(518d) Stochastic Programming Model for Supply Chain Design Under Uncertainty

Authors: 
Harjunkoski, I., ABB Corporate Research
Grossmann, I., Carnegie Mellon University



Supply chain (SC) design mainly involves decisions about where to place the assets (factories and distribution centers) considering a long term horizon planning. However, since operational performance is greatly influenced by the SC design, a responsive supply chain can only be guaranteed when an effective inventory management, as well as an appropriate distribution and storage structure are planned together. Furthermore, rising transport costs are key factors in a companies’ economy and managing inventory has become a major target in order to simultaneously reduce costs and improve customer service. For that reason, over the last few years, there has been an increasing interest in developing enterprise-wide optimization (EWO) models to solve problems that are broad in scope and integrate several decision levels (Grossmann, 2005).

When long term planning is involved, demand uncertainty must be taken into account In fact, demand uncertainty might have a relevant influence on warehouses capacity requirement. In that sense, if the plan for storage capacity does not consider demand uncertainty, it might be infeasible to provide the products as required.

In order to cope with demand uncertainty, there are two main approaches to consider. The first one uses a stochastic programming model where uncertainty is considered directly using a scenario based approach (Sahinidis, 2004). The main disadvantage of this method is that the model size tends to increase rapidly with the number of scenarios considered. The second approach is to use chance constraint in which each uncertain parameter is treated as a random variable with a given probability distribution (Charnes and Cooper, 1963). Even though this approach does not involve scenarios, the model gives rise to non-linearities in the formulation.

The optimization model designs a multi-echelon supply chain of multiple products. Long term decisions involve new installations, expansions and elimination of warehouses. Tactical decisions include deciding inventory levels to satisfy the uncertain demand in distribution centers and customer plants, as well as the connection links between the supply chain nodes. Capacity constraints are also considered when planning inventory levels.  Chance constraint approach has been previously applied to the integrated problem of inventory management and supply chain design under demand uncertainty by Rodriguez et al. (2012).

Regarding the stochastic programming model, each scenario is associated with certain probability of occurrence and represents one possible realization for the uncertain parameter. In general, there are two or more stages in the decision process. One of the main challenges is to define the different stages of this process and how scenarios are generated accordingly. In the first stage, ‘here and now’ decisions have to be made before the uncertain parameter realization is known. In this problem, investment decisions such as new assets, expansions and shut-downs are included as first stage variables. In the second, ‘wait and see’ decisions involve a recourse action because they can be made after the random parameter is known. In the case of this supply chain, inventory levels are decided in each scenario according to the expected demand. However, these inventory levels must also take into account the capacity of storage defined in the first stage of the process.

In this work, we reformulate the problem applying two-stage stochastic programming with the aim at comparing both approaches and analyze advantages and drawbacks of them in this specific problem.

References

Charnes, A. and Cooper, W.W. Deterministic equivalents for optimizing and satisfying under chance constraints. Operations Research. 1963, 11, 18–39.

Grossmann, I.E. Challenges in the New Millennium: Product Discovery and Design, Enterprise and Supply Chain Optimization, Global Life Cycle Assessment. Computers and Chemical Engineering, 2005, 29, 29-39.

Rodriguez, M.A., Grossmann, I.E., Vecchietti, A.R. and Harjunkoski, I. Optimal Supply Chain Redesign in the Electric Motors Industry. 2012, AICHE Annual Meeting, Pittsburgh, PA, USA.

Sahinidis, N.V. Optimization under uncertainty: state-of-the-art and opportunities. Computers and Chemical Engineering. 2004, 28, 971–983.