25146884

A Multiperiod Dynamic Model of Taxi Services with Endogenous Service IntensityAuthor(s): Hai Yang, Min Ye, Wilson Hon-Chung Tang and Sze Chun WongSource: Operations Research, Vol. 53, No. 3 (May - Jun., 2005), pp. 501-515Published by: INFORMSStable URL: http://www.jstor.org/stable/25146884 .

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Operations Research infflJuTfl. Vol. 53, No. 3, May-June 2005, pp. 501-515

ISSN0030-364XI eissn 1526-5463105 15303 10501 DQI 10.1287/opre.l040.0181 ?2005 INFORMS

A Multiperiod Dynamic Model of Taxi Services with

Endogenous Service Intensity

Hai Yang Department of Civil Engineering, The Hong Kong University of Science and Technology,

Clear Water Bay, Kowloon, Hong Kong, China, [email protected]

Min Ye China Academy of Urban Design and Planning Institute, Beijing 100044, China, [email protected]

Wilson Hon-Chung Tang Department of Civil Engineering, The Hong Kong University of Science and Technology,

Clear Water Bay, Kowloon, Hong Kong, China, [email protected]

Sze Chun Wong Department of Civil Engineering, The University of Hong Kong, Hong Kong, China, [email protected]

This paper presents a spatially aggregated multiperiod taxi service model with endogenous service intensity. The whole

day service period is divided into a number of subperiods; during each subperiod, taxi supply and customer demand

characteristics are assumed to be uniform. Customer demand is period-specific and described as a function of waiting time and taxi fare. Taxi operating cost for each work shift consists of two components: one component a function of total

service time and the other component period-dependent. Each taxi driver can work for one or more shifts each day and

freely chooses the starting and ending time of each shift. Equilibrium of taxi services is obtained when taxi drivers cannot

increase their individual profits by changing their individual working schedules. A novel clock network representation is proposed to characterize the multiperiod taxi service equilibrium problem. The problem of interest is formulated as a

network equilibrium model with path-specific costs and arc-capacity constraints, which can be solved using conventional

nonlinear network flow optimization methods. The proposed model can ascertain at equilibrium the service intensity and

utilization rate of taxis and the level of service quality throughout the day. The information obtained is useful for the

prediction of the effects of alternative government regulations on the equilibrium of demand and supply in the urban taxi

industry.

Subject classifications: transportation: models, taxis/limousines; networks/graphs: applications, flow algorithms. Area of review: Transportation.

History: Received December 2002; revision received January 2004; accepted April 2004.

1. Introduction

1.1. Previous Static Analysis of the Taxi

Service Industry

Largely because of their 24-hour-a-day availability and pro vision of door-to-door service, taxis are an important com

plement to regular scheduled services provided by other

forms of public transport. Taxi services are particularly valuable to less mobile groups in the community, such as elderly and disabled people. It is important that such

services are efficiently provided, meet users' needs, and are appropriately priced through government regulation. Taxi regulations are mainly concerned with two primary issues: price and entry controls. First, what restrictions?

if any?should be placed on the supply of taxis? Second, what kind of control should be placed on fares? In an

attempt to understand the manner in which the demand and supply are equilibrated in the presence of such reg ulations, economists have examined these two questions

in various ways using highly aggregate models, without

considering the spatial structure of the market and the

time-of-day dynamic nature of taxi supply and customer

demand (Douglas 1972, De vany 1975, Shrieber 1975, Abe and Brush 1976, Foerster and Gilbert 1979, Schroeter

1983, Frankena and Pautler 1986, Arnott 1996). Cairns and

Liston-Heyes (1996) made an interesting extension of the

conventional taxi service models by treating the service

intensity of licensed taxis as an endogenous variable. They

developed a simple analytical model by assuming that each

taxi driver can choose the number of service hours per day, but customer demand and taxi supply are still assumed to

be distributed uniformly throughout the day. In view of the fact that demand for and supply of taxi

services take place over space, Yang and Wong and their

colleagues have recently modeled urban taxi services in a

network context. A realistic method has been proposed to

describe vacant and occupied taxi movements in a road

network and taxi drivers' search behavior for customers

501



Yang et al.: A Multiperiod Dynamic Model of Taxi Services with Endogenous Service Intensity 502 Operations Research 53(3), pp. 501-515, ? 2005 INFORMS

(Yang and Wong 1998). A few extensions have been made

to deal with the demand elasticity, multiclass taxi servipes with service area regulation, and congestion effects as well

as development of efficient solution algorithms (Wong and

Yang 1998; Wong et al. 2001, 2003; Yang et al. 2001, 2002,

2003). These developments have important implications for

both an assessment of road traffic congestion due to taxi

movements and a precise understanding of the equilibrium nature of taxi services.

1.2. Hourly Variation of Customer Demand and

Taxi Service Intensity The existing taxi service models, as mentioned above, involve a single static time period over which taxi supply and customer demand are distributed uniformly. The real

situation is that, unlike regular transit such as trains, trams, and buses, taxis are usually operated by a large number of

private firms, mostly owned and operated by individuals.

While the taxi industry is generally subject to the regu lation of both entry restriction and fare control, the ser

vice intensity of a licensed taxi is generally uncontrolled

(in fact, service intensity is difficult to monitor in practice). With such a market setting, individual taxi drivers or firms

can freely choose their working schedule in response to

the market profitability and operating cost as well as the

opportunity cost of being in service in different times of

a day. As a result, taxi services in most of the large cities

exhibit a remarkable hourly variation of customer demand

and taxi service intensity throughout the 24 hours of a day. In Hong Kong (Transport Department, 1986-2002), a taxi

driver generally owns his own taxi or hires one from a

taxi owner, and the majority of taxis are cruising taxis.

Taxi drivers vary the extent of service by changing their

working hours and working schedules in a day. In general, taxi drivers work on a shift basis, with eight hours per shift,

most of the taxis operate on two or three shifts by more

than one driver every day, and the shift schedule varies across individual taxis or drivers. Although the three eight hour shifts span the whole 24-hour demand period, not all

taxis operate for three shifts, and during each shift a taxi

driver may not work for eight hours. Taxi drivers have to

spend time for other activities, such as a meal break. Taxi

service hours or service intensity are strongly governed by the market profitability. When the business is nonprofitable, the owners may simply park their taxis. Profitability of the

market depends on the actual utilization of taxis in service, while taxi utilization depends on the customer demand and

taxi supply; all of these variables are endogenous variables

that exhibit substantial time of day variations. As an exam

ple, Figure 1 depicts the profiles of the number of (empty and occupied) taxis that travel through the three Hong Kong cross-harbor tunnels over a typical 24-hour period. These

profiles show the hourly taxi flow rates and highlight the

morning and evening peak-hours.

1.3. Necessity to Model Period-Dependent Taxi Services

The essential hourly fluctuation of customer demand and

taxi supply calls for the development of a dynamic taxi

service model that explicitly takes into account how the ser

vice intensity of individual taxis varies with the customer

demand in each subperiod, thereby helping to understand

Figure 1. Hourly variations of the numbers of taxis through the three cross-harbor tunnels between Hong Kong Island

and Kowloon (Transport Department 1998. The Annual Traffic Census).

2,000-j-j

-O- Cross Habour Tunnel

? ? Eastern Habour Crossing / ^-^

??r- Western Habour Crossing / V

0 I i" i ̂ ""T" i-i 'i' *-* i i-1-1-1-1-1-i-1-1-1-1-1-1-1-1-1-1-1-1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hourly Period of the Day



Yang et al.: A Multiperiod Dynamic Model of Taxi Services with Endogenous Service Intensity

Operations Research 53(3), pp. 501-515, ?2005 INFORMS 503

precisely the nature of the market equilibrium. The devel

opment of a multiperiod taxi service model has useful

application in practice. In most large cities, taxi indus

tries are regulated almost everywhere, and the regulation is likely to remain a reality. A common rationale for regu

lating the industry has been to make transport available at

times when demand is low and in areas where population is dispersed. Hence, it is important to predict the impact of both price and entry regulations. What can be observed

is the number of taxis operating, the ratio of occupied to

empty (unoccupied but in service) taxis, trips made, and

waiting times. These four quantities have been prominent in discussion of the trade. Indeed, in Hong Kong, annual

roadside observation surveys are conducted to gather infor

mation regarding the percentage of occupied/vacant taxis

and the average taxi headway between vacant taxis on

some major selected road locations (Transport Department,

1986-2002). The former index is intended to reflect the uti

lization level of taxis (a higher proportion of occupied taxis

would indicate a higher utilization rate of taxis), while the

latter index is used as a measurement of the availability of

vacant taxis (a shorter time headway between vacant taxis means a higher availability or a higher flow rate of vacant

taxis), thereby reflecting the level of taxi service. Obser

vation surveys on customer/taxi waiting times at operative taxi stands are also carried out as well. The results of these

surveys provide important information for government in

policy formulation on taxi services. For instance, the survey results are used for the evaluation of whether taxi service

has been improved, the identification of whether or not the

number of taxis should be increased and if so by how many, and whether or not the taxi fare should be adjusted upward or downward. All these decisions may affect the resultant

service quality and taxi utilization.

The average performance measures from the survey results are, however, imperfect. Taxi utilization rates are

biased because owners take taxis off the street when they

anticipate lower rates of utilization due to a lower demand. The remaining taxis then operate at utilization rates that are

higher than would be otherwise the case. Taxi availability shares similar feature as well. All these characteristics point to the necessity and importance of developing a taxi ser

vice model capable of analyzing the multiperiod dynamic characteristics, thereby acquiring more useful measures of

service quality and market profitability.

1.4. Objective of the Study This paper makes a further extension of the earlier large number of economic studies of taxi services. The exist

ing spatially aggregated single-period models (Yang et al.

2003) are extended to a more realistic multiperiod taxi ser

vice situation where period-dependent demand and supply characteristics are modeled. In the next section, we describe

the essential elements required to describe the multiperiod characteristics of urban taxi services that are common to

most major cities. In ?3, we formulate the aggregate mul

tiperiod dynamic taxi service model and prove that the

model does lead to dynamic taxi service equilibrium with

endogenous service intensity in a competitive market under

given entry and fare regulations. Section 4 shows how the

model can be solved efficiently by an interior penalty func

tion method. Section 5 provides a numerical experiment to

demonstrate the characteristics of the model and investi

gate the impacts of alternative taxi regulations on system

performances. Finally, general conclusions are given in ?6.

2. Network Representation, Demand, Cost, and Revenue

2.1. Network Representation

We now propose a clock-network or multiperiod dynamic method to model the hourly variation and equilibrium of

customer demand and taxi service intensity throughout the

day. As shown in Figure 2, we divide a typical day of

Figure 2. The proposed clock network for equilibrium characterization of multiperiod taxi services

within a whole day.



Yang et al.: A Multiperiod Dynamic Model of Taxi Services with Endogenous Service Intensity 504 Operations Research 53(3), pp. 501-515, ?2005 INFORMS

taxi services into T periods of equal length, over which

customer demand varies. Typically, 24 periods of an equal one-hour length is preferable {T

= 24). Within each period,

the taxi supply and customer demand are assumed to be

uniform, and the customer demand is period-specific and

described as a function of waiting time for a given fare

structure. There are three groups of arcs connecting a sin

gle origin-destination pair. Along the clock, the number of

each node corresponds to a clock time (called clock node); each time period t {t

= 1,2,..., T) is represented by an

arc at with start node (time) nt_x and end node (time) nt9

respectively, with one exception of arc 1, for which the start

node is 24 corresponding to 0 o'clock. This group of arcs

along the clock is termed as clock arcs. The second group of arcs consists of entry arcs connecting the origin node 25

and each of the clock nodes, nt {t = 1, 2,..., T), denoting

the starting or entry time t of a shift of taxi services. The

third group of arcs consists of exit arcs connecting each

of the clock nodes, nt {t = 1, 2,..., T) and the destination

node 26, denoting the ending or exit time r of a shift of

taxi services. Clearly, a path flow in the network from ori

gin 25 to destination 26 represents a shift of taxi services

during the day. For example, a path flow along the nodes, 25 -> 8 ->- 9 ->-> 15 -? 26, represents a shift starting at 8:00 a.m. and ending at 3:00 p.m. In addition, flow on

a clock arc represents the total number of taxis in service

during the relevant time period.

2.2. Customer Demand

Let

t=\,2,...,T (1)

be the number of customers per unit time (one hour)

demanding taxi rides in period t over the whole service

area, where Ft is the expected fare or price of a taxi ride, and Wt is the expected customer waiting time for engaging a taxi during service period t. We naturally assume that

Qt{Ft9 Wt) ?> 0 as Wt

-> +oo. For example, one may adopt

the following period-specific negative exponential demand

function:

Qt =

Qttxp{-a{Ft + TxWt + T2lt)}, f=l,2,...,7\ (2)

where Qt is the potential customer demand in period t; ltis the average taxi ride time in period t9 which is assumed, on average, of a constant duration during each time period

(a given period-specific constant); rx and t2 are the values

of waiting time and in-vehicle time of customers, respec

tively; and a {a > 0) is a demand sensitivity parameter. Note that, even though taxi and/or customer queues build

ing up in period t might carry over into period t + 1, the

use of period-specific demand function or the assumption of demand independence between periods is reasonable.

As long as the period is reasonably long (e.g., one hour

considered here), the transient number of demand is negli

gible in comparison with the total demand in each period. Let N be the number of licensed taxis and vt the average

number of taxis actually in service in period t. Clearly,

vt^N for all f = l,2,...,7\ (3)

Given a reasonably large number of taxis, there will be no differentiation of individual taxis, the average rides per hour of each taxi in service in a given period is

qt = ^9 t=\92,...,T. (4)

Given the average taxi ride time lt (h), the average number

of vacant taxis, iV/\ available (in service but not occupied) at any given instant in period t is

N? =

vt-ltQt, t = \929...9T. (5)

Note that Equation (5) still holds if modeling periods of

a fraction of one hour are used. Furthermore, as N* ^ 0, we must have Qt < vt/lr Here a taxi in service can be

either vacant or occupied by a customer, and the vacant or

unoccupied time here is with respect only to a taxi currently in service.

Assuming that customer waiting time is negatively related to the number of vacant taxis available (equal to

the total vacant taxi-hours as each one hour subperiod is

considered);

Wt = Wt{N?)9 w; = dWJdN? < 0,

t=l,2,...,T. (6)

Furthermore, we require that Wt ?> +oo as

Ntv -> 0. For

example, the following customer waiting time function is

appropriate and adopted later:

w,-k-^iA- ,ml-2.T- m

where /3 {(3 > 0) is a positive parameter common to all time

periods, whose value depends on the size of the service

area and the distribution of taxi stands over the service area.

Substitute Equation (5) into (6), and then into the demand

function (1). We have

Qt =

Qt{Ft9Wt) =

Qt{Ft,Wt{vt-ltQt))9

r=l,2,...,r. (8)

When the specific demand function (2) and customer wait

ing time function (7) are used, Equation (8) takes the fol

lowing specific form:

a = a?Iip{-?(i,+Tl-^-+ra//)J.

(9)





For a given number of taxis vt in service and a given

period-specific fare Ft and average taxi ride time lt for a

taxi ride, demand is an implicit function of itself. Define

HQ,) =

G, -

Q,(Fn wt(vt -

ltQ,)) = 0, 0 < Q, < vt/l?

t=l,2,...,T. (10)

Note that <J>(<2,) is a continuous function of Q e (0, vjl,), and

^(a)sBl_?a(./r)^ =

1+/,^L^>1.o>o W'> dWt^ <>dNt

^ ?aW/ ^

because ?^

^<0, ^<0. dWt dNt

Furthermore, from the assumptions on the demand and cus

tomer waiting time functions, we can easily verify that

\im?{Qt) =

-Qt{Ft9Wt{vt))<0 and

lim *(G,) = r>0 Gr-?r/'i lt

We thus conclude that for a given taxi fare Fn taxi fleet

size vt9 and average taxi ride time lt in period r, there exists one and only one equilibrium value or solution denoted as

Q* = Q*{Ft9 vn /,), t = 1, 2,..., T for Equations (8) or (9)

within the feasible domain {09vt/lt). Note that the negative

exponential demand function (2) and the waiting time func

tion (7) meet all the necessary assumptions stated above, and thus Equation (9) has a unique solution of Q* in the

interval (0, vt/lt) for given Ft, vt, and lt. As expected, from Equation (8), we have

(11) d-& = Q* <0,

M = G^;(,o_^)

or

aa = _GH1^_>0 (12)

dv, l + /,Gww; '

where QFt =

dQ,/dFt and Qw> =

dQJdW,.

2.3. Average Revenue

Now we look at the average revenue per taxi, Rt, in each

service period:

F,Q, fl,

=

-^, t = l,2,-..,T, (13)

?_5&(?i_lj0)_54(,f_lj(,. (14)

where e^f

is the elasticity of the customer demand with respect to the total taxi-hours supplied in period r

(f = 1,2,..., T). Likewise, the average taxi utilization rate

Ut, measured as the fraction of occupied service time, is

given as

Ut=1-^, f = l,2,...,r (15)

and

^ =

^#(<-l-0), r = l,2,...,r. (16)

It is self-evident that, for a given fare and a given taxi

ride duration, average taxi revenue is directly proportional to taxi utilization. It is generally expected that S?1 < 1.0,

which means that both the taxi utilization rate Ut and aver

age taxi revenue Rt are monotonically decreasing functions

of the number of taxis in service (or fleet size of taxis in

service). Nevertheless, this is not always true. As found in a number of studies (De vany 1975, Manski and Wright 1976, Schroeter 1983, Arnott 1996, Wong et al. 2001), taxi

utilization rate and hence average revenue increase initially with taxi fleet size.

To look at the above increasing return-to-scale phe

nomenon, we consider a particular period r with the

following parameter values: /3 = 400.0 (veh-h/km2) and

Z, = 0.3 (h) for Equation (7); a = 0.03 (1/HKD), rx = 60

(HKD/h), r2 = 35 (HKD/h), and Qt = 1.0 x 105 (trip) for

Equation (9). These parameter values are given with ref erence to the calibration study carried out by Yang et al.

(2002). Figure 3 shows the results generated from the simu

lation with the above data. The average taxi utilization and revenue do increase initially with fleet size at all three alter

native taxi fares, and the increasing range varies slightly with the taxi fare charged. The reason is that as entry ini

tially occurs, average customer waiting time (and hence

average cost per trip) falls more than proportionately, so

the customer demand increases by more than the supply of taxis. As a result, entries of new taxis into service will both increase expected utilization rate and hence revenue

of taxis already in service, and decrease expected customer

waiting time at the initial stage. But this intriguing result of

increasing return to scale generally corresponds to an unsta

ble situation with a small fleet size and seldom emerges in a realistic taxi market in equilibrium.

From the above analysis, hereinafter we assume that

dRt/dvt < 0, t =

1,2,..., T, are always satisfied in our

subsequent discussions.

2.4. Cost Structure

Suppose that the total variable operating cost of each taxi

shift consists of two components: one a function of the

total number of service hours of the shift and the other

additive period-dependent costs. Let ts and te be the start

ing and ending time of a shift corresponding to a path in the clock network: 25 -? nt -> -> n. -> 26, and h\s ls le ls

be the corresponding duration (number of hours) of the




Figure 3. Variation of the average taxi revenue and average taxi utilization with respect to taxi fleet size.

500-j-r 1.0

. Average revenue for fare = 40

450 - ... -

Average revenue for fare = 60 ~

0.9

jr N. - - - Average revenue for fare = 80

400- /^? ^\ "?-8 if ^*v \

- Utilization rate for fare = 40

O 350- // \^ ^v - Utilization rate for fare = 60 -0.7

? if ^\ \ N. mmm^mm Utilization rate for fare = 80 g

X 300- If

\ \ ^v -0.6 *? ? f \ N. \v Fare: HKD/trip j3

8 250-1 \ N. ^v. -0.5 5

< 150- !/' n;;--.. ^s^ ^^-*^_ '?-3

Q_J-,-,-,-,-n-,-,-,-,-\- QQ 0 2 4 6 8 10 12 14 16 18 20

Number of Taxis (103 veh)

shift: h\\ = te-ts if ts < te ̂ 24 and h\\

= 24 - {te

- ts)

if te < ts ̂ 24. Let ct {ct ^ 0) be the period-dependent cost associated with period t {t

= 1,2,..., T) and c\\ be

their sum over the shift: c\\ =

Y!te=ts+\ ct ^ h < *e ̂ 24 and

ctes =

??=,5+i cr + Eti c, if r, < t* ̂ 24. Then, the total

shift or path cost is given as

c;; = c0 + ̂ /i;;+^2(^;r + c;;, (17)

where c0 (c0 ̂ 0) is the constant entry cost and /jlx, /jl29 and

y are all constant parameters. It is assumed that /jlx > 0,

jjl2 ̂ 0, and y > 1.0. The above specification of taxi operating cost is justified

below. c0 represents a constant entry cost that is indepen

dent of individual shifts. For example, it could represent the cost for driving a taxi to enter or leave the service area.

The second and third terms represent the taxi operating cost that is monotonically increasing with the number of

service hours of a shift. This assumption stems from the

fact that the cost of keeping a taxi in service will surely rise rapidly as a driver works continuously for more than a certain number of hours. Such cost includes the mount

ing maintenance cost and the gasoline consumption cost, as well as the imputed value of the owner's time that rises as he/she drives more hours per day, leaving less time for

other uses. The last term is added to the cost function to

capture the period-specific costs that are not captured by the second and third terms. This is true in reality because

taxi drivers might prefer certain periods of a day to others.

Everything else being equal, a driver might prefer to work

in daytime rather than at midnight.

Finally, we remark that with the given cost structure, the

taxi service cost c?

for a given shift 25 -* nt$

-> ->

ntg -> 26 is a predetermined constant independent of the

aggregate taxi workflow. Namely, we have a uniquely defined flow-independent, but path-specific (arc nonaddi

tive) cost on the clock network from the origin 25 to the

destination 26.

3. Model Formulation

Now we treat the constructed clock network in Figure 2

with a single origin-destination pair as a usual network and

adopt the normal network notation. Let Aa, Ac9 and Ae be

the subsets of access (entry), clock (service), and exit arcs

in the network, respectively, and let A = Aa U Ac U Ae and

P be the set of paths (shifts) in the network. Given path flow fp9 p eP9 arc flow va, aeA can be expressed as

?? = E/A' a A> (18)

PeP

where 8ap equals 1 if path p uses arc a, and 0 otherwise.

With reference to the cost structure given in Equation (17), we let each entry arc be associated with an identical, constant entry cost: ca

= c0, a e Aa\ and each service

arc with an arc-specific but constant service cost: ca =

ct9

a e Ac if arc a corresponds to service period f, t = 1,

2,..., T\ and for all exit arcs, we let ca = 0, a Ae. Finally,

each path is associated with a path-specific constant cost:

cp =

iLxh\ + ti2{hl)y, p e P if p =: 25 -* ntg

-> ->

ntg -? 26, in addition to the costs of arcs constituting that

path.

In a competitive market, all taxi drivers can be regarded as competitive firms, each driver can work for one or more

shifts each day and freely chooses starting and ending time

of each shift. The equilibrium of taxi services is obtained





when taxi drivers cannot increase individual profits by alter

ing individual working schedule (extending service time or

shifting service periods). To derive such an equilibrium, we now consider the following mathematical programming model:

minF(f) = ? f?{ca

- /??} da> + ? c0va + ? cpfp

aeAc ?

aeAa peP

(19a)

subject to

va^N9 aeAc9 (19b)

fP>0, peP, (19c)

where va, aeA is defined by Equation (18), and f =

{fp9peP) and \ =

{va9 aeA).

Forming the following Lagrangian:

L((,tt)=J: {ca-Ra(a>)}da>

aeAa \peP / peP

+ Z*a\T,fp8ap-NY

(2?) aeAc I peP J

where it = {ira9a e Ac) is a collection of the Lagrange

multipliers that are associated with constraints (19b). Under

the assumption introduced in ?2 that average taxi revenue

Ra9 a e Ac is a monotonically decreasing function of the

number of taxis in service va9 problem (19) is a convex

programming problem with a convex objective function

and linear constraints. THus, from the Karush-Kuhn-Tucker

conditions (Bazaraa et al. 1993), (f, tt) is a solution if and

only if

/p>0^^>=0, (21a)

/, = <>= ^?I>>0,

(21b) JP

d-^<0=>*a=0, (21c)

*Ê)=0 = *.>(>. (21d)

d7Ta

Differentiating the Lagrangian (20),

^?^ =

E {ca -

K(va)}8ap + E c0sap Jp aeAc aeAa

+ cp+EÂP, (22) aeAc

dJ^ = ZfPK-N. (23)

d7ra pep

Note that each path uses only one entry arc in the clock

network. Thus, J2aeAa coâp ? co f?r a^ P e p- Therefore,

HaeAc ca8ap + ZaeAa "câp + cp is the total taxi operating

cost of a path (or shift) p e P, denoted as Cp. Further

more, Y,aeAc Ra(va)8ap *s me total revenue of a path p e P9

denoted as Rp. As a result, from Equations (21a), (21b),

and (22) we have

fp>0 =>

Rp-Cp=Y,*a8ap>? aeAc

because 7ra ̂ 0 Va e Ac9 (24a)

fp = 0 => Rp-Cp^J2 7Ta8ap, (24b)

aeAc

where Rp ?

Cp is the net profit of a path p eP9 and it is

denoted here as Hp9 Up

= Rp

? Cp.

On the other hand, from (21c), (21d), and (23) we find

that

va<N = 7ra

= 0, (24c)

va = N =v 77fl^0. (24d)

The above necessary conditions (24a)-(24d) characterize

the multiperiod temporal equilibrium conditions of taxi ser

vices. First, 7Ta can be interpreted as the "net profit" earned

in a time period corresponding to service arc a. This net

profit on service arc a can be regarded as the arc revenue

minus the arc cost if we regard the arc cost as the sum

of the period-specific cost plus the cost that the arc por

trays for the constant entry cost and the path-specific cost.

Let Ac =

{a \ va = N9 a e Ac}. If taxi service in a time

period is profitable {ira > 0) after taking into account all

possible costs (including period-specific cost and the cost

due to extended service hour), taxi drivers will choose or

adjust their shift schedules so that all taxis will provide service in this profitable period, namely va = N. Consider a case in which taxi supply / in a shift p e P is pos itive {fp

> 0). There are two possibilities for this shift: one is that this path or shift does not go through any arc

a e Ac or period of positive net profit; namely, for any a

on path p, va < N and ira ? 0, the net profit for this shift is

Up =

Rp ?

Cp =

YlaeAc ââp = 0- ̂ ^ means that no addi

tional taxi driver has incentive to adjust or change his/her

schedule or increase service intensity to cover or join this

service shift, even though some taxis are idle throughout this shift period. The other possibility is that this path or

shift goes through some arcs a e Ac or periods of strictly

positive net profit. Namely, there exists arc a on path p such that va = N and 7ra > 0. Then, the net profit for this

shift is Up =

RF -Cp =

J2aeAc ̂ a8ap > ?- * means that

all licensed taxis are put in service in the strictly profitable time periods and taxis working for the said shift make a

positive profit. Note that there are also two possibilities for a shift with

fp = 0. One is that path p does not contain any arc a e Ac,




and of course the profit Up =

Rp -

Cp < 0. The other pos

sibility is that path p contains one or more arcs a e Ac and the profit Tlp

= Rp

- Cp ^ Y,aeAc ira8aP> and hence

both Ilp < 0 and Up ^ 0 are possible along this path

because Y^aeAc 7TaSap > 0. For the case with fp

= 0

and np < 0, no taxi driver works for this shift simply

because of an operating loss; whereas, for the case with

fp = 0 and np ^ 0, no taxi driver works for this shift

because taxi drivers already choose to work for other, more profitable shifts even though the current shift is

profitable. In summary, suppose that there are a large number of

competing individual taxi drivers, each operating indepen

dently of the others. Then, Equations (24a)-(24d) hold irre

spective of individual taxis. No taxi drivers can increase

individual profits by shifting their service periods or

increasing service intensity. Therefore, we conclude that

the proposed clock network model (19a)-(19c) does lead

to a multiperiod dynamic equilibrium of competitive taxi

services.

4. Solution Algorithm The minimization problem (19) with side constraints (19b) can be solved by an inner penalty function method in con

junction with the well-known Frank-Wolfe algorithm. The

method has been successfully applied to solve the capacity constrained traffic equilibrium problem (Yang and Yagar

1994). The penalty function (Bazaraa et al. 1993) is used

to transform a constrained problem into an unconstrained

or partially constrained problem. In the present case, the

inner penalty function method is applied to incorporate the

side constraints (19b) into the objective function (19a) as

a penalty term. After appropriate transformation, the aug

mented objective function becomes

._, r ZpeP fp ?ap

H^P)=E {ca-Ra(a>)}da>

+pJ2 \t,? \d<?

aeAa \peP '

peP

where p is a penalty parameter. Consequently, the mini

mization problem (19) is transformed into the following

program:

minF(f, p) (26a)

subject to

fP>0, peP. (26b)

As parameter p -> 0, we have

Pt;?-- * ?, aeAc9 (27) N-va

and the corresponding solution (f, v) will converge to the

optimal solution of the original network equilibrium prob lem (19). The Lagrange multipliers ira9 a e Ac can thus be

calculated using relation (27). We now consider how the Frank-Wolfe method can be

used to solve problem (26) for a given penalty parameter p. The Frank-Wolfe method consists of two major steps

(Bazaraa et al. 1993): finding a descent direction and imple

menting a line search. Clearly,

wff.ri vl ? , , . _?_1.

+ E C08ap + Cp aeAa

= E Ca8ap + E C08ap + S

~ E RaM8ap

aeAc a^Aa a Ac

+ E -f^8ap aeAc iV Ua

= Cp-Rp + np. (28)

Thus, the descent direction for problem (26) can be found

by solving the following linear programming (LP) problem for given current solution {f^} at iteration k:

minZ<*>(g) = ?(C<*> ~Rf + Kpk))gp (29a)

peP

subject to

O^gp^N, peP, (29b)

where g = {gp,pzP) and gp < N9 p e P is added because

total taxi fleet size is clearly an upper bound of the path flow.

Finding the optimal solution of problem (29) requires

assignment of the taxi supply N to the path p for which

(Cf ?

i?f + nf) is negative, and 0 otherwise. Namely,

gp = N if (Cf

- RM + nf) < 0 and gp

= 0 if (Cf -

#f + nf) > 0. In view of its special structure, all paths of the clock network can be simply enumerated (in total, a fixed number of paths 24 x 24 = 576 is considered by

ignoring the shifts of more than 24 hours), and thus the

set of paths with negative value of (Cf

- Rf

+ nf)

can

be easily identified at the current solution (vf ,<z e Ac).

Figure 4 formally describes the algorithm for path identifi

cation and LP solution for this purpose. In application of the Frank-Wolfe algorithm for solving

problem (26), we have to modify the range of the one

dimensional search to ensure that none of the side con

straints is violated. The algorithm for solving problem (26)





Figure 4. Algorithm for solving the decent-direction

finding LP problem.

Algorithm decent-direction-finding

For every a e {1,2,..., 24} do

begin rik)<-Ra{v(ak))

^k)^p/{N-v^) end

for ts = 1 to 24 do

begin for te

= ts + 1 to 24 & te

= 1 to ts - 1 do

begin

<ae.Ac where p =: 25 -> nt ->- nr -> 26

If (Cp -*?> + nf) < 0 then gp <-N.

Otherwise gp ?- 0

end

end

return g

is described in Figure 5. In the algorithm, </>(f(n),p(n)) is

the penalty term in the objective function (25) given by

^-)>pW) =

p(.)2^^{_l_j</fll. (30)

The algorithm is judged to have converged if the penalty term is sufficiently small compared to the augmented objec tive function value.

5. Numerical Experiments

5.1. Experiment Setting

Here we provide a numerical experiment with our pro

posed model and algorithm and show how the taxi service

intensity or working schedule is determined, and how the

restriction of entry and fare affects the market profitabil

ity and dynamic equilibrium of taxi supply and customer

demand. Here we utilize the demand function specified in Equation (2) and the customer waiting function in

Equation (7) together with the same values of parameters

given in ?2.3: j3 = 400.0 (veh-h/km2), a = 0.03 (1/HKD), rx = 60 (HKD/h), and t2 = 35 (HKD/h). For simplic

ity, we assume that lt = 0.3 (h) for all time periods t =

1,2,..., T. In line with the observed hourly variation in

the pattern of taxi traffic reported by the Transport Depart ment (1986-2002), we simply suppose that the potential customer demands Qt,t? \,2,... ,T exhibit a similar tem

poral characteristic and take the hourly value depicted in

Table 1. As to the cost function given by Equation (17), we assume that c0 = 5.0 (HKD), fix = 20.0 (HKD/h), 7 = 3, and p,2 = 0.10 (HKD/h3), respectively. The addi

tional period-specific opportunity costs ct are presented in

Table 2.

Figure 5. The Frank-Wolfe convex combination algo rithm for solving the multiperiod taxi service

equilibrium problem.

Algorithm Frank-Wolfe Method

input

gj, s2: predetermined convergence tolerance

p?: initial value of penalty factor

cr (0 < a < 1): penalty-updating parameter initialization: Calculate constant path costs:

c? ̂ ? cA? + EVv + c?, p e/>; aeAc aeAc

and choose a feasible interior point:

while |</>(v<"?,p<"))/F(vW,p("))| ^ ?2, do

begin

... fE^r'-w*^ H while { ?-?prr?^? } ^ e,, do I EP?>(02 1

begin call Algorithm decent-direction-finding to obtain

{gp,p e P]

A* <- argmin F(hw + A(g -

h(/c)), p(n)), oaamax

where

Amax = min 11, min I (k) l

aeA,ZpePSpsap<Epephp &ap

N-HP,phfdap 1

^) =

^) + A*(^-/z^),/7eP *<-*-|-l

end

ttn) =

hpk+l\peP p(n+l) <_ a .

p{n) n <- n+ 1

end

return (/p(n+1),p(n+1))

We note that solving problem (19) requires evaluation of

the objective function (19a), which needs a numerical inte

gration of the first term. This numerical integration, in turn,

requires the value of the average taxi revenue and thus the realized customer demand Qt in Equation (13) for given vt and Ft. As Equation (9) associated with the demand func tion (2) and the customer waiting time function (7) does not

entail a closed-form solution, we have to resort to a numer

ical method (bisection method is used in our calculation) to find a unique solution Q* for given vt and Fr

Table 1. Period-specific customer potential demands.

Potential Potential Potential

Time demand Time demand Time demand

period (trips) period (trips) period (trips) 1 43,500 9 98,600 17 81,200 2 34,800 10 89,900 18 81,200 3 29,000 11 81,200 19 82,650 4 29,000 12 81,200 20 69,600 5 29,000 13 81,200 21 66,700 6 34,800 14 82,650 22 58,000 7 69,600 15 82,650 23 58,000 8 95,700 16 82,650 24 58,000




Table 2. The additional period-specific opportunity cost

of taxi services ct (HKD).

Time ct Time ct Time ct

period (HKD) period (HKD) period (HKD)

1 10.0 9 0.0 17 0.0 2 15.0 10 0.0 18 0.0 3 15.0 11 0.0 19 0.0 4 15.0 12 0.0 20 0.0 5 15.0 13 5.0 21 0.0 6 10.0 14 0.0 22 5.0

7 5.0 15 0.0 23 5.0 8 5.0 16 0.0 24 5.0

5.2. Results and Discussion

We first briefly look at the convergence of the solution

algorithm. Table 3 shows the changes in the values of

the penalty term given by Equation (30) and the objective function (19a) with decreasing value of the penalty factor

p in the case: AT = 1.0 x 104 (veh), F = 50 (HKD/trip) for all periods, sx

= 5.0 x 10~5 and cr = 0.5. We can

observe that the objective function value almost stabilizes

after four outer iterations, and the value of the penalty term

declines and becomes sufficiently small in comparison with

the absolute value of the objective function.

Figure 6 shows the relationship between the taxi arc

workflows and the corresponding Lagrange multipliers (an indicator of arc net profit) at the converged solution for the

above case. When the arc flows tend to approach the fleet

size (total number of licensed taxis), the Lagrange multipli ers will converge to certain positive values (from 7:00 a.m.

to 24:00 p.m.). This implies that all licensed taxis are in

service in the market during the time periods of strictly

positive profits. Note that here all taxis in service from 7:00 a.m. to 24:00 p.m. does not mean that individual work

ing shifts extend over the whole duration; it can comprise more than one shift operated by two or more taxi drivers.

When the number of taxis in service is less than the fleet

size, the corresponding Lagrange multiplier nearly equals zero. This implies that the incentive for additional taxis or

drivers to offer service disappears in the time periods of nil

profits.

Table 3. The convergence of the Frank-Wolfe convex

combination algorithm.

Number of Value of Value of Value of iteration penalty factor penalty term objective function

1 1,000.0 116,188.2 -7,948,190 2 500.0 62,921.7 -7,965,850 3 250.0 34,241.9 -7,978,010

4 125.0 18,956.1 -7,980,910 5 62.5 9,948.4 -7,981,763

6 31.3 5,322.4 -7,982,051 7 15.6 2,900.7 -7,982,314

8 7.8 1,493.9 -7,982,841 9 3.9 784.3 -7,982,903

Figure 7 displays the net profit (revenue minus cost) associated with the shift duration starting at different times of the day. If there are entries at a given hour, a taxi driver

who enters the market at that time will exit the market at

the time corresponding to the maximum shift profit curve.

It can be seen that the early morning working shifts are

generally longer and earn more profit than the middle night

working shifts.

Now we look at the central results of the total net profits and the social welfares resulting from the various combina

tions of taxi fleet size (N) (number of licensed taxis) and

daily uniform taxi fare {F)9 which are of primary concern in

taxi service regulation. Note that the path cost incurred in

each individual working shift is the short-run variable cost, based on which individual taxi drivers choose service inten

sity and working schedule. Here we have to add a constant

long-run fixed cost of holding a taxi. This cost includes the capital cost (depreciation) of purchasing the taxi vehi

cle and the cost for taxi parking facility, and so on. After

appropriate transformation, a daily taxi holding cost of

250 (HKD/day) is assumed in our calculation. Inclusion of this long-run fixed cost will aid precise understanding of the long-run equilibrium taxi supply (entry to the indus

try) in a completely competitive market with free entry, and the price (or the medallion value) of a taxi licence in a competitive market with binding entry restriction. Here

the medallion value or license price of a taxi should be

regarded as the outcome of the industry profitability rather

than as part of the taxi holding cost. With this long-run taxi

holding cost included, the social welfare is defined as

social welfare = total consumer surplus+total taxi revenue ? total taxi operating cost ? total taxi holding cost.

Calculation of consumer surplus is given in Yang et al.

(2003). Figures 8 and 9 portray the principal operational charac

teristics of the taxi market in a fleet {N) - fait {F) space.

The incremental intervals used for generating the graphs are 10 (HKD) for taxi fare and 103 (veh) for taxi fleet size.

Note that here fleet size refers to the number of licensed

taxis rather than the number of taxis in service. The var

ious combinations of fleet size and fare would give rise

to various social welfares, taxi profits, customer demands,

and customer waiting times (averaged over all time peri

ods). These are represented by iso-social welfare, iso-profit, iso-demand, and iso-waiting time contours, respectively.

A number of representative solutions with and without fare

and/or entry controls in a competitive market can be iden

tified and discussed from the figures. The dynamic oper ational diagrams here with endogenous service intensity

generally resemble those obtained from the static models

and a detailed similar discussion can be carried out as in

Yang et al. (2002). In the following, we merely highlight some key points of interest and the major differences in the

diagrams between the dynamic and static models.



Yang et al.: A Multiperiod Dynamic Model of Taxi Services with Endogenous Service Intensity Operations Research 53(3), pp. 501-515, ? 2005 INFORMS 511

Figure 6. The relationship between arc flow and its corresponding Lagrange multiplier (taxi fleet size =1.0 x 104 veh, fare = 50 HKD/trip).

80-i-rl2

M Lagrange Multiplier ^+? Arc Flow

70

60-1

w If *? *

I <?

V / 6 I ? \ / \

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hourly Period of the Day

Figure 7. The net profit associated with entry time and shift duration (taxi fleet size =1.0 x 104 veh, fare = 50 HKD/trip).

24 |

-. .. ..' .. , ,.. ,

*'**^^^^*~L7L^LTTLmLW5!!L^^^L^L^Zl^^^^*^^^mm^mm^*mi^

6 ll^V \\ \l Ma^^ShiftProfit "" ? ^

/^^*X \

0 2 4 6 8 10 12 14 16 1ft tt -iv* ^%#

Entry Time (o'clock) :

f '^ .'




Figure 8. The iso-profit and iso-surplus contours in a fleet (TV)-fare (F) two-dimensional space.

I II j - Profit (106 HKD)

I I X / \-Social Welfare (106 HKD)

xl m F hi ^^ ^ fc 8 X

.is g \ is 8 \, *

120 y-050 ^V^ \^

\ Competitive Solution I J y^-flfe \

2 / ^^^--fso^\\Hr\-- Competitive Solution with-^^j-oo* ^J^v S -f f

' "^XXXrV \ Minimum Waiting Time

"^^oo^T^->^ ""^

80 I I /Y/^3' ""^^

?

I V\ Maximum Profit Solution ^-^^^ /A / y

^^^^^^^^^^^ Quasi-Second Best Solution SE Quasi-First Best Solution

"^

0 4 8 12 16 20

Taxi Fleet Size (103 veh)

As seen from Figure 8, the zero-profit curve is a closed

curve within which profit is positive. The competitive

equilibrium solution is governed by the break-even condi

tion and indicated by the bold zero-profit contour in the

figures. In a fully unregulated competitive market, if the

initial supply of taxis is within the interior of the zero-profit curve, then there will be positive profits in the industry.

Supply will be expanded toward zero-profit contours until

the attractiveness of this action disappears?that is, until

profits become zero. No service will be offered outside the

zero-profit contour. While there is no compelling evidence

that any combination of fare and fleet solution values in the

zero-profit curve will occur (in particular, the competitive

zero-profit solutions at small taxi fleet sizes are unstable, as

mentioned in ?2.3, where the assumption of demand elas

ticity in total taxi-hours less than one is not satisfied), it is

conceivable that the most probable long-run stable competi tive equilibrium occurs at point E9 at which the competitive fleet size is maximized, and the incentive for all individual

firms to change fare and/or for new taxi firms to enter the

market disappears (refer to Yang et al. 2002 for a detailed

explanation in the static case). This observation indicates

that a competitive market will eventually operate at point E if the market is fully deregulated.

The positive profit domain can be realized in a competi tive market with binding entry limit (and/or fare controlled at appropriate level). In the case with entry limit, the market becomes protected and thus operates in the positive profit

region at the given entry limit. The fare, if not fixed, could

probably take any value in between the lowest and the high est zero-profit fares at the given entry limit, but it tends to

self-adjust upward or downward and eventually stays at the

average profit-maximizing point at the entry limit. When ever the industry is operating in the positive profit domain

due to active entry limit, the competitive firms will earn

supernormal profits. As a result, a positive taxi license price or a positive medallion market value will arise.

In a partially regulated competitive market where only the regulator sets fare, the solution of taxi fleet size is

unique and given by the right-hand-side point located at

the zero-profit curve corresponding to the regulated fare

(the small taxi fleet size at the left-hand side of the zero

profit curve is unstable, as mentioned in ?2.3). Thus, the point at which the market would operate depends on

the fare set by the regulator. Here the social welfare

maximizing point B on the zero-profit curve is partic

ularly worth mentioning. This welfare-maximizing point is referred to as the "quasi-second-best solution," which

gives a maximum social welfare along the zero-profit curve





Figure 9. The iso-demand and iso-waiting time contours in a fleet (N)-fare {F) two-dimensional space.

/ / ///// , - Consumer Demand (trip)

f <&' */ / / / / ^ ^-Waiting Time (h)

16?- / / / / //V - ""*" - ^

?* ""^

/^^St Competitive Solution_ _0.06?^^^.

K >^ I ^^A Competitive Solution with _?

I / \ \^-^"^sT\?1 Minimum Waiting Time f4'000 ^?? * 80 " / >C^^ \ ^V\\ \\ 1 ? '-?

?sV \/ \^s^*^ \ ^^N \ A Stable Competitive Solution""

/ \ / \ jr v x ^^<T^^^^ ?-*^A^-3e?-V"?10,00?

3 \ Vi-s * ̂ --^--^-i ( \ / J*\ s^*?16,000? [A v. ^ Maximum Profit Solution, \ / \^/f S*^^ N .,

. X \\J \ yx Quasi-Second Best Solution I Quasi-First Best Solution

0 4 8 12 16 20

Taxi Fleet Size (103 veh)

(as seen in Figure 8). Here we use the term "quasi-second best" because the maximum welfare under the break-even

constraint is achieved by regulating taxi fare, but with

endogenous service intensity and working schedule. The

service intensity and working schedule selected by indi

vidual taxi drivers for self-profit-maximization in a com

petitive equilibrium are not necessarily consistent with the

socially optimal ones. Thus, in addition to fare control, attainment of the "second-best" solution requires regulation

of service intensity and schedule of licensed taxis so that

the time-varying customer demand is served with socially

optimal service intensity and working schedule. Of course, in reality, it is extremely difficult, if not impossible, to reg ulate working schedule of licensed taxis.

We point out that the quasi-second-best point also leads

to a maximum realized customer demand, as confirmed in

Figure 9. This means that the social welfare and customer

demand are maximized simultaneously under zero-profit

constraint in this case. As the fare is lowered below the

quasi-second-best point B, the equilibrium utilization ratio

rises, implying that the number of vacant taxis falls and the

waiting time rises, thus taxi demand declines because the

increased waiting time more than nullifies the advantage of the lower fare for every customer. Similarly, as the fare

is raised above point B, customer demand also declines,

because the reduction in waiting time is insufficient to off set the higher fare for every customer. The extent of this

trade-off depends on the value of time of customers. There

fore, efficient fare regulation would require that fare be set

at point B so that a feasible quasi-second-best solution is

realized. It is noteworthy from the diagram that this quasi second-best solution can be attained only by fixing fare;

optimal taxi entry will come about as a result of fare regu

lation. This optimal regulation would produce a medallion value of zero. Finally, we point out that while the maxi

mum fleet size point E on the zero-profit curve is a long-run stable equilibrium point in a fully unregulated competitive

market, this point is inefficient from the society point of

view because the maximum offered quantity of taxi-hours

of service is not utilized efficiently. This results in an inten

sive, but wasteful, competition among taxi firms.

Figures 8 and 9 also locate the quasi-first-best solution of

social optimum (for the same reason mentioned above, the term "quasi-first-best" is used). From the point of view of

the economy as a whole, the efficient allocation would be to choose a fleet-fare combination to maximize total social

welfare. Like the static taxi operation model, it is found that

the quasi-first-best social optimum is located at the right side of the zero-profit contour. The quasi-first-best taxi pric

ing entails taxi operation at a loss. Therefore, attainment

of the quasi-first-best solution would require introduction




of a mechanism that subsidizes taxi service to cover the

operational deficit.

We now turn to the average customer waiting times (plot ted in Figure 9), which are regarded as a proxy of service

quality. It can be seen that the maximization of service

quality (minimum average customer waiting time) occurs

at point H on the zero-profit curve, with a somewhat higher fare than at point E. This service quality-maximizing point

certainly gives rise to a very low taxi utilization. Note that

the pattern of iso-waiting time contours is quite different

from that in the static taxi service model (Yang et al. 2002). In the present dynamic model, the average customer wait

ing time tends to be nearly constant (contour locus becomes

approximately horizontal) because taxi fleet size is large, whereas in the static taxi model customer waiting time at a

given fare is always declining with taxi fleet size (although with a decreasing change rate). This phenomenon arises

from the different assumptions embodied in these two types of models. The single-period static model assumes that all

licensed taxis are in service, and thus adding new taxis

will always lead to an increase in vacant taxi-hours and

thus a decrease in customer waiting time. In contrast, the

multiperiod dynamic model supposes that all taxi drivers

optimize the time-of-day use of individual taxis for service

in response to the market temporal conditions. It is self

evident from our clock network model that, as taxi fleet

size increases and reaches a certain level, the net profit for

all paths or shifts with positive taxi workflow will even

tually become zero. Any increase in taxi service intensity or increase in the number of licensed taxis beyond this

limit level will merely suffer further losses. This means that

with increase in the number of licensed taxis, taxi drivers

will simply park their cars instead of offering unprofitable service. As a result, at large fleet size, further increase in licensed taxis does not lead to much improvement in

service quality.

Finally, we look at the sensitivity of the social-welfare

maximizing and profit-maximizing fares and taxi fleet

sizes with respect to the period-specific potential customer

demands. Keeping the same temporal change pattern, we

uniformly scale up and scale down the customer potential demands presented in Table 1. For given scaled potential customer demands, we use the Hooke-Jeeves pattern search

method (Bazaraa et al. 1993) to find the optimal taxi fare

and fleet size for maximizing profit and social welfare,

respectively. Figure 10 displays the change of the optimal fares and fleet sizes with the demand-scaling factor. As

observed from this figure, the optimal fare for social wel

fare maximization keeps quite stable around 20 (HKD/trip), while the optimal fare for profit maximization increases

with increasing potential demand and eventually tends to

stabilize at extremely high demand. On the other hand, both

the profit-maximizing and the welfare-maximizing fleet

sizes grow quickly as the demand-scaling factor increases, and then tend to be constant as the scaling factor goes

beyond a certain limit. The welfare-maximizing fleet size

Figure 10. Scale effect of potential customer demand on optimal taxi fare and fleet size.

20-1- -r 100 -*- Taxi fleet size for maximum social welfare

18 ~?" Taxi fleet size for maximum profit - 90

-*- Fare for maximum social welfare 16 - - Fare for maximum profit

" 80

? 14-

^-*-* -70

2 12- ^y^

r60 q ^ i / ^^-"-ii- ^

? 10- S *^ \w S

"5 I / -7^ I Js

i 8*' / s H h 61 / ^^^ ^a-^-"***-n-n r30

4- wn* + m**ik ?*-*-*-*-4 h-20

2 1/

0-1-,-,-,-,-,-,-,-,-,-,-|-o 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Scaling Factor

is generally larger than the profit-maximizing one. The gap between the two fleet sizes enlarges quickly as the demand

scaling factor increases, and keeps constant as the scaling factor exceeds a certain limit.

6. Conclusions

This paper extended the conventional aggregate taxi service

models for a single uniform period to a multiperiod dynamic situation. Our extension explicitly incorporates the time-of

day variations of customer demand into a taxi modeling framework with endogenous service intensity. Our proposed

model thus added substantial new realism to the work to

date by a large number of economists by considering the

time dimension of services. The extension is especially

meaningful for precise understanding of the operational characteristics of a realistic taxi industry. Indeed, our numer

ical experiments offered some interesting insights into the

temporal nature of the taxi industry equilibrium. The aggregate multiperiod dynamic model can be fur

ther extended in at least two directions. First, the aggregate

dynamic model can be integrated with the multiclass net

work taxi model for service area regulation developed by

Wong et al. (2003). In this case, each service area is asso

ciated with a clock network, and the spatially aggregated

demand-supply model on each clock arc of a clock net

work is replaced by the spatial taxi model on the physical road network in a particular service area. Such integration

will help acquire more useful measures of service qual

ity and market profitability at times when demand is high or low and at locations where the population is concen

trated or dispersed, as well as congestion effects due to

both occupied and vacant taxi movements. The time and

location information of services is useful for regulating

temporal and spatial fare differentiation. Second, the clock




Operations Research 53(3), pp. 501-515, ? 2005 INFORMS 515

network equilibrium model can be applied to determine an

optimal temporal fare structure using a bi-level program

ming approach (Yang and Bell 2001), where the upper level problem is to ascertain period-specific fares to maxi

mize social welfare or attain a quasi-second-best optimum,

and the lower-level problem would be the multiperiod ser

vice equilibrium problem on the clock network. Finally, we

remark that the clock network equilibrium model is gener

ally applicable for modeling time-of-day utilization of gen eral public service facilities such as public library, public

swimming pool, ski slopes, and public parking facilities.

These facilities can be shared subject to congestion.

Acknowledgments The authors thank Robin Lindsey of the University of

Alberta, an anonymous associate editor, and two anony

mous referees for their helpful comments on an earlier

version of this paper. This study was supported by a

research grant from the Research Grants Council of the

Hong Kong Special Administrative Region (Project No.

HKUST6107/03E).

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