One of the main topics discussed in a supply chain is the production-distribution problem. Producing and distributing the products plays a key role in reducing the costs of the chain. To design a supply chain, a network of efficient management and production-distribution decisions is essential. Accordingly, providing an appropriate mathematical model for such problems can be helpful in designing and managing supply chain networks. Mathematical formulations must be drawn close to the real world due to the importance of supply chain networks. This makes those formulations more complicated. In this study, a novel multi-objective formulation is devised for the production-distribution problem of a supply chain that consists of several suppliers, manufacturers, distributors, and different customers. Also, a Mixed Integer Linear Programming (MILP) mathematical model is proposed for designing a multi-objective and multi-period supply chain network. In addition, grey flexible linear programming (GFLP) is done for a multi-objective production-distribution problem in a supply chain network. The network is designed for the first time to cope with the uncertain nature of costs, demands, and capacity parameters. In this regard, due to the NP-hardness and complexity of problems and the necessity of using meta-heuristic algorithms, NSGA-II and Fast PGA algorithm are applied and compared in terms of several criteria that emphasize the quality and diversity of the solutions.