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450 Chapter 12 Urban Runoff and Combined Sewer Overflow Management
eliminate the subjective comparisons required in other matrix approaches. These criteria
should be:
• Nondominant—no criterion should be dominant.
• Complete—no pertinent information should be left out.
• Scorable—criteria cannot be vague, since they must be weighted clearly.
• Independent—criteria should not overlap each other
Weights are then generated for each decision factor. These weights must have a com-
mon scale, and the relative importance of each factor to the decision should be reflected in
the weights. An example is the BMP selection approach in Case Study 12.2 at the end of
this chapter. The major difference between this approach and the matrix approach outlined
above is that, in this approach, the decision factors must be quantitative. Therefore, subjec-
tive comparison terms, such as good or fair, cannot be utilized. The decision factors must
be able to be described by values that can be summed. Variations on this type of approach
and various decision support software can facilitate the conduct of these analyses.

Optimization. Optimization, a widely used method of quantitative decision making, in-
volves formulating a problem as the maximization (or minimization) of an objective func-
tion, subject to a series of constraints. In linear optimization, both the objective function and
the constraints must be linear functions of the decision variables. Various methods are avail-
able for finding the optimum set of decision variables and several software packages can
perform the analyses. These methods are summarized in basic textbooks on optimization.
For plan selection, the objective function can be cost or a more complicated function
of cost, benefits, and detriments. Examples of benefits that could be included are gallons of
discharge removed, pounds of pollutants removed, and days of beach closure avoided. A
multifactor objective function can account for trade-offs among costs, benefits, and detri-
ments by incorporating relative weights for each factor:
F a y (12.1)
i i
where
F objective function
a i weight and conversion factor
y i cost/benefit factor.
All terms in the preceding equation must have the same dimension (e.g., dollars) so
that weights also incorporate a conversion factor. The optimization process then consists of
maximizing the objective function, by optimally selecting the values of the decision vari-
ables on which the different factors depend. Then, each cost/benefit factor, y , must be ex-
i
pressed linearly in terms of each of the decision variables x:
y b x (12.2)
i
i i
where b is a different weight or conversion factor.
i
This relationship is relatively easily established for cost (such as life-cycle cost), but
more difficult for other factors, such as pounds of pollutant removed or days of beach clo-
sure. For these types of factors, models need to be applied with different values of the de-
cision variable and then straight line fitted to the result. Constraints must also be estab-
lished as linear functions of the decision variables. Possible constraints are the maximum
number of excursions of standards per year or the maximum amount of pollutant reduction
achievable given background conditions. Once the objective function and constraints are
defined, various algorithms and software packages are available to determine the combina-
tion of decision variables maximizing the objective function.