This dissertation focuses on formulating and solving multi-stage decision problems in uncertain environments using stochastic programming and robust optimization approaches. These approaches are applied to the design of closed-loop supply chain (CLSC) networks, which integrate both traditional flow and the reverse flow of products. The uncertainties associated with this application include forward demands, the quantity and quality of used products to be collected, and the carbon tax rate. The design decisions include long-term facility configurations as well as short-term contracts for transportation capacities by various modes that differ according to their variable costs, fixed costs, and emission rates.

This dissertation consists of three papers. The first paper develops a multi-stage stochastic program for a CLSC network design problem with demands and quality of return uncertainties. The second paper focuses on robust optimization; particularly, the question of whether an adjustable robust counterpart (ARC) produces less conservative solutions than the robust counterpart (RC). Using the results of the second paper, a three-stage hybrid robust/stochastic program is proposed in the third paper, in which an ARC is formulated for a mixed integer linear programming model of the CLSC network design problem.

In the first paper, a multi-stage stochastic program is proposed for the CLSC network design problem where facility locations are decided in the first stage and in subsequent stages, the capacities of transportation of different modes are contracted under uncertainty about the amounts of new and return products to transport among facilities. We explore the impact of the uncertain quality of returned products as well as uncertain demands with dependencies between periods. We investigate the stability of the solution obtained from scenario trees of varying granularity using a moment matching method for demands and distribution approximation for the quality of returns. Multi-stage solutions are evaluated in out-of-sample tests using simulated historical data and also compared with two-stage model. We observe an instance of overfitting, in which a scenario tree including more outcomes at each stage produces a dramatically different solution that has slightly higher average cost, compared to the solution from a less granular tree, when evaluated against the underlying simulated historical data. We also show that when the scenarios include demand dependencies, the solution performs better in out-of-sample simulation.

In the second paper, the ARC of an uncertain linear program extends the RC by allowing some decision variables to adjust to the realizations of some uncertain parameters. The ARC may produce a less conservative solution than the RC does but cases are known in which it does not. While the literature documents some examples of cost savings provided by adjustability (particularly affine adjustability), it is not straightforward to determine in advance whether they will materialize. We establish conditions under which adjustability may lower the optimal cost with a numerical condition that can be checked in small representative instances. The provided conditions include the presence of at least two binding constraints at optimality of the RC formulation, and an adjustable variable that appears in both constraints with implicit bounds from above and below provided by different extreme values in the uncertainty set.

The third paper concerns a CLSC network that is subject to uncertainty in demands for both new and returned products. The model structure also accommodates uncertainty in the carbon tax rate. The proposed model combines probabilistic scenarios for the demands and return quantities with an uncertainty set for the carbon tax rate. We constructed a three-stage hybrid robust/stochastic program in which the first stage decisions are long-term facility configurations, the second stage concerns the plan for distributing new and collecting returned products after realization of demands and returns but before realization of the carbon tax rate, and the numbers of transportation units of various modes, as the third stage decisions, are adjustable to the realization of the carbon tax level. For computational tractability, we restrict the transportation capacities to be affine functions of the carbon tax rates. By utilizing our findings in the second paper, we found conditions under which the ARC produces a less conservative solution. To solve the affinely adjustable version, Benders cuts are generated using recent duality developments for robust linear programs. Computational results show that the ability to adjust transportation mode capacities can substitute for building additional facilities as a way to respond to carbon tax uncertainty. The number of opened facilities in ARC solutions are decreased under uncertainty in demands and returns. The results confirm the reduction of total expected cost in the worst case of the carbon tax rate by increasing utilization of transportation modes with higher capacity per unit and lower emission rate.