This paper aims to propose a multi-period multi-product supply chain network design which takes the sustainability dimensions into consideration in both strategic and operational decisions. Several critical issues in the planning of supply chain networks are considered in the model such as the capacity of facilities, the minimum acceptable rate for the social score of manufacturing plants and distribution centers, the maximum coverage radius, and the limited budget availability. In order to obtain an effective and efficient network design, different categories of uncertainty are also considered, including the provider-side uncertainty reflected in the capacity of constructed facilities, as well as the economic, environmental, social, and technical parameters, the receiver-side uncertainty reflected in the demand, and the in-between uncertainty reflected in the transportation costs and the maximum coverage radius. To deal with different sources of uncertainty in the concerned problem, a chance constrained fuzzy programming approach is employed. Several test problems are used to analyze the characteristics of the proposed problem. The computational results can help decision makers to design supply chain networks from economic, environmental, and social perspectives.