A utility maximizing model of the demand for multi-destination non-work travel

Transpn Rer-8 Vol 14B. pp. 369-386 0 Pergamon Press Ltd , 1980. Printed in Great Britam

A UTILITY MAXIMIZING MODEL OF THE DEMAND FOR MULTI-DESTINATION NON-WORK TRAVEL JOEL. HOROWITZ U.S. Environmental Protection Agency, Washington, (Received

17 August

1979: in

revised form

DC 20460, U.S.A.

28 January

1980)

Abstract-In this paper, a structural model of the demand for multi-destination non-work travel is developed. and an empirical illustration of the structural model is presented. The structural model is based on the principle of utility maximization and differs from previous utility maximizing models of multidestination non-work travel in several important ways. In contrast to most previous models, the model presented here incorporates travel frequency. destination choice and mode choice for both single and multi-destination travel into a unified utility-maximizing framework The model includes a representation of the demand for travel between individual origin-destination pairs but avoids the need for enumerating complete travel patterns. Finally, the model incorporates the concept that current travel decisions depend on past travel decisions and future travel.plans. as well as on current conditions. Empirical tests of the model have produced encouraging results concerning the model’s structural validity. The empirical tests also have indicated that there is a need to develop improved sets of explanatory variables for non-work travel.

In recent years utility maximization has emerged as a fundamental behavioral principle of urban passenger travel demand modeling. According to this principle, an individual’s preferences for the travel options he faces can be described by a utility function, and each individual chooses the option that maximizes his utility. In typical applications, the principle leads to models of the logit and probit type (McFadden, 1974; Domencich and McFadden, 1975; Hausman and Wise, 1978). Although utility maximizing demand models have been developed for both work and non-work travel, the most widespread and successful use of these models has been for forcasting work trip mode choice (Spear, 1977; Ben-Akiva and Atherton, 1977; Parody, 1977; Ben-Akiva and Richards, 1976). Accordingly, multinominal logit is now included among the mode choice modeling techniques available in standard computer software packages for transportation systems analysis (Urban Mass Transportation Administration, 1976). The development of satisfactory utility maximizing models of the demand for non-work travel has proved to be considerably more difficult than the development of work trip mode choice models. This is due mainly to the variety and complexity of the travel options available to non-work travelers. These options typically include the frequency, destination and mode of travel, among other factors. In addition, non-work trips can be linked with each other or with work trips to form multi-destination tours. Choices concerning trip frequency and multidestination travel have been particularly difficult to treat in utility-maximizing models. For example, the models of Domencich and McFadden (1975). Adler and Ben-Akiva (1976) and Charles River Associates (1976) permit non-work travel frequencies of only 0 to I trips per household per day and do not permit multi-destination travel at all. The Markov and semiMarkov process models of Ben-Akiva, Sherman and Kullman (1979) and Lerman (1979) permit multidestination travel but treat choices of travel frequency outside of the utility maximizing behavioral framework. Moreover, these models require the simplifying assumption that current travel decisions are independent of past travel decisions or future travel plans. The model of Sheffi (1979) treats the choice of trip frequency in a utility maximizing framework, but it does not deal with multi-destination travel or choices of destination and mode. Two models have been developed that attempt to incorporate choices concerning the frequency and destination of non-work travel, including multidestination travel, into a unified utility-maximizing framework. Adler (1976) and Adler and Ben-Akiva (1979) have described a utility maximizing model of the demand for daily travel patterns-i.e. the ordered sequences of trips made during the day by members of a household. Choices concerning travel frequency, destination, mode, and multi-destination tours all are subsumed in this model. However, the set 369

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JOEL HOROWITZ

of travel patterns that are, in principle, available to households is virtually infinite, and many topologically distinct travel patterns are likely to be closely related behaviorally. The Adler and Ben-Akiva travel pattern model provides no means for distinguishing the travel patterns actually considered by households from the ones that are in principle available. Nor does it address the statistical and computational issues associated with enumerating large sets of closely related alternative travel patterns and incorporating them into the modeling framework. In an attempt to avoid the difficulties of the travel pattern model, Horowitz (1979) developed a utility maximizing model of the demand for non-work tours (i.e. complete home-to-home round trips to non-work destinations). This model treats travel frequency, destination choice and multi-destination travel within the utility maximizing framework. However, the mode! does not treat mode choice, and it contains no representation of the demand for travel between individual origin-destination pairs. Consequently, the model cannot be used to predict traffic volumes within geographic subareas of a region or on individual links of a roadway network. This paper presents the theoretical development and an empirical realization of a utilitymaximizing model of non-work travel demand that alleviates some of the problems associated with previous models. The model described here incorporates travel frequency, destination choice and mode choice for both single-and multi-destination travel into a unified utility-maximizing framework. The model includes a representation of the demand for travel between individual origin-destination pairs but avoids the need for enumerating complete travel patterns. In addition, the model incorporates the concept that current travel decisions may depend on past travel decisions and future travel plans. The remainder of this paper is organized as follows. The mathematical derivation of the demand model is presented in Section 2. In Section 3, certain implications of the model’s structure are explored, and the relation of the present model to conventional models of non-work travel demand is discussed. An empirical realization of the model, based on data obtained from the Washington, D.C., area transportation survey, is presented in Section 4. The results of diagnostic tests of the empirical model are described in Section 5, and the conclusions of this research are presented in Section 6. !.MATHEMATICALDERIVATIONOFTHEMODEL The behavior being modeled consists of decisions by members of households that result in non-work person trips by the household members. For the purposes of this paper, attention is directed to trips that take place as components of tours from home to one or more non-work destinations to home without intervening stops at work. Non-work travel that originates or terminates at work is not included in the model. Because the members of a household may interact closely in the process of reaching travel decisions, it is convenient to treat the household as a collective decision-making unit. Accordingly, households are the basic decisionmaking units of the model. Let A be the set of non-work trips that are available to the members of a household. The elements of A are identified by their points of origin and destination and their modes. Thus, the trip ijm is an element of A if i and j are non-work locations (possibly including home) available to the household, and m is a mode that is available to the household for travel between i and j. It is not required that i and j be distinct. For example if trip origins and destinations are defined in terms of traffic zones, then a trip could reasonably originate and terminate in the same zone. Let t be the time of day, and let At be a time interval sufficiently short that the members of the household can begin at most one trip during the time period t to t + At. Ultimately, At will be taken to be an infinitesimal interval. If two or more household members travel together, assume that their trips begin at times that are separated by very small, empirically insignificant, intervals of time. Let A, c A be the set of trips that household members can start during t to I + At. In general, A, will be a proper subset of A, as the trip ijm can be an element of A, only if a household member is at location i at time t. The travel options available to the household during t to t + At are: Option 1: Begin a person trip from origin i to destination j by mode m (ijm E A,) as part of a tour from home to one or more non-work destinations to home. Option 2: Do not begin non-work travel that is part of a home-non-work-home tour. A utility value is associated with each of these options. The household is assumed to choose the

A utility

maximizing

371

model of the demand for multi-destination non-work travel

option with the highest utility. The utilities are given the following functional Option 1: U;j, (x, S, Z, NC),t, At) + &ijm Option 2: UO(s, N,, t, At) + Ed, where:

representations:

CJ = x = s = z=

deterministic component of utility a vector of transportation level-of-service variables relevant to the choice ijm a vector of household characteristics a vector of destination characteristics, other than transportation level-of-service, that are relevant to the choice ijm. N, = number of person trips to non-work destinations other than home begun by members of the household at times other than 1 to t + AL F = random component of utility. No is included in the utility function in order to capture the effects of past travel decisions and future travel plans on current travel decisions. The presence of N, is based on the hypothesis that households have limited travel resources (e.g. time, money, automobiles) and hence, that the decision to begin a trip during t to t + At may depend on the number of trips taken at other times of day.t The random utility component E includes such unobserved variables as the identities of the destinations that household members may have visited or be planning to visit at times other than t to t + At, as well as unobserved characteristics of travel modes, transportation levels of service, locations, and individuals. To deduce the structure of a demand model from the utility functions and the principle of utility maximization it is necessary to specify the probability distribution of the random utility components F. The simplest specification that can be made, and the one that is adopted here, is that the random components are independently and identically distributed with the Gumbel Type I extreme value distribution:

F(F) = exp [- exp(- &)I,

(1)

where F is the cumulative probability distribution function of the random variable E.S This specification leads to the multinomial logit model of demand (McFadden, 1974), in which the probability that a household member chooses to begin the trip ijm E A, during t to t + At is:

P(iimIN,,6 A 6 At) = exp( Uij,)/[exp(uo) + ,,,$ A, exp( U,,,)l.

(2)

The members of a household can make only a finite number of trips during a day. Therefore, it is reasonable to suppose that as At approaches zero, the probability of travel during t to t + A t also approaches zero and that for small At, the probability of the trip ijm E A, is proportional to At: P(ijrn(N,, t, At, A,) = P(ijmlN,,

This implies that for small At the deterministic Uijm (x,

S,

Z,

of utility can be represented

No, t, At) = Vi, (x, S, Z, No, t) + log At

*More detailed representations of the effects of travel time spent on such trips might be included

effects of travel outside the period t to

components

t, A,) At.

(3) as:

(4)

travel at other times of day are obviously

possible. For example, the total in the utility function. However, the use of N,, alone to represent the t + At minimizescomplexity and leads to a model that has useful properties.

*Whether this specification of the distribution of E is appropriate as an empirical matter depends on the population of travelers being modeled and on the specification of the deterministic component of utility (McFadden, Train and Tye, 1977). Although this matter is not addressed in detail here, it is shown in Section 4 that the specification (I) leads to a reasonably beh,aved empirical model in the case that is considered.

312

JOEL HOROWTZ

Thus, for small At

P(ijmINo,t, A,) At = [exp ( Vii, - V,)] At.

(6)

The notation can be simplified by absorbing V, into Vi,, thereby obtaining:

P(ijmlN,, t, A,) At = [exp (Vii,)] At.

(7)

The conditioning on A, of the probability of choosing trip ijm can be removed by noting that ijm cannot be chosen if ijmE A,. This leads to the following marginal probability of choosing ijm : P(ijm~N,,t)At=P(ijm~N,,t,A,)Pr(ijm

E A,)At

(8)

or P(ijm(N,, t) At = (exp Vi,) Pr(ijm E A,) At

(9)

The functional form of Pr(ijm E A,) is not known a priori. However, it is possible to impute certain intuitively plausible properties to Pr(ijm E A,) and to use a simple function that has these properties to approximate Pr(ijm E A,). First, let i be home. Travel from home is possible only if one or more household members is located at home. The probability that one or more household members is at home at a given time t can reasonably be expected to decrease as N,, the number of trips to non-work destinations begun at other times of day, increases. Accordingly, Pr(ijm E A,) should be a decreasing function of N, if i is home. This property of Pr(ijm E A,/i = home) can be represented simply by the following functional specification: Pr(ijm E A,li = home) = B(s, t)[l - K(s, t)N,],

(10)

where B and K are functions of the household characteristics s and time of day t. Now let i be a non-home location. The trip ijm at time t is possible only if at least one household member is located at i at time t, an event that can occur only if one or more household members has traveled to i at a time other than t. In addition, relatively few non-work travelers change modes of travel during non-work tours.? Accordingly Pr(ijm E A$# home) can reasonably be expected to be an increasing function of Nim.0,the number of trips household members make to i by mode m at times other than t. This property of Pr(ijm E A,Ji# home) can be represented simply by the functional specification: Pr( ijm E A, )i # home) = Li, (x, S, Z) Nim.0,

(I 1)

where Li,(X, s, z) is an as yet unspecified function. Later in this paper it will be shown that internal consistency of the non-work travel demand model requires Li, to be the inverse of the time-integrated probability that a household member who is at location i and using mode m departs from i during a time interval of length At. Thus, in the specification (1 l), Pr(ijm E A,li# home) increases as the number of trips to i by mode m increases, and decreases as the average likelihood of having departed from i increases. Although the specifications (10) and (11) are intuitively plausible and simple, they clearly are not the only possible specifications of Pr(ijm E A,). The ultimate justification for these specifications and for the other functional specifications and approximations used in the development of the non-work travel demand model lies not in their a priori plausibility, but in their ability to produce an analytically tractable, empirically valid model. As will be seen later in this section, the specifications (10) and (11) do produce an analytically tractable model. Some preliminary investigations of the model’s empirical validity are described in Section 5.

tL.ater in this section the approximation

is made that no non-work travelers change modes during non-work tours.

A utility

maximizing

model of the demand for multi-destination

non-work travel

313

Given the specifications (10) and (1 l), eqn (9) can be rewritten as:

P(ijm (No, t) At =

kw Vijm)LimNim.oAt; if home B(exp Vij,)( I- K N,)At;

i = home.

(12)

The conditioning on N, can be removed by forming the marginal probability P(ijmlt)At = C Pr(N,) P(ijmJN,,, t) At, N,,

(13)

where Pr(N,) is the probability distribution function of N,, and the sum extends over all possible values of N,. Combining (12) and (13): At L;, P(ijm It)At =

!

2 N;,,, Pr(N,,, N,,,,)(exp Vi,); N,,.N,w,

At E Pr(N,) B( 1 - KN,)(exp Vii,);

if home

i = home

N,,

(14)

where Pr(N,, N,,,,) is the joint probability distribution function of N, and Nim.0. To obtain a tractable expression for P(ijm(t) it is necessary to approximate the summands in (14). The approximation that is used here consists of replacing P(ijm IN,, t) by a linear function that coincides with P(ijm IN,, t) when N, = 0 and 1. This approximation is motivated by the observation that most households make at most one non-work sojourn per day (Adler, 1976). Significant consequences of the approximation are discussed in Section 3. Using the linear approximation: Ni,,o,Li, (exp V$);

P(ijmIN,, 0 =

iZ home

B(exp V$$J{ 1 + N, [(.l - K) . exp (V!!) ) - I]}. r,m- VW [Jm

i = home

(15)

where V$, and V&!Aare, respectively, the values of Vii, when N, is 0 and 1. Note that the functions B(s, t) and K(s, t) can be absorbed into the utility functions V{yAand V$, when the approximation (15) is made. Thus, B and (1 -K) can be dropped from the right-hand side of eqn (15) to obtain: Ni,,J,i,(exp V$,Q; if home p(ijm IN” t, = 1(exp V$,){ 1+ N,[exp (Vf,& - V$,) - I]}; i = home

(16)

Equation (16) can be substituted into eqn (13) and the summation performed. This produces: P(ijmlt)At =

AtNi,.,Li,

(exp V&); i# home At (exp V$$,){l+ N,[exp (V$, - V$,) - l]}; i = home

(17)

where N, and Y?;,,,, are, respectively, the average values of N, and Nim,o.As At approaches zero, &, approaches N, the average value of total daily person trips to non-work destinations other than home. Similarly, as At approaches zero, Nim.0approaches Ni,, the average value of total non-work trips to destination i(i# home) by mode m. Thus, At NimLim(exp V$,); i# home P(ijm’t) At = At (exp V$,){l + N[exp (V$#, - V$,) - 11); i = home.

(18)

Because at most one non-work trip can begin during t to t + At, P(ijm(t)At is equal to the average numbers of trips from i to j by mode m that start during t to t + At. The average number of trips per day from i to j by mode m, Nijm, therefore can be obtained by integration: Nijm =JP(ijmlt)dt

(19)

JOEL HOROWITZ

314

where the integral extends over a day. Note that Niimand N are related by: N = Z Nijm,

(20)

ijm j#h

where the subscript h signifies home. For r equal to 0 or 1, define W$, by: W!$A= log

exp ( V$

dt . I

(21)

W$, is a time-integrated utility of travel from i to j by mode rn, given that r other trips to non-work destinations are made during the day. Using equations (19), (20) and (21), integration of (18) over a day yields. Nij~ = Ni, Lim (exp W$,$); i# home

(22a)

Nhjm= (exp W$$,J{l + N[exp ( Wj,$, - W&,) - l]}.

(22b)

If a household has J non-work destinations and M modes available to it, then (20), (22a) and (22b) define a system of MJ(J + 1) + 1 equations in the MJ(J t 3) t 1 unknown quantities N, Nrj,, Nim, N+, and Li,, where i, j and m range over the available destinations and modes. To obtain a unique solution for the unknown quantities, it is necessary to add 2MJ equations to the system. The equations can be obtained by requiring travelers to make tours that are spatially connected. In other words, the number of trips that members of a household make to a given location must equal the number of trips that the members of the household make from that location. In addition, it is assumed that travelers do not change modes during tours, so that the numbers of trips to and from a given location by a given mode are equa1.t The tour connectivity requirements add the following 2MJ equations to the system represented by (20), (22a) and (22b): Ni, = 2 Nijm; i # home

(23)

C Njim= C Nijm; j# home.

(24)

i

8

I

Equations (20) and (22a) through (24) constitute a system that is solvable for N, Nijm,Nhjm.NiI, and LimeTo display the solution of these equations, first define the matrix R, to be inverse of the matrix whose ij element (i,j# home) is:

(R,‘), = 8, - (exp WC)/: exp W$!,

(25)

where 6, is the Kronecker delta, and the sum in the denominator includes k = h. For any locations p and 4 not equal to home, define R,,4,mto be the pq element of the matrix R,. Finally, define D by D = 1+ jz [exp W$L, - exp Wk!k,,lR,m.

(26)

j,k#h

t Although this assumption obviously is only approximately true, the approximation appears not to be severe. For example, in preparing the estimation data set for the empirical model described in Section 4, it was found that fewer than 2 per cent of the households examined made non-work tours in which there was a change of mode. Ben-Akiva et al. (1979) report similar findings.

315

A utility maximizing model of the demand for multi-destination non-work travel

Then, as can be verified by substitution, the solution to the system of eqns (20) and (22a) through (24) is: Li, = l/X exp W$;

Nhjm= [exp W$$,,+ Z

if home

(27)

RPq.rexp ( WIp,',+ W$&)

(28)

PW Pzqfh

-

1 Rpq,,exp(W&i, + W$,,)]/D

PV

m#h

Nij~ = L,InI(exp

W'!' urn) C NhkmR km; i# h.

(29)

kfh

Ni, is equal to the summation term on the right-hand side of eqn (29) and N is obtained from (28) and (29) by means of (20). Equations (26), (28) and (29) can be simplified greatly if it is assumed that the utility function W(,,!, can be written in the form:

W!O = WI

fij, (X,&Z) + H) (S); i,j# h Fhjm(x,s,z) + Gr (s) . i=h

j=h

Eh,,,(X,S,Z)+H,(S);

(30)

In eqn (30) r is the number of non-work trips made at times of day other than the current time. Thus, eqn (30) states that the time integrated utility function W can be represented as the sum of two components: a component F that depends on transportation levels of service, household characteristics and destination characteristics but that is independent of past travel decisions and future travel plans, and a component G or H that represents the effect on the time-integrated utility of current travel of non-work trips that have been or will be made at other times of day. Because travel behavior depends only on the differences between the utilities of travel options, there is no loss of generality in assuming that H?(s) is zero for all r and s. Using the utility specification (30), eqns (26), (28) and (29) become:

D = 1+ bp G(S)- expGds)l jE Rkj,m exp&km

(31)

j,kth

Nhjm= D-’ exp [Fhjm+ G,(s)] Nij~= De’ Li,,,(eXp I$,) ,z, Rki,mexP [Fhkm + GO(S

Nhm= De’ &,{eXP i&m + Hds)ll

(32) (33)

i,j# h

,z,Rki,mexP [Fhkm+ GO(S)]; if

h.

(34)

In addition, the total number of sojourns at non-work locations, which can be obtained by summation from eqns (32) and (33), is N = D-’

j;mR,m exp [Fhkm+ G&)1.

(35)

j.k#h

Equations (32)-(35) together with the definitional relations (27) and (31) constitute the desired model of non-work travel demand. Specifically, eqn (35) for total non-work sojourns is analogous to a trip-generation equation, whereas eqns (32) (33) and (34) are analogous to direct

JOELHOROWITZ

376

demand models of frequency, home-based trips.

destination

choice, and mode choice for home-based

3. PROPERTIES

OF THE

and non

MODEL

In this section the structure of the model represented by eqns (32)-(35) is interpreted in qualitative terms, and the relation of this model to some previous disaggregate models of non-work travel demand is explored. Several easily identified qualitative properties of eqns (32)-(35) are shown to be consistent with reasonable intuitive ideas about non-work travel behavior. If home-based trip frequency is known, then eqns (32)-(35) imply that home-based mode and destination choice are given by a conventional logit model:

NhjmI 2 Nhpr= (expFiji)/ 2P’ exp pr

If non home-based trip frequency and mode choice are known, tination choice also is given by a logit model:

NJ

(36)

Fhpr. then non home-based

x Nik,,,= (exp fi~,,d/[exp (fihrn + HI) + c exp&,,,I; k#h

k

jZ h

des-

(37)

and

N/vnl~ Nikm= [eXP(Fi/,m.+ k

H,)l/[exP

(Fihm

+

HI)

+

,fh

eXp

F;.km].

(38)

Non home-based mode choice is the same as home-based mode choice, owing to the assumption that travelers do not change modes during non-work tours. Therefore, if trip frequencies are known, eqns (32)-(35) can be reduced to a set of logit models for non-work mode and destination choice.? If households are permitted to make at most one non-work trip per day as in the models of Adler and Ben-Akiva (1976) and Domencich and McFadden (1975), then H,(s) = 30 and G,(s) = --co. Under these conditions, R, is the identity matrix and eqns (32)-(34) become:

Nhjm= [exp (Fhjm + Go)]/ [ 1+ km2 exp (Fhkm t Go)] Nijm~0;

j# h

Njhrn = Nhjm

(39)

(40) (41)

Equations (39)-(41) are mathematically equivalent to the models of Adler and Ben-Akiva (1976) and Domencich and McFadden (1975). It is shown in the Appendix that Rkj,m (k# j) is the average number of sojourns at location j that follow a sojourn at location k during a tour that uses mode m. If k = j, then the interpretation of Rkj,m is the same, except that the sojourn at k is included in the average. It

f Equations (37) and (38) imply that non home-based destination choice occurs without regard to the number of sojourns that have been or will be made on the current tour or on other tours. In other words, non home-based destination choice is governed by a Markov process. This propertyof the model is a consequence of the linear approximation to P(ijmIN,, t) that was made in eqn (15). If this approximation is not made or if a higher order approximation is made, then non home-based destination choice depends on NO and is not Markovian. Although this presumably increases the realism of the representation of travel behavior, it is not possible to develop closed-form equations for travel frequency and destination choice if the linear approximation is not used. The linear approximatinn is, therefore, essential to achieving a tractable travel demand model. Note that in contrast to the models of Ben-Akiva er al. (1979) and Lerman (1979), the Markovian property of the present destination choice model arises from a mathematical approximation, rather than a behavioral assumption.

A utility

follows that C R,,

maximizing

model of the demand for multi-destination

non-work

travel

371

is the average number of sojourns on a non-work tour that uses mode m

and whose l&t sojourn is at location k. In addition, eqns (33), (34) (37) and (38) imply that Li,(exp &,,) and Li,[exp (eh,,, + H,)], respectively, are the probabilities that a trip originating at location i and using mode m terminates at location j and home. Given these interpretations of R,,, Li, (exp P&,) and Li, exp [P& + H,(s)], a straightforwerd, intuitive connection between eqn (32) for home-based travel demand and eqns (33)-(35) for non home based and total non-work travel demand can be established. Note that Nhjmrepresents not only the number of trips from home to location j by mode M but also the number of non-work tours that use mode m and have location j as their first sojourn. Therefore, eqn (35) states simply that the total number of non-work sojourns is the sum over all locations k and modes m of the number of non-work tours whose first sojourn is at k multiplied by the number of sojourns on a tour that uses mode m and whose first sojourn is at location k. Similarly, it can be seen that the summation terms in eqns (33) and (34) equal Ni,, the number of trips that arrive at and depart from location i by mode m. Thus, eqns (33) and (34) state that the average number of mode m trips from location i to locations j and h is equal to the average number of mode m trips departing from location i multiplied by the probabilities that such trips terminate at destinations j and h. An interpretation of the quantity Li, can be developed as follows. From eqn (7) the probability that a traveler who is located at i and using mode m departs for another destination during the time period t to t + At is:

P;,(t)At = T z Pr(N,,)(exp Vijm)At. 0

(42)

Using the approximation of eqn (l5), exp Vij, is approximated by exp V$$,. With this approximation eqn (42) becomes: Pim(t

= X (exp V$,) At. jti

(43)

From eqn (21), the time-integrated value of the right-hand side of eqn (43) is X (exp W$,). jti Therefore, from eqn (27) Li, is the inverse of the time-integrated probability that a traveler located at i and using mode m departs from i during a time interval of length At. This is the interpretation of L,, that was given in connection with eqn (1 I). Equations (31) and (35) imply that N

-c(exp G&exp G,,- exp G,).

(44)

Thus, a household’s average non-work sojourn frequency is bounded from above by an expression that depends on household characteristics. This bound is a consequence of the hypothesis, made in developing eqns (32)-(35), that households have limited travel resources and, therefore, that current travel decisions depend on past travel decisions anti future travel plans. If G,(s) is independent of r, implying that there are no resource constraints, then the upper bound on N is infinite. Thus, households whose travel decisions are unconstrained by resource availability can undertake an unlimited amount on non-work travel. Finally, differentiation of eqns (32)-(35) with respect to the utility components F shows the model to have the following intuitively reasonable properties: (1) Increasing the utility of travel between home and non-work locations tends to increase average tour and sojourn frequencies but tends to decrease the average number of sojourns per tour. (2) Increasing the utility of travel between non-work locations other than home tends to increase average sojourn frequencies and to decrease average tour frequencies, thereby increasing the average number of sojourns per tour.

JOEL HOROWITZ

378 4. AN EMPIRICAL

REALIZATION

OF THE

MODEL

To illustrate the model of eqns (32)-(34) and to test its performance in an empirical context, an empirical realization of the model was estimated using data from the Washington, D.C., area transportation survey. The estimation data set contained travel and socio-economic information for 780 households located in 13 traffic districts around the Washington area. To reduce the computational effort involved in this initial estimation of the model, the 780 households were selected from traffic districts where transit service was unavailable for non-work travel. Thus, the empirical model does not include mode choice; it does include destination choice and travel frequency. The model was estimated in two stages. In the first stage, the utility function parameters relevant to destination choice were estimated using eqns (36)-(38). These equations constitute a multinomial logit model of destination choice, and the estimation was carried out by means of the maximum likelihood method. In the second stage of estimation, the utility function parameters obtained from the first stage were used to compute estimates of the quantities R,. These estimates were substituted into eqn (39, and the remaining utility function parameters (i.e., those relevant to travel frequency but not to destination choice) were estimated from eqn (39, using nonlinear regression. In terms of the functions F, G and H of eqns (32)-(35) the parameters of F and HI were estimated in the first stage, and the parameters of Go and G, were estimated in the second stage. This two stage estimation procedure yields consistent estimates of the utility function parameters. The first estimation stage was carried out using the 510 households out of the full sample of 780 that reported making one or more non-work trips on the survey day. To avoid biasing the estimation results through possible serial correlations in sequences of destination choices by members of the same household, the first stage estimation data set included only a single, randomly chosen observation of destination choice for each of the 510 households. In addition to this observed destination, which consisted of a traffic zone, each household was assigned a set of alternative destinations consisting of a random sample of Washington area traffic zones. Home was included in the set of alternative destinations for trips originating away from home.t The explanatory variables included in the first stage of estimation (i.e. the explanatory variables of the destination choice model) consist of destination attraction variables, and travel times and costs. The destination attraction variables are retail employment, service employment, and population, all according to traffic zone. The first of these variables is used to characterize attraction for shopping trips; the other two variables characterize attraction for non-shopping trips. These attraction variables are admittedly crude. Their choice was dictated by the contents of the Washington survey, zonal level population and employment data being the only information pertaining to attraction that is contained in the survey. The travel time and cost variables included in the destination choice model are the automobile travel times and operating costs associated with travel between a trip’s origin and its potential destinations, and between the potential destinations and the traveler’s home. The origin-todestination travel times and costs characterize the generalized cost of origin-to-destination trip links and are standard variables of disaggregate destination choice models. The destination-tohome travel times and costs are included in the model on the hypothesis that in making destination choices for multi-destination travel, travelers consider not only the generalized cost of reaching a particular destination, but also the generalized cost of returning to home from that destination. In other words, the destination-to-home travel time and cost variables incorporate into the destination choice model the concept that travelers think ahead when deciding on the structures of tours. The complete set of explanatory variables used in the destination choice model is shown in Table 1. The particular functional forms shown in the table were selected through experimentation, the tabulated forms having been found to yield the best estimated model. The utility function conponents were specified as linear-in-parameters functions of the explanatory variables. Thus, if {Xi}represents the set of explanatory variables and {ai} is the corresponding set of parameters, then fij (j# home) and & + HI have the form Z (YiXi. 1

t Estimation of a logit destination choice model with an alternative set that is generated in this manner requires use of a modified form of the logit likelihood function, see McFadden (1979).

A utility maximizing model of the demand for multi-destination Table I. Explanatory Variable

non-work travel

319

variables of the destination choice model Definition

LNTIMI

Natural logarithm of travel time (in minutes) from the origin of a trip to a potential destination.

CT/INC

Automobile operating cost (in cents) of travel from the origin of a trip to a potential destination, divided by household income in hundreds of dollars per year.

LNTIMF

Natural logarithm of travel time (in minutes) from a non-home potential destination of a trip to the traveler’s home. Zero if the potential destination is home.

CF/INC

Automobile operating co.?.t (in cents) of travel from a non-home potential destination of a trip to the traveler’s home, divided by household income in hundreds of dollars per year. Zero if the potential destination is home. Retail employmenti’ a non-home potential destination of a trip. Zt o if the potential destination is home.

POP

Population at a non-home potential destination of a trip. Zero if the potential destination is home.

svc

Service employmentat a non-home potential destiZero if the potential destinanation of a trip. tion is home.

RETLO

Retail employmentat a non-home potential destination of a trip if retail employmentat that destination is less than 100. Zero if retail employment is 100 or nwre, or if the potential destination is home:

SVCLO

Service employmentat a non-home potential destination of a trip if service employmentat that destination is less than 100. Zero if service employmentis 100 or more, or if the potential destination is home.

DHOMF

Home-specific dumrmy variable equal to one for home and zero for all other potential trip destinations.

The first stage (destination choice) estimatio&esults are shown in Table 2. All of the estimated coefficients have signs that are consistent with intuition, and all are statistically significantly different from zero at normal significance levels. The statistical significance of the destination-to-home travel time and cost variables tends to confirm the hypothesis that travelers consider the generalized cost of returning home from a destination as well as the generalized cost of reaching the destination when deciding on tour structures. Tests of the hypotheses that the two travel time coefficients are equal and that the two cost coefficients are equal failed to reject these hypotheses at the 0.10 level, thus indicating that travelers may value current origin-to-destination travel times and costs and future destination-to-home travel times and costs equally. The parameters of the functions Go and GI, which affect travel frequency but not destination choice, were estimated in the second estimation stage. The explanatory variables included in these functions are listed in Table 3. For computational reasons, the denominator of eqn (35) was written for estimation purposes as: D = I+ (exp GJ[ I- exp (G, - Go)] Z$ Rkj exp Fhk

(45)

j,k+h

and the functions G,, and G, - Go were estimated, rather than Go and G,. Both Go and G, - Go were specified as linear-in-parameters functions of the tabulated explanatory variables. Thus, if {Xi}is the set of explanatory variables and {ai} is a set of parameters, G,, and G, - G,, each are

380

JOEL HOROWITZ Table 2. Coefficients Variable

of the destination

choice

model

Coefficient

t-statistic

LNTIM

-1.91

-4.75

CT/INC

-3.81

-3.46

LNTIMF

-2.63

-6.13

CF/INC

-5.37

-4.83

RET

0.00976

11.8

POP

0.000314

11.5

svc

0.00171

REnO

0.0640

13.9

SVCLO

0.0646

10.7

DHOME

3.11

Observations

510

Alternatives (the sum over all households of the number of alternative destinatiom assigned to each household) Log likelihood with all coefficients equal

to

2.66

13297

-3556.5

zero

Log likelihood

with the estimated values of the coefficients

Table 3. Variables

Variable

5.82

-878.3

of the functions

G,, and G, - G,,

Definition

PFO

Number of persons aged 5 years or more in a household.

ALD

Number of automobiles owned by members of a household multiplied by the number of licensed drivers in the household.

INC

Household income in hundreds of dollars per year.

of the form 2 Qi Xi, although the numerical values of the parameters of the two functions may # differ. The second stage estimation results are shown in Table 4. Owing to the severe nonlinearity of eqn (35), meaningful estimates of t-statistics for the estimated coefficients are not available (Beale, 1960). However, it is possible to perform likelihood ratio tests of hypotheses concerning the coefficients (Horowitz, 1979). Using these tests, it was found that all of the estimated coefficients except those of PFO and INC in Go are statistically significantly different from zero at the 0.10 level. However, the hypothesis that the coefficients of PFO and INC in G, are both zero was rejected at the 0.01 level. The statistical significance of the coefficients of G, - G,, is particularly noteworthy. These coefficients result from the hypothesis that households have limited travel resources and, therefore, must consider both past travel decisions and future travel plans when making current travel decisions. The finding that the coefficients of G, -Go are statistically significantly different from zero tends to confirm this hypothesis. The signs of the coefficients of Go and G, - G,, imply that increases in the numbers of persons aged 5 years and over, automobiles and licensed drivers in a household tend to increase sojourn frequencies, whereas increases in income may either increase or decrease sojourn frequencies. These results are consistent both with intuition and the results of previous investigations (Horowitz, 1979).

A utility maximizing model of the demand for multi-destination Table 4. Coefficients of the functions

non-work travel

G0 and G, - G,,

Coefficient for G"

Variable

381

Coefficient for G, - G,,

COllStailt

5.46

PFO

3.41

0.0515

ALD

0.779

0.0159

INC

-0.0264

-1.02

0.000939

780 Observations

82 -

0.172

5. DIAGNOSTIC

TESTS

OF THE

MODEL

The empirical model described in Section 4 was subjected to a number of diagnostic tests in order to investigate its validity and identify ways in which it might be improved in future research. The first test was of the hypothesis implicit in the destination choice model of eqns (36)-(38) that destination choice is independent of the number of non-work trips that household members have made prior to the current one. In particular, the decision to continue a multi-destination tour, rather than returning to home, does not depend on the number of prior trips. An alternative hypothesis, which is inconsistent with the structural model developed in Section 2, is that the probability of choosing home as the destination of a trip that originates away from home increases as the number of prior non-work trips that household members have made increases. Acceptance of this alternative would suggest that the variable N, in the option 1 and 2 utility functions of Section 2 should be replaced by a variable equal to the number of non-work sojourns preceding the current one. Behaviorally, this would suggest that travelers, in deciding on tour structures, tend to be influenced more by travel that has already occurred than by travel that is planned for the future. The hypothesis that destination choice is independent of the number of prior non-work trips was tested by including in the utility function component for travel to home, Fib,the variable NTRIPS, which equals the number of non-work trips preceding the current one. The estimation results are shown in Table 5. The coefficient of NTRIPS is not statistically significantly different from zero at the 0.10 level, thus tending to confirm both the hypothesis that destination choice is independent of the number of prior non-work trips and the structural model developed in Section 2. A second test of the destination choice model was obtained by re-estimating the model using a sample of households that were drawn from different traffic districts and were assigned different sets of alternative destinations from those used in the original estimation. This re-estimation provides a test of the hypothesis that destination choice can be explained with a logit model. In both the original and the re-estimated destination choice models, households were sampled from subsets of the full population, and the sets of alternative destinations assigned to households are subsets of the full set of available destinations. If the logit model of destination choice is correct, then as a consequence of the independence-of-irrelevant-alternatives property of the logit model, destination choice models estimated from different subpopulations of households and different subsets of the full set of alternative destinations will yield consistent estimates of the same utility function coefficients (McFadden, Train and Tye, 1977). Accordingly, the two sets of estimated coefficients should not be statistically significantly different from one another. If the coefficients are significantly different from one another, this can be taken as an indication that the logit model with the given utility-function specification is incorrect. In applying this test procedure to the current destination choice model, it is necessary to compensate for the strong likelihood that the destination attraction terms of the utility function are incorrectly specified and, therefore, may not be transferable to destinations other than the ones used in estimating the model. There are two reasons for this. First, variables such as

JOEL

382

HOROWITZ

Table 5. Coefficients of the destination choice model with the variable NTRIPS included* Variable

Coefficient

t-

Statistic

LNTIMl

-1.91

-4.79

CTIINC

-3.80

-3.53

LNTIMF

-2.62

-6.18

CFiINC

-5.40

-4.94

RET

0.00975

11.9

POP

0.000314

11.5

WC

0.00172

RETLO

0.0640

13.9

SVCLO

0.0646

10.8

5.95

DHOME

3.70

2.96

NTRIPS

0.134

0.885

Observations

510

Alternatives

13297

Log likelihood with all coefficients equal to zero

-3556.5

Log likelihood with the estimated values of the coefficients

-877.9

*

NTRIPS is defined as the number of non-work trips made by household members prior to the current trip.

employment and population are, at best, crude proxies for the characteristics of destinations that travelers actually consider when making destination choices. The relation between these proxies and the true attraction variables may be different for different sets of destinations. Second, because the destinations used in the model are traffic zones, whereas travelers visit individual stores, offices and dwellings, the attraction variables used in the model are aggregate measures of attraction. In general, the aggregate attraction variables will depend on the distributions within zones of the attraction measures of individual destinations. These distributions are likely to be different in different sets of traffic zones. Accordingly, the aggregate attraction measures may not be transferable among different sets of alternative destinations. Because of the potential non-transferability of the attraction terms of the destination choice utility function, in re-estimating the model the attraction terms were respecified as needed to achieve satisfactory estimation results with the new data set. The coefficients of the time, cost and home-specific dummy variables in the original and re-estimated models then were compared to test the validity of the logit structure of the destination choice model.? The coefficients of the re-estimated model are shown in Table 6. As expected, the specification of the destination attraction variables is significantly different from the original specification. Use of the original specification was found to yield the counter-intuitive result that increases in retail employment reduce a destination’s attractiveness for non-work trips. As in the original model, the coefficients of destination-to-home travel time and cost are statistically significantly different from tThis test is obviously weaker than a test based on a transferable specification of the attraction variables would be. However, the procedure used here provides a means of testing for such violations of the logit structure as random variations in the coefficients of the time, cost, and home-specific dummy variables, and omission from the utility function of variables that may be correlated with the time, cost and home-specific dummy variables.

A utility Table

maximizing

6. Coefficients

model of the demand for multi-destination

of the destination

Variable

choice

non-work

travel

model estimated from the second data set

Coefficient

t- Statistic

LNTIMT

-2.29

-6.03

CTIINC

-1.10

-1.10

LNTIMF

-2.34

-6.53

CF/INC

-2.24

-2.22

RET

0.000359

1.51

POP

0.000301

8.98

svc

0.00403

12.8

NEWRETLO*

0.00812

10.6

NEWSVCLO*

0.00278

6.42

DHOME

4.38

6.20

Observations

507

Alternatives

13190

383

LOS likelihood with all coefficents equal to zero

-3543.51

Log likelihood with the estimated values of the coefficients

-996.2

*

NEWRETLO is defined as retail employment at a potential non-home destination if retail employment at that destination is less than 600. NEWRETLO is zero for home and for non-home destinations where retail emolovment is 600 or more. NEWSVCLO is defined as service employment . , at a potential non-home destination if service employment at that destination is less than 900. NEWSVCLO is zero for home and for non-home destinations where service employment is 900 or more.

zero at the 0.05 level, thus reconfirming the hypothesis used in developing the model that travelers think ahead when structuring tours. Also, as in the original model, it is not possible to reject the hypothesis that the re-estimated coefficients of the two travel time variables are equal and that the re-estimated coefficients of the two cost variables are equal. In tests of the hypotheses that the coefficients of the time, cost, and home-specific dummy variables in the original and re-estimated models are equal, it was found that the hypothesis of equality of the coefficients of the time and home-specific dummy variables could not be rejected at the 0.10 level. The hypothesis of equality of the cost variables was rejected at the 0.10 level but not at the 0.02 level. Thus, the hypothesis that these coefficients are equal and that, apart from errors in the specification of the attraction variables, the logit structure of the destination choice model is correct can neither be accepted nor rejected conclusively. In interpreting this result, it is useful to take two additional considerations into account. First, in both estimations of the destination choice model, it was found that the cost coefficients were sensitive to the specification of the attraction variables. Thus, the test of the ability of the logit model to satisfactorily describe destination choice may be confounded by uncertainty as to the correct specification of the attraction terms of the utility function. Second, it was found that the differences between the attraction terms of the original and re-estimated models have a much greater effect on destination choice probabilities computed from the two models than do the differences in the coefficients of the time, cost, and home-specific dummy variables. When destination choice probabilities were computed using a number of different sets of alternative destinations, it was found that the differences between the two sets of attraction terms caused the choice probabilities to change by as much as 0.60, whereas the differences between the two sets of time, cost, and home-specific dummy terms caused changes of at most 0.12. Thus, although the

JOEI.HOROWITZ

384 Table 7. Coefficients

of the functions

Variable

COnSt.3!3t

Gn and Cl-G,) estimated from the second data set

Coefficient for GO

-0.987

Coefficient for G1 - GO

-0.654

PFO

2.23

0.0428

ALD

0.763

-0.00287

INC

0.00153

0.000426

775 Observations R2= 0.088

tests described here have not given a clear indication of whether multinomial logit is the correct structural model for destination choice forecasting, it does seem clear that errors and uncertainties in the specifications of the destination attraction variables are likely to be sources of substantial forecasting error. The sojourn frequency model (eqn (35)) also was re-estimated using the second data set. The estimation results are shown in Table 7. A test of the hypothesis that the coefficients of the original and re-estimated sojourn frequency models are equal resulted in rejection of the hypothesis at the 0.01 level. The lack of equality of the two sets of coefficients is associated mainly with the constant terms of the function Go and G, -G, and, to a much lesser degree, with the coefficient of ALD in G, - G,,. Specifically, the hypothesis that all of the coefficients of the original and re-estimated models except the constant terms are equal was not rejected at the 0.05 level. The hypothesis that all of the coefficients except the constants and the coefficient of ALD in G, - G, are equal was not rejected at the 0.10 level. Moreover, when sojourn frequencies were computed using the two models, it was found that the differences between the two models’ constant terms caused changes in estimated sojourn frequencies of roughly a factor of two, whereas the differences among the remaining coefficients caused changes of less than 15%. These results can be interpreted by noting that the constant terms in Go and G, represent the composite mean contributions to the travel utility function of variables that affect sojourn frequency but that are not included among the explanatory variables of the sojourn frequency model. If the values of the omitted variables are substantially different in two data sets, then the constant terms of sojourn frequency models estimated from the two data sets also will differ, even if the general structure of the sojourn frequency model (i.e. eqn 35) is correct. On the other hand, if the structure of the sojourn frequency model is seriously erroneous, then changing the estimation data set will tend to cause the estimated values of all of the coefficients, not only the constant terms, to change. Therefore, the finding that the constant terms of Go and G, - Go are the main sources of differences between the original and re-estimated destination choice models suggests that the structural model represented by eqn (35) may be satisfactory but that one or more variables that significantly affect sojourn frequency and that vary systematically among households have been omitted from the utility function specification. This underspecification of the utility function is a potentially serious source of forecasting error. 6CONCLUSIONS

In this paper a structural model of the demand for multi-destination non-work travel has been developed, and an empirical illustration of the model has been presented. The structural model is based on the principle of utility maximization and differs from previous utility maximizing models of multi-destination non-work travel in several important ways. In contrast to most previous models, the model presented here incorporates travel frequency, destination choice and mode choice for both single and multi-destination travel into a unified utility-maximizing framework. The model includes a representation of the demand for travel between individual origin-destination pairs but avoids the need for enumerating complete

A utility maximizing model of the demand for multi-destination non-work travel

385

Finally, the model incorporates the concept that current travel decisions depend on past travel decisions and future travel plans, as well as on current conditions. The empirical tests of the model have produced encouraging results. However, the power of these tests to distinguish between correct and incorrect models is highly variable; the tests can reject models that make relatively minor errors while failing to reject models that make serious errors. Thus, the test results by themselves do not provide a basis for reaching firm conclusions concerning the validity of the model’s structure. However, the test results together with the model’s theoretical merits and analytical tractability clearly indicate that further experimentation with the model, possibly including its application in practical forecasting, is warranted. The empirical tests also have indicated that inadequacies in the sets of destination attraction and household socioeconomic variables that were used in the empirical model have considerable influence on the forecasts of travel behavior that the model produces. The variables that were used here are typical of those normally available in transportation data sets. Their inadequacy indicates that modeling non-work travel behavior will remain a difficult task until improved sets of explanatory variables are available. There is a need to identify and develop data concerning destination attraction variables that are more closely associated with travelers’ choice processes than the zonal aggregate population and employment variables that currently are in widespread use owing to their availability. There also is a need to identify additional explanatory variables for travel frequency, the empirical tests performed here having shown that there are large systematic differences among households’ non-work travel frequencies that cannot be explained by the standard household size, income, and automobile availability variables, even when differences in households’ accessibilities to attractive non-work destinations are taken into account. travel patterns.

Acknowledgements-I

thank Franklin Ching and Joseph Ossi for their assistance in preparing the data and performing the computations associated with the empirical model. Much of the work described in this paper was performed while I was a Visiting Associate Professor in the Department of Civil Engineering of the Massachusetts Institute of Technology. The views expressed in this paper are mine and are not necessarily endorsed by the Environmental Protection Agency.

REFERENCES Adler T. J. (1976)Modeling Non-Work Travel Patterns. Ph.D. dissertation, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts. Adler T. and Ben-Akiva M. (1979)A theoretical and empirical model of trip chaining behavior. Transpn Res. 138,243-257. Adler T. J. and Ben-Akiva M. (1976)Joint choice model for frequency, destination and travel mode for shopping trips. Transpn Res. Rec. No. 569, pp. 136-150. Beale E. M. L. (l%O) Confidence regions in nonlinear estimation. J. R. Stat. Sot. B22,41-76. Ben-Akiva M. and Atherton T. J. (1977)Methodology for short-range travel demand predictions. J. Transport Eon. Policy 2, 226261.

Ben-Akiva M. and Richards M. G. (1976)Disaggregate multimodal model for work trips in the Netherlands. Transp Res. Rec. No. 569, 107-123. Ben-Akiva M., Sherman L. and Kullman B. (1979)Disaggregate travel demand models for the San Francisco Bay Area: non-home-based models. Transpn Res. Rec. No. 673, 128-133. Charles River Associates (1976). Disaggregate Travel Demand Models, Vol. 2. Report prepared for the National Cooperative Highway Research Program, Transportation Research Board, Washington, D.C. Domencich T. A. and McFadden D. (1975) urban Travel Demand. North Holland/American Elsevier, New York. Hausman J. A. and Wise D. A. (1978) A conditional probit model for qualitative choice: discrete decisions recognizing interdependence and heterogeneous preferences. Economettica 46.403-426. Horowitz J. (1979) Disaggregate demand model for non-work travel. Transpn Res. Rec. No. 673.5671. Lerman S. R. (1979)The use of disaggregate choice models jn semi-Markov process models of trip chaining behavior. Transpn Sci. 13, 273-291.

McFadden D. (1979) Quantitative methods for analyzing travel behavior of individuals: some recent developments. Behavioral Travel Modelling (Edited by Hensher D. A. and Stopher P. R.). Croom Helm, London. McFadden D. (1974) Analysis of qualitative choice behavior. Frontiers in Econometrics (Edited by P. Zarembka). Academic Press. New York. McFadden D., Train K. and Tye W. B. (1977) An application of diagnostic tests for the independence from irrelevant alternatives property of the multinomial logit model. Trunspn Res. Rec. No. 637, 39-46. Parody T. E. (1977) Analysis of the predictive qualities of disaggregate modal choice models. Transpn Res. Rec. No. 637, 51-56. Sheffi Y. (1979) Estimating choice probabilities among integer alternatives. Trunspn Res. 138, 189-205. Spear B. D. (1977) Applications of New Travel Demand Forecastina Techniaues to Transoortation Plannina. U.S. Department of Transportation, Washington, D.C. Urban Mass Transportation Administration (1976) Urban Transportation Planning System: Reference Manual. Washing-

ton, D.C.

JOEL HOROWITZ

386

APPENDIX

Interpretation

of Rkj.m In eqn (25) R, is defined as the inverse of the matrix whose ij element is

&j

-

(exp W$J/ f exp Wi&.

(25)

Define A,,, to be the matrix whose ij element is A,,

2 (exp

W$J/ Z exp WQ,,. k

641)

and note that A,, is the probability that a trip originating at location i and using mode m terminates at location j. Using (Al), R,,, can be defined by the matrix equation R, =(1-A,)-‘,

W)

where I is the identity matrix. Thus, R, can also be expressed as the series R, = Z + : (A,)“.

ll=I

(A3)

However, A,,, also can be interpreted as the transition matrix of a Markov chain. The single-step ij transition probability of the chain is the probability that a trip originating at location i and using mode m terminates at location j. It follows that Rkj,, is the conditional mean occupation time of state j of the Markov chain, given that state k was the initial state. Because the states of this Markov chain correspond to geographical locations and the transitions correspond to trips by mode m, Rkj,, is the mean number of sojourns at location j that follow a sojourn at location k during a tour that uses mode m. The presence of the term I in eqn (A3) implies that this mean includes the initial sojourn at location k if j = k.