39, 11487–11493. [CrossRef] 56. Liu, P.; Teng, F. An extended TODIM method for multiple attribute group decision-making based on 2-dimension uncertain linguistic Variable. Complexity 2014, 21, 20–30. [CrossRef] 57. Tosun, O.; Akyu, G. A fuzzy TODIM approach for the supplier selection problem. Int. J. Comput. Intell. Syst. 2015, 8, 317–329. [CrossRef] 58. Gomes, L.F.A.M.; Machado, M.A.S.; Da Costa, F.F.; Rangel, L.A.D. Criteria interactions in multiple criteria decision aiding: A Choquet formulation for the TODIM method. Procedia Comput. Sci. 2013, 17, 324–331. [CrossRef] 59. Fan, Z.P.; Zhang, X.; Chen, F.D.; Liu, Y. Extended TODIM method for hybrid multiple attribute decision making problems. Knowl. Based Syst. 2013, 42, 40–48. [CrossRef] 60. Krohling, R.A.; Pacheco, A.G.C.; Siviero, A.L.T. IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making. Knowl. Based Syst. 2013, 53, 142–146. [CrossRef] 61. Wang, J.Q. TODIM method with multi-valued neutrosophic set. Control Decis. 2015, 30, 1139–1142. [CrossRef] 62. Zhang, M.; Liu, P.; Shi, L. An extended multiple attribute group decision-making TODIM method based on the neutrosophic numbers. J. Intell. Fuzzy Syst. 2016, 30, 1773–1781. [CrossRef] 63. Ji, P.; Zhang, H.; Wang, J. A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput. Appl. 2016. [CrossRef] 64. Xu, D.S.; Wei, C.; Wei, G.W. TODIM method for single-valued neutrosophic multiple attribute decision making. Information 2017, 8, 125. [CrossRef] 65. Liu, P.D.; Wang, Y.M. Multiple attribute decision-making method based on single valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl. 2014, 25, 2001–2010. [CrossRef] 66. Liu, P.D.; Tang, G.L. Some power generalized aggregation operators based on the interval neutrosophic numbers and their application to decision making. J. Intell. Fuzzy Syst. 2016, 30, 2517–2528. [CrossRef] 67. Liu, P.D.; Wang, Y.M. Interval neutrosophic prioritized OWA operator and its application to multiple attribute decision making. J. Syst. Sci. Complex. 2016, 29, 681–697. [CrossRef] 68. Liu, P.D. The aggregation operators based on Archimedean t-conorm and t-norm for the single valued neutrosophic numbers and their application to decision making. Int. J. Fuzzy Syst. 2016, 18, 849–863. [CrossRef] 69. Liu, P.D.; Chu, Y.C.; Li, Y.W.; Chen, Y.B. Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int. J. Fuzzy Syst. 2014, 16, 242–255. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Information Multidisciplinary Digital Publishing Institute

NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment

Download PDF
21 pages

Loading next page...
 
/lp/multidisciplinary-digital-publishing-institute/nc-todim-based-magdm-under-a-neutrosophic-cubic-set-environment-watFjwLU3B

References (67)

Publisher
Multidisciplinary Digital Publishing Institute
Copyright
© 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
ISSN
2078-2489
DOI
10.3390/info8040149
Publisher site
See Article on Publisher Site

Abstract

information Article NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment 1 , 2 2 2 ID ID Surapati Pramanik * , Shyamal Dalapati , Shariful Alam and Tapan Kumar Roy Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District–North 24 Parganas, West Bengal 743126, India Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah, West Bengal 711103, India; [email protected] (S.D.); [email protected] (S.A.); [email protected] (T.K.R.) * Correspondence: [email protected]; Tel.: +91-947-703-5544 Received: 19 October 2017; Accepted: 14 November 2017; Published: 18 November 2017 Abstract: A neutrosophic cubic set is the hybridization of the concept of a neutrosophic set and an interval neutrosophic set. A 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 has been reported in the current literature. In the present paper, we propose the score and accuracy functions for neutrosophic cubic sets and prove their basic properties. We also develop a strategy for ranking of neutrosophic cubic numbers based on the score and accuracy functions. We firstly develop a TODIM (Tomada de decisao interativa e multicritévio) in the neutrosophic cubic set (NC) environment, which we call the NC-TODIM. We establish a new NC-TODIM strategy for solving multi attribute group decision making (MAGDM) in neutrosophic cubic set environment. We illustrate the proposed NC-TODIM strategy for solving a multi attribute group decision making problem to show the applicability and effectiveness of the developed strategy. We also conduct sensitivity analysis to show the impact of ranking order of the alternatives for different values of the attenuation factor of losses for multi-attribute group decision making strategies. Keywords: neutrosophic cubic set; single valued neutrosophic set; interval neutrosophic set; multi attribute group decision making; TODIM strategy; NC-TODIM 1. Introduction 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 with uncertainty, Zadeh [1] left an important mark to represent and compute with imperfect information by introducing the fuzzy set. The fuzzy set fostered a broad research community, and its impact has also been clearly felt at the application level in MADM [2–4] and MAGDM [5–9]. Atanassov [10] incorporated the non-membership function as an independent component and defined the intuitionistic fuzzy set (IFS) at first to express uncertainty in a more meaningful way. IFSs have been applied in many MADM problems [11–13]. Smarandache [14] proposed the notion of the neutrosophic set (NS) by introducing indeterminacy as an independent component. Wang et al. [15] grounded the concept of the single valued neutrosophic set (SVNS), an instance of the neutrosophic set, to deal with incomplete, inconsistent, and indeterminate information in a realistic way. Wang et al. [16] proposed the interval neutrosophic set (INS) as a subclass of neutrosophic sets in which the values of truth, indeterminacy, and falsity membership degrees are interval numbers. Theoretical development Information 2017, 8, 149; doi:10.3390/info8040149 www.mdpi.com/journal/information Information 2017, 8, 149 2 of 21 and applications of SVNSs and INSs are found in [17–37] for MADM or MAGDM. Some studies on MADM in single valued neutrosophic hesitant fuzzy set environments are found in [38–41]. NS and INS are both capable of handling uncertainty and incomplete information. By fusing NS and INS, Ali et al. [42] proposed the neutrosophic cubic set (NCS) and defined external and internal neutrosophic cubic sets, and established some of their properties. In the same study, Ali et al. [42] proposed an adjustable strategy to NCS-based decision making. Jun et al. [43] also defined NCS by combining NS and INS. In decision making process, the advantage of NCSs is that the decision makers can employ the hybrid information comprising of INSs and SVNSs for evaluating and rating of the alternatives with respect to their predefined attributes. However, there are only a few studies in the literature to deal with MADM and MAGDM in the NCS environment. Banerjee et al. [44] established grey relational analysis (GRA) [45–47] based on the new MADM strategy in the NCS environment. In the same study, Banerjee et al. [44] proposed the Hamming distances for weighted grey relational coefficients and ideal grey relational coefficients, and offered the concept of relative closeness coefficients for presenting the ranking order of the alternatives based on the descending order of their relative closeness coefficients. Similarity measure is an important mathematical tool in decision-making problems. Pramanik et al. [48] at first defined similarity measure for NCSs and proved its basic properties. In the same study, Pramanik et al. [48] developed a new MAGDM strategy in the NCS environment. Lu and Ye [49] proposed cosine measures between NCSs and established their basic properties. In the same study, Lu and Ye [49] proposed three new cosine measures-based MADM strategies under a NCS environment. Due to little research on the operations and application of NCSs, Pramanik et al. [50] proposed several operational rules on NCSs, and defined Euclidean distances and arithmetic average operators of NCSs. In the same study, Pramanik et al. [50] also employed the information entropy scheme to calculate the unknown weights of the attributes, and developed a new extended TOPSIS strategy for MADM under the NCS environment. Zhan et al. [51] proposed a new algorithm for multi-criteria decision making (MCDM) in an NCS environment based on a weighted average operator and a weighted geometric operator. Ye [52] established the concept of a linguistic neutrosophic cubic number (LNCN). In the same study, Ye [52] developed a new MADM strategy based on a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator under a linguistic NCS environment. In the literature, there are only six strategies [44,48–52] for MADM and MAGDM in NCS environment. However, we say that none of them is generally superior to all others. So, new strategies for MADM and MAGDM should be explored under the NCS environment for the development of neutrosophic studies. TODIM (an acronym in Portuguese for interactive multi-criteria decision making strategy named Tomada de decisao interativa e multicritévio) is an important MADM strategy, since it considers the decision makers’ bounded rationality. Firstly, Gomes and Lima [53] introduced the TODIM strategy based on prospect theory [54]. Krohling and Souza [55] defined the fuzzy TODIM strategy to solve MCDM problems. Several researchers applied the TODIM strategy in various fuzzy MADM or MAGDM problems [56–58]. Fan et al. [59] introduced the extended TODIM strategy to deal with the hybrid MADM problems. Krohling et al. [60] extended the TODIM strategy from fuzzy environment to intuitionistic fuzzy environment to process the intuitionistic fuzzy information. Wang [61] introduced TODIM strategy into multi-valued neutrosophic set environment. Zhang et al. [62] proposed the TODIM strategy for MAGDM problems under a neutrosophic number environment. Ji et al. [63] proposed the TODIM strategy under a multi-valued neutrosophic environment and employed it to solve personal selection problems. In 2017, Xu et al. [64] developed the TODIM strategy in a single valued neutrosophic setting and extended it into interval neutrosophic setting. Neutrosophic TODIM [64] is capable of dealing with only single-valued neutrosophic information or interval neutrosophic information. Information 2017, 8, 149 3 of 21 NCS can be used to express the interval neutrosophic information and neutrosophic information in the process of MADGM. It seems that TODIM in NCSs has an enormous chance of success to deal with group decision making problems. In the NCS environment, the TODIM strategy is yet to appear. Motivated by these, we initiated the study of TODIM in the NCS environment, which we call NC-TODIM. However, NCSs comprise of hybrid information of INSs and SVNSs simultaneously, which are more flexible and elegant for expressing neutrosophic cubic information. To apply NCSs to MADGM problems, we introduce some basic operations of neutrosophic cubic (NC) numbers and the score, and accuracy functions of NC numbers, and the ranking strategy of NC numbers. In this paper we develop a TODIM strategy (for short, NC-TODIM strategy) for MAGDM in the NCS environment. The proposed NC-TODIM strategy was proven to be capable of successfully dealing with MAGDM problems by solving an illustrative example. What is more, a comparative analysis ensured the feasibility of the proposed NC-TODIM strategy. The remainder of the paper is divided into seven sections that are organized as follows: Section 2 presents some basic definitions of NS, SVNS, INS, and NCS. Section 3 presents comparison strategy of two NC-numbers. Section 4 is devoted to present the proposed NC-TODIM strategy. Section 5 presents an illustrative numerical example of MAGDM in the NCS environment. Section 6 is devoted to analyzing the ranking order with different values of attenuation factors of losses. Section 7 presents a comparative analysis between the developed strategy and other existing strategies in the NCS environment. Section 8 presents the conclusion and the future scope of research. 2. Preliminaries In this section, we review some basic definitions which are important to develop the paper. Definition 1. [14] NS. Let U be a space of points (objects) with a generic element in U denoted by u, i.e., u 2 U. A neutrosophic set R in U is characterized by truth-membership function t , indeterminacy-membership function i , and falsity-membership function f , where t , i , f are the functions from U to ] 0, 1 [ i.e., t , R R R R R R i , f : U!] 0, 1 [ that means t (u), i (u), f (u) are the real standard or non-standard subset of ] 0, R R R R R 1 [. The neutrosophic set can be expressed as R = {<u; (t (u), i (u), f (u))>: 8u2U}. Since t (u), i (u), R R R R R + + f (u) are the subset of ] 0, 1 [, there the sum (t (u) + i (u) + f (u)) lies between 0 and 3 , where 0 = 0 " R R R R and 3 = 3 + ", " > 0. Example 1. Suppose that U = {u , u , u , . . .} is the universal set. Let R be any neutrosophic set in U. 1 2 3 1 Then R expressed as R = {<u ; (0.6, 0.3, 0.4)>: u 2U}. 1 1 1 1 Definition 2. [15] SVNS. Let U be a space of points (objects) with a generic element in U denoted by u. A single valued neutrosophic set H in U is expressed by H = {<u, (t (u), i (u), f (u))>; u2U}, where t (u), i (u), H H H H H f (u)2[0, 1]. Therefore for each u2U, t (u), i (u), f (u)2[0, 1] and 0  t (u) + i (u) + f (u)  3. H H H H H H H Definition 3. [16] INS. Let G be a non-empty set. An interval neutrosophic set G in G is characterized by truth-membership function t , the indeterminacy membership function i and falsity membership function f . e e e G G G For each g2G, t (g), i (g), f (g)  [0, 1] and G defined as e e e G G G + + + G = f< g; [t (g), t (g)], [i (g), i (g), ], [f (g), f (g)] : 8g 2 Gg. e e e e e e G G G G G G Here, + + + + t (g), t (g), i (g), i (g), f (g), f (g) : G !] 0, 1 [, e e e e e e G G G G G G and + + + + 0  sup t (g) + sup i (g) + sup f (g)  3 . e e G G G Information 2017, 8, 149 4 of 21 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), t (g), i (g), i (g), f (g), f (g) : G !] 0, 1 [. e e e e e e G G G G G G Here, + + + t (g), t (g), i (g), i (g), f (g), f (g) : G ! [0, 1]. e e e e e e G G G G G G e e Example 2. Suppose that G = {g , g , g , . . ., g } is a non-empty set. Let G be an INS. Then G is n 1 1 1 2 3 expressed as G = f< g ; [0.39, 0.47], [0.17, 0.43], [0.18, 0.36] : g 2 Gg. 1 1 e e Definition 4. [42,43] NCS. A NCS in a non-empty set G is defined as © = {<g; G(g), R(g)>: 8g2G}, where G and R are the INS and NS in G respectively. NCS can be presented as an order pair © = <G, R>, then we call it as a neutrosophic cubic (NC) number. Example 3. Suppose that G = {g , g , g , . . ., g } is a non-empty set. Let © be any NC-number. Then © n 1 1 1 2 3 can be expressed as © = {<g ; [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4)>: g 2G}. 1 1 Some operations of NC-numbers: i. Union of any two NC-numbers e e Let © =< G , R > and © =< G , R > be any two NC-numbers in a non-empty set G. 1 1 1 2 2 2 Then the union of © and © denoted by © [ © and defined as 1 2 1 2 e e © [ © =< G (g) [ G (g), R (g) [ R (g)8g 2 G >, 1 2 1 1 1 2 + + + e e where G (g) [ G (g) = {<g, [max {t (g), t (g)},max {t (g), t (g)}], [max {i (g), i (g)}, max {i (g), 1 1 e e e e e e e G G G1 G G G G 1 2 2 1 2 1 + + + i (g)}], [min {f (g), f (g)}, min {f (g), f (g)}]>: g2G} and R (g) [ R (g) = {<g, max {t (g), t (g)}, 1 2 R R e e e e e 1 G G G G G 2 1 2 1 2 max {i (g), i (g)}, min {f (g), f (g)}>:8g2U}. R R R R 2 2 1 1 Example 4. Let © and © be two NC-numbers in G presented as follows: © =< [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4) > and © =< [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6) > . Then © [ © =< [0.56, 0.70], [0.27, 0.43], [0.15, 0.26], (0.7, 0.3, 0.4) > . 1 2 ii. Intersection of any two NC-numbers Intersection of two NC-numbers denoted and defined as follows: e e © \ © =<G (g) \ G (g), R (g) \ R (g)8g 2 G >, 1 2 1 1 1 2 + + + e e where G (g) \ G (g) = {<g, [min {t (g), t (g)},min {t (g), t (g)}], [min {i (g), i (g)}, min {i (g), 1 1 e e e e e e e G G G1 G G G G 1 2 2 1 2 1 + + + i (g)}], [max {f (g), f (g)}, max {f (g), f (g)}]>: g2G} and R (g) \ R (g)= {<g, min {t (g), t (g)}, 1 2 R R e e e e e 1 2 G G G G G 2 1 2 1 2 min {i (g),i (g)}, max {f (g), f (g)}>:8g2U}. R R R R 1 2 1 2 Information 2017, 8, 149 5 of 21 Example 5. Let © and © be any two NC-numbers in G presented as follows: 1 2 © =< [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5) > and © =< [0.67, 0.75], [0.22, 0.44], [0.17, 0.21], (0.8, 0.4, 0.4) > . Then © \ © =< [0.45, 0.57], [0.22, 0.33], [0.18, 0.46], (0.7, 0.3, 0.4) > . 1 2 iii. Compliment of a NC-number e e Let © =< G , R > be a NCS in G. Then, the compliment of © =< G , R > denoted by 1 1 1 1 1 1 c c © = {<g, G (g), R (g)>: 8g2G}. 1 1 c + + + Here, G = {<g, [t (g), t (g)], [i (g), i (g)], [f (g), f (g)]>: 8g2G}, where, c c c c e c e c e e e e G G G G G G 1 1 1 1 1 1 + + + + t (g) = f (g), t (g) = f (g), i (g) = {1} i (g), i (g) = {1} i (g), f (g) = t (g), e c e e c e e c e e c e e c e G G G G G G G G G G 1 1 1 1 1 1 1 1 1 1 + + c + c c f (g) = f (g) and t (g) = f (g), i (g) = {1 } i (g), f (g) = t (g). R R´ R R R 1 R e c e 1 1 1 G G 1 1 1 1 Example 6. Assume that © be any NC-number in G in the form: © =< [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5) > Then compliment of © is obtained as © =< [0.18, 0.46], [0.73, 0.67], [0.45, 0.57], (0.5, 0.7, 0.7) > . Definition 5. Score function. Let © be a NC-number in a non-empty set G. Then, a score function of © , 1 1 denoted by Sc (© ) is defined as: 1 2 + a + a 2b 2b c c 1 + a 2b c 1 2 1 2 1 2 Sc (© ) = [( ) + ( )] (1) 2 4 2 where, © = <[a , a ], [b , b ], [c , c ], (a, b, c) > and Sc (© )2[–1, 1]. 1 1 2 1 2 1 2 Proposition 1. Score function of two NC-numbers lies between 1 to 1. Proof. Using the definition of INS and NS, we have all a , a , b , b , c , c , a, b, and c [0, 1]. 1 2 1 2 1 2 Since, 0  a  1, 0  a  1 1 2 0  a + a  2, (2) 1 2 ) 2  2 + a + a  4 1 2 0  b  1 ) 0  2b  2 and 0  b  1 ) 0  2b  2 2 2 1 1 ) 2  2b  0 (3) ) 2  2b  0 ) 4  2b 2b  0 1 2 0  c  1 ) 1  c  0 1 1 0  c  1 ) 1  c  0 (4) 2 2 ) 2  c c  0 1 2 Information 2017, 8, 149 6 of 21 Adding Equations (2)–(4), we obtain ) 4  2 + a + a 2b 2b c c  4, 1 2 1 2 1 2 (5) 2 + a + a 2b 2b c c 1 2 1 2 1 2 ) 1   1 Again, 0  a  1) 1  1 + a  2 , (6) 0  b  1 ) 0  2b  2, 0  c  1, (7) ) 0  2b + c  3, ) 3  2b c  0 Adding (6) and (7), we obtain 2  1 + a 2b c  2, (8) 1 + a 2b c ) 1   1 Adding (5) and (8) and dividing by 2, we obtain 1 2 + a + a 2b 2b c c 1 + a 2b c 1 2 1 2 1 2 1  [( ) + ( )]  1 2 4 2 Sc (© ) 2 [1, 1]. Hence the proof is complete. Example 7. Let © and © be two NC-numbers in G, presented as follows: 1 2 © =< [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4) > and © =< [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6) > . Then, by applying Definition 5, we obtain Sc (© ) = 0.01 and Sc (© ) = 0.07, In this case, we can say 1 2 that © > © . Definition 6. Accuracy function. Let © be a NC-number in a non-empty set G, an accuracy function of © is 1 1 defined as: 1 1 Ac(© ) = /2 [ /2(a + a b (1 a ) b (1 a ) c (1 b ) c (1 b ) + a b(1 a) c(1 b)] (9) 1 1 2 2 2 1 1 2 1 1 2 Here, Ac (© )2[–1, 1]. When the value of Ac (© ) increases, we say that the degree of accuracy of the NC-number © increases. 1 1 Proposition 2. Accuracy function of two NC-numbers lies between 1 to 1. Proof. The values of accuracy function depend upon f (a + a b (1 a ) b (1 a ) c (1 b ) c (1 b )) andf a b(1 a) c(1 b)g 1 2 2 2 1 1 2 1 1 2 The values of f (a + a b (1 a ) b (1 a ) c (1 b ) c (1 b ))g 1 2 2 2 1 1 2 1 1 2 2 Information 2017, 8, 149 7 of 21 and f a b(1 a) c(1 b)g lie between 1 to 1 from [37]. Thus, 1  Ac (© )  1. Hence the proof is completed. Example 8. Let © and © be two NC-numbers in G presented as follows: © =< [0.41, 0.52], [0.10, 0.18], [0.06, 0.17], (0.48, 0.11, 0.11) > and © =< [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > . Then, by applying Definition 6, we obtain Ac (© ) = 0.14 and Ac (© ) = 0.30. In this case, we can say 1 2 that alternative © is better than © . 2 1 With respect to the score function Sc and the accuracy function Ac, a strategy for comparing NC-numbers can be defined as follows: 3. Comparison Strategy of Two NC-Numbers Let © and © be any two NC-numbers. Then we define comparison strategy as follows: 1 2 i. If Sc(© ) > Sc(© ), then © > © . (10) 1 2 1 2 ii. If Sc(© ) = Sc(© ) and Ac(© ) > Ac(© ), then © > © . (11) 1 2 1 2 1 2 iii. If Sc(© ) = Sc(© ) and Ac(© ) = Ac(© ), then © = © . (12) 1 2 1 2 1 2 Example 9. Let © and © be two NC-numbers in G, presented as follows: 1 2 © =< [0.23, 0.29], [0.37, 0.46], [0.34, 0.42], (0.26, 0.26, 0.26) > and © =< [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > . Then, applying Definition 5, we obtain Sc (© ) = 0.13 and Sc (© ) = 0.13. Applying Definition 6, 1 2 we obtain Ac(© ) =0.20 and Ac (© ) =0.18. In this case, we say that alternative © > © . (Score values 1 2 1 and Accuracy values taking correct up to two decimal places). Definition 7. Let © and © be any two NC-numbers, then the distance between them is defined by 1 2 ¶(© , © ) = [ a d + a d + b e + b e + c f + c f + a d + b e + c f ] (13) j j j j j j j j j j j j j j j j j j 1 2 1 1 2 2 1 1 2 2 1 1 2 2 where, © = <[a , a ], [b , b ], [c , c ], (a, b, c)> and © = <[d , d ], [e , e ], [f , f ], (d, e, f)>. 1 1 2 1 2 1 2 2 1 2 1 2 1 2 Example 10. Let © and © be two NC-numbers in G presented as follows: 1 2 © =< [0.66, 0.75], [0.25, 0.32], [0.17, 0.34], (0.53, 0.17, 0.22) > 1 Information 2017, 8, 149 8 of 22 ∂ (©1, ©2) = [ a − d + a − d + b − e + b − e + c − f + c − f + a − d + b − e + c − f ] 1 1 2 2 1 1 2 2 1 1 2 2 (13) where, ©1 = <[a1, a2], [b1, b2], [c1, c2], (a, b, c)> and ©2 = <[d1, d2], [e1, e2], [f1, f2], (d, e, f)>. Example 10. Let ©1 and ©2 be two NC-numbers in G presented as follows: = <[0.66, 0.75], [0.25, 0.32], [0.17, 0.34], (0.53, 0.17, 0.22)> and = < [0.35, 0.55], [0.12, 0.25], [0.12, 0.20], (0.60, 0.23, 0.43)> © 2 Then, applying Definition 7, we obtain ∂ (©1, ©2) = 0.12. Information 2017, 8, 149 8 of 21 − + − + − + {<[ , ],[ , ],[ , ],(t,i,f ) >} Definition 8. Let ©ij = be any neutrosophic cubic value. ©ij used to evaluate t t i i f f ij ij ij ij ij ij and i-th alternative with respect to j-th criterion. The normalized form of ©ij is defined as follows: © =< [0.35, 0.55], [0.12, 0.25], [0.12, 0.20], (0.60, 0.23, 0.43) > − + t t ⊗ ij ij ©ij ={< [ , ] , Then, applying Definition 7, we obtain ¶ (© , © ) 1= 0.12. 1 1 2 m m − 2 + 2 − 2 + 2 2 2 ( ( ) + ( ) ) ( ( ) + ( ) )  t t  t t ij ij ij ij i=1 i=1 + + Definition 8. Let © = f< [t , t ], [i , i ], [f , f ], (t, i, f) >g be any neutrosophic cubic value. ij ij ij ij ij ij ij © used to evaluate i-th alternative with respect to j-th criterion. The normalized form of © is defined as follows: − + ij ij i i ij ij [ , ] , 1 1 m m − 2 + t 2 − 2 + 2 t 2 2 ij ij ( ( ) + ( ) ) ( ( ) + ( ) )   i i i i ij ij ij ij © = f< [ , ], m m 1 1 ij i=1 2 i2 =1 2 2 + + 2 2 ( S (t ) +(t ) ) ( S (t ) +(t ) ) ij ij ij ij i=1 i=1 (14) i i − + ij ij f f [ ij , ij ], m m 1 1 2 2 2 2 [ + , + ] 2 2 ( S (i ) +(i ) ) ( S (i ) +(i ) ) 1 1 ij ij ij ij m m i=1 i=1 − 2 + 2 − 2 + 2 2 2 (14) ( ( ) + ( ) ) ( ( ) + ( ) )  f f  f + f ij ij ij ij f f ij ij i=1 i=1 [ , ] m 1 m 1 2 2 2 2 + + 2 2 ( S (f ) +(f ) ) ( S (f ) +(f ) ) ij ij ij ij i=1 i=1 ij iij f ij t i f ij ij ij [ , , ] >}. [ , , ] >g. 1 1 1 m m m 1 1 1 m m m 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 ( S (t ) +(i ) +(f ) ) 2 ( S (t ) +(i ) +(f ) ) 2 ( S (t ) +(i ) +(f ) ) 2 ij ij ij ij ij ij ij ij ij ( ( ) + ( ) + ( ) ) ( ( ) + ( ) + ( ) ) ( ( ) + ( ) + ( ) )  t i f  t i f  t i f i=1 ij ij ij i=1 ij ij ij i=1 ij ij ij i=1 i=1 i=1 A conceptual model of the evolution of the neutrosophic cubic set is shown in Figure 1. A conceptual model of the evolution of the neutrosophic cubic set is shown in Figure 1. Figure 1. Evolution of the neutrosophic cubic set. Figure 1. Evolution of the neutrosophic cubic set. 4. NC-TODIM Based MAGDM under a NCS Environment 4. NC-TODIM Based MAGDM under a NCS Environment Assume that A = {A , A , . . . , A } (m  2) and C = {C , C , . . . , C } (n  2) are the discrete set m n 1 2 1 2 of alternatives and attributes respectively. W = {W , W , . . . , W } is the weight vector of attributes 1 2 n C (j = 1, 2, . . . , n), where W > 0 and S W = 1. Let E = {E , E , . . . , E } be the set of decision makers j j j 1 2 r j=1 and g = fg , g , . . . , g g be the weight vector of decision makers, where g > 0 and S g = 1. 1 2 r k k k=1 NC-TODIM Strategy Now, we describe the NC-TODIM strategy to solve the MAGDM problems with NC-numbers. The NC-TODIM strategy consists of the following steps: Information 2017, 8, 149 9 of 21 Step 1. Formulate the decision matrix k k k k Assume that M = (© ) be the decision matrix, where © = <G , R > is the rating value ij ij ij ij m n provided by the k-th (E decision maker for alternative A , with respect to attribute C . The matrix i j k) form of M is presented as: 0 1 C C . . . C 1 2 n k k k B C A © © . . . © B 11 12 1n C B C k k k k M = A © © © (15) B C 21 22 2n B C @ A . . . . . . k k k A © © . . . © m1 m2 mnj Step 2. Normalize the decision matrix The MAGDM problem generally consists of cost criteria and benefit criteria. So, the decision matrix needs to be normalized. For cost criterion C , we use the Definition 8 to normalize the decision matrix (Equation (15)) provided by the decision makers. For benefit criterion C we don’t need to normalize the decision matrix. When C is a cost criterion, the normalized form of decision matrix (see Equation (15)) is presented below. 0 1 C C . . . C 1 2 n B C A © © . . . © B C 11 12 1n B C M = A © © © (16) B C 21 22 2n B C @ A . . . . . . A © © . . . © m1 m2 mnj Here © is the normalized form of the NC-number. ij Step 3. Determine the relative weight of each criterion The relative weight W of each criterion is obtained by the following equation. ch W = (17) ch where, W = max {W , W , . . . , W }. h 1 2 Step 4. Calculate score values Using Equation (1), calculate the score value Sc (© ) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) of © if C ij ij is a cost criterion. Using Equation (1), calculate the score value Sc ((c) ) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) ij of © if C is a benefit criterion. ij Step 5. Calculate accuracy values Using Equation (9), calculate the accuracy value Ac (© ) (I = 1, 2, . . . , m; j= 1, 2, . . . , n) of ij k k © if C is a cost criterion. Using Equation (9), calculate the accuracy value Ac (© ) (I = 1, 2, . . . , m; j ij ij j = 1, 2, . . . , n) of © if C is a benefit criterion. ij Step 6. Formulate the dominance matrix Calculate the dominance of each alternative A over each alternative A with respect to the criteria i j C (C C , . . . , C ), of the k-th decision maker E by the following Equations (18) and (19). 1, 2 n k Information 2017, 8, 149 10 of 21 (For cost criteria) Ch Y (A , A ) = ( ¶(© , © ) , if © > © i j n c ic jc ic jc S W ch > c=1 = 0, if © = © (18) ic jc S W > ch 1 c=1 k ; = ( ¶(© , © ) , if © < © ic jc ic jc Ch (For benefit criteria) W k k k k k Ch Y (A , A ) = ( ¶(© , © ) , if © > © > i j n c ic jc ic jc S W ch c=1 k k = 0, if © = © (19) ic jc S W > ch 1 c=1 k k k k = ( ¶(© , , © ) , if © < © W ic jc ic jc Ch where, parameter represents the attenuation factor of losses and must be positive. Step 7. Formulate the individual overall dominance matrix Using Equation (20), calculate the individual total dominance matrix of each alternative A over each alternative A under the criterion C . j j k k = (A , A ) = S Y (A , A ) (20) i j c i j c=1 Step 8. Aggregate the dominance matrix Using Equation (21), calculate the collective overall dominance of alternative A over each alternative A . (A , A ) = S g  (A , A ) (21) i j k i j k=1 Step 9. Calculate global values We present the global value of each alternative as follows: n n S  (A , A ) min ( S  (A , A )) i j i j j=1 1 i m j=1 W = (22) n n max ( S  (A , A )) min ( S  (A , A )) i j i j 1 i m j=1 1 i m j=1 Step 10. Rank the priority Sorting the values of W provides the rank of each alternative. A set of alternatives can be preference-ranked according to the descending order of W . The highest global value corresponds to the best alternative. A conceptual model of the NC-TODIM strategy is shown in Figure 2. Information 2017, 8, 149 11 of 22 Step 10. Rank the priority Sorting the values of Ω provides the rank of each alternative. A set of alternatives can be preference-ranked according to the descending order of Ω . The highest global value corresponds Information 2017, 8, 149 11 of 21 to the best alternative. A conceptual model of the NC-TODIM strategy is shown in Figure 2. Figure 2. A flow chart of the proposed neutrosophic cubic set (NC)-TODIM strategy. Figure 2. A flow chart of the proposed neutrosophic cubic set (NC)-TODIM strategy. 5. Illustrative Example 5. Illustrative Example In this section, a MAGDM problem is adapted from the study [18] under the NCS environment. In this section, An investme a MAGDM nt company pr wants to se oblem is lect the best al adapted tfr ernat om ive the amostudy ng the set [18 of ]fea under sible altthe ernatNCS ives. environment. The feasible alternatives are An investment company wants to select the best alternative among the set of feasible alternatives. 1. Car company (A1) The feasible alternatives are 2. Food company (A2) 3. Computer company (A3) 1. Car company (A ) 4. Arms company (A4). 2. Food company (A ) The best alternative is selected based on the following criteria: 3. Computer company (A ) 1. Risk analysis (C1) 2. Growth analysis (C2) 4. Arms company (A ). 3. Environmental impact analysis (C3). The best alternative is selected based on the following criteria: An investment company forms a panel of three decision makers {E1, E2, E3} who evaluate four alternatives in decision making process. The weight vector of attributes and decision makers are 1. Risk analysis (C ) 1 T T considered as W = (0.4, 0.35, 0.25) γ = (0.32, 0.33, 0.35) respectively. 2. Growth analysis (C ) The proposed strategy is presented using the following steps: 3. Environmental impact analysis (C ). An investment company forms a panel of three decision makers {E , E , E } who evaluate four 1 2 3 alternatives in decision making process. The weight vector of attributes and decision makers are T T considered as W = (0.4, 0.35, 0.25) g = (0.32, 0.33, 0.35) respectively. The proposed strategy is presented using the following steps: Step 1. Formulate the decision matrix Formulate the decision matrices M (k = 1, 2, 3) using the rating values of alternatives with respect to three criteria provided by the three decision makers in terms of NC-numbers. Assume k k that the NC-numbers © = <G , R > present the rating value provided by the decision maker E for ij ij k ij alternative A with respect to attribute C . Using these rating values © (k = 1, 2, 3; i = 1, 2, 3, 4; j = 1, i j ij k k 2, 3), three decision matrices M = (© ) (k = 1, 2, 3) are constructed (see Equations (23)–(25)). ij 43 Decision matrix for E 0 1 C C C 1 2 3 B C A < [0.41, 0.52], [0.10, 0.18], [0.06, 0.17], (0.48, 0.11, 0.11) > < [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > < [0.22, 0.27], [0.41, 0.52], [0.41, 0.52], (0.31, 0.31, 0.31) > B 1 C B C M = BA < [0.35, 0.46], [0.18, 0.27], [0.17, 0.34], (0.43, 0.16, 0.21) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.57, 0.12, 0.12) >C (23) B C @ A A < [0.23, 0.29], [0.36, 0.45], [0.34, 0.42], (0.26, 0.26, 0.26) > < [0.34, 0.45], [0.20, 0.30], [0.19, 0.39], (0.44, 0.16, 0.22) > < [0.22, 0.27], [0.41, 0.52], [0.41, 0.52], (0.31, 0.31, 0.31) > A < [0.17, 0.23], [0.45, 0.55], [0.42, 0.59], (0.21, 0.32, 0.37) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.57, 0.12, 0.12) > Decision matrix for E 0 1 C C C 1 2 3 B C A < [0.17, 0.23], [0.46, 0.55], [0.42, 0.59], (0.21, 0.32, 0.37) > < [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > < [0.34, 0.43], [0.13, 0.27], [0.13, 0.27], (0.49, 0.11, 0.11) > B 1 C B C M = BA < [0.23, 0.29], [0.37, 0.46], [0.34, 0.42], (0.26, 0.26, 0.26) > < [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > < [0.34, 0.43], [0.13, 0.27], [0.13, 0.27], (0.49, 0.11, 0.11) >C (24) B C @ A A3 < [0.41, 0.52], [0.10, 0.18], [0.10, 0.17], (0.48, 0.11, 0.11) > < [0.44, 0.57], [0.10, 0.17], [0.10, 0.17], (0.51, 0.11, 0.11) > < [0.19, 0.24], [0.53, 0.67], [0.53, 0.67], (0.27, 0.27, 0.27) > A < [0.35, 0.46], [0.20, 0.28], [0.17, 0.34], (0.42, 0.16, 0.21) > < [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > < [0.34, 0.43], [0.13, 0.27], [0.13, 0.27], (0.49, 0.11, 0.11) > 4 Information 2017, 8, 149 12 of 21 Decision matrix for E 0 1 C C C 1 2 3 B C A < [0.22, 0.27], [0.42, 0.52], [0.42, 0.52], (0.28, 0.28, 0.28) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.41, 0.52], [0.10, 0.18], [0.10, 0.17], (0.48, 0.11, 0.11) > B 1 C B C M = BA < [0.22, 0.27], [0.42, 0.52], [0.42, 0.52], (0.28, 0.28, 0.28) > < [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > < [0.23, 0.29], [0.36, 0.45], [0.34, 0.42], (0.26, 0.26, 0.26) >C (25) B C @ A A < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.50, 0.11, 0.11) > < [0.34, 0.45], [0.20, 0.30], [0.19, 0.39], (0.44, 0.16, 0.22) > < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.50, 0.11, 0.11) > A < [0.38, 0.49], [0.10, 0.21], [0.10, 0.21], (0.50, 0.11, 0.11) > < [0.22, 0.28], [0.40, 0.50], [0.39, 0.48], (0.28, 0.28, 0.28) > < [0.17, 0.23], [0.45, 0.54], [0.42, 0.59], (0.21, 0.32, 0.37) > Step 2. Normalize the decision matrix Since all the criteria are benefit type, we do not need to normalize the decision matrix. Step 3. Determine the relative weight of each criterion Using Equation (17), we obtain the relative weight vector W of criteria as follows: ch W = (1, 0.875, 0.625) . ch Step 4. Calculate score values The score values of each alternative relative to each criterion obtained by Equation (1) are presented in the Tables 1–3. Table 1. Score values for M . C C C 1 2 2 A 0.56 0.54 0.06 A 0.40 0.09 0.54 A 0.50 0.38 0.06 A 0.03 0.09 0.54 Table 2. Score values for M . C C C 1 2 2 A 0.03 0.13 0.49 A 0.13 0.13 0.49 A 0.56 0.60 0.04 A 0.39 0.13 0.49 Table 3. Score values for M . C C C 1 2 2 A 0.07 0.09 0.56 A 0.07 0.52 0.13 A 0.51 0.37 0.39 A 0.51 0.09 0.03 Step 5. Calculate accuracy values The accuracy values of each alternative relative to each criterion obtained by Equation (9) are presented in Tables 4–6. Table 4. Accuracy values for M . C C C 1 2 2 A 0.14 0.30 0.24 A 0.12 0.23 0.32 A 0.20 0.09 0.24 A 0.38 0.23 0.32 4 Information 2017, 8, 149 13 of 21 Table 5. Accuracy values for M . C C C 1 2 2 A 0.38 0.18 0.21 A 0.20 0.18 0.21 A 0.14 0.36 0.21 A 0.12 0.18 0.21 Table 6. Accuracy values for M . C C C 1 2 2 A 0.24 0.23 0.41 A 0.24 0.30 0.20 A 0.26 0.09 0.12 A 0.26 0.23 0.38 Step 6. Formulate the dominance matrix Using Equation (19), we construct dominance matrix for = 1. The dominance matrices are represented in matrix form (See Equations (26)–(34)). 1 1 The dominance matrix Y , the dominance matrix Y 1 2 0 1 A A A A 1 2 3 4 B C A 0 0.18 0.30 0.35 B 1 C B C Y = (26) BA 0.46 0 0.58 0.30C B C @ A A 0.74 0.23 0 0.19 A 0.88 0.74 0.47 0 0 1 A A A A 1 2 3 4 B C A 0 0.29 0.18 0.28 B 1 C B C Y = A 0.82 0 0.69 0 (27) B C B C @ A A 0.51 0.24 0 0.29 A 0.81 0 0.65 0 1 2 The dominance matrix Y , the dominance matrix Y 3 1 0 1 A A A A 1 2 3 4 B C A 0 1 0 1 B 1 C B C Y = A 0.25 0 0.26 0 (28) B C B C @ A A 0 1 0 1 A 0.25 0 0.26 0 0 1 A A A A 1 2 3 4 B C A 0 0.46 0.88 0.74 B 1 C B C Y =BA 0.18 0 0.75 0.58C (29) B C @ A A 0.35 0.09 0 0.04 A 0.30 0.23 0.19 0 4 Information 2017, 8, 149 14 of 21 2 2 The dominance matrix Y , the dominance matrix Y 2 3 0 1 A A A A 1 2 3 4 B C A 0 0 0.84 0 B 1 C B C Y =BA 0 0 0.84 0 C (30) B C @ A A 0.29 0.29 0 0.29 A 0 0 0.84 0 0 1 A A A A 1 2 3 4 B C A 0 0 0.26 0 B 1 C B C Y =BA 0 0 0.26 0 C (31) 3 2 B C @A 1 1 0 1A A 0 0 0.26 0 3 3 The dominance matrix Y , the dominance matrix Y 1 2 0 1 A A A A 2 3 1 4 B C A 0 0 0.78 0.78 B 1 C B C Y =BA 0 0 0.78 0.78C (32) B C @ A A 0.31 0.31 0 0 A 0.31 0.31 0 0 0 1 A A A A 1 2 3 4 B C A 0 0.83 0.65 0 B 1 C B C Y = (33) B A 0.29 0 0.18 0.29C B C @ A A 0.23 0.51 0 0.23 A 0 0.83 0.65 0 The dominance matrix Y 0 1 A A A A 1 2 3 4 B C A 0 0.94 0.59 1.1 B 1 C B C Y = A 0.23 0 0.73 0.15 (34) B C B C @ A A 0.59 0.18 0 0.23 A 1.1 0.58 0.94 0 Step 7. Formulate the individual overall dominance matrix The individual overall dominance matrix is calculated by the Equation (20) and the dominance matrices are represented in matrix form (see Equations (35)–(37)). First decision maker ’s overall dominance matrix 0 1 A A A A 1 2 3 4 B C A 0 0.53 0.47 0.37 B 1 C B C = BA 1 0 1 0.30 C (35) B C @A 1.3 0.53 0 0.52 A A 1.5 0.74 0.86 0 4 Information 2017, 8, 149 15 of 21 Second decision maker ’s overall dominance matrix 0 1 A A A A 1 2 3 4 B C A 0 0.46 1.5 0.74 B 1 C B C = BA 0.18 0 1.3 0.58C (36) B C @ A A 0.36 0.62 0 0.67 A 0.30 0.23 0.39 0 Third decision maker ’s overall dominance matrix 0 1 A A A A 1 2 3 4 B C A 0 1.8 2 1.9 B 1 C B C =BA 0.52 0 1.3 0.34C (37) B C @ A A 0.05 0.02 0 0.46 A 0.79 1.1 1.6 0 Step 8. Aggregate the dominance matrix Using Equation (21), the aggregate dominance matrix  is constructed (see Equation (38)) as follows: 0 1 A A A A 1 2 3 4 B C A 0 0.94 1.1 0.53 B 1 C B C = A 0.10 0 1.23 0.22 (38) B C B C @ A A 0.54 0.38 0 0.23 A 0.64 0.55 0.96 0 Step 9. Calculate global values Using Equation (22), we calculate the values of W (i = 1, 2, 3, 4) and represented in Table 7. Table 7. Global values of alternatives. A A A A A i 1 2 3 4 W 0.49 0.61 1 0 Step 10. Rank the priority Since W > W > W > W , alternatives are then preference ranked as follows: A > A > A > A . 3 2 1 4 3 2 1 4 Hence A is the best alternative. From the illustrative example, we see that the proposed NC-TODIM strategy is more suitable for real scientific and engineering applications because it can handle hybrid information consisting of INS and SVNS information simultaneously to cope with indeterminate and inconsistent information. Thus, NC-TODIM extends the existing decision-making strategies and provides a sophisticated mathematical tool for decision makers. 6. Rank of Alternatives with Different Values of Table 8 shows that the ranking order of alternatives depends on the values of the attenuation factor, which reflects the importance of the attenuation factor in the NC-TODIM strategy. Information 2017, 8, 149 16 of 21 Table 8. Global values and ranking of alternatives for different values of . Values of Global Values of Alternative (W ) Rank Order of A i i Information 2017, 8, 149 17 of 22 W = 0, W = 0.89, W = 1, W = 0.46 1 2 3 4 0.5 A > A > A > A 3 2 4 1 W > W > W > W 3 2 4 1 Ω Ω Ω = 0, = 0.72, Ω = 1, = 0.44 W = 0.49, 1 W = 0.61, 2 W = 1, W3 = 0 4 1 2 3 4 1 A > A > A > A 1.5 3 2 1A3 > A 4 2 > A4 > A1 W > W > W > W 3 2 1 4 Ω Ω Ω Ω > > > 3 2 4 1 W = 0, W = 0.72, W = 1, W = 0.44 1 2 3 4 1.5 A > A > A > A 3 2 4 1 Ω Ω Ω = 0, = 1, Ω = 0.81, = 0.38 W > W > W > W 1 2 3 4 3 2 4 1 2 A2 > A3 > A4 > A1 W = 0, W = 1, W = 0.81, W = 0.38 1 2 3 4 Ω Ω Ω > Ω > > 2 A > A > A > A 2 3 4 1 2 3 4 1 W > W >W > W 2 3 4 1 Ω Ω Ω W = 0, W = =0, 0.56, W = 0. = 5 1,6W , Ω = 0.45 = 1, = 0.45 1 1 2 2 3 4 3 4 3 A > A > A > A 3 2 4 1 3 A3 > A2 > A4 > A1 W > W > W > W 3 2 4 1 Ω Ω Ω Ω > > > 3 2 4 1 Analysis on Influence of the Parameter to Ranking Order Analysis on Influence of the Parameter α to Ranking Order The impact of parameter on ranking order is examined by comparing the ranking orders taken The impact of parameter on ranking order is examined by comparing the ranking orders with varying the different values of . When = 0.5, 1, 1.5, 2, 3, ranking order are presented in taken with varying the different values of α . When α = 0.5, 1, 1.5, 2, 3, ranking order are presented Table 8. We draw Figures 3 and 4 to compare the ranking order for different values of . When = 0.5, in Table 8. We draw Figures 3 and 4 to compare the ranking order for different values of . When = 1.5 and = 3, the ranking order is unchanged and A is the best alternative, while A is the worst 3 1 α = 0.5, α = 1.5 and α = 3, the ranking order is unchanged and A3 is the best alternative, while alternative. When = 1, the ranking order is changed and A is the best alternative and A is the worst 3 4 A1 is the worst alternative. When α = 1, the ranking order is changed and A3 is the best alternative alternative. For = 2, the ranking order is changed and A is the best alternative and A is the worst 2 1 and A4 is the worst alternative. For α = 2, the ranking order is changed and A2 is the best alternative. From Table 8, we see that A is the best alternative in four cases and A is the worst for 3 1 alternative and A1 is the worst alternative. From Table 8, we see that A3 is the best alternative in four four cases. We can say that ranking order depends on parameter . cases and A1 is the worst for four cases. We can say that ranking order depends on parameter α . Figure 3. Global values of the alternatives for different values of attenuation factor = 0.5, 1, 1.5, 2, 3. Figure 3. Global values of the alternatives for different values of attenuation factor = 0.5, 1, 1.5, 2, 3. Information 2017, 8, 149 17 of 21 Information 2017, 8, 149 18 of 22 Figure 4. Ranking of the alternatives for = 0.5, 1, 1.5, 2, 3. Figure 4. Ranking of the alternatives for α = 0.5, 1, 1.5, 2, 3. 7. Comparative Analysis and Discussion 7. Comparative Analysis and Discussion On comparing with the existing neutrosophic decision making strategies [26–29,33–35,64–69], On comparing with the existing neutrosophic decision making strategies [26–29,33–35,64–69], we see that the decision information used in the proposed NC-TODIM strategy is NC numbers, we see that the decision information used in the proposed NC-TODIM strategy is NC numbers, which comprises of interval neutrosophic information and single-valued neutrosophic information which comprises of interval neutrosophic information and single-valued neutrosophic information simultaneously; whereas the decision information in the existing literature is either SVNSs or INSs. simultaneously; whereas the decision information in the existing literature is either SVNSs or INSs. Since NC numbers comprises of much more information, the NC numbers based on the TODIM Since NC numbers comprises of much more information, the NC numbers based on the TODIM strategy proposed in this paper is more elegant, typical and more general in applications, while the strategy proposed in this paper is more elegant, typical and more general in applications, while the existing neutrosophic decision-making strategies cannot deal with the NC number decision-making existing neutrosophic decision-making strategies cannot deal with the NC number decision-making problem developed in this paper. problem developed in this paper. The first decision making paper in NCS environment was studied by Banerjee et al. [44]. The first decision making paper in NCS environment was studied by Banerjee et al. [44]. On On comparison with existing GRA-based NCS decision making strategies [44], we observe that comparison with existing GRA-based NCS decision making strategies [44], we observe that the the proposed NC-TODIM strategy uses the score, and accuracy functions, while the decision proposed NC-TODIM strategy uses the score, and accuracy functions, while the decision making-strategy in [44] uses Hamming distances for weighted grey relational coefficients and standard making-strategy in [44] uses Hamming distances for weighted grey relational coefficients and (ideal) grey relational coefficients, and ranks the alternatives based on the relative closeness coefficients. standard (ideal) grey relational coefficients, and ranks the alternatives based on the relative Hence, the proposed NC-TODIM strategy is relatively simple in the decision making process. closeness coefficients. Hence, the proposed NC-TODIM strategy is relatively simple in the decision On comparing with cosine measures of NCSs [49], we observe that the proposed NC-TODIM making process. involves multiple decision makers, while in [49] only a single decision maker is involved. This shows On comparing with cosine measures of NCSs [49], we observe that the proposed NC-TODIM that [49] cannot deal with group decision making, while the proposed NC-TODIM strategy is more involves multiple decision makers, while in [49] only a single decision maker is involved. This sophisticated as it can deal with single as well as group decision making in the NCS environment. shows that [49] cannot deal with group decision making, while the proposed NC-TODIM strategy On comparison with extended TOPSIS [50] with neutrosophic cubic information, we observe is more sophisticated as it can deal with single as well as group decision making in the NCS that nine components are present in NCSs. Therefore, by calculation of a weighted decision matrix, environment. a neutrosophic cubic positive ideal solution (NCPIS), and a neutrosophic cubic negative ideal solution, On comparison with extended TOPSIS [50] with neutrosophic cubic information, we observe the distance measures of alternatives from NCPIS and NCNIS (NCNIS,) and entropy weight, and use of that nine components are present in NCSs. Therefore, by calculation of a weighted decision matrix, a an aggregation operator are lengthy, time consuming, and hence expensive. The proposed NC-TODIM neutrosophic cubic positive ideal solution (NCPIS), and a neutrosophic cubic negative ideal solution, strategy is free from different kinds of typical aggregation operators. The calculations required for the distance measures of alternatives from NCPIS and NCNIS (NCNIS,) and entropy weight, and the proposed strategy are relatively straightforward and time-saving. Therefore, the final ranking use of an aggregation operator are lengthy, time consuming, and hence expensive. The proposed obtained by the proposed strategy is more conclusive than those produced by the other strategies, NC-TODIM strategy is free from different kinds of typical aggregation operators. The calculations and it is evident that the proposed strategy is accurate and reliable. required for the proposed strategy are relatively straightforward and time-saving. Therefore, the On comparison with the strategy proposed by Zhan et al. [51], we see that they employ score, final ranking obtained by the proposed strategy is more conclusive than those produced by the accuracy, and certainty functions, and a weighted average operator and weighted geometric operator other strategies, and it is evident that the proposed strategy is accurate and reliable. of NCSs for decision making problem involving only a single decision maker. This reflects that the On comparison with the strategy proposed by Zhan et al. [51], we see that they employ score, strategy introduced by Zhan et al. [51] is only applicable for decision making problems involving accuracy, and certainty functions, and a weighted average operator and weighted geometric single decision maker. However, our proposed NC-TODIM strategy is more general as it is capable of operator of NCSs for decision making problem involving only a single decision maker. This reflects dealing with group decision-making problems. that the strategy introduced by Zhan et al. [51] is only applicable for decision making problems involving single decision maker. However, our proposed NC-TODIM strategy is more general as it is capable of dealing with group decision-making problems. Information 2017, 8, 149 18 of 21 A comparative study is conducted with the existing strategy [48] for group decision making under a NCS environment (See Table 9). Since the philosophy of two strategies are different, the obtained results (ranking order) are different. At a glance, it cannot be said which strategy is superior to the other. However, on comparison with similarity measure-based strategies studied in [48], we observed that ideal solutions are needed for ranking of alternatives but in a real world ideal solution, this is an imaginary case, which means that an indeterminacy arises automatically, whereas in our proposed NC-TODIM strategy we can calculate the rank of the alternatives based on global values of alternatives. So, the proposed NC-TODIM strategy is relatively easy to implement and apply for solving MAGDM problems. Table 9. Ranking order of alternatives using three different decision making strategies in the neutrosophic cubic set (NCS) environment. Proposed NC-TODIM Strategy Similarity Measure [48] W = 0, W = 0.89, W = 1, W = 0.46  = 0.20,  = 0.80,  = 0.22,  = 0.19 1 2 3 4 1 2 3 4 Ranking order: A > A > A > A Ranking order: A > A > A > A 3 2 4 1 2 3 1 4 8. Conclusions NCSs can better describe hybrid information comprising of INSs and NSs. In this study, we proposed a score function and an accuracy function, and established their properties. We developed a NC-TODIM strategy, which is capable for tackling MAGDM problems affected by uncertainty and indeterminacy represented by NC numbers. The standard TODIM, in its original formulation, is only applicable to a crisp environment. Existing neutrosophic TODIM strategies deal with single valued neutrosophic information or interval neutrosophic information. Therefore, proposed NC-TODIM strategy demonstrates the advantages of presenting and manipulating MAGDM problems with NCSs comprising of the hybrid information of INSs and NSs. Furthermore, NC-TODIM strategy that considers the risk preferences of decision makers, is significant to solve MAGDM problems. The proposed NC-TODIM strategy was verified to be applicable, feasible, and effective by solving an illustrative example regarding the selection problem of investment alternatives. In addition, we investigated the influence of attenuation factor of losses on ranking the order of alternatives. The contribution of this study can be concluded as follows. First, this study utilized NCSs to present the interval neutrosophic information and neutrosophic information in the MAGDM process. Second, the NC-TODIM strategy established in this paper is simpler and easier than the existing strategy proposed by Pramanik et al. [48] for group decision making with neutrosophic cubic information based on similarity measure and demonstrates the main advantage of its simple and easy group decision making process. Third, TODIM strategy was extended to the NCS environment. Fourth, we defined the NC number. Fifth, we defined the score and accuracy functions and proved their basic properties. Sixth, we developed the ranking of NC numbers using score and accuracy functions. Therefore, two functions namely, score function, accuracy function, and proofs of their basic properties, ranking of NC numbers, and NC-TODIM strategy for MAGDM are the main contributions of the paper. Several directions for future research are generated from this study. First, this study employs the NC-TODIM strategy to deal with MAGDM. In addition to MAGDM, MAGDM problems in a variety of other fields can be solved using the NC-TODIM strategy, including logistics center selection, personnel selection, teacher selection, renewable energy selection, medical diagnosis, image processing, fault diagnosis, etc. Second, this study considers the risk preferences of decision makers i.e., the essence of TODIM, while the interrelationship between criteria are ignored. In future research, the NC-TODIM strategy will be improved to address this deficiency. Third, the proposed strategy can only deal with crisp weights of attributes and decision makers, rather than NCS, which reflects its main limitation. This limitation will be effectively addressed in our future research. Fourth, in our illustrative Information 2017, 8, 149 19 of 21 example, three criteria are considered as an example. However, in real world group decision making problems, many other criteria should be included. A comprehensive framework for MAGDM problem comprising of all relevant criteria should be designed based on prior studies and the proposed NC-TODIM strategy in future research. Finally, we conclude that the developed NC-TODIM strategy offers a novel and effective strategy for decision makers under the NCS environment, and will open up a new avenue of research into the neutrosophic hybrid environment. Acknowledgments: The authors would like to acknowledge the constructive comments and suggestions of the anonymous referees. Author Contributions: Surapati Pramanik conceived and designed the problem; Shyamal Dalapati solved the problem; Surapati Pramanik, Shariful Alam and Tapan Kumar Roy analyzed the results; Surapati Pramanik and Shyamal Dalapati wrote the paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–356. [CrossRef] 2. Chen, S.J.; Hwang, C.L. Fuzzy Multiple Attribute Decision-Making, Methods and Applications; Lecture Notes in Economics and Mathematical Systems; Springer-Verlag: Berlin/Heidelberg, Germany, 1992; Volume 375. 3. Chang, T.H.; Wang, T.C. Using the fuzzy multi-criteria decision making approach for measuring the possibility of successful knowledge management. Inf. Sci. 2009, 179, 355–370. [CrossRef] 4. Pramanik, S.; Mondal, K. Weighted fuzzy similarity measure based on tangent function and its application to medical diagnosis. Int. J. Innovat. Res. Sci. Engg. Tech. 2015, 4, 158–164. [CrossRef] 5. Chen, C.T. Extension of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst. 2000, 114, 1–9. [CrossRef] 6. Zhang, G.; Lu, J. An integrated group decision-making method dealing with fuzzy preferences for alternatives and individual judgments for selection criteria. Group Decis. Negot. 2003, 12, 501–515. [CrossRef] 7. Krohling, R.A.; Campanharo, V.C. Fuzzy TOPSIS for group decision making: A case study for accidents with oil spill in the sea. Expert Syst. Appl. 2011, 38, 4190–4197. [CrossRef] 8. Xia, M.; Xu, Z. A novel method for fuzzy multi-criteria decision making. Int. J. Inf. Technol. Decis. Mak. 2014, 13, 497–519. [CrossRef] 9. Mehlawat, M.K.; Guptal, P. A new fuzzy group multi-criteria decision making method with an application to the critical path selection. Int. J. Adv. Manuf. Technol. 2016, 83, 1281–1296. [CrossRef] 10. Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [CrossRef] 11. Xu, Z. Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optim. Decis. Mak. 2007, 6, 109–121. [CrossRef] 12. Mondal, K.; Pramanik, S. Intuitionistic fuzzy multi criteria group decision making approach to quality-brick selection problem. J. Appl. Quant. Methods 2014, 9, 35–50. 13. Dey, P.P.; Pramanik, S.; Giri, B.C. Multi-criteria group decision making in intuitionistic fuzzy environment based on grey relational analysis for weaver selection in Khadi institution. J. Appl. Quant. Methods 2015, 10, 1–14. 14. Smarandache, F. A unifying field in logics. In Neutrosophy: Neutrosophic Probability, Set and Logic; American Research Press: Rehoboth, DE, USA, 1999. 15. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Single valued neutrosophic sets. Multispace Multistruct. 2010, 4, 410–413. 16. Wang, H.; Smarandache, F.; Zhang, Y.Q.; Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing; Hexis: Phoenix, AZ, USA, 2005. 17. Kharal, A. A neutrosophic multi-criteria decision making method. New Math. Nat. Comput. 2014, 10, 143–162. [CrossRef] 18. Ye, J. Similarity measures between interval neutrosophic sets and their multi criteria decision-making method. J. Intell. Fuzzy Syst. 2014, 26, 165–172. 19. Ye, J. Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single valued neutrosophic environment. J. Intell. Fuzzy Syst. 2014, 27, 2927–2935. 20. Mondal, K.; Pramanik, S. Multi-criteria group decision making approach for teacher recruitment in higher education under simplified neutrosophic environment. Neutrosophic Sets Syst. 2014, 6, 28–34. Information 2017, 8, 149 20 of 21 21. Biswas, P.; Pramanik, S.; Giri, B.C. Entropy based grey relational analysis method for multi-attribute decision making under single valued neutrosophic assessments. Neutrosophic Sets Syst. 2014, 2, 102–110. 22. Biswas, P.; Pramanik, S.; Giri, B.C. A new methodology for neutrosophic multi-attribute decision-making with unknown weight information. Neutrosophic Sets Syst. 2014, 3, 44–54. 23. Biswas, P.; Pramanik, S.; Giri, B.C. Cosine similarity measure based multi-attribute decision-making with trapezoidal fuzzy neutrosophic numbers. Neutrosophic Sets Syst. 2014, 8, 46–56. 24. Mondal, K.; Pramanik, S. Neutrosophic decision making model for clay-brick selection in construction field based on grey relational analysis. Neutrosophic Sets Syst. 2015, 9, 64–71. 25. Mondal, K.; Pramanik, S. Neutrosophic tangent similarity measure and its application to multiple attribute decision making. Neutrosophic Sets Syst. 2015, 9, 85–92. 26. Pramanik, S.; Biswas, P.; Giri, B.C. Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment. Neural Comput. Appl. 2017, 28, 1163–1176. [CrossRef] 27. Sahin, R.; Küçük, A. Subsethood measure for single valued neutrosophic sets. J. Intell. Fuzzy Syst. 2015, 29, 525–530. [CrossRef] 28. Ye, J. An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers. J. Intell. Fuzzy Syst. 2015, 28, 247–255. 29. Biswas, P.; Pramanik, S.; Giri, B.C. TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput. Appl. 2016, 27, 727–737. [CrossRef] 30. Biswas, P.; Pramanik, S.; Giri, B.C. Value and ambiguity index based ranking method of single-valued trapezoidal neutrosophic numbers and its application to multi-attribute decision making. Neutrosophic Sets Syst. 2016, 12, 127–138. 31. Biswas, P.; Pramanik, S.; Giri, B.C. Aggregation of triangular fuzzy neutrosophic set information and its application to multi-attribute decision making. Neutrosophic Sets Syst. 2016, 12, 20–40. 32. Smarandache, F.; Pramanik, S. (Eds.) New Trends in Neutrosophic Theory and Applications; Pons Editions: Brussels, Belgium, 2016; pp. 15–161, ISBN 978-1-59973-498-9. 33. Sahin, R.; Liu, P. Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput. Appl. 2016, 27, 2017–2029. [CrossRef] 34. Sahin, R. Cross-entropy measure on interval neutrosophic sets and its applications in multi criteria decision making. Neural Comput. Appl. 2017, 28, 1177–1187. [CrossRef] 35. Sahin, R.; Liu, P. Possibility-induced simplified neutrosophic aggregation operators and their application to multi-criteria group decision-making. J. Exp. Theor. Artif. Intell. 2017, 29, 769–785. [CrossRef] 36. Biswas, P.; Pramanik, S.; Giri, B.C. Multi-attribute group decision making based on expected value of neutrosophic trapezoidal numbers. New Math. Natural Comput. 2015. [CrossRef] 37. Sahin, ¸ R. Multi-Criteria Neutrosophic Decision Making Method Based on Score and Accuracy Functions under Neutrosophic Environment. arXiv, 2014. [CrossRef] 38. Biswas, P.; Pramanik, S.; Giri, B.C. GRA method of multiple attribute decision making with single valued neutrosophic hesitant fuzzy set information. In New Trends in Neutrosophic Theory and Applications; Smarandache, F., Pramanik, S., Eds.; Pons Editions: Brussels, Belgium, 2016; Volume 1, pp. 55–63, ISBN 978-1-59973-498-9. 39. Biswas, P.; Pramanik, S.; Giri, B.C. Some distance measures of single valued neutrosophic hesitant fuzzy sets and their applications to multiple attribute decision making. In New Trends in Neutrosophic Theory and Applications; Smarandache, F., Pramanik, S., Eds.; Pons Editions: Brussels, Belgium, 2016; Volume 1, pp. 27–34, ISBN 978-1-59973-498-9. 40. Sahin, R.; Liu, P. Distance and similarity measure for multiple attribute with single–valued neutrosophic hesitant fuzzy information. In New Trends in Neutrosophic Theory and Applications; Smarandache, F., Pramanik, S., Eds.; Pons Editions: Brussels, Belgium, 2016; Volume 1, pp. 35–54, ISBN 978-1-59973-498-9. 41. Sahin, R.; Liu, P. Correlation coefficient of single-valued neutrosophic hesitant fuzzy sets and its applications in decision making. Neural Comput. Appl. 2017, 28, 1387–1395. [CrossRef] 42. Ali, M.; Deli, I.; Smarandache, F. The theory of neutrosophic cubic sets and their applications in pattern recognition. J. Intell. Fuzzy Syst. 2016, 30, 1957–1963. [CrossRef] 43. Jun, Y.B.; Smarandache, F.; Kim, C.S. Neutrosophic Cubic sets. New Math. Natural Comput. 2017, 13, 41–54. [CrossRef] 44. Banerjee, D.; Giri, B.C.; Pramanik, S.; Smarandache, F. GRA for multi attribute decision making in neutrosophic cubic set environment. Neutrosophic Sets Syst. 2017, 15, 60–69. [CrossRef] Information 2017, 8, 149 21 of 21 45. Deng, J.L. Introduction to grey system theory. J. Grey Syst. 1989, 1, 1–24. 46. Pramanik, S.; Mukhopadhyaya, D. Grey relational analysis based intuitionistic fuzzy multi criteria group decision-making approach for teacher selection in higher education. Int. J. Comput. Appl. 2011, 34, 21–29. 47. Chang, K.H.; Chang, Y.C.; Tasi, I.T. Enhancing FMEA assessment by integrating grey relational analysis and the decision making trial and evaluation laboratory approach. Eng. Fail. Anal. 2013, 31, 211–224. [CrossRef] 48. Pramanik, S.; Dalapati, S.; Alam, S.; Roy, T.K.; Smarandache, F. neutrosophic cubic MCGDM method based on similarity measure. Neutrosophic Sets Syst. 2017, 16, 44–56. [CrossRef] 49. Lu, Z.; Ye, J. Cosine measures of neutrosophic cubic sets for multiple attribute decision-making. Symmetry 2017, 9, 121. [CrossRef] 50. Pramanik, S.; Dey, P.P.; Giri, B.C.; Smarandache, F. An extended TOPSIS for multi-attribute decision making problems with neutrosophic cubic information. Neutrosophic Sets Syst. 2017, 17, 20–28. 51. Zhan, J.; Khan, M.; Gulistan, M. Applications of neutrosophic cubic sets in multi-criteria decision-making. Int. J. Uncertain. Quantif. 2017, 7, 377–394. [CrossRef] 52. Ye, J. Linguistic neutrosophic cubic numbers and their multiple attribute decision-making method. Information 2017, 8, 110. [CrossRef] 53. Gomes, L.; Lima, M. TODIM: Basics and application to multicriteria ranking of projects with environmental impacts. Found. Comput. Decis. Sci. 1991, 16, 113–127. 54. Kahneman, D.; Tversky, A. Prospect theory: An analysis of decision under risk. Econ. J. Econ. Soc. 1979, 47, 263–291. [CrossRef] 55. Krohling, R.A.; De Souza, T.T.M. Combining prospect theory and fuzzy numbers to multi-criteria decision making. Expert Syst. Appl. 2012, 39, 11487–11493. [CrossRef] 56. Liu, P.; Teng, F. An extended TODIM method for multiple attribute group decision-making based on 2-dimension uncertain linguistic Variable. Complexity 2014, 21, 20–30. [CrossRef] 57. Tosun, O.; Akyu, G. A fuzzy TODIM approach for the supplier selection problem. Int. J. Comput. Intell. Syst. 2015, 8, 317–329. [CrossRef] 58. Gomes, L.F.A.M.; Machado, M.A.S.; Da Costa, F.F.; Rangel, L.A.D. Criteria interactions in multiple criteria decision aiding: A Choquet formulation for the TODIM method. Procedia Comput. Sci. 2013, 17, 324–331. [CrossRef] 59. Fan, Z.P.; Zhang, X.; Chen, F.D.; Liu, Y. Extended TODIM method for hybrid multiple attribute decision making problems. Knowl. Based Syst. 2013, 42, 40–48. [CrossRef] 60. Krohling, R.A.; Pacheco, A.G.C.; Siviero, A.L.T. IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making. Knowl. Based Syst. 2013, 53, 142–146. [CrossRef] 61. Wang, J.Q. TODIM method with multi-valued neutrosophic set. Control Decis. 2015, 30, 1139–1142. [CrossRef] 62. Zhang, M.; Liu, P.; Shi, L. An extended multiple attribute group decision-making TODIM method based on the neutrosophic numbers. J. Intell. Fuzzy Syst. 2016, 30, 1773–1781. [CrossRef] 63. Ji, P.; Zhang, H.; Wang, J. A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput. Appl. 2016. [CrossRef] 64. Xu, D.S.; Wei, C.; Wei, G.W. TODIM method for single-valued neutrosophic multiple attribute decision making. Information 2017, 8, 125. [CrossRef] 65. Liu, P.D.; Wang, Y.M. Multiple attribute decision-making method based on single valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl. 2014, 25, 2001–2010. [CrossRef] 66. Liu, P.D.; Tang, G.L. Some power generalized aggregation operators based on the interval neutrosophic numbers and their application to decision making. J. Intell. Fuzzy Syst. 2016, 30, 2517–2528. [CrossRef] 67. Liu, P.D.; Wang, Y.M. Interval neutrosophic prioritized OWA operator and its application to multiple attribute decision making. J. Syst. Sci. Complex. 2016, 29, 681–697. [CrossRef] 68. Liu, P.D. The aggregation operators based on Archimedean t-conorm and t-norm for the single valued neutrosophic numbers and their application to decision making. Int. J. Fuzzy Syst. 2016, 18, 849–863. [CrossRef] 69. Liu, P.D.; Chu, Y.C.; Li, Y.W.; Chen, Y.B. Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int. J. Fuzzy Syst. 2014, 16, 242–255. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Journal

Information – Multidisciplinary Digital Publishing Institute

Published: Nov 18, 2017

There are no references for this article.