a b s t r a c tThe concern about significant changes in the business environment (such as customer demands and transportation costs) has spurred an interest in designing scalable and robust supply chains. This paper proposes a robust optimization model for handling the inherent uncertainty of input data in a closed-loop supply chain network design problem. First, a deterministic mixed-integer linear programming model is developed for designing a closed-loop supply chain network. Then, the robust counterpart of the proposed mixedinteger linear programming model is presented by using the recent extensions in robust optimization theory. Finally, to assess the robustness of the solutions obtained by the novel robust optimization model, they are compared to those generated by the deterministic mixed-integer linear programming model in a number of realizations under different test problems.

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This paper presents a multi-objective possibilistic programming model to design a second-generation biodiesel supply chain network under risk. The proposed model minimizes the total costs of biodiesel supply chain from feedstock supply centers to customer centers besides minimizing the environmental impact (EI) of all involved processes under a well-to-wheel perspective. Non-edible feedstocks are considered for biodiesel production. Variable cultivation cost of non-edible feedstock is assumed to be non-linear and dependent upon the amount of cultivated area. New formulation of possibilistic programming method is developed which is able to minimize the total mean and risk values of problems with possibilistic-based uncertainty. To solve the proposed multi-objective model, a hybrid solution approach based on flexible lexicographic and augmented ɛ-constraint methods is proposed which is capable to find appropriate efficient solutions from the Pareto-optimal set. The performance of the proposed possibilistic programming method as well as the developed solution approach are evaluated and validated through conducting a real case study in Iran. The outcome of this study demonstrates that high investment cost is required for improving the environmental impact and risk of sustainable biodiesel supply chain network design under risk. Decision maker preferences are required for suitable trade-off among total costs, risk values and environmental impact.

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