(518f) Supply Chain Design Under the Risk of Disruptions At Capacitated Distribution Centers

The impact of supply chain disruptions in the performance of companies has received increasing attention in recent years. The management of disruptions is an important topic for every company with a complex supply chain. Intuitively, it is well understood that reactive measures are not enough to deal with disruptive events. Therefore, the design of resilient supply chains has become a central objective for business stability.

The classical formulation of the supply chain design is based on the facility location problem. Given a number of candidate locations for distribution centers (DCs), it is desired to select the locations that minimize the sum of investment and distribution cost during a finite period of time. When supply chain resilience under disruptions is included in the problem, aspects such as selection of DCs capacity and inventory management must be considered in the formulation in order to hedge against uncertain disruptions.

In this research we address the optimal design of supply chains under the risk of disruptions at DCs. The problem is formulated as a mixed-integer linear programing problem (MILP). The decision variables include the selection of DCs among candidate locations, the capacities of DCs for different commodities, and the assignments of customer demands to available DCs considering penalties for unsatisfied demands. The inclusion of capacity constraints in the distribution strategy provides a fully realistic problem setting but implies a great challenge. In particular, for any possible combination of active and disrupted locations, the allocation of customer demands to available capacity at DCs must be found.

Given the two-stage nature of the supply chain design under the risk of disruptions, the problem is formulated in the framework of stochastic programming. The first-stage decisions include selecting DCs among a set of candidate locations and establishing their capacities for each commodity. The second-stage decisions are the demand assignments in the scenarios given by the possible combinations of active and disrupted locations. Assuming that DCs have independent probabilities of being disrupted, the number of scenarios grows exponentially with the number of candidate DCs. This makes the full-space problem intractable even for a small number of candidate DCs. However, if the probability of being disrupted for individual DCs is relatively small, the independence of disruptions implies that the scenarios with several disruptions occurring simultaneously are very unlikely. Therefore, their impact in the objective function is limited and can be bounded. Despite this property, many supply chains require a very large number of scenarios to achieve good bounds on the optimal solution.

The standard approach to solve this kind of problems is to apply decomposition strategies to overcome the large dimensionality. By using Benders decomposition, the optimal value of the objective function can be found by iteratively improving lower and upper bounds on it. The lower bound on the optimal cost can be found from a master problem that approximates the distribution cost of the scenarios in the space of the first-stage variables. The upper bound is found by fixing the first-stage variables and solving for the scenarios. The convergence of the algorithm is achieved by improving the lower bounding approximation using the information obtained in the upper bounding subproblems. The convergence can be accelerated by refining the lower bounding approximation with multiple cuts added to the master problem in every iteration. For the proposed supply chain optimization, individual cuts for every scenario and every commodity can be added in each iteration. The disadvantage of this approach is that the MILP Benders master problem becomes very difficult to solve due to the extremely large number of cuts that are added after a few iterations. On the other hand, if only a small number of cuts are added in every iteration, an unpractical number of iterations are required to converge.

In order to overcome this limitation, we propose a modification to the master problem that includes more detailed information about the cost of the second-stage decisions. In particular, we include in the master problem some scenario independent assignments that provide good lower bounds on the distribution costs. These assignments represent the level of preference of customers for DCs as presented in the reliable location model ^[1],[2]. By using this approach and adding Benders cuts per commodity, the master problem predicts much stronger lower bounds within few iterations.

The proposed methodology has been successfully applied to the design of an industrial supply chain under the risk of disruptions at DCs. The modification of the master problem has proved to be very effective and has allowed addressing the number of scenarios required to obtain tight bounds on the full-space problem. Furthermore, the results obtained show the importance of considering disruptions in supply chain design and the role of capacitated DCs in the distribution strategies.