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The fuzzy-TOPSIS, a well-known MCDM method, was developed by [87]. Fuzzy-TOPSIS is extensively used in the various fields for ranking where decision makers give judgement in the form of crisp value especially for qualitative indicators. In qualitative approaches, the decisions are usually expressed in the form of linguistic terms (e.g., low, high, very high), and due to the possibility of vagueness in a human decision, uncertainty is always associated with the result. Fuzzy combined with the MCDM method has proven to be useful in handling qualitative indicators with associated uncertainties [53, 54].

Before elaborating the steps of fuzzy-TOPSIS. The basic theory of fuzzy is explained as follows:

 (i) The membership function for a triangular fuzzy number à given by (a1, b1, c1) is defined as:(2.9)

 (ii) The distance between two triangular fuzzy numbers à = (a1, b1, c1) and is given by Eq. (2.10):(2.10)

 (iii) The multiplication of fuzzy triangular numbers is given by Eq. (2.11):(2.11)

 (iv) The addition of fuzzy triangular numbers is given by Eq. (2.12):(2.12)

Based on the basic fuzzy theory explained above. The fuzzy-TOPSIS method steps are described as below:

1 (a) Establish a decision matrix choosing the linguistic values for alternatives (i = 1, 2, … m) with respect to indicator (j = 1, 2,…n).

2 (b) Choose the suitable linguistic variables for the weights of the indicators.

3 (c) The fuzzy linguistic value are in range of [0, 1]; thus, normalization is not required.

4 (d) The weighted 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.