Doubly uncertain transportation network: Degradable capacity and stochastic demand

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European Journal of Operational Research 191 (2008) 166–181 www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Doubly uncertain transportation network: Degradable capacity and stochastic demand Barbara W.Y. Siu, Hong K. Lo

*

Department of Civil Engineering, The Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong Received 7 November 2006; accepted 10 August 2007 Available online 4 September 2007

Abstract This study developed a methodology to model doubly uncertain transportation network with stochastic link capacity degradation and stochastic demand. We consider that the total travel demand comprises of two parts, infrequent travelers and commuters. The traffic volume of infrequent travelers is stochastic, which adds to the network traffic in a random manner based on fixed route choice proportions. On the other hand, the traffic volume of commuters is stable or deterministic. Commuters acquire the network travel time variability from past experiences, factor them into their route choice considerations, and settle into a long-term habitual route choice equilibrium in which they have no incentive of switching away. To define this equilibrium, we introduce the notion of ‘‘travel time budget’’ to relate commuters’ risk aversion on route choices in the presence of travel time variability. The travel time budget varies among commuters according to their degrees of risk aversion and requirements on punctual arrivals. We then developed a mixed-equilibrium formulation to capture these stochastic considerations and illustrated its properties through some numerical studies.  2007 Elsevier B.V. All rights reserved. Keywords: Transportation network reliability; Degradable network; Stochastic demand; Uncertainty modeling

1. Introduction Traditionally, the transportation network performance is modeled via a deterministic approach, which assumes that both transportation supply (network link capacity) and origin–destination (OD) demand are perfectly known. In reality, both supply and demand are subject to stochastic variations. Link capacity degradations can be caused by major events, such as earthquakes, which involve catastrophic damages to the system for a prolonged period of time. In this case, the problem to address is mainly on network connectivity. On the other hand, link capacity degradations can also be caused by relatively minor events, in which the system is not severely damaged but nevertheless its capacity is degraded to different degrees. Such events may include vehicle breakdown, accident, etc. These relatively minor events occur quite frequently in our daily commutes,

*

Corresponding author. Tel.: +852 2358 8742. E-mail address: [email protected] (H.K. Lo).

0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.08.026

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causing stochastic link capacity degradations, which constitute the source of capacity variations modeled in this study. On the demand side, its stochastic variation on transportation network performance is equally important but is generally not as well covered in the literature. Instead, demand is typically modeled as either fixed or a deterministic function of average travel time (Nicholson and Du, 1997; Chen et al., 2002; Lo and Tung, 2003; Lo et al., 2006). Recently, Clark and Watling (2005) and Sumalee et al. (2006) considered the effect of stochastic demand on network reliability. To our knowledge, there were no prior studies that include both stochastic capacity degradation and stochastic demand in transportation network analysis. One key issue to be addressed in this study is to integrate the consideration of stochastic demand with stochastic capacity degradation. We consider the travel demand as consisting of two groups of trip makers: (i) commuters, who travel regularly between the same OD pair on a daily basis, and (ii) infrequent travelers, who make infrequent or irregular trips, such as tourists and people on special errands. The traffic volume of commuters is fairly stable, whereas that of infrequent travelers constitutes higher stochastic variations. We postulate that these two groups of travelers behave differently in their route choices. Commuters are keenly aware of the alternatives and would select routes to minimize their generalized travel costs. Infrequent travelers, on the other hand, not knowing the traffic condition or alternatives better, would select routes based on shortest distance or spread themselves randomly among a few reasonable routes. In surveys, it was found that the routes that travelers use in general constitute a limited set (Cascetta et al., 1997). Due to stochastic variations in both transportation supply and demand, travel time becomes uncertain. Travel time reliability plays an important role in travelers’ route choice behavior. Jackson and Jucker (1981) indicated that the perceived reliability is an important component of their route choice decisions. In another empirical study, Abdel-Aty et al. (1995) found that travel time reliability was one of the most important factors for route choice considerations. Recently, Lo et al. (2006) modeled commuters’ route choice behavior under stochastic link capacity degradation by the notion of ‘‘travel time budget’’, i.e., the travel time allocated for a trip between an origin–destination pair. This concept is similar to the idea of ‘‘headstart’’ discussed in Gaver (1968). In the presence of travel time variations, to hedge against uncertainty, commuters generally add a travel time margin to the expected trip time to form their travel time budget. In forming their routing plans, commuters select routes to minimize their travel time budget, just as they do to lower their mean travel time, forming a habitual travel pattern. Indeed, our preliminary study shows that a close relationship exists between travelers’ travel time budget and their schedule departure time (Siu and Lo, 2007). How to integrate travelers’ departure time and route choices in the presence of travel time uncertainty definitely is an important topic for further studies, but will require more sophisticated dynamic traffic assignment models (e.g., among others, Lo and Szeto, 2002; Szeto and Lo, 2004, 2006). In this study, with the purpose of laying out the analysis framework, we adopt the simpler static model via the concept of travel time budget for the analysis and save the problem of simultaneous route and departure time choices under travel time uncertainty to a future study. In fact, a similar static treatment of congestion (i.e., congestion delay is time invariant) was adopted in previous travel time reliability studies (e.g., Noland et al., 1998; Bates et al., 2001; Noland and Polak, 2002). Here, our main objective is to develop a framework to study network performance under both stochastic supply and demand. The outline of this paper is as follows. Section 2 depicts the formulation. Section 3 describes the numerical studies to illustrate the properties of the formulation. Finally, Section 4 contains some concluding remarks. 2. Formulation In the following, Section 2.1 defines the main notations in this paper. Sections 2.2 and 2.3 depict the link and route travel time distributions by considering both stochastic link capacity degradation and stochastic demand. Section 2.4 discusses the travel time budget and associated equilibrium conditions. The notion of travel time budget is extended from Lo et al. (2006); for readability, we skip the derivation details but include only key results there. Section 2.5 formulates the travel time equilibrium problem as a Nonlinear Complementarity Problem (NCP), with its solution existence proved by the equivalent Variational Inequality (VI) formulation in Section 2.6. Finally, Section 2.7 provides the solution procedure via a gap function.

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2.1. Notations Random variables are expressed in capital letters and lower-case letters are used for instances of random variables or deterministic variables: a Ca ca xa Ya Xa rs i ki drs d irs Ers Prs p fp fpi bip pirs qi Gp Fp dpa wrs p lY a rY a SY a Ta t0a Tp rT p ST p ha h x

link index, a 2 A capacity of link a design capacity (upper bound) of link a commuter flow on link a infrequent traveler flow on link a total link flow on a: Xa = xa + Ya OD pair index, rs 2 RS type i commuter, with different degree of risk aversion, i 2 I degree of risk aversion of type i commuters commuter OD demand between pair rs type i commuter OD demand between pair rs infrequent traveler OD demand between pair rs path set between OD pair rs Path index, p 2 Prs commuter path flow on p type i commuter path flow on p the travel time budget of type i commuter on p the shortest OD travel time budget for type i commuter the within-budget time reliability of type i commuter infrequent traveler path flow on route p total path flow on p: Fp = fp + Gp link-path incidence parameter; 1 if link a is on path p, zero otherwise P proportion of infrequent travelers between OD pair rs use path p; p2P rs wrs p ¼1 mean volume of infrequent travelers on link a standard deviation of volume of infrequent travelers on link a standard normal variate of volume of infrequent travelers on link a travel time on link a free-flow travel time on link a travel time on route p standard deviation of travel time on route p standard normal variate of route travel time degree of worst-degraded capacity for link a degree of worst-degraded capacity for all links in the network proportion of commuter traffic

2.2. Link and route flows of commuters and infrequent travelers We consider that infrequent travelers on OD pair rs, Ers, select routes independently of the traffic condition (as they do not know the traffic condition better), and spread themselves among a few reasonable routes with given proportions wrs p . Accordingly, Ya can be written as X X p X X p Ya ¼ ðda Gp Þ ¼ ðda Ers wrs ð1Þ p Þ: rs2RS p2P rs

Ers wrs p

rs2RS p2P rs

The product on the RHS of (1) represents the flow on route p between rs, i.e., Gp. By applying the route-link incidence variable dpa and summing through all the route flows in the network, one obtains the corresponding link flows on each link a. Since the route flows of infrequent travelers are stochastic, so are the corresponding link flows. For networks with many OD pairs, assuming that each term Ers wrs p is independent

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and bounded, which is sensible as infrequent travelers are not fully aware of the differences among the few selected reasonable alternatives, then the Lyapunov condition under the Central Limit Theorem is satisfied (see Theorem 27.3 and Example 27.4, Bilingsley, 1995). Hence, Ya defined in (1) follows the normal distribution and can be expressed in terms of its mean ðlY a Þ, standard deviation ðrY a Þ and standard normal variate ðS Y a Þ: Y a ¼ lY a þ rY a S Y a :

ð2Þ

The normal distribution is used for convenience here. The random variable Ya may be modeled with the lognormal distribution for the analysis, which ensures non-negative link flows. For networks with a limited number of OD pairs, by assuming each Ers as independent and normally distributed, (2) is still valid. On the other hand, the correlation between different Ers’s can be introduced by modeling the OD demands as multivariate normal distributions. Then, the link flows of these OD demands follow another set of multivariate normal distributions, with each Ya being univariate normal. The mean and variance of Ya can be readily computed, albeit more laboriously, due to the covariance matrix, which could be a future extension of this study. For the sake of simplicity, we consider for the rest of our discussion that Ers is independent and bounded. 2.3. Link and route travel time distribution To model congestion, we use the commonly adopted Bureau of Public Roads (BPR) link performance function:   n  xa 0 ta ðxa ; ca Þ ¼ ta 1 þ b ; ð3Þ ca where subscript a refers to a particular link; t0a ; ca ; ta , respectively, are link a’s free-flow travel time (which is deterministic), capacity, and travel time with flow xa; b and n are deterministic parameters associated with the BPR travel time function. In the case of stochastic demand, the link flow xa is replaced by Xa. In addition, as the link capacity is subject to stochastic degradation, ca is replaced by the random variable Ca. Then in (3), the link travel time ta becomes a random variable:   n  xa þ Y a 0 T a ðxa ; Y a ; C a Þ ¼ ta 1 þ b ð4Þ Ca with its mean and variance expressed as    n   n  xa þ Y a xa þ Y a 0 0 0 ¼ ta þ bta E ; EðT a Þ ¼ E ta 1 þ b Ca Ca 2

varðT a Þ ¼ E½ðT a Þ   E2 ðT a Þ:

ð5Þ

The variability of Ya and Ca arises from difference sources, which need to be accounted for separately, with their combined effect modeled jointly through (4) and (5). Assuming that the capacity degradation associated with Ca is independent1 of the amount of traffic on it (i.e., xa + Ya), and knowing that the free-flow travel time t0a is deterministic (i.e., Eðt0a Þ ¼ t0a and varðt0a Þ ¼ 0Þ, E(Ta) and E[(Ta)2] can be simplified as   1 n 0 0 EðT a Þ ¼ ta þ bta E½ðxa þ Y a Þ E n ; Ca !   1 1 2 n 2n 2 0 2 0 2 0 2 E½ðT a Þ  ¼ ðta Þ þ 2bðta Þ E½ðxa þ Y a Þ E n þ b ðta Þ E½ðxa þ Y a Þ E : ð6Þ Ca C 2n a The expressions in (6) allow of the expectation and variance of link travel time, provided  the  calculation  1 1 that these four terms, E Cn , E C2n , E[(xa + Ya)n], E[(xa + Ya)2n], can be evaluated. The first two terms a

a

depend on the probability distribution of the link capacity Ca; whereas the last two terms relate to the prob1

Lo and Tung (2003) provides the generalization of the capacity random variable as a function of link flow.

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ability distribution of the stochastic link flow demand Ya. To simplify the exposition, this study assumes that link capacity follows a uniform2 distribution with the design capacity of the link as its upper bound and the worst-degraded capacity as its lower bound. Furthermore, we consider the lower bound to be a fraction ha of the design capacity, ca , which can be link and/or network specific. For uniformly distributed link capacities, with n > 1, the mean and variance of 1=C na can be derived as by the following:  E

1 C na



1 E C 2n a

¼

Z

ca

ha ca

! ¼

Z

1 1 1  h1n a dca ¼ n ;  n ca ð1  ha Þð1  nÞ ca ðca  haca Þ ð7Þ

ca

ha ca

1 1 1  ha12n dc :  ¼ a c2n c2n ðca  haca Þ a a ð1  ha Þð1  2nÞ

For the special case of n = 1 (i.e., travel time varies linearly with link flow), the same procedure of finding the expectations still applies, but the integration of the term 1/c involves the logarithm function (Lo and Tung, 2003). The fraction ha indicates the degree of link capacity degradation: ha = 1 refers to the ideal case of a non-degradable link or when capacity degradation is ignored. If ha is statistically independent among links, then the link capacity Ca is also independent of each other. On the other hand, one may consider situations where network link capacity degradations are correlated. In this case, we let C some joint distribution fC(Æ) of the capacities of neighboring links. In this case, the term a follows  1 E Cn can be evaluated as a #   Z ca "Z 1 1 fc ð:Þ dcAna dca ; ð8Þ E n ¼ Ca cna Ana 0 where the inner integral is taken over the domain of all link capacities except for link a. The other two terms E[(xa + Ya)n], E[(xa + Ya)2n] are related to demand variations. Using (2) and the binomial expansion, we obtain this general expression for the expected link flow volume: n

E½ðxa þ Y a Þ  ¼ ðxa þ lY a Þ

n

n   X n i¼0

i

rY a xa þ l Y a

i

E½ðS Y a Þi :

ð9Þ

The term E½ðS Y a Þi  can be obtained through Mellin Transform as depicted in Lo and Tung (2003). In a similar fashion, we obtain the expanded expression for the term E[(xa + Ya)2n] in (6). In short, the four terms related to capacity degradation and stochastic demand can all be evaluated for the purpose of calculating the expectation and variance of link travel time. Based on (6), according to different modeling assumptions of commuters’ responses to travel time variability, the analysis can be separated into four scenarios, listed below in descending order of complexity. Case A is the most complete specification, including considerations in both stochastic capacity and stochastic demand, which represents the ‘‘true’’ behavior. On the other hand, the Base Case is the most simple, completely ignoring the stochastic effects. Cases B and C neglect, respectively, the effect of stochastic demand and that of stochastic capacity. The purpose here is to illustrate the relative importance of incorporating these two stochastic considerations, as discussed in the section of numerical studies. 2.3.1. Case A: Stochastic supply, stochastic demand (SS–SD) In Case A, commuters consider both stochastic variations in supply (or capacity) and demand in their route choice selection. This is the most complete and realistic among the four scenarios. To obtain the travel time mean and variance expressions for Case A, we substitute (7) and (9) into (6):

2

Generalizing the consideration to other probability distributions can be accomplished with the technique Mellin Transform as discussed in Lo and Tung (2003).

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EðT a Þ ¼

t0a

þ

bt0a ðxa

n X 1  h1n a þ lY a Þ n ca ð1  ha Þð1  nÞ i¼0 n

varðT a Þ ¼ b2 ðt0a Þ2 ðxa þ lY a Þ2n

1h12n a c2n a ð1ha Þð12nÞ

2n P i¼0

! n  i !

2n  i

rY a xa þ lY a

rY a xa þlY a

i

i

171

E½ðS Y a Þi ; ( i

E½ðS Y a Þ  

1h1n a cna ð1ha Þð1nÞ

n P i¼0

! n  i

rY a xa þlY a

i

)2 ! i

E½ðS Y a Þ 

:

ð10Þ

Note that in (10), the effect of stochastic demand is reflected through the mean and standard deviation of the random demand Ya; whereas the effect of stochastic capacity is reflected through the level of capacity degradation ha and its underlying distribution. Both of these stochastic variations affect the mean and variance of travel time. 2.3.2. Case B: Stochastic supply, deterministic demand (SS–DD) In Case B, only the effect of stochastic capacity is considered in modeling commuters’ route choices, with the effect of stochastic demand ignored or that the random demand Ya and its associated statistics vanish, i.e., S Y a ¼ rY a ¼ lY a ¼ 0. Consequently, (10) is simplified to EðT a Þ ¼ t0a þ bt0a varðT a Þ ¼ b

2

1  ha1n xn ; cna ð1  ha Þð1  nÞ a

2 ðt0a Þ

 2 ! 1  ha12n 1  h1n a  n x2n a : c2n  ð1  h Þð1  2nÞ ð1  h Þð1  nÞ c a a a a

ð11Þ

2.3.3. Case C: Deterministic supply, stochastic demand (DS–SD) In Case C, only the effect of stochastic demand is captured in modeling commuters’ route choices. Thus, the stochastic terms involving Ya in (10) remain, whereas those associated with Eð1=C na Þ and Eð1=C 2n a Þ in (6) are simplified to their deterministic forms, respectively, as 1=cna and 1=c2n . Consequently, (10) can be simplified to a     n i n bt0 ðxa þ lY a Þ X n rY a i EðT a Þ ¼ t0a þ a E½ðS Y a Þ ; cna x þ l i a Ya i¼0 ð12Þ  n   2 !   2n 2 0 2 i i 2n b ðta Þ ðxa þ lY a Þ P P 2n n i i rY a rY a varðT a Þ ¼ : E½ðS Y a Þ   E½ðS Y a Þ  xa þlY a xa þlY a c2n i i a i¼0 i¼0 2.3.4. Base case: Classical deterministic user equilibrium (DUE) In this base case, stochastic variations in both supply and demand are ignored. Commuters are assumed to select routes as if the network and demand were perfectly deterministic. In this case, E(Ta) is simply represented by the BPR function in (3), with the travel time variance set to zero. The base case degenerates into the classical deterministic user equilibrium (DUE) formulation. These four scenarios are set up to compare the accuracy gained vis-a`-vis the complexity of the modeling effort. Scenario A is the most realistic and complete as it captures both types of stochastic variations, whereas the base case is the status quo, also the simplest among the four. This study assumes that link capacity (and hence link travel time) distributions are independent, as is customary in this type of study. This assumption is reasonable for relatively minor network disruptions this study aims to model. The route travel time variable can thus be expressed by summing the corresponding link travel time variables (which are themselves random): X p Tp ¼ da T a ; ð13Þ a

dpa

where is the route-link incidence variable whose value is one if a is on p; zero otherwise. Regardless of link travel time distributions, applying the Lyapunov condition for the Central Limit Theorem, Tp defined in (13) follows the normal distribution. The route travel time mean and standard deviation can thus be written as

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T p  N ðEðT p Þ; rT p Þ; X p ½da  EðT a Þ; EðT p Þ ¼ a

rT p

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X p ¼ ½da  varðT a Þ:

ð14Þ

a

Combining the link travel time and variance expression derived above into (13) and (14), we obtain the expected route travel time and its variance or standard deviation. The assumption of independent link time distribution can be relaxed to allow for correlation between link travel times. Clark and Watling (2005) and Sumalee et al. (2006) analytically derived the moments of total network travel time due to Poisson OD demand and route choices while incorporating the correlation between link flows and that between link travel times. Some of the results there are useful for this relaxation. In this study, the assumption of independent link time distribution is adopted for simplification and mathematical clarity in the model development effort. For actual implementation, as adeptly noted by Watling (2006), estimating link correlations is a difficult problem; if they can be assumed to be small as compared with the variances, it might be justifiable to neglect them. Certainly, this issue of link correlation is an important topic on its own and deserves further studies. 2.4. Definition of travel time budget Link capacity degradations and stochastic demand cause link and route travel time variability. Commuters, therefore, do not know their exact a priori travel times. Some commuters would depart earlier to allow for additional time, or add a travel time margin to the expected trip time, in order to avoid late arrivals. In other words, commuters allow for a longer travel time budget to hedge against travel time variability. The reserve margin under travel time uncertainty was proposed by Knight (1974) and referred to as ‘‘headstart’’ in Gaver (1968). In Lo et al. (2006), we defined the travel time budget as the sum of the expected travel time plus a travel time margin as in (15) and related the travel time budget to the probability of punctual arrival as explained below. ½Travel Time Budget ¼ ½Expected Travel Time þ ½Travel Time Margin:

ð15Þ

Mathematically, the travel time budget associated with route p, bip , is expressed as bip ¼ EðT p Þ þ ki rT p

8p 2 P rs 8rs 2 RS 8i 2 I:

ð16Þ

The second term on the right hand side of (16) is analogous to the ‘‘planning cost’’ in Noland et al. (1998). Nonetheless, ki is not the same as the relative weight in mean-variance or mean-standard deviation models discussed in Jackson and Jucker (1981) or those summarized in de Palma and Picard (2005). Here, the parameter ki is defined as the requirement of punctual arrivals or the probability of arriving at the destination within the travel time budget by different types of commuters, expressed as n o P T p 6 bip ¼ EðT p Þ þ ki rT p ¼ qi : ð17Þ qi is referred to as the within-budget time reliability (WBTR) for type i commuter. Eq. (17) can be rearranged into:   T p  EðT p Þ P ST p ¼ 6 ki ¼ qi ; ð18Þ rT p where S T p is the standard normal variate associated with Tp. According to (17) or (18), a large ki results in a large travel time budget bip , which in turn maintains the WBTR qi at a high level. Commuters with a value of ki > 0 (ki < 0) are regarded as ‘‘risk averse’’ (‘‘risk prone’’). Commuters with ki = 0 are ‘‘risk neutral’’, who are concerned with the mean travel time only and is categorized as the mean-time (MT) commuters in this study. Specifically, commuters with ki = 1.64 results in a WBTR of 95% (assuming travel time follows normal distribution) and is categorized as high reliability (HR) commuters in this study. In fact, the coefficient ki is formally related to the early and late schedule delay penalties of travelers within a broader modeling framework (Siu and Lo, 2007).

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There is a vast volume of studies on measuring the disutility of travel time uncertainty (e.g., de Palma and Picard, 2005; Abdel-Aty et al., 1995; Jackson and Jucker, 1981; Noland et al., 1998; Liu et al., 2004; Noland and Polak, 2002; and Bates et al., 2001). Virtually all of these studies attempt to measure the relative importance between travel time and its uncertainty with respect to trip-related attributes. The coefficient ki defined in this study, on the other hand, is related directly to the reliability of punctual arrivals through (16)–(18). To demonstrate the applicability of this new interpretation, we conducted a survey to relate the risk aversion parameter ki with travelers’ requirements on punctual arrivals with respect to their socio-economic characteristics such as gender, age, occupation, and income. The results are summarized in Siu and Lo (2006b), showing the consistency between these two measures, i.e., between ki and the requirements on punctual arrivals. The travel time budget formulation developed here is also different from the bi-criteria traffic assignment model (e.g., Dial, 1996, 1997). The bi-criteria traffic assignment model considers the value-of-time (VOT) as a random variable to indicate how travelers value travel time relative to monetary cost. In this model, we are after the variability in travel time itself, not the variability associated with the VOT. In our model, the risk aversion coefficient ki is associated with a particular type of commuters. Different types of commuters are pre-defined and combined in the multi-class traffic assignment platform to be discussed in Section 2.5. 2.5. Habitual travel time budget equilibrium In modeling route choices of commuters, we draw upon the notion of a long-term habitual equilibrium. That is, after experimenting and exploring their route choices for an extended period of time, commuters settle into a set of fixed route choices day in and day out, which they do not have incentive to switch away. This set of route choices is specific to the type of commuters, notated as type i, dependent on their level of risk aversion, ki. Formally, this long-term habitual equilibrium route choice pattern is analogous to the Wardropian principle: For each OD pair, the flow of type i commuters, fpi , on route p is positive if the travel time budget on p is equal and minimal; all unused routes have an equal or higher travel time budget. Denoting pirs as the minimum travel time budget linking OD pair rs for type i commuters, the travel time budget equilibrium conditions for them can be described by ð19Þ fpi ðbip  pirs Þ ¼ 0 8p 2 P rs 8rs 2 RS 8i 2 I; bip  pirs P 0 8p 2 P rs 8rs 2 RS 8i 2 I; X f i ¼ d irs 8rs 2 RS 8i 2 I; p2P rs p

ð20Þ

fpi P 0

ð22Þ

8p 2 P rs 8rs 2 RS 8i 2 I:

ð21Þ

(19), if > 0, then ðbip  pirs Þ ¼ 0, or that the travel time budget for type i commuters is equal travel time budget pirs . Condition (20) asserts that pirs is the minimum travel time budget for all fpi

According to to the minimum routes on OD pair rs. Condition (21) is the demand constraint and (22) constitutes the non-negativity condition. 2.6. Equivalent formulations The mixed-equilibrium conditions (19)–(22) can be expressed as a nonlinear complementarity problem (NCP) (Aashitiani, 1979): xT  FðxÞ ¼ 0;

ð23Þ FðxÞ P 0; x P 0;   f i where x ¼ is a column vector with f ¼ ðfpi Þ a column vector  f ofroute flows; p ¼ ðprs Þ a column vector of p F ðxÞ the minimum OD travel time budget; and FðxÞ ¼ , where Ff(x) is a column vector Fp ðxÞ P  i i of ðbip  pirs 8rs 2 RS 8p 2 P rs 8i 2 IÞ and Fp(x) a column vector of f  d 8rs 2 RS 8i 2 I . We ders rs p p2P n2 fine X ¼ fXf : f 2 Rn1 þ ; Xp : p 2 Rþ g as the solution set to the above NCP, in which n1 is the number of routes

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multiplied by the number of commuter types and n2 is the number of OD pairs in the network multiplied by the number of commuter types and n = n1 + n2. To take advantage of recent advancements in solution procedures and analysis introduced by variational inequality (VI), the NCP (23) can be rewritten as the following VI: VIðx; XÞ : find a vector x such that ðx  x Þ  FðxÞ P 0 8x 2 X;

ð24Þ

where x, F(x) and X are as defined in (23). In general, the equivalence between VI and NCP is well established (Nagurney, 1993). The existence of solution of VI(x, X) requires that (i) F(x) is a continuous function of x and (ii) X is a non-empty compact convex set (Theorem 1.4, Nagurney, 1993). Condition (i) holds if route cost is a continuous function of route flow (Szeto and Lo, 2005), which is generally the case for static traffic assignment. In addition, condition (ii) also holds for the problem here, as Xf is bounded by the maximum OD demand and Xp is bounded by the highest travel time budget among all the paths which is finite. Therefore, the solution space X can be reduced to Br(0), which is a closed ball centered at 0 with radius f = max{A, B}, where A ¼ maxp;i fbip g, B = maxrs{drs}. 2.7. Solution approach A number of projection methods can be used to solve the VI in (24) (e.g., Han and Lo, 2002; Han and Lo, 2004). To draw upon the efficiency of algorithms developed for nonlinear mathematical programs, one can reformulate the VI to a minimization program by converting the complementarity conditions (19)–(22) to a gap function as addressed in Lo and Chen (2000a,b). The gap function adopted for the solution procedure is based on the following simple, two-variable, convex function proposed by Fischer (1992): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ða; bÞ ¼ a2 þ b2  ða þ bÞ: ð25Þ Note that /(a, b) = 0 () a P 0, b P 0, ab = 0, or when /(a, b) = 0, the complementarity condition between a and b are fully satisfied. The proposed gap function is formed by squaring /(a, b)(Facchinei and Soares, 1995): u(a, b) = (1/2)/2(a, b). Let GðxÞ ¼

n X

uðxk ; F k ðxÞÞ

ð26Þ

k¼1

be the gap function for solving the NCP (23), where k represents the kth element in the vector x. This function G(x) satisfies the following three conditions required of a gap function: (i) G(x) P 0. (ii) G(x) = 0 () x 2 X. (iii) minx2WG(x) = 0 is a global minimum, where W = {x P 0, F(x) P 0}. Basically, when G(x) attains the minimum gap at zero, the complementarity formulation is solved. Reformulating the UE conditions as this gap function is advantageous from a computational point of view, as it enables a smooth and unconstrained optimization on min G(x) (Lo and Chen, 2000a). For a general network which comprises of I classes of heterogeneous risk aversions, the mixed-equilibrium can be obtained by the following unconstrained mathematical program (MP): !  X X X X X  X i i i i i i min G¼ u fp ; bp  prs þ u prs ; fp  d rs : ð27Þ i i fp ;prs

i2I

rs2RS p2P rs

i2I

rs2RS

p2P rs

In this formulation, the decision variables include the route flows fpi and the minimum OD budget time pirs . The function G refers to the overall gap to capture the equilibrium conditions for the I classes of commuters, as in (19)–(22). The travel time budget bip for class i commuter follows from (16) with the corresponding WBTR value of ki. As discussed earlier, the network travel time statistics are derived (Section 2.3) as functions of both the stochastic network link capacities, route choices of different types of commuters, as well as the

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stochastic demand of infrequent travelers. The solution of this nonlinear mathematical program provides the route choice pattern and the corresponding travel time budget that satisfy the equilibrium complementarity conditions and the within-budget time reliability requirements. The unconstrained MP (27) can be solved by a number of commercially available packages. 3. Numerical studies To demonstrate the formulation, we apply it to the Nguyen and Dupuis network (1984) in Fig. 1, which consists of 13 nodes, 19 links, and 4 OD pairs. The base demand for each OD pair is 20 units and is adjusted by a total demand multiplier to model the effect of congestion on the system performance. The free-flow travel time, design capacity, and degradation parameter for each link are shown in Table 1. The parameters for the link performance function in (4) are b = 0.15, n = 4. The 4 OD pairs are served by a total of 25 routes. We consider the commuter OD demands as comprised of equal portions of MT and HR commuters with

Origin 2

1

12 18

1

17

Origin 3

4

5

5

6

6

4

7

12

8 11

10

8

9

9

7

14

10

11

15

2 Destination

16

13 19 13

3 Destination

Fig. 1. The Nguyen and Dupuis network.

Table 1 Link parameters Link

Free-flow travel time

Design capacity

Degradation parameter (ha)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

10 10 10 20 10 10 10 10 10 10 10 10 20 10 10 10 10 30 10

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9 0.5 0.7 0.9

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ki = 0, 1.64 (WBTR = 50%, 95%), respectively. The mean demand of infrequent travelers constitutes a fraction of the total OD demand, with a standard deviation equal to half of the mean commuter demand, and distributes themselves among all the available routes as specified. 3.1. Importance of modeling supply and demand uncertainty 3.1.1. Effect of congestion The four modeling scenarios discussed in Section 2.3 are analyzed under three different congestion levels: slightly congested (50% of the base demand), congested (100% of the base demand), and heavily congested (150% of the base demand). We solve the unconstrained MP in (27) with different travel time variability specifications according to Cases A–C in Section 2.3. As a reminder, Case A is the most complete representation of the actual traffic as both stochastic supply and stochastic demand are jointly incorporated, which represents the ‘‘true equilibrium’’ under the static congestion assumption without departure time choices (as will be discussed in Section 4); whereas Cases B and C are ‘‘incomplete models’’ with certain aspects of the stochastic behavior of the network ignored; and DUE is the classical approach in analyzing traffic equilibrium. To demonstrate the effects of neglecting certain aspect of the stochasticity, we solve the formulation with the travel time expressions corresponding to each simplification and obtain the resultant route flows. Subsequently, we simulate the performance of these route flow patterns in Scenario A (i.e., the most realistic representation). For instance, in Case B, we solve the unconstrained MP (27) with the travel time expressions in (12) and substitute the resultant route flows according to the travel time expressions for Case A, i.e., (10). In this way, we simulate the performance of the resultant route flows from Case B in the presence of the most complete representation of the system modeled herein. The performance measure used for comparison is the average travel time budget, as normalized by that of Case A. In Fig. 2, the four models are compared considering that commuters constitute 70% of the network traffic and ha is chosen arbitrarily among 0.5, 0.7 and 0.9, as shown in Table 1. For DUE and Case C, the effect of stochastic demand is ignored. Commuters are assumed to select routes based on the commuter traffic alone while ignoring the presence of infrequent travelers. For DUE and Case B, stochastic capacity degradation is ignored. Commuters are assumed to select routes based on the perfect network, i.e., ha = 1 "a 2 A. Of course, the scenarios of DUE, Cases B and C are incomplete models and cannot reproduce the correct result. From Fig. 2, it can be seen that the discrepancies of the simplifications strongly depend on the congestion levels. When traffic is light, all cases produce similar average travel time budgets, implying that one does not lose too much accuracy by ignoring the stochastic behavior of the system. However, as the congestion level increases, their differences become pronounced. For example, simulating the DUE route flows in the most complete representation of Case A, the average travel time budget can go up by 35%. That is, ignoring the stochastic effect can result in a sizeable underestimation of the actual travel time budget. This is sensible: when the network is not congested, spare capacity exists as a buffer to absorb small surges in demand or drops in 1.35

Average travel time budget normalized by Case A

1.30 1.25

Case A (SS-SD) Case B (DS-SD) Case C (SS-DD) DUE (DS-DD)

1.20 1.15 1.10 1.05 1.00 0.95 Slight

Moderate Congestion Level

Heavy

Fig. 2. Comparison of the average travel time budgets among the four modeling scenarios.

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link capacity. As the congestion worsens, such spare capacity simply does not exist. Hence, commuters typically underestimate their travel time budgets by ignoring stochastic supply and demand in their route choice considerations. 3.1.2. Effect of stochastic supply and demand In this section, we study the sensitivity of model errors in omitting stochasticity in supply and demand. For illustration purposes, we set the link degradation parameter ha to be uniform across the whole network, as denoted by one single h. We vary h (stochastic supply) and the proportion of infrequent travelers (stochastic demand) and plot the errors associated with ignoring their effects. For exposition, we choose DUE in this analysis and define the error to be the deviation, GAP, from the correct travel time budget equilibrium as determined in Case A, computed by  X X X   GAP ¼  pi;A ð28Þ fpi;DUE ðbi;DUE p rs Þ: i2I

rs2RS p2P rs

In (28), the notations follow from previous definitions with some superscripts added. fpi;DUE refers to the route flow as determined by DUE; bi;DUE the resultant travel time budget associated with fpi;DUE when simup lated with the link and path time specifications as in Case A; and pi;A rs the minimum OD travel time budget determined in Case A. In Fig. 3, GAP is plotted as contours. The x-axis refers to h, plotted in log scale; when h = 1, the network is perfect without any degradation. On the other hand, h = 0 represents a highly degradable network. The y-axis represents the proportion of commuters (x), again plotted in log scale; when x = 1, the network consists of purely commuters with no stochastic infrequent travelers. On the other hand, x = 0 represents a network with entirely stochastic demand, rendering the equilibrium analysis meaningless. In summary, a network with h = x = 1 is identical to the DUE case. The parameters h, x, with values between 0 and 1, represent the stochastic supply and demand nature of the network. Examining the GAP values in Fig. 3, for a network with even slight stochastic behavior (the upper right corner: x = h = 0.9), the GAP value is in the hundreds, meaning that the DUE route flows is away from the true equilibrium solution of Case A. Moreover, as x and h get lower, toward the lower left corner, the GAP values increases substantially and quickly. A GAP value in the order of ten of thousands essentially indicates that the DUE route flows are far from the route flow pattern of Case A. In Fig. 3, moving from point A

B

0.9

A

50

15 00 0 20 00 0

F

5000

00 100

0.3 0.3

C

D

15000 20000

00 250 00 300 0 00 35 000 00 40 0 45 0000 0 5 500 5

Commuter Portion ω

0.5

10 00 0

00

0.7

E 0.5 Network θ

0.7

0.9

Fig. 3. GAP values of the DUE route flows under heavy congestion.

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to point C (i.e., holding h = 0.9 while lowering x from 0.9 to 0.5), the GAP value increases from 120 to 2000+. On the other hand, moving from A to B in Fig. 3 (i.e., holding x = 0.9 while lowering h from 0.9 to 0.5), the GAP value increases from 120 to 620. Apparently, at relatively high values of x and h, or that the stochastic nature of the network is not pronounced, the effect of x is a bit higher – capturing the effect of stochastic demand is more important. Repeating the analysis from D to E in Fig. 3 (i.e., holding h = 0.5 while lowering x from 0.5 to 0.3), the GAP value increases from 9000 to 14,000. As for the case of shifting from D to F in Fig. 3 (i.e., holding x = 0.5 while lowering h from 0.5 to 0.3), the GAP value changes drastically from 9000 to 42,000. At relatively low values of x and h, the effect of h dominates the result. That is, capturing the effect of stochastic capacity degradation is critically important. Summarizing these results, albeit they are network specific, it appears that for a network that is subject to neither severe stochastic capacity degradations nor heavy stochastic demand (i.e., both h and x are close to 1), capturing the effect of stochastic demand is more important than capturing the effect of stochastic capacity degradation. On the other hand, for a network that is subject to both more severe stochastic capacity degradation and stochastic demand (i.e., h and x are around 0.5), capturing the effect of capacity degradation far outweigh that of stochastic demand. 3.2. Sensitivity of travel time budget equilibrium The travel time budget equilibrium solved in Case A depends on the extent of stochastic variations in supply and demand. Fig. 4 plots the mean travel time budgets of commuters in Case A by averaging over MT and HR commuters for all OD pairs. Similar to Fig. 3, the upper right corner gives the DUE results (deterministic network, deterministic demand). The results for Case B (deterministic network, stochastic demand) are shown along the vertical bolded line on the right. The results for Case C (stochastic supply, deterministic demand) are along the top bolded line. By reading the graphs vertically, the overall mean travel time budget of the network always decreases with higher portions of commuters (or lower proportions of stochastic demand). Note that the mean total network demand is constant for this analysis. Commuters, by selecting routes to minimize their own travel time budgets while accounting for supply and demand uncertainties, improve the overall mean travel time budget. Conversely, stochastic demand worsens the network performance. Tracing along the vertical bolded line (i.e., Case B), when the commuter portion decreases from 1.0 to 0.5, the overall mean travel time budget increases from 56.7 to 62.5, an increase of about 10%. On other hand, tracing along the horizontal bolded line (i.e., Case C), Case C 1.0

0.8 Case B

60

70

80

90

0.7

100

Commuter Portion ω

60

70

80

90

0.9

0.6 70

80

90

100

110

0.5 0.5

0.6

0.7

0.8

0.9

1.0

Network θ

Fig. 4. Average travel time budget under moderate congestion.

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decreasing the network degradation parameter from 1.0 to 0.5, the overall mean travel time budget increases from 56.7 to 92.2, or 63%. Here, we observe that the effect of stochastic network degradation is much more pronounced. Actually, toward the left side of the plot, as the network degradation worsens, the contours are inclined toward the vertical direction, showing that the network performance, in terms of overall mean travel time budget, is dominated more and more by network degradation. Combining the two stochastic effects, with both the network degradation parameter and commuter proportion at 0.5, the overall mean travel time budget increases from 56.2 to about 119, or a substantial increase of 112%. This shows that the two stochastic factors incur more than additive detriments on the network performance. In particular, when h < 0.7, the contours are packed more closely together as compared with the case of h > 0.7. This seems to indicate that any sparse capacity the network originally might have is fully utilized; any further decrease in h would lead to a relatively large increase in the overall mean travel time budget. What is surprising is the abruptness of this change in pattern, almost suggesting that there is a threshold h value below which there is a fundamental shift in performance. Of course, all these observations are based on this limited experiment. Whether this pattern shall always hold should be subject to further analysis. 3.3. Link improvements

0

Change in mean travel time

-50 -100 -150 -200 -250 -300 -350 -400 Link DUE

A

Fig. 5. Sensitivity of average mean travel time with respect to link capacity enhancement.

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

The models developed can be used to identify the optimal locations for link improvements while considering the stochastic nature of the problem. Specifically, by conducting a sensitivity analysis of the reduction in mean travel time of the network per each link improvement, we find out the most productive link improvement. For illustration purposes, we conduct the numerical study with the network consisting of 70% commuters (and, therefore, 30% stochastic infrequent travelers) and ha = 0.5 for all links. We then introduce the marginal link capacity enhancement by raising each link’s ha by 0.01 to 0.51 sequentially and simulate their resultant reduction in mean travel time. In Fig. 5, the x-axis represents the link where improvement is introduced; whereas the y-axis is the corresponding change in mean travel time. The results for Case A are shown by the lightly shaded bars. Based on these results, one can identify that Link 13 is the most productive location for improvement; whereas the next candidate is Link 2. The models that ignored the stochastic nature of the problem, on the other hand, would come up with a very different set of results, sometimes misidentifying the most beneficial improvement locations. Say, in the case of DUE, the model suggests that the most productive link for improvement is Link 14 (dark bar);

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however, in the most realistic representation of Case A, Link 14 (lightly shaded bar) would bring about a much lower reduction in mean travel time of the network. This result again illustrates the importance of capturing the stochastic aspects of this problem. Ignoring them could lead to erroneous prediction and investing resources in a sub-optimal way. 4. Concluding remarks This study developed a methodology to study doubly uncertain transportation networks with stochastic link degradation and stochastic demand. We consider that the total travel demand comprises of two parts, infrequent travelers and commuters. The traffic volume of infrequent travelers is stochastic. On the other hand, the traffic volume of commuters is stable or deterministic. Commuters acquire the network travel time variability from past experiences and settle into a long-term habitual route choice equilibrium in which they have no incentive of switching away. Any routes used by commuters on the same OD pair have the same and minimal travel time budget. The travel time budget varies among commuters according to their degrees of risk aversion and requirements on punctual arrivals. A mixed-equilibrium model is developed to capture these stochastic considerations. This study illustrated possible errors associated with neglecting stochastic behaviors of the network. The results showed that, as consistent with our expectation, both capacity degradation and stochastic demand play important roles, especially under high network congestion and stochasticity. Both causes of travel time variability addressed in this paper show significant impact on model predictions and, therefore, should not be ignored. We contend that the notion of travel time budget adopted in this study is more relevant and important than the notion of travel time in modeling travelers’ behavior in trip planning. Indeed it is also a broader concept. In fact, the concept of travel time budget can be related formally to early and late schedule delay penalties as discussed in Small (1982, 1992) and Arnott et al. (1998). In our ongoing study, we developed a theoretical framework to relate the two approaches and established the relationship between the within-budget time reliability to the coefficients of early and late penalties (Siu and Lo, 2007). Currently, we are developing empirical studies to verify their relationship. How to model travelers’ combined departure time and route choices in the presence of travel time uncertainty is our present research focus. In this paper, we considered that all commuters have perfect knowledge about the travel time variability of all routes associated with their daily commutes, i.e., perception errors do not exist. Relaxing this assumption by performing equilibrium stochastic traffic assignment on top of the travel time budget equilibrium formulation to cater for perception errors can be an interesting extension (Siu and Lo, 2006a). We also assumed that the risk aversion of individuals is constant with respect to the expected route travel time or its variance. It is interesting to investigate the relationship between risk aversion and travel time statistics and enrich the formulations with these behavioral extensions. Acknowledgements This research was sponsored by the Competitive Earmarked Research Grant HKUST6033/01E from the Hong Kong Research Grant Council. We appreciate the effort of X.W. Luo at the initial phase of this study. The constructive comments of the anonymous referees are gratefully acknowledged. References Aashitiani, H., 1979. The multi-modal traffic assignment problem. Ph.D. Thesis, Operations Research Centre. MIT Press, Cambridge, MA. Abdel-Aty, M.A., Kitamura, R., Jovanis, P.P., 1995. Investigating effect of travel time variability on route choice using repeatedmeasurement stated preference data. Transportation Research Record 1943, 39–45. Arnott, R., de Palma, A, Lindsey, R., 1998. Recent developments in the bottleneck model. In: Bulton, K.J., Verhoef E.T. (Eds.). Road Pricing, Traffic Congestion and the Environment: Issues of Efficiency and Social Feasibility. pp. 79–110. Bates, J., Polak, J., Jones, P., Cook, A., 2001. The valuation of reliability for personal travel. Transportation Research 37E, 191–229. Bilingsley, P., 1995. Probability and Measure, third ed. John Wiley and Sons, Inc., New York.

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