Due to the lack of consideration of heterogeneity in the traditional choice model based on regret theory, there may be errors in interpreting the real choice behavior. Traditional regret functions do not account for the perception of different alternatives and individual socioeconomic characteristics. This paper utilizes Weber’s law to explain the heterogeneity of travelers’ perceptions regarding alternative attributes. It introduces new parameters to consider individual socioeconomic characteristics to improve the classic random regret minimization (RRM) model. Then, these two improvements are incorporated into the model together. Different choice models are established based on random utility maximization (RUM) and RRM, respectively. This paper then takes taxi-hailing choice behavior in Chengdu as an empirical study. The results suggest that the calibration results of different models are consistent, and the overall goodness of fit and hit rate of models under RRM are better than models under RUM. The improved RRM model considering both perception heterogeneity using Weber’s law and socioeconomic characteristics has the best model evaluation indexes. Thus, the improved model could better explain and predict travelers’ choice behavior.

1. Introduction

The travel mode choice model is an important component of the travel demand analysis. Improving the accuracy of the travel mode choice model to reflect travelers’ characteristics accurately is always the key point in travel behavior research. The theory and structure of the model are the foundation for improving the accuracy of the model. In traditional travel choice models, individual preferences are generally assumed to be homogeneous. However, due to the heterogeneous influence of individual preferences, risk tolerance, and personal economic and social attributes, individuals often show different preferences when facing the same choice [1]. If the assumption of homogeneity is imposed when there is heterogeneity, the parameter estimation and the choice probability results will be biased and inconsistent [2]. More and more studies show that considering the influence of heterogeneity in travel behavior studies can improve models’ goodness of fit and explanatory ability and better analyze travelers’ choice behavior [2, 3].

The concept of heterogeneity was first put forward in genetics, which means that the heterogeneity in genetic material can lead to heterogeneity in inherited traits of subsequent generations. Subsequently, heterogeneity was introduced into the field of economic research. In the Nobel Lecture in Economics in 2000, Mcfadden pointed out that different decision-makers significantly differ in taste and preference [4]. In the field of travel behavior research, heterogeneity has also attracted the attention of more and more researchers. Many studies have also shown that the heterogeneity can be represented by travelers’ preference for choice of travel modes to better explain the travel decision-making behaviors of different types of travelers and improve the goodness of fit and explanatory ability of models. Thus, the travel choice behaviors of different groups can be described more accurately [5].

In the travel choice behavior, traveler’s heterogeneity mainly includes preference heterogeneity and response heterogeneity. The preference heterogeneity refers to the difference between different travelers in observable individual socioeconomic attributes and unobservable psychological characteristics [3, 6]. Usually, observed personal attributes such as gender, age, and income can be obtained directly through questionnaires. The unobserved psychological characteristics mainly include travelers’ perceptions, values, travel habits, and lifestyle preferences. Many studies have shown a significant heterogeneity in the psychological characteristics of different travelers, and these psychological factors can significantly affect the behavior of travelers choosing their trip [7, 8]. The response heterogeneity refers to the difference in evaluating different travelers on the attributes of the alternatives, including the observable and unobservable kinds [9, 10]. The observable response heterogeneity refers mainly to the difference in the degree of impact of the same influencing factors in different travelers. For example, the travel time has a negative effect on travelers’ mode choice behavior. However, its negative effect on middle-income and low-income groups may be less than that on the high-income group. The unobservable response heterogeneity refers to the assumption that travelers’ evaluation of alternative option attributes follows a continuous distribution or that travelers are grouped according to the combination of different psychological variables. Capturing individuals’ preference and response heterogeneity is important in modeling travel mode choice. As a first step in constructing the model, the analyst should strive to attribute as much preference and response heterogeneity to systematic variations as possible [6]. However, it is unlikely that the analyst could be aware of or have all information on all individual factors that affect intrinsic biases and responsiveness to level-of-service variables. In particular, there may be appreciable taste variation even after developing the best possible systematic specification for the preference and response heterogeneity [7]. Consequently, we must consider superimposing the random heterogeneity on the best systematic specification [6].

As a complementary behavioral modeling paradigm to random utility maximization (RUM), random regret minimization (RRM) has attracted much interest in recent years. Some studies use error component models to allow for correlated alternatives or estimated models in which a subset of attributes is subject to RUM, and another subset is subject to RRM [11–13]. However, these studies assume that the preference for each attribute is homogeneous across the entire population, although different attributes may be translated into the preference in different forms. Thus, most studies using the RRM paradigm have assumed a preference homogeneity among decision-makers and estimated models of a multinomial logit (MNL) form [14].

Some studies have attempted to account for the heterogeneity in the decision-making process by incorporating the RRM model into a different framework. Chorus, Rose, and Hensher proposed a hybrid model combining the RRM and RUM models. The hybrid RUM-RRM model allows different choice processes based on a hybrid random utility-regret function. The hybrid function consists of a random utility part and a random regret part. Moreover, the hybrid function suggests that attributes in the utility specification are processed based on the utility maximization rule. In contrast, other attributes in the regret specification are processed based on the regret minimization rule [12]. Hess et al. studied the heterogeneity in decision-making rules based on a latent class approach. They suggested different decision rules, such as RUM and RRM, should be used in different classes [4, 13]. Boeri, Scarpa, and Chorus applied this approach in an empirical study of traffic calming schemes. Moreover, another approach, called the RRM-based hybrid choice model, is incorporating an RRM model into a hybrid choice model framework [15]. To investigate the relationship between people’s latent needs and their choices in terms of leisure activity, Kunhikrishnan and Srinivasan used the hybrid RUM-RRM model as the discrete choice model of the hybrid choice model framework [16]. Furthermore, there are some other studies on RRM-based hybrid decision-making rules [17–19]. The studies above attempted to provide more elaborated model structures to analyze heterogeneity in the decision-making process based on the RUM and RRM models.

Several studies have previously researched the random preference heterogeneity under the RRM framework. Chorus, Rose, and Hensher focused on a single attribute that was not statistically significant and a random regret threshold that was statistically significant [12]. Chorus, Arentze, and Timmermans allowed for the additional random taste heterogeneity when estimating models. The results suggested that significant variations were only observed in the stand-alone mixed MNL (MMNL) under RUM and in the RUM component of a combined model (MMNL and RRM) but not in the stand-alone RRM model or the RRM component of the combined model [20]. Boeri and Maserio estimated MMNL models under RUM and RRM to research freight transport. The comparison result showed noticeable differences in mean elasticity estimates and market shares. They suggested that regret became an important paradigm of choice when a negative change in the reference point was introduced [21]. Van Cranenburgh et al. introduced a scale under fixed parameters to build a μRRM model and concluded that this noticeably improved the relative fit of RRM compared to RUM models. However, the authors were unaware of any RRM modeling with the accommodated scale and the taste heterogeneity [22]. Jinhee et al. introduced a hybrid choice model based on RRM to study the effects of satisfaction and uncertainty in car-sharing decisions. However, none of the individual socioeconomic characteristics were considered in the model [23]. Hensher established MNL and MMNL models based on RUM and RRM, respectively, to study residents’ travel choices after a new urban rail line was built in Sydney. The results showed that the goodness of fit of the RUM-MMNL model and the RRM-MMNL model were significantly improved compared to the RUM-MNL model and the RRM-MNL model. However, the difference in goodness of fit between the MNL and the MMNL models based on different theories is slight. Moreover, this was the first study to obtain mixed logit elasticities for RRM. This study estimated several models with socioeconomic characteristics such as age, gender, and income. However, it is a pity that none of these variables were statistically significant in any modal alternatives. Furthermore, Hensher suggested that more empirical studies were needed to analyze the applicability of different theories and heterogeneous modeling methods [14]. Jang et al. suggested that regret may be based on the perception of physical attribute differences. In their study, regret was defined as the ratio of attribute differences to absolute attribute values of the considered alternative instead of attribute differences between the considered choice alternative and the non-chosen choice alternative. Thus, Weber’s law and generalized Weber’s law were introduced into the regret function regarding perception heterogeneity. The suggested models were tested based on two data sets. One data set concerned the stated choice of shopping center, and the other data set concerned revealed preference (RP) of mode choice. The goodness-of-fit model results and the K-fold validation tests suggested that the improved models performed better. However, individual heterogeneity was missing because none of the socioeconomic characteristics was considered in the models [24].

2. Motivation and Contribution

As mentioned above, most existing studies on heterogeneity based on regret theory focus on the decision-making process. These studies usually proposed a hybrid model combining the RRM and RUM models. However, they did not consider any preference heterogeneity or individual variables. Individual socioeconomic characteristics were also missing in several studies that considered preference heterogeneity or psychological characteristics. Thus, there is a lack of consideration of perception heterogeneity and individual variables together in travel choice behavior. Generally speaking, the regret functions of different traditional RRM models cannot capture the heterogeneity of travelers. Thus, the choice model ignores the heterogeneity among travelers, which can cause the model analysis results to be inconsistent with the real choice behavior. Therefore, to make the model describe and explain the behavior of the real travel choice more accurately, heterogeneity must be considered to improve the traditional RRM model.

This paper considers adding heterogeneity into the model to improve the classic RRM model. This paper attempts to enhance the accuracy and explanatory ability of the model from a theoretical perspective to describe travelers’ real travel choices and characteristics more accurately. Moreover, this paper further expands the application of regret theory in studying urban travel choice behavior and provides theoretical support for urban transportation planning and operation management. In this paper, Weber’s law in psychophysics is used to consider the heterogeneity of the traveler’s perception of the attributes of various travel modes to improve the Classic RRM model. Subsequently, because the random regret function is a convex function with asymmetric and nonlinear form, the attributes of the individual’s socioeconomic characteristics cannot be directly added to the regret function. Therefore, a new parameter is introduced into the regret function to reflect the heterogeneity of the socioeconomic characteristics of the individual. Then, both improvements are considered together to establish the choice model. Finally, the taxi-hailing choice behavior in Chengdu is taken as an empirical study, and the improvement effects are analyzed based on survey data from the questionnaire. Meanwhile, policy discussion is analyzed in the end.

3. Model Structure

As mentioned above, there is a lack of consideration for heterogeneity in the studies of traditional choice models under RRM. In this section, we will frame the model structure to consider perception heterogeneity and individual heterogeneity. This paper develops the model structure based on MNL under RUM and RRM, respectively. Two models under RUM contain the basic comparison model without individual socioeconomic characteristics and the model with individual socioeconomic characteristics. As for the models under RRM, the basic comparison models also do not consider heterogeneity. Moreover, the rest of the models respectively consider perception only, individual socioeconomic characteristics only, and both perception and individual socioeconomic characteristics. Weber’s ratio is introduced in the regret function to consider the perception of attributes. Moreover, new parameters are introduced in the regret function to consider the impact of individual socioeconomic characteristics.

3.1. MNL Models Under RUM

Choice models under RUM are most widely used in research on travel choice behavior. The RUM theory assumes that travelers are completely rational when making a decision and would choose the alternative with the maximum utility.

According to the RUM theory, the probability of alternative i will be chosen by traveler n can be expressed as follows:

()

where U_ni, U_nj are the utility of alternative i and alternative j for traveler n respectively. C_n is the set of alternatives. The utility consists of observable fixed term V_ni and unobservable random error term ε_ni, thus:

()

where ASC_i is the constant of alternative i. x_nik is the attribute k of alternative i. And θ_k is the corresponding parameter of attribute k. When considering individual socioeconomic characteristics in the model, the V_ni can be written as follows:

()

where x_nil is the individual attribute l of traveler n, and θ_il is the corresponding parameter of alternative i.

When random error ε_ni is assumed to follow the Gumbel distribution, the general formula of the RUM-MNL model can be written as

()

To facilitate comparison of the results of the follow-up model, we will label the model under RUM without individual socioeconomic characteristics as RUM-MNL and the model under RUM with individual socioeconomic characteristics as iscRUM-MNL.

3.2. MNL Models Under RRM

In the travel behavior study, the regret theory pays attention to the psychological state of travelers’ decision-making process, which assumes that travelers follow the limited rational decision-making principle and pursue the alternative with regret minimization. This paper develops choice models based on the Classic RRM model, which assumes that travelers would compare each attribute of the selected alternative to each attribute of the unselected alternative to pursue the minimum regret.

When traveler n faces a travel choice, the number of alternatives is I, each alternative i(i = 1, 2, ⋅⋅⋅I) is influenced by attribute m(m = 1, 2, ⋯M). According to Classic RRM model, when making a choice, the decision maker would aim to minimize random regret. The random regret value R_rni of alternative i is composed of a deterministic term R_ni and an independent distributed random error ε_ni:

()

The deterministic term R_ni is the sum of all attribute regret values obtained by comparing alternative i with another alternative j(i ≠ j), thus:

()

where R_nijm is attribute regret value determined by comparing results of attribute m for alternative i and alternative j; x_nim and are the values of attribute m for alternative i and alternative j respectively; β_m is corresponding parameter.

The random regret value is equivalent to its opposite number. Assuming that the random error obeys the distribution of Gumbel, the probability P_ni of traveler n choosing alternative i can be described as follows:

()

In the traditional RRM-MNL choice model, travelers would compare each attribute of the chosen alternative with other alternatives to choose the alternative that minimizes regret. Moreover, the absolute size of attribute values for different alternative (x_njm − x_nim) is used to consider the comparison performance. However, in the comparison process, the perception of various attributes of travelers is quite different. Considering people’s perception in real choice situations, the same difference in attributes may be perceived as more minor when the size of the attribute is larger. For instance, when a traveler chooses a travel mode that costs $20, he would feel a certain amount of regret if another non-chosen mode costs $10. However, if the cost of the chosen alternative is $110, and the cost of another non-chosen alternative is $100, would the traveler feel the same amount of regret? Based on traditional regret models, the answer would be yes because regret only depends on differences between attributes. However, the difference in the exact $10 travel cost would likely be perceived by the traveler in an entirely different way in these two choice situations. That is perception heterogeneity. Traditional RRM cannot account for this kind of heterogeneity.

In order to consider perception heterogeneity in the RRM-MNL model, this paper introduces Weber’s law into the regret function. In 1834, the German physiologist Weber, in studying people’s sense of weight, found that only when two heavier objects than the weight of two lighter objects have a more significant difference can people feel as two objects of unequal weight. It is more difficult to distinguish two heavier objects with a more significant difference threshold. The difference threshold, also known as the minimum perceptible difference, refers to the amount of stimulus change that can cause different feelings in people. Weber found that the difference threshold of the stimulus varied linearly with the intensity of the stimulus itself, and the amount of stimulus required to produce a minimum perceptible difference was always equal to the product of the current stimulus and a fixed fraction. In the experiment, Weber also found that a fraction can express the minimum perceptible difference. Although this fraction would change with the different sensory channels tested, it is unchanged for the same sensory channel. Hence, he pointed out that it is possible to determine a constant minimum perceptible difference fraction for each human sense. Weber’s law is an important psychophysical theorem that suggests the relationship between psychological quantity and physical quantity; that is, the difference threshold of sense changes with the change in the quantity of the original stimulus. Moreover, the change tends to a certain regularity: The ratio of stimulus increment to original stimulus value is constant. The formula of Weber’s law is as follows:

()

where Φ is original stimulus quantity, ΔΦ is difference threshold, C is fixed value of a particular sensation, which is a constant. Although Weber’s law is based on psychophysical research that has traditionally focused on the perception of change in physical stimulus, such as weight and sound, the perception of other stimulus in travel choice research may be driven by similar perceptual mechanisms. Travel behavior could be regarded as a kind of special commodity or service, and travelers are buyers. Therefore, in the study of travel mode choice behavior, the attributes of various travel modes could be regarded as the stimulus that affects travelers’ choices.

When Weber’s law is introduced into the regret function, the deterministic term R_ni can be described as follows:

()

In equation (9), the (x_njm − x_nim) in equation (7) is replaced by (x_njm − x_nim)/x_nim, which is the Weber’s ratio of attribute m for alternative i and alternative j.

Assuming that traveler n has the number of L individual’s socioeconomic attributes, each attribute l(l = 1, 2 · ⋅⋅L) would affect the choice behavior. Specifically, travelers with different attribute l have different evaluations on the performance of alternative attribute m. In traditional RRM-based models, these individual socioeconomic attributes cannot be introduced directly into the regret function. To describe such different subject evaluations of level-of-service attributes, parameter β_lm is introduced into the regret function, and the corresponding attribute value is x_nl. Hence, replacing β_m with β_m + β_lm·x_nl in regret function, so the deterministic term R_ni can be described as follows:

()

where β_lm reflects that traveler with different individual socioeconomic attribute l has different preferences for alternative attribute m. In other words, the random regret value generated by travelers in the alternative i is not only related to the attribute performance of the alternative attribute, but also affected by the individual socioeconomic characteristics.

When considering both perception heterogeneity and individual socioeconomic characteristics in the choice model, the deterministic term R_ni can be described as follows:

()

In order to facilitate the comparison of the results of the follow-up model, we will label the basic model under RRM without any heterogeneity as RRM-MNL, the model under RRM with perception as pRRM-MNL, the model under RRM with individual socioeconomic characteristics as iscRRM-MNL, the model under RRM with both perception and individual socioeconomic characteristics as p&iscRUM-MNL.

4. Empirical Setting and Data

4.1. Survey Design

In this paper, we adopt the RP and the stated preference (SP) in Chengdu to obtain the results on the choice behavior of taxi-hailing. Chengdu is a megacity and national central city in China. With the rapid development in recent years, Chengdu’s urban transportation system has become increasingly rich and perfect. On the one hand, Chengdu has overtaken Beijing in terms of the number of motor vehicles in China. On the other hand, Chengdu has a developed public transportation system. In recent years, online taxi-hailing has gradually become an important travel mode in residents’ daily lives. As an emerging mode of travel, online taxi-hailing has brought a certain impact on other traditional modes, especially the traditional taxi. Gito et al. used demand-supply and dynamic models to evaluate the demand for taxis and online taxi-hailing in the Jakarta Greater Area, Indonesia. The results suggested that taxi-hailing service quality was better than traditional taxis based on variables such as waiting time, travel time, and travel cost [25, 26]. In order to give full play to the advantages and functions of various transportation modes in the urban transportation system, it is necessary to explore and analyze the transportation mode choice behavior of different residents according to their travel characteristics. Therefore, we choose the choice behavior of taxi-hailing in Chengdu as a case study. We choose the corridor between the southwest Jiaotong University cluster and the Tianfu Square cluster as the research project. The southwest Jiaotong University cluster is a typical residential area in the north of Chengdu. Moreover, Tianfu Square is in the middle of Chengdu and is the center of business, shopping, and culture.

The alternative sets in this paper include five travel modes along the corridor: bus, subway, traditional taxi (taxi), taxi-hailing, and private car (car). Moreover, attributes of alternatives include cost, waiting time, and in-vehicle time. For the alternatives of bus and subway, the cost is the ticket price of each mode, the waiting time is the time from travel to the station to get on the bus or subway, and the in-vehicle time is the whole trip time in the bus or subway. For the alternatives of traditional taxi and taxi-hailing, the cost is the fee of the car trip, the waiting time is the time from a traveler beginning hailing a car to get on the car, and the in-vehicle time is the whole trip time in the car. As for the alternative of car, the cost is the fuel fee and parking fee, the in-vehicle car time is also the whole trip time in the car. Besides, the waiting time is zero because travelers do not need to wait to drive their own car. The attributes are illustrated in Table 1, and all the attribute levels are based on the real level.

Table 1. Level of alternative attributes.

Travel alternatives	Attributes	Level 1	Level 2
Bus	Cost	¥1	¥2
	Waiting-time	5 min	10 min
	In-vehicle time	40 min	60 min

Subway	Cost	¥3	¥5
	Waiting-time	2 min	5 min
	In-vehicle time	30 min	40 min

Taxi-hailing	Cost	¥25	¥30
	Waiting-time	2 min	5 min
	In-vehicle time	25 min	35 min

Traditional taxi	Cost	¥30	¥35
	Waiting-time	5 min	10 min
	In-vehicle time	25 min	35 min

Car	Cost	¥40	45
	Waiting-time	0	0
	In-vehicle time	25 min	35 min

The complete factorial design has been shown to negatively influence the survey because the number of attribute combinations is enormous. Thus, the orthogonal design was chosen to achieve appropriate combinations of attributes. The orthogonal design would ensure the independence of each attribute and keep off incorrect results translated from the rooted multicollinearity problem of the attribute. The accuracy of model results can be effectively improved. The article applies orthogonal design to obtain 16 combinations of mutually orthogonal attributes for each questionnaire. In addition, each questionnaire contains an RP survey with taxi-hailing characteristics. Individual socioeconomic attributes are also collected. An example of a questionnaire is shown in Figure 1. The RP survey method could objectively reflect the market supply and demand situation and be reliable and confident. However, the most obvious shortcoming is the inability to use RP surveys to study travelers’ choices of alternatives that do not exist yet. In addition, using RP surveys may lead to a lack of diversity of attribute levels and the existence of unselected alternatives. These problems would make the model analysis difficult and inaccurate. Compared to the RP survey method, the SP survey method could analyze the alternatives that do not exist yet. Moreover, SP surveys could reduce the sample size and improve investigation efficiency. Although the SP survey method has many outstanding advantages, the hypothetical bias cannot be ignored. Because respondents are faced with a series of designed hypothetical situations or alternatives, this kind of choice in the hypothetical situation and the real choice is biased. Hypothetical bias implies that respondents’ real choices may differ completely from their choices when making a hypothesis. In order to mitigate the potential biases, we need to minimize the sampling bias. Moreover, this paper uses an SP survey to obtain the respondents’ hypothetical choice and an RP survey to obtain taxi-hailing characteristics and individual socioeconomic variables. In the process of data denoising, the SP choice and RP data of each respondent are combined to clean the samples with apparent contradictions and finally get valid samples.

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

An example of the questionnaire.

4.2. Descriptive Profile of the Data

We conducted face-to-face surveys in the area of the southwest Jiaotong University cluster and the Tianfu Square cluster in Chengdu, China. The first survey was conducted from August to September 2019, before the pandemic condition. The second survey was conducted from March to April 2024, after the pandemic condition. Through random sampling, investigators asked 1000 respondents to fill in the questionnaire in the first survey. After selecting incomplete questionnaires and questionnaires with obvious filling errors, the total number of valid samples acquired from the first survey is 928, and the effective rate is 92.8%. The statistical results of the valid surveyed sample population are shown in Table 2. The age distribution of the survey sample is consistent with the sampling plan. The gender rate between males and females is 0.97, and the average income is ¥5863, consistent with the 2019 citywide census (0.98, ¥5931). The above indicates that the sampling bias is minimal. In the second survey, 543 respondents filled out the questionnaire. The total number of valid samples acquired from the second survey is 496, and the effective rate is 91.3%. The statistical results of the valid surveyed sample population are shown in Table 3. The age distribution of the survey sample is consistent with the sampling plan. The gender rate between males and females is 0.98, and the average income is ¥7166, consistent with the latest citywide census (0.98, ¥7654). The above also indicates that the sampling bias is minimal. A sample of the SP survey is shown in Table 4.

Table 2. Statistical results of the surveyed sample population before the pandemic condition.

	Levels	Sample size	Percentage (%)
Gender	Male	458	49.35
Gender	Female	470	50.65

Age	18–25	24	2.59
	26–30	144	15.52
	31–35	241	25.97
	36–40	208	22.41
	41–45	163	17.56
	46–50	84	9.05
	51–55	52	5.60
	55以上	12	1.29

Income	< 3000	156	16.81
	3000–5000	237	25.54
	5001–8000	337	36.31
	8001–10,000	146	15.73
	> 10,000	52	5.61

Education	Below undergraduate	426	45.91
	Undergraduate	406	43.75
	Master or above	96	10.34

Whether have private cars	Yes	375	40.41
Whether have private cars	No	553	59.59

Table 3. Statistical results of the surveyed sample population after the pandemic condition.

	Levels	Sample size	Percentage (%)
Gender	Male	246	49.60
Gender	Female	250	50.40

Age	18–25	24	2.91
	26–30	144	15.63
	31–35	240	25.86
	36–40	206	22.20
	41–45	162	17.46
	46–50	84	9.05
	51–55	52	5.60
	55 that’s all	12	1.29

Income	< 3000	156	16.81
	3000–5000	237	25.54
	5001–8000	337	36.31
	8001–10,000	146	15.73
	> 10,000	52	5.61

Education	Below undergraduate	426	45.91
	Undergraduate	406	43.75
	Master or above	96	10.34

Whether have private cars	Yes	375	40.41
Whether have private cars	No	553	59.59

Table 4. A sample of SP survey.

Alt	Choice	Cost	Waiting time	In-vehicle time	Gender	Edu	Age	Inc
1	0	1	10	40	1	2	3	3
2	1	3	5	30	1	2	3	3
3	0	45	0	35	1	2	3	3
4	0	30	5	35	1	2	3	3
5	0	30	5	35	1	2	3	3

Note: Alt 1, 2, 3, 4, 5 are alternative bus, subway, car, traditional taxi, and taxi-hailing, respectively. Choice = 1 is the selected mode. Edu is the education background, and Inc is the monthly income.

In addition, statistical analysis is conducted on the use characteristics of online taxi-hailing before and after the pandemic condition, and the results are shown in Figure 2. The results suggest that the popularity rate of online taxi-hailing among Chengdu residents has increased. Only three respondents said they had never used the taxi-hailing, accounting for only 1% of the total sample. 38% of respondents said they would have at least one taxi-hailing trip every month, which is a significant increase compared to which is 29% before the pandemic condition. As for the trip purpose of using taxi-hailing, the proportion of each purpose increased after the pandemic. Combined with the use frequency, it suggested that online taxi-hailing has become a travel choice for more residents after the pandemic condition. After the pandemic condition, price discount is still the most important reason for travelers to choose taxi-hailing, and the proportion has increased significantly. Moreover, a comfortable in-vehicle environment became the secondary reason for travelers to choose taxi-hailing based on a significant increase. This may be because the fact that the economy has not fully recovered from the pandemic condition; the residents are more sensitive to prices. Meanwhile, residents are more concerned about a comfortable and clean in-vehicle environment. The proportion of people choosing non-cash payment as an influential factor has decreased significantly, which may be related to the fact that all other travel modes now support online payment. In terms of the substitution of taxi-hailing for other modes, it is basically consistent before and after the pandemic conditions. However, more people who do not have private cars said that taxi-hailing has replaced their private car travel and bus travel after the pandemic condition.

5. Model Results and Discussion

5.1. Model Calibration

All six models have been estimated using the Nlogit version 6, with two models under RUM and four under RRM [27]. The alternative car is the base reference alternative. According to the statistical results of the sample population surveyed listed in Tables 2 and 3, the level and coding of the individual attribute variables are shown in Table 5. All individual attribute variables are binary indicator variables.

Table 5. Individual attribute levels and coding.

Attribute	Levels and coding
Gender	Male = 1
Age1	18–35, yes = 1
Age2	36–50, yes = 1
Age3	> 50, yes = 1
Income1	< 5000, yes = 1
Income2	5001–10,000, yes = 1
Income3	> 10,000, yes = 1
Education1	Below undergraduate, yes = 1
Education2	Undergraduate, yes = 1
Education3	Master or above, yes = 1
Whether have private cars	Yes = 1

To compare the models regarding their overall fit and complexity, the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), which are widely used, are calculated in this paper. The AIC and BIC can be calculated, respectively, as follows [28, 29]:

()

where ln L is the maximum value of the log-likelihood m is the number of estimated parameters, and N is the number of samples.

5.2. Model Results

The results of three models that do not consider individual attributes are summarized in Table 6 (before the pandemic condition) and Table 7 (after the pandemic condition). The results of the other three models considering individual attributes are summarized in Table 8 (before the pandemic condition) and Table 9 (after the pandemic condition). Tables 8 and 9 only show the individual attribute parameters that are statistically significant.

Table 6. Estimation results for models without individual attributes before the pandemic condition (t-values in parentheses).

Attribute	RUM-MNL	RRM-MNL	pRRM-MNL
Bus constant	−4.2132 (−3.721)	−3.1254 (−3.342)	−3.2341 (−3.623)
Subway constant	4.4368 (3.932)	2.9320 (4.358)	3.4312 (4.521)
Taxi constant	−3.7654 (−2.345)	−2.1323 (−2.931)	−2.2315 (−3.013)
Taxi-hailing constant	5.6743 (4.231)	3.9325 (5.643)	4.1281 (5.877)
Waiting time	−0.1165 (−2.852)	−0.0903 (−2.832)	−0.1004 (−2.931)
In-vehicle time	−0.1070 (−2.134)	−0.0732 (−1.998)	−0.0837 (−2.004)
Cost	−0.1698 (−3.305)	−0.1253 (−3.237)	−0.1431 (−3.536)
Log-likelihood (no parameters)	−1145.43
Log-likelihood at convergence	−869.38	−843.04	−821.27
R²	0.241	0.264	0.283
AIC	1752.76	1700.08	1656.54
BIC	1786.59	1733.91	1690.37
Hit rate	57.5%	67.3%	71.5%

Table 7. Estimation results for models without individual attributes after the pandemic condition (t-values in parentheses).

Attribute	RUM-MNL	RRM-MNL	pRRM-MNL
Bus constant	−4.2214 (−3.814)	−3.1306 (−3.395)	−3.2375 (−3.877)
Subway constant	4.287 (3.687)	2.9248 (3.882)	3.4275 (4.246)
Taxi constant	−3.7674 (−2.387)	−2.1387 (−3.204)	−2.2364 (−3.324)
Taxi-hailing constant	5.4201 (3.987)	3.9108 (5.114)	4.1042 (5.523)
Waiting time	−0.1148 (−2.785)	−0.0887 (−2.801)	−0.0989 (−2.746)
In-vehicle time	−0.1062 (−2.104)	−0.0721 (−1.971)	−0.0801 (−1.985)
Cost	−0.1756 (−3.498)	−0.1289 (−3.487)	−0.1456 (−3.872)
Log-likelihood (no parameters)	−1259.86
Log-likelihood at convergence	−953.714	−931.037	−903.32
R²	0.243	0.261	0.283
AIC	1921.43	1876.07	1820.64
BIC	1955.26	1909.90	1854.47
Hit rate	56.9%	68.5%	72.2%

Table 8. Estimation results for models with individual attributes before the pandemic condition (t-values in parentheses).

Attribute	iscRUM-MNL	Attribute	iscRRM-MNL	p&iscRRM-MNL
Bus constant	−3.8342 (−3.214)	Bus constant	−2.7138 (−2.975)	−2.8214 (−3.102)
Subway constant	4.0134 (3.431)	Subway constant	2.5153 (3.317)	2.4231 (3.421)
Taxi constant	−3.2637 (−2.135)	Taxi constant	−1.9422 (−1.981)	−1.8421 (−2.003)
Taxi-hailing constant	5.1412 (3.748)	Taxi-hailing constant	3.8246 (4.054)	3.935 (4.423)
Cost	−0.1093 (−2.623)	Cost	−0.086 7 (−1.984)	−0.0926 (−2.134)
Waiting time	−0.0971 (−2.041)	Waiting time	−0.0891 (−2.721)	−0.0952 (−2.321)
In-vehicle time	−0.1643 (−3.211)	In-vehicle time	−0.1187 (−2.972)	−0.1162 (−3.321)

Individual attribute		Individual attribute

Bus-age3	0.0638 (1.982)	Waiting-time-gender	−0.0131 (−1.983)	−0.0156 (−2.215)
Bus-income1	0.0763 (2.213)	Waiting-time-age1	−0.0128 (−1.978)	−0.0139 (−1.994)
Bus-income3	−0.0832 (−2.547)	Waiting-time-age3	0.0154 (2.140)	0.0183 (2.368)
Bus-car ownership	−0.0461 (−2.236)	Waiting-time-income3	−0.0241 (−2.822)	−0.0256 (−2.948)
Taxi-income2	0.0739 (2.524)	Waiting-time-education3	—	−0.0112 (−1.988)
Taxi-income3	0.0758 (2.573)	In-vehicle time-age1	−0.0173 (−2.231)	−0.0245 (−2.586)
Subway-income1	0.0682 (2.138)	In-vehicle time-age3	0.0166 (2.226)	0.0184 (2.289)
Taxi-hailing-age1	0.0804 (2.422)	In-vehicle time-income3	−0.0146 (−1.982)	−0.0191 (−2.468)
Taxi-hailing-age3	−0.0662 (−1.991)	In-vehicle time-car ownership	—	−0.0144 (−2.037)
Taxi-hailing-income2	0.0712 (2.438)	Cost-age3	−0.0193 (−2.562)	−0.0294 (−2.912)
Taxi-hailing-car ownership	0.0684 (2.035)	Cost-income1	−0.0274 (−2.833)	−0.0293 (−3.138)
		Cost-income3	0.0235 (2.643)	0.0256 (2.655)
		Cost-education3	—	0.0162 (2.211)
		Cost-car ownership	0.0196 (2.329)	0.0265 (2.728)
Log-likelihood (no parameters)	−1145.43
Log-likelihood at convergence	−840.75		−805.24	−780.04
R²	0.266		0.297	0.319
AIC	1735.50		1654.48	1604.08
BIC	1865.99		1760.81	1710.41
Hit rate	62.6%		71.0%	74.8%

Table 9. Estimation results for models with individual attributes after the pandemic condition (t-values in parentheses).

Attribute	iscRUM-MNL	Attribute	iscRRM-MNL	p&iscRRM-MNL
Bus constant	−3.8389 (−3.334)	Bus constant	−2.7174 (−3.022)	−2.8230 (−3.197)
Subway constant	3.8551 (3.207)	Subway constant	2.3876 (3.187)	2.2578 (3.281)
Taxi constant	−3.2688 (−2.198)	Taxi constant	−1.9480 (−2.032)	−1.8442 (−2.233)
Taxi-hailing constant	4.9537 (3.576)	Taxi-hailing constant	3.7467 (3.887)	3.687 (4.136)
Cost	−0.1658 (−3.218)	Cost	−0.1004 (−2.727)	−0.1088 (−3.025)
Waiting time	−0.0982 (−2.087)	Waiting time	−0.0791 (−2.514)	−0.0922 (−2.284)
In-vehicle time	−0.1238 (−2.839)	In-vehicle time	−0.0867 (−2.189)	−0.1017 (−2.561)

Individual attribute		Individual attribute

Bus-age3	0.0657 (1.976)	Waiting-time-gender	−0.0140 (−1.988)	−0.0151 (−2.182)
Bus-income1	0.0693 (2.167)	Waiting-time-age1	−0.0133 (−1.982)	−0.0145 (−2.011)
Bus-income3	−0.0884 (−2.778)	Waiting-time-age3	0.0150 (2.1277)	0.0174 (2.284)
Bus-car ownership	−0.0419 (−2.098)	Waiting-time-income3	−0.0257 (−2.874)	−0.0259 (−2.957)
Taxi-income2	0.0788 (2.617)	Waiting-time-education3	—	−0.0132 (−1.992)
Taxi-income3	0.0812 (2.832)	In-vehicle time-age1	−0.0168 (−2.217)	−0.0241 (−2.566)
Subway-income1	0.0677 (2.083)	In-vehicle time-age3	0.0169 (2.244)	0.0192 (2.300)
Taxi-hailing-age1	0.0841 (2.491)	In-vehicle time-income3	−0.0159 (−1.998)	−0.0178 (−2.247)
Taxi-hailing-age3	—	In-vehicle time-car ownership	—	−0.0148 (−2.114)
Taxi-hailing-income2	0.0775 (2.554)	Cost-age3	−0.0215 (−2.757)	−0.0305 (−2.988)
Taxi-hailing-car ownership	—	Cost-income1	−0.0285 (−2.908)	−0.0323 (−3.326)
		Cost-income3	0.0206 (2.239)	0.0241 (2.498)
		Cost-education3	—	0.0151 (2.009)
		Cost-car ownership	0.0182 (2.087)	0.0243 (2.543)
Log-likelihood (no parameters)	−1145.43
Log-likelihood at convergence	−928.52		−880.64	−850.41
R²	0.263		0.301	0.325
AIC	1736.50		1805.08	1744.81
BIC	2041.53		1911.61	1851.14
Hit rate	60.9%		73.4%	75.2%

5.3. Discussion

5.3.1. Constant and Generic Parameters

According to the model results, all four constants are significant, and the signs of the parameters are the same, which suggests that unobserved factors have different impacts. As for alternative attribute variables, three generic parameters are statistically significant with negative signs. It suggests that waiting time, in-vehicle time, and cost significantly influence travelers’ choice behavior. In addition, we can find some differences between the RUM and RRM models. The absolute values of the coefficient and the t-test of the models under RRM are smaller than those under RUM, which may reflect the different decision-making principles of RUM and RRM. Comparing the RRM-MNL and the pRRM-MNL model, the iscRRM-MNL model, and the p&iscRRM-MNL mode, respectively, the absolute values of parameters are increased in both models, which consider perception. This seems reasonable since the attribute difference is divided by the attribute value, so more extensive parameters are needed to represent similar attribute-level regret.

Comparing the model results before and after the pandemic condition, the absolute values of negative constant parameters and t-test (bus and taxi) after the pandemic condition increased. At the same time, the absolute values of positive constant parameters and t-test (subway and taxi-hailing) decreased. This may indicate that compared to private cars, the attractiveness of the other four modes decreased after the pandemic condition. As for three generic variables, the absolute values of cost parameters and t-test are increased, while those of waiting time and in-vehicle time are decreased. It probably suggests that residents have become more sensitive to costs after the pandemic condition because of the economic situation.

5.3.2. Individual Parameters

The different results of the individual attribute estimation reflect the different impacts of socioeconomic characteristics on travelers’ choice behavior. When we compare the iscRUM-MNL model with the iscRRM-MNL model, the individual attributes of the two models are quite different because of the different construction of models under RUM and RRM.

In the iscRUM-MNL mode, for alternative buses and subways, the income1 parameters are both statistically significant. The parameter signs are positive. It suggests that low-income people would prefer cheaper public transport. Regarding alternative transport modes such as bus and taxi, the age3 parameter is statistically significant. The parameter signs for the bus are positive, while those for the taxi are negative. Meanwhile, the age1 parameter for taxi-hailing is positive. It suggests that older people prefer public buses, while taxis are more popular among young people. This may be because the elderly are usually thrifty and do not value travel time specifically, so they prefer a public bus. Meanwhile, the elderly may not be skilled in using smartphones, so they have a low utilization rate of online taxi-hailing. As for the income2 attribute, the parameter signs for taxi and taxi-hailing are positive. This suggests that middle-income groups have a certain preference for traveling by car. The results of the income3 parameters for alternative buses and taxis show that high-income people prefer the taxi instead of the bus. Finally, the attributes of car ownership for bus and taxi-hailing are statistically significant, and the parameter signs for bus are negative while for taxi-hailing is positive, which suggests that taxi-hailing is more attractive compared with public bus for people possessing private cars.

In the models under RRM, the parameters of individual attributes reflect travelers with different socioeconomic characteristics, whether paying attention to the different mode alternative attributes. The attribute of gender for waiting time is statistically significant, and the parameter sign is negative, which suggests that male travelers are more sensitive than female travelers to the increased waiting time. This may be because women are generally more patient than men. The attributes of age1 and age3 for waiting time and in-vehicle time, as well as the attributes of age3 for cost, are statistically significant. The different parameter signs suggest that older people are sensitive to the increase in travel cost instead of the travel time. Generally, the elderly are relatively rich in time because they are not commuters, while they are relatively thrifty and pay more attention to the cost. On the contrary, young people would pay more attention to travel time. As for the attributes of income, the parameters of income3 for three alternative attributes are statistically significant, and the parameter signs for the waiting time and the in-vehicle time are negative. In contrast, the sign for the cost is positive. This indicates that higher-income travelers are more sensitive to travel time than travel costs. Groups of higher income usually pay more attention to the value of time and the quality of the trip rather than the cost. Meanwhile, the parameter sign of income1 for cost is negative, which suggests low-income people are more sensitive to the cost. Finally, car owners also show less sensitivity to increased travel costs.

In conclusion, analyzing individual attributes mainly reflects the influence of different socioeconomic characteristics on travelers’ preference for mode choice in the RUM-MNL model. In contrast, the influence of individual attributes is reflected in the sensitivity to alternative attributes of different modes of travelers with different socioeconomic characteristics in the RRM-MNL model. Furthermore, when comparing the results of the iscRRM-MNL model and the p&iscRRM-MNL model, we can see that the absolute values of parameter values and t-test values for individual attributes are more significant in the model considering both perception heterogeneity and socioeconomic characteristics. Meanwhile, the attributes of education3 for the waiting time and the cost and the attribute of the car ownership for the in-vehicle time become statistically significant. Travelers with higher education are more sensitive to the increased waiting time. Moreover, car owners are more sensitive to travel time than travel costs. This may be because travelers pay more attention to road congestion when driving cars than to fuel and parking fees. It can be concluded that considering the heterogeneity of travelers’ perception of the attributes of the travel mode could improve the ability of the model to capture the impact of socioeconomic characteristics on travelers’ sensitivity to each alternative attribute of the mode.

The model results of individual parameters are consistent before and after the pandemic conditions. In the iscRUM-MNL model, the variable of taxi-hailing–age3 became not significant. It may be because taxi-hailing has become an important travel mode, and more elders have learned to use a smartphone to hail a car. What is more, the difference in “cost–income” variables after the pandemic condition suggests that cost has more impact on travelers’ mode choice behavior after the pandemic condition. This is consistent with the analysis of the above generic parameters.

5.3.3. Model Comparison

The comparison of the goodness of fit (R²), AIC and BIC, and hit rate are summarized in Tables 10 and 11. The results are consistent whether before or after the pandemic condition.

Table 10. Model evaluation index before pandemic condition.

	RUM-MNL	RRM-MNL	pRRM-MNL	iscRUM-MNL	iscRRM-MNL	p&iscRRM-MNL
R²	0.241	0.264	0.283	0.266	0.297	0.319
AIC	1752.76	1700.08	1656.54	1735.50	1654.48	1604.08
BIC	1786.59	1733.91	1690.37	1865.99	1760.81	1710.41
Hit rate	57.5%	67.3%	71.5%	62.6%	71.0%	74.8%

Table 11. Model evaluation index after pandemic condition.

	RUM-MNL	RRM-MNL	pRRM-MNL	iscRUM-MNL	iscRRM-MNL	p&iscRRM-MNL
R²	0.243	0.261	0.283	0.263	0.301	0.325
AIC	1921.43	1876.07	1820.64	1736.50	1805.08	1744.81
BIC	1955.26	1909.90	1854.47	2041.53	1911.61	1851.14
Hit rate	56.9%	68.5%	72.2%	60.9%	73.4%	75.2%

According to the tables, the overall goodness of fit and hit rate of the models under RRM are better than those under RUM. Meanwhile, the values of AIC and BIC of models based on RRM are smaller than those of models based on RUM. These suggest that RRM models have certain advantages over RUM models in this study. As we can see, the goodness of fit and hit rate of the pRRM-MNL model is better than the RRM-MNL model, which suggests that the improved RRM-MNL model considering perception heterogeneity by Weber’s law has a particular advantage in describing and explaining the behavior of real choice. The AIC and BIC values of the pRRM-MNL model are smaller than those of the RRM-MNL model, suggesting the same result. Furthermore, when considering individual socioeconomic characteristics, the goodness of fit and hit rate of both models under RUM and RRM are better than those without individual attributes, suggesting that considering socioeconomic characteristics can improve the explanatory and predictive ability of the choice models. However, the values of AIC and BIC have different results. Both AIC values in the iscRUM-MNL model and the iscRRM-MNL model are smaller than those in models without socioeconomic characteristics. However, both values of BIC in the iscRUM-MNL model and iscRRM-MNL model are larger. More parameters likely make the model more complex. The p&iscRRM-MNL model has the best goodness of fit and hit rate among the six models. Meanwhile, the value of AIC is the smallest, and the value of BIC is only larger than that of the pRRM-MNL. It suggests that simultaneously considering both perception and individual socioeconomic characteristics in the model has a specific effect on improving the model under RRM. On the one hand, the model evaluation indexes are overall better than other models. On the other hand, the improved model considers the influence of travelers’ individual socioeconomic characteristics. Moreover, considering the perception makes more variables significant. We can do more detailed analysis and research based on the model results. Therefore, the improved p&iscRRM-MNL model could better explain and predict the real choice behavior of travelers.

In this study, we only considered the heterogeneity of perception. Other heterogeneities, such as travel habits, lifestyle preferences, etc., which could influence travelers’ choice behavior, could be considered in the regret function to improve the model in future research. In addition, in order to study heterogeneity under RRM, the model structure could be extended to some other model forms, such as the mixed logit model and the latent class model. What is more, the pandemic condition has brought certain impact on residents’ travel choice behavior. More future research is needed to analyze travel behavior after the pandemic condition.

5.3.4. Policy Discussion

Online taxi-hailing has become a new travel choice for residents in China. Thus, taxi-hailing has become an important part of the urban traffic system. However, the survey results and model results in this paper indicated that taxi-hailing has brought a certain impact on other means of transportation, especially the traditional taxi and public bus. Meanwhile, urban traffic congestion is becoming more and more serious with the rapid development of the complex urban traffic system. Therefore, there are some possible policy recommendations based on the results of both the RUM and RRM models in this paper. According to the model results, the travel cost, in-vehicle time, and waiting time have a significant impact on travelers’ choice behavior. As for public buses, subway, and taxi-hailing, the cost is the most important factor. Meanwhile, middle-income and low-income people and the elderly are generally cost-sensitive; they are more sensitive to travel costs than travel time. While the high-income people care more about the timeliness of travel instead of cost. Therefore, during peak commuting hours with high travel demand and road traffic pressure, we can reduce the discount on taxi-hailing or even raise the price of taxi-hailing on the one hand and the other hand, reducing the discount on public transportation for the elderly, and giving more discounts to the elderly in other periods. Meanwhile, give more price discounts to commuters. Besides, public transportation should improve service efficiency to reduce waiting time and in-vehicle time. In this way, the trips of the elderly can be reduced to some extent during peak hours, and certain pressures can be relieved. At the same time, public transportation could retain the existing low-income commuters and attract some middle-income and high-income people. In addition, it is also possible to add more high-occupancy vehicle (HOV) lanes to improve the service of taxi-hailing and traditional taxis to attract some high-income car owners to reduce private car trips.

6. Conclusions and Future Work

In this paper, regret is not defined in terms of differences in attributes between the chosen alternative and other non-chosen alternatives but rather as the Weber ratio of attribute differences to the absolute attribute values of the chosen alternative. Additionally, new parameters were introduced into the regret function to describe different subject evaluations of level-of-service attributes. The empirical study results of taxi-hailing choice behavior in Chengdu indicated that the improved model, which considered both perception heterogeneity and socioeconomic characteristics together, had the best goodness of fit and hit rate. The AIC and BIC values also provided a consistent result. Therefore, the improved model, considering perception heterogeneity using Weber’s law and considering socioeconomic characteristics, performed better in describing and predicting travelers’ choice behavior.

Based on this paper, the following research is the regret-based choice model structure under mixed logit and latent class frameworks and whether improving perception heterogeneity with Weber’s law applies to the new models. Furthermore, the behavior of residents’ travel choices after the pandemic condition is another area for future study.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

No funding was received for this research.

Open Research

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Enhancing Random Regret Minimization With Perception and Demographic Heterogeneity Insights: A Taxi-Hailing Case Study in Chengdu, China

Abstract

1. Introduction

2. Motivation and Contribution