Supply Chain Network Design with Concave Costs: Theory and Applications

Many practical decision models can be formulated as concave minimization problems. Supply chain network design problems (SCNDP) that explicitly account for economies-of-scale and/or risk pooling often lead to mathematical problems with a concave objective and linear constraints. In this thesis, we propose new solution approaches for this class of problems and use them to tackle new applications. \n \nIn the first part of the thesis, we propose two new solution methods for an important class of mixed integer concave minimization problems over a polytope that appear frequently in SCNDP. The first is a Lagrangian decomposition approach that enables a tight bound and a high quality solution to be obtained in a single iteration by providing a closed-form expression for the best Lagrangian multipliers. The Lagrangian approach is then embedded within a branch-and-bound framework. Extensive numerical testing, including implementation on three SCNDP from the literature, demonstrates the validity and efficiency of the proposed approach. The second method is a Benders approach that is particularly effective when the number of concave terms is small. The concave terms are isolated in a low dimensional master problem that can be efficiently solved through enumeration. The subproblem is a linear program that is solved to provide a Benders cut. Branch-and-bound is then used to restore integrality if necessary. The Benders approach is tested and benchmarked against commercial solvers and is found to outperform them in many cases. \n \nIn the second part, we formulate and solve the problem of designing a supply chain for chilled and frozen products. The cold supply chain design problem is formulated as a mixed-integer concave minimization problem with dual objectives of minimizing the total cost, including capacity, transportation, and inventory costs, and minimizing the global warming impact that includes, in addition to the carbon emissions from energy usage, the leakage of high global-warming-potential refrigerant gases. Demand is modeled as a general distribution, whereas inventory is assumed managed using a known policy but without explicit formulas for the inventory cost and maximum level functions. The Lagrangian approach proposed in the first part is combined with a simulation-optimization approach to tackle the problem. An important advantage of this approach is that it can be used with different demand distributions and inventory policies under mild conditions. The solution approach is verified through extensive numerical testing on two realistic case studies from different industries, and some managerial insights are drawn. \nIn the third part, we propose a new mathematical model and a solution approach for the SCNDP faced by a medical sterilization service provider serving a network of hospitals. The sterilization network design problem is formulated as a mixed-integer concave minimization program that incorporates economies of scale and service level requirements under stochastic demand conditions, with the objective of minimizing long-run capacity, transportation, and inventory holding costs. To solve the problem, the resulting formulation is transformed into a mixed-integer second-order cone programming problem with a piecewise-linearized cost function. Based on a realistic case study, the proposed approach was found to reach high quality solutions efficiently. The results reveal that significant cost savings can be achieved by consolidating sterilization services as opposed to decentralization due to better utilization of resources, economies of scale, and risk pooling.