(234a) A New Robust Scenario Approach to Supply Chain Optimization Under Uncertainty

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:

                                                    
(1a)

,                                                                      
(1b)

where the first-stage variables are represented by
, the second stage cost for uncertainty realization
 is
represented by,

,

where
,
,
,
,
 are known
constants. This two-stage model is often approximated by the following scenario
formulation for tractable solution  [1]:

       
                                                     (2a)

 
                                                                                         (2b)

    
,        (2c)

where
 are
sampled uncertainty realizations, called scenarios, and
 are the
probabilities associated with the scenarios. The major shortcoming of the
scenario formulation is that, a large number of scenarios are usually needed,
to ensure that the solution is feasible for the original two-stage formulation
and that the predicted expected cost is close to the true expected cost. To
overcome the shortcoming of the scenario formulation, a robust scenario
formulation has been proposed [2-3] based on partitioning the uncertainty
region
 rather
than sampling a finite number of uncertainty realizations from it. The
resulting formulation is

                                  
(3a)

                                                                                 
(3b)

    (3c)

where scenario
 is
associated with uncertainty subregion
 rather
than a uncertainty realization and
 is
required to ensure that the solution to Formulation (3) is feasible for the
original problem. The second stage decision for scenario
 is assumed
to be an affine function of the uncertainty realization in
, i.e.,
. As a result, the coefficients of the affine function
(
) rather than
 serve as
the second-stage variables in the optimization problem.

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.