ABSTRACT
The only remaining step is to write down an objective function which contains both kinds of trip terms, and this is done in this example using the group consumers' surplus criterion based on random util ity theory eso the s-wei ghts appear on the util ity terms). We also, to illustrate yet another way of writing these things, follow Coelho and Williams in including the attractiveness terms within the entropy function rather than as a separate 'benefit' term. Formally, this does not make any difference in this particular model but there are possible differences in interpretation: in the form given below, the attractiveness terms can be taken as representing prior
probabilities with the entropy term taking the Kullback form. (See the references for further details on this.) It can be argued that it is important to deal with intrinsic properties of zones, such as size in this way, but this turns on whether service centres are to be interpreted as directly related to zone size or as 'points' within a zone. In the residential case, it may be more reasonable to associate attractiveness partly with size. However, this debate takes us beyond what we need for present purposes and is not pursued further here. The objective function is therefore:
- - 1: T.. (log ---!l.. - 11 ij lJ W~es
1) - 1: T..c .. ij lJ lJ
S .. lJ W iJ' lJ lJ1J j Thus, the problem is to maximise Z with respect to T.. and S..•R lJ lJ E., E. and P. are determined by the constraints (Al.47)-(Al.51)J J 1 which must be satisfied simultaneously. In their formulation,
{W~es} and {W.} are being used more passively than hitherto as 1 J
intrinsic and exogenous attractiveness weights. A somewhat more elgant form of the model, and one which
gives us new insights, can be obtained by eliminating the variables P., E. and E~ from Equations (Al.47)-(Al.51) which
1.2 i lJ nEj
where 1.1i and 1.2 are defined by
Ali n/pClO i
1.2 nip
(Al.53)
(Al.54 )
(Al.55)
(Al.56)
When the model is used in this form, the variables which have been eliminated can always of course then be calculated from the constraint equations in their original form.
