Novel Linguistic Intuitionistic Fuzzy Score and Accuracy functions for Linguistic Intuitionistic Fuzzy TOPSIS Method

Abstract

This work proposes various Linguistic Intuitionistic Fuzzy Score and Accuracy functions (LIF-Score and LIF-Accuracy), including linguistic degree, membership, non-membership, and hesitancy degree, for Linguistic Intuitionistic Fuzzy Sets (LIFSs) in Multiple Attribute Group Decision Making (MAGDM) problems. The proposed functions are integrated into the Linguistic Intuitionistic Fuzzy-Technique of Order Preference by Similarity to Ideal Solution (LIF-TOPSIS) MAGDM approach to rank the alternatives, and a numerical example demonstrating their use in ranking alternatives is shown in the paper.

Introduction

I. INTRODUCT?ON

Whenever attributes involving differences of opinion are involved in a decision system, one of the DSS's working principles is to rank the best option among those that are available, and if the distance is sought with the ideal solution, then the method is called TOPSIS technique. Based on ranking techniques that gauge proximity to either the positive or negative ideal answer, the TOPSIS method is where the decision-making problem will focus its methodology [3,5,6,7,8,9,11] in choosing the appropriate alternative which will better suit the problem solution. In recent days, intuitionistic fuzzy data proposed in [1,2]  has gained the attention of researchers to a large extent. Score and accuracy functions are extremely important in Fuzzy Decision Making situations [4,10,12], especially in ranking the final alternatives. In this work, we have suggested several score and accuracy measures for Linguistic Intuitionistic Fuzzy Sets (LIFSs), and we employ them in the Linguistic Intuitionistic Fuzzy TOPSIS method for both attribute weight determination and ranking of the best alternatives from the available ones. Utilizing the suggested score and accuracy functions, various calculations are carried out, and new decision algorithms are established based on the proposed measures. According to the study, our novel score and accuracy functions which are new to the field of Linguistic Intuitionistic Fuzzy MAGDM problems are observed to be one of the effective measures in ranking of the alternatives as well as in producing weight vectors for the aggregation process in the MAGDM problems.

II. LINGUISTIC INTUITIONISTIC FUZZY SETS

Definition 2.1 [11] Let and  be two LIFNs and . Then the operations of LIFNs are defined as:

, 

,

, and   .

III. SCORE FUNCTIONS FOR LINGUISTIC INTUITIONISTIC FUZZY SETS

1) Definition 3.1 Let  for j=1,2,…,n be a collection of Linguistic Intuitionistic Fuzzy numbers, where  denotes the linguistic degree and, denotes the membership degree and denotes the non-membership degree.

The proposed LIF-Score functions are as follows:

• 
• ,                
• where π(σj)=1-α(σj)+(γ(σj)  is the hesitation degree.
• 
• 
• 
• 
• .
• .

2) Definition 3.2 Let  and be two linguistic intuitionistic fuzzy numbers, where , Then,

• If and , then
• If and , then
• If and , then

3) Theorem 3.1 Let σj,j=1,2,.....,n,  be a collection of Linguistic Intuitionistic Fuzzy Numbers. Then the Score function  is

(i) Bounded and (ii) Monotonic.

Proof:

1. Boundedness: From definition 3.1 (a),

Since ,

And since

1. Monotonicity: If then

and  ,

.

1. Theorem 3.2 Let σj,j=1,2,.....,n,  be a collection of Linguistic Intuitionistic Fuzzy Numbers. Then the Score function  is bounded.

Proof:

1. Boundedness: From definition 3.1 (b),  

.

Since, ;

And since

.

1. Theorem 3.3 Let  σj,j=1,2,...,n,  be a collection of Linguistic Intuitionistic Fuzzy Numbers. Then the Score function

Siiiσj=θj.ασj-γσj+πσj2,Siiiσj∈-1,1  is bounded.

Proof:

1. Boundedness: From definition 3.1 (c),  

Since ,

And since

.

1. Theorem 3.4 Let σj,j=1,2,.....,n,  be a collection of Linguistic Intuitionistic Fuzzy Numbers. Then the Score function   

Sivσj∈-1,0.5  is bounded.

Proof:

1. Boundedness: From definition 3.1 (d),  

Since ,

And since

.

1. Theorem 3.5 Let σj,j=1,2,.....,n,  be a collection of Linguistic Intuitionistic Fuzzy Numbers. Then the Score function  

Sv(σj)∈0,1   is (i) Bounded, (ii) Monotonic.

Proof:

1. Boundedness: From definition 3.1 (e),

               Sv(σj)=θj(λ.α(σj)+(1-λ)(1-γ(σj))),λ∈0,1

Since ,

1. Monotonicity: If then Siσ1>Siσ2

  and  ,

 ⇒(1-λ)θ1-θ1.γ(σ1)>(1-λ)θ2-θ2.γ(σ2)

.

IV. ACCURACY FUNCTIONS FOR LINGUISTIC INTUITIONISTIC FUZZY SETS

This section proposes new Accuracy functions for Linguistic Intuitionistic Fuzzy numbers.

Definition 4.1     Let  for j=1,2,…,n be a collection of Linguistic Intuitionistic Fuzzy numbers, where  denotes the linguistic degree and, denotes the membership degree and denotes the non-membership degree.

The LIF-Score functions are proposed as follows:

Theorem 4.1 Let σj,j=1,2,.....,n,  be a collection of Linguistic Intuitionistic Fuzzy numbers. Then the Score functions:

are bounded.

Proof:

• From definition 4.1 (a), .

Since ,

• From definition 4.1 (b), 

Since ,

• From definition 4.1 (c), 

,

Since ,

• From definition 4.1 (d), 

Since ,

• From definition 4.1 (e), 

Since ,

.

V. The MAGDM with LIF-TOPSIS method using Score and Accuracy function

Let E=  be a set of experts, L=(L1,L2,...Lm)  be a discrete set of alternatives, C=(C1,C2,...Cn)  be the set of attributes, and W=(ω1,ω2,...ωn)T be the weighting vector of the attributes,  Let λ=(λ1,λ2,...λp)  be the expert’s weighting vector,  k=1pλk=1.  Suppose that  is the decision matrix, where tkij=akij,(αijk,γijk)  takes the form of the Linguistic Intuitionistic number, given by the decision maker , for alternative  with respect to attribute . Rank the alternatives by using the steps below:

Step 1: Normalize the given matrices and make the integrated matrix

Integrate the matrix given by decision maker  into the integrated matrix :tij=k=1pλktkij ,  where, .

Step 2: The integrated matrix is converted into a crisp matrix using Score function/ Accuracy function

The score matrix is , where Stkij   is the score value of the alternative  with respect to attribute .

Step 3: Evaluate the attribute weights

For the attribute Cj , the deviation values of alternative  to all the other alternatives can be defined as  where  indicates the total deviation values of all alternatives to the other alternatives for the attribute Cj .   represents the deviation of all attributes to all alternatives. The optimum model is built as follows:

&maxD(ωj)=j=1ni=1ml=1mStkijωj&s.tj=1nωj2=1,ωj≥0,j=1,2,...n

and:  

Step 4: To rank the alternatives, use the LIF-TOPSIS method.

The fundamental principle of TOPSIS is that the chosen alternative ought to be closest to the positive ideal solution and most far away from the negative optimal solution.

a) Construct the weighted Score matrix:

,

b) Decide the separation Score measures  based on Euclidean distance from the positive and negative ideal solution:

c) Compute the relative closeness coefficient as follows:       

d) To rank the alternatives, apply the relative closeness coefficient for all alternatives. The bigger , the better the alternative.

VI. NUMERICAL ILLUSTRATION

Assume there are four industries (alternatives)  to be weighed against certain criteria. Evaluate industries in terms of their technological innovation capability, evaluating 'factors' such as resource ability for digitalization , organizational innovation , Innovation Centers , and Innovative products . Consider a group of experts whose weights are given as λ=(0.4,0.32,0.28) . The Experts assessment of the four industries are listed in Tables 1, 2, and 3.

Table 1: Decision Matrix I

Table 2: Decision Matrix II

Table 3: Decision Matrix III

Step 1: Normalize the given matrices and make the integrated matrix:

The following matrices are obtained by following the normalization of the linguistic values for the above mentioned decision matrices.

Table 4: Normalized Decision Matrix I

Table 5: Normalized Decision Matrix II

Table 6: Normalized Decision Matrix III

For the first value, the calculation is

Similarly, calculations can be done for all the elements in the matrices, and the collective matrix is given as follows:

Step 2: Using score function of definition 3.1 (a), the score matrix is obtained as follows:

Score value of l0.780,(0.169,0.670) is 0.780×(0.169-0.670)=-0.131 .

Similarly, all Score values are computed and listed below.

Calculate the attribute weights using   

. Similarly, all the remaining weights are computed.

 Hence

To rank the alternatives, use the TOPSIS method:

a) Make the weighted score matrix:

Here, .

   Similarly, all the elements of the Weighted Score matrix can be computed.

b) Decide the separation measure,

,

Similarly, all values are calculated and are listed below:

, , .

, , .

c) The relative closeness coefficient is calculated as follows:

Similarly, closeness coefficient is calculated for all the alternatives as:

,

Hence, the ranking of the best alternative is  Based on the order of ranking, L3?L2=L4?L1,   is observed to be the best alternative.

Table 6.1: Weight Vector obtained from different Score and accuracy functions and the ranking of alternatives for TOPSIS.

From the above table, it can be observed that is the best alternative using many of the proposed functions.

VII. RESULTS

Various score and accuracy functions are proposed in this research work which are in turn used for the data interpretation in this study. Various theorems which proved the properties of the proposed functions are given in detail. The weights required for aggregation in the TOPSIS technique are mostly determined by these suggested Score and accuracy functions. A Decision Support System (DSS) called TOPSIS is used in decision systems by which the attribute-based differences are determined and finally, the best option in the MAGDM involved is obtained.

Conclusion

In this paper, New Score functions for LIFNs are proposed and in the theorems, it is proved that the proposed score functions satisfy the conditions namely Boundedness and Monotonic properties. New Accuracy functions for LIFNs are proposed and in the theorems, it is proved that the proposed Accuracy functions satisfy the conditions namely Boundedness. LIF-TOPSIS algorithm using the proposed Score and Accuracy function is constructed. Every Score function is applied in the proposed LIF-TOPSIS for computing weights and in the final ranking of the alternatives. Numerical illustration is given to prove the consistency of the proposed Score and Accuracy functions applied in the TOPSIS method of solving MAGDM problems.

References

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Copyright

Copyright © 2024 A. Leonishiya, P. John Robinson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.