This paper presents a joint discretecontinuous model for activitytravel time allocation by employing the ordered probit model for departure time choice and the hazard model for travel time prediction. Genetic algorithm (GA) is employed for optimizing the parameters in the hazard model. The joint model is estimated using data collected in Beijing, 2005. With the developed model, departure and travel times for the daily commute trips are predicted and the influence of sociodemographic variables on activitytravel timing decisions is analyzed. Then the whole time allocation for the typical daily commute activities and trips is derived. The results indicate that the discrete choice model and the continuous model match well in the calculation of activitytravel schedule. The results also show that the genetic algorithm contributes to the optimization and thus the high accuracy of the hazard model. The developed joint discretecontinuous model can be used to predict the agenda of a simple daily activitytravel pattern containing only work, and it provides potential for transportation demand management policy analysis.
The time allocation of individuals for trip making is an important determinant of the temporal pattern of traffic demand on a transportation network. An analysis of individual time allocation decision is, therefore, important for the practical work of transportation planning and management as well as the theoretical work about travel behavior analysis and modeling. Practically, understanding individuals’ time allocation decisions is essential for (a) planning the development and construction of new transportation infrastructure by providing predicted temporal travel demands, (b) examining the potential responses to improved operational measures (such as realtime information), (c) assessing the effectiveness of timespecific transportation demand management policies, such as compressed working week [
In the time dimension, a daily activitytravel pattern includes all the timing and duration of all the activities and trips in a day. The need to analyze the daily activity pattern makes it essential to consider time of day and activitytravel durations together. However, to date, timing and duration have been largely treated separately in the literature [
One important objective of transportation planning is to relief congestion by improving the level of service during the peak hours on the transportation network. Peak periods’ traffic demands, or the source of congestion, are largely contributed by commute trips. For example, based on one survey conducted in Beijing, China, in 2005, over 32% of all the trips in both morning and evening peak periods are commute trips. Therefore, commute trips are at the core of many recent studies, such as Habib et al. [
As stressed above, the study on commute trips and daily time allocation is of great importance for learning the travel behavior during the peak hours as well as obtaining the daily activity pattern, both of which are essential component of for transportation planning and management. However, to the authors’ knowledge, there is no study that considered the overall daily time allocation of commute activitytravel pattern as a whole and developed a model system to predict it. Therefore, this paper focuses on predicting timing and duration of daily commute trips, expecting that commuter’s typical travel schedule can be obtained based on the developed model. Using data from a 2.5% sample household survey in Beijing, China, a joint discretecontinuous model system for prediction of daily commute time allocation was developed and estimated, including ordered probit models for departure time analyzing and hazard models for travel time forecasting.
The choice of parameters is of great importance for the estimation efficiency and prediction accuracy of the models. As there are a lot of potential factors affecting traveler’s decision about time allocation, genetic algorithms (GAs) will be employed in parameters optimization. Being one of the heuristic algorithms, GA has been successfully applied in various optimization problems [
The remainder of this paper is organized as follows: Section
With respect to the two dimensions of time allocation, timing and duration, existing studies can be classified into three categories: those only looked at departure time, those only looked at travel time, and those looked at both.
For the first direction, timing, the major method employed is discrete choice model. For example, Bowman and BenAkiva [
For the second aspect (duration analysis), the most widely used model is the hazard model. The hazard model recognizes the dynamics of travel or activity durations by considering the conditional probability of event termination, usually as a function of covariates (explanatory variables) [
Comparing with the first two categories of studies, those using the joint analysis of timing and duration are more helpful for the modeling of daily activity schedule, by contributing to an insight into the influence and connection among duration and time of day choice as well as activities and trips. The examples include the study done by Janssens et al. [
There is also a kind of conjunct model that was used in the analysis of timing and duration, which is the discretecontinuous choice model. Pendyala and Bhat [
Moreover, the study of Pendyala and Bhat [
In this paper, work location refers to the usual work location for a worker and the usual school location for a student. As shown in Figure
Basic time allocation of daily commute activitytravel pattern.
This study employs the ordered probit model (belonging to the discrete choice models) for departure time forecasting and the hazard model (belonging to the continuous models) for travel time analysis. Four models, hometowork departure time choice model, hometowork travel time estimation model, worktohome departure time choice model, and worktohome travel time estimation model will be developed and the values of
Modeling process of the commute activitytravel time allocation.
This study uses data from a largescale daily travel survey conducted in Beijing, China, in 2005. A facetoface interview was given to a sample of 54,398 households, and activity/travel information of all household members on one particular working day is collected. The study area is 1,368 km^{2} (covering all 18 districts of Beijing) and had more than 30 millions’ population in 2005. In addition to weekday OD information, the survey also collected information regarding household (size, car ownership, home location, monthly income, and mobility), people (age, gender, driving license, and occupation), and trips (departure time, arrival time, purpose, mode, transit path, etc.). With records containing missing values eliminated, our final sample consists of 37,842 commute trips of 37,842 individual workers/students from 28,382.
Based on a preliminary correlation test, 15 sociodemographic and trip characteristics variables were selected from the survey, shown in Table
Variables in the departure time choice models.
Factors  Variables  Values 

Gender  Gender  Male: 1, female: 0 
Age  Age  Continuous values 
Below 1500 RMB: 1  
1501–2500 RMB: 2  
2501–3500 RMB: 3  
Month income  Income  3501–5500 RMB: 4 
5501–10000 RMB: 5  
10001–20000 RMB: 6  
20001–30000 RMB: 7  
Over 30001 RMB: 8  
 
Vocation  
Bluecollar worker (manufacture, construction, maintenance, etc.)  Occub  Yes: 1, No: 0 
Administration  Occua  Yes: 1, No: 0 
Education  Occue  Yes: 1, No: 0 
Services  Occus  Yes: 1, No: 0 
Health care  Occuh  Yes: 1, No: 0 
 
Travel mode  
Walk  Modew  Yes: 1, No: 0 
Bike  Modebi  Yes: 1, No: 0 
Bus  Modebu  Yes: 1, No: 0 
Auto  Modea  Yes: 1, No: 0 
 
Travel distance  Distance  Continuous value (meter) 
 
Dummy variable of going to work  Work  Yes: 1, no: 0 (going to school) 
The statistics of the variables based on the sample data are shown in Table
Statistics index of the variables.
Variables  Percentage of each value  

1  0  
Gender  50.96%  49.04% 
Occub  42.44%  57.56% 
Occua  3.48%  96.52% 
Occue  30.29%  69.71% 
Occus  9.19%  90.81% 
Occuh  2.49%  97.51% 
Modew  42.51%  57.49% 
Modebi  43.89%  56.11% 
Modebu  0.31%  99.69% 
Modea  4.67%  95.33% 
Work  72.79%  27.21% 
1: 13.46%  5: 11.86%  
Income  2: 24.90%  6: 1.41% 
3: 24.18%  7: 0.16%  
4: 23.92%  8: 0.10%  
 
Continuous variables  
 
Variables  Mean  Standard deviation 
 
Distance  2806.1  3416.91 
Age  32.93  15.32 
The reported hometowork morning departure time and the worktohome evening departure time cover the time period of 4:00 am–12:00 am and 15:00 pm–22:00 pm, respectively, in our sample. Moreover, we observed that the hometowork morning and the worktohome evening peak hours in Beijing are 6:00 am–9:00 am and 16:00 pm–19:00 pm using our sample data. In order to reduce the number of alternatives in the models’ choice set, we divided departure times into onehour segments in peak periods and segments of two or three hours in offpeak periods. Table
Alternatives in the departure time choice models.
Hometowork departure time  Worktohome departure time  Coded value 

(4:00, 6:00]  (15:00, 16:00]  1 
(6:00, 7:00]  (16:00, 17:00]  2 
(7:00, 8:00]  (17:00, 18:00]  3 
(8:00, 9:00]  (18:00, 19:00]  4 
(9:00, 12:00]  (19:00, 22:00]  5 
As shown in Table
The ordered multiple choice model assumes the following relationship:
The ordered probit model, which assumes standard normal distribution for
Estimation results of the departure time choice models.
Variables  Hometowork departure time choice model  Worktohome departure time choice model  

Coef. 

Coef. 


Income  0.09  19.14*  0.06  16.18* 
Age  −0.01  −10.49*  −0.01  −29.41* 
Occuw  —  —  0.43  32.00* 
Occua  −0.14  −4.60*  0.35  11.44* 
Occue  −0.72  −27.09*  −0.25  −12.96* 
Occus  —  —  0.27  11.68* 
Occuh  −0.48  −12.96*  0.33  9.24* 
Gender  −0.11  −9.34*  0.08  8.36* 
Modew  0.18  7.89*  0.42  35.49* 
Modebi  −0.41  −19.30*  —  — 
Modebu  —  —  0.59  4.76* 
Modea  0.10  3.12*  0.41  15.29* 
Distance  0.000067  −32.89* 

−3.11* 
Work  0.39  12.92*  —  — 

−2.42  —  −1.30  — 

−0.98  —  0.01  — 

0.58  —  0.97  — 

1.60  —  2.07  — 
Hit ratio  64.36%  61.00%  

0.0000  0.0000  

37842 
The estimation results indicate that highincome travelers are more likely to depart from home or work location later than travelers with low income. Older persons tend to have earlier departure times than younger ones. Commuters whose occupations are administration or health care are more likely to depart from home earlier but back to home (from work location) later than those with other occupations. The probability that students or teachers have early departure times both from home and school is high. Workers and servers tend to depart from workplace late. Concerning gender, men are more likely to leave home earlier but leave work later than women. Regarding travel modes, commuters choosing walk or auto have late hometowork and worktohome departure times. Bikers tend to leave home earlier, while commuters taking bus depart from workplace later. Longdistance trips tend to be made later from home while earlier from work location. Workers are more likely to leave home later than students.
According to the travel survey data in Beijing, the average travel time for the hometowork commute is 19.36 minutes, with a maximum of 205 minutes and a minimum of 1 minute; the average duration for the worktohome commute is 18.36 minutes, with a maximum and a minimum of 168 minutes and 1 minute, respectively.
Treating travel times as natural continuous variables, one can use the hazard model to predict both the hometowork and the worktohome travel times. Hazardbased duration models are ideally suited to modeling duration data [
Let
Let
Thus, the hazard function can be further expressed as
The distribution of the hazard can be assumed to be one of many parametric forms or to be nonparametric. Because the distribution of the travel time is unknown, one of the nonparametric methods, the KaplanMeier (KM) product limit estimator, is conducted to explore the covariate effects and the potential distribution.
As a nonparametric method, the KM estimator produces an empirical approximation of survival and hazard but hardly takes any covariate effects into consideration. It is similar to an exploratory data analysis. Denoting the distinct failure times of individuals
By using the KM estimator, the survival function curves of the hometowork and the worktohome travel time are estimated, which are shown in Figures
Survival curve of the hometowork travel time.
Survival curve of worktohome travel time.
The AFT model is one of the popular parametric forms of hazard model. It permits the covariates to affect the duration dependence. The survival function of AFT model is given as
The AFT model can be expressed as a loglinear model:
Assuming the random error
The parameters in the AFT models will influence the estimation efficiency and prediction accuracy of the models greatly, especially for largescale or realtime feature practice application. Therefore, this paper attempts to find the appropriate parameters in AFT models by using GA. GA is a part of evolutionary computing, which is a rapidly growing area of artificial intelligence. The process of GA is as follows.
GA is started with a set of solutions (represented by chromosomes) called population. The individuals comprising the population are known as chromosomes. In most GA applications, the chromosomes are coded as a series of zeroes and ones, or a binary bit string. For the travel time forecasting models, some parameters are continuous valued (like
Crossover is a reproduction technique that takes two parent chromosomes and produces two child chromosomes. A commonly used method for crossover called onepoint crossover [
Crossover diagram.
Mutation is a common reproduction operator used for finding new points in the search space to evaluate. A genetic mutation operation [
Assume a chromosome is
The function
To deal with the problem that the mutation may violate the parameters constraints, we will assign a relatively high weight to reduce their probability of being selected in the following search [
There are four GA parameters, namely,
Figure
Survival curves of the hometowork travel time impacted by several factors.
Travel mode
Dummy variable of going to work
Departure time
Income
Gender
Occupation
Two AFT models are estimated, each of which assumes the random error in (
Estimation results of the hometowork travel time model.
Variables  Weibull distribution  Exponential distribution  

Coef. 

Coef. 


Constant  3.11  163.26  2.84  73.09 
Income  −0.01  −4.82  −0.01  −2.65 
Age  0.01  22.49  0.00  7.00 
Occue  −0.20  −19.86  −0.10  −4.49 
Gender  −0.04  −7.38  −0.04  −3.72 
Modew  −0.18  −17.28  −0.14  −7.65 
Modebi  0.07  7.04  0.15  8.93 
Modebu  0.20  4.28  —  — 
Modea  −0.04  −2.97  —  — 
Distance  0.00013  100.58  0.00012  57.51 
Departure time  −0.09  −29.84  −0.09  −13.06 
Work  −0.32  −29.82  −0.19  −7.36 

0.50  —  1  — 
Prob > chi^{2}  0.0000  0.0000  

37842 
The mean absolute percentage error (MAPE), which looks at the average percentage difference between predicted values and observed ones, is adopted to examine the accuracy of the developed hometowork travel time model. MAPE is calculated as
Goodness of fit index and estimated distribution statistics of the hometowork travel time model.
Model statistics  Weibull distribution  Exponential distribution 

MAPE value  0.4449  0.3882 
Mean (min)  18.46  13.66 
Maximum (min)  6.02  5.09 
Minimum (min)  158.64  114.08 
According to the results shown in Table
The estimation results indicate that the most essential factor of travel time is distance. The longer the distance from home to work is, the more time it will take. Comparing with other travel modes, the travel times of walk are 14% lower while those of bike are 15% higher. The reason is that, at first short distance encourages short travel time according to the estimation results, and then distance has influence on mode decisions, that is, walking usually belongs to shortdistance travel comparing to biking. As for the factor of departure time, the results show that the later the departure times are, the longer the travel times will be. In the above three parameters, the factors of distance and mode are related to the transportation network, while the real traffic condition is considered by using departure time as a factor, because the traffic condition depends regularly on departure time. For instance, if a traveler departs from home in the morning peak time, the probability that he/she encounters traffic congestion is much larger than that in the nonpeak time. The results also show that the higher travelers’ income is, the less their travel time will be. The travel times of students or teachers are about 10% less than those of other travelers. The older the travelers are, the longer the travel time will be. Women have longer travel time than men. The travel times of travelers whose occupation is administration are 19% longer than that of education.
Figure
Estimation results of the worktohome travel time model.
Variables  Weibull distribution  Exponential distribution  

Coef. 

Coef. 


Constant  2.65  164.15  2.53  79.98 
Income  −0.01  −8.13  −0.01  −3.57 
Age  0.0034  19.23  0.0033  8.80 
Occuw  −0.03  −6.20  −0.03  −2.22 
Occue  0.05  6.56  0.07  4.25 
Modew  −0.06  −5.53  −0.06  −3.51 
Modebi  0.05  4.38  0.08  −3.51 
Modea  0.05  3.33  —  — 
Distance  0.00015  119.81  0.00015  66.12 
Departure time  −0.06  −23.50  −0.06  −11.96 

0.4835  —  1  — 
Prob > chi^{2}  0.0000  0.0000  

37842 
Goodness of fit index and estimated distribution statistics of the worktohome travel time model.
Model statistics  Weibull distribution  Exponential distribution 

MAPE value  0.4541  0.3905 
Mean (min)  18.50  13.36 
Maximum (min)  8.44  6.17 
Minimum (min)  269.11  172.80 
Survival curves of the worktohome travel time impacted by several factors.
Travel mode
Departure time
Income
Occupation
Same as the hometowork model, the AFT model of worktohome travel times with Exponential distribution is better than that with Weibull distribution. Therefore, the former is selected. The hazard function and survival function are as follows:
The estimation results indicate that the travel time of highincome travelers is 1% lower than that of lowincome travelers. Moreover, old persons are likely to spend longer time in workhome trip. Regarding commuter’s occupation, bluecollar workers are likely to spend shorter time for evening commute trip, while the travel times for teachers or students are longer. Comparing with other modes, walking trips have shorter time, while cycling trips have longer time. Longdistance trip takes longer travel time. The later the commuters depart from work, the longer the travel times will be.
As explained in Section
Observed commute activitytravel pattern and time allocation.
Based on the developed models, his daily commute time allocation is predicted and shown in Figure
Predicted commute time allocation.
Comparing with the observed values, the errors of the predicted results can be calculated as follows.
The errors of hometowork departure time: the maximum error is 45 minutes, the minimum error is 15 minutes.
The error of hometowork travel time: 5.47 minutes.
The errors of arrival time at work location: the maximum error is 51 minutes, and the minimum error is 0 minute.
The errors of worktohome departure time: the maximum error is 60 minutes, and the minimum error is 0 minute.
The error of worktohome travel time: 4.29 minutes.
The errors of arrival time at home: the maximum error is 65 minutes, and the minimum error is 5 minutes.
By comparing the predicted values of the developed daily time allocation model with the observed values of our sample, the maximum errors for all the departure times and the activitytravel durations are as follows: error(
Results also show that the errors of departure and arrival time are much larger than that of the travel time. The main reason is that we divided the natural continuous departure time into discrete time interval artificially, which reduces the predictive accuracy of the model. It has been tested that the smaller the interval is, the higher the predictive accuracy will be. However, as there are already five alternatives for both of the departure time choice models, at least halfanhour interval will make the number of the alternatives double. Then the model will be more complex and the efficiency of the model will be lower.
In this paper, we have formulated and estimated a joint model of departure time choice and travel duration for commuters’ daily activitytravel time allocation. Two ordered probit models have been employed to forecast the hometowork and worktohome commute departure time. By doing so, we were able to recognize the natural temporal ordering among the departure time alternatives and address the IIA limitation of the standard MNL model. Furthermore, two AFT models were built and estimated to predict hometowork and worktohome travel times by using GA as parameters optimization. Then timing choice of a simple daily activitytravel pattern has been calculated.
Comparing with the previous studies, this paper developed a joint discretecontinuous model system to predict all the departure times and the activitytravel durations of a typical daily commute activity pattern. Results of this study not only contribute to developing a fullscale daily activity pattern forecasting model but also provide useful insights in the influence of sociodemographic variables on activitytrip timing decisions as well as the time constraint between daily activities and trips. Moreover, GA contributes to the optimization and thus the high accuracy of the travel time prediction model. In addition, this analysis of daily commute time allocation can be applied to a wide range of TDM policies, especially the measures aimed at adjusting the commute times, such as flexible work and compressed working week. For examining the effects of the traffic demand strategies, the developed model cannot only describe the overall change of the daily activity schedule caused by the strategies but also explore the time tradeoff between the connected trips as well as trips and activities. Besides evaluating the effects of the transportation demand management strategies, this study is also essential for planning the development and construction of new transportation infrastructure as well as examining the potential responses to improved traffic operational measures.
The results of this paper confirm that the discrete choice models and the continuous models can match well in the calculation of a wholeday activitytravel schedule, although comparing with the continuous models, the predictive accuracies of the discrete choice models are a little lower, as they divide the naturally continuous time into artificially defined time periods. Similar studies were also found to employ the discretecontinuous methods to model coupled mode and commute timing choice [
It should be pointed out that only the typical commute activitytravel pattern, comprising two commute trips and one work activity, has been considered in this paper. In reality, it is also common to observe other commute activitytravel patterns such as those including workbased subtour or homebased nonwork trips, or those having stops during commute travel. Further study may be done to model such daily activitytravel patterns. It will also be very important to exam one’s activitytravel patterns over multiple days if the multiday travel survey data can be obtained.
The research is funded by the National Natural Science Foundation of China 50908099 and 51108053 and China Postdoctoral Science special Foundation 201104526.