assignment with variable destination costs

Transpn. Rw.-B.Vol. 27B, No. 3, pp. 207-217, Printed in Great Britain.

1993

0191-2615/93 s6.cxl + .@I 0 1993 Pergamon Press Ltd.

EQUILIBRIUM TRIP DISTRIBUTION/ASSIGNMENT WITH VARIABLE DESTINATION COSTS NORBERT OPPENHEIM Institute for Transportation Systems, City University of New York and Department of Civil Engineering, The City College of New York, New York, NY 10031, U.S.A. (Received 20 June 1991;in revised form 15 March 1992) Abstract-A “user-equilibrium” trip distribution/assignment model is developed, which includes an endogenous destination cost for the conduct of a given activity at the destination. Destination costs are determined endogenously, as increasing functions of the number of trips to the destination. Link travel costs are increasing functions of link volumes. Travelers are assumed to choose the itinerary of deterministically perceived least cost. Activity performers are assumed to choose among alternative destinations so as to minimize the randomly perceived total cost of activity and travel. It is shown that the model always possesses a unique solution, when destination costs reflect either positive or negative externalities. An algorithm for obtaining that solution is described. The model is well suited to the analysis of a variety of issues concerning the effects of road pricing policies on urban travel demand and, conversely, of toll-based traffic management strategies on economic performance.

1. INTRODUCTION: THE DISTRIBUTION/ASSIGNMENT PROBLEM WITH VARIABLE ZONAL ATTRACTIVENESS

Given the numbers of trips Oi that originate at various locations i, link travel time functions g,,(.) for each of the A links of the (unimodal) transportation network, and measures of attractiveness Aj for each of the alternative destinations j (measured in units of travel time), the standard combined distribution/assignment problem is formulated as (Sheffi, 1985)

ga(X)dX -

C Ci Aj j

J’tj

(1)

. . ,n

(2)

such that c Y, = oi i

c F;

i = 1,2,. i,j=

= Y,

1,2 ,...,

a = 1,2,. V

n

(3)

. . ,A

(4)

6 j, r

where the unknowns are the origin/destination flows between origin zone i and destination zone j, Yg; the flows on link a, x0; and implicitly, the flows on route r connecting i and j, Fur.liaru is equal to one if link a is part of route r between i and j, and zero otherwise. The behavioral principle that underlies this model is that at equilibrium, a traveler starting a trip from a given zone i chooses the destination j that offers the minimum total cost of travel minus the destination’s attractiveness, while at the same time choosing the itinerary of least cost to that destination. In the above formulation, link travel times are variable, whereas destination attractiveness is assumed fixed. In general, however, zonal attractiveness may depend on destination trip ends. For instance, congestion at the destination may create delays that add to the total trip time and thus to travel cost. Moreover, travel is often undertaken to 207

208

N. OPPENHEIM

perform some activity (i.e. shopping, recreation, obtaining a service, etc) at the destination, for which a price must be paid. In turn, the destination cost, which may include the price charged for the conduct of the activity, may be a function of demand for it. In such circumstances, destination attractiveness will not be fixed but rather a function of the number of trip ends Yj at the destination. In a previous paper (Oppenheim, 1989), the author developed a trip distribution model, with variable travel and destination costs, but trip assignment was not considered. The purpose of this paper was to develop a combined distribution/assignment model with endogenous route and destination costs, so as to extend the formulation outlined above. 2. ASSUMPTIONS OF THE MODEL

In this section, the various assumptions underlying the model are formulated. 2.1. Criterion for the endogenous determination of destination costs Because the fundamental difference with the standard model above is the fact that now the destination costs, Ris, are determined endogenously as a function of trip ends, Yj, the destination “cost function” must be specified. Thus, we define the destination cost function as

Rj =fiCrj)

(5a)

where Rj is the destination cost and Yj is the endogenous number of trip ends to j.

rj = cj r,

(5b)

The nature of functions fj ( Yj) may translate either negative or positive externalities. An example of a negative externality is when destination costs result from congestion at the destination (i.e. queuing to obtain some service such as parking). In this case, functions J are strictly increasing in their argument Yj. The specific expression of functions fi may be derived from queuing theory and will reflect the nature of the arrival process, the number of “servers” and other characteristics of the queuing process. For instance, when destination demands are Poisson distributed and travelers are served by a single server, the destination cost function may be expressed by the Pollaczek-Kintchine formula:

for GYj < 1 and 00 otherwise, where rj and -rj are the first and second moments, respectively, of the distribution of service time at j. When destination costs reflect the price charged for the activity conducted at the destination (such as buying a retail commodity), function fi may be specified as the “supply price” function at location j, which specifies the unit price the supplier would charge at a given level of demand. If the supplier increases the unit price with demand, the price function would again reflect a negative externality similar to congestion effects (Varian, 1984). If, on the other hand, the supplier can afford, due to economies of scale, for instance, to lower the unit price with increasing volume, then consumers/travelers would benefit from positive externalities. More generally, positive externalities might be present if any other destination attribute became more attractive with increasing destination volumes. This might be the case in retailing, for instance, because it might mean a higher turnover and consequently more recent products. In any case, positive externalities imply that functions fi are strictly decreasing in their argument Yj. It is not necessary to specify which type of externality effects are present, as it turns out (as will be seen in Section 4) that it does not affect the properties

Equilibrium trip distribution/assignment

209

of the model, in particular the uniqueness of its solution, as long as only one type of externality is present. 2.2. Formulation of the traveler’sbehavior Next, we must also specify how destination costs affect traveler behavior, specifically the choice of a destination. We assume that travelers choose a destination so as to minimize its disutility, which is equated to the generalized cost of travel to the destination. Destinationjs utility for a traveler whose trip originates in location i is then UC = - (aj + w Rj + 7 tij)

(6)

where tu is the travel time between origin i and destination j and Rj is the destination cost, as described above. The values of these respective variables are not known but must be given their equilibrium values. Parameter T represents the value of time when traveling. Parameter w translates the importance of the destination cost relative to travel cost. Parameter aj is a constant that represents the monetary value of all other unspecified destination attributes, and may be positive or negative. We assume that the values of all parameters are known, for instance, from model calibration using observational data. We further assume that costs Rj are perceived randomly, but that the to’s are not perceived randomly. Thus

Rj = fi(Yj) + E

(64

where the random term E is assumed to have a Gumbel distribution, with a mean of zero and a variance d. This assumption implies that a given destination ~3 total disutility is perceived randomly: Vu = - (aj +

ORj + T tQ) + E

(6b)

Following well-known results of random utility theory (McFadden, 1974), the probability Pi,, that a traveler starting a trip in zone i chooses to go to destination j is given by the standard logit function

Pj,i =

exp {-/3 (aj + WRj + du)} (aj + WRj + Ttu)}

Cj exp (-0

for i,j = 1,. . . ,n

(7)

Given the number of trips 0; originating at the various locations i, Y, equals Yij =

oi

exp { -/3 (ai + WRj + Ttv)} Cj exp {-P (aj + WRj + Ttu)}

for i,j = 1,. . . ,n

(W

The value of parameter /3 is related to the value of the standard deviation u of the random utility terms e in formula (6a). Specifically p =

?r(a&)-’

All quantities Yti,Rj and t, are to be determined endogenously, combined distribution/assignment problem.

(7b) i.e. are unknowns in the

3. FORMULATION OF THE MODEL

It is possible, in the usual manner, to construct a nonlinear optimization problem whose solution represents equilibrium trip distribution and assignment with all of the above required features. The problem is the following:

210

N. OPPENHEIM

(8) c

C(QjYj+W j

+

s

Yij

d

fi (x)dx)

The constraints are i = 1,2,

C Yij = Oi

cF,‘=

Y,

. . . , tl

CW

i,j=

1,2 ,...,

n

(9b)

W, i,j=

1,2 ,...,

n

(W

r

cHi,‘=

In addition, the following definitional relationships hold: q=CY, X, =

ccc *

i

(F;

Vj

So,/

+ H;)

(94

a = 1,2,.

. . ,A

I

CW

Functions fi(.) represent the destination cost functions, as defined in eqn (5a). Functions g,(.) represent the link travel time functions. It is assumed that these functions are also strictly increasing. Constants W, represent origin/destination flows of travelers who do not participate in the given activity (i.e. general traffic) and are given. Variables Hi represent the corresponding path flows and are implicit unknowns. All other symbols have the same meaning as in program (1) to (4). In addition, all variables are nonnegative, or positive, as indicated.

Parameters fl and T are positive and are assumed given, for instance, from model calibration to observational data: p > 0;

7 >

0

In order to show that the solution of program (8) and (9) does indeed provide the required equilibrium flows, we follow the usual approach and write the Karush-KuhnTucker conditions (KKT) with respect to the origin/destination flows Yii and the path flows Fi,’ and HUT.These conditions are ~‘-~~p,--$[~F;aFuk i j F;,$u

+-&;1

~-~Ce,&[CH;-W,j~O; i II u

Yo)zO;

Yuj] =O; u

vi,j,k

v

i, j, k

r vi,j,k r

(10) (11)

UW

Equilibrium trip distribution/assignment

211

(lla)

(12)

(13) First, the derivatives of Z with respect to the activity-related route flows are equal to (14) Remembering that g(.) is the link travel time function and given the definition of &rU above, it is apparent that the summation in eqn (14) represents the travel cost dor on route r between origin i and destination j. Conditions (10) and (11) may then be written, for all i and j: rtiir 2 pii;

V

F; (rtor - &

i, j, r

= 0;

V

(15) i, j, r

(16)

Equation (15) implies that pU represents the minimum travel cost 7ti/’ on all routes between i and j. Equation (16) implies that if there is no activity-related traffic on route r between i and j, then the travel cost on the route is at least as large as the minimum travel cost pU, and that conversely, if the route is used by activity-related traffic, the travel cost on it is the minimum travel cost between i and j. These conditions represent the standard “user-equilibrium” conditions with respect to activity-related trips. In a similar manner, conditions (1Oa) and (1 la) may be written V

rtii’ 2 8,;

i, j, r

Hur ( rtiir - Cl,) = 0;

V

Wa) i, j, r

(W

These conditions state that general trips, i.e. for purposes other than the given activity, are also routed according to user-equilibrium. Furthermore, these conditions imply that 8, = ~1~for all i and j. Next, the derivatives of Z with respect to the origin/destination flows YUare equal to

az w-0-

’ 108

Fj

+

aj

+

Wfi

C

;

Yu

(i

Vi, j

(17)

>

in which, according to eqn (5a) fi(&Y-,)

= &( yi) = Rj

Conditions (12) and (13) may then be written 1 log YQ + aj + w Rj + /.LU 2 Ai; B Yo

’

1

log Yu + aj +

w

1

Vi, j

Rj - Xi + /Q = 0;

Y. is strictly positive, as specified in condition (9f), left side of eqn (19) must be equal to zero, which implies

so

(18) Vi,j

(19)

that the second factor on the \ .

N. OPPENHEIM

212

log Y, = -0

(aj + WRj + /LO- Xi)

which in turn implies that inequality (18) may be dismissed. Because pij is equal to the minimum travel cost rto between i and j, as per eqn (15), we have y,, =

e-

B (aj+oRj+rf,,-Ai)

e-8

=

(aj+wR,+rf&

e@j

!J

(20)

From eqn (9a), we have c

e-8 (aj+wRj+rf,,) @Xi = ,0x, c

i

i

c-8 U,+WRj+rt,,)= 0

and

so that

yti

(21)

=

Thus, as was required, the expression for the Yi;s conforms to eqn (7a), in which t, is equal to user-equilibrium travel time between i andj, and Rj is the equilibrium destination cost as given by the value of the cost function. In summary, the solution of program (8) and (9) describes a combined trip distribution/assignment that possesses all of the required properties: 1. All travelers minimize travel time. 2. Activity performers minimize the perceived destination cost, which includes the activity cost plus the travel cost to the destination. 3. Destination costs depend on the level of demand at the destination. 4. The activity/travel system is in user equilibrium, i.e., no traveler may unilaterally decrease his/her total costs. In the case of activity performers, this includes destination costs. In the next section, it will be demonstrated solution. 4. EXISTENCE

AND UNIQUENESS

that the model always has a unique

OF THE MODEL SOLUTIONS

4. I. Existence First, it is clear that because the feasible region Q in the multi-dimensional space (F, H, Y), where F is the vector of path flows for activity-related trips, H is the vector of path flows for general trips and Y is the vector of activity-related interzonal flows, is defined by the linear equality constraints (9a) to (9e), Q is convex, closed and bounded. Also, it is clear that for any given set of values 0, Q is nonempty, as it contains, for instance, the point Y, = cjk =

0,/J; Y/K,;

Hij” = WV/Ku;

Vj v

i, j, k

V i, j, r

where Ku represents the number of paths between locations i and j. Therefore, the feasible

Equilibrium trip distribution/assignment

213

region Q in terms of the origin/destination and path flow variables F, H and Y, respectively, is not empty. Because the correspondence between path flows F and H and link flows X is linear, the feasible region in terms of the variables X and Y is also convex, compact and not empty. Consequently, program (8) and (9) possesses at least one solution. 4.2. Uniqueness Because, as shown above, the feasible region Q defined by eqn (9) in the space (X, Y) is compact, convex and nonempty, if the objective function 2(X, Y) in eqn (8) is everywhere strictly convex, then the solution to program (8) and (9) will be unique (Rockafellar, 1970). We can, without loss of generality, write 2(X, Y) as

zw,

Y) = Z,(X) + Z,(Y)

(22)

with

(224 (22b) 1

J

i

i

It is well known that because the functions g,(x), which define the link travel times, are assumed to be strictly increasing, Z, (X) is strictly convex. We must then examine whether Z,( Y) is convex. We begin with the case of negative externalities, i.e. when the functions fi( q) defined in eqn (5a) are increasing in the yi’s. In this case, each of the terms

in formula (22b) is a function Fj(CiYu) of the variable Yi/‘s. Because the functions&(x) are positive and strictly increasing, functions Fj(x) are increasing and strictly convex. Furthermore, because the argument CiYuof these latter functions is a linear, and therefore also convex, function of the YU’s,the function Fj(CiYg) is convex in the YG’s.(See, for instance, Berge, 1963, p. 191). In addition, each of the terms aj c L hence the function

YU is obviously convex, being a linear function, I

is itself convex. Finally, it is very easy to show, for instance, by computing its Hessian, that because P > 0, the first term in formula (22b)

is a strictly convex function of the YU’s. Thus, function Z, is the sum of a strictly convex function and a convex function, and is therefore strictly convex. Consequently, the solution to problem (8) and (9) is unique. It may be noted, however, that the solution in terms of the path flows (F*, H*, Y*) may not be unique, because function Z,(F*, H*) is a convex, but not strictly convex, function. iR(B)

27:3-o

N.

214

OPPENHEIM

In the case of positive externalities, i.e. when the functions&( Yj) defined in eqn (5a) are decreasing in the Y;s (but are of course still positively valued), the convexity properties of function Z,(Y) are best investigated by examining its Hessian, i.e. the matrix with general element

a*z*( Yij) ar, ar,,

vii =

The second partial derivatives of 2, ( Yu) are, respectively,

a2z

1

qcr,)

ar,

2=ar,+ aYij

or, in more compact notation

a*z -= a Y$ Ykj a22 ----= a r-,a Y,,

0

Thus, the Hessian is a diagonal block matrix, where the element M, is equal to i,j =

l,j

2,j...

n, i

m,j...

1.i

f;

2, j

fj

. . . . . . . . ,.............

f;. ........ .............

i,i ....

n,j

..............

.....

f;. ........ ..............

f; ...

fj. .... .............. &,+fj

fj. . . . . .

f;. . .

fj

..............

“I

i

and where the off-diagonal blocks are square matrices of size n x n, with all entries equal to zero. Function Z, will be strictly convex if and only if its Hessian is positive definite (Rockafellar, 1970). All matrices M,, are symmetric. In turn, a diagonal block matrix where all submatrices are positive-definite is itself positive-definite. A sufficient condition for positive-definiteness of symmetric matrices [a,,] is

au>

cjei

laul

Wa)

and because by hypothesis fJ is negative for all j, the above sufficient condition applied to M,, is then

l +f;(v ar, or

>

c Ifp-j,I

j+i

= (n - 1) lf,‘(~,l;

i,j=

1,2 ,...,

n

Equilibrium trip distribution/assignment

1

Y, <

i,j=

BnIfi((I;:)l;

1,2 ,...,

215

n

(25)

Because of constraint (9a), i=

Y, I oi; Therefore,

1,2,...,n

constraint (25) will be observed if Maxi{ Oi} < Min (yj, la &J,l;

v&j

Typically, decreasing cost functions h will be such that functions fj are increasing and that fj (0) # 0. (This would, for instance, be the case if functions & were approximated by negative exponent& or convex parabolas.) In such a case, constraint (26) would be met if the magnitude of the largest of the 0,‘s was set such that Maxi

< (P n Max]{ If,! (0) I ) I-’

(27)

In this case, inequality (24a) will hold everywhere, i.e. for all values of YU;all submatrices everywhere; the Hessian of Z, ( Y) will be everywhere positivedefinite; and Z,( Y) will be everywhere strictly convex. Consequently, the optimal solution (P, rC) will be unique. Finally, it is easy to see that if the destination cost is a sum of several different costs, some increasing in the destination volume and others decreasing, it is no longer possible to show strict convexity theoretically and thus guarantee uniqueness of the solution. Of course, uniqueness is still possible but would have to be shown experimentally. In conclusion of this section, the model will always have a unique solution, under the assumption of negative externalities at the destination, as well as (provided that the origin demands are scaled appropriately) under the assumption of positive externalities. In the next section, an algorithm for obtaining solutions to problems (8) and (9) is described. Mu will be positive-definite

5. ALGORITHM FOR THE MODEL SOLUTION

An algorithm directly based on Evans’s partial linearization method (Evans, 1976) may be used to solve problem (8) and (9). At iteration k, the sub-problem for the descent direction is

subject to constraints (9). It may be shown that this problem is a convex problem that therefore may be solved directly from its first-order conditions. Leaving the demonstrations of these respective statements to the interested reader, as they are fairly standard, the first-order conditions are @uj+Rjk+r

wok = oi

c

e-f?(aj+R;+r

t,,“, I;,

(29)

and

0-W The optimal size of the move along the descent direction is then obtained by minimizing, with respect to 8, the function

N. OPPENHEIM

216

T(Xk + e(yk - Xk),Yk + B(Uk - Yk))

(31)

Once the optimal value of 0 is found, the new, improved solution is X k+l

Y

k+l

= xk + e(yk - Xk) =

yk + e(uk - yk)

(3W (3W

This basic algorithmic step is then iterated until convergence is detected with an appropriate criterion. 6. SUMMARY AND MODEL EXTENSIONS

In this paper, an equilibrium trip distribution/assignment on a congestible network was formulated in which an endogenous destination cost that is a function of the destination’s volume is incurred in addition to the cost of travel to the destination. Travelers are assumed to choose among alternative destinations so as to minimize the sum of the randomly perceived destination cost and the deterministically perceived travel cost, and to choose a route to the destination so as to minimize the travel cost. It was shown that such an equilibrium may be obtained as the solution of a nonlinear minimization problem. It was also shown that as long as destination costs functions are either all increasing or all decreasing in the destination volume, the equilibrium always exists and is always unique. An algorithm for solving the model was also suggested. The model is well suited to the analysis of a variety of issues concerning the effects of road pricing policies on urban travel demand and, conversely, of toll-based traffic management strategies on the economy. For instance, the effects of tolls imposed on travelers to a given destination (e.g. zonal centroid) on congestion and its negative impacts (e.g. on air quality), as well as on business activity, may be analyzed. The model lends itself to several important extensions. First, because the model is structurally similar to the standard model with fixed-destination attractiveness described in the introduction, it would be relatively straightforward to extend it along the dimensions of modal choice, and/or variable trip generation, in a fashion similar to the case of fixed destination costs. Also, the assumption that travelers assess (route) travel costs deterministically may be relaxed and the network equilibrium formulated as a stochastic user equilibrium through an appropriate change in the objective function in eqn (8). In practical terms, however, the solution algorithm would be significantly more cumbersome, as now stochastic assignment methods must be used. Perhaps the most important generalization of the present model concerns the specific case in which travelers consume some commodity at the destination (e.g. retail goods). In such a case, as seen above, the destination cost will depend on the commodity’s local price. However, the commodity will in general be produced in, and transported from, other zones than the consumer’s destination. For instance, retailers buy goods from wholesalers who are typically not located in the same zone as the retailers. In such a case, commodity prices at the destination will now be a function of the cost of transporting the commodity from its production, or storage, location to the consumption location (e.g. store). In addition, if, as is mostly the case in urban contexts, the same network is used for both personal travel and goods movements, the commodity’s transport may itself significantly affect network congestion and thus consumer/traveler costs. A model that integrates the present model with a goods movement model is presented in another paper (Oppenheim, 1993). Acknowledgement-The author is grateful to Tony E. Smith for providing a better, less restrictive proof of uniqueness in the case of negative externalities than he was able to devise. The usual disclaimer applies. REFERENCES Berge C. (1963) Topological Spaces. MacMillan, New York. Evans S. (1976) Derivation and analysis of some models for combining trip distribution Transp. Res., lOB,37-57.

and assignment.

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McFadden D. (1974) Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in Econometrics. Academic Press, New York. Oppenheim N. (1989) Discontinuities in equilibrium travel distributions: The Harris-Wilson model revisited, with some numerical experiments. Trunspn. Res. 25B, 225-242. Oppenheim N. (1993) Combined, equilibrium model of urban personal travel and goods movements. Transpn. Sci., forthcoming. Rockafellar R. (1970) Convex Analysis. Princeton University Press, Princeton, NJ. Sheffi Y. (1985) Urban Transportation Networks. Prentice-Hall, Englewood Cliffs, NJ. Varian H. (1984) Microeconomic Analysis, 2nd ed. Norton, New York.