i j
hii/M (13)
then this would be correct only if the number of trips leaving each origin, i, per unit of time were the same. But of course that removes the question of elastic demand for travel by assumption. The relation-ships between gi and fii in the case of multiple purpose trips are enormously complex. The solution as outlined here is pragmatic rather than theoretically rigorous.
An alternative approach to multiple purpose trips should also be mentioned. This is to capture the externality benefits of multiple purpose trips directly in the specification of fii. fii typically includes a measure of the in situ attractiveness of the facility at j. If the definition of in situ attractiveness is extended to encompass the closeness of j to other complementary opportunities that might be used during a multiple purpose trip, then this may account for some of the multiple trip effects. This is analogous to capturing elastic travel demand by extending the definition of origin generating effects to include accessibility of destinations.
An example
For the case of single purpose trips, which is all that will be treated in the following sections, then the interaction model, from (1), (2) and (9), becomes:
Iii = Oig(cpJ · q>j1
· fii . (14)
The function g(cpJ could be derived theoretically from microeconomic considerations. However, given the dubious nature of assuming perfect information and decision making .characteristics in real situations g ( cpJ would be better estimated empirically, taking due consideration of customer type, and other in situ origin specific factors that would influence the demand for use of the facility. Note that:
g(cp;) =
(~Iii)
q>j1 0;-1 . (15)440 E.S. Sheppard
In particular, if we assume the functional form in equation (3):
Iii = aOiq>r-1 • ~i (16)
a case which is curiously similar to the origin constrained Alonso
«theory of movement» (Alonso, 1978; Ledent, 1980). To call this an origin constrained interaction model would be a loose use of the term, however, since the number of trips is not fixed.
2. Users' benefits
2.1. The Neuburger approach
The case of consumers' surplus as a measure of user benefit in the case of transport improvements has been treated by Neuburger (1971) who argues that the change in benefits due to improvements is equal to the perceived net benefits of the new system plus the fall in transport costs incurred under changed user behavior on the new system. However, he restricts himself to the case where in situ
destination attractivities are held constant while travel costs are changed on some routes. He is thus able to use travel costs as the measure of benefits. Fig. 1 reproduces an example of a demand curve of this type. Benefits perceived by the user are the increase in demand as perceived costs fall from PC1 to PC2 (ABDE); perceived costs are the extra perceived travel cost of the new trips (BCHG). Actual benefits to the system are the cost reduction on old trips (IJKL) less the incurred cost of new trips (LMHG). The total benefit is ABDE
+
IJKL - BCHG - LMHG.However, in the case where destination attractivities are changed, which by definition occurs in facility location problems, this is
inadequate. Consider
an
addition of attractiveness to all destinations in such a way that hii is unchanged ¥ ij . Then more trips will occur at no greater cost, perceived or actual, per trip. Then PC1 = PC2 ,AC1 = AC2; ABDE = IJKL = 0; and BCHG
+
LMHG>
0. Thus user benefits are negative, despite the construction of new facilities and the new trips generated by these.Analytically, Neuburger defines consumer surplus by:
.LlS1
2 cij
I I
~
lu dcu Pii(c)I J I
cij
(17)
in which CD is the old travel cost, c/i is the new travel cost, and Pii ( c) is the path of integration in cost space between the initial and final matrices of transport costs (Beaumont, 1980). If Iii = Oiaie-~c•i/I ak e-~c•k, where ai is the in situ attractiveness of site j, then k
olu
Io
cim = o limIo
cu (18)Public facility location with elastic demand 441
implying that integration is independent of the path. Pii ( c) may then be dropped from (17), and
[
I
ai exp { -~
cii}l
as, ~ +Lo,
log~a,
exp {-~cU J
However, if the travel costs are unchanged, but facility sizes are increased, (17) is not a valid index, as only changes in travel costs,
.A
cost
d
PC1 .
PCz
o'
AC, i'i 11111111111-1
AC2
IK
....L_~~~~~~G~~'~~~IH:....-~~~~----,~;-:-::-:-:~~
t2 =!fa of trips
(19)
cii,
t1 [Z'.'.'.2J perceived user benefit
~ perceived user cost
[ I I ] ac~ual decrease in trip com
E:3
actual incre3se in trip costsPC1 = Perceived cost before change PCz =Perceived cost after change AC1 =Actual cost before change AC2 =Actual cost after change t1 = total trips before change tz = total trips after change
1 I
D - 0 = aggregate demand cul'\le
Figure I Consumer surplus after Neuburger
442 E.S. Sheppard
are allowed for. If these do not change, (17) must be zero. If
changes in the in situ attractivenesses ai are also to be considered, then (17) should be integrable with respect to ai as well as cii. But this implies that the integrability conditions (18) should be also expanded to include the effects of am and ai on Iu and Im. If this is done, then the definition of Iii that will satisfy (18) must also be changed, implying that (19) .is no longer correct.
Coelho and Williams (1978) have made one such extension in the following manner. Define:
Iu = a · oi ·
r;/(I f;k)
k
implying an inelastic demand for travel, a. Then define surplus as:
AS1
F?i
I I
~
Iii dFu Pu(F)i j I
Fu
where Fu = 1 n (f;J Now it may be shown that: