NC-TODIM Based MAGDM under Neutrosophic 1 Cubic Set Environment 2

1 Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District –North 24 4 Parganas, Pin code-743126, West Bengal, India; sura_pati@yahoo.co.in 5 2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic 6 Garden, Howrah-711103, West Bengal, India; salam50in@yahoo.co.in (S.A.); roy_t_k@yahoo.co.in (T.K.R.) 7 * Correspondence: shyamal.rs2015@math.iiests.ac.in; Tel.: +91-9804234197 8


Introduction 23
While modelling multi attribute decision making (MADM) and multi attribute group decision 24 making (MAGDM), it is often observed that the parameters of the problem are not precisely known. 25 The parameters often involve uncertainty. To deal uncertainty, Zadeh [1] left an important mark to 26 represent and compute with imperfect information by introducing fuzzy set. Fuzzy set fostered a 27 broad research community, and their impact has also been clearly felt at the application level in Similarity measure is an important mathematical tool in decision-making problems. 48 Pramanik et al. [29] at first defined similarity measure for neutrosophic cubic sets and proved its 49 basic properties. In the same study, Pramanik et al. [29] developed a new MAGDM method in 50 neutrosophic cubic set environment. Lu and Ye [30] proposed cosine measures between 51 neutrosophic cubic sets and proved their basic properties. In the same study, Lu and Ye [30] 52 proposed a new cosine measures-based MADM method under a neutrosophic cubic environment. 53 Due to little research on operations and application of neutrosophic cubic sets, Pramanik et al. 54 [31] proposed several operational rules on neutrosophic cubic sets and defined Euclidean distance 55 and arithmetic average operator in neutrosophic cubic sets environment. Pramanik  In this paper we develop a TODIM method (for short, NC-TODIM method) for MAGDM in 83 neutrosophic cubic set environment. We solve an illustrative numerical example of MAGDM 84 problem in neutrosophic cubic set environment to show the applicability and effectiveness of the 85 proposed NC-TODIM method. 86 Remainder of the paper is divided into five sections that are organized as follows: Section 87 2 presents some basic definition of neutrosophic sets, interval-valued neutrosophic sets, 88 neutrosophic cubic sets. Section 3 is devoted to present the proposed NC-TODIM method. Section 4 89 presents an illustrative numerical example. Section 5 is devoted to analyse the ranking order with 90 different values of attenuation factor of losses. Finally, Section 6 presents conclusion and future 91 scope of research. 92

Preliminaries 93
In this section, we review some basic definitions which are important to develop the paper. 94

Definition 1. [14] Neutrosophic set (NS) 95
Let U be a space of points (objects) with a generic element in U denoted by u i.e. u∈ U. A 96 neutrosophic set R in U is characterized by truth-membership function tR , indeterminacy-97 membership function iR and falsity-membership function f R , where tR , iR , f R are the functions from

98
U to ]  Let G be a non-empty set. An interval neutrosophic set G in G is characterized by In real problems it is difficult to express the truth-memberships function, 112 indeterminacy-membership function and falsity-membership function in the form of tG − (g), tG Here, tG − (g), tG , gn} be a non-empty set. Let G 1 be any interval neutrosophic set.  iii.
Compliment of a NC-number 150 be any neutrosophic cubic set in G. Then compliment of

152
Here, Example 6. 156 Assume that ©1 be any NC-number in G in the form: Let ©1 be a NC-number in a non-empty set G. Then, a score function of ©1 , 161 ) © ( Sc 1 is defined as:

210
The values of accuracy function depend upon 211 Hence complete the proof.
Step 2. Normalize the decision matrix 291 MAGDM problem generally consists of cost criteria and benefit criteria. So, the 292 decision matrix needs to be normalized. For cost criterion Cj we use the Equation (7) to 293 normalize the decision matrix (Equation (3.1)) provided by the decision makers. For benefit 294 criterion Cj we don't need to normalize the decision matrix. When Cj is a cost criterion, the 295 normalized form of decision matrix (see Equation (3.1)) is presented below. 296

2) 297
Here © k ij ⊗ is the normalized form of NC-number. 298 Step 3. Determine the relative weight of each criterion 299 Relative weight Wch of each criterion is obtained by the following equation. where, Wh = max {W1, W2, …, Wn}.

302
Step 4. Calculate score values 303  Where, parameter ' α ' represents the attenuation factor of losses and α must be positive.

318
Step 7. Formulate the individual total dominance matrix 319 Using Equation (3.6), calculate the individual total dominance matrix of each alternative Ai over each 320 Step 8. Aggregate the dominance matrix 323 Using Equation (3.7), calculate the collective overall dominance of alternative Ai over each 324 Step 9. Calculate global values 327 Using the Equation (3.8), we calculate global value of each alternative 328 Step 10. Rank the priority 330 Sorting the values of i Ω provides the rank of each alternative. A set of alternatives can be preference 331 ranked according to the descending order of i Ω . Highest global value corresponds to the best 332 alternative. 333 Figure 2. Step 1: Formulate the decision matrix Decision makers

A conceptual model of the proposed approach is shown in
Step 2: Normalize the decision matrix Step 3: Determine the relative weight of each criterion Step 4: Calculate score values Step 5: Calculate accuracy values Step 6: Formulate the dominance matrix Step 7: Formulate the individual total dominance matrix Step 8: Aggregate the dominance matrix Step 9: Calculate global values

Stop
For cost type criteria For benefit type criteria Step 10: Rank the priority

Illustrative example 360
In this section, a MAGDM problem is adapted from the study [18]  The proposed method is presented using the following steps: 375 Step 1. Formulate the decision matrix 376 Formulate the decision matrices Decision matrix for E1 383  Step 2. Normalize the decision matrix 390 Since all the criteria are benefit type, we do not need to normalize the decision matrix. 391 Step 3. Determine the relative weight of each criterion 392 Using Equation (3.3), we obtain the relative weight of criteria Wch as follows: 393 Wch = (1, .875, .625) T .

394
Step 4. Calculate score values 395 The score values of each alternative relative to each criterion obtained by Equation (2.1)   Step 6. Formulate the dominance matrix 410 Using Equation (3.5), we construct dominance matrix for α = 1 The dominance matrixes are 411 represented in matrix form (See Equations (4.4), (4.5), (4.6), (4.7), (4.8), (4.9), (4.10), (4.11), and (4.12)). 412 (4.8) (4.10) (4.11) 420 The dominance matrix Ψ 3 (4.12) 422 Step 7. Formulate the individual overall dominance matrix 423 The individual overall dominance matrix is calculated by the Equation (3.6) and The dominance 424 matrixes are represented in matrix form (see Equations (4.13), (4.14), and (4.15)). 425 First decision maker's overall dominance matrix λ (4.14) 429 Third decision maker's overall dominance matrix λ 3 430  (4.16) 435 Step 9. Calculate global values 436 Using Equation (3.8 ) we calculate the values of Ωi (i = 1, 2, 3, 4) and represented in Table 7. 437 Table 7. Global values of alternatives 438 .61 1 0 Step 10. Rank the priority 439 Ω , alternatives are then preference ranked as follows: Hence A3 is the best alternative. 442 From the illustrative example, we see that the proposed NC-TODIM method is more suitable for real 443 scientific and engineering applications because it can handle hybrid information consisting of INS 444 and SVNS information simultaneously to cope indeterminate and inconsistent information. Thus, 445 NC-TODIM extends the existing decision-making methods and provides a sophisticated 446 mathematical tool for decision makers.  Table 8 shows that the ranking order of alternatives depends on values of attenuation factor, which reflects the 449 importance of attenuation factor in NC-TODIM method. 450 451  The impact of parameter α on ranking order is examined by comparing the ranking orders taken 454 with varying the different values of α . When α = .5, 1, 1.5, 2, 3, ranking order are presented in 455 Table 8. We draw Figure 3 and Figure 4 to compare the ranking order for different values of α .

456
When α =.5, α = 1.5 and α =3 the ranking order is unchanged and A3 is the best alternative, A1 is 457 the worst alternative. When α = 1, the ranking order is changed and A3 is the best alternative and A4 458 is the worst alternative. For α = 2, the ranking order is changed and A2 is the best alternative and A1

459
is the worst alternative. From Table 8 we see that A3 is the best alternative in four cases and A1 is the 460 worst. We can say that ranking order depends on parameter α and A3 is the best alternative and A1 461 is the worst alternative.