Hong Chen,Zhanguo Song, Hanqi Wang, Ruichang Ning and Fangjing Xu

(School of Highway, Chang’an University, Xi’an 710064, China)

Abstract: In order to better describe the commuter’s travel decision-making behavior under different travel environment, heterogeneous commuters and types are defined, and the commuters are divided into three types, including conservative type, neutral type and adventure type, respectively, analysis on the travel environment supply and the travel environment demand. Suppose the travel demand obeys the gamma distribution and the capacity obeys the beta distribution, and the travel time function of different commuter type is deduced, the travel decision model based on the cumulative foreground theory is established. Analyze the example results, compared with the fluctuation of travel demand, the degradation of traffic capacity has a more significant impact on travel decision-making behavior; and different types of commuters cause different disturbances to travel decision-making behavior.

Keywords: traffic engineering; travel decision behavior; cumulative prospect theory; heterogeneous commuters; travel environment

1 Introduction

Travel decision-making behavior has been identified as one of the hot issues to be tackled for developing traffic management. It is not only related to travel environment, but also related to heterogeneous traveler types. Heterogeneous traveler refers to combining the ordinary people’s decision-making behavior and individual psychological perception factors, and producing different decision-making behavior under different travel conditions, there are three categories about heterogeneous travelers, including conservative type, neutral type, and adventurous type. Finally citing and optimization of the cumulative prospect theory, analysis describes the travel decision- making behavior of heterogeneous travelers under different travel environment.

For the current research about travel decision analysis based on the cumulative prospect theory, most domestic and foreign scholars have unfolded the relevant aspects of research. Among them, to better describe the decision-making behavior of travelers, Refs.[1-2] simulated comparative analysis of the expected utility theory and cumulative prospect theory, resulting in two kinds of results, and the decision model based on the cumulative foreground theory is more consistent with the traveler’s actual travel behavior; in Ref.[3], through the actual investigation, Jon found that when the traveler selected the departure time and path, the decision-making results depended on their own reference point; Kitsikopoulos[4]also found that travelers with different reference points in path selection experiments showed different risk attitudes. Most domestic research is based on research carried out abroad, Li Xiaojing[5]considered both the value and the reliability change, researched the attitude of the commuter’s behavior choice and the attitude of dealing with risk; Gan Zuoxian[6]introduced the arrival time of perceived value, researched the change of traveler travel risk through the setting of the variable information intelligence board; Wang Qian[7]considered the characteristics of travel bounded rationality, based on the cumulative prospect theory, established the user equilibrium model that considered the problem of road capacity degradation, Lu Biao[8]expanded the user equilibrium model, taking into account the random changes in demand and road capacity.

Through analysison the domestic and international research results, although in commuter travel decision-making behavior, many models are proposed, but the results of such models are concentrated in a specific travel environment, and have not yet carried out an in-depth analysis in the travel behavior of heterogeneous travelers. In view of this, this article from the point of view of traveler heterogeneity, the travel decision model in the uncertain environment is established, and the travel decision behavior of the heterogeneous commuters is analyzed.

2 Travel Time Budget of Uncertain Environment

Actually the road traffic network is random, influenced by many random factors, thus has an impact on travel behavior of travelers’ decisions. These factors can be divided into two aspects: demand and supply. The uncertainty of demand is manifested as the fluctuation of travel demand. The uncertainty of supply is manifested as a random degradation of the bottleneck capacity, both of which eventually led to the travel uncertainty.

2.1 Travel Time Function of UncertainEnvironment

Commuter travel is affected by purpose and weather, resulting in fluctuations in travel demand; the supply capacity affected by the accident and the signal control factor is gradually degraded. Therefore, in order to better describe this complex and changeable phenomenon, suppose that the travel demand follows the Gamma distribution, the supply obeys the Beta distribution.

2.1.1Fluctuationsintraveldemand

Adopt normal distribution or lognormal description travel demand, and this form of distribution is too simple. While the Gamma distribution describes the complex and volatile travel demand fluctuations, so it is more prepared to describe the demand fluctuations.

Assumed travel demand distribution obedience Gamma distribution form, namely

N′=N·φ

(1)

whereNshowing car travel demand,φobedience shape parameterαand a scale parameterβgamma distributed random variables.

Gamma probability density distribution

(2)

2.1.2Capacitydegradation

In fact, due to the presence of various random factors, such as road construction, weather, traffic accidents, and other control signals, road capacity will be randomly degraded, leading to changes in travel time. Therefore, by setting flexible beta distribution parameters to describe the link capacity distribution.

Suppose the linkτcapacity is

Cτ=Cmax·χ

(3)

whereCmaxrepresents the maximum capacity of sections,χrepresents obedience parameterslandmbeta distributed random variables.

Beta distribution probability density.

(4)

Select the free-flow time as the base point, and compared travel time with the base point time, to determine the size of the loss of travel time. Adopt BPR function described travel time function.

Tτ=T(Vτ,Cτ)=tτ0[1+p(Vτ/Cτ)n]

(5)

wheretτ0is freedom of travel time on roadτ;Cτis capacity on roadτ;n,pare deterministic parameters.

SupposetτandεTτare the mean and variance of travel time:

(6)

(7)

εTτ=Var(Tτ)=E((Tτ)2)-(E(Tτ))2

(8)

By the Eq.(1), (2) available

(9)

(10)

By the Eq.(3), (4) available

(11)

(12)

Substitute the Eqs.(9)-(12) into Eqs.(6)-(8), obtain thenmean and variance of travel time function.

(13)

(14)

In the actual road network, a correlation exists between theroad from each other. To simplify the study, this article assumes independent sections, the path of travel time functionTrmean and variance istrandεTr.

(15)

(16)

whereδis correlation coefficient of passing-road, whenτ∈r,δis 1; otherwiseδis 0.

2.2 Preference Travel Time Budget

In an uncertain environment, due to the restricted number of factors, commuter travel treat the attitude is not the same. In this paper, commuters are heterogeneous, and different attitudes toward risk according to commuters classification. Commuter’s preferred travel time is not a certain value, but a rough range, that the maximum and minimum estimated travel time.

According to the Central Limit Theorem[9], the path consists of many independent sections, so it can be assumed that the path travel time approximates the normal distribution, itsTr∈N(tr,εTr). Due to the different attitude toward risk commuters, travel time for its reliability is not the same. In a certain degree of reliability[10], the travel time valuetsup(γ) under ideal conditions is the minimum estimated travel time , and it can be expressed as:

tsup(γ)=supTePrTe≤tr≤1-γ

(17)

Eq.(17) can be expressed as:

maxTe

(18)

Derivation available,Te=tr-σTrΦ-1(γ).

where Φ-1(·) is the normal function of the inverse function.

Pessimistic value of travel timetinf(β) is the greatest estimated travel time, and it can be expressed as:

tinf(β)=infTlPrTl≥tr≤1-γ

(19)

Eq.(19) can be expressed as:

(20)

After derivedTl=tr+σTrΦ-1(γ).

Therefore, the range of preference travel timeT0is:

[tr-σTrΦ-1(γ),tr+σTrΦ-1(γ)]

(21)

2.3 Reference Point and the Risk Appetite

(22)

(23)

Although there is a point in between, this point is clearly not true point of reference, but the pseudo-reference point.

Risk appetite is at risk of commuters travel choice environment manifested mental attitude and behavioral intention, different characteristics of commuters have different risk preferences.But the appetite for risk factors are used by Kahneman[11]experiments calibration is 0.88. Obviously, this cannot be expressed in different types of risk appetite commuters correct. Because of the level of risk appetite and expectations of the level of the reference point. Therefore, the coefficient is associated with the risk appetite point of reference each other, according to reference document[12], the definition of various types of commuters risk appetite factor is:

(24)

3 Travel Decision Model

Cumulative prospect theory is developed on the basis of Kahneman’s prospect theory. With the ideology about accumulation weight, it is often used to analyze the travel decision under uncertain conditions and the cumulative foreground theory value is determined by the value function and the decision weight function. The decision-making process is divided into two parts, including the editing stage and the evaluation stage. At the editing stage, based on the previous set of commuter reference points, the value functionV(xk) and the cumulative weight functionπ(p) are established, these functions are used to describe the various options possible outcomes; at the evaluation stage, estimate the prospective value of travel options based on the value function and the cumulative weight function.

3.1 Value Function

Suppose commuter best preference arrival time istb, the earliest estimated arrival time iste, accept the latest arrival time istl, as shown in Fig.1, two reference pointste,tland pseudo reference pointtb, and divided into four time (-,te], (te,tb], (tb,tl], (tl,+). If commuters actually arrive earlier thanteor later thantl[7], then the time function value is negative, the loss is considered; if commuters arrival time is betweenteandtl, then the time function value is positive, is considered income.

(25)

whereβiis the loss aversion, calibration results using Kahneman, when earningsβ=1, lossesβ=2.25;θiis risk attitude factor, and 0<θi≤1.

Fig.1 Curve of value function

3.2 Weighting Function

In this paper, used the weight function expression given by Kahneman. The concept of probability of decision event is described by “capacity”, the probability weight is defined, and the subjective probability of decision event is represented by a numerical value, which is divided into two forms of gain and loss probability weight function.

When policy makers earn:

(26)

When policy makers lose:

(27)

where, according to Kahneman calibration results,λ=0.61,δ=0.69.

3.3 Cumulative Weight Function Value andProspects

The foreground value is determined by the value function and the decision function, and the path decision information is determined by the foreground value. According totbpoint, were divided into two prospects point values, they aref(xl,pl) andf(xr,pr), circumstances are present on both sides of gains and losses. It should be calculated separately. In the left part of the prospect another example is that:

(28)

(29)

(30)

Left foreground value

(31)

Also available on the right foreground is:

(32)

The overall outlook for the value of the expression

(33)

4 Case Analysis

4.1 Model Algorithm

Commuter travel decision model solution as follows:

Step1Determine the distribution of travel demand and capacity, according to the Eqs.(1)-(16) to obtain the travel time function meantrand varianceεTr;

Step2According to all kinds of commuter attitudes toward riskωcalculated reliabilityγ, according to Eqs.(17)-(21) calculated reach time various types of commuters;

Step3According acceptable earliest arrival time and the latest arrival time, according to Eqs.(22)-(24) to determine the reference point for all kinds of commuters and risk appetite;

Step4According to Eq.(25) to determine the value of various types of commuter function, according to Eqs.(26)-(30) calculates the cumulative probability weighting function;

Step5According to Eq.(31) - prospects calculate all kinds of different travel options for commuters (33) value, select the maximum prospects of mobility options.

4.2 Case Analysis

Road network exists a point OD, travel demand isN=3 000 pcu/h, commuter departure from the starting point O to reach the terminal D has three paths, they are O→1→2→D(Path 1), O→1→3→5→D(Path 2), O→4→5→D(Path 3). As shown in Fig.2, the figure in parentheses indicates the content (travel time (h), capacity (pcu / h)).

Fig.2 Network structure diagram

Assume that the attitude of the commuter to deal with the risk obeysω∈U(0,3) distribution, due to the commuter’s time value is different, so the time to reach the reliability and risk factors are also different, commandγ=ln(1.7+0.8/ω), reliability can be solved to obtain arrival time. According commuter different attitudes toward risk, select three commuters, they areω=0.9( conservative),ω=1.5( neutral),ω=2.7( adventure). Suppose commuter best time of arrival istb=8:50, calculate the parameters of heterogeneous commuters according to Steps 1-3, as shown in Table 1.

Table 1 Parameters of heterogeneous commuters

Because risk attitude is different, commuters estimated the earliest arrival time and the latest arrival time are not the same. As can be seen from Table 1, the commuter ofω=0.9 arrival time interval is 10 minutes, this type of commuter’s arrival time is more reliable and the preference coefficient is 0.56, so this type commuter shows the characteristics of risk aversion in the trip; while the commuter ofω=2.7 arrival time interval is 4 minutes, so this type of commuter arrival time is higher, its risk preference coefficient is 0.93, and showing the pursuit of travel in the trip.

According to the model algorithm, the results obtained in the travel decision under capacity and travel demand constant commuters. Among them, the greater the negative foreground value, indicating that the more the loss of travel time, the more positive foreground values, indicating that the greater the benefits for the best travel program, shown in Fig.3.

1) Commuter ofω=2.7 will depart at 8:15, select the path 3 to reach the destination; commuter ofω=1.5 will depart at 8:10, select the path 1 to reach the destination; commuter ofω=0.9 will depart at 8:15, select the path 3 to reach the destination;

2) Comparison of different types of commuter prospects of value can be seen, commuter ofω=0.9 conservative in the travel, the larger the acceptable arrival time is, the larger the probability of arriving at the destination within this range is, therefore, there are more alternative travel solutions, and commuters ofω=2.7 who travel in the aggressive attitude in the travel risky, less so the choice of mobility options.

(a)ω=2.7

(b)ω=1.5

(c)ω=0.9

Fig.3TraveldecisionsresultsofHeterogeneouscommuters

The following analysis shows the travel decision results of commuter, when the travel demand and capacity change.

①Fluctuations in travel demand

In the capacity unchanged, takeα=1,β=0.9 andα=1,β=1.1 in both cases, get travel demand fluctuation coefficientsφto be 0.9 and 1.1, reduce or increase travel demand analysis on commuter travel decision results, as shown in Table 2.

a. When reducing travel needs, increasing the range of acceptable arrival time, increased commuter alternative mobility options, travel decision result of three commuter have small changes, all choose the path 3, aggressive type of commuter departure time constant, neutral and conservative commuter departure time will be postponed to 8:20.

b. When travel demand increases, the travel decision results of three commuters to changed greatly, aggressive type of commuters still choose the path 3, but the early departure time to 8:05; conservative commuters adjustment path from the path 2 to 3, while the earlier departure time to 7:55; neutral commuter departure time only made minor adjustments.

c. When the travel demand fluctuates, the commuters are usually only adjusting departure times, rather than adjusting the travel path. When the travel demand is decreased, take the deferred departure mode; when the travel demand is increased, take an early departure mode. Compared with the decrease in travel demand, the increase in travel demand has a greater impact on the decision-making results of commuter travel.

Table 2 Fluctuation in travel demand

②Capacity degradation

In the case of travel demand constant, take two conditionsl=10,m=90 andl=10,m=30, obtain capacity degradation coefficientsχwere 0.9 and 0.75, analyze the capacity degradation on commuter travel decision results, as shown in Table 3.

Table 3 Degeneration in capacity

a. When smaller capacity degradation degree, three commuter travel decision results of a small change, aggressive type of commuters still choose the path 3, while the departure time to 8:05 earlier; conservative commuters departure time will advance to the 7:55, and the path is adjusted from path 3 to path 1; neutral commuters without any adjustments.

b. When capacity is large degree of degradation, three commuter travel decision results of changed greatly, aggressive type of commuters opting abandon the path 3 instead of path 1, and the departure time to 8:00 earlier; neutral commuters departure time will be 10 minutes earlier; conservative commuters will not only adjust the path 1 to path 2, and the departure time to 7:55 earlier.

c. With the capacity degradation was intensified, commuters prior to adjusting the departure time, and adjust later travel route, the severer the degree of degradation of capacity is, the greater the impact on the results of commuters travel decisions is.

③Travel demand and capacity at the same time change

In the case of travel demand and capacity change simultaneously, select four cases and analyze the change of commuter travel decision results, as shown in Table 4.

Table4Fluctuationintraveldemandanddegenerationincapacity

φ, χω=2.7PathTimeω=1.5PathTimeω=0.9PathTime0.9,0.7538:0518:0528:000.9,0.938:1518:1018:151.0,1.038:1518:1038:151.1,0.918:1018:0528:001.1,0.7518:0027:5527:50

a. When travel demand decreases and capacity degrades, due to reduced travel demand partially offset by the impact of the capacity degradation, so the impact on commuters travel decision result is not significant; aggressive type and neutral type of commuters do not adjust the travel route, instead of choosing an early departure; conservative commuter adjusted from path 2 to path 1, and departure time to 8: 00 advancedly.

b. With the increase in travel demand and capacity degradation, since the impact of increased travel demand and capacity degradation caused by superposition, so the impact on commuters travel decision result is more significant; three commuters caught abandon choose the path 3, aggressive and conservative type of commuter travel path selection unchanged, but others departure time advancedly; neutral commuter route adjust from path 1 to the path 2, while departure time to 7:55 earlier.

c.The impact of traffic capacity degradation on travel decision-making is greater than travel demand fluctuation. For conservative commuters, the impact of travel demand and capacity changes is more significant, but for the adventure and neutral type commuter, the impact is not significant.

5 Conclusions

This paper starts from the perspective of uncertain environment, establishes the travel time function in uncertain environment,and the commuters are classified based on different risk attitudes. The travel decision model is constructed based on the cumulative foreground theory, with examples, the travel decision-making behavior of commuters in different situations is analyzed, these different situations include travel demand fluctuations, traffic capacity degradation and changes in both at the same time. The results show that the model can accurately describe the travel decision behavior in uncertain environment of heterogeneous commuters, more in line with reality.

In this paper, the classification of commuters is mainly based on the different risk attitudes,but in practice, commuters will be affected by other factors, and the classification will be more complex. Since the travel time function of the path is composed of independent sections, ignoring the interconnection between the sections. So in the future, it is necessary to further study the clustering analysis to refine the commuter classification and determine the path travel time function composed of interconnected sections.