A new way of handling multi-attribute group decision making problems

1 Introduction

The decision making problems of the real life do not provide enough information to make the decision. Smarandache [1] gave the idea of the neutrosophic set which is a much better concept to express that kind of information. After that Wang et al., [2, 3] initiated idea of the interval neutrosophic set and single valued neutrosophic set (SVNS) and considerable supplementary possessions of SVNSs and interval neutrosophic sets. Chi and Liu [5] inflamed a TOPSIS method to interval neutrosophic multi-attribute group decision making (MAGDM) to vigorous substitutes. Ye [4] introduced both the Euclidean and Hamming distances between interval neutrosophic sets, the distance-based resemblance events, and related them to decision-making in interval neutrosophic situation. Ye proposed the idea of a simplified neutrosophic sets (SNs), and he also explored the operators: simplified neutrosophic weighted arithmetic averaging (SNWAA) operator and simplified neutrosophic weighted geometric averaging (SNWGA) operator, and applied them to the MAGDM under simplified neutrosophic situation. Sometimes the decision-makers cannot spring a detailed numeric or vague price for characteristics given to the condition. So in that state, linguistic rapports are an actual valuable tool to fast the rational of decision-maker, particularly for qualitative data, for example, the presentation of a fabric business. The decision-maker can effortlessly rapid his rational in linguistic terms for decent results. LA Zadeh initially created the improper for linguistic variables and its usage in fuzzy reasoning.

Zadeh [6] introduced the notion of a linguistic variable and employed it to the fuzzy cognitive. Herrera et al. [8] established a perfect of accord in decision making beneath linguistic analysis. Later on, Herrera-Viedma and Herrera [9] extended a linguistic decision survey for resolving decision making problems with linguistic data. Moreover, Xu [10] presented linguistic hybrid arithmetic averaging operator. Wang and Li [11] combined the linguistic variable with IFS and presented the idea of intuitionistic linguistic fuzzy number (ILFN). After that Jun Ye [7] defined interval neutrosophic linguistic number and also defined some aggregation operator and applied that operator to the MAGDM. To look for improvement of interval neutrosophic linguistic number based sets and relevant operators is a natural motivation for this research work. Therefore, some linguistic interval valued intuitionistic neutrosophic fuzzy sets (LIVINFSs) are introduced and a new TOPSIS method based on these sets is presented to solve MAGDM problems in this paper.

A linguistic interval-valued intuitionistic neutrosophic fuzzy set is an extension of the linguistic intuitionistic fuzzy set. The intuitionistic fuzzy set (IFS) deals with membership association and non-membership association but an intuitionistic neutrosophic set (INS) deals with truth association, false association and uncertain association. So it is better to employ INS as compared to IFS. That is the reason in this paper intuitionistic neutrosophic set is the focus of interest instead of IFS to get more accurate information. After defining the LIVINFS, some geometric operators are introduced as LIVINF weighted geometric operator, LIVINF ordered weighted geometric operator, and LIVING hybrid geometric operator. The MAGDM problem, based on these novel concepts, is also discussed in this paper. Moreover, score functions are calculated to determine the ranking. Then an example is presented for illustration of the proposed new method by using the given information. Furthermore, our proposed method is compared with a good existing method to establish which one is better.

The paper is organized as follows. There are six more sections. Section 2 contains some fundamental thoughts and properties of basic concepts. Section 3 presents LIVINFNs and their operational laws. In section 4, steps of TOPSIS method based on the LIVINFNs are discussed. In section 5, stepwise application process of the developed method in a MAGDM problem is elaborated by presenting an illustrative example. Section 6, comparison analysis of the newly developed method and existing approach is given. Finally, section 7 concludes this research work.

2 Basic concept

Definition 2.1. Suppose X is a nonempty set. Then, the formula γ ={ 〈μ_γ(x)〉 : x ∈ X } defines a fuzzy set, where μ_γ(x) represents a function from X to that defines membership of an element x in X.

Definition 2.2. Suppose X represents a universal set and consider an element x ∈ X. An IVNS Z in X is Z ={ x (A _A  (x) , B _A  (x) , C _A  (x))  x ∈ X }, where C _A  (x), B _A  (x), A _A  (x) are the falsity membership, indeterminacy-membership and the truth-membership function separately, for all x in X we have that 0 ≤ sup A A ( x ) + sup B A ( x ) + sup C A ( x ) ≤ 3 .

Definition 2.3. Let S_[0, h] be a continuous linguistic term set and X be a fixed set. Then a LIFS Z is defined as Z ={ (u, s_{α(u), s β (u)}) |u ∈ X }, where s _α , s _β  ∈ S_[0, h] such that 0 ≤ α + β ≤ h and s _α  (s _β ) represents the linguistic membership (non-membership) degree. The linguistic indeterminacy s _ϖ is defined as s _ϖ  = s_h-α-β. The pair (s _α , s _β ), usually, is represented by γ and called LIFN, where s _α , s _β  ∈ S_[0, h] such that 0 ≤ α + β ≤ h . The LIFN is called original if s _α , s _β  ∈ S and actual otherwise.

3 LIVINFNs and their operational laws

Definition 3.1. Let A = { [ s w 1 l , s w 1 u ] , [ s Γ 1 l , s Γ 1 u ] , [ s α 1 l , s α 1 u ] , [ s β 1 l , s β 1 u ] , [ s c 1 l , s c 1 u ] , [ s d 1 l , s d 1 u ] } and

B = { [ s w 2 l , s w 2 u ] , [ s Γ 2 l , s Γ 2 u ] , [ s α 2 l , s α 2 u ] , [ s β 2 l , s β 2 u ] [ s c 2 l , s c 1 u ] , [ s d 2 l , s d 2 u ] } be two LIVINFNs and λ be any real number. Then we have A ⊕ B = { [ s w 1 l + w 2 l - w 1 l + w 2 l g , s w 1 u + w 2 u - w 1 u + w 2 u g ] , [ s Γ 1 l + Γ 2 l - Γ 1 l + Γ 2 l g , s Γ 1 u + Γ 2 u - Γ 1 u + Γ 2 u g ] , [ s α 1 l α 2 l g , s α 1 u α 2 u g ] , [ s β 1 l β 2 l g , s β 1 u β 2 u g ] , [ s c 1 l c 2 l g , s c 1 u c 2 u g ] , [ s d 1 l d 2 l g , s d 1 u d 2 u g ] } , A ⊗ B = { [ s w 1 l w 2 l g , s w 1 u w 2 u g ] , [ s Γ 1 l Γ 2 l g , s Γ 1 u Γ 2 u g ] [ s α 1 l + α 2 l - α 1 l + α 2 l g , s α 1 u + α 2 u - α 1 u + α 2 u g ] , [ s β 1 l + β 2 l - β 1 l + β 2 l g , s β 1 u + β 2 u - β 1 u + β 2 u g ] , [ s c 1 l + c 2 l - c 1 l + c 2 l g , s c 1 u + c 2 u - c 1 u + c 2 u g ] , [ s d 1 l + d 2 l - d 1 l + d 2 l g , s d 1 u + d 2 u - d 1 u + d 2 u g ] } , λ A = { [ s g ( 1 - ( 1 - w 1 l g ) λ ) , s g ( 1 - ( 1 - w 1 u g ) λ ) ] , [ s g ( 1 - ( 1 - Γ 1 l g ) λ ) , s g ( 1 - ( 1 - Γ 1 u g ) λ ) ] , [ s g ( α 1 l g ) λ , s g ( α 1 u g ) λ ] , [ s g ( β 1 l g ) λ , s g ( β 1 u g ) λ ] , [ s g ( c 1 l g ) λ , s g ( c 1 u g ) λ ] , [ s g ( d 1 l g ) λ , s g ( d 1 u g ) λ ] } , A λ = { [ s g ( w 1 l g ) λ , s g ( w 1 u g ) λ ] , [ s g ( Γ 1 l g ) λ , s g ( Γ 1 u g ) λ ] , [ s g ( 1 - ( 1 - α 1 l g ) λ ) , s g ( 1 - ( 1 - α 1 u g ) λ ) ] , [ s g ( 1 - ( 1 - β 1 l g ) λ ) , s g ( 1 - ( 1 - β 1 u g ) λ ) ] , [ s g ( 1 - ( 1 - c 1 l g ) λ ) , s g ( 1 - ( 1 - c 1 u g ) λ ) ] , [ s g ( 1 - ( 1 - d 1 l g ) λ ) , s g ( 1 - ( 1 - d 1 u g ) λ ) ] } .

Example 3.2. Let A = { [ s 1 , s 3 ] , [ s 2 , s 4 ] , [ s 3 , s 4 ] , [ s 1 , s 2 ] , [ s 2 , s 3 ] , [ s 2 , s 4 ] } and B = { [ s 1 , s 2 ] , [ s 2 , s 3 ] , [ s 3 , s 4 ] , [ s 2 , s 4 ] , [ s 3 , s 5 ] , [ s 2 , s 5 ] } be two LIVINFNs and λ be any real number. Then we have A + B = { [ s 1.833 , s 4.58 ] , [ s 3.666 , s 6.416 ] , [ s 0.75 , s 1.33 ] , [ s 0.166 , s 0.1666 ] , [ s 0.5 , s 1.25 ] , [ s 0.33 , s 1.66 ] } , A ⊗ B = { [ s 0.083 , s 0.5 ] , [ s 0.33 , s 1 ] , [ s 5.5 , s 7.33 ] , [ s 2.75 , s 5.5 ] , [ s 4.58 , s 7.33 ] , [ s 3.66 , s 8.25 ] } , λ = 1 λ A = { [ s 1 , s 3 ] , [ s 2 , s 4 ] , [ s 3 , s 4 ] , [ s 1 , s 2 ] , [ s 2 , s 3 ] , [ s 2 , s 4 ] } , A λ = { [ s 1 , s 3 ] , [ s 2 , s 4 ] , [ s 3 , s 4 ] , [ s 1 , s 2 ] , [ s 2 , s 3 ] , [ s 3 , s 4 ] } .

Definition 3.3. Let α t = { [ s w 1 l , s w 1 u ] , [ s Γ 1 l , s Γ 1 u ] , [ s α 1 l , s α 1 u ] , [ s β 1 l , s β 1 u ] , [ s c 1 l , s c 1 u ] , [ s d 1 l , s d 1 u ] } , (t = 1, 2, . . . , n) be the collection of LIVINFNs and the weight vector is w = (w₁, w₂, . . . , w _n )  ^T of α _t (t = 1, 2, . . . , n) such that w t > 0 , ∑ t = 1 n w t = 1 . The GLIVINFWG operator is a map GLIVINFWG: Ω ⁿ  - > Ω define by GLIVINFWG GLIVINFWG ( α 1 , α 2 , . . . . α n ) = ⊗ j = 1 n α t w t = { [ s g ( ∏ j = 1 n ( ( w j l g ) w ) λ ) 1 λ , s g ( ∏ j = 1 n ( ( w j u g ) w ) λ ) 1 λ ] , [ s g ( ∏ j = 1 n ( ( Γ j l g ) w ) λ ) 1 λ , s g ( ∏ j = 1 n ( ( Γ j u g ) w ) λ ) 1 λ ] , [ s g ( ( 1 - ∏ j = 1 n ( 1 - α j l g ) w ) λ ) 1 λ , s g ( ( 1 - ∏ j = 1 n ( 1 - α j u g ) w ) λ ) 1 λ ] , [ s g ( ( 1 - ∏ j = 1 n ( 1 - β j l g ) w ) λ ) 1 λ , s g ( ( 1 - ∏ j = 1 n ( 1 - β j u g ) w ) λ ) 1 λ ] [ s g ( ( 1 - ∏ j = 1 n ( 1 - c j l g ) w ) λ ) 1 λ , s g ( ( 1 - ∏ j = 1 n ( 1 - c j u g ) w ) λ ) 1 λ ] , [ s g ( ( 1 - ∏ j = 1 n ( 1 - d j l g ) w ) λ ) 1 λ , s g ( ( 1 - ∏ j = 1 n ( 1 - d j u g ) w ) λ ) 1 λ ] }

If w = ( 1 n , 1 n , . . . , 1 n ) t , the GLIVINFWG operator reduced to GLIVINFG operator expressed as GLIVINFG ( α 1 , α 2 , . . . α n ) = ( α 1 ⊗ α 2 ⊗ , . . . , ⊗ α n ) 1 n .

4 LIVINF TOPSIS method

LIVINF TOPSIS is a technique to be presented in this section to address decision making problems. Main purpose of the technique is to find the best choice of solution that is closest to the optimal solution. In fact, LIVINFNs are employed in TOPSIS method to develop this new extension of the existing method. It is done through following steps.

Step 1: To consider a LIVINF TOPSIS decision making problem with multiple attributes and construct its decision matrix. Suppose the problem contains n decision attributes and m students. Then its LIVINF decision matrix is given as below. β = { [ s w 11 l , s w 11 u ] , [ s Γ 11 l , s Γ 11 u ] , [ s α 11 l , s α 11 u ] , [ s β 11 l , s β 11 u ] , [ s c 11 l , s c 11 u ] , [ s d 11 l , s d 11 u ] [ s w 12 l , s w 12 u ] , [ s Γ 12 l , s Γ 12 u ] , [ s α 12 l , s α 12 u ] , [ s β 12 l , s β 12 u ] , [ s c 12 l , s c 12 u ] , [ s d 12 l , s d 12 u ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ s w n 1 l , s w n 1 u ] , [ s Γ n 1 l , s Γ n 1 u ] , [ s α n 1 l , s α n 1 u ] , [ s β n 1 l , s β 11 u ] , [ s c n 1 l , s c n 1 u ] , [ s d n 1 l , s d n 1 u ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ s w m 1 l , s w m 1 u ] , [ s Γ m 1 l , s Γ m 1 u ] , [ s α m 1 l , s α m 1 u ] , [ s β m 1 l , s β m 1 u ] , [ s c m 1 l , s c m 1 u ] , [ s d m 1 l , s d m 1 u ] [ s w m 2 l , s w m 2 u ] , [ s Γ m 2 l , s Γ m 2 u ] , [ s α m 2 l , s α m 2 u ] , [ s β m 2 l , s β m 2 u ] , [ s c m 2 l , s c m 2 u ] , [ s d m 2 l , s d m 2 u ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [ s w mn l , s w mn u ] , [ s Γ mn l , s Γ mn u ] , [ s α mn l , s α mn u ] , [ s β mn l , s β mn u ] , [ s c mn l , s c mn u ] , [ s d mn l , s d mn u ] }(1)

Calculate the GLIVINFWG operator.

Step 2: To construct a normalized LIVINF TOPSIS decision matrix R = [β _ij ].

The normalized value β _ij is calculated as: β = { [ s w l Σ i = 1 n ( s w ij l ) 2 , s w u Σ i = 1 n ( s w ij u ) 2 ] , [ s Γ l Σ i = 1 n ( s Γ ij l ) 2 , s Γ u Σ i = 1 n ( s Γ ij u ) 2 ] , [ s α l Σ i = 1 n ( s α ij l ) 2 , s α u Σ i = 1 n ( s α ij u ) 2 ] , [ s β l Σ i = 1 n ( s β ij l ) 2 , s β u Σ i = 1 n ( s β ij u ) 2 ] , [ s c l Σ i = 1 n ( s c ij l ) 2 , s c u Σ i = 1 n ( s c ij u ) 2 ] , [ s d l Σ i = 1 n ( s d ij l ) 2 , s d u Σ i = 1 n ( s d ij u ) 2 ] }(2)

Step 3: To construct the weighted normalized LIVINF TOPSIS decision matrix by multiplying the normalized LIVINF TOPSIS decision matrix by its associated weights. The weighted normalized value is given by v _ij w _j  = B _j

Step 4: To find the +ve LIVINF ideal solution and the -ve LIVINF ideal solution. It is shown as under: α i + = { [ max i s w ij l , min i s w ij u ] , [ max i s Γ ij l , min i s Γ ij u ] , [ max i s α ij l , min i s α ij u ] , [ max i s β ij l , min i s β ij u ] , [ max i s c ij l , min i s c ij u ] , [ max i s d ij l , min i s d ij u ] }(3) α i - = { [ min i s w ij l , max i s w ij u ] , [ min i s Γ ij l , max i s Γ ij u ] , [ min i s α ij l , max i s α ij u ] , [ min i s β ij l , max i s β ij u ] , [ min i s c ij l , max i s c ij u ] , [ min i s d ij l , max i s d ij u ] }(4)

Step 5: Separation of each candidate from the positive LIVINF ideal solution q i + 〈 [ B - , B + ] , η 〉 that is given as under: q i + 〈 [ B - , B + ] , η 〉 = 〈 1 18 ( | s w ij l - s w j l | + | s w ij u - s w j u | + | s α ij l - s α j l | + | s β ij l - s β j l | + | s c ij l - s c j l | + | s d ij l - s d j l | ) 〉 .(5)

Separation of each candidate from the negative LIVINF ideal solution q i - 〈 [ B - , B + ] , η 〉 that is given as under: q i - 〈 [ B - , B + ] , η 〉 = 1 18 ( | s w ij l - s w j l | + | s w ij u - s w j u | + | s α ij l - s α j l | + | s β ij l - s β j l | + | s c ij l - s c j l | + | s d ij l - s d j l | ) .(6)

Step 6: To compute closeness relative to the ideal solution. Relative closeness to the ideal solution is comprehended by the Equation. Z i = q i - 〈 [ B - , B + ] , η 〉 q i - 〈 [ B - , B + ] , η 〉 + q i + 〈 [ B - , B + ] , η 〉 .

In this section, the developed technique is employed to assess criteria framework of cleaner generation counting three criteria built up concurring to the particular characteristic features of gold mines. Consider a committee of choice creators to perform the evaluation and to choose the foremost appropriate cleaner generation, among the three circulation centers A₁, A₂ and A₃. The choice producer assesses the cleaner generation concurring to three criteria, which are given as follows:

Management level (C₁): It demonstrates the administration level of cleaner generation, which contains the integrity of cleaner generation controls and execution of cleaner generation regulations.

Step 1: To construct the decision matrices. The Tables 1 and 2 represent these matrices.

Calculate the GLIVINGWG operator (Table 3) and w = (0.2, 0.3, 0.5)

Step 2: To construct a normalized LIVINF TOPSIS decision matrix as shown in Table 4.

Step 3: Construction of weighted normalized LIVINF TOPSIS decision matrix as shown in Table 5.

Step 4: To find positive LIVINF ideal solution { [ s 1.1159 , s 4.4210 ] , [ s 4.1169 , s 5.2813 ] , [ s 5.1760 , s 8.2682 ] , [ s 2.1538 , s 3.2895 ] , [ s 1.1538 , s 4.3126 ] , [ s 3.1760 , s 5.2682 ] } , { [ s 6.1159 , s 5.3427 ] , [ s 1.1407 , s 1.3366 ] , [ s 2.1977 , s 4.2657 ] , [ s 4.1784 , s 5.3126 ] , [ s 5.2000 , s 7.3349 ] , [ s 4.1538 , s 9.3126 ] } , { [ s 11.1159 , s 9.3257 ] , [ s 12.1407 , s 17.3366 ] , [ s 0.1760 , s 12.2682 ] , [ s 9.1309 , s 0.2632 ] , [ s 5.1538 , s 7.3349 ] , [ s 4.1760 , s 5.2682 ] }

To find negative LIVINF ideal solution { [ s 0.1812 , s 0.3095 ] , [ s 0.1416 , s 0.3005 ] , [ s 0.1331 , s 0.3308 ] , [ s 0.1275 , s 0.8187 ] , [ s 0.2325 , s 0.3333 ] , [ s 0.2629 , s 0.38171 ] } , { [ s 0.4224 , s 2.2579 ] , [ s 2.0974 , s 2.6854 ] , [ s 4.1874 , s 0.3066 ] , [ s 0.1874 , s 5.8001 ] , [ s 0.4359 , s 4.2844 ] , [ s 0.3103 , s 0.2527 ] } , { [ s 0.4382 , s 0.5175 ] , [ s 0.1223 , s 0.5395 ] , [ s 0.5113 , s 0.5348 ] , [ s 0.5513 , s 0.6755 ] , [ s 0.1574 , s 0.5307 ] , [ s 0.4638 , s 0.5817 ] }

Step 5: To separate every candidate from +ve LIVINF ideal solution. It is as under: q 1 + = s 0.1234 , q 2 + = s 0.2223 , q 3 + = s 0.3451 .

To separation each candidate from -ve LIVINF ideal solution. It is as under: q 1 - = s 0.2223 , q 2 - = s 0.1212 , q 3 - = s 0.1098 .

Step 6: To calculate closeness of the solution with respect to the ideal solution. The closeness of solution of the problem under consideration is given by the following equations. Z 1 = s 0.2567 , Z 2 = s 0.7834 , Z 3 = s 0.8712 .

Then from the rankings s₃ (a) > s₂ (a) > s₁ (a) it is clear that s₃ (a) is the best. Three cases are shown in Figs. 3, 4 and 5, and Fig. 6 shows all cases.

OPEN IN VIEWER

Fig. 1

History of advancements in fuzzy sets.

OPEN IN VIEWER

Fig. 2

(Proposed method).

OPEN IN VIEWER

Fig. 3

(Case 1).

OPEN IN VIEWER

Fig. 4

(Case 2).

OPEN IN VIEWER

Fig. 5

(Case 3).

OPEN IN VIEWER

Fig. 6

(Allcases).

OPEN IN VIEWER

Fig. 7

(Score values LIVAIFS).

OPEN IN VIEWER

Fig. 8

(Comparsion analysis).

To validate and establish effectiveness of the suggested method, its comparison with another method dependent upon linguistic interval-valued Atanassov intuitionistic fuzzy (LIVAIF) sets [12] is presented with an illustrative example. Other methods are special cases of our proposed method that is based on LIVINFN to the same illustrative example.