Analysis of the suitability of the solar panels for hospitals: A new fuzzy decision-making model proposal with the T-Spherical TOP-DEMATEL method

3.1 General information about spherical fuzzy sets and DEMATEL

Decision making problems are becoming more and more complex. As a result, these problems have become very difficult to solve. This situation has made it necessary to make new improvements to decision-making techniques. In this context, many researchers have preferred to integrate these techniques with fuzzy numbers. On the other hand, the very serious increase in complexity in decision-making problems also necessitated the generation of new fuzzy numbers. Spherical fuzzy numbers have also been developed for this purpose [13]. In these numbers, experts can assign hesitation information for the criteria. In other words, the degrees of membership, non-membership and hesitancy can be taken into consideration [14]. This situation is accepted as the main advantage of Spherical fuzzy sets.

On the other side, DEMATEL technique is used to weight different items. In this process, the evaluations of the experts are considered to create direct relation matrix. After that, this matrix is normalized, and total relation matrix is created [12]. The main difference of DEMATEL from other techniques is that impact relation map can be created. The threshold value in the total relation matrix is taken into consideration to understand the causal directions among the alternatives. This situation is the main superiority of the DEMATEL technique by comparing with other approaches [9].

3.2 T-spherical fuzzy DEMATEL

In this study, a new model is created by using T-Spherical fuzzy TOP-DEMATEL technique for the purpose of evaluating different factors that affect the solar panel investment decision for the hospitals. Thus, in this title, firstly, the steps of T-Spherical fuzzy DEMATEL are detailed. Equation (1) gives information about the condition for the fuzzy sets based on t values [16].

0 ⩽ μ A t ( u ) + η A t ( u ) + υ A t ( u ) ⩽ 1(1) In this equation, t can be any natural number. A refers to a cluster in the discourse universe (U). On the other side, Equation (2) gives information about the degree of refusal (r). In this framework, the higher value of t leads to increase the value of r [17].

r A ( u ) = 1 - ( μ A t ( u ) + η A t ( u ) + υ A t ( u ) ) t(2)

If the value of 2 is taken as t, the Spherical fuzzy number is obtained. In addition, if t = 1 is taken, the Pythagorean fuzzy number is formed. On the other hand, if η=0, the q-rung orthopair fuzzy sets (q-ROFSs) value is reached. Moreover, if η=0 and t = 1 conditions are met, intuitionistic fuzzy sets (IFSs) are obtained. Only μ value is taken, other components are accepted as 0, and if t = 1, it becomes the classical fuzzy number. Equation (3) shows the T-Spherical soft-weighted arithmetic mean (TSFAM) [18].

TSFWA M W ( A ˜ S 1 , A ˜ S 1 , … A ˜ Sn ) = { [ 1 - ∏ i = 1 n ( 1 - μ A ˜ Si t ) w i ] 1 t , ∏ i = 1 n η A ˜ Si w i , ∏ i = 1 n ν A ˜ Si w i }(3) where ∑ i = 1 n w i = 1 , w i ∈ [ 0 , 1 ]

In this proposed model, T-Spherical fuzzy sets are integrated with DEMATEL methodology for the purpose of reaching more effective results. The calculation steps of this technique are demonstrated as following.

Step 0: Expert opinions are obtained, and expert evaluation matrix is created for each opinion. In this process, the linguistic expression equivalents in Table 1. are used.

The experts make evaluations by considering these five different scales stated in Table 1. After that, these values are converted into membership (μ), non-membership (η) and hesitancy (υ) degrees given in this table. Additionally, Equation (4) indicates the expert evaluation matrix [16].

D i = [ 0 ⋯ ( μ 1 n i , η 1 n i , ν 1 n i ) ⋮ ⋱ ⋮ ( μ n 1 i , η n 1 i , ν n 1 i ) ⋯ 0 ](4)

Step 1: Average value of the expert evaluations (D) are calculated as in Equation (5).

D = [ 0 ⋯ ( μ 1 n d , η 1 n d , ν 1 n d ) ⋮ ⋱ ⋮ ( μ n 1 d , η n 1 d , ν n 1 d ) ⋯ 0 ](5)

Step 2: D matrix is divided into three submatrices as μ, η, υ. The three submatrices are then normalized with Equations (6) and (7).

X = sD(6)

s = min [ 1 max i ∑ j = 1 n | d ij | , 1 max j ∑ i = 1 n | d ij | ](7) With the above normalization, three matrixes are obtained as in Equation (8) [17].

X μ = [ 0 ⋯ μ 1 n ⋮ ⋱ ⋮ μ n 1 ⋯ 0 ] X η = [ 0 ⋯ η 1 n ⋮ ⋱ ⋮ η n 1 ⋯ 0 ] X ν = [ 0 ⋯ ν 1 n ⋮ ⋱ ⋮ ν n 1 ⋯ 0 ](8)

Step 3: The total relationship matrix (T) is calculated over the X submatrices as in Equation (9).

T = X * ( 1 - x ) - 1(9)

At this stage, the obtained 3 matrixes are subjected to the Euclidean normalization process to preserve the T-Spherical fuzzy number property. Thus, normalized submatrices are obtained.

Step 4: These three normalized submatrices are combined as the total relationship matrix (T) with Equation (10).

T = [ 0 ⋯ ( μ 1 n T , η 1 n T , ν 1 n T ) ⋮ ⋱ ⋮ ( μ n 1 T , η n 1 T , ν n 1 T ) ⋯ 0 ](10)

Step 5: The row and column sums (R and C) of matrix T are calculated. Equations (11) and (12) give information about these details.

r i = ∑ i = 1 n ( μ ij T , η ij T , ν ij T )(11)

c j = ∑ j = 1 n ( μ ij T , η ij T , ν ij T )(12)

Equation (13) is taken into consideration when calculating the sum of two different T-Spherical numbers (T _u ,  T _v ).

T u + T v = ( μ T u t + μ T v t - μ T u t * μ v t t , η Tu η Tv , ν T u ν T v )(13)

Step 6: R and C matrixes are defuzzified. For this purpose, Equation (14) is applied to both matrixes [37].

Score = μ t - η t - ν t(14)

Step 7: C + R is calculated from the scores of R and C matrixes and their order of importance (W) is determined with Equation (15).

W i = ( C + R ) i ∑ j = 1 n ( C + R ) j(15)

The DEMATEL method is based on normalizing the decision matrix, calculating the total relationship matrix with the help of the normalized matrix, and taking the row and column sums of the total relationship matrix. Then, criteria weights are calculated over the row and column totals. This weighting calculation, which is mentioned in the seventh step of the model proposed in this study and is frequently used for DEMATEL, has been criticized in the literature [10–12]. The reason for the criticism of this situation is that if the sum of the symmetrical values around the diagonal remains equal, the weights become equal. In other words, even if the experts think that the importance weights of the criteria are not equal, because of this step in the DEMATEL method, the importance weights of the factors specified are computed incorrectly equal. This situation also hinders the correct assessment. In the classical DEMATEL approach, the row and column sums (R and C) of matrix T are calculated in step 5 and these values are normalized in step 6 by Equation (14).

To overcome this weakness in the DEMATEL model, T matrix is normalized firstly with the help of Equation (14) instead of R and C values. After that, Equations (16)–(22) are taken into consideration within the concept of TOP-DEMATEL. These seven different equations are the steps of TOPSIS approach [40]. Because this method is created by integrating the steps of both DEMATEL and TOPSIS, this new technique is named as TOP-DEMATEL.

C * j = ∑ i = 1 n ( t i - max j t i ) 2 j = 1 , 2 , . . n(16) In Equation (16), C^* represents the sum of the distance to the positive ideal of the criteria on a column basis. Additionally, t _i values are the elements of the T matrix that is calculated by Equation (14). Moreover, max j t i demonstrates the maximum value of the jth column in T matrix [41].

C - j = ∑ i = 1 n ( t i - min j t i ) 2 j = 1 , 2 , . . n(17)

In this equation, C^- gives information about the sum of the distance to the negative ideal of the criteria on a column basis. On the other side, min j t i explains the minimum value of the jth column in T matrix [42].

R * i = ∑ j = 1 n ( t j - max i t j ) 2 i = 1 , 2 , . . n(18) In Equation (18), R^* identifies the sum of the distance to the positive ideal of the criteria on a row basis [40].

R - i = ∑ j = 1 n ( t j - max i t j ) 2 i = 1 , 2 , . . n(19) Within this framework, R^- shows the sum of the distance to the negative ideal of the criteria on a row basis [42].

S i * = C i * + R i *(20) In this equation, S* demonstrates the sum of the ideal positive degree distance sums based on rows and columns.

S i - = C i - + R i -(21) In Equation (21), S indicates the sum of the ideal negative degree distance sums based on rows and columns.

W i = S i - S i - + S i *(22)

On the sum of ideal positive and ideal negative sums calculated by Equation (20-21), criterion weights values (W) are calculated with the help of Equation (22) [41].

Step 1: By taking the average values of the expert evaluations, the D matrix is formed. The details of this matrix are given in Table 4. In this process, Equation (5) is taken into consideration.

Step 2: In this process, firstly, D matrix is divided into three submatrices with the name of μ, η, υ. After that, Equations (6) and (7) are considered to normalize them. Table 5 indicates the details of the normalized matrix. In this matrix, X^M, X^U and X^L give information about the sub matrixes obtained after the normalization process.

Step 3: While using Equation (9), total relationship matrix (T) is generated as three different submatrices. The details are shown in Table 6.

Step 4: By combining these submatrices, the final total relation matrix (T) can be formed with the help of Equation (10). Table 7 indicates the details of this matrix.

Step 5: Equations (11) and (12) are considered to compute the row and column sums (R and C) of matrix T as in Table 8.

Step 6: The defuzzified values of R and C matrixes are computed by Equation (14) as in Table 9.

Step 7: The weights (W) of the criteria are computed by considering the values of D + C. In this context, Equation (15) is taken into consideration. Table 10 demonstrates the details of the weights of the items.

Table 10 gives information about the weights of the criteria when t value equals to 1. In this table, C and R demonstrate the sums of the row and column whereas W indicates the weights. Based on these results, it is determined that generating own energy in the long run is the most crucial factor while making decision about the solar panel investments to the hospitals. Additionally, improving the image is the second most important item in this context. Similarly, minimizing the price risk in the energy market is also significant factor for this situation. The findings demonstrate that the advantages of the solar panels have the higher weights in comparison with the drawbacks of these investments. Hence, it is understood that it is quite appropriate to make solar panel investments for the hospitals. Additionally, for the purpose of measuring the coherency of the findings, analysis results are also calculated for different t values. Comparative analysis results based on various t values are indicated in Table 11.

Table 11 states that the first three criteria that show the advantages of the solar panels have the greater weights by comparing with the last three factors which define the drawbacks. This situation gives information about the coherency and reliability of the findings. On the other side, it is also identified that when t value increases, the significance of the criterion 1 goes up as well. Therefore, it is understood that in case of high uncertainties, generating own energy in the long run plays more critical role for the hospitals.

In this study, a new technique is created by the name of TOP-DEMATEL while integrating the critical steps of DEMATEL and TOPSIS methods. Therefore, it is aimed to overcome the criticisms regarding the weighting calculation steps of classical DEMATEL approach. Within this framework, firstly, T matrix is Table 7 is normalized by the help of Equation (14). The details of the defuzzified T matrix are demonstrated in Table 12.

In the next stage, Equations (16)-(22) implemented to the defuzzified total relation matrix. Table 13 states the critical values of this new technique.

In Table 13, C^* and ^C show the sums of the distance to the positive and negative ideal of the criteria on a column basis. On the other side, R^* and ^R explain these sums on the row basis. Moreover, S* demonstrates the sum of the ideal positive degree distance sums based on rows and columns. Similarly, ^S indicates the sum of the ideal negative degree distance sums based on rows and columns. Finally, W demonstrates the weights. Additionally, In the final step, the weights of the criteria are also computed for different t values. The weighting results are indicated in Table 14.

Table 14 explains that generating own energy in the long run is the most crucial item for all different t values. Because this result is the same for all two different analyses, it is understood that the findings are reliable. This situation gives information that the solar panels should be taken into consideration for the hospitals because they will minimize energy dependency of the hospitals. However, the results of T-Spherical fuzzy TOP-DEMATEL indicate that the high initial investment cost is the second most critical factor in this respect. This finding is quite different by comparing with the results of T-Spherical fuzzy DEMATEL. Hence, it is seen that cost effectiveness should also be taken into consideration for the decision of generating the solar panels in the hospitals.