INCORPORATING ENVIRONMENTAL CONCERNS IN SUPPLY CHAIN OPTIMIZATION - Environmentally Conscious Materials Handling - page 128

is often used. In this method, the decision maker specifies fractional weights for
each individual objective. The AOF combines the weighted individual objectives
into a single function that can be solved in the same manner as a traditional,
single-objective problem. A challenge is to arrive at meaningful weights for the
various objectives (Ehrgott 2005). Shue (2006) employs a MOP to optimize both
generation operations and associated induced-waste reverse logistics at a nuclear
facility. The AOF to be maximized aggregates the positively weighted power
supply chain profits and the negatively weighted reverse chain costs. A menu of
solutions is generated for varying values of weights.

5.2.2 Constraint Method
Apart from the weighted-sum approach, other techniques to solve multicrite-
ria optimization problems also exist (e.g., -constraint method, elastic constraint
method, hybrid method, etc.). Popular among these techniques is the -constraint
method. Unlike the weighted-sum approach, no aggregation of criteria is neces-
sary. Instead, one criterion is chosen as the primary criterion to be optimized and
all other criteria are transformed into constraints, such as in Subramanian et al.
(2008), where emissions are minimized subject to a reservation level of profit.
The Greek letter denotes the RHSs (or reservation levels) of the constraints cor-
responding to the nonprimary objectives. For convex optimization problems (i.e.,
where the decision and criteria spaces are convex), the weighted-sum method is
guaranteed to find efficient and weakly efficient solutions, while the -constraint
method only guarantees weakly efficient solutions (Ehrgott 2005, p. 98). How-
ever, the -constraint method works for even nonconvex optimization problems.
MOPs are also well-suited for representing and solving optimization problems
that span multiple firms or networks of firms and involve objectives at various
levels. Sabri and Beamon (2000) develop a multiobjective optimization model
for a traditional supply chain, allowing for metrics (derived not only from cost
but also from other performance criteria such as customer satisfaction) to be
assessed across the entire supply chain network rather that at the individual firm
level. We suggest that such models can be expanded to incorporate green metrics,
affording a supply-chain-wide treatment of environmental considerations.

5.3 Dynamic Models

Static (or myopic) SCO approaches are incapable of suitably accommodating
the dynamism inherent in environmental factors. As mentioned in the preceding
sections, factors such as legislative changes to emissions limits and the decreasing
availability of certain resources render dynamism to an SCO exercise.
Several solution methods exist for complex dynamical systems. Among them
are differential equations and dynamic programming. As an example, Cruz (2008)
develops a dynamic framework for modeling a multilevel supply chain network.
In particular, each decision maker in the network faces a multicriteria (including