In today’s growing world, the Green Supply Chain (GSC) is a new approach to include environmental impacts and economic goals in a supply chain network. This paper continues previous research studies by designing a new green supply chain network considering different social, economic, environmental, service level, and shortage aspects. This study introduces a fresh, comprehensive tradeoff model that considers factors such as overall expenses, quality of service, environmental pollution levels, and societal impacts within a sustainable supply chain. The proposed model is formulated as a multi-product multi-objective mixed-integer programming model to assist in planning a green supply chain. The suggested model has three objective functions: maximizing social responsibility, minimizing the cost of carbon dioxide (CO2) emissions, and minimizing economic costs. The model allows for shortages in the form of backorders and seeks to maximize service level in addition to the mentioned objective functions. Robust Possibilistic Programming (RPP) was employed to deal with the problem's uncertain input parameters in the solution approach. Also, a multi-objective model of the problem was solved using Fuzzy Goal Programming (FGP). To examine and evaluate the model in a simple framework, the proposed mathematical model of the problem was implemented in an industrial unit in the real world, and the results obtained from it were analyzed. Among the results that the output of the model provides to managers and decision-makers, it is possible to mention the determination of the optimal amount of production of each product in the manufacturing plants, quantity of products and parts transported between facilities, and also the determination of the of network's carbon emissions which is equal to 51.59 tons.