Robust optimisation with applications in supply chain management

We consider a production planning problem in supply chain management, namely the lot sizing problem under uncertainty. Lot sizing problems contain a fairly high degree of uncertainty in real life. Thus pre-decided optimal production plans are no longer of practical use. Current methods to address the uncertainty in the lot sizing problem either make strong assumptions or increase the complexity of the deterministic model significantly. In this study, we propose lot sizing models that use robust optimization to deal with uncertainty in demand and production costs. These models provide a good trade-off between the robustness of the solution, i.e., to what extent the solution is feasible, and the quality of the objective function value.
Four robust methodologies are used, namely, the minimax regret criterion, the uncertainty budget, the uncertainty range, and the bw-robustness criterion. With the minimax regret criterion, the robust lot sizing model aims to achieve a production plan that minimizes the maximum surplus between the total cost of the production plan and that of the optimal production plan after revealing the uncertain data. With the uncertainty budget and the uncertainty range, the robust lot sizing model minimizes the maximum total cost that could occur due to the uncertain data. However, the size of the total uncertainty is bounded by the uncertainty budget and the scope of the allocation of the uncertainty data is bounded by the uncertainty range. The bw-robust lot sizing model identifies a production plan that guarantees an objective function value of at most w in all data circumstances and maximizes the probability that the objective function is below b.
While the main priority of most classic robust optimization approaches is to deal with feasibility, the quality of the objective function value is equally important to the decision maker in the process of production planning. In this thesis, we focus more on the quality of the objective function and, besides being robust, we model explicitly how the uncertain parameters affect the total production cost in the lot sizing problem. As far as the author is aware, the minimax regret lot sizing study is the first to consider uncertainty in both the coeficients of the objective function and a parameter that defines the feasible region, and the bw-robust lot sizing study is the first to deal with a parameter that defines the feasible region. With the popular idea of uncertainty budget, we redefine the worst case caused by data uncertainty as when it harms the objective function value the most, and extend this idea by further proposing the uncertainty range parameter which helps to provide a better trade-off between robustness and the quality of the objective function value.
Efficient polynomial time algorithms are proposed for the minimax regret uncapacitated lot sizing model, the robust lot sizing model with uncertainty budget, and the robust lot sizing model with uncertainty range. For the bw-robust lot sizing model with demand uncertainty, we propose Mixed Integer Nonlinear Programming formulations, which are reformulated as Mixed Integer Linear Programming formulations and solved with a commercial package. Numerical results are provided with various settings of parameters to show the effectiveness of the solutions of the proposed robust lot sizing models, in terms of feasibility and quality of the objective function.

Advisor: Morales, D