Annals of Operations Research 135, 111–125, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

A Stochastic Model for CombinedActivity/Destination/Route Choice Problems

HAI-JUN HUANG hjhuang@mail.nsfc.gov.cnSchool of Management, Beijing University of Aeronautics and Astronautics, Beijing 100083,People’s Republic of China

WILLIAM H.K. LAM ∗

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon,Hong Kong, People’s Republic of China

Abstract. In travel behavior modeling, an important topic is to investigate what drives people to travel.A systematic analysis should examine why, where and when various activities are engaged in, and howactivity engagement is related to the spatial and institutional organization of an urban area. In view of this,this paper presents a stochastic model for solving the combined activity/destination/route choice problem.It is a time-dependent model for long-term transport planning such as travel demand forecasting. Theactivity/destination choices are based on multinomial logit formulae and, the route choice is governed bystochastic user equilibrium principle. The solution algorithm is proposed together with a numerical examplefor demonstration. It is shown that the proposed modeling approach provides a powerful tool for fullyunderstanding and predicting the complex travel behavior at strategic level.

Keywords: travel behavior modeling, combined activity/destination/route choice, multinomial logitformula, stochastic user equilibrium

1. Introduction

Most of the traditional models, whether they are used to study the route choice of travel-ers, departure time choice, transport mode choice, or the combined ones of these choices,belong to trip-based approach, in which the trip by origin-destination (OD) pair is con-sidered as the unit of analysis and the trips in a trip chain made by a traveler are treatedas separate, independent entities in the analysis. For saving travel time, however, moreand more people tend to chain trips together, stopping at retail stores or giving a ride oftheir children to schools on the way to or from work. All trips in a trip chain made by anindividual are dependent on each other. One trip may be delayed or even cancelled

∗The work described in this paper was substantially supported by the grants from the National NaturalScience Foundation of China (Project No. 79825101), the Chinese Academy of Sciences (MADIS ResearchProject) and the Research Grants Council of the Hong Kong Special Administrative Region (Project No.PolyU5077/97E).

112 HUANG AND LAM

due to the postponement of the previous trips. The traditional trip-based approachoften fails to recognize the existence of linkages among trips, leading to an inappro-priate representation of travel behavior relationships (Jones, Koppelman, and Orfueil,1990). Travel demand is a demand derived from various social and economic activi-ties. To fully understand and predict the travel demand for planning and managementpurposes, it is necessary to investigate what drives people to travel. Instead of focusingonly on trips and networks, the analysis must examine why, where and when activitiesare engaged in, and how activity engagement is related to the spatial and institutionalorganization of an urban area.

In contrast with traditional trip-based approach, the activity-based approach studiestravel patterns in the context of structure of activities, of the individual or household,with a framework emphasizing the importance of time and space constraints (Goodwin,1983). Comprehensive review to this timely approach can be found in Damm (1983) andJones, Koppelman, and Orfueil (1990).

Only a few activity-based travel demand forecast models have been presented in theliterature. Axhausen (1990) combined an activity chain simulation model with a trafficflow simulation model, to achieve a simultaneous simulation of activity chains and trafficflows. Fellendorf et al. (1995) presented an activity-based demand forecasting processthat is a planning package’s sub-model. In this process, the trip chains are generatedfrom given activity chains and the travel demands are allocated to the various trans-port modes by a multinomial logit model. Kitamura et al. (1996) proposed a dynamicand integrated micro-simulation forecasting system, the sequenced activity mobilitysimulator.

Concerning the trip chaining problems, there is a set of studies to investigate theinclusion of non-work activities (i.e., the ones carried out on the way to or from work)in the basic home-work-home trip chain. Oster (1979) emphasized the “second role” ofthe work trip, that is, providing the opportunity to link non-work trips. Damm (1982)developed a theoretical model to investigate whether someone participates in a non-workactivity and if so, for how long. Kondo and Kitamura (1987) investigated whether acommuter will make the non-work stop during the commuting trip or, alternatively, willpursue the non-work activity by making a separate trip chain from home. Nishii, Kondo,and Kitamura (1990) made an empirical analysis of trip chaining behavior.

However, speaking frankly, the current development of this approach lacks rigidlystructured, quantitative methods or analysis procedures (Lam and Yin, 2001; Lam andHuang, 2002). In addition, most of the current studies are static and deterministic. Clearly,the dynamic or time-dependent and stochastic activity-based models may represent moreaccurately the real time traffic conditions in a transportation network. Kitamura et al.(1996) stated that although the development of dynamic versions is a challenge, advancesin computer and methodology promise that truly travel behavioral modeling is possible.Following this direction, in this paper we present a conceptual model for time-dependentand stochastic activity/destination/route choice problems. The ultimate aim of our studyis to assess coherently the effects of activity/destination/route choices for long-termstrategic planning purpose.

STOCHASTIC MODEL 113

Section 2 formulates the model and Section 3 gives the solution algorithm ofthe model. In Section 4, a numerical example is presented to illustrate the model andalgorithm. Section 5 concludes the paper.

2. Model Formulation

In this section, the assumptions adopted for the proposed model are firstly discussed. Itfollows with formulation of the multinomial logit-based activity/destination choice prob-lem. The formula is then developed to involve the stochastic user equilibrium conditionfor route choice. Finally, an equivalent mathematical programming is proposed for thetime-dependent and stochastic activity/destination/route choice problem.

2.1. Assumptions

A general urban transportation network consists of nodes and directed links. Let a rep-resent a link and p a path (or route). An origin node, which generates trips, may also bea destination node that attracts trips from other origin nodes, and vice versa. For a speci-fied time period, a path that connects an origin r and a destination s,is simply an acyclicordered set of links. In order to facilitate the essential ideas without loss of generality,the following assumptions are made in this paper.

(i) The activity/destination/route choice problem is investigated in a fixed study hori-zon, T , generally a whole day. The T is divided equally into a number of timeperiods, k (= 1, 2, . . . , K ). Since the time-dependent model proposed in this paperis used for strategic planning purpose, the period duration is set to be reasonablylong, such as one hour, so that most path flows departing from their origins in atime period can finish their journey in the same period. For tractability of the model,this paper does not consider the flow interaction between neighbor periods (sometravelers departing at a given period may carry over their journeys to the next period).

(ii) In each period, people face alternatives of choice: what to do (activity choice, i),where to do it (destination choice, s), and by which path to go (route choice, p).The transport mode choice is not considered in this paper in order to simplify thepresentation of the essential ideas and the car occupancy is fixed and equal to oneperson per car.

(iii) Only one behaviorally homogeneous group is considered. However, different classesof commuters may have different perception for evaluating various activities/traveltimes.

(iv) A computable utility is used to measure the degree of satisfaction from performingan activity or the degree of intensity that an activity is going to be performed. Theutility of an activity is related to the physiological needs for the activity and may be

114 HUANG AND LAM

time-dependent. How to measure the activity utility so as to collect it in real world,is still one of the research topics in utility-maximization approach and is out of theconcerns in this paper. For more discussions on activity utility, see Supernak (1990),Axhausen and Garling (1992), Kitamura and Supernak (1997).

(v) Commuters have no perfect knowledge of traffic conditions throughout the wholenetwork and make their route choice decisions according to logit-based stochasticuser equilibrium principle so as to minimize their perceived travel times (Huang,1995; Huang and Bell, 1998).

(vi) The time to traverse a link in period k,is simply given by a BPR (Bureau of PublicRoad) function: ta(k) = t0

A[1 + 150(ua(k)/sa)4], where t0a is the free-flow travel

time (minutes) on link a, sa is the link capacity (veh/hr) and ua(k) is the traffic flowon link a in period k.

2.2. Activity/destination choice problem

Define Ursi (k) as the total utility obtained by a person who departs in period k from zone

r to destination s,to perform activity i.We can write

Ursi (k) = Vi (k) − α · trs(k) + V s + V s

i + εs + εsi , ∀r, s, i, k (1)

where Vi (k) is the utility value of activity i that is intended in period k, trs(k) is theexpected travel time (minutes) from r to s for departing in period k, α is a parameterfor converting travel time to disutility, Vs represents the systematic component of utilitycommon to all elements with destination s (such as area, population, employment andparking facilities, which are independent of activities), V s

i represents the systematiccomponent of utility depending on both activity i and destination s (such as businessarea, food quality and service level, for an eating activity in a particular location), εs isthe destination choice-related random term, and εi

s is the activity choice-related randomterms associated with the specific destination.

Note that if s = r , then trs(k) = 0. It implies that activity i is performing at ori-gin zone. The attractiveness of other opportunities found after arriving at destination scan also be incorporated into Eq. (1). But, this will lead to more burdensome compu-tation in the model and is neglected in this paper in order to present the essential ideaseasily.

Suppose that all the random terms are separately identically and independentlyGumbel distributed variables with mean zero and identical standard deviation, then thejoint probability of selecting destination s and activity i can be estimated by a multinomiallogit formula (Ben-Akiva and Lerman, 1985), i.e.,

Prsi (K ) = exp

[θ(Vi (k) − αtrs(k) + V s + V s

i

)]

∑s ′

∑i ′ exp

[θ(Vi ′(k) − αtrs ′(k) + V s ′ + V s ′

i ′)] , ∀r, s, i, k (2)

STOCHASTIC MODEL 115

where θ (>0) is a logit parameter reflecting the degree of uncertainty. As the θ -valueapproaches infinity, the uncertainty vanishes. However, when the θ -value is close to zero,people would choose activities and destinations randomly. Equation (2) is a special caseof the hierarchical or nested logit model that usually has different logit parameters subjectto the destination and activity choices. Note that the effects of departure time and routechoices are not directly reflected in the above equation. However, the departure time androute choices would affect the destination and activity choices via the expected traveltimes between different OD pairs. This will be discussed below.

The marginal choice probability that destination s is chosen, can be written as

Prs(K ) = exp[θ (V s − αtrs(k) + V s(k))]∑

s ′ exp[θ (V s ′ − αtrs ′(k) + V s ′(k))], ∀r, s, k (3)

where V s(k) = (1/θ ) ln∑

i exp[θ (Vi (k) + V si )].

So, the activity/destination choice problem has been formulated in (2) and (3).Equation (2) is related to the probability that individuals choose destination s to performactivity i while Eq. (3) is the probability of individuals to choose destination s to performthe activities of concerns, but not necessary confined to a particular activity.

Assume that within each origin zone r , at the end of period k −1 there are a numberof potential travelers, Nr (k − 1), who will choose their destinations and complete theirtravels in the next period k, according to the probability given by Eq. (3). Of course,perhaps the destination chosen by them is the origin zone itself. This means that theywill stay at home continuously. Hence the aggregate departure flow in period k fromorigin r to destination s can be expressed as

drs(k) = Nr (k − 1)Prs(k) = Nr (k − 1)exp[θ (V s − αtrs(k) + V s(k))]

∑s ′ exp[θ (V s ′ − αtrs ′(k) + V s ′(k))]

, ∀r, s, k

(4)

where Nr (0) is a predetermined constant representing the number of seed individuals inzone r at the initial time of the study horizon. These people will constitute the potentialtravelers in the coming sequential periods through the period-by-period trip distribu-tion/assignment process (presented in the next sub-section).

Equation (4) indicates that the travel demand by origin-destination pair in period kis dependent on the utility function and the expected travel time associated with periodk. The expected travel time, in turn, is dependent on the OD travel demand because ofthe traffic assignment effect on the network. In addition, there is an inter-relationshipbetween the trip distribution/assignment results in a period and the number of potentialtravelers available for the next period. Hence, at a macroscopic level, the proposedmodel can be regarded as a travel demand analysis tool to forecast the time-dependentOD demands according to the space-time distribution of activity utilities and the trafficconditions.

116 HUANG AND LAM

2.3. Combined activity/destination/route choice problem

In this sub-section, the above formulated activity/destination choice problem is integratedwith the logit-based stochastic user equilibrium (SUE) traffic assignment problem in thestudy network. We adopt such stochastic user equilibrium that, for each OD pair thetravel times perceived by travelers departing in the same period are equal and minimal.This says for each period, at SUE no traveler would believe his or her perceived traveltime to be improved by unilaterally switching paths. Let trs

p (k) and T rsp (k) be the unit

travel time and unit perceived travel time on path p from r to s in period k, respectively.According to the basic requirement for generating a logit-based SUE assignment, wehave

T rsp (k) = trs

p (k) + 1

βξ rs

p , ∀r, s(�= r ), p, k (5)

where ξ rsk are identically and independently distributed (i.i.d.) Gumbel variations for all

paths and O-D pairs. β is the dispersion parameter relating to variance of path traveltime perception errors (Huang, 1995). It can be shown that E[T rs

p (k)] = trsp (k) and

Var[T rsp (k)] = π2/6β2. From the utility maximization principle of path choice, the

probability of selecting path p is

Prsp (k) = Pr

(T rs

p (k) ≤ T rsl (k), ∀l

) = exp( − βtrs

p (k))

∑l exp

( − βtrsl (k)

) , ∀r, s(�= r ), p, k (6)

and the path flow is given by

f rsp (k) = drs(k)Prs

p (k), ∀r, s(�= r ), p, k (7)

Clearly, the Eq. (7) leads to∑

p

f rsp (k) = drs(k), ∀r, s(�= r ), k (8)

f rsp (k) ≥ 0, ∀r, s(�= r ), p, k (9)

The set of path flows satisfying conditions (6)–(7) is called logit-based SUE assignment.In this paper we assume that link travel times are continuous and strictly increasing

functions of link flows and the period duration is large enough such that it covers thewhole journey of each path flow for a given period. So, we expect that the path travel timefor each time period is depending on the path flows in the current period only. Hence, foreach period and a given OD demand we are in fact dealing with a static logit-based SUEtraffic assignment problem. It is easy to show that such assignment can be characterizedas a solution to the following convex mathematical programming

min1

β

∑

r

∑

s(�=r )

∑

p

[f rs

p (k) ln f rsp (k) − f rs

p (k)] +

∑

a

∫ ua(k)

0ta(w)dw, ∀k (10)

STOCHASTIC MODEL 117

subject to (8)–(9) and

trsp (k) =

∑

a

ta(ua(k))δrsap, ∀r, s(�= r ), p, k (11)

ta(k) = t0a [1 + 150(ua(k)/sa)4], ∀a, k (12)

ua(k) =∑

r

∑

s(�=r )

∑

p

f rsp (k)δrs

ap, ∀a, k (13)

where δrsap = {1, if path p from r to s contains link a; 0, otherwise}. After finishing the

logit-based SUE assignment, we can compute the path choice probabilities by Eq. (6),and get the expected OD travel times used in Eqs. (1)–(4), as follows

trs(k) =∑

p

trsp (k)Prs

p (k), ∀r, s(�= r ), k (14)

Now, we examine the flow conservation constraints for OD demand. Note that in(4) we allow for s = r . Clearly, Eq. (4) satisfies

∑

s(�=r )

drs(k) + drr (k) = Nr (k − 1), ∀r, k (15)

with the boundary condition

Nr (0) = Mr , ∀r. (16)

The variables in the left-hand side of (15) are solved from the current period’s tripdistribution/assignment process. As a result, drr (k) represents the number of trips whostill stay in origin r till the end of period k, and drs(k) represents the number of tripswho has already arrived at zone s by the end of period k. These persons are regarded asthe potential travelers who will continue to go for their activities and travel in the nextperiod, k + 1. Hence, the potential travelers for continuous travel at the end of period k,can be written as

Nr (k) = drr (k) +∑

r ′dr ′r (k), ∀r, k. (17)

Clearly,∑

r Nr (k) = ∑r Mr for all periods, i.e., at the end of each period all per-

sons gather at zones. During each period all individuals are engaged in activities and/ortravels.

Up until now, we have already formulated all the conditions required for the time-dependent stochastic model in which activity/destination/route choices are combined. Inthis model, the activity choice is governed by (2), the destination choice by (4), the routechoice by (6)–(7), and the evolution of travel potential by (15)–(17). It is easy to showthat seeking a solution { f rs∗

p (k), drs∗(k), Nr∗(k)} for all k, satisfying these conditions, is

118 HUANG AND LAM

equivalent to

min Z (·) =∑

k

∑

a

∫ ua(k)

0ta(w) dw + 1

β

∑

k

∑

r

∑

s(�=r )

∑

p

[f rs

p (k) ln f rsp (k) − f rs

p (k)]

+(

1

θα

) ∑

k

∑

r

∑

s

[drs(k) ln drs(k) − drs(k)]

− (1/α)∑

k

∑

r

∑

s

[V s + V s(k)] drs(k) (18)

subject to∑

p

f rsp (k) = drs(k), ∀r, s, (�= r ), k (19)

∑

s( �=r )

drs(k) + drr (k) = Nr (k − 1), ∀r, k (20)

Nr (0) = Mr , ∀r. (21)

Nr (k − 1) = drr (k − 1) +∑

r ′dr ′r (k − 1), ∀r, k(≥2) (22)

{f rs

p (k), drs(k), Nr (k)} ≥ 0 ∀r, s, p, k (23)

where ta(w) and ua(k) are defined by (12) and (13), respectively. Note that in the (18)the superscript s appearing in drs(k)–related terms covers r . The above mathematicalprogramming problem can be split into K one-period sub-problems and solved by thevariants of the standard convex combination algorithm (Sheffi, 1985). For each sub-problem, for example the one in period k, the travel potential within origin, Nr (k − 1) ,are given by the last period sub-problem’s solution, i.e., computed by (22).

3. Solution algorithm

The algorithm for solving the mathematical programming problem (18)–(23) is describedas follows. In this algorithm, the Step 2.3 for reducing objective function (18) is basedon path flow variables, so that all OD demand variables (except the case of s = r )arereplaced by path flow variables using (19).

Step 1. Let k = 1 and prepare Nr (0), ∀r .Step 2. Solve the logit-based user equilibrium singular-constrained OD estimation prob-

lem.

Step 2. 1. Set the iteration index n = 1 and choose an initial feasible OD demands{drs,n(k)}. On the base of path travel times corresponding to free-flow network,

STOCHASTIC MODEL 119

compute the initial path choice probabilities by (6) and then determine the initialfeasible path flows { f rs,n

p (k)}. by (7).Step 2.2. Set ua

n(k) = ∑r

∑s( �=r )

∑p f rs,n

p (k)δaprs . Compute link travel times and then

path travel times {trs,np (k)}.

Step 2.3. Find the direction for reducing the value of objective function (18). Solve

min Z (·) =∑

r

∑

s(�=r )

∑

p

∂ Z (·n)

∂ f rsp (k)

f rsp (k) +

∑

r

∂ Z (·n)

∂drr (k)drr (k)

=∑

r

∑

s(�=r )

∑

p

[trs,n

p (k) + 1

βln f rs,n

p (k) + 1

θαln drs,n(k)

− 1

α(V s + V s(k))

]f rs

p (k) +∑

r

[1

θαln drr,n(k) − 1

α(V r + V r (k))

]drr (k)

(24)

subject to∑

s( �=r )

∑

p

f rsp (k) + drr (k) = Nr (k − 1), ∀r (25)

f rsp (k) ≥ 0 drr (k) ≥ 0 ∀r, s. (26)

Denote the solution as{

f rs,np (k), drr,n(k)

}.Compute

drs,n(k) =∑

p

f rs,np (k), ∀r, s(�= r ). (27)

Step 2.4. In (18), Replace path flow variables by f rs,np (k) + λ( f rs,n

p (k) − f rs,np (k)),

OD flow variables by drs,n(k) + λ(drs,n(k) − drs,n(k)), and link flows by una(k) +

λ(una(k) − un

a(k)) where una(k) = ∑

r

∑s(�=r )

∑p f rs,n

p (k)δ pars and 0 ≤ λ ≤ 1. Solve

the resulted line search problem for generating an optimal step size, λn .Step 2.5. Let un+1

a (k) = una(k) + λn(un

a(k) − una(k)) for all a,

f rs,n+1p (k) = f rs,n

p (k) + λn( f rs,nP (k) − f rs,n

p (k)) for all r, s, p, and

drs,n+1(k) = drs,n(k) + λn(drs,n(k) − drs,n(k)) for all r, s.

Set n = n + 1, if some termination criterion is satisfied, go to Step 3; otherwise, goto Step 2.2.

Step 3. Compute

Nr (k) = drr,n(k) +∑

r ′dr ′r,n(k), ∀r. (28)

120 HUANG AND LAM

Step 4. if k = K , stop; otherwise, set k = k + 1 and go to Step 2.

In order to improve the convergence, a modified method can be used as follow. InStep 2.2, use (6) to compute the path choice probabilities, and compute the expected pathtravel times by (14), i.e., trs,n(k) = ∑

p trs,np (k)Prs,n

p (k). In Step 2.3, instead of solvinga linear programming problem, use the logit formulation to compute the auxiliary ODdemand, i.e.,

drs,n(k) = Nr (k − 1)exp[θ (V s − αt rs,n(k) + V s(k))]

∑s ′ exp[θ (V s ′ − αtrs ′,n(k) + V s ′(k))]

, ∀r, s (29)

and finish a logit-based SUE assignment by (6)–(7) so as to get the auxiliary path flows,i.e.,

f rs,np (k) = drs,n(k)Prs,n

p (k) = drs,n(k)exp

( − βtrs,np (k)

)

∑l exp

( − βtrs,nl (k)

) ∀r, s, p (30)

and the link flow una(k) = ∑

r

∑s( �=r )

∑p f rs,n

p (k)δrsap. All the other steps of the algorithm

remain unchanged. In solving the standard logit-based SUE problem, Sheffi (1985) pro-vided the proof for the convergence of this variant.

4. Numerical example

Figure 1 shows a simple transportation network consisting of 4 zones and 12 links. Foreach link, the free-flow travel time (minutes) and the capacity (vehicles per hour) are also

Figure 1. A simple example.

STOCHASTIC MODEL 121

Figure 2. Simplified temporal utility profiles of three types of activities.

Figure 3. Populations at locations by time of day (beginning of hour).

given in this figure. Four types of activities, namely staying home, working, shoppingand eating, can be performed within these four zones, respectively, with the temporalutility profiles shown in figure 2. Assuming that there are 1000 persons in the studynetwork. The study horizon is from 6 am to 8 pm and is divided into 14 hourly periods.The initial conditions are: N H = 1000, Nw(0) = 0, N E (0) = 0 and N s(0) = 0. Thethree parameters required are: α = 0.2, θ = 0.25 and β = 0.4.

The most essential results are presented in figures 3 to 5 for the population atlocations (i.e., the potential travelers) and departure flows (i.e., the actual travelers) bytime of day, respectively. From these three figures, we can investigate the people’s activitypattern and the OD travel pattern on the network temporarily and spatially. It is shownthat for the 6–8 am period, more people are leaving their homes and going to their offices,

122 HUANG AND LAM

Figure 4. Flows departing from zones H and E .

Figure 5. Flows departing from zones S and W .

due to the high utility of working; they work there till the end of 11 am. At noon, becauseof the high utility of having lunch, many people leave their offices to have lunch and thenreturn. There are some persons who go to shop after having lunch. At 6 pm, people startto go home, shop or have their dinners outside their homes. But, most people choosegoing home due to its higher utility.

It should be noted from figures 3 to 5 that the potential travel at each origin zone byeach hour period is always larger than the summation of the actual flows departing fromthis zone. This indicates that some persons choose staying in the origin zones, i.e., notmaking any actual travels, during the forthcoming period, after comparing the utility dif-

STOCHASTIC MODEL 123

Table 1Main results on the base and expanded networks.

Base network Expanded network

Average travel time (I) 3.073 2.577Average duration at home (II) 3.274 2.990Average duration in office (III) 6.213 6.477Average duration in restaurant (IV) 0.599 0.924Average duration in retail store (V) 0.841 1.032Total 14.00 14.00

Note. The unit (hours/person) is used for all data.

ferences between the two alternatives. Table 1 shows the average travel time and the aver-age duration of the four activities associated with the 1000 persons during the whole studyhorizon (i.e. 14-hour period). The formulations used for calculating these indexes are:

I =∑

k

∑

r

∑

s( �=r )

drs(k)t rs(k)/(1000 × 60)

II =∑

k

[dHH(k) +∑

r=E,W,S

dr H (k)(1 − t rH(k)/60)]/1000

III =∑

k

[dWW(k) +∑

r=E,H,S

dr W (k)(1 − tr W (k)/60)]/1000

IV =∑

k

[dEE(k) +∑

r=H,W,S

drE(k)(1 − t rE(k)/60)]/1000

V =∑

k

[dSS(k) +∑

r=H,E,W

drS(k)(1 − t rS(k)/60)]/1000

(31)

where we assume that travelers, after finishing their journeys, stay within the destinationzone in the remainder of one hour. On the scenario with the base network, each personspent 3.073 hours for traveling, 3.274 hours for staying at home, 6.213 hours for working,0.599 hours for eating and 0.841 for shopping. Assume that there is an expansion schemeof improving links from H to W (i.e., link 1 in figure 1) and from W to E (i.e., link 6 infigure 1) so that their capacities increase to 3000 veh/hr and their free-flow travel timesare shortened to 25 minutes. Traditionally, the benefits of such network expansion aremeasured by travel time saving, operation cost reduction and decrease of traffic accidents.However, how the time saved can be productively used is difficult to determine by thetraditional trip-based models. It is because the representative value of time used in theevaluation cannot practically reflect the opportunity cost of time. The main results ofthe above network expansion scheme are listed in Table 1, in comparison with that of thebase network.

Although the proposed modeling approach is conceptual, it can be used to evaluatesome transport policies for long term strategic planning. As shown in Table 1, it wasfound that in comparison with the base network, the expanded network reduces theaverage travel time by 16.14%. Because of the alleviated traffic congestion from H to

124 HUANG AND LAM

Figure 6. Total travel times (minutes) by time of day.

W , people can arrive at workplace with less travel times than before and then more peoplespent longer time in the working activities. This leads to the longer in-office duration(increase by 4.25%). On the other hand, the at-restaurant and in-store durations becomelonger as the traffic condition from W to E is improved (some people go to shop afterhaving lunch because S is very close to E). Based on this time-use distribution, a betterestimation of the benefit induced by travel time saving is not a difficult task. In this way,the more efficient and coherent evaluation of transport policies can be achieved.

Figure 6 compares the total network travel times by time period before and afterexpanding the network. It can be seen that the expansion scheme mainly improves thetraffic condition during three peak periods. This is another advantage of our model thatthe time-varying effect of an expansion scheme can be clearly explored.

5. Conclusions

In this paper, the concept of the temporal utility profile of activities is employed to studythe activity choice behavior. A mathematical programming model is proposed for solvingthe time-dependent, stochastic equilibrium activity/destination/route choice problem. Asolution algorithm is presented together with numerical example for illustration. It isshown that the modeling approach proposed in this paper provides a powerful tool forfully understanding and predicting the complex travel-related behavior at macroscopiclevel. By introducing the activity-based approach, we offer a new avenue of travel demandanalysis for transportation planning and policy evaluation.

The modeling approach proposed in this paper can be further extended to incorpo-rate the multiple behaviorally homogeneous commuter groups and the transport modechoice. Calibrating the logit parameters and the value of travel time from observed datais also interesting topic. In addition, the model proposed in this paper should be furtherimproved in order to deal with the flow interaction between neighbor periods.

STOCHASTIC MODEL 125

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