# Evaluation of Cloud Services: A Fuzzy Multi-Criteria Group Decision Making Method +
## 3 A Fuzzy Multi-Criteria Group Decision Making Method  
Evaluating the performance of the available Cloud services with respect to multiple, usually conflicting criteria in a specific situation is always challenging due to (a) the availability of multiple Cloud services; (b) the multi-dimensional nature of the decision making problem; (c) the involvement of multiple decision makers; and (d) the presence of subjectivity and imprecision involved in the decision making process [34]. To overcome these concerns, this paper presents a fuzzy multi-criteria decision making method based on the fusion of several concepts including (a) the TOPSIS method; (b) the Choquet integral operator; and (c) intuitionistic fuzzy numbers. Modelled as a multi-criteria group decision making problem, the performance evaluation of available Cloud services involves (a) discovering all the alternatives; (b) identifying the evaluation criteria; (c) assessing the alternatives' performance ratings and the criteria weightings by individual decision makers; (d) aggregating the alternative ratings and criteria weightings for producing an overall performance value for each alternative across all the criteria; and (e) selecting the best alternative in the given situation [35; 36]. To better model the subjectivity and imprecision of the human decision making process, interval-valued intuitionistic fuzzy numbers are used by the decision maker for assigning the weightings of the evaluation criteria. Interval-valued intuitionistic fuzzy numbers [10] are the generalization of the intuitionistic fuzzy numbers. The values of the membership function and non-membership function of interval-valued intuitionistic fuzzy numbers are represented as intervals rather than exact numbers. In many real situations, it is often difficult to define the membership grade of an element because decision makers often do not agree on the same membership grade for an element. To effectively deal with this situation, a hesitant fuzzy set is introduced [37] as a generalization of fuzzy sets. For the multi-criteria group decision making problem, let \(A=\{A_{1}\), \(A_{2}\),..., \(A_{n}\}\) be the set of \(n\) alternatives, \(C=\{C_{1}\), \(C_{2}\),..., \(C_{m}\}\) be the set of \(m\) criteria and \(D=\{D_{1}\), \(D_{2}\),..., \(D_{s}\}\) be the set of decision makers. The performance of the alternatives \(A_{i}\) (\(i=1\), \(2\),..., \(n\)) with respect to criteria \(C_{j}\) (\(j=1\), \(2\),..., \(m\)) which is assessed by individual decision makers \(D_{k}\) (\(k=1\), \(2\),..., \(s\)) is measured by an interval-valued intuitionistic hesitant fuzzy element [37]. In this paper, an interval-valued intuitionistic hesitant fuzzy Choquet integral operator [38] is introduced for dealing with the hesitant fuzzy multi-criteria decision making problem in an efficient and effective manner. The procedure for the adoption of the interval-valued intuitionistic hesitant fuzzy Choquet integral operator in the fuzzy multi-criteria group decision making process includes the following steps:  
Step 1. Construct an interval-valued intuitionistic hesitant fuzzy decision matrix [39]\(Z=\left(h_{ij}\right)_{n\times m}\) where \(h_{ij}=\left\{\alpha_{ij}\left|\alpha_{ij}\in h_{ij}\right.\right\}=\left\{ \left(\left[\mu_{a_{ij}}^{-},\mu_{a_{ij}}^{+}\right],\left[\
u_{a_{ij}}^{-}, \
u_{a_{ij}}^{+}\right]\right)\left|\alpha_{ij}\in h_{ij}\right.\right\}\) denotes an interval-valued intuitionistic hesitant fuzzy element, and alternative \(A_{i}\) is evaluated by each decision maker \(D_{k}\) with respect to criteria \(C_{j}\). The hesitant fuzzy decision making matrix for each decision maker \(Z_{k}\) can be represented as in (1). \[Z^{k}=\left[ \begin{array}{cccc}h_{11}^{k}&h_{12}^{k}&...&h_{1m}^{k}\\ h_{21}^{k}&h_{22}^{k}&...&h_{2m}^{k}\\...&...&...&...\\ h_{n1}^{k}&h_{n2}^{k}&...&h_{nm}^{k}\end{array} \right] \tag{1}\]  
Step 2. Obtain the overall interval-valued intuitionistic hesitant fuzzy decision matrix by averaging the fuzzy assessments made by individual decision makers as given in (1). \[Z=\left[