1 Introduction

Classical urban transportation planning system, in an aggregate approach, mainly consists of four sub-models: trip generation, trip distribution, modal split, and traffic assignment [1]. In the last stage, the problem is to allocate travel demand of each O–D pair to the network links and to find links’ flows. The solution is known as a network equilibrium state suggested by Wardrop. Wardrop’s principles are interpreted as two states of the network [1,2,3].

User equilibrium (UE) at user equilibrium state each driver’s travel time is minimum and no driver can improve his travel time by no-cooperatively route changing.

System optimum (SO) at system optimum state the traffic flows are in such a way that the total travel time of the whole system (network) is minimum.

Traffic assignment is basically the process of making decision for the route selection between origins and destinations. To solve such a problem, some assumptions should be made about travellers’ decision-making behavior. The basic assumption is that travellers behave rationally [4]. Beckman et al. introduced first mathematical model of network equilibrium based on Wardrop’s principles. It is still the base model of static traffic assignment. Static assignment leads to an equilibrium state that is constant over the time, while dynamic one leads to an equilibrium state varying over the time. Wardrop’s user equilibrium definition implies that all drivers are fully informed about the condition of the network and they always select the best route. These assumptions will not thoroughly match to the reality. Furthermore, drivers have different perception of cost and they want to optimize composite measures [1, 2]. To relax these assumptions and make the model more realistic, Daganzo and Sheffi offered the concept of stochastic user equilibrium (SUE) [5]. At the stochastic user equilibrium each driver’s perceived (anticipated) travel time is minimum and no driver believes that he can improve his travel time by changing the route.

Decision-making process (for route selection) may be simulated by applying discrete choice models. These models predict the probability of choosing a given alternative as a function of its attractiveness [1, 2, 6]. It means the selection process is stochastic, and therefore less attractive alternatives may have still the opportunity to be selected. So far, random utility theory is the most common concept that has been applied to define attractiveness function and generate discrete choice models [1, 2]. Utility models are based on this assumption that individuals desire to achieve the highest utility through making a decision. The utility function consists of two parts: a deterministic (observed) part and a random error part [7, 8]. The random error term indicates that individuals may have different perceptions of the same situation. There is a voluminous literature concerning random utility maximization (RUM) models, such as multinomial logit, nested logit, multinomial probit, generalized extreme value, mixed logit and multiple discrete–continuous extreme value [9].

Kahneman and Tversky presented several cases that selection pattern violates conventional expected utility axioms and they argued that utility theory is not an adequate descriptive model [10]. They have developed an alternative model, called prospect theory. Prospect theory describes the choosing process based on perceived gains and losses relative to a reference point rather than final state of wealth or welfare.

A number of studies have shown that when individuals choose an alternative they will take into account the consequences they would have been experienced by choosing other alternatives [11,12,13,14,15]. An empirical study accomplished by Chorus et al. indicates that degree of satisfaction of choosing a mode or route is dominantly influenced by the regret due to better performance of non-chosen alternatives [16].

Regret theory as an alternative to utility theory was developed by Bell, Fishburn and Loomes and Sugden independently [9]. Chorus et al. have presented a travel choice model based on the regret theory-coined random regret-minimization (RRM) [9]. RRM states that an individual makes a decision in such a way that minimizes his or her perceived regret, i.e., he or she wishes to avoid the situation that non-chosen alternatives bring about more satisfaction. It has been shown that regret-based model outperforms its utility-based rival model and it will be a promising alternative to model and analyze travel choice [16]. Chorus has provided a perfect tutorial on random regret-based discrete choice modeling [17]. He has also presented a Generalized Random Regret-Minimization model, which was more generalized and more flexible, by recasting a fixed constant in the attribute-specific regret functions of the conventional RRM model, into an attribute-specific regret-weight [18]. Ramos et al. have provided a comparison of theory and application of the expected utility theory, prospect theory and regret theory [19].

Charoniti et al. have focused on the effect of context and personality traits on decision-making under uncertainty, using route choice in an activity context as an example. They have estimated a latent class random regret-minimization model, which takes into account the travel time, personality traits, socio-demographic profiles and contextual factors, which increase or decrease travellers’ feelings of regret [20]. Ramos et al. have modelled the route choice problem as a multiagent reinforcement learning scenario. Using a regret minimization objective they have shown how agents can compute regret, as a linear combination of local (experience-based) information and global (app-based) one provided by a mobile (Waze-like) navigation app. They have shown that the system converges to approximate user equilibria [21]. Li and Huang using regret theory, have developed a new combined trip distribution and traffic assignment model, formulated as a variational inequality and solved by a path-based algorithm. It has been shown that regret aversion indeed influences the travellers’ route choice behavior [22]. Maia et al. have focused on the comparison of the random regret-minimization and mother logit models for analyzing the choice between alternatives having deterministic attributes using a random regret-based recursive logit-formulation for link-based route choice models [23]. Rasouli and Timmermans have made two contributions to the works of Chorus [9, 24], namely formal analyses to identify the parameter space where the logarithmic specification became theoretically inferior to the original specification, and an empirical stated choice study on the choice of shopping center to empirically test which specification best described stated choices [25]. The ideas presented in this study were later replied by Chorus [26].

The current study focuses on configuring a closed-form SUE formulation based on the concept of RRM. Proposing an alternative simpler definition of regret, and using it in a well-known stochastic user equilibrium formulation, a regret-based traffic assignment has been carried out. The rest of the paper is organized as follows: at first, route choice model based on regret theory is described and a new model is developed and discussed. Then a closed-form model of SUE is developed based on the regret model. Subsequently two numerical examples are presented and the last section concludes the study.

2 Route Choice Model Based on Regret Theory

Random regret-Minimization (RRM) first was proposed by Chorus et al. based on regret theory [9]. Regret theory states that preference of an alternative is not only based on its performance, but also on relative performance of the others. The model of Chorus et al. is also based on pairwise comparison of alternatives. If a decision maker faces a choice between k alternatives and each alternative can be described by M attributes, then anticipated regret of each alternative i (\({R_i}\)) will be:

$${\phi _m}({x_{mi}},{x_{mj}})=\hbox{max} \{ 0,{\beta _m}({x_{mi}} - {x_{mj}})\} ,$$
(1)
$${R_{ij}}=\sum\limits_{{m=1}}^{M} {{\phi _m}({x_{mi}},{x_{mj}})} ,$$
(2)
$${R_i}=\hbox{max} \{ {R_{i1}},{R_{i2}}, \ldots ,{R_{ik}}\} ,$$
(3)

where \({x_m}\) and \({\beta _m}\) are the value and the weight of the attributes m, respectively.

Chorus stated that this model faced two important limitations: first, it evaluated regret only with respect to the best alternative, while the presence of a better alternative, though worse than the best one, increased perceived regret. Second, the likelihood function was non-smooth [24]. He introduced a smooth function to define the regret of an alternative described in terms of M attributes xm:

$${R_i}=\sum\limits_{{j \ne i}} {\sum\limits_{{m \in M}} {\ln (1+\exp [{\beta _m} \cdot ({x_{jm}} - {x_{im}})])} } .$$
(4)

According to this definition, regret is a non-negative value, i.e., when an alternative outperforms the others, decision maker will sense no regret. Moreover, the total perceived regret of an alternative is the sum of the pairwise choice regret with respect to the other alternatives. By this definition, regret is considered only with respect to the outperformed alternatives and the fact that the chosen alternative may outperform some alternatives will not reduce the total perceived regret. In the current study, Eq. (5) is proposed to consider the performance of an alternative compared with all others simultaneously and with respect to only one attribute:

$${R_i}=\sum\limits_{{j \ne i}} {\beta {{({C_i} - {C_j})}^\alpha }} ,$$
(5)

where \(C_{i}^{{}}\) is the travel cost of the mode i, and α is an odd value to scale the influence of the magnitude of performance difference on the regret. In pairwise comparison of alternatives, when an alternative outperforms the other, regret of that alternative will be negative. If \(R_{l}^{{}}\) is the regret correlated to the alternative l having the minimum travel cost, the whole terms of the Eq. (5) would be negative and \(R_{l}^{{}}\) is negative, and If \(R_{u}^{{}}\) is the regret correlated to the alternative u having the maximum travel cost, the whole terms of the Eq. (5) would be positive and \(R_{u}^{{}}\) is a positive value. The regrets of any other alternatives are between this lower and upper bounds:

$${R_l} \leq {R_i} \leq {R_u},i=1, \ldots ,k.$$
(6)

The choice probability of the alternative i among k alternatives in multinomial logit-formulation is:

$$P(i)=\frac{{{{\text{e}}^{ - \left( {\sum\nolimits_{{j \ne i}} {\beta {{({C_i} - {C_j})}^\alpha }} } \right)}}}}{{\sum\nolimits_{{i=1}}^{k} {{{\text{e}}^{ - \left( {\sum\nolimits_{{j \ne i}} {\beta {{({C_i} - {C_j})}^\alpha }} } \right)}}} }}.$$
(7)

One of the properties of multinomial logit models (MNLs) is that it satisfies the IIA property (independence of irrelevant alternatives) [1, 2, 27,28,29]. However, despite this property, performance of other alternatives may affect the probability of choosing an alternative. Chorus has showed that his model will not satisfy the IIA property [9].

3 Modeling SUE Based on RRM

Bekhor et al. have applied RRM to develop a formulation of SUE problem. According to them, because the regret function proposed by RRM (Eq. (4)) is not separable, formulating an equivalent optimization program is not conceivable. They have proposed SUE as a variational inequality problem as follows [30]:

$${(f - {f^ * })^{\text{T}}} - ({f^ * } - P(c({f^ * })) \cdot q) \geq 0,\quad \forall f \in \Omega ,$$
(8)
$$f=q \cdot P,$$
(9)

where \(\cdot\) is the Hadamard product, f is the path flow vector, \({f^*}\) is a solution for SUE based on RRM, P is the vector of route choice probability, q represents the vector of O–D demands, and Ω is the feasible set.

The current study tries to develop a closed-form SUE model based on the alternative regret formulation presented in previous section (Eq. (5)). Assume G(N,A) a directed graph representing a network, N is the set of nodes and A is the set of arcs. R is the set of origin nodes and S is the set of destination nodes. K is the set of paths connecting origin node r (\(r \in R\)) to the destination one s (\(s \in S\)).

Sheffi has been offered a general closed form mathematical minimization program to model SUE [2]:

$$\hbox{min}\, Z= - \sum\limits_{{rs}} {{q_{rs}}{S_{rs}}\left[ {{C^{rs}}(x)} \right]} +\sum\limits_{x} {{x_a}{t_a}({x_a})} - \sum\limits_{a} {\int\limits_{0}^{{{x_a}}} {{t_a}(\omega ){\text{d}}\omega } } ,$$
(10)

where \({S_{rs}}\left[ {{C^{rs}}(x)} \right]\) is the satisfaction function for a model of choice between the alternative paths connecting O–D pair r–s, \({q_{rs}}\) is the demand between O–D pair r–s, \({x_a}\) is the flow of the link a, \({t_a}\) is the travel time on the link a, and \({C^{rs}}(x)\) is the travel cost between O–D pair r–s.

This model is general enough that allows applying it in every context of choice modeling if the Eq. (11) is satisfied.

$$\frac{{\partial {S_{rs}}\left[ {{C^{rs}}(x)} \right]}}{{\partial C_{i}^{{rs}}}}={P^{rs}}(i).$$
(11)

In route choice context, if travel cost is the only attribute describing an alternative, choice probability of alternative i based on Chorus model can be stated as Eq. (12) [24]:

$${P_i}=\frac{{\exp \left( { - \sum\nolimits_{{j \ne i}} {\ln \left( {1+\exp [\beta \cdot ({C_j} - {C_i})]} \right)} } \right)}}{{\sum\nolimits_{j} {\exp \left( { - \sum\nolimits_{{k \ne j}} {\ln \left( {1+\exp [\beta \cdot ({C_k} - {C_j})]} \right)} } \right)} }}.$$
(12)

By taking the probability as Eq. (12), no explicit satisfaction function will be derived that can satisfy Eq. (11). Taking choice probability as Eq. (7) and integrating P over the travel cost C will lead to achieve the satisfaction function. To make the problem simpler, α is assumed equal 1. By trying and test, Eq. (13) has been developed to describe the satisfaction function of choice between the alternative paths connecting O–D pair r–s:

$${S_{rs}}\left[ {{C^{rs}}(x)} \right]=\frac{1}{{k\beta }}\left( {\ln \left( {\sum\limits_{{i=1}}^{k} {{e^{k\beta \left( {\sum\limits_{{j \ne i}} {C_{j}^{rs}} } \right)}}} } \right) - \sum\limits_{{i=1}}^{k} {k\beta \,C_{i}^{rs}} } \right)$$
(13)

Substituting (13) in (10) configures a mathematical program that its answer is stochastic user equilibrium state of the network based on regret concept:

$$\hbox{min} \,Z= - \sum\limits_{{rs}} {{q_{rs}}\left( {\frac{1}{{{k^{rs}}\beta }}\left( {\ln \left( {\sum\limits_{{i=1}}^{{{k^{rs}}}} {{e^{{k^{rs}}\beta \left( {\sum\limits_{{j \ne i}} {C_{j}^{rs}} } \right)}}} } \right) - \sum\limits_{{i=1}}^{{{k^{rs}}}} {{k^{rs}}\beta C_{i}^{rs}} } \right)} \right)} +\sum\limits_{x} {{x_a}{t_a}({x_a})} - \sum\limits_{a} {\int\limits_{0}^{{{x_a}}} {{t_a}(\omega ){\text{d}}\omega } .}$$
(14)

It is an unconstrained minimization program. \(C_{i}^{rs}\) is the travel cost of path i connecting O-D pair r-s and is calculated as \(C_{i}^{rs}=\sum\nolimits_{a} {{C_a}({x_a})\delta _{{a,i}}^{{rs}}}\). \({C_a}({x_a})\) is the travel cost of the link a when flow xa passes through the link. \(\delta _{{a,i}}^{{rs}}=1\) if link a is a part of path i connecting O–D pair r–s, and \(\delta _{{a,i}}^{{rs}}=0\) otherwise.

4 Numerical Examples

In this section two networks are introduced to test the derived model, the well-known Sioux Falls network, and a smaller one. Both networks are the same as those used by Suwansirikul et al. [31]. The relevant computations have been done in Maple and Matlab softwares. To provide a comparison, the equilibrium condition of each network is obtained by two equilibrium models: utility theory-based model (UTBM), and regret theory-based model (RTBM). The UTBM is defined as Eq. (15) and RTBM is as Eq. (14).

$$\hbox{min}\, Z= - \sum\limits_{{rs}} {{q_{rs}}\left( {\frac{{ - 1}}{\beta }\ln \left( {\sum\limits_{{i=1}}^{{{k^{rs}}}} {{e^{\beta \left( {C_{i}^{rs}} \right)}}} } \right)} \right)} +\sum\limits_{x} {{x_a}{t_a}({x_a})} - \sum\limits_{a} {\int\limits_{0}^{{{x_a}}} {{t_a}(\omega ){\text{d}}\omega } } .$$
(15)

The small-scale network consists of six nodes, sixteen links, and two O–D pairs, as shown in Fig. 1. Travel demands for O–D pairs (1–6) and (6–1) are, respectively, five and ten. Because there is not enough data to calibrate the models, the models are solved by various values of β. The results are summarized in Table 1. In this table the results of the two types of the model, i.e., the traditional utility-based, and the proposed regret-based approach have been shown. The numbers presented in this table are the link flows for different values of β. The Z’s are the values of the objective function in each approach. The lower three rows are the similar results for deterministic traffic assignment based on the Beckman’s formulation.

Fig. 1
figure 1

The small-scale test network, (the link numbers are seen close to each link)

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Table 1 Equilibrium flows of the small-scale test network based on regret theory and utility theory
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It can be seen in Table 1 that by increasing the value of β in UTBM, the flows tend to deterministic user equilibrium flows and decreasing the value of that diminishes the sensitivity of the flow to travel time, i.e., the demand will divide equally on paths independent to the paths’ travel times. The results obtained from RTBM show that by increasing the value of β, some links reach the deterministic user equilibrium flow, but the others do not. In RTBM, the travel time of each path is compared with respect to the others, and so two parallel paths by different travel times but with low magnitudes with respect to the travel time of other paths can have nearly same preferences. In addition, it can be seen that to reach nearly the same flows by RTBM and UTBM, the amount of β in UTBM must be approximately ten times of it in RTBM.

The medium-scale test network is the well-known aggregated network of the city of Sioux Falls, Dakota, as shown in Fig. 2. It consists of 24 nodes and 76 links. Here the network is solved by two different values of β for each of regret-based and utility-based models. The results are presented in Table 2. In this table the results of the two types of the model, i.e., the traditional utility-based, and the proposed regret-based approach have been shown. The numbers presented in this table are the link flows for two different values of β. The Z’s are the values of the objective function in each approach. By taking β = 0.001 for RTBM and β = 0.1 for UTBM, the obtained flows of links will be nearly the same for these two models, and that is the same for the pair β = 1 for RTBM and β = 100 for UTBM. It implies that by scaling the value of β by an appropriate scale factor RTBM and UTBM can be equivalent. This observation can be easily proven.

Fig. 2
figure 2

The medium-scale test network

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Table 2 Equilibrium flows of the medium-scale test network based on regret theory and utility theory
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5 Conclusion

In this paper stochastic user equilibrium by regret-based route choice modeling approach is considered. It has been shown in the literature, conventional utility theory approach has shortcomings and an alternative approach is introduced called regret theory. Based on regret theory a random regret-minimization model has been developed. But RRM considers an option with respect to all outperforming options and this fact that the option may outperform some others has no effect on total regret of that. Also, RRM does not lead to a closed-form SUE. So an alternative definition of the regret of an option and a closed form SUE is developed based on it. Numerical results show that RTBM and UTBM may be equivalent and this is proved mathematically too. As β value increases, UTBM yields more deterministic answers, while RTBM tends to resist that, because the flows of some links still remain almost the same as the stochastic results.