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The Effect Function

Let tex2html_wrap_inline3173 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 tex2html_wrap_inline2733 be the number of compartments and tex2html_wrap_inline3177 be the pollutant mass in compartment k (tex2html_wrap_inline2737) at the point tex2html_wrap_inline3183 and time tex2html_wrap_inline2787. Furthermore, let tex2html_wrap_inline3187 be the number of cell populations in compartment k and tex2html_wrap_inline3191 be the number of cells of type i in compartment k (tex2html_wrap_inline3197) at the point tex2html_wrap_inline3183 and time tex2html_wrap_inline2787. Set tex2html_wrap_inline3203. The modeler has then to specify n + N real-valued weights tex2html_wrap_inline3207 (tex2html_wrap_inline2737, tex2html_wrap_inline3197) (one for each compartment and one for each cell population). Let tex2html_wrap_inline3213 be the vector of decision variables. The vector tex2html_wrap_inline3215 represents the location of the pollutant emitter, while tex2html_wrap_inline3217 is the width of the "smokestack" and tex2html_wrap_inline3219 is it's height. With the weights given as above, define the function
displaymath4037
which measures the agglomerated effect of the decision tex2html_wrap_inline3221 at the point tex2html_wrap_inline3183 at time tex2html_wrap_inline2787. Here, the dependance of tex2html_wrap_inline2765 and tex2html_wrap_inline3229 on tex2html_wrap_inline3221 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
 equation677
This approach has the advantage that
displaymath4038
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. tex2html_wrap_inline3239 for some points tex2html_wrap_inline3183 and decisions tex2html_wrap_inline3243. 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 up previous contents
Next: Alternatives Up: The Objective Function Previous: The Objective Function

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