Global warming and other environmental issues brought on by the usage of fossil fuels and the unsustainable urban population development have prompted energy industry policymakers to look for renewable sources to create clean energy. Biogas production from agro-livestock waste is a sustainable renewable bioenergy option for overcoming fossil fuel depletion. Since there are a few considerations of sustainable development goals, all air contaminants during the network activities, uncertainties in critical parameters, and crucial strategic and operational decisions in a cohesive framework at the same time in the design of bioenergy systems, this research intends to establish a bioenergy supply chain network by constructing biogas plants to produce both bioelectricity and bio-fertilizer from agricultural and livestock waste. In order to design a profit maximizing, environmentally friendly, and socially responsible supply chain, a multi-objective mathematical model is elaborated to decide strategic and operational decisions of the bioenergy supply chain with the objectives of maximizing network profits, minimizing the environmental impacts of transportation and production, and maximizing the job opportunities generated by the designed network. The uncertainty of key parameters such as costs, revenues, availability of agro-livestock waste, maximum allowable budget, and demands are considered. An approach based on the fuzzy programming method is used to handle the uncertainty. The mathematical model is solved by combining a group best-worst method with a fuzzy goal programming approach. The model is tested using actual data from the Iranian province of Fars and various agro-livestock waste, such as rice straw, wheat straw, maize straw, cow manure, and hen manure. According to the findings, 87.2% of the total network income comes from the sale of bioelectricity, which accounts for the biggest share of network revenues. Most of the network's overall costs are related to setting up facilities, supplying materials, and production. The biogas plant activities produce the majority of the expected network's emissions and the built-in biogas plants and their operation unquestionably generate the largest employment opportunities. In addition, although the system's profit is somewhat reduced by taking into account the sustainable development goals, the environmental and social objectives are much enhanced.