**A
new robust scenario approach to supply chain optimization under uncertainty**

Niaz Chowdhury, Xiang Li*

*Department
of Chemical Engineering, Queen's University, *

*19
Division Street, Kingston, ON, K7L 3N6, Canada*

The following two-stage stochastic programming models [1] are often used for supply chain optimization under uncertainty:

where the first-stage variables are represented by

where

where

where scenario

The construction of uncertainty subregions and the reformulation of Formulation (3) into a tractable deterministic optimization problem are the two key steps for applying the robust scenario method. In previous work [2-3], the uncertainty region bounded by the infinity-norm (i.e., box uncertainty) is considered, for which the construction of uncertainty subregions is straightforward (see Figure 1(a)). In addition, in this case the uncertainty subregions are also bounded by the infinity-norm, so Formulation (3) can be equivalently transformed into a deterministic linear programming problem. However, when the original uncertainty region has an arbitrary shape, it is not clear how to effectively partition the region to yield a set of uncertainty subregions such that Formulation (3) can be transformed into a tractable deterministic optimization problem (see Figure 1(b)).

Figure 1. Partition of uncertainty regions.

In this paper, we propose a novel robust scenario approach that constructs a sequence of box uncertainty subregions that over-estimate the original uncertainty region, and a sequence of box uncertainty subregions that under-estimate the original uncertainty region (as illustrated in Figure 2). The robust scenario formulations with the over- and under-estimation of uncertainty region can both be transformed into deterministic linear programming problems; when the number of scenarios (i.e., uncertainty subregions) increase, the over- and under-estimates of the uncertainty region become close and the optimal values of the two robust scenario formulations converge, leading to a good solution for the original two-stage stochastic programming problem. Case study of a real industrial supply chain optimization problem with ellipsoidal demand uncertainty demonstrates the advantages of the proposed robust scenario approach.

Figure 2. Over- and under-estimation of uncertainty region

**References**

[1] Birge, J. R.; Louveaux, F. *Introduction to stochastic programming*, Second Edition; Springer:
New York, 2011.

[2] McLean, K.; Li, X. "Robust
scenario formulations for strategic supply chain optimization under
uncertainty",* Industrial &
Engineering Chemistry Research*, 52, 5721-5734.

[3]
McLean, K.; Ogbe, E.; Li, X. "Novel formulation and efficient solution strategy
for strategic optimization of an industrial chemical supply chain under demand
uncertainty", *Canadian Journal of
Chemical Engineering*, DOI 10.1002/cjce.22173, 2015.

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