Environmental Engineering Reference
In-Depth Information
also often exist among the decision variables themselves. For example, produc-
tion decisions related to the mix between new and remanufactured products are
linked nontrivially to the product design decision of product remanufacturability.
Traditional, linear methods are not equipped to provide an adequate representa-
tion of such dependencies. Nonlinear methods, on the other hand, allow for more
realistic representations of managerial trade-offs and the interactions among deci-
sions and consequences. By definition, an optimization problem is nonlinear when
either the objective or any of the constraints is nonlinear.
Researchers have recently demonstrated the use of nonlinear programming
(NLP) methods to incorporate environmental considerations into SCO. For
example, Subramanian et al. (2008) develop a nonlinear optimization model for
a manufacturing firm attempting to integrate environmental considerations (such
as remanufacturing and product design) with traditional operations planning
considerations (such as production and inventory). The NLP approach allows
them to include dependencies between new and remanufactured products
such as the cannibalization of new products by remanufactured products and
competition for limited production capacity. Apart from the nonlinear objective,
other nonlinear elements in the optimization problem include the cost of product
design and consumer demand, which is a nonlinear decreasing function of
production quantities (again, decision variables). Relationships expressed in
the form of nonlinear expressions can realistically and flexibly characterize a
representative problem. Although the computational effort involved in solving
NLPs is significant, present-day desktop computers, together with commercially
available
software
tools,
can
be
used
to
solve
even
complex
NLPs
in
a
reasonable amount of time, allowing for what-if analyses.
5.2
Multiobjective Optimization
Environmental considerations often give rise to multiple, possibly conflicting,
objectives in SCO. For example, in the work by Subramanian et al. (2007), the
firm might want to maximize profits as well as minimize emissions. A multiob-
jective (or multicriteria) optimization problem (MOP) involves the simultaneous
optimization of two or more objectives subject to constraints. To date, the con-
cept has been employed effectively in scenarios when problems are characterized
by decisions that are conflicting in nature, which is often the case when incor-
porating environmental considerations. As an example, minimizing emissions is
often at odds with maximizing profit. Several solution methods exist to solve
multiobjective problems. This section discusses two such methods.
5.2.1 Weighted-Sum Method
One approach is to construct a single aggregate objective function (AOF), in
which all of the criteria being considered are combined into a single objective.
There are several ways to aggregate criteria; the weighted linear sum of objectives
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