A Unified Closed-form Expression of Logit and Weibit and its Application to a Transportation Network Equilibrium Assignment

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Transportation Research Procedia 7 ( 2015 ) 59 – 74

Selection and peer-review under responsibility of Kobe University doi: 10.1016/j.trpro.2015.06.004

ScienceDirect

21st International Symposium on Transportation and Traffic Theory

A unified closed-form expression of logit and weibit and its

application to a transportation network equilibrium assignment

Shoichiro Nakayama

a,

*, Makoto Chikaraishi

b

a _{Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan} b _{Hiroshima University, 1-5-1 Kagamiyama, Higashi-Hiroshima 739-8529, Japan}

Abstract

This study proposes a generalized multinomial logit model where heteroscedastic variance and flexible shape of utility function are allowed. The novel point of our approach is that, while the model is theoretically derived by applying a generalized extreme value distribution to the random component of utility, the model maintains its closed-form expression. Also, the weibit model, where the random utility is assumed to follow the Weibull distribution, is a special case of the proposed model. This is achieved by utilizing q-generalization method developed in Tsallis statistics. Then, the generalized logit model is incorporated into a transportation network equilibrium model. The network equilibrium model with the generalized logit route choice is formulated as an optimization problem under uncongested networks. The objective function includes Tsallis entropy, which is a type of generalized entropy. The generalization of the Gumbel and Weibull distributions, logit and weibit models, and network equilibrium model is made within a unified framework with q-analysis or Tsallis statistics.

Peer-review under responsibility of the Scientific Committee of ISTTT21.

Keywords: closed-form expression; logit; weibit; Tsallis entropy; transportation network equiblium assignment

1.Introduction

The multinomial logit model plays an important role in transportation network analysis as well as travel behavior analysis. A stochastic user equilibrium model with a logit-based route choice (logit-based network equilibrium model) is one of the most widely used network equilibrium models. The multinomial logit model is closed-formed and is easily applicable. Such a simple closed formulation is desirable, especially considering the embedment of route choice model into network equilibrium analysis. The calculation of route choice probabilities is iterative, and

* Corresponding author. Tel.: +81-(0)76-234-4614; fax: +81-(0)76-234-4644 E-mail address: [email protected]

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requires significant computational costs in the network equilibrium models. The closed-form logit formulation is derived from the assumption of independent identical Gumbel distributions for error terms.

Castillo et al. (2008) proposed a closed-form discrete choice model with the Weibull-distributed utility. This is

called weibit model. Fosgerau & Bierlaire (2009) considered a multiplicative error term, and derived a closed-form

model similar to the weibit model of Castillo et al. (2008). Li (2011) extended the logit and weibit models for other

distributions, and offered other alternative error distributions for discrete choice models. Kitthamkesorn & Chen (2013, 2014) proposed a stochastic user equilibrium model with the weibit route choice. The weibit model considers heterogeneous perceived variances with respect to different travel costs, while the (multinomial) logit model has the property of homogeneity in the variance of the error terms. Bhat (1995, 1997), DeShazo & Fermo (2002), Caussade et al. (2005), and Koppelman & Sethi (2005) considered an additive error term or scale parameter to relax the homogeneity in the variance of the error terms.

There is a possibility to integrate the logit model discourse and weibit model discourse under a closed-form formulation, since both Gumbel and Weibull distributions are in a family of extreme value distributions. A “generalized extreme value distribution” in the field of probability/statistics may play an important role for the integration. The generalized extreme value (GEV) distribution consists of the Gumbel-type, Fréchet-type and Weibull-type extreme value distributions, and has a greater variety in shape than Gumbel or Weibull distribution. The Gumbel-type extreme distribution refers to the Gumbel distribution. Note that the above GEV distribution is different from the GEV distribution that derives the nested logit model and other more elaborate nested logit models (e.g., cross-nested logit model) in the travel behavior analysis. Recently, to avoid confusion, the latter GEV distribution has been referred to as the multivariate extreme value distribution.

Nakayama (2013) proposed a discrete choice model with the GEV-distributed utility. This previous model has a complicated utility function, and it results in unstability of parameter estimation. In this study, we improve the previous model, and propose a more simplified and elaborate formulation of generalized logit model with the GEV-distributed utility. This generalization can avoid one of the limitations of logit model, i.e., the homogeneity of the utility’s variance. Furthermore, the generalized logit model includes the (multinomial) weibit model as well as multinomial logit model as special cases, because the GEV distribution combines the Gumbel-type, Fréchet-type and Weibull-type extreme value distributions. Thus, the proposed model unifies the logit and weibit models in a single closed-form expression. On the other hand, the weibit models proposed previously do not include the logit model. The generalized logit model is incorporated into the transportation network equilibrium model as a route choice model. The network equilibrium model with the generalized logit route choice is formulated as an optimization problem under uncongested networks. The objective function includes Tsallis entropy, which is a type of generalized entropy. Finally, the relationship between Tsallis entropy and generalized logit model with the GEV-distributed utility is examined, and its mathematical framework is elucidated.

2. Extreme value distribution and q-exponential function

The logit model has the Gumbel-distributed utility (or error term). The Gumbel distribution is a type of extreme value distribution, and the generalized extreme value (GEV) distribution includes the Gumbel distribution.The

cumulative distribution function (CDF) of GEV, G~(x), is expressed as

° ¿ ° ¾ ½ ° ¯ ° ® » ¼ º « ¬ ª ¸ ¹ · ¨ © § J T P J 1 1 exp ) ( ~ x x G (1)

where P, T (> 0), and J are parameters (e.g., Johnson et al., 1995). When J = 0, G~(x) = exp[íexp{í(x íP)/T)}]

because limUo0(1 + U x)1/U = exp(x). This is the CDF of the Gumbel distribution. Thus, the GEV distribution includes

the Gumbel distribution as a special case.

Tsallis (1994, 2009) proposed a type of generalization of Boltzmann–Gibbs statistical mechanics and thermodynamics. A core concept in his study is Tsallis entropy, which is the generalization of Boltzman–Gibbs (or

Shannon) entropy. Such a generalization is sometimes called “q-generalization.” The basic operations of the q

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follows:

>

_q _x

@

q x q 1 1 ) 1 ( 1 : ) ( exp (2)

where the domain of the above function is

^

x1(1q)xt0

`

. Recently, the above-generalized exponential function

has been called the q-exponential function (e.g., Umarov et al., 2008). When q = 1, exp1(x) = exp(x) because

limUo0(1 + U x)1/U = exp(x) as stated above. Thus, we confirm that the q-exponential function is a type of

generalization of the exponential function. The q-logarithm function is also defined as follows:

q x x q q 1 1 : ) ( ln 1 (3)

where x > 0. When q = 1, ln1(x) = ln(x). Therefore, the q-logarithm function includes a (standard) logarithm function

as a special case. Furthermore, lnq(expq[x]) = x.

Let q = J + 1. Then, using the q-exponential function in Eq. (2), the CDF of the GEV distribution is rewritten as

°¿ ° ¾ ½ °¯ ° ® »¼ º «¬ ª » ¼ º « ¬ ª ¸ ¹ · ¨ © § _q x q x q x G 1 1 1 1 exp exp exp ) ( ˆ T P T P (4)

The domain of the above (q-form) GEV distribution is

^

x1

1q

xP

Tt0

`

. The CDF of Gˆ(x) should be

within the range [0.0, 1.0] and be increasing. Therefore, T t 0. When q = 1, P = 0, and T = 1, the above GEV

distribution is the standard Gumbel distribution, whose CDF is exp[íexp(íx)]. As stated above, the CDF of the

Gumbel distribution is exp[íexp{í(x íP)/T)}]. It is found that switching one of the exponential functions of the

Gumbel distribution’s CDF to the q-exponential function yields that of the GEV distribution. Thus, the GEV

distribution is a type of q-generalization of the Gumbel distribution.

Fig. 1 shows the probability density function (PDF) of the (q-form) GEV distribution with q = 1 (and P = 0), that

is, the Gumbel distribution. The domain of the Gumbel distribution is from negative infinity to positive infinity. The

distribution becomes flatter as T increases. As will be stated below, T ordains the distribution’s variance.

Fig. 2 presents the PDF of the (q-form) GEV distribution with q = 1/2 and T = 1. The distribution with q = 1/2

leans to the right, while the Gumbel distribution (GEV with q = 1) leans to the left. The domain with q = 1/2 and T =

1 is xd 2 + P. When P = í2, the domain is xd 0. A change of P translates the distribution in the x direction.

Figs. 3 and 4 show the PDFs of the (q-formed) GEV distribution with T = 1 and P = 0. The distribution with q < 1

leans to the right, as shown in Fig. 3, while that with q > 1 leans to the left, is illustrated in Fig. 4. Thus, the (q

-formed) GEV distribution has various shapes according to the value of q and other parameters. This flexibility helps

the distribution to fit the data.

The mean of the (q-form) GEV distribution is

Fig. 1 PDF of GEV distribution with q = 1 and P = 0 Fig. 2 PDF of GEV distribution with q = 1/2 and T = 1 ș= 1 ș= 2 ș= 4 probability density 0 1 2 í1 í2 í3 í4 0.1 0.2 0.3 0.4 0.5 4 3 x probability density 0 1 2 ȝ= 0 ȝ= í2 í1 í2 í3 í4 0.1 0.2 0.3 0.4 0.5 4 3 x

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° ¯ ° ® z * 1 1 and 2 1 1 ) 2 ( q q q q q T K P T P (5)

where K = 0.572216 (Euler constant) and *() is a gamma function. When q > 2, the distribution does not have a

mean. Note that the (q-form) GEV distribution is not symmetric, and the mean is not generally equal to the mode.

The mode of the (q-form) GEV distribution is P, and the variance of the (q-form) GEV distribution is

>

@

° ° ¯ ° ° ® z * * 1 6 1 and 2 3 1 ) 2 ( ) 2 3 ( 2 2 2 2 2 q q q q q q T S T (6)

When qt 3/2, the distribution has no variance. As the value of q increases, the tail of the distribution becomes fat.

In the field of finance, the fat tail distribution is important for considering risks. Similarly, it is useful in the field of transportation. One of the well-known issues which involve the fat tail distribution is modelling route choice behaviour. Sheffi (1985) pointed out that the normality assumption may not be appropriate for modelling the distribution of the perceived travel times, and a distribution with long tail to the right (positive skewness) could be more appropriate. Note that the distribution with long tail to the left (negative skewness) should be applied for the error term in route choice models, since the utility is usually defined as a negative value of travel time. The above

(q-form) GEV distribution is negatively skewed with a long left tail when q < 1 as shown in Fig. 3. Thus, the (q

-form) GEV distribution would be useful as a utility distribution for route choice.

3. q-Generalized logit model

In the multinomial logit model, the random utility consists of the systematic utility and error term. The separativeness of systematic utility and error term of independent utility results in homogeneity in the variance of the error terms. On the other hand, the random utility in the proposed generalized logit model cannot be decomposed into the systematic utility and error term. Thus, the key point of the current model development is to derive a model

with the expression of whole random utility, Uij, without the decomposition into the deterministic utility and error

term, where Uij is the random utility of route j (= 1, 2,…, Ji) between OD pair i ( = 1, 2,…, I). Let vij denote the

mean of Uij, that is, E[Uij] = vij, where E[] is the operator of taking the expectation. Furthermore, assume vij =

6

Kk 1

Dikyijk, where Dik is parameter k between OD pair i, yijk is explanatory variable k on route j between OD pair i, and K

is the total number of explanatory variables. In the case where the parameters are common among the OD pairs, vij is

Fig. 3. PDF of GEV distribution with q < 1, T = 1, and P = 0 Fig. 4. PDF of GEV distribution with q > 1, T = 1, and P = 0 probability density 0 1 2 í1 í2 í3 í4 0.1 0.2 0.3 0.4 0.6 4 3 x 0.5 q = 1/3 q = 1/2 q = í1/2 probability density 0 1 2 í1 í2 í3 í4 0.1 0.2 0.3 0.4 0.6 4 3 x 0.5 q = 3/2 q = 2 q = 5

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6

Kk 1Dkyijk. Thus, instead of systematic utility, the mean utility is given by the utility function in the generalized

logit model.

To meet the assumption that the mean of random utility of route j between OD pair i, which follows the (q-form)

GEV distribution, is equal to vij, that is, E[Uij] = vij, the CDF of Uij, which is denoted by Gij(x), should be the

following:

^

`

» » » » » ¼ º « « « « « ¬ ª ¸ ¸ ¸ ¸ ¸ ¹ · ¨ ¨ ¨ ¨ ¨ © § * ¿ ¾ ½ ¯ ® * » » ¼ º « « ¬ ª * ) 2 ( ) 1 ( 1 ) 2 ( ) 1 ( 1 1 1 1 exp exp ) ( exp ) 2 ( ) ( exp exp ) ( 2 ₁₍ ₁₎ i ij i i ij i i q q q i ij q ij q v q q v q q x x q v x G i i i i , (7) By substituting _» ¼ º « ¬ ª * (2 ) ) 1 ( 1 1 1 1 i ij i i q v q q P and ) 2 ( ) 1 ( 1 i ij i q v q *

T into Eq. (5), we can confirm E[Uij] = vij. As Eq.

(5) states, qi < 2 is required in order that the mean of random utility distribution exists. Furthermore,

0 ) 1 (

1 qi vijt (8)

should be met to define exp2íqi(vij) if qi1. When qi= 1, such a condition is not needed.

The probability of choosing route j between OD pair i is the probability that the utility on route j between OD pair

i is greater than those on any other routes between OD pair i. That is, the utility on route j between OD pair i is the

maximum between OD pair i. Therefore, the probability of choosing route j between OD pair i is given by

³

: c z c ! i i x i ij ij ij iJ j i j j ij ij dx x G x G x g x G x G U U p ) ( ) ( ) ( ) ( ) ( )] ( max Pr[ 1 1 1 ) ( " " (9)

where pij is the probability of choosing route j between OD pair i, Pr[] and max() are the operators that determine

the probability and the maximum, respectively, gij(x) is the PDF of the utility on route j between OD pair i, Ji is the

number of routes between OD pair i, and :i is the domain of the CDF between OD pair i.

To simplify the following equation expansion, let )

( exp

: x

zi q_i . (10)

The PDF of the utility on route j between OD pair i is

>

@

dx dz x G q v x G dx d x g i ij q i ij q ij ij i _i ( ) ) 2 ( ) ( exp ) ( ) ( 2 ₁₍ ₁₎ * (11) because dzi/dx = í[expqi(íx)]

qi. Substituting the above into Eq. (9) yields

>

@

>

@

>

@

>

@

¦

³

¦

³

f : : » » ¼ º « « ¬ ª °¿ ° ¾ ½ °¯ ° ® * » » ¼ º « « ¬ ª * * * i i i i i i i i i i i i i i i i i i i i i J j q ij ij q J j ij q q i i J j ij q ij q i z J j q ij q i i q i ij q i z J j ij q i ij q ij v v v q z v v dz v q z q v dz x G q v p 1 2 2 0 1 2 ) 1 ( 1 1 2 2 ˆ 1 2 ) 1 ( 1 ) 1 ( 1 2 ˆ 1 ) 1 ( 1 2 ) ( exp ) ( exp ) ( exp ) 2 ( exp ) ( exp ) ( exp ) ( exp ) 2 ( exp ) 2 ( ) ( exp ) ( ) 2 ( ) ( exp (12)

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where :ˆ_iis the domain of zi. Thus, the q-generalized logit model is given by

¦

i i i J j ij q ij q ij v v p 1 2 2 ) ( exp ) ( exp (13)

This closely resembles the choice probability equation of the multinomial logit model. When qi = 1, the route choice

probability of Eq. (13) is

¦

i J j ij ij ij v v p 1 ) exp( ) exp( (14)

because exp₁(x) exp(x). This is the multinomial logit model equation. The above q-generalized logit model,

which is a discrete choice model with the (q-form) GEV-distributed utility, includes the multinomial logit model as

a special case.

Eq. (13) is also expressed as

>

@

>

@

_¦

¦

¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © § i i i i _i i i i i i i i J j ij ij J j q i ij q i ij J j q ij i q ij i J j ij q ij q ij v v q v q v v q v q v v p 1 1 1 1 1 1 1 1 1 1 1 1 2 2 ~ ~ 1 1 1 1 ) 1 ( 1 ) 1 ( 1 ) ( exp ) ( exp [ [ , (15)

where v~_ij v_ij1(q_i1) D_i01(q_i1)D_i1y_ij1D_i2y_ij2"D_iKy_ijK, [i 1(qi1), yijk is a variable, K is the

number of variables and Dik is a parameter. This is the weibit model (see e.g. Castillo et al., 2008). The q

-generalized logit model thus involves both logit and weibit as special cases.

The CDF of the maximum of Uij (j=1, 2,…, Ji), which is denoted byGimax(x), is given by

^

`

» » » » ¼ º « « « « ¬ ª *

¦

exp ( ) ) 2 ( ) ( exp exp ) ( ) ( 1 ₁ 2 1 max _x q v x G x G _i i i i i q q J j q ij J j ij i . (16)

Thus, max[Uij|j] follows the GEV distribution. Comparing the above equation with Eq. (7), we obtain that the

mean maximum utility as follows:

>

@

» » ¼ º « « ¬ ª

_¦

i i i J j q ij q ij j v U 1 2 2 exp ( ) ln | max E . (17)

We can confirm this by substituting ln2íqi[

6

jexp2íqi(vij)] into Eq. (7) yields Eq. (16). When qi =1, max[Uij |j] =

ln[

₆

jexp(vij)]. Thus, the above equation is the generalization of log-sum mean maximum utility of logit model.

It is natural that the variance of the utility becomes large as the lengths of routes increase. We confirm this in the

q-generalized logit model as follows. The CDF of random utility is given by Eq. (7). From Eq. (6), the variance of

the above random utility is expressed as

>

@

° ° ¯ ° ° ® ¸¸ ¹ · ¨¨ © § * z ¸¸ ¹ · ¨¨ © § * * * 1 ) 2 ( ) 1 ( 1 6 1 and 2 3 ) 2 ( ) 1 ( 1 1 ) 2 ( ) 2 3 ( 2 2 2 2 2 i i ij i i i i ij i i i i q q v q q q q v q q q q S . (18)

As the above equation shows, the variance links to vij. Therefore, as the absolute value of mean, vij, gets larger, the

variance increases. When qi < 1 and vij = ícij, the variance of the utility on route j between OD pair i becomes large

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utility variance is relaxed in the q-generalized logit model.

Suppose that two OD pairs in the network are connected by two pairs of non-overlapping routes. The route utility

follows the (q-form) GEV distribution in the q-generalized logit model. The route disutility, that is, íUij, can be

interpreted as the generalized travel cost. Set c11 = 10, c12 = 20, c21 = 50, and c22 = 60. The difference between the

two travel costs is 10 between each OD pair. Therefore, the route choice probabilities between the two OD pairs are equal in the standard logit model. Figs. 5 and 6 show the distributions of the disutilities (or generalized travel costs),

íU11, íU12, íU21, and íU22, for the two OD pairs when q1 = q2 = 0.5. The figures show that the variances of the

route disutilities (or generalized travel costs) between OD pair 2 are much larger than those between OD pair 1. The

probability of choosing route 1 between OD pair 1, p11, is 0.771, and that of route 2, p12, is 0.229. On the other hand,

the probability of choosing route 1 between OD pair 2, p21, is 0.587, and that of route 2, p22, is 0.413. Thus, the

homogeneity of the utility variance is relaxed, and more intuitive route choice probabilities for different OD pairs

are given by the q-generalized logit model.

4. Parameter estimation of q-generalized logit model

In this section, some properties of parameter estimation in the q-generalized logit model is discussed.

4.1. An alternative view: a logit model with a flexible utility function

For simplicity, qiis assumed to be common for all OD pairs, that is, qi=q, in this section. As stated in the

previous section, the q-generalized logit model, where the GEV distribution is assumed, is given by Eq. (13). Here,

we introduce an alternative way to view the model, i.e., a logit model with flexible utility function. This alternative view is useful for practical applications, i.e., the empirical estimation of the model. Concretely, the model can be rewritten as

¦

c c i J j ij ij ij v f v f p 1 )] ( exp[ )] ( exp[ (19) where f(v_ij) ln

>

exp₂_q(v_ij)

@

(20)

Thus, the q-generalized logit model can also be understood under the standard logit model framework, i.e., the logit

model with the non-linear utility function defined as Eq. (20), where the standard Gumbel distribution is assumed

for error term. The similar transformation can be done for the weibit model as well. According to Castillo et al.

(2008), when the random utility is Weibull-distributed with the mean of utility vij = *(1+1/[) + ] where [ and ] are

the parameters, f(vij) = ln(vijí[).

In the previous model of Nakayama (2013), f(vij) =ln

>

expq{vij (s[1q]vij)}

@

, where s is a parameter to be

specified or estimated. Table 1 summarizes the differences between the two models. The difference is only in the

definition of systematic utility, and behavior of f(vij) are quite similar. In fact, when s is assumed to be one, both

Fig. 5. PDFs of route disutilities between OD pair 1 Fig. 6. PDFs of route disutilities between OD pair 2

0 20 40 60 80 100 120 0.02 0.04 0.06 0.08 Route 1 Route 2 pro ba bi lity d en sity 0 20 40 60 80 100 120 0.02 0.04 0.06 0.08 Route 1 Route 2 pr ob ability de ns ity

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models are identical. In other words, by employing the mean of the GEV as a systematic utility (instead of mode of the GEV), we can reduce one parameter that needed to be specified in the previous model. The important point is

that it contributes to simplifying the model structure without loss of theoretical foundation. In the function f(vij) in

the previous model, the systematic utility is divided by the function of systematic utility, causing the unstability of parameter estimation. Thus, some simplifications were needed in the previous model, but we had no theoretical

rationale to assume s is equal to one. From the empirical perspective, our current work provides the theoretical

rationale for the simplification of the previous model.

In this section, the systematic utility, vij, is defined as íĮ(yij1+ ȕyij2), where Įandȕ are unknown parameters, and

yij1 and yij2 are travel cost and time, respectively. Under such settings, ȕ can be directly understood as the value of

travel time. Let Wij denote the generalized cost of route j between OD pair i, that is, Wij = yij1+ ȕyij2. Fig. 7 shows how

the utility function changes with the changes in parameter q. As the figure shows, the smaller q shows higher

concavity of utility f(vij).

The parameters q and Į in the proposed model could play a similar role under the certain conditions as shown in

Fig. 8, possibly causing a model identification issue. This problem is mainly due to the fact that both q and Į are

related to both the scale of utility f(vij) and the degree of concavity of the utility f(vij): the scale of f(vij) is

monotonically increasing with respect to q while monotonically decreasing with respect to Į, and the degree of

concavity of f(vij) is monotonically decreasing with respect to q while monotonically increasing with respect to Į.

To avoid this identification issue, an alternative model introduced in Chikaraishi and Nakayama (2015) could be

used, since only parameter q is related to the elasticity of marginal utility with respect to the generalized cost Wij in

their model. The alternative model shown in Chikaraishi and Nakayama (2015) is the case with f(vij) = íDlnq(Wij).

This model is also a natural extension of conventional logit model and includes logit and weibit models as special

cases: f(vij) = íD(Wij í 1) when q = 0 (i.e., logit model), and f(vij) = íDln(Wij) when q = 1 (i.e., weibit model).

However, its utility does not follow the GEV. The comparison analysis of different generalized logit models will be done in Chikaraishi and Nakayama (2015).

Fig. 7. Effect of q value on the utility function f(vij)

Table 1. Differences in model formulas between the current and previous q-generalized logit models Distribution

assumption Systematic utility

Utility function after logit

transformation f(vij) fc(vij) fcc(vij) Proposed model Uij ~ GEV vij = mean of the GEV ln

>

exp2q(vij)

@

ij v q 1) ( 1 1

^

1 ( 1)

`

2 1 ij v q q

Nakayama (2013) Uij ~ GEV vij = mode of the GEV ln

>

expq

^

vij (s

>

1q

@

vij)

`

@

ij v q s ( 1) 1

^

( 1)

`

2 1 ij v q s q

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Fig. 8. Combination effects of q and Įon the utility function f(vij)

4.2. Simulation analysis

In the simulation analysis, a simple route choice problem is considered as shown in Fig. 9. In total, we prepare 7

datasets for simulation analysis as shown in Table 2. We first set the true parameter values for Į, ȕ, and q, and then

generate the utility value on each route for 10,000 drivers by generating random numbers for yij1, yij2, and the

Gumbel-distributed error, İij. We assume that drivers choose the route that has the maximum utility.

Fig. 9. Route choice problem considered in the simulation analysis Table 2. Simulation datasets used in this study

Utility function Parameter value_Į _ȕ _q yij1 yij2 İij Sample _size

Dataset 1 Proposed: ln[exp2íq(vij)] 2.0 1.5 0.0 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000 Dataset 2 Proposed: ln[exp2íq(vij)] 2.0 1.5 0.1 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000 Dataset 3 Proposed: ln[exp2íq(vij)] 2.0 1.5 0.3 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000 Dataset 4 Proposed: ln[exp2íq(vij)] 2.0 1.5 0.5 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000 Dataset 5 Proposed: ln[exp2íq(vij)] 2.0 1.5 0.7 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000 Dataset 6 Proposed: ln[exp2íq(vij)] 2.0 1.5 0.9 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000 Dataset 7 Proposed: ln[exp2íq(vij)] 2.0 1.5 1.0 Uniform[0.1, 1.0] Uniform[0.1, 1.0] Standard _Gumbel 10,000

㻻㻰

㻾㼛㼡㼠㼑㻌㻝㻾㼛㼡㼠㼑㻌㻞

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The model in this study is much simpler than the previous model of Nakayama (2013), and we can estimate the parameters in all datasets. Tables 3 and 4 show the estimation results of three different models: logit model, weibit model, and the proposed q-generalized logit model. Table 3 indicates that the model performances of logit/weibit models measured by final log-likelihood are almost similar with those of proposed model under certain values of q

(particularly when q = 0.0 for weibit model, and when q = 1.0 for logit model), but the proposed model is superior than logit and weibit models especially when q is between 0.3 and 0.7. However, we also confirm that when the q is underestimated, Į is consistently overestimated as shown in Table 4, implying that these two parameters may substitute for each other.

5. q-Generalized logit traffic assignment and Tsallis entropy

5.1. q-Generalized logit traffic assignment

The transportation network equilibrium is modeled with the q-generalized logit route choice. In this study, this is called the q-generalized logit traffic assignment. The q-generalized logit traffic assignment is formulated as a fixed point problem, in which the following equation is satisfied for any OD pair and any route:

> @

>

@

>

@

¦

c c c c i i i i i i J j q ij ij q J j q ij ij q ij c c v v p 1 2 2 1 2 2 ) ( exp ) ( exp exp exp p p D D ₍₂₁₎

where p is the vector of route choice probabilitiescij(p) is the travel cost function on route j between OD pair i, vij = Dcij(p) and D is a positive parameter.

Clearly, cij(p) > 0, and vij < 0 in the traffic assignment of Eq. (21). Therefore, qid 1 is assumed according to Eq. (8).

Table 3. Estimation results of logit, weibit, and proposed models

logit model weibit model proposed model (q-generalized logit model)

Į ȕ Į ȕ Į ȕ q1)

param t-value param t-value param t-value param t-value param t-value param t-value param t-value2) _t-value3) dataset1 í0.54 í11.29 1.65 9.66 í0.69 í21.59 1.65 9.39 í2.06 í5.59 1.62 9.81 0.00 0.00 0.05 dataset2 í0.58 í12.09 1.64 10.34 í0.73 í23.01 1.64 10.07 í2.61 í4.90 1.62 10.46 0.00 0.00 0.35 dataset3 í0.68 í14.07 1.65 12.08 í0.86 í26.54 1.66 11.69 í4.49 í1.28 1.65 11.94 0.04 0.05 1.00 dataset4 í0.85 í17.25 1.61 14.82 í1.04 í31.45 1.65 14.06 í3.14 í2.27 1.62 14.55 0.33 2.46 4.91 dataset5 í1.12 í22.20 1.56 19.12 í1.32 í38.31 1.61 17.71 í2.33 í4.44 1.57 18.81 0.64 8.11 4.56 dataset6 í1.58 í29.51 1.52 25.94 í1.81 í47.63 1.58 23.75 í2.28 í7.55 1.53 25.70 0.85 11.38 2.05 dataset7 í1.99 í35.09 1.51 31.68 í2.24 í53.51 1.56 28.71 í2.09 í11.38 1.51 31.63 0.98 2.82 0.05

1) q = exp(qq)/(1+exp(qq)) and, qq was estimated; 2) standard deviation was calculated based on delta method (null: q = 0); 3) standard deviation was calculated based on delta method (null: q = 1)

Table 4.Final log-likelihoods of logit, weibit, and proposed models 㻌 Final log-likelihood

logit weibit proposed

dataset 1 10742.7 10733.6 10733.8 dataset 2 10708.6 10698.4 10698.4 dataset 3 10608.0 10596.4 10595.6 dataset 4 10435.0 10425.6 10422.4 dataset 5 10112.3 10116.4 10102.7 dataset 6 9470.9 9506.6 9465.5 dataset 7 8844.2 8935.3 8844.1

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5.2. Non-congested network case

The multinomial logit equation of Eq. (14) can be obtained by maximizing the Shannon (or Boltzman–Gibbs)

entropy. The Shannon entropy for route choice probabilities between OD pair i is í

₆

jpij ln pij. As stated in section 2,

the q-logarithm function is given as lnq(x) = (x1q í1)/(1íq), and ln1(x) = ln(x) when q = 1. Using the q-logarithm

function, the Tsallis entropy is defined as

) ( ln 1 1 ) ( 1 1 ij q J j q ij i J j q ij i q p p q p S i i _i i i i

¦

p (22)

where pi is the vector whose component is pij ( j = 1, 2,…, Ji) and Sqi(pi) is the Tsallis entropy. When qi = 1, S1(pi) =

í

6

jpij ln pij, and is the standard entropy (Shannon entropy). Thus, the Tsallis entropy is the q-generalized entropy,

and includes the standard entropy. The following constrained maximization problem of the Tsallis entropy yields the

q-generalized logit equation of Eq. 21:

) ( . max ₂ _q_i _i i S p p (23) 1 . . 1

¦

i J j ij p t s (24)

In the case of a non-congested network, the q-generalized logit traffic assignment can be formulated as follows:

>

@

) ..., , 2 , 1 ( 1 . . ) ( ln . min 1 1 1 2 2 I i p t s p v p i i i i J j ij I i J j ij q ij q ij

¦

¦¦

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When qi = 1, the objective function of the above problem is D

6

i

6

jpijcij+

6

i

6

jpijlnpij, because vij = íDcij. This is

identical to the objective function of Fisk’s optimization problem (Fisk, 1980) for logit-type stochastic user equilibrium under noncongested networks (e.g., see Oppenheim, 1995, p. 170 for Fisk’s problem). Thus, the above problem is a generalized optimization problem of traffic assignment with multinomial logit route choice (multinomial logit-based stochastic user equilibrium).

Define the following Lagrangean function:

>

@

_¸¸ ¹ · ¨ ¨ © §

_¦

¦¦

_ln ₍ ₎ ₁ ) ( 1 1 1 1 2 2 i i i i J j ij I i i I i J j ij q ij q ij v p p p L p O (26)

The condition necessary for solving the above minimization problem is to find the solution of wL/wpij = 0 for any i

and j. Then,

>

ln ( )

@

1 0 ) 2 ( 1 2 w w i ij q ij q ij i ij p v p q p L i i _O ₍₂₇₎

because dlnq(x)/dx = xq, since lnq(x) = (x1q í 1)/(1 íq). The above is organized as

>

i ij

@

i i i q ij v q q q p i ) 1 ( 1 ) 2 ( ) 1 ( 1 1 O ₍₂₈₎

According to Eq. (27), Oi have to be greater than 0, because lnq(pij)!0, vij0, and qid 1. Therefore, 2íqi,

1 + (1íqi)Oi, and 1 + (qií1)vij, are all positive, and, then,

>

@

1 1 2 1 1 1 1 2 ) 1 ( 1 ) ( exp 2 ) 1 ( 1 ) 1 ( 1 » ¼ º « ¬ ª » ¼ º « ¬ ª i i i q i i i ij q q i i i q ij i ij q q v q q v q p O O (29)

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1 1 2 2 ) 1 ( 1 ) ( exp 1 » ¼ º « ¬ ª

¦

i i i i _q i i i J j q ij J j ij q q v p O (30)

Combining Eqs. (29) and (30) gives

¦

c c i i i J j q ij ij q ij v v p 1 2 2 ) ( exp ) ( exp (31)

Thus, the minimization problem of Eq. (25) solves the q-generalized logit traffic assignment with Eq. (21).

5.3. Congested network case

The Fisk’s optimization problem with logit route choice network equilibrium can also be applied to the congested network (Fisk, 1980). Prashker & Bekhor (2001) and Bekhor & Prashker (2001) formulated the network equilibrium problem with the cross nested logit (CNL) and generalized nested logit (GNL) route choice, respectively. More

recently, Zhou et al. (2012) considered the C-Logit route choice in the traffic assignment. Kitthamkesorn & Chen

(2013, 2014) formulated an optimization problem of network equilibrium with weibit route choice, but they

assumed the logarithmic route travel cost structure. It is, however, difficult to formulate the q-generalized logit

traffic assignment for a congested network as an optimization problem with a single integral and standard route travel cost structure. Aashtiani & Magnanti (1981) introduced a nonlinear complementarity problem to the traffic assignment problem. Likewise, the nonlinear complementarity problem is adopted in this paper. Compared with the weibit network equilibrium formulation of Kitthamkesorn & Chen (2013, 2014), our formulation uses the nonlinear complementarity problem, but assume the standard travel cost structure. Let

>

ij q ij

@

i q ij i p ij q p v p f i i

_O

₍ ₎ _ln ₍ ₎ ₁ ) 2 ( ) , (pȜ 1 p ₂ (32) and

¦

i J j ij i p f 1 1 ) (p O ₍₃₃₎

where

Ȝ

is the vector of Lagrangean multiplier Ȝi. The complementarity problem of the q-generalized logit traffic

assignment is to find p and

Ȝ

subject to

0 p f 0 Ȝ 0 Ȝ p f 0 p p f Ȝ Ȝ p f p, p( , ) , O( ) 0, t , p( , )t , t , O( )t _{, (34)}

where fp(p,Ȝ)_and_fO₍_p₎_{are the vector-valued functions whose component functions are} p₍_p_,_Ȝ₎

ij

f and O(p)

i

f ,

respectively, ¢, ² denotes the inner product, and 0 is the zero vector.

As stated in the previous section, if Eq. (27), that is, p(p,Ȝ) 0

ji

f , and O(p) 0

i

f , holds, then the generalized

logit model is obtained. Clearly, the necessary condition of fjip(p,Ȝ) 0 and f_iO(p) 0 under p, Ȝt 0 is the above

problem. The sufficient condition is proven using reductio ad absurdum. If there existed pi such that

¦

J_ji₁pij1!0,

0

i

O

. Clearly, there existed pijc > 0. Even if pijc > 0, fjipc(p,Ȝ)!0 because vij0 and

O

i 0 . Then,

0 ) , ( ! c c pȜ p ji ji f

p . This contradicts the above complementarity problem. Thus,

¦

Jji1pij1 0 for any OD pair. If

0 ) , (pȜ p ji

f , pij = 0. Then, fijp(p,Ȝ) 1Oi!0, that is, Oi < 1. There should be (at least one) pijc > 0 because of

0 1 1 t

¦

Ji j pij and p t 0. Even if pijc > 0, ln ( ) ˆ 0 1 1 _! c c c c i ji q ji i ji q q ji p q p qv p i i i _and

Oi < 1, as stated above. Then,

0 1 ) , ( i! p ij f pȜ O and p(p,Ȝ)!0 ij ijf

p . This also contradicts the above complementarity problem. Therefore,

0 ) , (pȜ p ji

f for any i and j. Consequently, the sufficient condition is proven, and the above complementarity

problem solves the q-generalized logit traffic assignment.

The q-generalized logit traffic assignment model can also be formulated as a fixed point problem of Eq. (21). The

problem is to find pij (0 dpijd 1) i, with j subject to Eq. (21). The vector p is in the finite closed convex set for

which 0 d pijd 1 (i, j). The right-hand side of Eq. (21) is also in the same closed convex set. Clearly, it is

continuous in the set. According to the Brouwer’s fixed point theorem, the existence of a solution to the problem is guaranteed.

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If the Jacobian of [fp(p, Ȝ), fȜ₍_p_)]T_{is positive definite, the solution is unique, where}T_{is the transpose of matrix or}

vector. For any [pc, Ȝc]T_,

p f p Ȝ p f f f f Ȝ p p Ȝ p Ȝ p _c c » ¼ º « ¬ ª c c » » ¼ º « « ¬ ª » ¼ º « ¬ ª c c T p p _T _p O O (35) because _ȜfO 0 and pc Ȝf Ȝc Ȝc pfOpc T p T _{. If}_p _f _p p c

cT p >0 for any pc z 0, the q-generalized traffic

assignment has a unique network flow. The Jacobian, pfp, is given by

° ° ° ° ¯ ° ° ° ° ® z c z c c w w c c » » ¼ º « « ¬ ª w w w w c c c i i if j j and i i if p v p q j j and i i if p v p v q p q p f j i ij q ij i ij ij ij ij i q ij j i p ij i i 0 ) 2 ( ) 1 ( 1 ) 2 ( ) , (p Ȝ 1 (36)

If

>

cij(p)cij(pc)

@

ppc

!0, then

>

vij(p)vij(pc)

@

ppc

!0. In this case, pfp is positive definite, and the

uniqueness of the solution of the complementarity problem is guaranteed. However, the condition of [cij(p) cij(pc)]

(p pc) > 0 may be restrictive. The condition is not necessary and sufficient, but is just necessary. There is some

room for relaxing the uniqueness condition, and the realization of this relaxed uniqueness condition will be a goal of our future study.

There are various ways of solving the complementarity problem. One of them is to reformulate the

complementarity problem using quadratic Fischer–Burmeister functions. The Fischer–Burmeister function I(x,y), is

x + yí _x2_y2 _{(Fischer & Jiang, 2000), and Lo & Chen (2000) introduced it to the traffic assignment formulation.}

The function is (always) non-negative, I(x,y) t 0, and I(x,y) = 0 is identical to xt 0, yt 0, and xy = 0. Therefore,

the complementarity problem of solving xf(x) = 0 s.t. x t 0 and f(x) t 0 is reformulated as min. I(x,f(x)). The

solution of minimizing I(x,f(x)) without constraints is identical to that of the original complementarity problem.

However, the Fischer–Burmeister function, I(x,y), is not differential at (x, y) = (0,0). In this study, the following

quadratic Fischer–Burmeister function is adopted:

>

@

>

@

¦

I i I i i i J j p ij ij f f p L i 1 1 2 1 2 ) ( , 2 1 ) , ( , 2 1 ) , (p Ȝ p Ȝ O p O I I (37)

Clearly, L(p,Ȝ) t 0. A solution of the unconstrained optimization problem to minimize L(p,Ȝ) is identical to that of

the above complementarity problem of Eq. (34). Many algorithms for unconstrained optimization problems have been developed. For example, a conjugate gradient method with the Polak–Ribiere formula, which guarantees its convergence, solves the above optimization problem.

5.4. Example

The q-generalized logit traffic assignment is applied to the simple example network shown in Fig. 10. This is one

of the simplest networks with multiple OD pairs and multiple routes. The network has two OD pairs, and each OD pair has two routes. OD pair 1 is between nodes 1 and 3 and OD pair 2 is between nodes 2 and 3. Route 1 between OD pair 1 consists of links 1 and 2, and route 2 comprises links 1 and 3. On the other hand, route 1 between OD pair

2 is link 2 and route 2 comprises link 3. The demands between OD pairs 1 and 2 are both 150. Set q1 = q2 = q and Į

= 2. The travel time functions on the three links are

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where ta() is the travel time function on link a (a = 1, 2, 3).

When q = 1, the q-generalized logit traffic assignment becomes the standard (multinomial) logit traffic

assignment. Then, ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( 3 2 2 12 11 11 11 t t t c c c p (40) ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( 3 2 2 22 21 21 21 t t t c c c p (41) Thus, p11 = p21 and p12 = p22 when q = 1.

As stated above, this example considers the case of qd 1. Then, the route choice probabilities are given by

) 2 ( exp ) 2 ( exp ) 2 ( exp 12 2 11 2 11 2 11 c c c p q q q ₍₄₂₎ ) 2 ( exp ) 2 ( exp ) 2 ( exp 22 2 21 2 21 2 21 c c c p q q q ₍₄₃₎

In general, p11 zp21 and p12 zp22. Fig. 11 illustrates p11 and p21 of solved q-generalized logit traffic assignment

problem with different values of q. Note that the parameter q in the horizontal axis decreases in Fig. 11. When q = 1,

p11 = p21 = 0.425. Because of homogeneity of variance in the standard multinomial logit model, p11 = p21 if q = 1.

Although c11íc12 = c21íc22, c11 or c12 > c21 or c22, that is, the two route travel times between OD pair 1 are longer

than those in OD pair 2. It is natural that the influence of the travel cost difference in OD pair 1 is smaller and p11 >

p21, as discussed above, even if the differences of the two route travel times are equal between the two OD pairs. As

Fig. 11 shows, p11 z p21 when q z 1, and we can confirm that the q-generalized logit model alleviates the

homogeneity of variance. Initially, as q decreases until about 0.9, the variance of the utility on route 1 between OD

pair 1 becomes increasingly different from that between OD pair 2. In other words, p11 and p21 differ. Then, the

difference decreases gradually.

6. Conclusions

The Gumbel-distributed utility in the multinomial logit model is restrictive, especially in the route choice behavior and network equilibrium analysis, because of the property of homogeneity of variance, although it is mathematically convenient. Especially in application to the modelling of route choice behaviour and network equilibrium analysis, the homogenous variance of logit model can cause serious biases in the analysis. In this study, the GEV distribution, which includes the Gumbel distribution as a special case, is incorporated into the discrete

Fig. 10. Example network

1 link1 2 3

link2

link3 ODpair1

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choice model. The GEV distribution is given by switching the standard exponential function to the q-exponential

function (which is a type of generalized exponential function) in the CDF. The q-generalized logit model with the

GEV-distributed utility allows heteroscedastic variance and a flexible shape, and includes the multinomial logit and weibit models as special cases. Maximizing the Tsallis entropy, which is a type of generalized Shannon entropy,

gives the q-generalized logit model (with the GEV-distributed utility) as well, while maximizing the Shannon

entropy produces the standard multinomial logit model.

The parameter estimation of q-generalized logit model was also examined. An identification problem in

parameter estimation might be occurred in the limited certain condition. The results of example parameter estimation with the simulated data indicate its applicability.

The generalized logit model with the GEV-distributed utility is incorporated into the transportation network equilibrium model. The network equilibrium model with generalized logit route choice is formulated as an optimization problem under uncongested networks. The objective function includes the Tsallis entropy, which is a type of generalized entropy. For congested networks, it is formulated as the complementarity problem. The existence of equilibrium flows is proven, and a uniqueness condition is examined. In this study, the Gumbel

distribution, logit model, and network equilibrium model are considered in a unified framework of q-generalization

with q-analysis, which includes the operation of q-exponential and q-logarithm functions, or Tsallis statistics. In our

future study, more relaxed conditions for unique equilibrium network flow will be examined. Also, more empirical works would be needed to fully understand the properties of the proposed model. In order to do this, an efficient link-based algorithm should be developed for large-scale networks. To alleviate the overlapping problem in route choice, the commonality factor can be introduced in the model. The simplest way to incorporate the commonality factor would put it into eq. (20) in an additive way. This will be discussed in a later paper.

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