**4. Adopted MCDM Methodology**

The most commonly used multi-criteria decision-making approaches [68–70] include the analytic hierarchy process (AHP), elimination and choice-translating algorithm (ELEC-TRE), technique for order of preference by similarity to the ideal solution (TOPSIS), and Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE). Compared to AHP, TOPSIS, and ELECTRE, Visual PROMETHEE has the capability of incorporating decision making via positive and negative preference flow. Visual management of performance using the PROMETHEE technique is appealing in the assessment of alternatives due to concepts like preference flow, weights, geometrical analysis for interactive aid (GAIA) plane, and sensitivity analyses. Partially and completely ranking the options also aids in determining the preferred alternative. However, those who make decisions are frequently interested not just in rating options, but also in determining the superiority of one over another (if such a superiority exists). The adopted methodology for an effective selection of solar panel cooling systems is explained step-by-step below.

Step 1: Decision matrix [70]:

Supposing there are i alternative solar panel cooling systems and j evaluation measures in outranking these cooling panels. PMij is the j performance measure's value for solar panel cooling system i. The decision matrix's structure is shown in Table 1 below. There are j performance measures and i alternative solar panel cooling systems, and Wj is the amount of significance assigned to each assessment criterion j.


**Table 1.** Decision matrix: alternatives, performance measures, and their weights.

Step 2: Performance measure weightage (Wj) [71–73]:

Wj estimates might be subjective or objective. Variations in assessment metrics are employed in the accepted strategy to evaluate the divergence in the ranking of various solar panel cooling systems. The adopted method takes into account both sorts of weights. The objective weights technique employs mathematical models like entropy calculation, such as in [74]; details of this are explained below. The decision matrix values of the jth performance measure for the ith solar panel cooling system are standardized using Equation (1) if the objective of the performance measure is maximization. In contrast, Equation (2) is opted for if the performance measure objective is minimization, wherein Sij is the standardized value for the jth performance measure of the ith solar panel cooling system; PMij is the jth

evaluation measure's value for the ith solar panel cooling system (refer to Table 1). After standardization of all performance measures, the decision matrix is expressed in matrix form in Equation (3), as seen below.

$$\mathbf{S}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}} = \left[ \frac{\mathbf{PM}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}} - \min\_{\mathbf{j}} \mathbf{PM}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}}}{\max\_{\mathbf{j}} \mathbf{PM}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}} - \min\_{\mathbf{j}} \mathbf{PM}\_{\overline{\mathbf{i}}\overline{\mathbf{j}}}} \right] \tag{1}$$

$$\mathbf{S}\_{\overline{\mathbb{I}}} = \left[ \frac{\max\_{\substack{\text{j} \\ \text{j}}} \underset{\text{ij}}{\text{p}}}{\max\_{\substack{\text{j} \\ \text{j}}} \text{PM}\_{\overline{\mathbb{I}}} - \min\_{\substack{\text{j}}} \text{PM}\_{\overline{\mathbb{I}}}}} \right] \tag{2}$$

$$\mathbf{S}'\_{\overline{\mathbb{N}}} = \begin{bmatrix} \mathbf{S}\_{11} & \mathbf{S}\_{12} & \dots & \mathbf{S}\_{1\overline{\mathbb{N}}} \\ \vdots & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{S}\_{\overline{\mathbb{N}}1} & \mathbf{S}\_{\overline{\mathbb{N}}2} & \dots & \mathbf{S}\_{\overline{\mathbb{N}}} \end{bmatrix} \tag{3}$$

Entropy Ej, according to its definition, is determined using the following Equation (4), and Wj, the performance measure objective weight, is determined by using Equation (5).

$$\mathbf{E}\_{\mathbf{j}} = -\frac{\sum\_{i=1}^{m} [\mathbf{S}\_{\bar{\mathbf{i}}\bar{\mathbf{j}}} \* \ln \left( \mathbf{S}\_{\bar{\mathbf{i}}\bar{\mathbf{j}}} \right)]}{\ln \left( \mathbf{m} \right)} \tag{4}$$

$$\mathbf{W}\_{\mathbf{j}} = \frac{1 - \mathbf{E}\_{\mathbf{j}}}{\left[1 - \sum\_{j=1}^{n} \mathbf{E}\_{\mathbf{j}}\right]} \tag{5}$$

In comparison, subjective weights [71] refer to the relative importance of performance measures in a multi-criteria decision making (MCDM) problem and are determined based on the judgment or opinion of the decision maker. In other words, the weights are not calculated mathematically but are assigned based on the subjective perception or expertise of the decision maker. The subjective weights are often obtained through surveys, interviews, expert opinions, or other qualitative methods.

Step 3: Outranking flow estimation [75]:

To start, we initially constructed a generalized preference function PFj <sup>i</sup>=a,i=b, where (a, b) is a pair of solar panel cooling systems and j is the performance measure. Each PFj <sup>i</sup>=a,i=<sup>b</sup> lies between 0 and 1. For given performance measure j, if 'i = a' solar panel cooling system is evaluated over 'i = b', then any one of the following preferences occurs based on the PFj <sup>i</sup>=a,i=<sup>b</sup> function value. If the PF<sup>j</sup> <sup>i</sup>=a,i=<sup>b</sup> function value is exactly equal to zero, then there is no preference for option a over alternative b. If the PFj <sup>i</sup>=a,i=<sup>b</sup> function value is close to zero, then the option 'a' solar panel cooling system has a weak preference over the 'b' solar panel cooling system. If the PFj <sup>i</sup>=a,i=<sup>b</sup> function value is close to one, there is a substantial preference for the option 'a' solar panel cooling system over 'b'. Lastly, if the PFj <sup>i</sup>=a,i=<sup>b</sup> function value is exactly equal to one, then there is a stringent preference for the option 'a' solar panel cooling system over 'b'.

Subsequently, using these preference function values, the preference index PIi<sup>=</sup>a,i=b, which has a value range of 0 to 1, is calculated for each pair of choices using Equation (6), as below.

$$\text{PI}\_{\text{i=a,i=b}} = \left[ \sum\_{j=1}^{j} \left( \mathbf{W}^{j} \times \text{PF}\_{\text{i=a,i=b}}^{j} \right) \right] \div \left[ \sum\_{j=1}^{j} \mathbf{W}^{j} \right] \tag{6}$$

In Equation (6), W<sup>j</sup> is the weight associated with each solar cooling system evaluation measure j, and the preference index PIi<sup>=</sup>a,i=<sup>b</sup> displays a preference for the option 'a' solar panel cooling system over option b, considering all j performance measures (j 1 to j). If PIi<sup>=</sup>a,i=<sup>b</sup> equals perfectly zero, then there is zero preference for alternative a over b; if PIi<sup>=</sup>a,i=<sup>b</sup> equals approximately zero, then there is a low preference for a over b; if PIi<sup>=</sup>a,i=<sup>b</sup>

equals approximately to one, then there is a high preference for alternative a over b, and if PIi<sup>=</sup>a,i=<sup>b</sup> is exactly equal to zero, then there is a perfect preference for alternative a over b. Finally, using the preference index, the outranking flows F<sup>+</sup> <sup>a</sup> and F<sup>−</sup> <sup>a</sup> are quantified using the following Equations (7) and (8), respectively, where i is the number of alternatives (i 1 to i), excluding alternative i = a.

$$\mathbf{F}\_{\mathbf{a}}^{+} = \frac{1}{\mathbf{i} - 1} \sum\_{i=1}^{\mathrm{i}} \mathrm{PI}\_{\mathbf{a},\mathrm{i}} \tag{7}$$

$$\rm{F}^{-}\_{\rm a} = \frac{1}{i-1} \sum\_{i=1}^{i} \rm{PI}\_{i,a} \tag{8}$$

Step 4: Calculation of net outranking flow and final ranking [76]:

The aforementioned predicted outranking flows F<sup>+</sup> <sup>a</sup> and F<sup>−</sup> <sup>a</sup> for each option are used to determine each alternative's dominance over the others. Positive ranking flow quantifies the 'a' solar panel cooling system's dominance over the other alternative solar panel cooling systems, whereas negative ranking flow quantifies alternative a's dominance over the other alternatives. F<sup>+</sup> <sup>a</sup> , <sup>F</sup><sup>+</sup> <sup>b</sup> , F<sup>−</sup> <sup>a</sup> , and F<sup>−</sup> <sup>b</sup> estimated outranking flows for each option a and b and were utilized to determine which option is dominant over the others. By understanding the outranking flow for any two choices, outranking relations can be inferred. Thus, a partial ranking is determined based on the outranking relations between any two choices as follows: if {[(F<sup>+</sup> <sup>a</sup> <sup>&</sup>gt; <sup>F</sup><sup>+</sup> <sup>b</sup> ) and (F<sup>−</sup> <sup>a</sup> < F<sup>−</sup> <sup>b</sup> )] or (F<sup>+</sup> <sup>a</sup> <sup>≥</sup> <sup>F</sup><sup>+</sup> <sup>b</sup> ) or (F<sup>−</sup> <sup>a</sup> ≤ F<sup>−</sup> <sup>b</sup> )},then solar panel cooling system 'a' has preference over 'b'; if {(F<sup>+</sup> <sup>a</sup> <sup>=</sup> <sup>F</sup><sup>+</sup> <sup>b</sup> )} and or {(F<sup>−</sup> <sup>a</sup> = F<sup>−</sup> <sup>b</sup> )}, then alternative 'a' has preference over 'b'; and if the information is otherwise inconsistent, then alternative a is incompatible with b. Net outranking flow for alternative a is obtained using Fa= F<sup>+</sup> <sup>a</sup> − F<sup>−</sup> <sup>a</sup> ; while net outranking flow for alternative b is obtained using Fb= F<sup>+</sup> <sup>b</sup> − F<sup>−</sup> b . A complete ranking is subsequently obtained as follows: if Fa > Fb, then alternative a has complete preference over alternative b; if Fa = Fb, then alternative a has complete indifference compared to alternative b. The complete ranking is obtained by ordering the alternatives in decreasing order of their net flow scores. The net flow score is a measure of the overall performance of an alternative, which takes into account the positive and negative outranking flows with respect to all other alternatives. The PROMETHEE I method is used to obtain the positive and negative outranking flows, whereas the PROMETHEE II method is used to obtain a complete ranking, and it considers all the alternatives and criteria involved in the decision-making problem. The final ranking obtained using the PROMETHEE provides a global view of the alternatives and facilitates the selection of the best option. The application of the above methodology for an efficient ranking of solar panel cooling systems is presented in the subsequent section.

#### **5. Application of the MCDM Approach**

The presented approach extends considerable support to comparing and ranking solar panel cooling systems along with their validation and sensitivity analyses. The step-wise application of the proposed multi-criteria decision-making approach for selecting solar panel cooling systems is presented as follows.

Step 1: This step is used to identify potential alternative solar panel cooling systems for evaluation. Each solar panel cooling system is evaluated using eleven performance measures; to start, equal weights are assigned to these measures. For the decision matrix on hand, refer to Table 2.


**Table 2.** Sample data for each alternative obtained via expert/published research.

Note: for # and \$, refer to Figure 1.

In Table 2, the solar panel finned air-cooling system (A1) has an average of 3.5% energy efficiency at an estimated cooling cost per square meter of a solar panel of USD 58, and this panel cooling system is highly reliable. The effects of carbon emissions on the environment are highly alarming. In the opinion of researchers [92,93], the alternative A1 has an extremely low effect as well as very little usage of electrical equipment in the cooling system network and is very supportive in terms of corrosion resistance. Similarly, Franklin et al. [77] stated that the finned air-cooling system is good and practical for installation and maintenance due to reduced physical deterioration and leakage difficulties. In light of numerous assessment parameters, one can, therefore, obtain comparable readings for any other alternative solar panel cooling system networks. Thus, for a given performance measure, one can see differences in values corresponding to alternative cooling systems. Not a single alternative outperforms others in all eleven measures. For selected performance measures, the selective solar panel cooling system score is better than the others. In other words, no single solar panel cooling system yields the best overall performance measures. Considering the multiple-attribute measure decision situation presented above, the application of the approach adopted to evaluate and rank these solar panel cooling system is presented here below.

Step 2: At this step, six sets of performance measure weights (equal weights, objective weights using the entropy technique, and four subjective weights) are chosen. Set 1 signifies an equal weighting of all performance measures, Set 2 is objective weights using the entropy approach, and Set 3 to Set 6 are subjective weights; refer to Table 3 below.




**Table 3.** *Cont.*

Note: \$ refer to Figure 1. # Set 1: equal weightage to all performance measures; Set 2: weightage to all performance measures using entropy approach; Set 3: 65% weightage to reliability, with others having 4% each; Set 4: 60% weight to cost, with others having 4% each; Set 5: 50% weightage to emission and others having 5% each; and Set 6: 60% weightage to efficiency and others having 4% each.

Step 3: In this step, as significantly large mathematical computations are needed, it is preferred to adopt the Visual PROMETHEE soft tool. So, using it, two preference flows (F<sup>+</sup> <sup>i</sup> and F<sup>−</sup> <sup>i</sup> ) and the net outranking flow (Fi) were obtained for each alternative solar panel cooling system i, as presented in Table 4 as follows.

**Table 4.** Partial and complete outranking flows for each alternative solar cooling system vs. the sets of weights.


Note: for \$, refer to Table 3, and for #, refer to Figure 1. Fi: net preference flows; F<sup>+</sup> <sup>i</sup> : positive preference flows; and F− <sup>i</sup> : positive preference flows.

In Table 4, there are two preference flows (F<sup>+</sup> <sup>i</sup> and F<sup>−</sup> <sup>i</sup> ), and these values help to draw a partial ranking. It also shows incomparability between solar panel cooling system alternatives when both (F<sup>+</sup> <sup>i</sup> and F<sup>−</sup> <sup>i</sup> ) preference flows have conflicting rankings. Similarly, in Table 4, Fi is the net preference flow, and it is a complete ranking of solar panel cooling system alternatives. For example, in Table 4, for set 1, when the corresponding values for alternatives A1 and A2 are compared using positive outranking flow (F<sup>+</sup> <sup>i</sup> ), it is evident that F+ A1 = 0.5655 > F+ A2 = 0.3053 ; however, when A1 and A2 are compared using negative outranking flow (F− <sup>i</sup> ), it is evident that F− A1 = 0.1557 < F− A2 = 0.0.2456 . This demonstrates that option A1 has a stronger preference than alternative A2. Similarly, when all alternatives are compared based on overall outranking flow (Fi), FA1 = 0.4098 and has the highest value when compared to the remaining eight alternatives, whereas alternative A7 has the lowest net outranking value, FA7 = −0.2095, implying that A1 is the first preference and A7 is the last preference.

Step 4: Using the previously calculated step 3, outranking flows, a ranking network diagram, and a geometrical analysis for the interactive plane are derived. Details are presented below.

Figure 2 shows the positive outranking flow F<sup>+</sup> <sup>i</sup> in the left column and the negative outranking flow F− <sup>i</sup> in the right column for each alternative i. Outranking flows are arranged in such a way that the best are projected at the top of the column. The center column represents the net outranking flow Fi. For each alternative, a representative line is drawn from its F<sup>+</sup> <sup>i</sup> to the corresponding F<sup>−</sup> <sup>i</sup> score. For any given two alternatives, if the representative lines are parallel, the alternative representing the top line is preferred. On the other hand, if the two lines intersect, the corresponding alternatives are incomparable. By correlating Table 4 and Figure 3, for outranking flow F<sup>+</sup> <sup>1</sup> , alternative A1 dominates all other alternatives; for outranking flow F<sup>+</sup> <sup>7</sup> , alternative A7, the water-sprayed solar panel cooling system, highly underperforms compared to all other alternatives; for outranking flow F− <sup>1</sup> , alternative A1, the finned air sink solar panel cooling system, dominates all other alternatives; and for outranking flow F− <sup>3</sup> , alternative A3 highly underperforms compared to others. Generally, these positive and negative outranking flows induce two different rankings. In order to circumvent this scenario, a complete ranking based on net flow F was obtained and is presented in Figure 3 as follows.

**Figure 2.** Partial ranking: as equal weightage to all performance measures (Set 1) (note: for notations, refer to Figure 1).

**Figure 3.** Complete ranking as equal weightage to all performance measures (Set 1) (note: for notations, refer to Figure 1).

From Figure 3, it is evident that alternative A1, the 'finned air-cooling system', surpasses all alternatives. Alternative A6, the forced air solar panel cooling system, performs as the next best option. Both of these options are best suited for dry, arid environments, while alternative A7, the water-sprayed solar panel cooling system, is least preferred over other alternatives. Subsequently, a network diagram was drawn (refer to Figure 4) in which each alternative is represented by a 'node' and its preference over other alternatives by an 'arrow'.

For example, in Table 4 for set 1, when the corresponding net outranking flow (Fi) values for all alternatives are compared, it is evident that (FA1 = 0.4098) > (FA6 = 0.2003) > (FA2 = 0.0597) > (FA4 = −0.0283) > (FA5 = −0.0668) > (FA3 = −0.0854) > (FA8 = −0.1195) > (FA9 = −0.1603) > (FA7 = −0.2095). Considering this net outranking relationship, the network diagram is drawn as presented in Figure 4. In Figure 4, it is evident that alternative A1, a passive cooling approach, is preferred over A6, which is an active cooling approach. When comparing all three active cooling approaches, A2, A4, and A3, approach A2 is preferred over both A4 and A3, whereas cooling approaches A4 and A3 are not comparable. Similarly, it is evident that passive cooling systems are outperforming active cooling systems, with the exception of forced air cooling system A6.

The Visual PROMETHEE represents the results in the GAIA (geometrical analysis for interactive assistant) plane (refer to Figure 5).

**Figure 4.** Network diagram as equal weightage to all performance measures (Set 1) (note: for notations, refer to Figure 1).

**Figure 5.** GAIA plane (note: for notations, refer to Figure 1).

In the GAIA plane (Figure 5), it is evident that solar panel cooling systems A1 and A3 are scoring opposite to A4 and A9; similarly, it is also apparent that the solar panel cooling systems A2 and A6 are scoring opposite to A5, A8 and A7. Solar panel cooling approach A1 scores better for measures PM02, PM05, and PM11 (refer to Figure 1). It is also observed that, as far as solar panel cooling systems A5, A8, and A7 are concerned, these cooling systems perform best for only one measure, i.e., panel temperature dropping. Similarly, when comparing active cooling approach A6 and passive cooling approach A2, both perform satisfactorily for the PM01, PM08, PM09, and PM10 performance measures. However, from Figure 5, it is evident that A6 scores better than A2. Thus, decision makers not typically having any pre-determined weights in mind warrants the need for sensitivity analysis. Hence, a feature of the Visual PROMETHEE software is adopted for this purpose. The details of the sensitivity analysis and results are presented here in the following section.

#### **6. Sensitivity Analysis**

Sensitivity analysis is performed to evaluate the deviation in the ranking of alternative solar panel cooling systems. Six sets of weights (equal weights, objective weights using the entropy approach, and four subjective scalings) of performance measure weights (refer to Table 3) are opted for in the sensitivity analysis. In subjective scaling, four scenarios are considered. Scenario 1 is based on technical experts who have knowledge about the workings of solar panel cooling systems; they prefer highly reliable solar cooling systems and so suggested having the highest importance of 65% to measure the reliability measure (PM02) compared to other measures. On the other hand, as in scenario 2, the management is keen on minimizing cooling system operating costs (PM03) as an economic concern and sets 60% weightages to cooling costs. In comparison, 60% weightages to environmental measures are seen in scenario 3, which is based on green and sustainable energy management; decision makers in this scenario are keen on minimizing carbon emissions by installing, operating, and maintaining environmentally friendly solar cooling systems. The last scenario, scenario 4, is that of an operational manager, whose target is to lower the amount of energy used for cooling and raise the overall effectiveness of the solar panel system, and suggested having 60% weightage to energy efficiency as crucial for the solar panel cooling system. On the contrary, objective weighting eliminates manmade disturbances and makes results accord more with the facts. The objective weights method makes use of mathematical models, such as entropy analysis [94]. Accordingly, the sensitivity analysis was performed relying on assigned entropy weights in the Visual PROMETHEE. The outcomes of the sensitivity analysis are presented below in Figure 6a–d.

**Figure 6.** (**a**) Partial ranking, (**b**) complete ranking, (**c**) network, and (**d**) GAIA plane for performance weightages in Set 2 (refer to Table 3) (note: for notations, refer to Figure 1).

Similarly, sensitivity analysis was performed using subjective weightage Set 3. Below, Figure 7a–d represent corresponding outcomes representing partial ranking, complete ranking, the network diagram, and the GAIA plane, respectively.

**Figure 7.** (**a**) Partial ranking, (**b**) complete ranking, (**c**) ranking network, and (**d**) GAIA plane for performance measure weightages in Set 3 (refer to Table 3) (note: for notations, refer to Figure 1).

In Figure 7, the decision maker's objective is to have a reliable solar cooling system to mitigate any level of operation risk. From Figure 7c, it is evident that A3, i.e., the phase change material cooling approach, is a more reliable solar panel cooling system compared to A2 (heat pipe cooling) and thermosiphon cooling with and without the material fluid pot (A2 and A4) in the category of passive cooling systems. However, A7, A8, and A9, active cooling systems, are the least reliable.

In the same way, Figure 8a–d below represent analysis corresponding to partial ranking, complete ranking, the network diagram, and the GAIA plane, respectively, for subjective weight applications. But, in this case, the highest weightage of 60% is set to cost

measure (refer to Set 4 in Table 3). Here, when the management objective is to have a cost-effective solar cooling system, A3 is the least preferred cooling system compared to A2, A4, and A5 in the category of passive cooling systems. This is opposite to the finding from Figure 7. However, when there is management that has economic concerns, alternative A1 dominates all other alternatives on hand.

**Figure 8.** (**a**) Partial ranking, (**b**) complete ranking, (**c**) ranking network, and (**d**) GAIA plane for performance measure weightages in Set 4 (refer to Table 3) (note: for notations, refer to Figure 1).

Likewise, the following Figure 9a–d represent analysis corresponding to partial ranking, complete ranking, the network diagram, and the GAIA plane, respectively, for the highest subjective weightage of 60% to an evaluation measure PM04 (refer to Set 5 in Table 3). As the management objective is to have an environmentally friendly solar cooling system, the passive cooling approaches A2, A3, and A4 are the least preferred solar panel cooling systems compared to A1 and A5 in the same category. However, in this case, the nanomaterial fluid cooling system is still outranked by the finned air cooling system.

**Figure 9.** (**a**) Partial ranking, (**b**) complete ranking, (**c**) ranking network, and (**d**) GAIA plane for performance measure weightages in Set 5 (refer to Table 3) (note: for notations, refer to Figure 1).

Lastly, Figure 10a–d represent analyses corresponding to partial ranking, complete ranking, the network diagram, and the GAIA plane, respectively, for the highest subjective weightage of 60% set to an evaluation measure, PM01 (refer to Set 6 in Table 3). Here, the management objective is to have energy-efficient solar panel cooling systems where passive cooling systems outrank active cooling systems.

Thus, from the analysis, it is evident that for each scenario, the solar panel cooling system performance is sensitive to variation in performance measure weightages. The overall ranking for each alternative solar panel cooling system corresponding with multiple sets of weights assigned to performance measures is presented here in Table 5. Subsequently, the complete outranking flow (refer to Table 4) for each alternative is aggregated to drive overall ranking. Thus, after the sensitivity analysis, it is evident that solar panel cooling system alternatives A1 and A6 are found to be the best choice over other alternatives.

**Figure 10.** (**a**) Partial ranking, (**b**) complete ranking, (**c**) ranking network, and (**d**) GAIA plane for performance measure weightages in Set 6 (refer to Table 3) (note: for notations, refer to Figure 1).

**Table 5.** Ranking for each alternative solar panel cooling system corresponding with multiple sets of weights assigned to performance measures.


Note: For \$, refer to Table 3, and for #, refer to Figure 1.
