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

Abstract

In this study, a new fuzzy decision-making model is created to evaluate whether the solar panels are efficient to minimize energy costs of the hospitals. The weights of the criteria are calculated by considering T-Spherical fuzzy decision-making trial and evaluation laboratory (DEMATEL) method. Moreover, for the purpose of measuring the coherency of the findings, analysis results are also calculated for different t values. Additionally, by making improvements to some criticisms to the classical DEMATEL method, a new technique is created by the name of TOP-DEMATEL while integrating some steps of technique for order preference by similarity to ideal solution (TOPSIS) to the DEMATEL technique. The main novelty of this study is that it is analyzed whether the solar panels are effective in reducing the costs of hospitals with an original decision-making model. It is concluded that generating own energy in the long run is the most crucial item according to both T-Spherical fuzzy DEMATEL and TOP-DEMATEL methods. The analysis results are quite similar for different t values. This situation gives information about the coherency and reliability of the findings. 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. On the other side, 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 TOP-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.

Keywords

T-Spherical fuzzy sets TOP-DEMATEL solar energy health industry

1 Introduction

Providing health services to the people is quite significant [1]. Controlling these costs is essential to ensure sustainability of these services. The highest cost item for healthcare businesses is personnel expenses and the purchase of machinery and equipment. However, it is very difficult to intervene in these cost items. One of the controllable cost items for hospitals is energy [2]. The health sector is one of the sectors that consume the most energy in the category of buildings. According to the study conducted in Germany, energy expenses for hospitals constitute 2.5% of the total hospital expenses [3]. According to another study, the sum of fuel, electricity and water expenses constitute more than 10% of the total expenses of hospitals [4].

One reason why energy costs are so high is foreign dependence on energy [5]. If hospitals have the renewable energy sources, such as solar and wind energy, energy dependency will be reduced, sustainability will be ensured, and hospitals will be able to produce their own energy [6]. In addition, the use of environmentally friendly energy will give hospitals a positive image and increase the preference of the hospital in question by the patients. In this way, hospitals will both save energy by using the renewable energy and increase their income owing to the positive image it has increased [7].

The positive as well as the negative aspects of the solar panel installation are mentioned. For this reason, it remains unclear whether the solar panels will be applied in hospitals. New studies are needed to clarify the issue of installing the solar panels in hospitals. In future studies, both positive and negative aspects of the solar panels should be taken into consideration at the same time. In this way, it can be possible to determine the right strategies to reduce the energy costs of hospitals.

In this study, it will be focused on whether solar panel installation reduces energy costs in hospitals. With the help of this analysis, it can be possible to determine which factor stands out in the installation of the solar panels in hospitals. In this context, a new fuzzy decision-making model is proposed. In this framework, first, six different criteria that are effective on solar energy investments are determined with the help of literature review. Three of these criteria show the positive aspects of the solar panels, while the others explain the disadvantages of solar energy investments. After that, the importance weights of the criteria are determined using the T-Spherical fuzzy decision-making trial and evaluation laboratory (DEMATEL) method. In addition, to test the coherency of the findings, analysis results are also calculated for different t values. Furthermore, while making improvements to some criticisms to the classical DEMATEL method, a new technique is created while integrating some steps of technique for order preference by similarity to ideal solution (TOPSIS) to the DEMATEL technique. This new technique is called as TOPSIS based DEMATEL (TOP-DEMATEL) method. Thanks to this analysis, it will be clear whether the energy costs of hospitals can be reduced by installing the solar panels.

The main novelty of this study is that it is analyzed whether the solar panels are effective in reducing the costs of hospitals with an original decision-making model. There are also some superiorities of the proposed model in comparison with other previous ones. In this model, a new technique is created by the name of TOP-DEMATEL. There are many different techniques considered in the literature in calculating criterion weights. The biggest advantage of the DEMATEL method compared to others is that the causal relationship between the factors can be determined [8, 9]. There may also be a causal relationship between the factors for the installation of the solar panels. For example, the ability of hospitals to produce their own energy in the long run can contribute to the improvement of the image of hospitals. Therefore, considering the DEMATEL method while analyzing these elements may offer some advantages.

However, the final steps of the DEMATEL technique are criticized in the literature because if the sum of the symmetrical values around the diagonal remains equal, the weights become equal [10]. 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 [11, 12]. To overcome this weakness in the DEMATEL model, the final steps of the TOPSIS method are integrated to the DEMATEL methodology. Because of the integration of the steps of these two techniques, this new approach is named as TOP-DEMATEL. With the help of this new technique, the criticized issues of the DEMATEL can be improved. This situation provides an important superiority to the proposed model.

Considering Spherical fuzzy sets provide some advantages to the proposed model [13]. These sets include both membership and non-membership degrees and hesitancy parameter [14]. With the help of this issue, a larger domain can be taken into consideration. This situation helps to reach more accurate findings [15]. Using T-Spherical fuzzy sets increase the superiority and originality of the proposed model. First, the cases of other fuzzy numbers are also covered according to the value of the t expression [16]. On the other hand, t value can be defined as the level of inclusion of uncertainty. Therefore, when the value of t increases, the uncertainty included in the concept also increases [17]. Thus, the scope can be much more perceptive. The installation of the solar panels in hospitals is also a subject that is widely discussed in the literature. Therefore, some of the criteria include negative aspects and some include positive aspects. In this context, the uncertainties within the determined concepts will be handled more clearly thanks to T-Spherical fuzzy sets.

Moreover, the uncertainty within the criteria can change over time. In other words, this may cause the criteria to include more elements. As a result, there may be changes in the importance weight of the criteria. With the help of T-Spherical fuzzy numbers, criterion weights can be calculated according to the t value [18]. In this way, the possibility of change in the weights of the criteria can be taken into consideration. The issue of installing the solar panels in hospitals also includes some uncertainties. It is possible that some of the negativities in this process can be reduced over time, especially thanks to the developing technology. In other words, the importance weight of a negative factor may vary over time. Therefore, it is understood that the model established with the T-Spherical fuzzy number is very suitable for the installation of the solar panels in hospitals.

The following section gives information about the literature review. The details of the methodology are explained in the third section. The fourth section includes analysis results. Discussions and conclusions are given in the final two sections.

2 Literature review

The use of the solar panels is still a matter of debate in the literature. Some studies focused on the significance of the initial costs of these investments. Zhang et al. [19] made an integrated hesitant 2-tuple Pythagorean fuzzy analysis to evaluate the costs of the renewable energy projects. They stated that the initial installation cost of the solar panels is very high. Jadhao et al. [20] also examined the performance of the solar panels. They discussed that the solar panels have high initial costs. Therefore, these costs should be handled effectively to improve the performance of the solar panels. Piotrowski and Farret [21] made a feasibility analysis regarding the performance of photovoltaic panels. It is identified that the problem of high initial costs should be solved with the help of technological improvements. Myyas et al. [22] defined that high initial cost problem moves hospitals away from the renewable energy investment.

In addition, maintenance and repairs of the solar panels need to be done periodically. This situation has a decreasing impact on the preference of solar panel investments. Cole et al. [23] explored the cost implications of the renewable energy projects in the United States. It is identified that for the sustainability of these projects, necessary actions should be taken to minimize maintenance and repair costs. Barker et al. [24] also focused on the ways to increase the performance of the solar panels. They discussed that high quality materials should be selected for the solar panels so that maintenance and repair costs can be reduced. Similarly, Chowdhury et al. [25] and Aboagye et al. [26] pointed out the significance of the maintenance and repairs for the solar panels. They claimed that this is reflected in the hospital as extra time and cost.

Moreover, the efficiency to be obtained from the solar panels also depends on the climate factor. In other words, because of the different climatic conditions, the instability occurs in the amount of electricity produced. Ibrahim et al. [27] evaluated the renewable energy systems in Malaysia. They made a risk assessment with respect to the solar photovoltaic systems in this country. It is determined that different climatic conditions represent one of the most important risks for these investments. Ganti et al. [28] made environmental impact analysis regarding the solar panels. In this study, it is aimed to find critical factors affecting the photovoltaic energy utilization. They reached a conclusion that different climate factors play a crucial role for this issue. Similarly, Kennedy et al. [29] stated that in locations with low annual sunshine duration, this is also a matter of debate.

However, there are also some significant advantages of considering the solar panels for the hospitals. For instance, if hospitals decide to invest in the solar panels, they will have produced their own energy. Snow et al. [30] focused on the energy market in Australia. They identified that for the purpose of minimizing energy dependency, companies should create the solar panels. AlQallaf et al. [31] aimed to examine solar energy systems. They identified that solar photovoltaic panels can reduce the external dependency of the companies regarding the energy needs. Biswal and Asija et al. [32] concluded that considering the solar panels provides the opportunity to generate the own energy for the companies. This situation has a positive impact on the cost reduction in a long-term period.

At the same time, because the solar panels generate clean energy, it will positively affect the image of the companies. Brunet et al. [33] focused on the impacts of using solar energy in different countries. They discussed that the image of the companies can be improved on the eyes of the customers with the help of the generation of the green energy. Additionally, Nematchoua [34] also studied the new energy policies of the Madagascar island. It is defined that using clean energy increases the competitive power of the companies because of the positive image. Similarly, Arslan et al. [35] focused on the energy strategies of the companies. They reached a conclusion that companies that use environmentally friendly energy will be preferred more by customers.

Apart from this, the volatility problem in energy prices can be handled more effectively by the help of solar energy. There are lots of different factors that affect the energy prices, such as political issues, energy supply and demand. In this regard, when companies cannot create their own energy, they will be affected by the uncertainties in the energy market. Hemrit and Benlagha [36] stated that solar panel projects provide sustainable services without being affected by fluctuations in energy prices. Tsao et al. [37] and Alao and Cuffe [38] also concluded that solar energy helps to minimize the price risk of energy. In this way, it will be possible to prevent radical cost increases in the long run [39].

The literature review indicates that the solar panels have both advantages and disadvantages. Due to this issue, it remains unclear whether the solar panels will be applied in hospitals. Most of the studies in the literature focused on these issues of the solar panels separately. However, there are limited studies that make effectiveness analysis for the solar panel installation in the hospitals. Hence, there is a need for a new study that considers both positive and negative aspects of the solar panels at the same time. Owing to this situation, it can be possible to determine the right strategies to reduce the energy costs of hospitals. Accordingly, in this study, it is aimed to evaluate whether the solar panels are efficient to minimize energy costs of the hospitals by considering six different factors. Three of these criteria show the positive aspects of the solar panels, while the others explain the disadvantages of solar energy investments. With the help of this issue, it can be understood whether the solar panels are efficient to minimize energy costs of the hospitals.

3 Methodology

In this study, a new technique is generated with the name of TOP-DEMATEL by integrating TOPSIS and DEMATEL methods. With the help of this situation, a novel model is proposed by using TOP-DEMATEL approach based on T-Spherical fuzzy sets. Hence, firstly, general information is given about Spherical fuzzy sets and DEMATEL. Next, the steps of T-Spherical fuzzy DEMATEL are detailed. Finally, new enhancements to this model and the development of the T-Spherical TOP-DEMATEL method will be explained.

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].

$r_{A} (u) = \sqrt[t]{1 - (μ_{A}^{t} (u) + η_{A}^{t} (u) + υ_{A}^{t} (u))}$ (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].

$\begin{matrix} TSFWA M_{W} ({\tilde{A}}_{S 1}, {\tilde{A}}_{S 1}, \dots {\tilde{A}}_{Sn}) \\ = {{[1 - \prod_{i = 1}^{n} {(1 - μ_{{\tilde{A}}_{Si}}^{t})}^{w_{i}}]}^{\frac{1}{t}}, \prod_{i = 1}^{n} η_{{\tilde{A}}_{Si}}^{w_{i}}, \prod_{i = 1}^{n} ν_{{\tilde{A}}_{Si}}^{w_{i}}} \end{matrix}$ (3) where $\sum_{i = 1}^{n} w_{i} = 1, w_{i} \in [0, 1]$

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.

Table 1

Scales and degrees

Scales	μ	η	ν
4	,85	,15	,45
3	,6	,2	,35
2	,35	,25	,25
1	0	,3	,15
0	0	0	0

μ: membership, η: non-membership, ν: hesitancy.

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].

$D^{i} = [\begin{matrix} 0 & \dots & (μ_{1 n}^{i}, η_{1 n}^{i}, ν_{1 n}^{i}) \\ ⋮ & ⋱ & ⋮ \\ (μ_{n 1}^{i}, η_{n 1}^{i}, ν_{n 1}^{i}) & \dots & 0 \end{matrix}]$ (4)

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

$D = [\begin{matrix} 0 & \dots & (μ_{1 n}^{d}, η_{1 n}^{d}, ν_{1 n}^{d}) \\ ⋮ & ⋱ & ⋮ \\ (μ_{n 1}^{d}, η_{n 1}^{d}, ν_{n 1}^{d}) & \dots & 0 \end{matrix}]$ (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)

$s = min [\frac{1}{max_{i} \sum_{j = 1}^{n} | d_{ij} |}, \frac{1}{max_{j} \sum_{i = 1}^{n} | d_{ij} |}]$ (7) With the above normalization, three matrixes are obtained as in Equation (8) [17].

$\begin{matrix} X^{μ} = [\begin{matrix} 0 & \dots & μ_{1 n} \\ ⋮ & ⋱ & ⋮ \\ μ_{n 1} & \dots & 0 \end{matrix}] X^{η} = [\begin{matrix} 0 & \dots & η_{1 n} \\ ⋮ & ⋱ & ⋮ \\ η_{n 1} & \dots & 0 \end{matrix}] \\ X^{ν} = [\begin{matrix} 0 & \dots & ν_{1 n} \\ ⋮ & ⋱ & ⋮ \\ ν_{n 1} & \dots & 0 \end{matrix}] \end{matrix}$ (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).

$T = [\begin{matrix} 0 & \dots & (μ_{1 n}^{T}, η_{1 n}^{T}, ν_{1 n}^{T}) \\ ⋮ & ⋱ & ⋮ \\ (μ_{n 1}^{T}, η_{n 1}^{T}, ν_{n 1}^{T}) & \dots & 0 \end{matrix}]$ (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.

$r_{i} = \sum_{i = 1}^{n} (μ_{ij}^{T}, η_{ij}^{T}, ν_{ij}^{T})$ (11)

$c_{j} = \sum_{j = 1}^{n} (μ_{ij}^{T}, η_{ij}^{T}, ν_{ij}^{T})$ (12)

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

$\begin{matrix} T_{u} + T_{v} \\ = (\sqrt[t]{{μ_{T_{u}}}^{t} + {μ_{T_{v}}}^{t} - {μ_{T_{u}}}^{t} * {μ_{v}}^{t}}, η_{Tu} η_{Tv}, ν_{T_{u}} ν_{T_{v}}) \end{matrix}$ (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).

$W_{i} = \frac{{(C + R)}_{i}}{\sum_{j = 1}^{n} {(C + R)}_{j}}$ (15)

3.3 T-Spherical fuzzy TOP-DEMATEL

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.

$\underset{j}{C^{*}} = \sqrt{\sum_{i = 1}^{n} {(t_{i} - max_{j} t_{i})}^{2}} j = 1, 2, . . n$ (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].

${C^{-}}_{j} = \sqrt{\sum_{i = 1}^{n} {(t_{i} - min_{j} t_{i})}^{2}} j = 1, 2, . . n$ (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].

$\underset{i}{R^{*}} = \sqrt{\sum_{j = 1}^{n} {(t_{j} - max_{i} t_{j})}^{2}} i = 1, 2, . . n$ (18) In Equation (18), R^* identifies the sum of the distance to the positive ideal of the criteria on a row basis [40].

${R^{-}}_{i} = \sqrt{\sum_{j = 1}^{n} {(t_{j} - max_{i} t_{j})}^{2}} i = 1, 2, . . n$ (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.

$W_{i} = \frac{S_{i}^{-}}{S_{i}^{-} + S_{i}^{*}}$ (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].

4 Analysis results

In this study, it is aimed to determine whether the solar panel investments are effective for hospitals. For this purpose, firstly, similar studies in the literature are evaluated. Based on this analysis, six different factors are identified as stated in Table 2.

Table 2
Selected criteria

Criteria Literature

Generating own energy in the long run (C1) [20, 31]

Improving the image (C2) [33, 34]

Minimizing the price risk in energy (C3) [38, 39]

High initial investment cost (C4) [19, 20]

Maintenance and repair considerations (C5) [23, 24]

Instability due to climatic conditions (C6) [28, 29]

Criteria	Literature
Generating own energy in the long run (C1)	[20, 31]
Improving the image (C2)	[33, 34]
Minimizing the price risk in energy (C3)	[38, 39]
High initial investment cost (C4)	[19, 20]
Maintenance and repair considerations (C5)	[23, 24]
Instability due to climatic conditions (C6)	[28, 29]

Table 1 indicates that the solar panels have both advantages and disadvantages. As a result of the literature review, six different factors (3 positive, 3 negative) are determined. The biggest positive effect of the solar panels on hospitals is the possibility of obtaining their own energy in the long term. In addition, thanks to the solar panels, hospitals can produce clean energy, which can improve the image of hospitals in a positive way. Furthermore, thanks to the fact that hospitals can produce their own energy, the risk of increase in energy prices can be minimized.

On the other hand, it is possible to talk about some of the disadvantages of the solar panels. First, the initial investment cost of the solar panels is high. This situation negatively affects the profitability of hospitals in the first place. In addition, periodic maintenance and repairs of the solar panels also create some difficulties. Finally, the solar energy can be adversely affected by climatic conditions. This situation causes the amount of energy obtained to be unstable.

In the next process, an expert team is created with three different decision makers. Two of these people are top managers in solar energy companies. They have joined many critical solar energy investment projects. Additionally, one of these people is the academician in energy investment area. She has lots of significant articles with respect to the investment decisions of the solar panels. All of these people have minimum 23-year of experience. Table 3 gives information about the evaluations of the expert team.

Table 3

Evaluations

DM1
Criteria	C1	C2	C3	C4	C5	C6
C1	0	4	4	3	2	3
C2	2	0	2	3	2	2
C3	2	3	0	2	2	2
C4	1	2	2	0	2	1
C5	1	2	2	1	0	2
C6	2	1	3	1	1	0
DM2
Criteria	C1	C2	C3	C4	C5	C6
C1	0	4	4	3	3	2
C2	2	0	3	2	2	2
C3	3	3	0	3	3	1
C4	2	2	3	0	2	1
C5	1	1	1	1	0	1
C6	3	1	2	1	2	0
DM3
Criteria	C1	C2	C3	C4	C5	C6
C1	0	3	4	2	3	3
C2	3	0	2	2	3	2
C3	3	2	0	1	2	2
C4	1	1	1	0	1	2
C5	1	1	2	1	0	2
C6	2	1	2	1	1	0

DM: decision maker; C: criterion.

In Table 3, DM represents decision-makers whereas C gives information about the criteria. In this table, five different scales stated in Table 1 are taken into consideration. After that, a new model is created by T-Spherical fuzzy TOP-DEMATEL for the purpose of weighting these criteria. In the following process, first, the analysis results of T-Spherical fuzzy DEMATEL will be shared when t value equals to 1. After that, a comparative evaluation is also made by using T-Spherical fuzzy TOP-DEMATEL.

4.1 Analysis results of T-Spherical Fuzzy DEMATEL

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.

Table 4
D Matrix

Criteria C1 C2 C3 C4 C5 C6

C1 ,00 ,00 ,00 ,79 ,17 ,41 ,85 ,15 ,45 ,53 ,22 ,31 ,53 ,22 ,31 ,53 ,22 ,31

C2 ,45 ,23 ,28 ,00 ,00 ,00 ,45 ,23 ,28 ,45 ,23 ,28 ,45 ,23 ,28 ,35 ,25 ,25

C3 ,53 ,22 ,31 ,53 ,22 ,31 ,00 ,00 ,00 ,36 ,25 ,24 ,45 ,23 ,28 ,25 ,27 ,21

C4 ,13 ,28 ,18 ,25 ,27 ,21 ,36 ,25 ,24 ,00 ,00 ,00 ,25 ,27 ,21 ,13 ,28 ,18

C5 ,00 ,30 ,15 ,13 ,28 ,18 ,25 ,27 ,21 ,00 ,30 ,15 ,00 ,00 ,00 ,25 ,27 ,21

C6 ,45 ,23 ,28 ,00 ,30 ,15 ,45 ,23 ,28 ,00 ,30 ,15 ,13 ,28 ,18 ,00 ,00 ,00

Criteria	C1	C2	C3	C4	C5	C6
C1	,00	,00	,00	,79	,17	,41	,85	,15	,45	,53	,22	,31	,53	,22	,31	,53	,22	,31
C2	,45	,23	,28	,00	,00	,00	,45	,23	,28	,45	,23	,28	,45	,23	,28	,35	,25	,25
C3	,53	,22	,31	,53	,22	,31	,00	,00	,00	,36	,25	,24	,45	,23	,28	,25	,27	,21
C4	,13	,28	,18	,25	,27	,21	,36	,25	,24	,00	,00	,00	,25	,27	,21	,13	,28	,18
C5	,00	,30	,15	,13	,28	,18	,25	,27	,21	,00	,30	,15	,00	,00	,00	,25	,27	,21
C6	,45	,23	,28	,00	,30	,15	,45	,23	,28	,00	,30	,15	,13	,28	,18	,00	,00	,00

C: criterion; D: average value of the expert evaluations.

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.

Table 5

Normalized matrix

X^M	C1	C2	C3	C4	C5	C6
C1	,00	,25	,26	,16	,16	,16
C2	,14	,00	,14	,14	,14	,11
C3	,16	,16	,00	,11	,14	,08
C4	,04	,08	,11	,00	,08	,04
C5	,00	,04	,08	,00	,00	,08
C6	,14	,00	,14	,00	,04	,00
X^U	C1	C2	C3	C4	C5	C6
C1	,00	,12	,11	,15	,15	,15
C2	,16	,00	,16	,16	,16	,18
C3	,15	,15	,00	,17	,16	,19
C4	,20	,19	,17	,00	,19	,20
C5	,21	,20	,19	,21	,00	,19
C6	,16	,21	,16	,21	,20	,00
X^L	C1	C2	C3	C4	C5	C6
C1	,00	,23	,25	,17	,17	,17
C2	,16	,00	,16	,16	,16	,14
C3	,17	,17	,00	,13	,16	,12
C4	,10	,12	,13	,00	,12	,10
C5	,08	,10	,12	,08	,00	,12
C6	,16	,08	,16	,08	,10	,00

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

Table 6

3 Submatrices for total relation matrix

T^M	C1	C2	C3	C4	C5	C6
C1	,40	,74	,70	,68	,63	,68
C2	,52	,25	,44	,51	,48	,47
C3	,57	,52	,27	,48	,49	,43
C4	,23	,27	,29	,13	,27	,23
C5	,10	,14	,18	,07	,08	,23
C6	,43	,18	,35	,16	,23	,16
T^U	C1	C2	C3	C4	C5	C6
C1	,29	,33	,33	,34	,34	,34
C2	,39	,35	,40	,40	,40	,40
C3	,39	,39	,35	,40	,40	,40
C4	,44	,44	,44	,39	,44	,45
C5	,46	,46	,46	,46	,41	,46
C6	,44	,45	,44	,45	,45	,39
T^L	C1	C2	C3	C4	C5	C6
C1	,41	,60	,59	,57	,55	,57
C2	,47	,31	,44	,46	,45	,45
C3	,48	,47	,32	,45	,45	,43
C4	,34	,35	,35	,24	,35	,34
C5	,31	,31	,31	,30	,21	,33
C6	,41	,34	,38	,34	,35	,25

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.

Table 7

Total relation matrix

Criteria	C1			C2			C3			C4			C5			C6
C1	,40	,29	,41	,74	,33	,60	,70	,33	,59	,68	,34	,57	,63	,34	,55	,68	,34	,57
C2	,52	,39	,47	,25	,35	,31	,44	,40	,44	,51	,40	,46	,48	,40	,45	,47	,40	,45
C3	,57	,39	,48	,52	,39	,47	,27	,35	,32	,48	,40	,45	,49	,40	,45	,43	,40	,43
C4	,23	,44	,34	,27	,44	,35	,29	,44	,35	,13	,39	,24	,27	,44	,35	,23	,45	,34
C5	,10	,46	,31	,14	,46	,31	,18	,46	,31	,07	,46	,30	,08	,41	,21	,23	,46	,33
C6	,43	,44	,41	,18	,45	,34	,35	,44	,38	,16	,45	,34	,23	,45	,35	,16	,39	,25

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.

Table 8

The values of the row and column sums (R and C)

	R			C
,998	,001	,026	,951	,004	,004
,973	,003	,006	,951	,004	,003
,977	,003	,006	,954	,004	,003
,804	,007	,001	,945	,004	,003
,581	,009	,001	,949	,004	,003
,836	,007	,002	,952	,004	,003

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

Table 9

Defuzzified values

Criteria	R Score	C Score
C1	,971192	,942692
C2	,963118	,943788
C3	,967289	,946543
C4	,79628	,937518
C5	,571326	,941924
C6	,827149	,944481

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

The weights of criteria

Criteria	C+R	W
Generating own energy in the long run (C1)	1,913885	,177981
Improving the image (C2)	1,906906	,177332
Minimizing the price risk in energy (C3)	1,913832	,177976
High initial investment cost (C4)	1,733798	,161234
Maintenance and repair considerations (C5)	1,51325	,140724
Instability due to climatic conditions (C6)	1,77163	,164752

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

Comparative analysis results based on different t values

t Values	C1	C2	C3	C4	C5	C6
1	,178	,177	,178	,161	,141	,165
2	,215	,191	,195	,147	,111	,141
3	,259	,193	,199	,134	,096	,119
4	,301	,190	,193	,125	,086	,104
5	,339	,187	,184	,118	,078	,095
6	,369	,185	,173	,113	,071	,089
7	,394	,186	,163	,108	,065	,085
8	,413	,188	,153	,105	,059	,082
9	,429	,192	,144	,101	,054	,080
10	,442	,197	,136	,097	,049	,079

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.

4.2 Analysis results of T-Spherical fuzzy TOP-DEMATEL

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.

Table 12
Defuzzified T matrix

T C1 C2 C3 C4 C5 C6

C1 -,3035 -,1896 -,2163 -,2307 -,2602 -,2217

C2 -,3447 -,4109 -,3966 -,3477 -,3734 -,3805

C3 -,3084 -,3422 -,3928 -,3644 -,3641 -,4100

C4 -,5546 -,5230 -,5011 -,4926 -,5198 -,5584

C5 -,6702 -,6280 -,5937 -,6928 -,5434 -,5585

C6 -,4173 -,6090 -,4771 -,6284 -,5643 -,4831

T	C1	C2	C3	C4	C5	C6
C1	-,3035	-,1896	-,2163	-,2307	-,2602	-,2217
C2	-,3447	-,4109	-,3966	-,3477	-,3734	-,3805
C3	-,3084	-,3422	-,3928	-,3644	-,3641	-,4100
C4	-,5546	-,5230	-,5011	-,4926	-,5198	-,5584
C5	-,6702	-,6280	-,5937	-,6928	-,5434	-,5585
C6	-,4173	-,6090	-,4771	-,6284	-,5643	-,4831

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

Table 13

Critical values of T-Spherical TOP-DEMATEL

	C*	C-	R*	R-	S*	S-	W
C1	,146	,185	,301	,764	,447	,949	,209
C2	,096	,104	,732	,724	,827	,829	,154
C3	,158	,139	,530	,621	,687	,761	,161
C4	,099	,102	,646	,836	,745	,938	,171
C5	,220	,234	,514	,513	,733	,748	,155
C6	,333	,305	,557	,535	,890	,840	,149

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

Comparative analysis results based on different t values

T	C1	C2	C3	C4	C5	C6
1	,209	,154	,161	,171	,155	,149
2	,204	,155	,161	,164	,153	,163
3	,198	,158	,163	,165	,154	,162
4	,193	,158	,164	,167	,157	,162
5	,188	,158	,165	,169	,159	,161
6	,184	,158	,166	,170	,162	,160
7	,180	,159	,166	,171	,164	,160
8	,177	,161	,167	,171	,166	,159
9	,173	,161	,167	,171	,167	,160
10	,180	,162	,164	,169	,163	,161

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.

5 Discussions

It is very important for these institutions to be financially successful to ensure the continuity of the hospitals. In this context, effective cost management will contribute significantly to achieving this goal. Energy costs have a very important place among the cost types of hospitals. Installing the solar panels in hospitals is considered a solution to reduce energy costs. However, this situation is frequently discussed in the literature and different opinions are put forward. The main reason for this is that the solar panels have both positive and negative aspects. Therefore, a new and comprehensive analysis is needed on this subject.

When the t value increases, the uncertainties included in the criteria increase. Therefore, in a situation where this uncertainty increases, it is more important for hospitals to be able to produce their own energy. Especially with the effect of globalization, political and economic relations between countries are increasing significantly. As a result, energy prices are affected by many different factors. Therefore, the scope of the contents of the criteria is also expanding. As this situation increases, energy prices become more volatile. To minimize this risk, it becomes important for hospitals to produce their own energy to ensure their sustainability. Installing the solar panels will also provide hospitals with such an advantage.

There are lots of different studies in the literature that reached similar conclusions. Pagliuca et al. [43] evaluated solar energy investments for European countries. They highlighted that solar energy projects play a very critical role to minimize energy dependency risk. Karakuş also aimed to generate appropriate strategies to improve energy dependency problems of the countries. It is identified that solar energy projects can be very helpful to overcome this problem. Abhinav et al. [45] also determined that solar energy is the most efficient way for the countries to minimize energy dependency.

The proposed model has also some superiorities in comparison with previous similar models in the literature. One of the most important advantages of this model is making evaluations for different t values. This situation provides opportunity to check the reliability of the findings according to different conditions. Yuan et al. [46], Yüksel et al. [47] and Nguyen et al. [48] generated novel fuzzy decision-making models to evaluate the performance of the energy investments. In these studies, Spherical fuzzy sets were taken into consideration. However, these models do not make examinations for different t values. In other words, the validity of the findings could not be checked because the evaluation was performed for only one t value.

Another important superiority of this proposed model is integrating the steps of DEMATEL and TOPSIS in the analysis process. As a result of this situation a new technique is created with the name of TOP-DEMATEL. There are some studies in the literature that criticized classical DEMATEL methodology. For example, Lin and Wu [49] and Govindan et al. [50] made criticism for this technique regarding the ineffectiveness to cope with uncertainties. Due to this issue, they considered DEMATEL methodology with different fuzzy sets. In this proposed model, the final steps of TOPSIS approach are integrated for the criticized steps of DEMATEL. Hence, this situation makes the proposed model superior by comparing with others.

6 Conclusion

It is concluded that generating own energy in the long run is the most crucial factor while making decision about the solar panel investments to the hospitals according to two different analyses. However, the results of T-Spherical fuzzy TOP-DEMATEL also indicate that the high initial investment cost is the second most critical factor in this respect. Additionally, for the purpose of measuring the coherency of the findings, analysis results are also calculated for different t values. The analysis results are quite similar for all different t values. This situation gives information about the coherency and reliability of the findings. Therefore, it is understood that in case of high uncertainties, generating own energy in the long run plays more critical role for the hospitals. Nevertheless, cost effectiveness should also be taken into consideration for the decision of generating the solar panels in the hospitals.

The main novelty of this study is that it is analyzed whether the solar panels are effective in reducing the costs of hospitals with an original decision-making model. With the help of this analysis, it can be possible to generate appropriate strategies for the hospitals to decrease energy costs. Hence, the efficiency and the effectiveness can be provided for the hospitals much easily. The performance improvement of the hospitals positively affects the quality of the health services. This situation has also positive contribution on both social and economic development of the countries. On the other side, the main limitation is making evaluation for only hospitals. The criteria can also be analyzed for different industries in the following studies, such as banking or automotive. In addition, different techniques can also be taken into consideration. Within this context, an evaluation can also be performed with SWARA methodology. Hence, this situation provides an opportunity to check the reliability of the findings. Another important limitation is that a case study could not be conducted in this study. Instead of this situation, only strategy recommendations are presented for the solar energy investors. Therefore, regarding the future research directions, cost-benefit analysis of the solar panels already installed in hospitals can be done.

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X^M	C1	C2	C3	C4	C5	C6
C1	,00	,25	,26	,16	,16	,16
C2	,14	,00	,14	,14	,14	,11
C3	,16	,16	,00	,11	,14	,08
C4	,04	,08	,11	,00	,08	,04
C5	,00	,04	,08	,00	,00	,08
C6	,14	,00	,14	,00	,04	,00
X^U	C1	C2	C3	C4	C5	C6
C1	,00	,12	,11	,15	,15	,15
C2	,16	,00	,16	,16	,16	,18
C3	,15	,15	,00	,17	,16	,19
C4	,20	,19	,17	,00	,19	,20
C5	,21	,20	,19	,21	,00	,19
C6	,16	,21	,16	,21	,20	,00
X^L	C1	C2	C3	C4	C5	C6
C1	,00	,23	,25	,17	,17	,17
C2	,16	,00	,16	,16	,16	,14
C3	,17	,17	,00	,13	,16	,12
C4	,10	,12	,13	,00	,12	,10
C5	,08	,10	,12	,08	,00	,12
C6	,16	,08	,16	,08	,10	,00

T^M	C1	C2	C3	C4	C5	C6
C1	,40	,74	,70	,68	,63	,68
C2	,52	,25	,44	,51	,48	,47
C3	,57	,52	,27	,48	,49	,43
C4	,23	,27	,29	,13	,27	,23
C5	,10	,14	,18	,07	,08	,23
C6	,43	,18	,35	,16	,23	,16
T^U	C1	C2	C3	C4	C5	C6
C1	,29	,33	,33	,34	,34	,34
C2	,39	,35	,40	,40	,40	,40
C3	,39	,39	,35	,40	,40	,40
C4	,44	,44	,44	,39	,44	,45
C5	,46	,46	,46	,46	,41	,46
C6	,44	,45	,44	,45	,45	,39
T^L	C1	C2	C3	C4	C5	C6
C1	,41	,60	,59	,57	,55	,57
C2	,47	,31	,44	,46	,45	,45
C3	,48	,47	,32	,45	,45	,43
C4	,34	,35	,35	,24	,35	,34
C5	,31	,31	,31	,30	,21	,33
C6	,41	,34	,38	,34	,35	,25

X^M	C1	C2	C3	C4	C5	C6
C1	,00	,25	,26	,16	,16	,16
C2	,14	,00	,14	,14	,14	,11
C3	,16	,16	,00	,11	,14	,08
C4	,04	,08	,11	,00	,08	,04
C5	,00	,04	,08	,00	,00	,08
C6	,14	,00	,14	,00	,04	,00
X^U	C1	C2	C3	C4	C5	C6
C1	,00	,12	,11	,15	,15	,15
C2	,16	,00	,16	,16	,16	,18
C3	,15	,15	,00	,17	,16	,19
C4	,20	,19	,17	,00	,19	,20
C5	,21	,20	,19	,21	,00	,19
C6	,16	,21	,16	,21	,20	,00
X^L	C1	C2	C3	C4	C5	C6
C1	,00	,23	,25	,17	,17	,17
C2	,16	,00	,16	,16	,16	,14
C3	,17	,17	,00	,13	,16	,12
C4	,10	,12	,13	,00	,12	,10
C5	,08	,10	,12	,08	,00	,12
C6	,16	,08	,16	,08	,10	,00

T^M	C1	C2	C3	C4	C5	C6
C1	,40	,74	,70	,68	,63	,68
C2	,52	,25	,44	,51	,48	,47
C3	,57	,52	,27	,48	,49	,43
C4	,23	,27	,29	,13	,27	,23
C5	,10	,14	,18	,07	,08	,23
C6	,43	,18	,35	,16	,23	,16
T^U	C1	C2	C3	C4	C5	C6
C1	,29	,33	,33	,34	,34	,34
C2	,39	,35	,40	,40	,40	,40
C3	,39	,39	,35	,40	,40	,40
C4	,44	,44	,44	,39	,44	,45
C5	,46	,46	,46	,46	,41	,46
C6	,44	,45	,44	,45	,45	,39
T^L	C1	C2	C3	C4	C5	C6
C1	,41	,60	,59	,57	,55	,57
C2	,47	,31	,44	,46	,45	,45
C3	,48	,47	,32	,45	,45	,43
C4	,34	,35	,35	,24	,35	,34
C5	,31	,31	,31	,30	,21	,33
C6	,41	,34	,38	,34	,35	,25

X^M	C1	C2	C3	C4	C5	C6
C1	,00	,25	,26	,16	,16	,16
C2	,14	,00	,14	,14	,14	,11
C3	,16	,16	,00	,11	,14	,08
C4	,04	,08	,11	,00	,08	,04
C5	,00	,04	,08	,00	,00	,08
C6	,14	,00	,14	,00	,04	,00
X^U	C1	C2	C3	C4	C5	C6
C1	,00	,12	,11	,15	,15	,15
C2	,16	,00	,16	,16	,16	,18
C3	,15	,15	,00	,17	,16	,19
C4	,20	,19	,17	,00	,19	,20
C5	,21	,20	,19	,21	,00	,19
C6	,16	,21	,16	,21	,20	,00
X^L	C1	C2	C3	C4	C5	C6
C1	,00	,23	,25	,17	,17	,17
C2	,16	,00	,16	,16	,16	,14
C3	,17	,17	,00	,13	,16	,12
C4	,10	,12	,13	,00	,12	,10
C5	,08	,10	,12	,08	,00	,12
C6	,16	,08	,16	,08	,10	,00

T^M	C1	C2	C3	C4	C5	C6
C1	,40	,74	,70	,68	,63	,68
C2	,52	,25	,44	,51	,48	,47
C3	,57	,52	,27	,48	,49	,43
C4	,23	,27	,29	,13	,27	,23
C5	,10	,14	,18	,07	,08	,23
C6	,43	,18	,35	,16	,23	,16
T^U	C1	C2	C3	C4	C5	C6
C1	,29	,33	,33	,34	,34	,34
C2	,39	,35	,40	,40	,40	,40
C3	,39	,39	,35	,40	,40	,40
C4	,44	,44	,44	,39	,44	,45
C5	,46	,46	,46	,46	,41	,46
C6	,44	,45	,44	,45	,45	,39
T^L	C1	C2	C3	C4	C5	C6
C1	,41	,60	,59	,57	,55	,57
C2	,47	,31	,44	,46	,45	,45
C3	,48	,47	,32	,45	,45	,43
C4	,34	,35	,35	,24	,35	,34
C5	,31	,31	,31	,30	,21	,33
C6	,41	,34	,38	,34	,35	,25