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## The Effect Function

Let be the ground surface of the modeled region. (We stress again that we are using a Cartesian coordinate system.) The discussion in the last chapters and the model derivation in Chapter 1 have shown that we are able to calculate not only the pollutant concentration in certain compartments, which can be anything from ecotrophic levels down to organs in individuals, but that we can also calculate the effect of a pollutant on certain cell populations in terms of toxicity and carcinogenicity. What remains is to find an agglomeration of the different effects on the different levels into one value. The minimization of all the quantified effects simultaneously is evidently a multicriteria problem for which the knowledge of some approximation of the efficient set would be of great use. However, the high computational demands to compute such an approximation as well as methodological difficulties force us at the present moment to globalize all the different effects by way of a utility function. The choice of such a utility is nontrivial, and it does not seem to be clear what effect a corresponding choice has on the solutions of the resulting single-criterion optimization problem. We prefer at the present moment to globalize all the different effects by way of a weighting function. It is clear that some of the efficient points of the abovementioned multicriteria problem will be missed in this way [9]. However, scalarizations with, e. g., the weighted max-norm, capable of computing all efficient points [16], lead to nondifferentiable problems or problems with a possibly large number of constraints, thereby adding computational costs which might well be able to prohibite the computation of a solution.

Let  be the number of compartments and be the pollutant mass in compartment k () at the point and time . Furthermore, let be the number of cell populations in compartment k and be the number of cells of type i in compartment k () at the point and time . Set . The modeler has then to specify n + N real-valued weights (, ) (one for each compartment and one for each cell population). Let be the vector of decision variables. The vector represents the location of the pollutant emitter, while is the width of the "smokestack" and is it's height. With the weights given as above, define the function

which measures the agglomerated effect of the decision at the point at time . Here, the dependance of  and  on has been skipped, as it has been done above. The agglomeration of effects over a time interval [0, T], T > 0, can be done in many ways. The method employed in OLAF is the time integral over the effect

This approach has the advantage that

The time variable t is already discretized by the air pollution model, and it is this equidistant discretization which is used in the numerical evaluation of the integral above. Note that this approach allows for hormetic values, i. e. for some points and decisions . In such a case, the pollutant effect has actually lead to a reaction of the individual's cell population residing at x which can be interpreted as an increase in health [2, 3].

Next: Alternatives Up: The Objective Function Previous: The Objective Function

Joerg Fliege
Wed Dec 22 12:25:31 CET 1999