Extended TOPSIS method based on the entropy measure and probabilistic hesitant fuzzy information and their application in decision support system

3.1 Distance measure for PHFs

Definition 7. Let T ˇ be be a univeral set and A ˘ = { A ˘ 1 , . . . , A ˘ n } and B • = { B 1 • , . . . , B n • } are any two PHFSs, where A ˘ ȷ = { g ˘ , h ˆ k ( α ȷ / p ȷ ) / g ˘ ∈ T ˇ } and B ȷ • = { g ˘ , h ˆ k ( β ȷ / p ȷ ) / g ˘ ∈ T ˇ } , ȷ = 1 , . . . , n , generalized distance measures between A ˘ and β is defined for ζ > 0 as, d ( A ˘ , B • ) = ( 1 2 n ∑ ȷ = 1 n | ∑ ȷ = 1 n h ˆ k ( α ȷ / p ȷ ) - ∑ ȷ = 1 n h ˆ k ( β ȷ / p ȷ ) | ζ ) 1 ζ .(13)

Definition 8. Let A ˘ = { A ˘ 1 , . . . , A ˘ n } and B • = { B 1 • , . . . , B n • } be any two PHFS sets, where A ˘ ȷ = { g ˘ , h ˆ k ( α ȷ / p ȷ ) / g ˘ ∈ T ˇ } and B ȷ • = { g ˘ , h ˆ k ( β ȷ / p ) / g ˘ ∈ T ˇ } , ȷ = 1 , . . . , n . Then, the weighted generalized distance measures between A ˘ and B^• is defined for ζ > 0 as, d ( A ˘ , B • ) = ( 1 2 n ∑ ȷ = 1 n w ȷ | ∑ ȷ = 1 n h ˆ k ( α ȷ / p ȷ ) - ∑ ȷ = 1 n h ˆ k ( β ȷ / p ȷ ) | ζ ) 1 ζ ,(14) where w_ȷ ∈ [0, 1] , with w_ȷ (ȷ =1, . . . , n) and ∑ ȷ = 1 n w ȷ = 1

If we put ζ = 1, then Equation (14) is called a weighted Hamming distance measure.

Definition 9. Let A ˘ ȷ = { g ˘ , h ˆ k ( α ȷ / p ȷ ) / g ˘ ∈ T ˇ } and B ȷ • = { g ˘ , h ˆ k ( β ȷ / p ) / g ˘ ∈ T ˇ } , be any two PHFS sets. Then, the generalized distance measure is defined as; d ( A ˘ , B • ) = ( 1 2 | h ˆ k ( α ȷ / P ȷ ) - h ˆ k ( β ȷ / p ȷ ) | ζ ) 1 ζ ,(15) then the distance measure defined in Equation (15), satisfies the below properties;

0 ≤ d ( A ˘ , B • ) ≤ 1 ,

Definition 10. [14] Let A ˘ = { A ˘ 1 , . . . , A ˘ n } be any PHFSs, where A ˘ ı = { h ˆ ( α ı / p ı ) } is a PHFN for ı=1, . . . , n . The entropy measure for PHFS by defined as; E ˘ ( A ˘ ) = 1 - 2 n ∑ ȷ = 1 n [ 1 k ∑ k h ˆ ( α ı / p ı ) - 1 2 ] .(16)

Properties of entropy measure;

Let A ˘ and B^• are the PHFSs in universal set T ˇ , the entropy measure E ˘ ( A ˘ ) and E ˘ ( B • ) satisfies the following properties;

( a ˘ 1 ) E ˘ ( A ˘ ) = 0 , iff A ˘ is a crisp set;

( a ˘ 2 ) E ˘ ( A ˘ ) = 1 , iff A ˘ is a most PHFs;

( a ˘ 3 ) E ˘ ( A ˘ ) = E ˘ ( A ˘ c ) ;

( a ˘ 4 ) E ˘ ( A ˘ ) ≤ E ˘ ( B • ) , if h ˆ ( α ı / p ı ) ≤ h ˆ ( β ı / p ı ) ≤ 1 / 2 or 1 / 2 ≤ h ˆ ( β ı / p ı ) ≤ h ˆ ( α ı / p ı ) .

F(o). llows form [14]. □

4.1 Probabilistic hesitant fuzzy MCGDM problem

Let S ˆ = { S ˆ 1 , . . . , S ˆ n } be the family of alternatives, and let C ={ C₁, . . . , C _m  } be the family of attribute. Let there also be a number of experts, E _k (k = 1, . . . , e), to express their opinion on n alternatives with respect to the m attributes by using PHFs H ˆ ı ȷ ( k ) = ( h ˆ k ( β ı / p ı ) ) ı ȷ . The decision matrix of the kth experts as. H ˆ ( k ) = [ H ˆ ı ȷ ( k ) ] m × n = [ ( h ˆ k ( β ı / p ı ) ) ı ȷ ] m × n , where H ^ ( k ) = S ^ 1 S ^ 2 . . S ^ n [ C 1 C 2 . . C n H ^ 11 ( k ) H ^ 12 ( k ) . . H ^ 1 n ( k ) H ^ 21 ( k ) H ^ 22 ( k ) . . H ^ 2 n ( k ) . . . . . . . . . . H ^ m 1 ( k ) H ^ m 2 ( k ) . . H ^ m n ( k ) ]

It is represent that the all the about given information the weight of experts and attribute both is unknown in the decision- making.

4.2 PHF-TOPSIS method

This section his main three steps. The first step, TOPSIS-based aggregation method for finding the weight of experts. The second step of the criteria given weight utilizing the entropy weight. The third step is a scoring process based on the grade of similarity with P • I ˜ S ˆ ( k ) and N I ˜ S ˆ to the ideal solution. The many steps were provided to solve the probabilistic hesitant MAGDM problem utilizing TOPSIS-based method:

Step 1. In the first place, the information presented by the group of experts is structured, and we have a decision matrix in the profit parameters used only.

H ˆ ı ȷ ( k ) = ( h ˆ k ( β ı / p ı ) ) ı ȷ , where k = 1, . . . , e, ı =1, . . . , m, ȷ =1, . . . , n,

The normalized decision matrices (NEs) are therefore presented as follows: D ˇ ( k ) = H ˆ ı ȷ ( k ) = = S ^ 1 S ^ 2 . . S ^ n [ C 1 C 2 . . C n H ^ 11 ( k ) H ^ 12 ( k ) . . H ^ 1 n ( k ) H ^ 21 ( k ) H ^ 22 ( k ) . . H ^ 2 n ( k ) . . . . . . . . . . H ^ m 1 ( k ) H ^ m 2 ( k ) . . H ^ m n ( k ) ] for k = 1, . . . , e .

Step 2. If the weights of expert are completely unknown, the final decision result can not be executed. The weights of the expert are therefore calculated as follows at first:

I . The expert opinion is similar to the group opinion (or I ˜ O ı ȷ ⋄ ), so the best I ˜ O ı ȷ ⋄ should be obtained by an aggregate of all expert opinions. Thus, I ˜ O ı ȷ ⋄ is obtained by taking probabilistic hesitant fuzzy weighted average of the alternatives’ decision values corresponding to the attribute given by the experts, considering the same weights of experts at the initial stage; I ˜ O ⋄ = [ I ˜ O 11 ⋄ I ˜ O 12 ⋄ ... I ˜ O 1 n ⋄ I ˜ O 21 ⋄ I ˜ O 21 ⋄ ... I ˜ O 2 n ⋄ . . . . I ˜ O m 1 ⋄ I ˜ O m 2 ⋄ ... I O m n ⋄ ] where, I ˜ O ı ȷ ⋄ = ∑ k = 1 e [ 1 e H ˆ ı ȷ ( k ) ] = { 1 - ∏ k = 1 e { 1 - H ˆ ı ȷ ( k ) } 1 e } II. Calculate the right ideal opinion ( R ˇ I ˜ O ∘ ) and the left ideal opinion ( L ´ I ˜ O • ). We have, R ˇ I ˜ O ı ° = [ R ˇ I ˜ O 11 ° R ˇ I ˜ O 12 ° ... R ˇ I ˜ O 1 n ° R ˇ I ˜ O 21 ° R ˇ I ˜ O 22 ° ... R ˇ I ˜ O 2 n ° . . ... . R ˇ I ˜ O m 1 ° R ˇ I ˜ O m 2 ° ... R ˇ I ˜ O m n ° ] and L ´ I ˜ O • = [ L ´ I ˜ O 11 • L ´ I ˜ O 12 • ... L ´ I ˜ O 1 n • L ´ I ˜ O 21 • L ´ I ˜ O 22 • ... L ´ I ˜ O 2 n • . . ... . L ´ I ˜ O m 1 • L ´ I ˜ O m 2 • ... L ´ I ˜ O m n • ] where R ˇ I ˜ O ı ȷ ∘ = { H ˆ ı ȷ ( k ) : max k [ S ( H ˆ ı ȷ ( k ) ) ] } and L ´ I ˜ O ı ȷ • = { H ˆ ı ȷ ( k ) : min k [ S ( H ˆ ı ȷ ( k ) ) ] } for ı=1, . . . , m, ȷ =1, . . . , n,

III . With Equation (13), determined the distances of each personal decision matrix D ˇ ( k ) from I ˜ O , R ˇ I ˜ O ∘ , and L ´ I ˜ O • , identified as D ˇ I ˜ O ⋄ ( k ) , D ˇ R ˇ I ˜ O ∘ ( k ) , and D ˇ L ´ I ˜ O • ( k ) , respectively, we have D ˇ I ˜ O ı ⋄ ( k ) = [ 1 2 n ∑ ȷ = 1 n { | ∑ ȷ = 1 n H ˆ ı ȷ ( k ) - ∑ ȷ = 1 n I ˜ O ı ȷ ⋄ | ζ } ] 1 ζ D ˇ R ˇ I ˜ O ı ∘ ( k ) = [ 1 2 n ∑ ȷ = 1 n { | ∑ ȷ = 1 n H ˆ ı ȷ ( k ) - ∑ ȷ = 1 n R ˇ I ˜ O ı ȷ ∘ | ζ } ] 1 ζ D ˇ L ´ I ˜ O ı • ( k ) = [ 1 2 n ∑ ȷ = 1 n { | ∑ ȷ = 1 n H ˆ ı ȷ ( k ) - ∑ ȷ = 1 n L ´ I ˜ O ı ȷ • | ζ } ] 1 ζ for ı=1, . . . , m and k = 1, . . . , e .

IV. The closeness indices ( C ˇ I ˜ s ) are used as below: C ˇ I ˜ k = ∑ ı = 1 m D ˇ R ˇ I ˜ O ı ∘ ( k ) + ∑ ı = 1 m D ˇ L ´ I ˜ O ı • ( k ) ∑ ı = 1 m D ˇ I ˜ O ı ⋄ ( k ) + ∑ ı = 1 m D ˇ R ˇ I ˜ O ı ∘ ( k ) + ∑ ı = 1 m D ˇ L ´ I ˜ O ı • ( k ) for k = 1 , . . . , e

V. The experts weights are calculated as follows: η k ∗ = C ˇ I ˜ k ∑ k = 1 e C ˇ I ˜ k

Step 3. The attributes weights are obtain by using the entropy measure as follows:

I. The revised ideal opinion ( REV I ˜ O ⋄ ) are calculate as; REV I ˜ O ı ȷ ⋄ = oplus k = 1 e [ η k ∗ . H ˆ ı ȷ ( k ) ] = { 1 - ∏ k = 1 e ( 1 - H ˆ ı ȷ ( k ) ) η k ∗ } II . The entropy measure relation to each attributes are obtain by which using Equation (16) which as; EM ȷ = E ( REV I ˜ O ı ȷ ⋄ , . . . , REV I ˜ O m ȷ ⋄ ) , ȷ = 1 , . . . , n . III . Using the entropy weighted, this the weight of attributes as follows; W ˆ C ˇ ȷ = 1 - EM ȷ n - ∑ ȷ = 1 n EM ȷ Thus, we have the following weights of the criteria as W ˆ C ˇ = ( W ˆ C ˇ 1 , … , W ˆ C ˇ n ) T

Step 4. Again, the TOPSIS method is utilized to classify the alternatives to each expert as follows through the evaluation of the attribute weights. We have

I. Obtained weighted NE ı ȷ k by the weight vector of the attributes calculated as: N E ȷ ı ( k ) = W ^ C ˇ ȷ . H ^ ȷ ı ( k ) = { 1 − ( 1 − H ^ ȷ ı ( k ) ) W ^ C ˇ ȷ } foreach k=1 , ... , e II. P • I ˜ S ˆ ( k ) and N I ˜ S ( k ) are calculated for each NE^(k), weighted for each E^(k) (k = 1, . . . , e) in the following method: P • I ˜ S ˆ ( k ) = { P • I ˜ S ˆ ȷ ( k ) } e × n = { NE ı ȷ ( k ) : max ı [ s ( NE ı ȷ ( k ) ) ] } for ȷ = 1 , . . . , n N I ˜ S ˆ ( k ) = { N I ˜ S ˆ ȷ ( k ) } e × n = { NE ı ȷ ( k ) : min ı [ s ( NE ı ȷ ( k ) ) ] } for ȷ = 1 , . . . , n III. Determined weights of the attributes are taken into consideration. Utilizing Equation (9), the weighted distances NE ı ȷ ( k ) from P • I ˜ S ˆ ( k ) and N I ˜ S ˆ ( k ) are shown by D ˇ I ˜ S ´ ı ∔ and D ˇ I ˜ S ´ ı - respectively, and are determined for each E _k , as follows: D ˇ I ˜ S ´ ı ∔ ( k ) = [ 1 2 n ∑ ȷ = 1 n W ˆ C ˇ ȷ { | ∑ ȷ = 1 n NE ı ȷ ( k ) - ∑ ȷ = 1 n P • I ˜ S ˆ k | ζ } ] 1 ζ D ˇ I ˜ S ´ ı - ( k ) = [ 1 2 n ∑ ȷ = 1 n W ˆ C ˇ ȷ { | ∑ ȷ = 1 n NE ı ȷ ( k ) - ∑ ȷ = 1 n N I ˜ S ˆ k | ζ } ] 1 ζ

IV. The following the revised closeness indices ( R ˇ C ˇ I ˜ ) of the alternatives are calculated for each E _k , as R ˇ C ˇ I ˜ ı k = D ˇ I ˜ S ´ ı - ( k ) D ˇ I ˜ S ´ ı - ( k ) + D ˇ I ˜ S ´ ı ∔ ( k )

Step 5. Using the calculated weights of experts η k ∗ , R ˇ C ˇ I ˜ s the final revised closeness index F R ˇ C ˇ I ˜ ı for each alternative is aggregated as; F R ˇ C ˇ I ˜ ı = ∑ k = 1 e η k ∗ . R ˇ C ˇ I ˜ ı k The greater F R ˇ C ˇ I ˜ ı is the most suitable alternative.