INCORPORATING ENVIRONMENTAL CONCERNS IN SUPPLY CHAIN OPTIMIZATION - Environmentally Conscious Materials Handling - page 129

environmental criteria related to corporate social responsibility) decision-making
problem. The dynamic system analysis aims to compute the trajectory of product
flows within the network, the levels of chosen social responsibility, and product
prices. The problem formulation involves differential equations that capture sys-
tem dynamics embedded within an optimization problem. They propose a solution
algorithm based on the Euler method. The Euler method has been applied effec-
tively to other dynamic supply chain network problems as well (Nagurney and
Dong 2002).
The rolling time-horizon approach, in which the start and the end of the
planning horizon are rolled forward as time progresses, has been employed for
dynamic decision making in the context of traditional supply chain decisions:
optimizing a vendor-managed inventory system (Al-Ameri et al. 2008), design-
ing flexibility contracts under variable demand (Walsh et al. 2007), and setting
safety-stocks in a multistage inventory system (Boulaksil et al. 2007), to name
a few. Such problems are solved using either traditional mathematical program-
ming techniques, simulation, or heuristic methods. Environmental considerations
that necessitate a rolling time-horizon approach can be accommodated in a similar
fashion.

5.4 Stochastic Programming and Robust Optimization

Decisions related to the environment often involve a high level of uncertainty.
The source of this uncertainty ranges from resource availability to future reg-
ulatory requirements. Deterministic optimization methods (such as linear pro-
gramming) assume that objective coefficients and constraints are known with
certainty. Deterministic methods may serve as good proxies if good estimates of
the unknown parameters exist. By employing sensitivity analysis, the decision
maker can determine the ranges of parameters over which a solution is feasi-
ble or even optimal. However, sensitivity analysis is not particularly helpful for
generating solutions that are robust to parameter changes.
The stochastic programming approach directly deals with uncertainty, in that
the uncertain parameters are modeled as random variables with known prob-
ability distributions. The goal, then, is to achieve the best objective value in
expectation. A popular method is the stochastic linear programming model with
recourse , in which corrective action can be taken after uncertainty is resolved.
For example, a two-stage recourse model has decision variables in each of the
two stages. The second-stage variables relate to recourses available after the real-
ization of uncertainty. In stochastic optimization, the problem is expressed using
probabilistic constraints — the assumption being that the distributions of problem
parameters are known. However, in practice, it may be difficult to obtain or even
approximate parameter distributions. In addition, stochastic optimization models
are often difficult to analyze. For more on stochastic programming in the context
of supply chains, see Shanthikumar et al. (2003).