(723d) Design of Supply Chains with Random Facility Disruptions

The impact of supply chain performance on companies’ competitiveness is widely recognized in the process industries. The formulation of the design and management of supply chains as an optimization problem allows selecting the alternatives that offer the best decisions based on a quantitative basis. However, the performance of any given supply chain is subject to uncertainties derived from product demand, operating costs, and reliability of facilities.

The design and management of supply chains under demand uncertainty has received significant attention over the past years. However, it is only recently that the possibility of disruptions in supply chains has been considered in the formulation of design problems [1]. Such disruptions, localized in specific facilities, can have a catastrophic impact on the entire supply chain as it might propagate through the system. In order to protect against the undesirable effects of such events, disruptions must be anticipated in the formulation of the design problem. In this work, the optimization of the supply chain design under random facility disruptions is formulated within the framework of multistage stochastic programming with recourse.

We consider the supply of a single commodity from a fixed set of production facilities, passing through distribution centers, and serving a fixed set of customers with deterministic demand. Lead times are considered between facilities. The objective is to select among the set of candidate distribution centers and design their capacity in order to minimize the investment cost and the expected operational cost of satisfying the customers demand. We develop a multi-period formulation where the first-stage variables correspond to the selection of distribution centers and their capacities; the recourse variables that are selected in the different time periods correspond to the inventory level at each distribution center, the amounts transported from production facilities to distribution centers, the amounts transported from distribution centers to customers, and penalties for unsatisfied demand.

At any instant in time, one or more disruptions can occur in the distribution centers according to given probabilities. The combination of disruption in a time instant is called a discrete stateof the system [2]. The transition of the system among discrete states through the different time periods can be modeled as a time-homogeneous Discrete-Time Markov Chain (DTMC).

Although the discrete nature of disruptions in the system determine a finite number of possible discrete states, the multi-period formulation with multi-stage recourse variables implies an exponential growth in the number of scenarios with respect to the time periods. In order to overcome this difficulty, a constant inventory level is maintained throughout all time periods. This policy eliminates the dependence of the system state from its history, avoiding the explosion in dimensionality that is caused by the multi-stage inventory recourse. The resulting problem is then equivalent to a two-stage stochastic programming problem where the scenarios correspond to the possible discrete states of the facilities.

In order to illustrate the application of the proposed model we present several example problems. Simulation results on the predicted supply chain designs show the economic advantages of the stochastic programming approach over the design based on deterministic supply when scenarios with random disruptions are considered. Furthermore, the periodic review of inventory is found to be particularly well suited for the design stage of supply chains. It is also shown that inflexibility on the inventory management offers a conservative estimate of the supply chain’s performance.

References

[1] Qi, L.; Shen, Z.-J. M. & Snyder, L. V. The Effect of Supply Disruptions on Supply Chain Design Decisions. Transportation Science, 2010, 44, 274-289.

[2] Terrazas-Moreno, S.; Grossmann, I. E.; Wassick, J. M. & Bury, S. J. Optimal design of reliable integrated chemical production sites. Computers & Chemical Engineering, 2010, 34, 1919 – 1936.