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decision matrix is calculated.
5 (e) Calculate the positive ideal (V+) and negative ideal solutions (V-) as below:J is the set of beneficiary indicators and J’ is of non-beneficiary indicators(2.13)(2.14)
6 (f) The distance of each alternative from V+ and V- is calculated as:(2.15)(2.16)
7 (g) Determine the corresponding closeness to the best solution as below:(2.17)
8 (h) Finally, according the value of Ri in descending order the alternatives are ranked.
2.5 Ranking of RE Technologies
2.5.1 The TOPSIS
As per methodology described in Section 2.5.3, Table 2.3 is input decision matrix. Following the steps (b) – (f) as mentioned in Section 2.5.3, the results obtained are presented as Table 2.4.
2.5.2 The Fuzzy-TOPSIS
To address the uncertainties associated with qualitative data fuzzy-TOPSIS is used for ranking. Table 2.3 will be the input decision matrix for fuzzy-TOPSIS. Since indicator’s ratings are not in the form of fuzzy linguistic variables, normalization in the range of [0, 1] is done using equations (2.18) and (2.19), and presented as Table 2.5.
(i) For the beneficial indicators:(2.18)
(ii) For the non-beneficial indicators:(2.19)
In the next step, decision matrix with a fuzzy linguistic variable has to be established using triangular membership functions defined as shown in Figure 2.2.
Table 2.5 is now transformed to Table 2.6 as described with an example. If the numeric rating of an indicator value is 0.50, the equivalent fuzzy linguistic value then will be “Medium”. Table 2.6 now is converted to Table 2.7 the fuzzy decision matrix using Figure 2.2. Table 2.7 presents the fuzzy decision matrix. As per step (e) of Section 2.4, V+ and V- are defined as Vj+ = (0,0,0) and Vj- = (1,1,1) for beneficial indicators and Vj- = (0, 0, 0) and Vj+ = (1, 1, 1) for non-beneficial indicators respectively. Table 2.8 presents the resulting fuzzy weighted decision matrix with their values ranges from the closed interval [0,1] and final ranking obtained following steps (f)-(h).
Table 2.4 The TOPSIS result.
| RE technology | I1 | I2 | I3 | I4 | I5 | I6 | I7 | I8 | I9 | I10 | Si+ | Si- | Ri | Ranking |
| Large hydropower | 0.0611 | 0.0822 | 0.0472 | 0.0379 | 0.0682 | 0.0179 | 0.0020 | 0.0115 | 0.0722 | 0.0674 | 0.0952 | 0.1589 | 0.6253 | 2 |
| Small hydropower | 0.0611 | 0.0164 | 0.0472 | 0.0483 | 0.0477 | 0.0001 | 0.0001 | 0.0344 | 0.0433 | 0.0405 | 0.0857 | 0.1587 | 0.6494 | 1 |
| Solar PV | 0.0114 | 0.0164 | 0.0189 | 0.0500 | 0.0341 | 0.0022 | 0.0022 | 0.0574 | 0.0144 | 0.0270 | 0.1030 | 0.1603 | 0.6088 | 4 |
| Onshore wind power | 0.0267 | 0.0164 | 0.0262 | 0.0403 | 0.0341 | 0.0109 | 0.0110 | 0.0574 | 0.0289 | 0.0135 | 0.0938 | 0.1505 | 0.6162 | 3 |
| Bioenergy | 0.0412 | 0.0493 | 0.0671 | 0.0459 | 0.0273 | 0.0978 | 0.0993 | 0.0459 | 0.0433 | 0.0539 | 0.1588 |
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