A Strategic User Equilibrium for Independently Distributed Origin-Destination Demands ·...

Computer-Aided Civil and Infrastructure Engineering 33 (2018) 316–332

A Strategic User Equilibrium for IndependentlyDistributed Origin-Destination Demands

Tao Wen & Chen Cai

School of Civil and Environmental Engineering, University of New South Wales, Sydney, Australia and Data61, TheCommonwealth Scientific and Industrial Research Organisation (CSIRO), Sydney, Australia

Lauren Gardner, Vinayak Dixit & S. Travis Waller*

School of Civil and Environmental Engineering, University of New South Wales, Sydney, Australia

&

Fang Chen

Data61, The Commonwealth Scientific and Industrial Research Organisation (CSIRO), Sydney, Australia

Abstract: This article proposes an extension of thestrategic user equilibrium proposed by Waller andcolleagues and Dixit and colleagues. The proposedmodel relaxes the assumption of proportional Origin-Destination (O-D) demand, as it accounts for users’strategic link choice under independently distributed O-D demands. The convexity of the mathematical formu-lation is proved when each O-D demand is assumed tofollow a Poisson distribution independently; link flowdistributions and users’ strategic link choice are alsoproved to be unique. Network performance measures aregiven analytically. A numerical analysis is conducted onthe Sioux Falls network. A Monte Carlo method is usedto simulate network performance measures, which arethen compared to the results computed from the analyti-cal expression. It is illustrated that the model is capable ofaccounting for demand volatility while maintaining com-putation efficiency.

1 INTRODUCTION AND BACKGROUNDS

Realism and computational complexity are both criticalfactors in transportation planning models and must beequally considered to determine a model’s practicalapplicability. A major complication in transportation∗To whom correspondence should be addressed. Email: [email protected].

modeling is the ability to properly account for the in-herent uncertainties such as demand (Axhausen et al.,2002; Richardson, 2003; Bellei et al., 2006; Stopheret al., 2008; Kim et al., 2009), capacity (Brilon et al.,2005; Wu et al., 2010), and connectivity (Iida and Wak-abayashi, 1989; Bell and Iida, 1997). Furthermore, ashas been noted, uncertainty surrounding these variablesdirectly impacts route choice behavior (Uchida andIida, 1993; Abdel-Aty et al., 1995; Lam and Small,2001; Brownstone et al., 2003; de Palma and Picard,2005). Researchers have focused on the stochasticuser equilibrium to account for the uncertainties in anetwork (Watling, 2002; Nakayama and Takayama,2003; Meng and Wang, 2008; Bekhor et al., 2009;Sumalee et al., 2011), which requires path enumerationfor all Origin-Destination (O-D) pairs. However, littleattention has been paid to the user equilibrium thatcan account for demand volatility while maintainingthe computation efficiency. It is, therefore, necessaryto incorporate these stochastic elements into modelsto ensure robust planning capabilities but to do so ina manner that maintains computational tractability. Inthis section, the review covers the following topics: (1)demand variability and its impact, (2) a brief introduc-tion to the strategic user equilibrium, (3) reliability intransportation, and (4) the maximum entropy methodin static user equilibrium.

C© 2017 Computer-Aided Civil and Infrastructure Engineering.DOI: 10.1111/mice.12292

The strategic user equilibrium 317

Improper consideration of demand variability inplanning models can result in gross underestimation oftravel time (Waller et al., 2001). Numerous researchefforts have focused on the impact of day-to-daystochasticity regarding demand through specific modelvariations. For example, Clark and Watling (2005) pro-posed an assignment model with stochastic demand todetermine the impact on variance in total system traveltime (TSTT). Additionally, Lo and Tung (2003) con-sidered stochastic capacity and proposed a probabilisticuser equilibrium. Castillo et al. (2013) further expandedthe probabilistic user equilibrium model under capacityuncertainty. Their model can be solved without pathenumeration and avoided to use the central limit theo-rem. The demand volatility can be modeled as variousstatistical distributions such as Poisson distribution(Bell, 1991; Watling, 2002; Hazelton, 2003), log-normaldistribution (Zhou and Chen, 2008; Kuang and Huang,2013), normal distribution (Shao et al., 2006b), or bi-nomial distribution (Nakayama and Takayama, 2003).Also, there have been many researches to account fordemand uncertainty in the network design problem(NDP) perspective (Sumalee et al., 2006; Ukkusuriet al., 2007; Gardner et al., 2008; Sharma et al., 2009;Ukkusuri and Patil, 2009; Yin et al., 2009; Chow andRegan, 2011). The concept of considering the strategicchoice in a user equilibrium context was proposedby Nguyen and Pallottino (1989), specifically in thecontext of transit assignment. This notion was laterexpanded in the dynamic traffic assignment (DTA)formulation (Hamdouch et al., 2004). Sumalee et al.(2009) proposed a new model for demand and capacityuncertainty where users’ strategic choices are obtainedvia a transformation of cumulative prospect theory.Dixit et al. (2013) proposed a strategic user equilibriumunder trip variability, which was further expandedin the linear programming formulation for DTA, bydividing it into strategic stage and realized demandstage (Waller et al., 2013); the link flow will thereforefluctuate corresponding to realized demand.

This article is an extension of the user equilibrium un-der trip variability proposed by Dixit et al. (2013), theassumption of fixed demand proportion in the originalformulation limits the applicability of the model. Thisarticle instead proposes a generalization of the strate-gic user equilibrium; here it is assumed that each O-Ddemand is independently distributed. The equilibriumassignment problem is to find the link flow distribu-tions that satisfy the user equilibrium criteria (Wardrop,1952). The equilibrium in this article is a situation whereusers equilibrate to minimize their expected travel costbased on the demand distribution, and the expectedtravel cost is less than the cost of any unused paths.The model is named “the strategic user equilibrium,”

because the model assumes that users make their routechoice strategy while knowing the day-to-day demandvolatility (e.g., demand is different on each Monday ofa week and is also volatile in each day of week), andthey tend to be “sticky” to this strategy, that is, theirroute choice strategy remains fixed regardless of thespecific day-to-day realized demand, thus the predictionof link flow will likely vary with the realized demand. Asa consequence, disequilibrium may be observed everyday, however, the expected link flow will be in the stateof equilibrium. Watling and Hazelton (2003) providean in-depth discussion on the definition of equilibrium,and also identify the existence of disequilibrium due tothe underlying variations such as demand. This modeltherefore captures the demand uncertainty and its im-pact on the travel time reliability, while maintaining thecomputation tractability and simplicity within the staticuser equilibrium.

It is due to these uncertainties in the network thathas led to the concerns with ensuring reliability. In part,this has come about due to the finding that road userstend to value reliability at about the same magnitudeas delays (Abdel-Aty et al., 1995; Bates et al., 2001;Brownstone et al., 2003; Asensio and Matas, 2008; Dixitet al., 2013; Uchida, 2015), hence the impact of traveltime variation should not be neglected. The reliabilityof travel time may be affected by capacity degradation,for example, Lo et al. (2006) extended the user equi-librium to a reliability-based user equilibrium (RUE)to account for travel time reliability by adding a safetymargin in the travel time budget. The RUE was furtherexpanded to incorporate multimodal transport (Shaoet al., 2006b; Fu et al., 2014), travelers’ perception er-rors (Clark and Watling, 2005; Shao et al., 2006b), andnetwork uncertainties (Shao et al., 2006b; Zhou andChen, 2008). Furthermore, extensive previous researchhas considered travel time reliability as a result of pathflow correlation (Clark and Watling, 2005; Lam et al.,2008; Shao et al., 2013), where a priori information onpath flows and path enumeration is required, whichmay be computationally complicated. Under the frame-work, the correlation between links is considered hencepath travel time is nonadditive (Fan et al., 2005; Dongand Mahmassani, 2009). Path travel time can be de-rived from link travel time covariance matrix (Sen et al.,2001; Xing and Zhou, 2011; Shahabi et al., 2013), tempo-ral and spatial correlations (Miller-Hooks and Mahmas-sani, 2003; Gao and Chabini, 2006), or simulation-basedapproaches (Ji et al., 2011; Huang and Gao, 2012; Zock-aie et al., 2013; Zockaie et al., 2014). Some researchalso focuses on the penalties due to late arrival andthe corresponding route choice (Watling, 2006; Chenand Zhou, 2010). Generally, the stochastic variations intransportation systems can be attributed to variations

318 Wen et al.

in demand, capacity, or driving behavior (Shao et al.,2006a; Siu and Lo, 2008; Van Lint et al., 2008), whichvariation is the dominant source of travel time varia-tion is still an open question. This article aims to ac-count for the demand uncertainty and travel time vari-ation caused by it. The risk of variation of travel de-mand and its effects on route choice are explained byUchida and Iida (1993), who developed a new assign-ment model to consider the impact of variation in traveltime. It should also be noted that the impact of not con-sidering these uncertainties leads to significant biasesand errors (Duthie et al., 2011). Hence, it is important toincorporate the critical realities of demand uncertaintiesin our transportation models for travel behavior, so thatconsistent decisions can be made based on cost-benefitanalysis associated to improving reliability by affectingvariations in demand. In this article, we account for thelink travel time reliability as a variable caused by de-mand uncertainty and formulate the strategic user equi-librium as a convex optimization problem, which can beefficiently solved by some numerical methods such asthe widely used Frank-Wolfe (F-W) algorithm.

The link flows are uniquely defined under the staticuser equilibrium framework originally formulated byBeckmann’s transformation (Beckmann et al., 1956),however, multiple path flow solutions are possible de-pending on the model assumptions and methodolo-gies (Larsson et al., 2001). Analyses based on an ar-bitrary choice among the infinite number of possibleroute flow solutions could cause inconsistencies in trans-portation planning models such as O-D matrix estima-tion, emission analysis, and many more. Rossi first sug-gested that the entropy-maximizing pattern is the mostlikely route flow pattern; the implication is to split flowsevenly across all user equilibrium (UE) paths (Rossiet al., 1989). Later, the stability and continuity of thisapproach are theoretically proved (Lu and Nie, 2010).However, the scalability of this approach remains an is-sue due to that the set of UE paths needs to be enu-merated. Janson (1993) provided a link-based problemwhich is equivalent to the maximum entropy approach,and proposed a modified F-W algorithm for the im-plementation of the model. However, the mathemati-cal proof of its equivalency to the original maximumentropy approach is pointed out to be inconsistent byAkamatsu (1997). The efficiency of the maximum en-tropy approach was later improved in various ways:Bar-Gera applied the condition of proportionality (Bar-Gera and Boyce, 1999; Bar-Gera, 2010) and found anapproximate set of UE paths (Bar-Gera, 2006). Somedual methods are exploited to find the most likely pathflows (Bell and Iida, 1997; Larsson et al., 2001), in ad-dition, Kumar and Peeta (2015) proposed an entropyweighted average method to minimize the expected Eu-

clidean distance from all other path flow solution vec-tors of the static user equilibrium. However, little at-tention has been paid to these two aspects: first, mostof the methods rely on Stirling’s approximation toconvert the entropy into a continuous function, andis subjected to the limitations of this approximation(Schrodinger, 1957); second, the path flow is treated asa deterministic variable, however, it is possible to definethe entropy of a probability distribution in informationtheory and to interpret this as a measure of uncertaintyassociated with that distribution. The maximum entropyof probability distribution method in this model has ad-dressed both issues by considering the entropy of pathflow distributions, and can eventually provide users’ O-D specific link choices, which will be referred to as thestrategic link choice in this article.

The article is organized as follows. Section 2 presentsthe mathematical model. The link flow distribution isproved to be unique under this framework, which isextremely useful for transport planning and policies.Users’ strategic link choice is also proved to be uniqueunder the further assumption of maximum entropy,which provides the possibility of applying this user equi-librium in many other transportation models such as O-D trip matrix estimation, environmental impact anal-ysis and fuel consumption analysis. The performancemeasures of a network are analytically derived with re-spect to the expected link flow; consequently, the impactof demand variability can be computed directly fromthe mathematical expression. An implementation algo-rithm is also demonstrated. In Section 3, the numeri-cal analysis demonstrates that these analytically derivedperformance measures are closely approximated to thesimulated results, so the model captures the demandvolatility while maintaining the simplicity and tractabil-ity of traditional deterministic traffic equilibrium meth-ods. The importance of incorporating uncertainty inmaximum entropy method is presented. Some discus-sions of model limitations and possible future researchare presented in Section 4.

The highlights of this article include:

(1) It accounts for demand uncertainty in users’ rout-ing mechanism, and its direct impact on link flowand link travel time variation.

(2) It introduces the maximum entropy of a randomvariable in the assignment model, instead of a de-terministic variable in the majority of the litera-ture.

(3) Network performance measures are provided an-alytically, which enhances computation efficiency.

(4) The uniqueness of link flow distribution andstrategic link choice are guaranteed, which ensuresits applicability in transportation planning process.


Table 1Summary of notations

N Link (index) set.M O-D pair (index) set.K Path set.ln Flow variable for link n.tn() The function of travel time on link n.tn The expected travel time on link n.tn f The free flow travel time on link n.Cn The capacity on link n.pm

k Users’ path choice on path k, connecting O-Dpair m.

dmn Users’ strategic link choice, which represents the

proportion of O-D pair demand Tm on link n.hm

k The flow on path k, connecting O-D pair m.Mθ

n The θ th raw moment of the link flow distributionon link n.

E() The expectation of a variable.var() The variance of a variable.G( ) The probability density function of a variable.qm Proportion of total trips that are between O-D

pair m; 1 = ∑∀m qm .

T m Demand variable for O-D pair m.λn The parameter of the Poisson distribution for

flow on link n[λ] = [λ1, . . . , λn]T .sm The parameter of the Poisson distribution for

O-D pair demand T m .g(ln ; λn) The probability mass function of a Poisson

distributed variable ln , defined by theparameter λn .

δmn,k Link-path indicator variable. δm

n,k ={1 i f link n is on path k between OD pair m0 otherwise

Km The path set for O-D pair m.K U E

m The shortest path set for O-D pair m.S() The entropy of a variable.α The parameter of the BPR function.∇z() The gradient of an objective function z().

(5) Users are assumed to stick to their route choiceregardless of day-to-day demand volatility, whichmimics the “sticky” behavior of travelers, espe-cially commuters.

(6) A solution algorithm is proposed to compute thelink flow parameter and the strategic link choice,without requiring path enumeration.

2 PROBLEM FORMULATION

This section defines the mathematical concept of thestrategic user equilibrium model (referred to as StrUEin this article). Table 1 shows a summary of notationsused in the article.

2.1 Model formulation

This model is an extension of the strategic user equilib-rium proposed by Dixit et al. (2013) and Waller et al.(2013), which was formulated as

min z([x]) =∑n∈N

∞∫0

xn∫0

tn (yT) G(T ) dydT (1)

subject to ∑k

pmk = 1, ∀m ∈ M (2)

pmk ≥ 0, ∀m ∈ M, k ∈ Km (3)

xn =∑

m

∑k

pmk δm

n,kqm , ∀n ∈ N (4)

In this formulation, users’ strategic choice is denotedas the link proportions xn , that is, the link flow dividedby the total demand T . The [x] represents a vector of allthe link proportions and y is a dummy variable for inte-gration. Each O-D demand is the fixed demand propor-tion qm multiplied by the total demand T , that is, eachO-D demand is proportional to the total demand, andthe proportions are constants. This implies that all O-D demands are perfectly correlated, that is, each O-Ddemand always inflates or deflates by the same percent-age. The total demand is assumed to follow a certaindistribution. The objective function is the sum of the in-tegrals of the expected value of the link cost functions;the link proportions are proved to be unique under theassumption of fixed demand proportions. Note that thelink proportion does not change in line with the real-ized total demand day-to-day, while link flow will varyevery day corresponding to the realized total demand.Further explanation and details can be found in Dixitet al. (2013).

However, the assumption of fixed demand proportionlimits the applicability of the model, therefore, this arti-cle proposes a generalization of the strategic user equi-librium, and here it is assumed that each O-D demand isindependently distributed. The equilibrium assignmentproblem is to find the link flow distributions that sat-isfy the user equilibrium (Wardrop, 1952) criterion. Dueto the introduction of demand volatility and expectedtravel time, in this article, it is defined as:

Definition 1. The strategic user equilibrium is definedsuch that the expected travel costs are equal on all usedpaths, and this commonly expected travel time is less thanthe actual expected travel time on any unused path. Inother words, given user equilibrium expected path cost,any deviation from the existing expected path flows can-not reduce the expected path cost.

320 Wen et al.

Based on Definition 1, the equilibrium condition canbe formulated by the following link-based mathematicalprogram:

min z ([l]) =∑n∈N

∫ ln

0

∫ +∞

0tn (w) G (ln) dlndw (5)

subject to ∑k∈Km

pmk = 1 ∀m ∈ M (6)

pmk ≥ 0 ∀m ∈ M, k ∈ Km (7)

ln =∑m∈M

∑k∈Km

pmk δm

n,k T m ∀n ∈ N (8)

where w is a dummy integration variable which repre-sents link flow. In this formulation, the objective func-tion is the sum of the integrals of the expected value ofthe link cost functions from 0 to expected link flow; itsequivalence to the variational inequality for user equi-librium will be proved later. The behavioral implicationand its equivalence of this formulation to user equilib-rium will be shown in this section. Equation (6) rep-resents a set of flow conservation constraints, that is,the sum of path flow proportions for every O-D pair mshould be equal to 1, which preserves the trips out ofand in each O-D centroid. Equation (7) indicates thatthe path flow must be non-negative. Equation (8) indi-cates the link flow in terms of path flows and O-D de-mand. The topology of link, path and O-D pairs of thenetwork is represented by Equation (8).

As δmn,k is dependent only on the network topology

and is a constant,∑

k∈Kmpm

k δmn,k can be integrated into

one term dmn which is defined here as the users’ strate-

gic link choice. It represents the proportion of O-D pairdemand Tm traversing link n, the strategic link choiceindicates users’ link choice disaggregated by O-D pairs,which is extremely important in many transportationmodels such as O-D matrix estimation, emission anal-ysis, and NDP. As aforementioned in this article, theusers will stick to this strategic link choice once it ismade. The strategic link choice will be discussed furtherlater in this article.

dmn =

∑k∈Km

pmk δm

n,k ∀m ∈ M, ∀n ∈ N (9)

Under the proposed framework, two further statisti-cal assumptions are made here.

A1. The actual O-D demand varies day-to-day andfollows a Poisson distribution independent of eachother (Bell, 1991; Hazelton, 2003; Clark and Watling,2005; Appiah, 2009; Bera and Rao, 2011) defined by

the probability mass function: T m ∼ g(T m ; sm), wheresm > 0. This assumption will be discussed later in thearticle.

A2. Conditional on the realized demand on any givenday, each user (driver) is assumed to choose inde-pendently between the alternative paths with a fixedstrategic path choice pm

k .

Before proceeding to the mathematical proof, someclarifications are made here.

Note that the uniqueness and equivalence conditionsare guaranteed by the assumption of Poisson distributeddemands, and the non-negativity of Poisson distributionis consistent with real-world positive demands. The ac-tual demand distribution may vary depending on net-work type, time frame, and many other factors; whichdistribution best fits the actual demand is still an openquestion. In some cases when demand is considered tonot follow the Poisson distribution, other distributionsare also applicable as long as they are non-negative andpreserve the uniqueness and equivalence conditions. Ifthe depicted distribution is not supported on the pos-itive semi-infinite interval, the truncated distributiontechniques may be applied, such as the truncated nor-mal distribution, which is defined on the domain of pos-itive real numbers.

Each O-D demand follows a Poisson distribution;however, each realized demand T ∗

m is a constant, thatis, on that day, there are T ∗

m travelers between O-D pairm. Each of the T ∗

m travelers will choose between the k al-ternative paths, each with a probability pm

k , note that thestrategic path choice should be treated as a probabilityinstead of a constant. One possible misunderstanding isto treat the strategic path choice as a constant; in thiscase, the path flow variable would become hm

k = pmk T m ,

namely, a constant multiplied by a Poisson variable. As0 ≤ pm

k ≤ 1, the path flow would not follow a Poissondistribution, which is inconsistent with our assumption.

Proposition 1. Given assumptions 1 and 2, theunconditional path flow hm

k follows a Poissondistribution g(hm

k ; pmk sm) independently. The uncon-

ditional link flow ln also follows a Poisson distribution.

Proof. Assumptions 1 and 2 together imply that foreach m ∈ M , conditional on a realized demand T ∗

m , eachuser has a probability of choosing path k. By defini-tion, the probability of observing a set of path flow[hm

1∗, hm

2∗, . . . , hm

k∗|m ∈ M] has a multinomial distribu-

tion

P(hm

1 = hm1

∗, hm2 = hm

2∗ . . . hm

k = hmk

∗ ∣∣T ∗m

)= T ∗

m!hm

1∗!hm

2∗! . . . hm

k∗!

(pm1 )hm

1∗(pm

2 )hm2

∗(pm

k )hmk

∗

∀m ∈ M, k ∈ Km

(10)


where∑k

1 hm∗k = T ∗

m . In StrUE, we are more interestedin the unconditional flows; here the path flow condi-tional on Tm = T ∗

m has a multinomial distribution, andTm follows a Poisson distribution, then from the rela-tionship of unconditional and conditional probabilitywe obtain

P{( hm1 = hm

1∗, hm

2 = hm2

∗ . . . hmk = hm

k∗ )

∩(

k∑1

hmk

∗ = T ∗m

)= P ( hm

1 = hm1

∗, hm

2 = hm2

∗

× . . . hmk = hm

k∗ ∣∣T ∗

m =k∑1

hmk

∗)

∗P

(T ∗

m =k∑1

hmk

∗)

=(∑k

1 hmk

∗)

!

hm1

∗!hm2

∗! . . . hmk

∗!

×(pm1 )hm

1∗(pm

2 )hm2

∗(pm

k )hmk

∗ e−sm s∑k

1 hmk

∗m(∑k

1 hmk

∗)

!,

∀m ∈ M, k ∈ Km (11)

Because∑

k∈Kmpm

k = 1, the above equation can berearranged as(∑k

1 hmk

∗)

!

hm1

∗!hm2

∗! . . . hmk

∗!(pm

1 )hm1

∗(pm

2 )hm2

∗(pm

k )hmk

∗ ∗ e−sm s∑k

1 hmk

∗m(∑k

1 hmk

∗)

!=

e−pm1 sm(

pm1 sm

)hm1

∗

hm1

∗!∗ e−pm

2 sm(

pm2 sm

)hm2

∗

hm2

∗!. . .

e−pmk sm (pm

k sm)hmk

∗

hmk

∗!

=k∏1

g (hmk ; pm

k sm) ∀m ∈ M, k ∈ Km (12)

Q.E.D.Note that the proof of Proposition 1 is also known

as “thinning of a Poisson process,” a general andmore detailed proof can be found in Corollary 9.17 inBoucherie and Serfozo (2001) or in Clark and Watling(2005) and Castillo et al. (2014). The right-hand sideof Equation (12) is a product of a series of Poissonvariables hm

k , whose parameters are pmk sm . So we have

proved that each unconditional path flow follows anindependent Poisson distribution; and note that theconclusion of independent path flow does not violatethe flow conservation constraints as the path flowconditional on a realized demand still must sum up toeach realized demand. From Equation (8), we have

ln =∑m∈M

∑k∈Km

pmk δm

n,k T m

=∑m∈M

∑k∈Km

hmk δm

n,k , ∀n ∈ N (13)

The link-path indicator variable can only be either 0or 1; therefore, link flow is the sum of several indepen-dent Poisson distributions, which also follows a Poissondistribution (Lehmann and Romano, 2006). The param-eter of link flow distribution is then defined by the equa-tions below

ln ∼ g (λn) (14)λn =

∑m∈M

∑k∈Km

pmk δm

n,ksm

=∑m∈M

∑k∈Km

E (hmk ) δm

n,k ∀n ∈ N (15)

Corollary 1. The flow conservation constraint also holdsfor expected path flow and variance of path flow.

Proof. Each O-D demand follows a Poisson distribu-tion with parameter sm > 0, which is the expected O-Ddemand. Multiply both sides of the flow conservationconstraints by the expected demand for O-D pair m, andas the summation is taken over k instead of m, Equation(6) can be rewritten as∑

k∈Km

pmk = 1 →

∑k∈Km

pmk ∗ sm = 1 ∗ sm

=∑

k∈Km

E (hmk ) ,∀m ∈ M (16)

Also, if we look at the variance of path flows, accord-ing to Proposition 1 and based on the basic property ofvariance, we have

V ar (Tm) = V ar

⎛⎝∑k∈Km

hmk

⎞⎠=∑

k∈Km

V ar( hmk ),∀m ∈ M (17)

Proposition 2. The parameters of the link flow distribu-tions are unique under A1 and A2. That is, there exists aunique solution to the mathematical program defined inEquations (5) to (8).

Use BPR (The Bureau of Public Roads) function asthe travel cost function

tn (ln) = tn f

(1 + α

(ln

Cn

)4)

∀n ∈ N (18)

Note that the travel cost is positive, and due tothe equilibrium conditions that all used paths haveminimum travel costs, a path with cycles will not be

322 Wen et al.

considered as the user equilibrium paths. Because thelink flow variable ln follows the discrete Poisson distri-bution with a probability mass function g(ln ; λn) whichis defined on positive integers, based on the aforemen-tioned proposition and corollary, the objective functionof StrUE in Equations (5) to (8) can then be rewrittenas

min z ([λ])

=∑n∈N

λn∫0

∞∑ln=0

[tn f

(1 + α

(ln

Cn

)4)]

g (ln ; λn) dλn

=∑n∈N

λn∫0

(tn f + αtn f

(1

Cn

)4

M4n

)dλn (19)

subject to∑k∈Km

E (hmk ) = sm ∀m ∈ M (20)

E (hmk ) ≥ 0 ∀m ∈ M, k ∈ Km (21)

λn =∑m∈M

∑k∈Km

E(hmk )δm

n,k ∀n ∈ N (22)

where M4n is the fourth raw moment of the correspond-

ing Poisson distribution for link n

M4n = λn

(1 + 7λn + 6λ2

n + λ3n

),∀n ∈ N (23)

The gradient of the objective function is

∇z ([λ]) =

⎡⎢⎢⎢⎢⎢⎢⎣

∂z ([λ])∂λ1

...

∂z ([λ])∂λn

⎤⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∞∑l1=0

[t1 f

(1 + α

(l1

C1

)4)]

g (l1; λ1) dλ1

...

∞∑ln=0

[tn f

(1 + α

(ln

Cn

)4)]

g (ln ; λn) dλn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

(t1 f + αt1 f

(1

C1

)4

M41

)...(

tn f + αtn f

(1

Cn

)4

M4n

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎣t1

...

tn

⎤⎥⎥⎦

(24)

where tn is the expected travel time on link n. Becausethe expected travel time depends only on the link flowdistribution

∂z ([λ])∂λaλb

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩∂

(tn f + αtn f

(1

Cn

)4M4

b

)∂λa

, if a = b

0 if a = b

∀a, b ∈ N (25)

The Hessian matrix of the objective function can beexpressed as:

Hessian

=

⎡⎢⎢⎢⎣1 + 14λ1 + 18λ2

1 + 4λ31 . . . 0

.... . .

...

0 . . . 1 + 14λa + 18λ2a + 4λ3

a

⎤⎥⎥⎥⎦a ∈ N (26)

As λa > 0, the Hessian matrix is positive definite, inaddition, all constraints are linear. Therefore, the ob-jective function is strictly convex and has a unique mini-mum with respect to link flow distribution. The unique-ness is extremely important to ensure stability of theproject rankings in planning models. Note that as theθ th raw moment of a Poisson distribution can always beexpressed in a polynomial form, other travel cost func-tions may also be used as long as they are monotonicallyincreasing with respect to link flow. For example, chang-ing the exponent parameter for BPR function to two orthere clearly does not change the positive-definitenessof the Hessian matrix of the objective function.

Q.E.D.

2.2 Proof of equivalency

In this part, we will demonstrate that the mathematicalformulation is equivalent to the notion of strategic userequilibrium as proposed by Definition 1.

Theorem 1. The mathematical program defined in Equa-tions (19) to (22) is equivalent to the strategic user equi-librium condition defined in Definition 1.

Proof. For convenience, here we number the pathsdistinctly and consecutively by dropping the subscriptsk and m and replacing them with e, that is, there aree = ∑

m∈M

∑k∈Km

k (i.e., the sum of all possible pathsacross all O-D pairs) distinct paths for the network. LetH denote the path flow vector and H represent the


corresponding expected flow vector. Equation (22) canthen be written in the following matrix form:

H =

⎡⎢⎢⎣h1

...

he

⎤⎥⎥⎦ , H =

⎡⎢⎢⎣E (h1)

...

E (he)

⎤⎥⎥⎦ ,

A =

⎡⎢⎢⎣a11 . . . a1e

.... . .

...

an1 . . . ane

⎤⎥⎥⎦ → [λ] = A ∗ H

(27)

where [λ] is a vector of λn(n ∈ N), let [λ∗] be the vec-tor of a local minimum, A is the link-path incidencematrix which represents whether link n is included inpath e or not, the entry of A is either 0 or 1. Equa-tion (27) indicates the constraints with respect to λn ;clearly, the constraints are convex with respect to λn .The convexity of the constraints implies that anothervector [ λ] = q[λ] + (1 − q)[λ∗] is also a feasible setfor 0 ≤ q ≤ 1. Thus,

[λ] − [λ∗] = q [λ] + (1 − q) [λ∗] − [λ∗]

= q ([λ] − [λ∗])(28)

This indicates a step move in a feasible direction. Fora q that is small enough, the convexity of the objectivefunction tells us that

q ([λ] − [λ∗]) ∇z ([λ∗]) ≥ 0 (29)

where ∇z([λ∗]) is the gradient of the objective function.Dividing both sides of Equation (29) by q yields

([λ] − [λ∗]) ∇z ([λ∗]) ≥ 0 (30)

As shown above in the calculation in Equation (24),the gradient of the objective function is a vector of theexpected travel cost on link n, and the expected cost ofany path e consists of the sum of the expected costs ofthe constituent links

O∗ =

⎡⎢⎢⎣o1

...

oe

⎤⎥⎥⎦ = AT ∗

⎡⎢⎢⎣t1

...

tn

⎤⎥⎥⎦ (31)

where O∗ represents the expected path cost vector.Therefore, from Equations (24), (30), and (31), we haveshown that

(H − H∗)T

O∗ = (H − H∗)T ∗ AT ∗

⎡⎢⎢⎣t1

...

tn

⎤⎥⎥⎦

= ([λ] − [λ∗])T ∗

⎡⎢⎢⎣t1

...

tn

⎤⎥⎥⎦ ≥ 0

(32)

Therefore, the formulation is equivalent to: givenuser equilibrium expected path cost, any deviation fromthe existing expected path flows cannot reduce the ex-pected path cost. Another way to say that the StrUEis reached when the expected travel times are equal onall used paths, and this common expected travel time isless than the actual expected travel time on any unusedpath.

The uniqueness of link flow is extremely important toensure the model’s applicability in transportation plan-ning process. Note that in the proposed framework,other distributions of O-D demand may also be as-sumed, provided that the gradient of the equivalent op-timization objective function represents the expectedtravel cost on each path, which is necessary to guar-antee the variational inequality of equilibrium. In ad-dition, the Jacobian of the link cost functions with re-spect to link flow must be positive definite to assure theuniqueness of expected link flow. That is, the cost func-tion should be monotonically increasing in terms of linkflow, and the dominant effect on the cost of a link shouldbe the flow. The convexity of constraints in the mathe-matical program is sufficient and necessary for the exis-tence of equilibrium. A further discussion of the varia-tional inequality and the equivalence conditions can befound in Cascetta (2009), which also explains anotherway to model variable demand.

2.3 Analytical expression

One of the strengths of this model is that we have tiedback the demand uncertainty to the mathematical ex-pression, which can substantially decrease computationsteps needed. Note that the link flow parameter λn andusers’ strategic link choice are derived from the numer-ical method and are treated invariant. They are substi-tuted into those mathematical expressions to calculatethose performance measures.

The expected travel time on a link is given by

tn = E[tn (λn) ]

324 Wen et al.

=∞∑

ln=0

[tn f

(1 + α

(ln

Cn

)4)]

g (ln ; λn) dλn

= tn f + αtn f

(1

Cn

)4

M4n ,∀n ∈ N (33)

The variance of travel time on a link is given by

var [tn (λn)] = E[t2n (λn)

]− E[tn (λn)]2

= α2 t2n f

(1C

)8 [M8

n − (M4n

)2],∀n ∈ N (34)

Mθn =

θ∑i=1

λin

{θ

i

},∀n ∈ N (35)

where the braces in Equation (35) denote the Stir-ling’s number of the second kind. Therefore, the rawmoment of link flow is monotonically increasing withrespect to λn , therefore, it can be characterized thatthe variance of travel time increases as the expectedtravel time increases, which is a commonly observedphenomenon on transportation networks around theworld (Van Lint et al., 2008; National Research Council,2013).

Once we have λn , the StrUE model can also estimatethe TSTT analytically, which are shown in the equationsbelow. Some characteristics of these expressions will bediscussed in Section 3.

E (TSTT) =∑n∈N

∞∑ln= 0

[lntn f

(1 + α

(ln

Cn

)4)]

g (ln ; λn) dλn

=∑n∈N

tn f M1n + αtn f

(1

Cn

)4

M5n

(36)

var (TSTT) = E[(TSTT − E [TSTT])2] = E(T ST T 2)

−E2 (TSTT) =∑n∈N

t2n f

[M2

n − (M1n

)2]

+ α2t2n f

(1

Cn

)8 [M10

n − (M5n

)2]+ 2αt2

n f

(1

Cn

)4 [M6

n − M1n M5

n

](37)

2.4 The most likely strategic link choice

Under the strategic user equilibrium, the link flows areuniquely defined, however, the strategic link choice ma-trix (sometimes referred to as the assignment map orO-D-specific link choice in other papers) is not unique.

That is, there might be multiple sets of strategic linkchoice that can produce the same link flows. Althoughif we run the F-W algorithm and store all the tem-porary shortest paths for each iteration, the F-W al-gorithm may provide different link choice each time,while producing the same link flows. The strategic linkchoice is extremely useful especially in transportationplanning models such as O-D estimation problem, emis-sion analysis, and so forth. In these cases, only hav-ing the aggregated link flows is not sufficient, hence auniquely determined strategic link choice is requiredto ensure the stability and applicability of this model.To address this issue, some researchers have proposedthe following maximum entropy optimization problemto determine the most likely path flows (Rossi et al.,1989; Janson, 1993; Larsson et al., 2001) (which can pro-vide strategic link choice subsequently), where entropyis defined as the number of possible route choice de-cisions made by individual travelers, and path flow andO-D demand are treated as deterministic variables, nor-mally only the expected path flow and O-D demand areconsidered

Ssystem =∏

m∈M

E (Tm)!∏k∈Km

E (hmk )!

(38)

In principle, entropy gives the number of possibleroute choice decisions made by individual travelerswithin a specific route flow solution. The objective is tomaximize the entropy of the system (the sum of the en-tropy of all shortest paths), which can be formulated asthe following equivalent mathematical program:

Max : −∑m∈M

∑k∈K U E

m

E (hmk ) ln [E (hm

k )] (39)

subject to∑k∈K U E

m

E (hmk ) = E (Tm) ∀m ∈ M (40)

hmk ≥ 0 ∀m ∈ M, k ∈ K U E

m (41)

ln =∑m∈M

∑k∈K U E

m

pmk E (hm

k ) ∀ n ∈ N (42)

Note that only the shortest paths for the strategic userequilibrium are considered; the problem is based on thatthe user equilibrium is solved and the equilibrium linkflow is obtained. Solving the Lagrangian program aboveshows that the most likely path flow follows the logitassignment, where the “link cost” is the Lagrangianmultipliers corresponding to Constraint 41 (Aka-matsu, 1996; Akamatsu, 1997). However, the aboveformulation may suffer from two limitations: First,


the transformation to the equivalent mathematicalprogram relies on Stirling’s approximation to convertthe entropy into a continuous function, and is subjectedto the limitations of this approximation (Schrodinger,1957). Second, the path flow is treated as a deterministicvariable hence it neglects the volatility in O-D demandand path flows. It is possible to define the entropy of aprobability distribution in information theory and to in-terpret this as a measure of uncertainty associated withthat distribution. In the formulation below, we will showthat if path flow and demand volatility are considered,the path flow will not follow the logit assignment, there-fore, Dial’s algorithm may not be applicable under thisconsideration.

Definition 2. The entropy of any path flow state (Herebyreferred to as path flow distribution) is the measure ofrandomness or uncertainty of locating an individual ran-dom network user. All possible states of this distribu-tion are considered. The higher the number of possi-ble states, the higher is the randomness or uncertaintyof locating an individual network user in that path flowstate.

Based on the definition, the following assumption ismade.

A.3. Under the strategic user equilibrium conditions,users make their strategic path choice (and thecorresponding strategic link choice), to maximize theprobability distribution entropy of the system.

In the aforementioned formulation, the strategic pathchoice is not uniquely defined, that is, there may be sev-eral sets of path flows that can provide the same linkflow distributions. However, the capability of estimatingpath flow uniquely may be important in various trans-portation models. Hence, extending the notion of en-tropy maximization method used in many previous re-searches (Rossi et al., 1989; Akamatsu, 1997; Kumar andPeeta, 2015), here the entropy of a random variable (in-stead of a deterministic value) which follows a certainstatistical probability distribution is defined as (Ochs,1976)

S (hmk ) = −

∞∑hm

k = 0

g (hmk ; pm

k sm) ln g (hmk ; pm

k sm)

∀m ∈ M, k ∈ K U Em (43)

where hmk is the path flow variable that follows a Poisson

distribution with parameter pmk sm . The base of the log-

arithm is not important as long as the same one is usedconsistently: change of base merely results in a rescalingof the entropy; here the natural logarithm is used. Giventhat the unconditional path flow follows a Poisson distri-

bution independently of each other, the entropy corre-sponding to a strategic path choice pm

k is

S (hmk ) = pm

k sm [1 − ln (pmk sm)]

+ e−pmk sm

∞∑hm

k = 0

(pmk sm)hm

k ln hmk !

hmk !

∀m ∈ M, k ∈ K U Em (44)

Note that the equilibrium path set for each O-D pairnormally only contains a small number of candidatepaths, leading to a relatively large pm

k sm . According toEvans et al. (1988), when pm

k sm > 4, the entropy func-tion can already be well approximated by the followingequation:

S (hmk ) ≈ 1

2ln (2πepm

k sm) ∀m ∈ M, k ∈ K U Em (45)

Considering all the O-D pairs, the objective is to finda set of strategic path choice [pm

k ] to maximize the en-tropy of the system

Max → zentropy ([pmk ]) =

∑m∈M

∑k∈K U E

m

12

ln (2πepmk sm) (46)

subject to ∑k∈K U E

m

pmk = 1, ∀m ∈ M (47)

pmk ≥ 0, ∀k ∈ K U E

m , m ∈ M (48)

λn =∑m∈M

∑k∈K U E

m

pmk δm

n,k sm ∀n ∈ N (49)

Proposition 3. The strategic path choice [pmk ] is unique

under the maximum entropy assumption.

Equivalently, the objective function can be trans-formed into the following minimization problem withthe same constraints:

min → −zentropy ([pmk ]) = −

∑m∈M

∑k∈K U E

m

12

ln (2πepmk sm) (50)

The objective function is the sum of several con-vex functions, and all three constraints are also con-vex, therefore, a unique optimal solution exists. As thestrategic link choice is the sum of several correspond-ing strategic path choices, it is also uniquely determined.The optimal solution can be obtained by various numer-ical methods such as the Newton’s method and the gra-dient descent method.

2.5 Implementation algorithms of the model

Note that the identification of the equilibrium path setis required when solving the maximum entropy problem

326 Wen et al.

above. Although this equilibrium path set is unique, it isdifficult to be obtained in practice due to computationalprecision limits. Hence, an approximation method sim-ilar to Larsson et al. (2001) is proposed here. In thismethod, all paths used in the all or nothing assign-ment procedure during the F-W algorithm are stored,until the strategic user equilibrium is reached. Then,the expected costs of all these paths are computed andthose paths whose expected costs are within a tolerancethreshold are saved as the approximated shortest paths,which will be used to solve the maximum entropy prob-lem here. The tolerance threshold is defined as a pro-portion of the shortest path cost.

t (hmk ) − t (hm

k∗)

t (hmk

∗)≤ T ol (51)

where t(hmk ) represents the expected cost of path k for

O-D pair m, t(hmk

∗) represents the shortest path cost forO-D pair m. It must be mentioned that by doing so, onlya subset of the equilibrium paths may be included, thiserror may be mitigated by setting a high relative gap(the difference between two consecutive iterations) forthe F-W algorithm. The choice of tolerance thresholdis also critical: low tolerance value may cause the opti-mization problem to be infeasible, high tolerance valuemay lead to inclusion of nonequilibrium paths. So thechoice of the tolerance threshold should be carefullymade.

The solution procedure of the model is demonstratedbelow:

Solution Algorithm:

Step 1: Initialization: Load free flow [λ]0 to the net-work, and find the free flow travel cost on eachlink.

Step 2: All or Nothing Assignment: Based on the travelcost, find a set of expected link flows [λ]1 →min z([λ]), subject to Equations (20) and (22),store the shortest path for each all or nothingassignment.

Step 3: Line search: Find the step size β →min z(β[λ]n+1 + (1 − β)[λ]n), subject to0 ≤ β ≤ 1.

Step 4: Repeat Steps 2 and 3 until [λ]n+1−[λ]n

[λ]n+1≤ ε, where

ε is the critical value which can be artificiallyset. In this step, the strategic user equilibrium isreached.

Step 5: Compute the expected cost of each stored pathbased on the equilibrium link cost.

Step 6: If t(hmk )−t(hm

k∗)

t(hmk

∗) ≤ T ol, save the path as a userequilibrium path.

Step 7: Solve the maximum entropy optimization pro-gram in Equations (46) to (49), a set of strate-

gic path choice [. . . , pmk . . . ] is obtained. Vari-

ous numerical methods, such as the Newton’smethod, may be applicable here.

Step 8: Calculate the strategic link choice from thestrategic path choice.

Step 1 provides an initial feasible solution to start thealgorithm, Step 2 builds the least cost paths set and loadscorresponding traffic on these paths. In Step 3, a stepsize parameter β is sought to minimize the objectivefunction, numerous methods are applicable here such asthe Golden section method or Bisection method. Step 4assesses the degree of convergence by computing therelative change in the expected link flow vector betweeniterations. Step 5 and Step 6 provide the equilibriumpath set which will be used to determine their corre-sponding strategic path choice. Step 7 solves the opti-mization maximum entropy problem. Step 8 computesthe strategic link choice from the strategic path choice.The F-W algorithm part can also be found in Cascetta(2009), the main difference is that in each all or nothingassignment iteration the shortest path will be stored inmemory to determine the equilibrium paths.

3 NUMERICAL EXAMPLES

The link flow volumes (and travel times) may differfrom the deterministic set of flows (and travel times)when demand variability is introduced. The strength ofthe proposed StrUE model is the incorporation of de-mand uncertainty into the user’s decision-making pro-cess; therefore, the remainder of the analysis considersonly stochastic demand conditions. Specifically, the de-mand follows a Poisson distribution with a prescribedparameter lambda. In the StrUE model, path propor-tions (and the corresponding link choice probabilities)are dependent on the demand distribution, and not anyparticular demand realization.

3.1 Monte Carlo simulation

To evaluate the StrUE model under demand variability,100 randomly selected demand scenarios for a given de-mand distribution curve (with a prescribed Poisson pa-rameter lambda) were generated through Monte Carlosampling, then for each realized demand sample, 100sets of multinomially distributed path flows are sam-pled based on the strategic path choices. As a result, wehave 10,000 sets of sampled path flows and 10,000 setsof corresponding sampled link flows, which representsthe unconditional distributions of link flow. These re-sults derived from the Monte Carlo method will be in-dicated as simulated results, and those ones computed


Fig. 1. The Sioux Falls network.

from the analytical equations will be indicated as es-timated results. Based on the simulation the followingperformance measures are computed.

(1) Expected total system travel time (ETSTT) andstandard deviation of total system travel time,(StdTSTT).

(2) Expected link travel times and standard deviationof link travel time.

(3) Expected link flow and standard deviation of linkflow.

The mathematical expression of expected travel timeon a link is given by the BPR function introduced inEquation (33). The F-W algorithm is implemented withthe modified expected link travel time function to de-termine the strategic choice, and the corresponding linkflow distributions, the relative gap for the F-W methodis set to 1*e−5. The following results are based on thenetwork depicted in Figure 1. The network has 24 nodesand 76 links. The capacity, free flow speed, length ofeach link, as well as other network attributes can befound in Bar-Gera (2012). The BPR parameters α andβ are taken to be 0.15 and 4, respectively.

Figure 2 illustrates the probability distribution of flowon several randomly chosen links, the x-axis denotesthe link flow and the y-axis represents the probabilityof observing the corresponding link flow. In total, thereare 10,000 sets of sampled link flows which are gener-ated from the 100 demand scenarios, these simulatedlink flows are plotted as a bar figure; the black curverepresents the standard Poisson distribution with theparameter lambda derived from the strategic user equi-librium. It is demonstrated that the standard Poissondistribution curve is well approximated by the simu-lated results. This is further validated by a Chi-squaretest on these four links, the corresponding p-values are

7700 7900 8100 8300 85000

2

4

6 x 10-3 link 5

Flow on link

Prob

abili

ty

4800 5000 5200 5400 56000

2

4

6

8 x 10-3 link 10

Flow on link

Prob

abili

ty

5000 5200 5400 5600 58000

2

4

6

8 x 10-3 link 31

Flow on link

Prob

abili

ty

6600 6800 7000 7200 74000

2

4

6 x 10-3 link 63

Flow on link

Prob

abili

ty

SimulatedStandard

Probability distribution of flow on different links

Fig. 2. The probability distribution of flow on several links.

328 Wen et al.

2 4 6 8 10 12 14 16 18 20 222

4

6

8

10

12

14

16

18

20

22

Estimated ETT

Sim

ulat

ed E

TT

Simulated vs estimated ETT

Rsquared=0.993

Fig. 3. Linear regression analysis of expected link travel time.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Estimated StdTT

Sim

ulat

ed S

tdT

T

Simulated vs estimated StdTT

Rsquared=0.973

Fig. 4. Linear regression analysis of standard deviation oflink travel time.

all equal to 1, which indicates that the simulated linkflow is not significantly different from the correspondingstandard Poisson distribution. Similar performance canbe observed on other links too. Therefore, if each O-D demand follows a Poisson distribution independently,the corresponding unconditional link flow estimated bythe model also follows a Poisson distribution, whose pa-rameter is determined by users’ strategic choice and de-mand distribution.

In Figures 3 and 4, the 10,000 sets of link flows aresubstituted in the BPR function to evaluate link traveltimes, which provide the expected link travel times

Table 2Network performance measures

Performancemeasures

Estimatedresults

Simulatedresults Relative error

ETSTT 7,481,223 7,582,740 1.3%StdTSTT 32,090.97 46,547.93 31.0%

(and respective standard deviation) for all the 76 links.In Figure 3, the x-axis represents the analytical traveltime estimated from Equation (33) and the y-axisrepresents the expected link travel time computed fromthe simulated results. It is shown in Figure 3 that theanalytically estimated expected link travel time closelyapproximates the simulated expected link travel times(as evident by the R2 = 0.99).

One of the strengths of the StrUE model is its capa-bility of estimating the link flow variation analytically.In Figure 4, the simulated standard deviations of linktravel times are compared with the analytically derivedvalues from Equation (34). Similar to the expected linktravel time, a linear regression analysis is done on the es-timated and simulated standard deviation of link traveltime on all the 76 links. The R squared value is veryclose to 1 despite it is smaller than that of the expectedlink travel time. Note that the simulated standard devi-ation of link flow may deviate from the estimated oneswhen the expected travel time is large due to the samplesize. Figure 4 demonstrates the model’s ability to cap-ture variability in travel time as a consequence of de-mand volatility.

Table 2 compares the system performance of ET-STT and StdTSTT of the network; the estimated re-sults are computed from Equations (36) and (37). Asdemonstrated by the table, the simulated ETSTT isvery close to the estimated ETSTT, as indicated bythe relative error of just 1.3%. However, the StdTSTTis moderately different, due to the Monte Carlo sam-pling process: first 100 realized demands are sampled,then 100 multinomially distributed path flows are sam-pled based on each realized demand, so the path flowsare actually correlated in each realized TSTT, how-ever, the analytical expressions of StdTSTT are derivedbased on the unconditional path flows which are inde-pendent of each other. This inconsistency (correlationand independency) leads to the difference in estimatedand simulated StdTSTT; therefore, the analytical ex-pression of StdTSTT should be applied cautiously. De-spite this, the ETSTT still presents a reliable approxi-mation; in addition, the estimated expected link flowsand the corresponding standard deviations satisfacto-rily match the simulated results, as shown in Figures 3and 4.


Table 3Path choices of MED and MEP for O-D pair (3-16)

Pathindex

Path (representedby a sequence of

nodes)

Path choiceprobability

(MED)Strategic path

choice for (MEP)

1 [3,4,5,6,8,16] 0.235 0.2342 [3,4,5,9,8,7,18,16] 0.093 0.0893 [3,1,2,6,8,16] 0.153 0.1574 [3,4,5,9,10,16] 0.166 0.1665 [3,4,5,6,8,7,18,16] 0.098 0.0996 [3,4,5,9,8,16] 0.17 0.177 [3,1,2,6,8,7,18,16] 0.085 0.085

3.2 The most likely strategic link choice

Sometimes the aggregated link flows would not suf-fice when O-D-specific information are required, suchas O-D matrix estimation, emission analysis and manyother transportation applications. The maximum en-tropy provides a way to find the “most likely” strate-gic link choice. In the analysis, the relative gap (termi-nation criterion) for the F-W method is set to 1*e−5,the tolerance threshold is set to 10%. In total, the F-W method provides 1,434 paths, among which 907 pathsare identified as the equilibrium paths. To demonstratethe difference between the traditional maximum en-tropy method (the deterministic case, hereby referred toas MED) and the maximum entropy of probability dis-tribution (where the entropy of path flow distribution isconsidered, hereby referred to as MEP), the results ofthese two methods are compared in Table 3. In MED,we first solve the Wardrop’s user equilibrium, whereall O-D demands are treated as deterministic variables.Then, the optimization program in Equations (39) to(42) is solved to provide the path choice probabilities. InMEP, we solve the optimization program in Equations(46) to (49) and the strategic path choice is obtained. InTable 3, O-D pair (3-16) is chosen to demonstrate thedifference between MED and MEP, the demand for thisO-D pair is 200. The differences in path choice can beseen in paths 1, 2, 3 and 5, and such difference may bemore significant if only few paths are considered as userequilibrium paths. No ground truth is found yet to provewhich method is better, but the results clearly indicatethe necessity and importance of accounting for uncer-tainties in entropy.

Table 4 demonstrates the strategic link choice on link24; the O-D pairs with no trips assigned to link 24 arenot presented in the table. If the strategic link choiceis one, it means all the users for that O-D pair will al-ways choose this link. Note that the strategic link choiceshould not be greater than one. The importance of thestrategic link choice lies in that it provides the link flow

Table 4The unique strategic link choice for link 24

From origin To destination Strategic link choice

3 7 0.3093 8 0.2703 16 0.2594 7 0.2074 8 0.1254 16 0.1464 18 0.3984 20 0.2505 7 0.3185 8 0.1725 16 0.2135 20 0.3989 7 1.0009 8 1.0009 16 0.6789 18 1.0009 20 1.000

10 8 1.00011 8 1.00012 7 0.13412 8 0.14812 16 0.15212 18 0.26813 7 0.05113 8 0.148

information disaggregated by O-D pairs. In real world,obtaining aggregated link flow is viable and efficient,and the strategic link choice can clearly present the“from-and-to” information on links.

4 CONCLUSION

In this article, we proposed a model which extends thenotion of the strategic user equilibrium originally pro-posed by Dixit et al. (2013) and Waller et al. (2013).The proposed model relaxed the assumption of propor-tional demand, hence improved the model fidelity. Inthis model, users equilibrate based on an expected con-dition as opposed to a deterministic cost. To capture thisbehavior, the model assumes that users rationally maketheir strategic link choice by considering all possibledemand scenarios (all the O-D demands are indepen-dent of each other) in a known distribution. This strate-gic link choice is then followed regardless of the real-ized travel demand in any given scenario. Therefore, thestate of equilibrium may not be observed on a given day.As such, the proposed model is illustrated to replicatethe behavior of observed link travel time variability. Inthe proposed model, link flow distributions and users’

330 Wen et al.

strategic link choice are proved to be unique mathemat-ically; network performance measures are given in ana-lytical expression, which reduces the computation bur-den of network performance prediction. The efficiencyand accuracy of these analytical expressions are demon-strated with a numerical example; the importance of ac-counting for probability distribution in entropy functionis also presented. Therefore, this model accounts for thedemand uncertainty and users’ strategic choice whilemaintaining computation simplicity and tractability.

However, every model has its limitations. In this arti-cle, we assumed that O-D demand follows a Poisson dis-tribution to ensure uniqueness, this forces the expecteddemand to be equal to the variance of demand, and thisassumption may limit the applicability of the model.

Many possibilities are still to be explored under themodel’s framework. O-D demands may be assumed tofollow some biparametric probability distributions if theuniqueness is guaranteed, which allows mean and vari-ance to be different. In addition, we may account forcapacity uncertainty by assuming the capacity also fol-lows a certain kind of distribution. Finally, integratingthis model into the O-D estimation problem would be astraight forward contribution.

ACKNOWLEDGMENTS

This research was supported (partially) by the Aus-tralian Government through the Australian ResearchCouncil’s Discovery Projects funding scheme (projectDP150104687). The views expressed herein are thoseof the authors and are not necessarily those of theAustralian Government or Australian Research Coun-cil. This research was in part made possible by thegenerous support of Data61. Data61 is funded by theCommonwealth Scientific and Industrial Research Or-ganisation (CSIRO), Australia.

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