With the growing demand and commercial availability of cloud services, the need for comparison of their functionality against different prices and performance has arisen. A relevant and fair comparison is still challenging due to diverse deployment options and dissimilar features of different services. This paper addresses a hybrid multi-criteria decision-making model involving the selection of cloud services among the available alternatives. The proposed methodology assigns various ranks to cloud services based on the quantified quality-of-service parameters using a novel extended Grey Technique for Order of Preference by Similarity to Ideal Solution integrated with analytical hierarchical process. Further, we analyse the proposed cloud service selection method in terms of sensitivity analysis, adequacy under change in alternatives, adequacy to support group decision-making, and handling of uncertainty. This analysis helps both researchers and practitioners for analysing more fruitful approaches for cloud service selection.