## Nonlinear optimisation for exponential preference behaviour

In both the Open Design methodology of Chapter 1 and the preference measurement as described in Chapter 5, linearity is implied. For instance, a finance-oriented stakeholder can specify the price per square meter he or she is prepared to pay for a parking lot. The realistic case mentioned before, that the stakeholder is prepared to pay more per square meter for a small parking lot for VIPs only, cannot be taken into account. Non-linear stakeholder preference behaviour is quite common. For instance:

• Price to be paid for a location related to the distance from a noisy factory;

• Price to be paid for percentage green in an urban area;

• Price to be paid for energy savings;

• Price to be paid for an airport island related to the distance to the shore;

• Unsupported length of buildings over the railways of a station related to the number of railway tracks;

• Price to be paid for shape, volume and area to enable the architect to create interesting architecture.

Figure 7.1 The price a decision maker is prepared to pay for a location at some distance from a noisy factory follows an exponential curve

CD O

Distance from noisy factory

Figure 7.1 The price a decision maker is prepared to pay for a location at some distance from a noisy factory follows an exponential curve

Non-linear preference behaviour tends to follow an exponential pattern (see Lootsma, 1999). For instance, the extra price the decision maker is prepared to pay for having a location at some distance from a noisy factory will be as indicated by the exponential curve in Figure 7.1 and not the linear one. The exponential curve can be expressed in the equation:

The unknown coefficients a, g and b can be determined if three points on the curve are given. The decision maker must, therefore, specify three instead of two equivalent alternatives. For the linear case, we could write:

from which the ratio wilw2 could be obtained. For the non-linear case, this becomes:

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