Transpn Res..B Vol. MB, pp. 243-257 @ Pergamon Press Ltd.. 1979. Printed in Great Britain

A THEORETICAL AND EMPIRICAL MODEL OF TRIP CHAINING BEHAVIOR

THOMAS ADLER

Resource Policy Center, Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, U.S.A.

MOSHE BEN-AKIVA

Transportation Systems Division, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.

(Received 21 April 1971; in revised form 19 December 1978)

Ahstraet-This paper addresses the theoretical and empirical issues involved in modeling complex travel patterns. Existing models have the shortcoming of not representing the interdependencies among trip links in trip chains with multiple non-home stops. A theoretical model based on utility theory and explicitly accounting for the trade-offs involved in the choice of multiple-stop chains is developed. Using this theoretical model, utility maximizing conditions for a household’s choice of a daily travel pattern are derived. The optimum travel pattern is described in terms of the number of chairs (tours) traveled on a given day and in terms of the number of stops (sojourns) made on each of those chains. For a given household, the form of the optimum pattern is a function of the transportation expenditures (time, cost) required to reach potential destinations. Constraints on the conditions of optimality due to the limited and discrete nature of travel pattern alternatives are also considered. Parameters of the general utility function were estimated empirically using actual travel data derived from a home interview survey taken in Washington, DC. The multinomial logit model is used to relate utility scores for the alternative travel patterns to choice probabilities. The resulting parameter estimates agree with theoretical expectations and with empirical results obtained in other studies. In order to demonstrate the empirical and theoretical implications of the model, forecasts for various transportation policies (e.g., gasoline price increases, transit fare reductions), as made by this model and by other less complex models, are compared. The results of these comparisons indicate the need for expanding the scope of existing travel forecasting models to explicit considerations of trip chaining behavior.

INTRODUCTION

Several studies have noted that households’ non-work travel is composed of relatively complex patterns of interdependent travel activities (Adler and Ben-Akiva, 1976; Vidakovic, 1972; Gilbert, 1972). A relatively large number (30%) of non-work trip links in one urban area were found to be components of travel tours which included more than one non-home sojourn. Different behavioral approaches to modeling these complex travel tours have been suggested but not yet empirically tested (e.g. Hensher, 1976, Hanson, 1977). Approaches that have been oriented toward empirical application (e.g. Kobayashi, 1976; Sasaki, 1976; Horowitz, 1976) have contained important simplifying assumptions as described in Adler, 1976. The models which are currently used to forecast travel demand, in particular, assume that multiple-sojourn tours can be represented in an overly simplified way. The Urban Transportation Modeling System (UTMS) has separate models for each type of trip link and, therefore, the individual trip links on a round trip tour are assumed to be chosen independently. Most of the disaggregate models developed to date (Adler and Ben-Akiva, 1976; Charles River Associates, Inc., 1972; Ben- Akiva, 1973) are estimated only for choices involving single-sojourn tours. However, in order to allow the formulation of reasonable transportation policies which affect non-work travel, it is important that the determinants of non-work travel patterns that include multiple-sojourn tours be better understood. In this paper, both behavioral theories and empirical models which were developed to further that understanding are described. The model is used in an empirical case study to demonstrate the effects of various simplifying assumptions on travel forecasts.

A BEHAVIORAL THEORY OF TRAVEL PATTERN CHOICES

Definitions Several terms describing travel activities are used in this paper. The most elemental unit of

travel activity is the sojdum, which is a visit at a place remote from home. The activity which an individual pursues during a sojourn can be either work or any of the many other

243

244 T. ADLER and M. BEN-AKIVA

non-work purposes; multiple activities may be pursued jointly on a single sojourn. The travel movement which carries an individual between his home and a sojourn or between temporally consecutive sojourns is called a trip link. A setof consecutive trip links which begin and end at an individual’s home is called a trip tour. Finally, the most inclusive description of travel activity is the term, travel pattern, which is used here to refer to the set of trip tours made by an individual or household within a fixed time period (usually 24 hr).

There are three basic factors which determine travel activity: the household, the trans- portation system, and the activity system. The household is defined as one or more individuals living at the same address and making joint economic decisions. While each element of the household’s travel pattern may involve a single individual, the travel activity is assumed to be motivated by needs (activities) which are allocated at the household level among individuals. Thus, the household is assumed to be the unit around which travel-related decisions are organized, and as such is referred to as a decision unit (with respect to travel).

The transportation system is described in terms of level-of-service attributes. Each trip link has a level-of-service which is a function of the time, cost, comfort, and other attributes of the mode used and of the route traveled.

The third component of travel is the activity system, used here to describe the pattern of economic and social activities of a region. Each sojourn is made at a spatial location, accessed by a transportation mode, and called an activity site or travel destination. The activity site is assumed to include all opportunities which are accessed without additional travel during a single sojourn.

Conceptual descriptions of travel activity Given a household with known socioeconomic characteristics and fixed mobility status,t the

problem to be addressed is how that household chooses a travel pattern; in particular, the specific choices of tour frequency, sojourn frequency, mode, and destination.

There is a considerable amount of literature attempting to explain households’ travel as related to the spatial distribution of needs (Hensher, 1976; Hanson, 1977; Kobayashi, 1976). One body of literature, the central place theories, uses hypotheses of travel behavior in order to provide an explanation for the observed spatial arrangement of supply outlets. The simplest assumption is, of course, that the household will seek to minimize travel costs in satisfying its temporally separated needs. If this assumption is used, and if individuals are allowed the alternative of combining sojourns into multiple-sojourn tours, the multiple-sojourn alternative would always dominate the other alternatives. In order to overcome this shortcoming, one study (Baumol and Ide, 1956) suggested the use of a value function in which the costs of travel, the probability that the consumer will find what he is seeking, and the opportunity costs of destinations which are passed, are weighted together. This type of formulation is appropriate in describing an individual’s pursuit of a single need or activity (e.g., shopping for a particular item of furniture) but does not account for the joint pursuit of more than one activity on a single travel tour (e.g. traveling to a grocery store and to a bank on the same tour).

A more recent study (Westelius, 1972) in which a descriptive simulation model of patterns of urban travel is developed, suggests an alternative representation of the household’s travel choice process. The household has a set of needs each of which accumulates at some rate over time. For each of these needs, there is a threshold level at which point a journey to accomplish that need is triggered. The threshold levels depend, however, on the travel times and costs required to satisfy the needs. Thus, a household which has poor access to goods and services is likely to allow needs to accumulate to higher levels than would a household with more convenient access. For a given household, the threshold level for any need can be affected by travel which is initially triggered by another need. For example, in traveling to work, an individual may be able to make a side trip with only very small additional expenditures of time and money. Thus, the individual may combine needs into a single tour in order to reduce the total travel required to satisfy those needs. The desire to combine needs into a single travel tour, however, is affected by the levels to which the needs have accumulated at that point in time.

tMobility status, as used here, refers to long-run household choices such as residence location, workplace location, and automobile ownership.

A theoretical and empirical model of trip chaining behavior 245

In terms of the dynamics of the decision-making process as described above, it is clear that many of the household’s travel choices may be made jointly. The choice of whether or not to pursue an activity on a given day depends on the choices of mode and destination for the trip. Choices involving the number of travel sojourns, the arrangement of those sojourns into tours, the location of activity sites for the sojourns, and the modes used to access the activity sites, are all affected by the level of travel expenditure required. In turn, the required travel expenditures vary across each of these dimensions of choice. Thus, the household’s travel pattern decisions are represented, by this behavioral theory, as being interdependent choices. Any assumptions of independence among the choices are, therefore, simplifications of this theory which should be subjected to careful examination.

Factors affecting the travel pattern choice The basic hypothesis is that a household develops needs for non-home activities, and

balances the desire to meet each need as it arises with the transportation expenditures required in travel. These trade-offs involved in the household’s comparison among alternative travel patterns can be expressed in terms of utility theory. A household is viewed as selecting the travel pattern from which it derives the greatest utility (or satisfaction) subject to time and money budget constraints. The utility to the household from a given travel pattern can be expressed as follows:

U (travel pattern) = f(SC, AT, DA, Z, SE)

where

SC = scheduling convenience of the arrangement of sojourns and tours AT = net non-home activity duration (excluding travel time)

Z = remaining income after travel expenses DA = attributes of the set of destinations (activity sites) in the travel pattern SE = socioeconomic characteristics of the household.

Scheduling convenience represents the degree to which a travel pattern fits the schedule of household activities. This can be expressed in terms of two different effects. The first comes from the way that a household allocates a given number of activities among activity sites. At one extreme a household may choose to engage in each activity at a separate activity site. At the opposite extreme, all activites may be pursued on a single sojourn at one activity site. For example, a grocery store, a bank, and a clothing store may be visited by a household either separately, in different areas of the city, or together in a single shopping center. Presumably, strictly from the perspective of scheduling convenience (i.e. independent of travel expen- ditures), maximum utility would be attained by pursuing each activity at the site most appropriate for that activity. A measure of the number of sojourns included in the travel pattern will, in general, capture this effect. A larger number of sojourns, indicating a higher degree of separation of activities, will be associated with a higher utility.

The second effect, which is represented by the schedule convenience term in the utility function, comes from the arrangement of sojourns into tours. A household may pursue each of its needs, as it arises, in a separate tour for that particular need alone. On the other hand, a household may attempt to temporally advance some needs and delay others in order to be able to make a single tour which includes sojourns for all of the needs. For example, at different times of the day (and possibly by different individuals) separate tours could be made to buy gasoline for the car, to visit a friend, and to purchase a hardware item. Alternatively, the three needs could be combined into a single tour at a time when all could be pursued conveniently. Presumably, and again strictly from the perspective of scheduling convenience, maximum utility would be attained in a travel pattern where each activity is pursued separately (as it arises in the course of time) in single-sojourn tours. There are two possible exceptions to this; (1) when two or more needs correspond exactly in space and time or (2) when activities are necessarily sequential, e.g. a trip to the bank to get money, followed by a shopping trip. The model developed here assumes that these represent only a small fraction of all travel.

244 T. ADLER and M. BEN-AKIVA

The amount of time spent at destinations away from home is an important component of the travel pattern’s utility. The demand for travel is derived from the desire to pursue activities away from home; non-home activity time represents the utility obtained from those activities. Income remaining after travel expenses represents money which can be used to purchase desired items or service.

The attributes of the set of destinations chosen in the travel pattern affect its utility. The set of destinations is assumed to be characterized by attributes such as the quality and relative costs of the services available. Higher utility is attained for travel patterns which include destinations that are perceived as the most appropriate for pursuing a given activity.

Socioeconomic characteristics of the household affect its needs and preferences for different travel alternatives. A household which does not own an automobile will generally not have a car available for a given trip. Similarly, residence location affects the availability of the transit mode. Households with higher incomes are less likely to be sensitive to travel costs than would lower income households. In similar ways, other attributes of the household such as its size, education level, life cycle, etc. could cause preferences which are different from those of other households.

Scheduling convenience can be represented by the two effects discussed above; the number of sojourns per tour indicating the relative disutility of combining sojourns into multiple- sojourns tours, and the number of sojourns in the travel patterns representing the additional utility gained from pursuing different activities at different spatial sites. Destination attraction can be expressed as a function of the number of activity sites visited, or the number of sojourns in the travel pattern.

For a given household the utility function can thus be simplified to:

U (travel pattern) = WT, S, AT, Z)

where

T = number of tours in the travel pattern S = number of sojourns in the travel pattern.

The household will derive greatest utility by choosing a travel pattern (characterized by S, T, AT, and Z) which maximizes this utility function subject to four basic constraints: S 3 T and T 2 1, definitional constraints; (2a, - bJT + b,S + AT = AB, travel time budget constraint; (2% - b,)T + b,S + Z = Y, income constraint; Y, is the total income of the household; TB, is the total allocated non-home activity duration (including travel time); a,, &, are the average travel times and travel costs, respectively, of home-based trip links;? bt, b, are the average travel times and travel costs, respectively, of non-home-based trip links. In the situation where only the time budget and income constraints are binding, the Kuhn-Tucker conditions can be used to determine the optimal choice of S and T.

The Lagrangian for this optimization problem is:

L = U(trave1 pattern) + p[(2at - b3T + b,S + AT] + h[(2a, - b,)T + b,S + Z]

g = 2 + ~(24 - b,) + h(2a, - b,) = 0

$=$+pb,+Ab,=O

tNote that the total number of home-based trip links in a travel pattern equals 2T while the number of non-home-based trip links equals S-T, so that the total time and cost of the travel pattern can be described as in the time budget and income constraints listed above.

A theoretical and empirical model of trip chaining behavior 241

Using these first-order conditions:

au au /

Ch - b,) + A(& - b,) aT bs=CL pb,+ Ab,

= p/A (2a, - bJ + (2% - b,) plAb,+b,

=2dAat+ac _1

plAb,+b, -

When U is held constant, this implies that the marginal rate of substitution is

This expression has a direct interpretation. The term p/A represents the marginal rate of substitution between time and money and thus (F/A&+ &) could be interpreted as the generalized cost of a home-based trip link. Similarly, (p/Ah, + b,) is the generalized cost of a non-home-based trip link. If the ratio of these generalized costs is greater than one-half, changes in T will be accompanied by opposing shifts in S. If the ratio is less than one-half, i.e. non-home-based trips are more than twice as expensive as home-based trips, changes in T and S will be in the same direction. Thus, depending on the relative generalized costs of home-based and non-home-based trip links, policy changes which affect the number of tours made in a household’s travel pattern may either increase or decrease the number of sojourns made by the household.

Some of the assumptions of this theoretical model are subject to criticism. For example, the relationship between travel time and travel cost varies considerably both across different modes and across the urban area. Particularly for transit systems which have “per-trip fares”, the relationship between travel times and travel costs will be quite different for different trips and thus cannot necessarily be represented by a single function of S and T. In general, the travel expenditure function would have to be quite complex in order to accurately represent the transportation service in a real urban setting. This does not invalidate the previous statements concerning the behavioral implications of a utility function as given above. Rather, it indicates that the results can be sensitive to a variety of factors in an urban area’s transportation service. Thus, it is important that the theory not be used as an absolute statement of expected travel behavior, but in all cases be subject to empirical verification. One such attempt at empirical verification of the theory is briefly described in the remainder of this paper.

The model A JOINT CHOICE MODEL OF HOUSEHOLD TRAVEL PATTERNS

The purpose of this model is to explain the household’s choice of a non-work travel pattern in terms of the attributes of the household and of the available travel pattern alternatives. It is assumed that the mobility choices of the household (i.e., residence location, work place. location, work mode choice, and auto ownership) are given; that is, the non-work travel choices are made in the short run and are conditional on the longer run choices of the household. Non-work travel is defined as travel for all purposes except to the workplace, to school, or to work-related business. Explicitly included in the non-work category is travel to a non-work sojourn (e.g. to shop) on the way to, or from a work activity.?

The basic choice alternative which is being modeled is the daily household travel pattern.

tFor non-work travel associated with the work trip, the mode is assumed to be constrained to that which is chosen, as a longer-run decision, for the journey to work. Travel times and costs are assumed to be those incurred above the times and costs of the simple home to work to home tour.

248 T. ADLER and M. BEN-AKIVA

The travel pattern is described by the number and characteristics of destinations chosen for non-work activities, the modes used to travel to those destinations, and by the number of tours used to travel to the set of destinations.The household is assumed to choose a single travel pattern based on its attractiveness relative to other possible travel patterns. While there are many individual choices which comprise the travel pattern choice (e.g. modes, destinations), they are assumed to be highly interdependent and are thus represented by the single joint choice of a complete travel pattern.

Assuming that the attractiveness of each alternative can be represented by a function Vtp, the choice probability of any single alternative, tp, from a set of alternatives, TP, can be expressed as:

P(tplTP) = eVtp

2 c”Q tp’ E TP

where P(tp(TP) is the probability that the alternative, tp, will be chosen from the set, TP. This mathematical form is called the multinomial logit model, discussed in detail by McFadden (1%8). There are, of course, other mathematical forms which associate choice probabilites with attributes of the alternatives (e.g. the linear probability model, the probit model). The multi- nomial logit model, however, has the advantage of representing reasonable (and, in most cases, testable) hypotheses about choice behavior while remaining tractable for empirical estimation and forecasting.

Assumptions of the travel pattern model The travel pattern model as developed

The most fundamental assumption is that which the alternatives that are considered

here incorporates several behavioral assumptions. travel arises from a household choice process in are complete daily travel patterns.This behavioral

perspective is discussed earlier. There are, however, alternative ways of I’epresnting the choice process. It would be possible to contruct a model based on choices of individuals, rather than of the household. Within this type of model framework, though, it is difficult to explicitly represent household level decisions such as the allocation of automobiles and of travel-related respon- sibilities.

Another alternative to the chosen approach would be to represent the travel pattern choice as a set of conditional choices. Using this approach, the household might be assumed to first choose the number of sojourns to be made, then, conditioned on that choice, the travel destinations, then the travel modes, and so on. Each choice would be modeled separately, resulting in a set of models of conditional and marginal choice probabilities which together could be used to express the choice probability for any type of travel pattern. Unfortunately, the set of models which assumes that the choices are made in one order will result in coefficient estimates which are different (and will result in different forecasts) from models which assume an alternative sequence of choices (Ben-Akiva, 1973). Thus, if such an approach is used the particular sequence of choices which is assumed must be defended over alternative sequences. From the earlier discussion, however, it should be clear that there is no natural sequence of component choices which lead to the travel pattern choice. Rather, choices, such as the number of sojourns, the number of tours, and the set of destinations to be visited, are interdependent and are compared jointly as travel pattern alternatives.

The choice set for estimation Obviously, a household has many possible travel patterns from which to make a choice.

Assume, for example, that a household decides to make a maximum of two sojourns on a given dayt and that it has twenty possible destinations, and three modes from which to choose. Assume further that a single mode is used for each tour, that all modes are available to all destinations, and that the order in which the destinations are visited is not important. The set of travel patterns which is available to this household consists of the following. First, the

tin the Washington, DC. data, over 30% of the households which make non-work trips choose more than two sojourns.

A theoretical and empirical model of trip chaining behavior 249

household may choose not to travel, so this constitutes one alternative. If the household decides to make only one sojourn, it can choose among 20 different destinations, each of which could be visited by any of three modes, making 60 possible travel pattern alternatives. If the household makes two sojourns, it can either do so using a single tour or using two separate tours. Using a single tour, the household has 180 two-destination combinationst from which to choose, and each tour can use any of the three modes, making a total of 540 possible alternatives. If the two sojourns are visited separately, each may use a different mode, and thus there are 3* modal combinations to correspond to each of the two-destination combinations, or 1620 possible alternatives. This means that, even given limitations on the choice set, the household has a total of 2171 possible travel patterns from which it can choose. While it is inconceivable that any household would systematically evaluate each of these alternatives, it is difficult to know a priori which of the alternatives would be eliminated from consideration by the household.

Fortunately, it is not necessary to enumerate the complete choice set in order to obtain consistent coefficient estimates for the multinomial logit model (Manski, 1973). All that is needed is information about the chosen alternative and any other relevant (i.e. available, non-zero probability) alternatives.

For this model, the household’s alternative set for estimation was constructed using the chosen travel patterns of other households. For each traffic district represented in the estima- tion sample, a pool of alternatives was formed consisting of the chosen travel pattetns of all surveyed households living in that district. Each household in the estimation sample was then allowed, as alternatives to its chosen travel pattern, a subset of the alternatives from the pool of travel patterns represented by its home district. Households not owning an automobile or without transit service were not allowed alternatives which required the unavailable mode. In addition, all households which made trips for non-work purposes as part of their recorded travel pattern were allowed the no-travel alternative.

Estimation resdts A summary description of the variables used in the model is given in Fig. 1. Three variables

are used to represent the schedule convenience of the travel pattern. The first two, the number of sojourns per tour and the total number of sojourns across the household’s travel pattern, have already been discussed. A third variable which has not been mentioned previously is formulated as the fraction of tours containing non-work sojourns which are associated with the journey to work. If a household makes only one non-work sojourn, and that sojourn is an intermediate stop on the way to a home from work, the variable takes its maximum value of one. If none of the non-work sojourns in the travel pattern are combined with the work trip, the variable takes on its minimum value of zero. The reason this variable is used is that non-work sojourns made on the work trip are generally constrained in both time-of-day and duration, due to the fixed work hours of most people. The travel times and costs which are used in the model for these sojourns, however, are computed as only incremental values beyond the normal times and costs of the work trip (which would have been made anyway). Based only on these relatively low times and costs for non-work trips ,associated with the work trip, the most attractive travel patterns would almost always be those with as many non-work sojourns as possible made on the work trip. However, the variable representing the fraction of such tours in the travel pattern should indicate their relative disutility in terms of schedule convenience.

Three variables are used in the model to capture the transportation levels-of-service of the travel pattern. Out-of-vehicle travel time is divided by travel distance so that its effect is diminished for higher travel distances, indicating less sensitivity to walking and waiting time for longer trips. The logarithmic transformation of total travel time is used in order to capture its assumed diminishing marginal utility. That is, as travel time increases, the added disutility of an additional unit of travel time is assumed to decrease.

The third element of level-of-service, out-of-pocket travel cost, is included in a composite variable. The composite variable is the estimated “remaining” income of the household. This is

there are 2d f 2 two-destination combinations, of which 20 involve two visits to the same destination and thus, by definition, do not have two sojourns.

250

Sojourns per tour

Sojourns

Fraction work tours

OVlT/Distsnce

Total travel time

Remaining income

Autos owned

Autos remaining

Fraction shared-ride

Fraction transit

Fraction auto

Retail employment density

Service employment density

Fraction park area

Fraction CBD

Area

No-travel Constant

Household income

Non-workers

Home zone retail employment density

Home zone park area

T. ADLER and M. BEN-AKIVA

the number of sojourns divided by the number of tours in the travel pattern

total number of sojourns in the travel pattern

fraction of total hours in travel pattern which have non-work sojourns on a work-based tour

out-of-vehicle travel time divided by total travel distance (minutes per hundredth of a mile)

out-of-vehicle travel time plus in-vehicle travel time WJTT) w.nutes)

gross yearly income minus: federal taxes, state taxes, $800 per auto, insurance costs for the autos, yearly work travel costs, and 250 time the out-of-pocket costs for the non-work travel pattern (i.e. yearly non-work travel costs) (dollars per year)

total number of automobiles owned by the household if the travel pattern uses auto; 0,othewise

autos owned minus autos used for work trips, if the travel pattern uses auto and if there are licensed non-wxkerqO, otherwise

fraction of total trip links in the travel p&tern made by shared-ride

fraction of total trip links made by transit

fraction of total trip links made by auto

number of retail employees divided by total land area of zones in travel pattern (employees per tenth acre)

number of service employees divided by total land area (employees per tenth acre)

fraction of total land area in zones devoted to public park land

fraction of total sojourns in travel pattern made to CBD area

total land area of zones in travel pattern (in tenths of an acre)

1, for no-travel alternative; 0, otherwise

total income (code*), for no-travel alternative; 0, otherwise

number of non-workers in the household, for no-travel alternative; 0, otherwise

retail employees per tenth acre in home zone, for no-travel alternative; 0, otherwise

fraction park area in home zone, for no-travel alternative; 0, otherwise

*Income code (in 1968$)

l= 0 - 2,999 4 = 6,000 - 7,999 7 = 12,000 - 14,999 10 = 25,000+ 2 = 3,000 - 3,999 5 = 8,000 - 9,999 8 = 15,000 - 19,999 3 = 4,000 - 5,999 6 = 10,000 - 11,999 9 = 20,000 - 24,999

Fig. 1. Variables used in model

computed by subtracting from the household’s gross income the following expenses: federal and state income taxes, auto ownership costs (insurance, registration, taxes), an eight hundred dollar living expense (for clothes, food, etc.) per household member, and work commutation costs. The resulting disposable income is the amount of money which is left to be allocated among discretionary activities such as non-work travel. Yearly travel costs for the travel pattern (based on 250 weekdays) are then subtracted from the disposable income figure and the logarithm of this value is used as the composite variable. Behaviorally, this variable represents the effects of non-work travel costs relative only to disposable income (instead of to gross income). In addition, use of the logarithm of that value assumes diminishing marginal utility of additional units of disposable income.

Several mode-specific variables are used in the utility function for the model. Three variables specify fractions of the links in the travel pattern which used the three major modes:

A theoretical and empirical model of trip chaining behavior 251

auto driven alone, auto shared ride, and transit. These variables reflect the added utility or disutility (not captured in the level-of-service variables) of those modes relative to each other and to the other possible modes (walk, taxi, etc.). Two additional variables are included in travel patterns using one of the household’s autos. The first one, auto ownership, reflects the overall availability of automobiles for non-work travel. A second variable is .added, however, to represent competition for the household’s automobiles during the daytime (when most non- work trips are made). Autos used for household work trips are subtracted from the total auto ownership in order to determine the number of automobiles left at home during the day. If a household has licensed drivers who do not work during the day, this value is used; otherwise the variable takes on a value .of zero.

Destination attraction is represented in the specification by five variables. Total land area of the destination zone (or of the set of destinations if there is more than one visited in the travel pattern) is used to capture the effects of grouping of spatial alternatives.? Other variables which are used to describe the destination attractiveness are: retail employment density, service employment density, and fraction of the total area devoted to local, state, or national parks. These represent three types of activity to which non-work trips might be attracted. In addition to these variables, the fraction of destinations in the travel pattern which are Central Business District (CBD) zones is included to represent the unique attraction of the Washington CBD area.

The remaining five variables in the base model relate to characteristics of the household which influence the choice between making and not making non-work trips on a given day. A constant, equal to one for the no-travel alternative and zero otherwise, accounts for the residual unmeasured utility of no-travel. Household income and the number of non-workers in the household are used to capture the effects of these characteristics on the likelihood of the no-travel alternative. Finally, two variables; retail employment density and fraction of park area in the home zone, are included to indicate the suitability of the residence location to substitution of intrazonal walk trips (which are not included in the travel diaries) for vehicular non-work trips.

Estimation results for this model are shown in Table 1. All of the parameters have the

Table 1. Estimation results-travel pattern model

Coeffi- Standard T-sta- Variable cient Error tistic

Sojourns per tour -.314 Sojourns .434 Fraction work tours -2.22 OVTT/distance -.536 In (total travel time) -2.05

In (remaining income) 1.15 Autos owned .616 Autos remaining .492 Fracti,on shared-ride -.146 Fraction transit 1.08

Fraction auto -.466 Retail employment density .0823 Service employment density .00962 Fraction park area 2.60 Fraction CBD -.133

In (Area) -616 No-travel constant 1.59 Household income .713 Non-workers -.447 Home zone retail employment density .843

Home zone fraction park area 1.54

Number of observations = 1,003 Number of alternatives = 16,421 % right = 65 P2 = .47 L*(O) = -1426 L*(8) = -756

.117

.0520

.301 153

:122

.0883 262

:312 .445 .441

.494

.0526 00377

:540 .398

.0866 891

:127 .121

1.26

2.24

-2.7 8.3

-7.4 -3.5 -16.8

13.0 2.4 1.6

-0.3 2.4

-0.9 1.6 2.6 4.8

-0.3

7.1 1.8 5.6

-3.7

0.7

0.7

tThe natural logarithm of this variable is used in order to make the model consistent for different levels of aggregation and spatial alternatives (Lerman, 1975).

252 T. ADLER and M. BEN-AKIV!,

expected signs and all but those of five relatively minor variables representing alternative- specific effects are significant at the 95% level. The coefficient of the sojourns per tour variable, as expected, is negative with a reasonably small standard error. Number of sojourns in the tour has a positive effect and its coefficient is highly significant. The fraction of total non-work tours associated with a work trip has the anticipated negative effect on the travel pattern utility, and its parameter estimate also has a very small standard error. Thus, the hypotheses concerning schedule convenience in the travel pattern choice seem to be supported by the empirical evidence.

The transportation level-of-service variables all have the expected effects. Out-of-vehicle travel time and total travel time both have negative coefficients. The logarithm of disposable income minus travel costs variable has a positive effect (and its coefficient is highly significant) indicating the utility of additional units of discretionary income and the disutility of additional travel costs. The auto-specific variables, auto ownership and autos remaining for use by licensed non-workers, both have the expected signs (and similar magnitudes) indicating their positive effects on auto mode choice and on the likelihood of making non-work trips.

Of the destination-related variables, the only insignificant one is the CBD variable, indicat- ing that outside of the effects measured in the utility function, the CBD is only slightly, if at all, different from the other destinations. The density variables of retail and service employment have positive coefficients of similar magnitudes. The fraction of park area at the destination also has a positive coefficient. These three variables indicate the increased attractiveness of destinations having the same land area but with higher levels of retail, service, and park facilities. The size variable, natural logarithm of area, has a positive coefficient which is statistically different from one.

Of the variables which are specific to the no-travel alternative, the two socioeconomic variables are the most significant according to these results. The number of non-workers in the household has a negative effect on the likelihood of the no-travel alternative. For a given household income level, the probability of no-travel increases as the household’s disposable income increases. Thus, the combined effect of the total income and disposable income variables may mean that increases in total income which result in increases in disposable income will decrease the likelihood of non-work travel on a given day. The two variables describing the residence location have coefficients which, although they have the expected signs, are not significantly different from zero.

Overall, the model parameter estimates seem quite reasonable, both with respect to the behavioral hypotheses outlined earlier and with respect to statistical considerations such as standard errors of the individual coefficients and goodness-of-fit of the set of coefficients.

AGGREGATE FORECASTS USING THE TRAVEL PATTERN MODEL

The method which is used to obtain aggregate forecasts from the disaggregate travel pattern model is similar to that described in Atherton et al. (1977). In that study, which uses disaggregate demand models to forecast effects of various carpool-related policies, a random sample aggregation technique is used. A random set of households is selected from a population, and the disaggregate model is used to forecast the choice probabilities of travel alternatives for each household. Population forecasts can then be derived by computing averages of the predicted probabilities over the households in the sample or by summing the probabilities and multiplying by the inverse of the sampling rate.

The random sample of 1003 households from the Washington metropolitan area on which the travel pattern model was estimated was used for this exercise. In order to compare the aggregate predictions of the travel pattern model to the known travel behavior of the sample, the model was first applied under base conditions (as used for model estimation). The actual travel of the sample is compared with the model predictions in Table 2. As can be seen, the model predictions replicate the actual data reasonably well with respect to a wide range of travel measures. t

I’Note that by including a sufficient number of alternative-specific constants in the model, it (or any logit-type model) could be made to perfectly replicate aggregate shares for each alternative in the data used for estimation.

A theoretical and empirical model of trip chaining behavior 253

Table 2. Comparison of base model predictions with actual travel

Actual Predicted BZISC? Base

Number of households not traveling 552 552 % 54.9 54.9

For households which made non-work trips:

Average sojourns per tour 1.509 1.487 Averaae number of tours 1.488 1.416 Average number of sojourns Average vehicle-miles traveled

Drive-alone mode share Shared ride mode share Transit mode share Other modes share Fraction of travelins households with:

1.0 = < sojourns/tour < 1.5 1.5 = < soiourns/tour < 2.0 210 = < so;ournsjtour < 2.5 2.5 = < sojourns/tour < 3.0 3.0 = < sojourns/tour < -35 3.5 = < sojourns/tour

-547 .097 .263 .018 .055 .015

591 :080 .231 .014 .051 .029

number of sojourns = 1 .417 .478 number of sojourns = 2 .276 .271 number of soiourns = 3 .146 .152

2.035 2.069 11.52 9.50 -625 -617 .136 .156 115

:124 .116 .lll

number of so;ourns = 4 .068 016 number of sojourns = 5 .026 :015 number of sojourns > = 6 .062 .064

A comparison of the base model predictions with forecasts for a variety of transportation policies is presented in Tables 3 and 4. The policies include the following:

(1) auto gasoline costs tripled, (2) a strict limitation of fifteen miles on the households’ average daily non-work auto vehicle-miles traveled,

(3) (4) (5) (6) (7) (8)

a halving of transit fares, a halving of transit out-of-vehicle waiting time, reduced transit in-vehicle times, a fixed transit fare with no transfer charges, free transit, and a combination of 1, 2, 4, 5 and 7.

In Table 3, forecasts for the different policies are described in terms of effects on the numbers of sojourns and tours made by each household. In general, the predictions correspond with the theoretical expectations described earlier. Policies which improve the transportation levels-of-service decrease the percentage of households not traveling on a given day and increase the average numbers of sojourns and tours for households which do make non-work trips. Policies, such as the VMT limitation and gasoline price increases, which reduce the households’ transportation levels-of-service, have the reverse effects. The forecasts in Table 3 indicate that, for all policies tested, the average number of sojourns per tour decreases from the base value. This result can be explained in terms of the theoretical model described earlier. The theoretical model indicates that increasing the times and costs of travel could cause either an increase or a decrease in the optimal number of sojourns per tour for a given in- dividual, depending on the number of tours in the travel pattern. In this case, policies represent-

ing both increases and decreases in travel times and costs result in decreases in the average number of sojourns per tour for the sample. The incentive policies cause individuals to choose a larger number of travel tours while disincentive policies result in fewer tours per person. Apparently, these changes in the number of tours are sufficient to cause a change in the sign of the partial derivative of the optimal value of sojourns per tour with respect to changes in times and costs.

Intuitively, the drop in sojourns per tour under travel disincentive policies can be explained in terms of the change (which results from those policies) in the distribution of the number of sojourns in households’ travel patterns. Of the households which choose to travel given disincentives, a higher fraction is predicted to choose travel patterns with only one or two

254 T. ADLER and M. BEN-AKIVA

Table 3. Forecasts of the distribution of households’ sojourns and tours

1 2 3 4 5 6 7 Base ?+UUtO Max. Transit Transit Transit Fixed

me- Cost "MT15 Cost 0"'I-T 1"TT Fare Free 1.2.4. dictions x3 Miles x.5 x.5 x.75 Transit Transit 5.7

Fraction of households

not traveling

For households which

make non-work trips:

Average sojourns

per tollr

Average number

of tours

riverage number

of sojourns

Fraction of traveling

households with:

Sojourns per tour

between

1.0 - 1.5

.549

1.487

1.416

2.069

.591

1.5 - 2.0 .OSO

2.0 - 2.5 .231

2.5 - 3.0 .014

3.0 - 3.5 -051

33.5 .029

Number of sojourns

equal to:.

1 .47S

.271

.152

-016

-015

.064

-551 .553 -548

(0.4) (0.7) (-0.2)

1.468 1.394 1.487 1.483 1.484

(-1.3) (-6.3) (0.0) (-0.3) (-0.2)

1.382 1.229 1.416 1.418 1.416

(-2.4) (-13.2) (0.0) (0.1) (0.0)

1.958 1.511 2.070 2.069 2.069

(-5.4) (-27.0) (0.0) (0.0) (0.0)

.596 .604 .591

(0.8) (2.2) W.0)

.076 .099 .079

(-5.0) (23.8) (-1.3)

.240 .244 .230

(3.9) (5.6) (-0.4)

.012 .009 .014

(-14.3) (-35.7) (0.0)

-047 .033 -052

(-7.8) (-35.3) (2.0)

.025 .006 -029

(-13.8) (-79.3) (0.0)

.486 .527 .478

(1.7) (10.3) (0.0)

.281 .296 -270

(3.7) (9.2) (-0.4)

-145 -134 .152

(-4.6) (-11.8) (0.3)

.015 -016 .016

(-6.3) (0.0) (0.0)

-013 .006 .014

(-13.3) (-60.0) (-6.7)

.055 -016 .064

(-14.1) (-75.0) (0.0)

.544 .548

L-0.9) l-0.2)

.595 .593

(0.7) (0.3)

.078 .079

(-2.5) (-1.3)

.228 .229

(-1.3) (-0.9)

.014 .014

(0.0) (0.0)

.053 -052

(3.9) (2.0)

.028 .029

(-3.4) (0.0)

.484 .478

(1.3) (0.0)

.267 .271

(-1.5) (0.0)

.152 -152

(0.0) (0.0)

.016 .016

(0.0) (0.0)

.013 .014

(-13.3) (-6.7)

.063 .064

(-1.6) (0.0)

.549

(0.0)

1.487 1.487 1.387

(0.0) (0.0) (-6.7)

1.416 1.415 1.214

(0.0) (-0.1) (-14.3)

2.070 2.070 1.479

(0.0) (0.0) (-28.5)

.591

(0.0)

.079

(-1.3)

.230

(-0.4)

.014

(0.0)

.052

(2.0)

.029

(0.0)

.47a

(0.0)

.270

(-0.4)

-153

(0.7)

.016

(0.0)

.014

(-6.7)

.064

(0.0)

.54S

(-0.2)

.591

(0.0)

.079 (-1.3)

.611

(3.4)

.092

(15.0)

.229

(-0.9)

.014

(0.0)

.243

(5.2)

.OOS

(-42.9)

.052 .037

(2.0) (-27.5)

.029 .004

(0.0) (-86.2)

.478 .537

(0.0) (12.3)

.270

(-0.4)

-153

(0.7)

.016

(0.0)

.014

(-6.7)

.295

(8.9)

.132

(-13.2)

.014

(-12.5)

.064

(0.0)

.004

(-73.3)

.013

(-80.0)

.545

(-0.7)

Note: Numbers in parentheses are percent changes from base values.

sojourns, while lower proportions choose those with larger number of sojourns (Table 3). The result of this decrease in sojourn frequency is that the travel patterns include fewer oppor- tunities for combining sojourns into multiple-stop tours. For example, with increasing costs of travel, a household, which initially chooses a travel pattern that consists of a single two-sojourn tour, may have to reduce its travel to only a single sojourn. In that case, the household has no other sojourns to combine into multiple-stop tours and thus reduces its sojourns per tour from 2 to 1. Similarly, households which initially choose larger number of sojourns face fewer opportunities for combining sojourns if they reduce the number of sojourns in their travel patterns. For example, if all households combined all of their sojourns into a single tour, the average number of sojourns per tour for the base case would be 2.13. By contrast, if the households were to combine all of the sojourns that they chose after the auto cost increases, the number of sojourns per tour could be only l.W,t a decrease of almost 7% from the base case. To summarize, travel disincentives cause individuals to reduce sojourn frequency as the primary response and, given these reductions, result in fewer opportunities to link the remaining sojourns.

tThese values are derived by multiplying the fraction of households having a given number of sojourns, as reported in Table 3, by the number of sojourns, (i.e. ah sojourns in one tour). For the category of sojourns 36, the average values of 8 for the base case and 7 for the auto cost increased case were used.

A theoretical and empirical model of trip chaining behavior

Table 4. Travel summary measures

255

1 2 3 4 5 6 7

Base Auto Max. Transit Transit Transit Fixed

P?X- cost VMT 15 cost ovrr IVTT Fae Free l-2,4, dictions x3 Miles x.5 x.5 x.75 Transit Transit 5.7

Number of trip links 1.571

Home-based 1.276

Non-hone-based .295

Number of person-miles 5.657

Auto driven-alone 3.649

Transit .308

Shared-ride 1.426

Other .274

Auto vehicle-miles 4.286

Auto driven-alone

mode share

Transit mode share

.617

-116

Shared-ride nude share .156

Other modes' share -111

A"erage travel pattern

distance (mi.)

Auto driven-alone

12.55

13.12

Transit 5.90

Shared-ride 20.28

Other 5.47

Average travel pattern

time (hrs.)

Auto driven-alone

.579

-563

Transit .441

Shared-ride .859

Other .296

1.500 1.226 1.573

(-4.5) (-22.0) (0.1)

1.242 1.100 1.278

(-2.7) (-13.8) (0.2)

.258 -126 .295

(-12.5) (-57.3) (0.0)

5.188 3.457 5.670

(-8.3) (38.9) (0.2)

3.362 1.793 3.651

(-7.9) (-50.9) (0.1)

.321 .317 .322

(4.2) (2.9) (4.5)

1.243 1.069 1.424

(-12.8) (25.0) (-0.1)

-265 -279 .273

(-3.3) (1.8) (-0.4)

3.914 2.266 4.279

(-8.7) (-47.1) (-0.2)

.611 .587 .616

(-1.0) (-4.9) C-0.2)

.121 .119 .119

(4.3) (2.6) (2.6)

-150 .161 .155

(-3.8) (3.2) (-0.6)

.118 .133 -110

(6.3) (19.8) C-0.9)

11.56 7.72 12.55

(-7.9) (-38.5) (0.0)

12.26 6.82 13.12

(-6.6) C-48.0) (0.0)

5.92 5.94 5.99

(0.3) (0.7) (1.5)

18.47 14.83 20.33

(-8.9) (-26.9) (0.2)

5.00 4.69 5.49

(-8.6) (-14.3) (0.4)

.541 .372 -580

(-6.6) (-35.8) (0.2)

.531 -310 .563

(-5.7) (-44.9) (0.0)

.445 -446 .450

(0.9) (1.1) (2.0)

.787 -621 .I360

(-8.4) (-27.7) (0.1)

.275 .254 .297

(-7.1) (-14.2) (0.3)

1.579 1.575

(0.5) (0.3)

1.284 1.280

(0.6) (0.3)

.295 -295

(0.0) (0.0)

5.614 5.677

(dO.8) (0.4)

3.612 3.644

(-1.0) (-0.1)

-357 .356

(15.9) (15.6)

1.383 1.411

(-3.0) (-1.1)

.264 .269

(-3.6) (-1.8)

4.231 4.263

(-1.3) (-0.5)

.600 .613

(-2.8) (-0.6)

.148 .127

(27.6) (9.5)

.145 .152

(-7.1) (-2.6)

-107 .108

(-3.6) (-2.7)

12.32 12.54

(-1.8) (-0.1)

13.21 13.13

(0.7) (0.1)

5.29 6.20

C-10.3) (5.1)

20.93 20.51

(3.2) (1.1)

5.42 5.51

(-0.9) (0.7)

.574 .582

(-0.9) (0.5)

.565 .563

(0.4) (0.0)

-394 .474

C-10.7) (7.5)

.S76 .866

(2.0) (0.8)

.290 .296

(-2.0) (0.0)

1.573 1.574 1.227

(0.1) (0.2) (-21.9)

1.278 1.278 1.106

(0.2) (0.2) (-13.3)

.295 .296 .121

(0.0) (0.2) C-59.0)

5.669 5.665 3.350

(0.2) (0.1) (-40.8)

3.651 3.648 1.678

(0.1) (0.0) f-54.0)

.320 .331 .485

(3.9) (7.5) (57.5)

1.421 1.418 .926

(-0.4) (-0.6) (-35.1)

-276 ,274 .264

(0.7) (0.0) (-3.6)

4.279 4.272 2.086

C-0.2) (-0.3) (-51.3)

.616 -615 .552

(-0.2) (-0.3) (-10.5)

.llS .121 .lSO

f.17) (4.3) (55.2)

.155 .154 .139

(-0.6) (-1.3) C-10.9)

.lll .llO -129

(0.0) (-0.9) (16.2)

12.55 12.54 7.35

(0.0) (-0.1) (-41.4)

13.12 13.13 6.67

(0.0) (0.1) (-49.2)

6.01 6.05 5.91

(1.9) (2.5) (0.2)

20.30 20.38 14.62

(0.1) (0.1) l-27.9)

5.50 5.51 4.49

(0.5) (0.7) (-17.9)

.580 .580 .364

(0.2) (0.2) (-37.1)

.563 .563 .304

(0.0) (0.0) (-46.0)

.451 .456 .456

(2.3) (3.4) (3.4)

.860 .862 .602

(0.1) (0.3) (-30.0)

.297 .297 .240

(0.3) (0.3) (-18.9)

Note : Numbers in parentheses are percent changes from base values.

In Table 4, the travel pattern model’s predictions are translated into travel summary measures. From these, it is possible to determine the effects that each policy would have on different components of the travel pattern choice. Looking, for example, at the auto vehicle- miles (VMT) tabulations, it can be seen that only three of the policies have appreciable effects. Tripling gasoline prices causes a decrease of almost 9%. It is interesting to note that this figure corresponds quite closely to the 12% reduction in VMT for non-work trips forecasted for a similar policy of tripling gasoline price as reported in a previous study (Atherton et al. 1977).

One of the most interesting aspects of these predictions is in the relationship between home-based and non-home-based links. It will be recalled, from Table 3, that for all of the policies which were tested, the average number of sojourns per tour either drops or remains constant. It was explained earlier that this is due to the fact that, in general, policies which reduce overall tripmaking also reduce the number of opportunities available for consolidating more sojourns into each tour. On the other hand, most policies which encourage more travel

2% T. ADLER and M. BEN-AKIVA

(e.g. by improving transportation levels-of-service) offer no incentive for making multiple- sojourn tours. The only policy which does offer such incentives, the fixed-fare transit service, results in an increase of 4% in the average number of sojourns per tour for transit trips (no increase across all modes). In Table 5, it is possible to see the effects that the changes in the numbers of sojourns and tours have on the number of home-based and non-home-based 1inks.t For the auto cost increase and VMT limitation policies (and for the combined policy), the percent change in non-home-based trips is greater than for home-based trips. For the transit policies, the change in non-home-based trips is less than for home-based trips.

If the number of home-based and non-home-based trip links were both assumed (as some forecasting models currently assume), in all cases, to change at the same rate as the home-based links change, the total number of trip links would be overpredicted, as shown in Table 5. As can be seen, the effects of the transit policies are small enough so that the predictions are only slightly affected by this assumption. For the auto cost increase, the VMT limitation, and the combined policy, however, the assumption that non-home-based trip links change at the same rate as home-based trip links results in significant overpredictions of both non-home-based and total trip links. Thus, for some policies, an explicit representation of the different rates of change of home-based and non-home-based trip links (as in the travel pattern model) seems to be quite important.

SUMMARY AND CONCLUSIONS

This paper describes a general theory of non-work travel behavior which accounts for the effects of transportation service levels on the use of multiple-sojourn tours. Non-work travel activities are seen as arising from a household’s balancing of a continuing accumulation of needs with the travel expenditures required for satisfying those needs. The degree to which a given travel pattern meets the substance and the temporal location of those needs, and the expenditures required for the travel, constitute the travel pattern’s utility or attractiveness to the household. Within this behavioral framework, the travel pattern’s utility can be described in terms of four sets of measurable attributes: those describing the “scheduling convenience” of the travel pattern, the travel times and costs associated with the travel pattern, attributes of the chosen destinations, and the socioeconomic characteristics of the households which determine the level and nature of non-work travel needs.

Existing travel demand models are based on a more simplified view of non-work travel behavior. Many of the choices which comprise the household’s travel pattern are assumed to be ‘independent of the others. For example, mode and destination choices for a tour are modeled as if they were independent of the number of stops on the tour. It is unlikely, however, that an individual would make either choice without first considering the structure of the planned travel tour. For policies which would affect the relative attractiveness of a multiple-stop tour, this type of.behavioral simplification could result in a model producing inconsistent forecasts.

Table 5. Over-predictions from assuming a single rate of change

1 2 3 4 5 6 7 Auto Max. TransitTransitTransitFixed- cost VMT 15 Cost OVTT IVTT fare Free 1.2.4. x3 Miles x .5 x .5 x .75 TransitTransit 5,7

Number of trip links: Travel pattern model 1.500 1.226 1.573 1.579 1.575 1.573 1.574 1.227

Equal rates of change 1.529 1.354 1.574 1.581 1,576 1.574 1.574 1.362

8 Over-prediction 1.9 10.4 0.1 0.1 0.1 0.1 0.0 11.0

Number of non-home- based trip links:

Travel pattern model .258 -126 .295 .295 .295 -295 .296 .121

equal rates of change .287 .254 .296 .297 .296 .296 .296 .256

% Over-prediction 11.2 101.6 0.3 0.7 0.3 0:s 0.0 111.6

tThe relationships between the numbers of home-based (HB) and non-home-based (NHB) links and the numbers of sojourns (S) and tours (T) for a given travel pattern are as follows: Total trip links = S + T; NHB = S-T; HB = 2T; NHB/HB = l/2 (S/T - 1).

A theoretical and empirical model of trip chaining behavior 257

The travel pattern model which is described in this paper attempts to model non-work travel patterns in their full complexity. The logit model that was estimated resulted in coefficient values which agree with the general behavioral theory advanced earlier. The coefficient estimates also seemed to be quite stable with respect to changes in the model specification and in the estimation sample.

When the model was used to produce forecasts for the behavior of an aggregate sample under varying policies, it was observed that disincentives on travel caused overall decreases in the average number of sojourns per tour. This was found to be caused by the general reduction in the number of sojourns per household, which results in fewer opportunities for a household to link trips together into multiple-sojourns. The only policy which resulted in a higher number of sojourns per tour (though only for transit tours) was the fixed-fare transit system. Under this operating policy, free transfers were allowed and, thus, multiple-sojourn tours by transit, as expected, became more attractive.

For the transit-oriented policies, shifts in the use of multiple-sojourn tours were not significant enough to result in major changes in the relative proportion of home-based and non-home-based trip links. For the other policies (tripling of gasoline price, VMT limitations), however, the number of non-home-based trips decreased at a substantially higher rate than did home-based trip links. Thus models which do not account for the varying effects of policy changes on the use of multiple-sojourn tours are likely, in some cases, to result in substantially biased forecasts of non-home-based and total trip links. In order to reduce those biases, different types of improvements to existing models should be explored along with entirely new

models such as the travel pattern model described here.

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TR-B Vol. 13B. No. 3-F