Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: A stochastic dominance based approach

Transportation Research Part B 80 (2015) 275–290

Contents lists available at ScienceDirect

Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: A stochastic dominance based approach Xing Wu ⇑ Department of Civil and Environmental Engineering, Lamar University, P.O. Box 10024, Beaumont, TX 77710, United States

a r t i c l e

i n f o

Article history: Received 13 October 2014 Received in revised form 17 June 2015 Accepted 15 July 2015 Available online 8 August 2015 Keywords: Mean-standard deviation Stochastic dominance (SD) Semi-standard deviation Time-dependent networks Travel time budget (TTB)

a b s t r a c t This paper studies a mean-standard deviation shortest path model, also called travel time budget (TTB) model. A route’s TTB is defined as this route’s mean travel time plus a travel time margin, which is the route travel time’s standard deviation multiplied with a factor. The TTB model violates the Bellman’s Principle of Optimality (BPO), making it difficult to solve it in any large stochastic and time-dependent network. Moreover, it is found that if path travel time distributions are skewed, the conventional TTB model cannot reflect travelers’ heterogeneous risk-taking behavior in route choice. This paper proposes to use the upper or lower semi-standard deviation to replace the standard deviation in the conventional TTB model (the new models are called derived TTB models), because these derived TTB models can well capture such heterogeneous risk-taking behavior when the path travel time distributions are skewed. More importantly, this paper shows that the optimal solutions of these two derived TTB models must be non-dominated paths under some specific stochastic dominance (SD) rules. These finding opens the door to solve these derived TTB models efficiently in large stochastic and time-dependent networks. Numerical examples are presented to illustrate these findings. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Optimal routing problems in a stochastic network have been intensively studied recently. The simplest assumption of travelers’ route choice behavior is that they will select the route which has the least expected travel time (i.e., the least mean travel time) (e.g., see Hall, 1986; Fu and Rilett, 1998; Miller-Hooks and Mahmassani, 2000). However, without considering travel time variability, this assumption which implies travelers are risk-neutral, may lead to a risky solution (Wu and Nie, 2011). How to define ‘‘travel risk’’ (or ‘‘travel reliability’’ on the other hand) in the routing problem varies. A common way to incorporate the trave time variability into the routing policy is to add a buffer index to the mean travel time, and the sum is called effective travel time or travel time budget (TTB) (Uchida and Iida, 1993; Lo et al., 2006; Shao et al., 2006; Lam et al., 2008; Xing and Zhou, 2011). Here the buffer index represents the travel risk (Lo et al., 2006). In this model, a path is optimal if it has the least TTB. The buffer is usually defined as the standard deviation of the travel time, multiplied by a

⇑ Tel.: +1 409 880 8757. E-mail address: [email protected] http://dx.doi.org/10.1016/j.trb.2015.07.009 0191-2615/Ó 2015 Elsevier Ltd. All rights reserved.

276

X. Wu / Transportation Research Part B 80 (2015) 275–290

factor, which reflects how a traveler regards the travel time variability: the more risk-averse a traveler is, a larger factor he/she will adopt (Lo et al., 2006). Therefore, this model is a typical mean-standard deviation model. The TTB model sees lots of applications in transportation for its easy-to-understand form because the mean and variance are two most common attributes of a random variable. Unfortunately, the TTB model exhibits a property of non-additivity (Xing and Zhou, 2011), and it violates the Bellman’s Principle of Optimality, making it difficult to apply the general dynamic programming approach to solve this model in a large network where it is not easy to enumerate all routes for a given OD pair. This can be demonstrated in the small network example shown in Fig. 1. There are two routes from node B to node D, and also two routes from node A to node D. Each link’s travel time distribution is given. Given a TTB model (mean + 1.8  standard deviation), route ABCD offers a smaller TTB than route ABD; on the other hand, however, as the sub-routes of routes ABCD and ABD, respectively, route BCD provides even larger TTB than route BD. Therefore, the Bellman’s Principle of Optimality is violated in this example. On the other hand, if considering the route with the least expected travel time, it is seen that the mean travel time of route BCD is less than that of route BD, and accordingly, the mean travel time of route ABCD is also less than that of route ABD. Therefore, the Bellman’s Principle of Optimality applies to the model of least expected travel time path. By assuming link travel times following normal distributions, analytical forms were developed to solve the TTB model (Lo et al., 2006; Shao et al., 2006) (it is called the conventional TTB model in the following). However, such an assumption violates the observations of link travel time distributions (Watling, 2006; Nie et al., 2012). Fu and Rilett (1998) proposed a Taylor-Expansion-based approximate method to calculate the variance of a route travel time in a stochastic and time-dependent network. However, it is difficult to apply this method in any large network due to the high complexity of calculation. To solve the conventional TTB model, Xing and Zhou (2011) proposed a Lagrangian substitution approximate method; Khani and Boyles (2015) proposed a two-step exact algorithm, in which the mean-standard deviation model is solved first to build the efficient path set and then the optimal solution to the conventional TTB model can be identified among those efficient paths; and considering the correlations between links, Mehrdad Shahabi and Unnikrishnan (2013) formulates the TTB model as a mixed integer nonlinear program and proposed an outer approximation solution approach. Meanwhile, recently, Chen and Nie (2013) proposed an approximate approach to solve the general non-additive models. However, these models are all built in static networks, where even through the model is not additive, the components such as the mean or variance of a path travel time are an additive sum of link travel time mean or variance. Note that given a stochastic and time-dependent network, not only the standard deviation of a path travel time but the mean travel time is non-additive. That is, the mean of a path travel time is not simply the sum of the mean travel times of its member links if link travel time distributions are time-dependent. This property largely increases the difficulty to solve the TTB model, so that the aforementioned existing approximate methods cannot be employed to solve this model. The travel risk can also be represented by a probability of on-time arrival (Frank, 1969; Fan et al., 2005; Nie and Wu, 2009): a route is optimal if it offers the smallest travel time which ensures the desired probability of not being late. Actually, this is a percentile value based on a probability distribution. Therefore, such travel time is called percentile travel time (PTT), which is aligned with the definition of value-at-risk (VaR) in finance. Chen and Zhou (2009) proposed a mean-excess travel time (METT) model which employs the concept of conditional value at risk (CVaR) that is widely used in finance, to define the optimal path: a path is optimal if it has the minimum METT, i.e., the minimum CVaR of travel time. The stochastic dominance (SD) theory is a widely used method to rank random variables based on their probability distributions. In this paper, we attempt to study the TTB model from the perspective of the SD theory. Though it is not applicable to solve the TTB model, the Bellman’s Principle of Optimality can be used to solve non-dominated paths under the first-, second- and third-order SD rules (Wu and Nie, 2011). Based on this point, an efficient label-correcting algorithm was proposed, which has been applied to a large regional transportation network for a case study (Nie et al., 2012). This algorithm was firstly used to solve the PTT-optimal paths as they must be non-dominated under the first-order SD. A revised version of this algorithm was found to be also efficient for large time-dependent networks (Wu, 2013). Similar to the TTB model, the METT model also exhibits a property of non-additivity (Toumazis and Kwon, 2013). Fortunately, Wu and Nie (2011) showed that the SD theory covers the METT model – any METT-optimal path must be non-dominated under the first-order SD. Therefore, it opens the door to solve the METT model efficiently even in large time-dependent networks. For this reason, it is of our interest to study whether the SD theory also covers the TTB model. If so (or under some mild conditions), we can find a general solution approach to the TTB model even for large stochastic and time-dependent networks. Moreover, our study shows that the conventional TTB model cannot explain the risk-taking behavior in route choice when probability distributions of path travel times are skewed (i.e., not symmetric about the mean). On the other hand, it is known that the SD rules are closely associated with a variety of risk-taking behavior (such as risk aversion and risk-proneness) defined in microeconomics (Wu and Nie, 2011). Therefore, it is of our interest to investigate from the perspective of SD theory, how the TTB model can better capture heterogenous risk-taking behavior in route choice for any probability distribution of path travel times. Recently, considering the state of user equilibria, Wang et al. (2014) found that path flow solutions of the TTB model must satisfy the bi-objective (to minimize mean and variance, respectively) user equilibrium condition (i.e., no one can reduce his/her mean or variance of travel time without worsening another by switching routes unilaterally). Tan et al. (2014) further expanded this relationship to the METT and PTT models (under some mild conditions) and called these non-dominated solutions (in terms of mean and variance) as Pareto-efficient solutions. They do not cover the solution methods to these models. This paper, however, studies the TTB model from the other side: we aim to investigate the property of a TTB-optimal path

277

X. Wu / Transportation Research Part B 80 (2015) 275–290

Travel Time 1 2 3 4 5 6

Network Topology

Probability mass of link travel me Link AB Link BC Link BD 0.6 0.3 0 0.4 0.4 0 0 0.3 0 0 0 0.2 0 0 0.8 0 0 0

Link CD 0.4 0.4 0.2 0 0 0

Mean, standard deviaon of each route’s travel me, as well as their travel me budgets (TTB) Route Mean ( ) Std ( ) Route Mean ( ) Std ( ) ABCD 5.2 1.18 7.33 3.8 1.08 5.74 BCD 4.8 0.40 5.52 ABD 6.2 0.63 7.34 BD Fig. 1. An example that violates the Bellman’s Principle of Optimality. Note: Considering a TTB model ðl þ 1:8rÞ, route BD is the better than route BCD between OD pair (B, D) as it offers the smaller TTB; however, on the other hand, route ABCD beats route ABD because ABCD has smaller TTB between OD pair (A, D).

(for an individual traveler), as well as its implication in terms of travelers’ risk-taking behavior given general path travel time distributions, and attempts to find an efficient method to solve the TTB model (or an alternative model derived from the conventional TTB model) in stochastic networks, especially in large time-dependent networks. A key to solve any traffic assignment (TA) problem is to find the ‘‘optimal’’ solutions of each OD pair in each iteration. In this paper, we do not move to the state of user equilibria, i.e., the traffic assignment problem is not covered here. In other words, this paper focuses on the route choice problem for a traveler given a stochastic network, instead of the interaction of all travelers’ route choice behavior which eventually may lead to an equilibrium state. The remaining paper is organized as follows. Section 2 reviews the TTB model, the SD theory, and the algorithm to solve the non-dominated paths in stochastic and time-dependent networks. Section 3 examines the relationship between the SD theory and the TTB model. Solution method and the test of this method in numerical examples are given in Section 4, and Section 5 concludes the paper. 2. Review of stochastic dominance (SD) and travel time budget (TTB) model The main utility of the stochastic dominance (SD) theory is helping decision-makers to rank and select random variables. The conventional SD theory is defined in the context of increasing utility functions, that is, decision makers always prefer more quantities of a random return (Hanoch and Levy, 1969; Hadar and Russell, 1971; Dentcheva and Ruszczynski, 2003). Ogryczak and Ruszczynski (1999) attempted to find a relationship between the conventional SD theory to several mean-risk models for the portfolio selection problem. Those models are defined in the context of increasing utility functions (i.e., a larger mean would be preferred). In our problem, however, travelers’ utility typically decreases with travel time in route choice (i.e., a smaller mean would be preferred). Therefore, the SD theory needs to be re-defined in the context of the routing problem. The first- and second-order SD (see Definitions 1 and 2 in the following) have already been re-defined in our previous research (Wu and Nie, 2011; Nie et al., 2012). For brevity, we first give the following notations and definitions. Readers are referred to Wu and Nie (2011) for details. In the following analysis, let X and Y be two random travel times of two routes of an OD pair. Therefore, both X and Y should be non-negative, and we assume there exists a finite upper-bound T for both X and Y, such that F X ðTÞ ¼ F Y ðTÞ ¼ 1, where F X ðÞ and F Y ðÞ are the cumulative distribution functions (CDFs) of X and Y, respectively. On the other hand, let f X ðÞ and f Y ðÞ be the probability density functions (PDFs) of X and Y, respectively. Finally, denote lX and lY as the means of X and Y, respectively; and r2X and r2Y the variances of X and Y, respectively. If no subscript X or Y appears, FðÞ; f ðÞ; l and r2 refer to the CDF, PDF, mean and variance of a general random variable, respectively. In the following analysis, we assume that f ðÞ is continuous in ½0; T and FðÞ is continuous and differentiable in ½0; T. Definition 1 (First order stochastic dominance (FSD), 1 ). X dominates Y in the first order, denoted as X 1 Y, if F X ðtÞ P F Y ðtÞ; 8t and F X ðtÞ > F Y ðtÞ for at least one t, where 0 6 t 6 T. Definition 2 (Second order stochastic dominance (SSD), 2 ). X dominates Y in the second order, denoted as X 2 Y, if

Z

T

F X ðwÞdw P

t

where 0 6 t 6 T.

Z t

T

F Y ðwÞdw; 8t and

Z t

T

F X ðwÞdw >

Z t

T

F Y ðwÞdw 9 t;

278

X. Wu / Transportation Research Part B 80 (2015) 275–290

Let

GðtÞ ¼

Z

T

F X ðwÞdw:

ð1Þ

t

Then Definition 2 can be rewritten as X 2 Y if and only if GX ðtÞ P GY ðtÞ; 8t and GX ðtÞ > GY ðtÞ for at least one t, where 0 6 t 6 T. In the following, function GðÞ will be frequently used in the analysis. The property of GðÞ was firstly investigated, and the following lemma was found. Lemma 1. GðÞ is a non-increasing, concave function with Gð0Þ ¼ T  l and GðTÞ ¼ 0. Moreover, a tangent line y0 ¼ T  l to the curve of GðtÞ exists at t ¼ 0 and another tangent line yT ¼ t þ T to GðtÞ exists at t ¼ T. Proof. The definition of GðtÞ (see Eq. (1)) yields G0 ðtÞ ¼ FðtÞ if GðtÞ is differentiable, and G00 ðtÞ ¼ f ðtÞ if FðtÞ is differentiable. Therefore, 1 6 G0 ðtÞ 6 0 and G00 ðtÞ 6 0. That is, GðÞ is non-increasing and concave. RT Meanwhile, GðTÞ ¼ T FðtÞdt ¼ 0 and

Gð0Þ ¼

Z 0

T

FðtÞdt ¼ tFðtÞjT0 

Z

T

t dFðtÞ ¼ T  l

ð2Þ

0

Given G0 ðtÞ 6 0; GðtÞ 6 Gð0Þ ¼ T  l. On the other hand, jG0 ðtÞj ¼ FðtÞ 6 1; 80 6 t 6 T, so GðtÞ must on the left side of yT ¼ t þ T. h Lemma 1 can be demonstrated by Fig. 2. The curve of function GðtÞ must be decreasing and concave, and it has two tangent lines: yT ¼ t þ T at Point A ðt; 0Þ and y0 ¼ T  l at Point C ð0; T  lÞ. Definitions 1 and 2 lead to Lemma 2. Lemma 2. If X 1 Y, then X 2 Y. On the other hand, X 2 Y does not necessarily mean X 1 Y. From Definition 1, X 1 Y mathematically means that PðX 6 tÞ P PðY 6 tÞ; 8t P 0. In the route choice problem, X 1 Y implies that for any time budget t; X always offers a larger or at least equal on-time arrival probability than Y. As to the SSD, it is related to the behavior of ‘‘risk-aversion’’. That is, even for some time budget, Y may provide a larger on-time arrival probability than X; Y is still not preferred at all for a risk-averse traveler if X 2 Y. Readers are referred to Wu and Nie (2011) for details. Given an OD pair rs, a non-dominated path is a path which is not dominated by any other path of this OD pair under the FSD/SSD rule. It is found that the Bellman’s Principle of Optimality is applicable to solve non-dominated paths under the FSD, js

is

SSD and even higher order SD. If a path k is dominated by other path(s) of OD pair jr, then path k must be also dominated is

js

given that k is made by extending k backwards along link ij. Therefore, it is not necessary to enumerate all paths but only keep those non-dominated paths when scanning a network backward from the destination. An efficient label-correcting solution algorithm was developed to solve non-dominated paths in a stochastic network. A brief description of the solution algorithm is given as follows. Readers are referred to Wu and Nie (2011) and Nie et al. (2012) for details. Algorithm SD-LC Step 0 Initialization. Let 0ss be a dummy path from destination s to itself. Initialize the scan list Q ¼ f0ss g. set pss 0 ¼ 0 with ss probability 1, where pss 0 is the random travel time of path 0 . At each node i where i – s, define a Pareto frontier pis ¼ 1 with probability 1. Note that pis is also a random variable with a probability distribution. js

Step 1 Select the first path from Q, denoted as l , and delete it from Q. is

js

Step 2 For any predecessor node i of j, create a new path k by extending l along link ij. js l

step 2.1 Calculate the distribution of p from the distribution of p by convolution integral, where pisk and pjsl refer is k

is

js

to the random travel time of paths k and l , respectively. is

step 2.2 Compare the travel time of new path k , i.e., pisk with current Pareto frontier pis . If the frontier is dominated by pisk under the FSD rule, let Pareto frontier pis ¼ pisk , drop all existing non-dominated paths at node i, and is

set Cis ¼ fk g, where Cis is the set of non-dominated paths from node i to destination s. Otherwise, further is

compare the distribution of the new path k with those of all existing non-dominated paths one by one is

is

under the FSD/SSD rule. If any of the existing path dominates k , drop k and go back to Step 2; otherwise, is

is

is

is

is

delete all paths (if any) that are dominated by k from C , set C [ fk g, and update Q ¼ Q [ fk g and is

Pareto frontier p based on the travel time distributions of all non-dominated paths in C . Step 3 If Q is empty, stop; otherwise go to Step 1. is

279

X. Wu / Transportation Research Part B 80 (2015) 275–290

y T

T-μ

y =T- μ y =-t+T y =G t

T

t

Fig. 2. Demonstration of Lemma 1. Note: l refers to the expected value of the random variable of interest.

Numerical examples have shown that the above algorithm is quite efficient, even applied to large regional networks (Nie et al., 2012). Further, this algorithm can be extended to solve non-dominated paths in time-dependent networks with a little revisions, including (1) adding the first-in-first-out (FIFO) condition to link travel time distributions and (2) making the departure time being well aligned with the time budget so that a one-dimension probability function urs k ðb  tÞ is able to rs

describe the probability of arriving at destination s using path k within budget b  t when departing origin node r at time t, where b is the maximum time budget – thus the time-dependent problem is reduced to the normal static-like problem by just comparing urs k ðb  tÞ of different paths (Wu, 2013). The TTB model described above is one of mean-standard deviation models. The TTB of random path travel time X, denoted as qX , is defined as

qX ¼ lX þ krX

ð3Þ

where k is a factor reflecting the risk preference, which can be positive (related to risk aversion) or negative (risk-proneness) (Lo et al., 2006). The TTB-optimal path is the one which has the minimum TTB. The more risk-averse a traveler is, the larger k he/she will take. As mentioned above, the TTB model is non-additive and violates the Bellman’s Principle of Optimality (Lo et al., 2006; Lam et al., 2008). This property makes it difficult to solve this model in large stochastic and time-dependent networks. In the next section, we attempt to solve the TTB model based on the SD relationship. More importantly, we aim to investigate the implication of the mean and variance of a random travel time from the prospective of the SD. 3. Relationship between SD and mean/variance 3.1. Implication of variance Since the TTB model has two key factors: mean and standard deviation. At first, this section aims to investigate the relationship between the mean of a random travel time and SD theory, as well as the relationship between its variance and SD theory. Then based on the finding of the first step, the relationship between the TTB model and SD theory will be studied. From Definition 1, it is not difficult to reach the following result. Proposition 1. X 1 Y )

lX < lY .

Proof. Given

lX ¼ EðXÞ ¼ Then

lX  lY ¼

Z

T

tfX ðtÞdt ¼

Z

0

RT 0

T

t dF X ðtÞ ¼ T 

0

tfX ðtÞdt 

RT 0

T

F X ðtÞdt;

0

tfY ðtÞdt ¼

RT

As to the SSD, a similar result was found. Proposition 2. X 2 Y )

Z

lX 6 lY .

0

ðF Y ðtÞ  F X ðtÞÞdt < 0 if X 1 Y, according to Definition 1. h

280

X. Wu / Transportation Research Part B 80 (2015) 275–290

Proof. Wu and Nie (2011) showed X 2 Y if and only if EðX  gÞþ 6 EðY  gÞþ ; 80 6 g 6 T and EðX  gÞþ < EðY  gÞþ , for at least one 0 6 g 6 T, where X; Y 6 T < 1 and xþ ¼ maxð0; xÞ. From this conclusion, it is easy to see that X 2 Y ) lX 6 lY just by letting g ¼ 0. h Propositions 1 and 2 show that some consistency does exist between the mean and the SD relationship. However, the consistency between the variance and the SD is not clear. After a series of mathematical manipulations, we have the following result.

rX < rY if

Proposition 3.

Z

T

GX ðwÞdw þ

0

where GX ðtÞ ¼

RT t

1 2 l > 2 X

Z

T

GY ðwÞdw þ

0

F X ðwÞdw and GY ðtÞ ¼

RT t

1 2 l 2 Y

ð4Þ

F Y ðwÞdw, and F X ðÞ and F Y ðÞ are the CDFs of X and Y, respectively.

Proof. Let l and r2 be respectively the mean and variance of a non-negative random variable which has a finite upper-bound T, then

r2 ¼

T

ðt  lÞ2 f ðtÞdt ¼

Z

0

T

ðt  lÞ2 dFðtÞ

0

Z T Z T Z T ðl2  2lt þ t 2 ÞdFðtÞ ¼ l2 dFðtÞ  2l tdFðtÞ þ t 2 dFðtÞ 0 0 0 0     Z T Z T FðtÞdt þ t2 FðtÞjT0  2 tFðwÞdw ¼ l2  2l tFðtÞjT0 

¼

Z

Z T

0

Note :

Z

ð5Þ

0

T

FðtÞdt ¼ Gð0Þ ¼ T  l; GðTÞ ¼ 0; see Eq:ð2Þ in the proof of Lemma1

0

¼ l2  2lðT  T þ lÞ þ T 2  2 ¼ T 2  l2  2

Z

T

tFðtÞdt ¼ T 2  l2 þ 2

0

Z

Z

T

t dGðtÞ

0

ð6Þ

T

GðtÞdt

0

Note that given GðtÞ ¼ CDF. Eq. (6) yields



r2X  r2Y ¼ 2

RT t

1 2 l þ 2 X

FðwÞdw ¼

Z

T

0

RT 0

FðwÞdw 

Rt 0

FðwÞdw, we have

dGðtÞ dt

¼ FðtÞ, and Fð0Þ ¼ 0 and FðTÞ ¼ 1 as FðÞ is a

   Z T 1 GX ðtÞdt þ 2 l2Y þ GY ðtÞdt 2 0

Therefore, Inequality (4) leads to

ð7Þ

rX < rY . h

Note that according to Definition 2, if X 2 Y, then GX ðtÞ P GY ðtÞ. While on the other hand, X 2 Y also leads to lX 6 lY . Therefore, X 2 Y does not necessarily lead to Inequality (4), implying that we cannot tell which variance is less between X and Y simply from X 2 Y. For one thing, it indicates that generally the SD theory is not compatible with the TTB model. However, Proposition 3 can lead to the following proposition: Proposition 4. If

rX P klX and rY P klY , where k > 0, then X 2 Y ) lX þ krX < lY þ krY . rX P klX and rY P klY , yield

Proof. At first these two conditions:

rX þ rY P klX þ klY ¼ kðlX þ lY Þ ) ðrX þ rY ÞðlY  lX Þ P kðlX þ lY ÞðlY  lX Þ:

ð8Þ

Note that X 2 Y ) lY > lX according to Proposition 1. By re-arranging Eq. (7), yields

1 2 1 ðr  r2Y Þ ¼ ðl2Y  l2X Þ  2 X 2 Then X 2 Y )

RT 0

GX ðtÞdt >

Z

RT 0

0

T

GX ðtÞdt 

Z



T

GY ðtÞdt

0

GY ðtÞdt (according to Definition 2) results in the following inequality:

r2X  r2Y < l2Y  l2X ) ðrX þ rY ÞðrX  rY Þ < ðlY  lX ÞðlY þ lX Þ

ð9Þ

X. Wu / Transportation Research Part B 80 (2015) 275–290

281

Combing Inequalities (8) and (9) together brings

1 ðl  lX ÞðrX þ rY Þ k Y

ðrX þ rY ÞðrX  rY Þ < ðlY  lX ÞðlY þ lX Þ 6 ) kðrX  rY Þ < lY  lX

given k > 0

lX þ krX < lY þ krY :

)

This concludes the proof. h Therefore, we can see that by adding some conditions, a relationship does exist between the SD theory and the conventional TTB model where k > 0. If r  l, then k would have a wide feasible range, so Proposition 4 could be very useful: if r  l for all paths between an OD pair, then the optimal path must be non-dominated one under the SSD rule given any 0 6 k 6 r=l for all routes. On the other hand, if r < l, especially if r  l, then the feasible range of k is very narrow, making this proposition minor. If r  l, then considering the mean travel time may be enough to make a decision of route choice for a traveler unless he/she is extremely risk-averse or risk-prone. The condition of r P kl, i.e., k 6 r=l limits the use of this proposition. In the real world, r  l could be a strong condition because it implies the road condition is very bad. Our previous research based on the real travel time data collected in the Chicago metropolitan area shows r < l for most links (Nie et al., 2012). Meanwhile, the selection of k is independent of the route conditions (this will be discussed in details in Section 3.3). If a traveler decides to use a k that is greater than r=l, then Proposition 4 becomes inapplicable. Note that the value of r=l is simply dependent on how wide the range of a travel time could be, but independent of what type distribution it follows. In the following text, two derived TTB models are proposed to fit the behavior of risk-averse and risk-prone route choice behavior, respectively, where the condition of k 6 r=l can be released. Assume X 2 Y, then based on Lemma 1, the solid and dashed curves in Fig. 3 were used to represent GX ðÞ and GY ðÞ, respectively. Note that GX ðÞ P GY ðÞ according to Definition 2; lX 6 lY according to Proposition 2; and RT Gð0Þ ¼ 0 FðtÞdt ¼ T  l (see Eq. (2)). Meanwhile, G0 ðtÞ ¼ FðtÞ results in jG0 ðtÞj ¼ FðtÞ 6 1. Thus, Line HL which connects two points: ð0; TÞ and ðT; 0Þ in Fig. 3, must be the tangent of both curves at Point L, i.e., ðT; 0Þ, and both curves must lie below this diagonal line HL. Inequality (4) can also be written as

Z

T

GX ðtÞdt 

0

Z

T

GY ðtÞdt > 0

1 2 1 2 l  l 2 Y 2 X

ð10Þ

The left side of Inequality (10) refers to the area between two curves, i.e., the shaded area of FGL in Fig. 3. On the other hand, l2X and 12 l2Y are actually the areas of Triangles HFM and HGN, respectively. Then the right side of Inequality (10) is the area of Trapezoid FGNM in Fig. 3 (as the marked trapezoid in the figure). Therefore, given X 2 Y, if the area between two curves is larger than that of Trapezoid FGNM, then rX < rY . The above analysis shows that the TTB model is not compatible with the SD theory. The reason comes from the variance. RT RT Actually, given that GðtÞ ¼ t FðwÞdw, manipulating r2 ¼ 0 ðl  tÞ2 f ðtÞdt yields:

1 2

r2 ¼

Z l

ðl  tÞ2 f ðtÞdt þ

Z

T

ðt  lÞ2 f ðtÞdt ¼

Z l

l

0

  Z l Z ¼ 2 ðT  lÞl  GðtÞdt þ 2

ðl2  2lt þ t 2 ÞdFðtÞ þ

0

½T  t  GðtÞdt

Note that

Z l

Z l

Z l tdFðtÞ þ t2 dFðtÞ 0 0 0 0 Z l Z l l l l ¼ l2 FðtÞj0  2ltFðtÞj0 þ 2l FðtÞdt þ t 2 FðtÞj0  2 tFðtÞdt 0 0 Z l Z l FðtÞdt þ l2 FðlÞ  2 tFðtÞdt ¼ l2 FðlÞ  2l2 FðlÞ þ 2l 0 0 Z l Z l Z l l ¼ 2l dGðtÞ þ 2 tdGðtÞ ¼ 2lGðtÞj0 þ 2lGðlÞ  2 GðtÞdt 0 0 0 Z l ¼ 2lðT  lÞ  2 GðtÞdt ðl2  2lt þ t 2 ÞdFðtÞ ¼ l2

dFðtÞ  2l

Z l

0

Similarly, we can see

Z l

T

ðl2  2lt þ t2 ÞdFðtÞ ¼ 2

Z l

T

ðl2  2lt þ t 2 ÞdFðtÞ

l

T

l

0

Z

T

½T  t  GðtÞdt

ð11Þ

282

X. Wu / Transportation Research Part B 80 (2015) 275–290

G (t) H

T

T - μX

M

F

N E

T - μY G C D

J 0

μX

K μY

L T

t

Fig. 3. Demonstration of Proposition 3. Note: the solid and dashed curves refers to GX ðtÞ and GY ðtÞ, respectively if X 2 Y.

RT l

Based on Eq. (11), by letting A ¼ ðT  lÞl 

Rl 0

RT Rl GðtÞdt and B ¼ l ½T  t  GðtÞdt, it yields 0 ðl  tÞ2 f ðtÞdt ¼ 2A and

ðt  lÞ2 f ðtÞdt ¼ 2B, respectively. Now, we have

r2 ¼

Z l

ðl  tÞ2 f ðtÞdt þ

0

Z

T

ðt  lÞ2 f ðtÞdt

ð12Þ

l

¼2A þ 2B

ð13Þ

In the following, the graphical interpretation of mean and variance is investigated, respectively. At first, as to mean, RT RT tf ðtÞdt ¼ 0 tdFðtÞ implies that the shaded area (ODC) in Fig. 4(a) (where the curve is the CDF) is exactly the mean 0 of a random variable. On the other hand, Eqs. (12) and (13) reveal the graphical interpretation of variance. The curve in Fig. 4(b) represents GðÞ for random variable X. If taking a closer look at the definitions of A and B; A and B are the area of DMC and MCE, respectively in Fig. 4(b) (given that Line HE’s function is y ¼ T  t). Therefore, the variance of X actually is the double of the area of DME in Fig. 4(b) (i.e., the sum of A and B).

l¼

3.2. TTB based on semi-standard deviation The above analysis indicates that the variance may be divided into two parts (See Eqs. (12) and (13)), which correspond to two separate areas (A and B) on either side of mean l, respectively. Each part refers to a semi-variance, and its applications can be found in finance and economics (e.g., see Bachmaier and Backes, 2008). Considering a random variable with a range ½0; T, we have: Definition 3 (Lower semi-variance,

r2L ). r2L is the lower semi-variance if r2L ¼

Rl 0

ðl  tÞ2 f ðtÞdt.

R

r2U ). r2U is the upper semi-variance if r2U ¼ lT ðt  lÞ2 f ðtÞdt. It is not difficult to see r2 ¼ r2L þ r2U . Correspondingly, rL and rU are called lower semi-standard deviation and upper 2 2 semi-standard deviation, respectively. Note that the semi-variance may also be denoted as rLX or rUX in the following text if a subscript is necessary. Therefore, r2L ¼ 2A and r2U ¼ 2B, where A and B refer to the areas indicated in Fig. 4(b). Definition 4 (Upper semi-variance,

Given that the SSD’s definition is related the integration from t to T (Definition 2), since we are interested in finding a relationship between the SD rule and the TTB model, the upper semi-variance is of our interest. The definition of upper semi-variance results in the following proposition. Proposition 5. If X 2 Y, then

lX þ rUX 6 lY þ rUY ; and if X 2 Y and lX – lY , then lX þ rUX < lY þ rUY .

Proof. This proposition can be proved using the two figures in Fig. 5. As shown in Fig. 5(a), given X 2 Y, the solid curve refers to GX ðtÞ and the dash curve represents GY ðtÞ, respectively. Then, according to Definition 4, Areas of MPE and NQE are the half upper semi-variances of X and Y, i.e.,

rUX 2 and rUY 2 , respectively.

283

X. Wu / Transportation Research Part B 80 (2015) 275–290

Fig. 4. Graphical interpretations of the mean and variance of a random variable.

G(t)

G(t)

T

H

T - μX

T

D

M

D

T-μ

P

T - μY

H

M

G

P

N

½ upper semi-variance (Area of MPE)

B

Q

F μX

0

G μY

E T

t

0

F μ

E T

t

Fig. 5. The graphical interpretations of the upper semi-variance of a random variable.

The shapes of GX ðtÞ and GY ðtÞ (both are non-increasing and concave according to Lemma 1) in Fig. 5(a) reveal the following inequality:

Area of NQE þ Area of Trapezoid MPQN P Area of MPE

ð14Þ

) Area of Trapezoid MPQN P Area of MPE  Area of NQE jMPj þ jNQj 1 2 1 2 ) ðlY  lX Þ P rUX  rUY 2 2 2

ð15Þ ð16Þ

Note that jMPj ¼ T  lX  GX ðlX Þ and jNQ j ¼ T  lY  GY ðlY Þ. Therefore, we have

Area of Trapezoid MPQN ¼

jMPj þ jNQ j ðlY  lX Þ 2

where jMPj represents the length of segment MP. Meanwhile, according to Fig. 4(b), Areas of MPE and NQE in Fig. 5(a) refer 2

2

to the half of rUX and rUY , respectively. On the other hand, Fig. 5(b) shows

1 2 1 r ¼ Area of MPE > Area of Triangle MPG ¼ jMPj2 2 U 2 Therefore, yields

)

rU > jMPj

rUX > jMPj and rUY > jNQ j (see Fig. 5(a)). Note that X 2 Y ) lX 6 lY (see Proposition 2). Then Inequality (16)

284

X. Wu / Transportation Research Part B 80 (2015) 275–290

rUX þ rUY

jMPj þ jNQ j 1 2 2 ðlY  lX Þ P ðlY  lX Þ P ðrUX  rUY Þ 2 2 2 ) ðlY  lX Þ P ðrUX  rUY Þ ) lY þ rUY P lX þ rUX

If

lX – lY , then the above inequality yields lY þ rUY > lX þ rUX . This concludes the proof. h

Define

qX ¼ lX þ krUX ; k > 0

ð17Þ

as the mean-upper-semi-standard deviation model. Here qX is still called as the TTB of random travel time X, so this model is regarded as a derived TTB model.1 Proposition 5 shows a connection between the SD rule and this derived TTB model. Even though the SD rule cannot be directly used to solve the conventional TTB model, it is still possible to use the SD rule to solve this derived TTB model based on the semi-standard deviation. More importantly, this derived model can well explain the route choice behavior of a risk-averse traveler when probability distributions of path travel times are skewed (such distributions are found to be skewed according to many empirical studies (Nie et al., 2012)); while the conventional TTB model (3) may fail to capture such risk-taking behavior in route choice given such skewed distributions (see Section 3.3 for details). Proposition 5 leads to the following important lemma. rs

Lemma 3. Considering a mean-upper-semi-standard deviation model qX ¼ lX þ krUX for any 0 < k 6 1, path k of an OD pair rs must not exhibit a smaller TTB than any other path of this OD pair, if it is dominated by other path(s) of this OD pair under the rs FSD/SSD rule, and unless it has the same mean to other(s), path k cannot be the optimal path. rs

rs

rs Proof. If path l dominates path k under the FSD/SSD, then we must have prs l 2 pk (note that the FSD also leads to the SSD rs rs rs rs as stated in Lemma 2), where pl and pk are the random travel time of paths l and k , respectively. Propositions 2 and 5 result in ll 6 lk and ll þ rUl 6 lk þ rUk , respectively, where lk and ll refer to the means of prs k and prsl , respectively, and rUk and rUl are the upper semi-standard deviations of prsk and prsl , respectively. Then, using the idea proposed by Ogryczak and Ruszczynski (1999), two inequalities are generated:

ð1  kÞll 6 ð1  kÞlk ; kll þ krUl 6 klk þ krUk ;

0
ð18Þ

0
ð19Þ

Adding Inequalities (18) and (19) together yields

ll þ krUl 6 lk þ krUk ; 0 < k 6 1 rs

ð20Þ rs

rs

rs

Therefore, if path l dominates path k in the first/second order, the TTB of path k cannot be smaller than that of path l . If ll – lk , then Inequality (19) leads to kll þ krUl < klk þ krUk ; 0 < k 6 1 according to Proposition 5, and thus Inequality rs rs (20) becomes ll þ krUl < lk þ krUk ; 0 < k 6 1, i.e., the TTB of path k must be strictly larger than that of path l . Therefore, rs path k cannot be the optimal path. h Lemma 3 then leads to the following corollary. rs

Corollary 1. Path k must be non-dominated under the SSD rule if it exhibits a unique minimum TTB, q, defined as q ¼ l þ krU , for any 0 < k 1. rs

rs

Proof. Suppose that there exists a path k which has a unique minimum TTB, but it is also dominated by path l in the secrs rs ond order. Then according to Lemma 3, the TTB of path k must not be smaller than that of path l . A contradiction. h Note that a non-dominated path under the FSD rule, may be still dominated by another path in the SSD rule (Wu and Nie, 2011). Therefore, the optimal path has to be the non-dominated paths under the SSD rule in Corollary 1. Proposition 5 gives us an important result. Even though the TTB model is not fully compatible with the SD theory, some specific relationship does exist as Corollary 1 indicates. Actually, Corollary 1 opens the door to efficiently solve the optimal paths of the mean-upper-semi-standard deviation model even in large stochastic and time-dependent networks: (1) at first identify the non-dominated paths of the OD pair of interest under the SSD rule using the existing label-correcting algorithm and then (2) identify the optimal one among these non-dominated paths identified in the first step. Readers are referred to Section 4 where this method is tested in numerical examples. However, Corollary 1 requires k 6 1, implying a kind of ‘‘mild’’ 1

In the following text, this model and the mean-lower-semi-standard deviation model are called derived TTB models if they are referred together.

X. Wu / Transportation Research Part B 80 (2015) 275–290

285

risk-taking behavior (see the discussion in Section 3.3). If someone picks a large k, the optimal path of this derived TTB model may not be non-dominated under the SSD rule at all. The FSD/SSD’s definition is dependent on random variables’ CDFs (see Definitions 1 and 2). The CDF of a random travel time, FðtÞ ¼ PðX 6 tÞ, indicates the probability of travel time not larger than t, i.e., the probability of ‘‘not being late’’, instead of the probability of travel time falling into a desired time range. On the other hand, the variance of a random travel time covers both sides (more or less) deviated from the mean. For this reason, the variance does not fit in the SD framework. It is already known that the METT and PTT models are well compatible with the SD theory (Wu and Nie, 2011) (i.e., the METT-optimal or PTT-optimal paths must be non-dominated paths under the FSD rule). The METT and PTT models are both related to a specific on-time arrival probability, and they only focus on a specific point or a section of the distribution larger than a specific point: the PTT is travel time for a specific on-time arrival probability; and the METT is the expected travel time which is larger than a percentile travel time based on a given on-time arrival probability. Here, the upper-semi-variance only focuses on the section of the distribution where travel time are larger than the mean, implying that a traveler only cares about ‘‘being late’’. The implication of the risk preference behind this derived TTB model is discussed in Section 3.3.

3.3. Risk-taking behavior in travel time margin selection in case of non-symmetric probability distribution of path travel time In the conventional TTB model ðq ¼ l þ krÞ, a traveler is said to be ‘‘risk-averse’’ if k is set to be positive, i.e., a risk-averse traveler wishes to reserve a positive travel time margin; while he/she is risk-prone if k < 0, i.e., a negative margin, implying that a risk-prone traveler may adopt a time budget that is even less than the mean travel time (Lo et al., 2006). By assuming the path travel time is normally distributed, Lo et al. (2006) found k ¼ U1 ðaÞ, where UðÞ refers to the CDF of a standard normal variable and a is the on-time arrival probability, so the larger a, the larger k is. Therefore, k is a parameter reflecting a traveler’s risk-taking behavior, as well as the purpose of a trip (e.g., a trip for a job interview may require a larger a than a trip to a friend’s party), but it is independent from routes. Note that k > 0 corresponds to a > 0:5, exactly indicating the travel time which is greater than the mean (considering a normal distribution); while k < 0 ) a < 0:5 entails the travel time which is less than the mean. However, if path travel times are not normally distributed, such one-to-one correspondence between k and a does not exist (Wu and Nie, 2011). Consider a general mean-standard deviation model:

qX ¼ lX þ kuX

ð21Þ

where uX can be the standard deviation or a derivative of the standard deviation of random travel time X, and uX is an exogenous factor only associated with a route’ travel time distribution, independent of travelers’ behavior and trips’ purposes. As discussed above, k can be regarded as an endogenous parameter that is independent of routes, but related to a traveler’s risk-taking behavior and the nature of a trip. For example, a traveler may adopt a larger k for a trip to a job interview, compared with a trip to a friend’s party. Then for the same route, different traveler may have different k, but the same u. On the other hand, given two alternative routes for a trip, one traveler may see different u though his/her k remains the same for two routes. For this reason, if the probability distributions of two path travel times are heavily skewed, but these two random travel times have similar or even identical variances, then the conventional TTB model that employs the standard deviation to describe the selection of travel time margin, may NOT be able to represent such discrepancy of u, which is usually an important concern for a risk-averse or risk-prone traveler when considering TTB for different routes. Let us consider the example of two routes. Table 1 reports the probability distributions of these two routes’ travel times, as well as their means, standard deviations, and upper and lower semi-standard deviations, respectively. Two routes have exactly the same mean and variance. Therefore, they always have identical TTBs for any k if using the conventional TTB model, implying that a traveler cannot make any difference between them (note that k is related to the traveler and independent of routes). However, obviously they are not identical: as shown in Fig. 6, the distribution of route 1 is positively skewed, but that of route 2 is negatively skewed. The variance cannot reflect such difference in skewness, so that if using the conventional TTB model, the travel time margin of two routes, i.e., kr, must be always identical for any k. However, a risk-averse traveler cares about such difference: he/she has to reserve a larger travel time margin (consequently a larger TTB) for route 1, because the travel time of route 1 could be as large as 6.5, but that of route 2 will never exceed 5. Though the likelihood of seeing 6.5 is quite low (only 5%), a risk-averse traveler does care about it. For this reason, route 1 becomes less attractive for a risk-averse traveler because a larger TTB is needed. The more heavily positively skewed a distribution is (like route 1), the larger the upper semi-variance it yields. Therefore, it seems that the derived TTB model (17) that employs the upper semi-standard deviation, can better reflect such discrepancy in route choice behavior of a risk-averse traveler. In this example, the upper semi-standard deviation of route 1 is 0.80; while that of route 2 is only 0.49, making it always having a smaller TTB for any k > 0. It is known that the SSD is related with ‘‘risk aversion’’ (Wu and Nie, 2011), which is defined based on the concavity of a utility function (Friedman and Savage, 1948), i.e., a risk-averse individual possess a concave utility function so that he/she prefers the expectation of a random return to this random return itself. According to Definition 2, the SSD looks at the tail of a distribution when comparing two distributions, implying larger travel time realizations are concerned by a risk-averse traveler. This is what the upper semi-variance focuses on. Therefore, even though the definition of risk aversion in the SD theory

286

X. Wu / Transportation Research Part B 80 (2015) 275–290

Table 1 The travel time probability distributions, means, standard deviation, and upper and lower semi-standard deviations of two routes. Route 1: Route 2:

Travel time t Prob. f 1 ðtÞ

2.5 0.3

3.5 0.35

4.5 0.2

5.5 0.1

6.5 0.05

Travel time t Prob. f 2 ðtÞ

1 0.05

2 0.1

3 0.2

4 0.35

5 0.3

l

r

rU

3.75

1.29

0.80

rL 0.49

l

r

rU

rL

3.75

1.29

0.49

0.80

Note: Here we assume the travel time realizations of two routes are discrete so that f 1 ðxÞ and f 2 ðxÞ are both the probability mass functions for two routes, respectively.

0.40

Route 1 Route 2

0.35

Probability

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

1

2

3

4

5

6

7

8

Travel me Fig. 6. Probability distributions of two routes’ travel times. Note: Since the travel time realizations of two routes are discrete, y-axis refers to probabilities instead of probability densities.

and TTB model differ, Proposition 5 shows that they do share some consistency for ‘‘mild’’ risk-averse travelers (as k 1), if the upper semi-standard deviation is adopted in the TTB model to reflect the travel time margin selection. On the other hand, a risk-prone traveler usually sets a TTB less than the mean travel time by adopting a negative travel time margin (note that a TTB is the mean travel time plus a travel time margin). Still using the two-route example discussed above, if using the conventional TTB model, a risk-prone traveler cannot tell any difference between these two routes either, and has to set the same travel time margin (which is negative) for both routes. It violates the behavior of a risk-prone traveler. The travel time of route 2 could be as small as 1 though there is only 5% chance (he/she would like to try his/her luck); while the travel time of route 1 will be at least 2.5. It makes route 2 still more preferred due to a smaller TTB needed. Such risk-prone behavior gives a rise of the mean-lower-semi-standard deviation model:

qX ¼ lX  krLX ; k > 0

ð22Þ

where qX is still regarded as a TTB of random travel time X, and rLX is the lower semi-standard deviation of X. Model (22) can be regarded as an alternative derived TTB model which reflects the route choice behavior of a risk-prone traveler. The more negatively skewed a distribution is, the larger the lower semi-variance it yields. In this two-route example, route 2 has a larger lower semi-standard deviation, and thus yields smaller TTB, making it also more attractive for a risk-prone traveler. Therefore, the mean-lower-semi-standard deviation model (22) well explains the route choice behavior of a risk-prone traveler. Actually, consider an alternative SSD: Definition 5 (Alternative second order stochastic dominance (Alternative SSD), 2A ). X dominates Y in the alternative second order, denoted as X 2A Y, if

Z 0

t

F X ðwÞdw P

Z

t

F Y ðwÞdw; 8t and

0

Z

t

F X ðwÞdw > 0

Z

t

F Y ðwÞdw 9 t;

0

where 0 6 t 6 T It is not difficult to see X 1 Y ) X 2A Y from Definitions 1 and 5. This definition is related with risk-proneness, which means that an individual prefers the random return to its expectation. Based on this definition, a risk-prone individual possesses a convex utility function (Wu, 2011). Using the similar approaches shown in Section 3.2 and (Ogryczak and Ruszczynski, 1999) yields the following result. Lemma 4. If X 2A Y, then

lX  rLX 6 lY  rLY ; and if X 2A Y and lx – lY , then lX  rLX < lY  rLY .

Lemma 4 further leads to

287

X. Wu / Transportation Research Part B 80 (2015) 275–290 rs

Corollary 2. Path k must be non-dominated under the alternative SSD rule if it exhibits a unique minimum TTB, q, defined as q ¼ l  krL , for any 0 < k 6 minð1; l=rL Þ. Note k < l=rL is required to ensure q > 0. The proof is similar to that of Corollary 1. Similarly, Lemma 4 and Corollary 2 show a consistency for ‘‘mild’’ risk-prone travelers (as 0 < k 6 1) between the SD and TTB model if the lower semi-standard deviation is used to define a TTB. 3.4. In case of symmetric probability distribution of path travel time It is seen that the upper or lower semi-variance well captures the risk-taking behavior of travelers in route choice if trave time distributions are heavily skewed. It is known that the path travel time follows normal distributions if the central limit theorem (CLT) is applicable, as assumed by Lo et al. (2006). If a path travel time follows a normal distribution or any other distribution that is symmetric about the mean, then the semi-variance is exactly the half of variance, i.e., r2L ¼ r2U ¼ 0:5r2 . In this case, the conventional TTB model (3) has no problem in reflecting the risk-taking behavior in the selection of travel time margin. Therefore, it is unnecessary to consider the mean-upper-semi-standard deviation model (17) and mean-lower-semi-standard deviation model (22). In this case, similar to Lemma 3, we have. Proposition 6. Given an OD pair rs where the probability distributions of all path travel times are symmetrical about their means, rs path k must not be optimal for the TTB model (l þ kr), where 0 6 k 6 1, if it is dominated by other path(s) under the FSD/SSD rule. Proof. Following Definitions 3 and 4, if PDF f ðÞ is symmetric about l, then

r2U ¼ r2L ¼ 12 r2 . Inequality (16) yields

2ðjMPj þ jNQjÞðlY  lX Þ P ðrX 2  r2Y Þ ) ð2jMPj þ 2jNQ jÞðlY  lX Þ P ðr2X  r2Y Þ

ð23Þ

Now it is necessary to explore jMPj and jNQ j, respectively. Actually, in Fig. 5(a) it is not difficult to see that RT RT jMPj ¼ T  lX  GX ðlÞ ¼ l ðlX  tÞf X ðtÞdt and jNQ j ¼ T  lY  GY ðlÞ ¼ l ðlY  tÞf Y ðtÞdt. The existing research has shown X Y RT RT that rX > 2 l ðlX  tÞf X ðtÞdt and rY > 2 l ðlY  tÞf Y ðtÞdt if both f X ðÞ and f Y ðÞ are symmetric about their means lX or X

Y

lY , respectively (Ogryczak and Ruszczynski, 1999). Therefore, that implies rX > 2jMPj and rY > 2jNQ j. If X 2 Y, then lX 6 lY according to Proposition 2 (note that X 1 Y ) X 2 Y). Thus, Inequality (23) yields ðrX þ rY ÞðlY  lX Þ > ðrX þ rY ÞðrX  rY Þ ) rs

lY þ rY > lX þ rX rs

rs

If path k is dominated, then there must exist a path l which travel time dominates that of k . Therefore, the above inequalities implies ll þ rl < lk þ rk . Then, using the similar procedure in the proof to Lemma 3, we have ll þ krl < lk þ krk ; 0 < k 6 1 if path krs is dominated by path lrs in the first/second order, i.e., at least one path’s TTB must rs be less than that of path k . It completes the proof. h Proposition 6 leads to. Corollary 3. Given an OD pair where the probability distributions of all path travel times are symmetrical about their means, path rs k must be non-dominated under the SSD rule if it is optimal for the TTB model l þ kr, where 0 < k 6 1. The proof is similar to that of Corollary 1. 4. Solution method and numerical examples Three TTB models are discussed in this paper: conventional TTB model (3), mean-upper-semi-standard deviation model (17) and mean-lower-semi-standard deviation model (22). Their properties and the implications in terms of risk-taking behavior are studied, and the connections between these models and the SD theory are also investigated, respectively. Corollary 1 for the mean-upper-semi-standard deviation model, Corollary 2 for the mean-lower-semi-standard deviation model and Corollary 3 for the conventional TTB model, open the door to efficiently solve this array of TTB models, respectively, in large static or dynamic networks, because these corollaries show that the optimal path must be non-dominated under the SSD or alternative SSD rule. Based on these findings, the general solution approach is stated as follows. Algorithm SD-MV Step 1 Solve all non-dominated paths for a given OD pair under the SSD or alternative SSD rule by using the label-correcting algorithm SD-LC described in Section 2 (see Wu, 2013 for details). Step 2 Compare the TTBs of non-dominated paths identified in Step 1, and then find the optimal path of the TTB model, and two derived TTB models using upper or lower semi-standard deviation.

288

X. Wu / Transportation Research Part B 80 (2015) 275–290

Table 2 Means, variances and upper semi-variances of non-dominated paths’ travel times (under the SSD rule) of four OD pairs in the CK and CMAP networks, respectively. Net-work

OD pair 47–933 (CK) or 1391–15,037 (CMAP) Non-dominated paths (SSD)

Mean

Var.

OD pair 880–933 (CK) or 6891–15,037 (CMAP) Upper semi-var.

Non-dominated paths (SSD)

Mean

Var.

Upper semi-var.

CK

Path Path Path Path Path

1 2 3 4 5

92.82 93.21 93.16 95.57 95.24

51.60 45.65 47.80 33.59 35.70

25.66 22.68 23.74 16.66 17.68

Path Path Path Path Path Path

1 2 3 4 5 6

67.84 67.69 68.25 68.51 70.01 71.70

41.63 40.22 36.85 35.15 26.86 18.27

20.71 20.04 18.35 17.51 13.34 9.07

CMAP

Path Path Path Path

1 2 3 4

335.82 335.70 336.01 335.88

307.01 302.31 303.25 298.10

151.18 149.39 149.30 147.03

Path Path Path Path

1 2 3 4

512.39 511.91 512.30 511.81

478.34 486.78 480.41 488.50

233.52 237.89 234.70 238.82

In the following, this solution method will be tested in two networks: Chicago Sketch (CK) network (922 nodes and 1950 links), and the much larger transportation network provided by Chicago Metropolitan Agency for Planning (CMAP) (15,037 nodes and 44,331 links). For brevity, we simply employed the same settings of link travel time distributions used in Wu (2013). The link travel time is assumed to be time-dependent but not spatially correlated. All experiments were run on a Windows 7 computer equipped with two Xeon 3.0 GHz CPUs and 2 GB RAM. For each network, two OD pairs were selected, and the non-dominated paths of these OD pairs under the SSD rule were found using the label correcting algorithm SD-LC presented in Section 2. The results are reported in Table 2. It is seen that the number of non-dominated paths is quite small (no larger than eight), considering the sheer size of networks, especially the second one. The algorithm is shown to be quite efficient: the CPU times are only 4.9 and 80.4 s for the algorithm to find all non-dominated paths in the CK and CMAP networks given a destination, respectively. Wu (2013) verified the accuracy of the results solved by this algorithm. It is seen from Table 2 that the standard deviations are much smaller than the mean travel times for all non-dominated paths. Therefore, Proposition 4 is not very useful here. Meanwhile, the fact that the standard deviation is much smaller than the mean (correspondingly, the upper/lower semi-standard deviation should be also very small) lets the mean dominate the route choice decision for mild risk-averse/risk-prone travelers who possess k not larger than 1. For example, path 1 of OD pair (47–933) in the CK network and path 2 of OD pair (1391–15,037) in the CMAP network are respectively the optimal paths for any 0 < k 6 1, no matter for the mean-upper-semi-standard deviation model or mean-lower-semi-standard deviation model. If someone is highly risk-averse/risk-prone, and picks a k which is greater than 1, we are not able to solve the optimal paths for him/her using the method proposed above, as the optimal paths may not be non-dominated at all under the SSD or alternative SSD rule; on the other hand, given that the path travel times may not be normally distributed, the conventional method (Lo et al., 2006) may not be applicable either. Approximately, we may still use the non-dominated paths under the SSD or alternative SSD rule as candidates to identify the optimal solutions even when k > 1. In this case, the optimal solution is just ‘‘locally’’ optimal since only the non-dominated paths are compared. The true optimal one may not be non-dominated at all under the SSD or alternative SSD rule. 5. Conclusions and discussions The TTB model is widely applied to the reliable route choice problem for its easy-to-understand formulation. However, the Bellman’s Principle of Optimality cannot be used to solve this model, making it very difficult to solve the optimal solution in any large stochastic and time-dependent network. This paper studies this TTB model from the perspective of the stochastic dominance (SD) theory, because (1) the SD theory is a common approach to rank random variables and the Bellman’s Principle of Optimality can be applied to solve the non-dominated paths efficiently under a variety of SD rules even in large stochastic and time-dependent networks (already conducted by our previous research); (2) the optimal paths of many existing reliability-based route choice models such as percentage travel time (PTT) and mean-excess travel time (METT) are found to be non-dominated paths under some specific SD rules; and (3) the SD theory is related with a variety of risk-taking behavior so that the TTB model may also be explained in the framework of such risk-taking behavior if a connection between them can be found. Findings of this paper are summarized as follows. 1. Given two random travel times X and Y, if X dominates Y in the second order (i.e., X 2 Y), then lX 6 lY . Unfortunately, X 2 Y does not imply r2X < r2Y . This paper studies the condition of r2X < r2Y based on the distributions of X and Y since the SD relationship is built upon the comparison of probability distributions of two random variables. It is found that the conventional TTB model is not compatible with any SD rule unless imposing some constraints between mean and variance.

X. Wu / Transportation Research Part B 80 (2015) 275–290

289

2. More importantly, it is found that if the probability distributions of path travel times are not symmetric (especially if they are largely skewed), then the conventional TTB model cannot reflect travelers’ heterogeneous risk-taking behavior in route choice. Alternatively, this paper shows that the derived TTB model where the standard deviation is replaced with the upper or lower semi-standard deviation can better reflect such behavior. Further, it is found that the optimal solutions to these derived TTB models (0 < k 6 1, i.e., for mild risk-averse/risk-prone travelers) must be non-dominated under the SSD or alternative SSD rule. This finding opens the door to solve these derived TTB models efficiently in large stochastic and time-dependent networks. 3. If the central limit theorem (CLT) can be applied to determine the path travel time distributions, the optimal paths for the conventional TTB model ð0 < k 6 1Þ must be non-dominated paths under the SSD rule, because the probability distribution of path travel times are normal, i.e., a symmetric distribution about the mean. Therefore, the optimal solution to any conventional TTB model ð0 < k 6 1Þ must be found in the set of non-dominated paths under the SSD rule. Our findings show that if a traveler is highly risk-averse or risk-prone, i.e., he/she would take a k larger than 1 in Eqs. (17) and (22), then the optimal path in terms of TTB may not be non-dominated under the SSD or alternative SSD rule at all. This makes the optimal TTB solutions untraceable for these travelers. It warrants further investigation in the future. This paper focuses on the route choice problem for an individual traveler, and does not consider the corresponding traffic assignment (TA) problem. Since the optimal solutions of these derived TTB models are non-dominated paths under some specific SD rules, in the future study it is promising to use a gradient projection (GP) based approach, which was proposed to solve the user equilibrium problem based on these non-dominated paths (Nie, 2011; Wu and Nie, 2013), to solve this TA problem where these derived TTB models serve as the route choice principles. Acknowledgements The author appreciates the comments and suggestions from three anonymous reviewers. The author also would like to thank Lauren Combs from Lamar University for her great help in editing and proofreading this paper. This research was supported by the Lamar University 2014 Research Enhancement Grant 420235. The author resumes sole responsibilities for the content expressed. References Bachmaier, M., Backes, M., 2008. Variogram or semivariogram? Understanding the variances in a variogram. Precision Agriculture 9 (3), 173–175. Chen, P., Nie, Y., 2013. Bicriterion shortest path problem with a general nonadditive cost. Transportation Research Part B 57, 419–435. Chen, A., Zhou, Z., 2009. A stochastic a-reliable mean-excess traffic equilibrium model with probabilistic travel times and perception errors. Transportation Research Part B 44 (4), 493–513. Dentcheva, D., Ruszczynski, A., 2003. Optimization with stochastic dominance constraints. SIAM Journal of Optimization 14 (2), 548–566. Fan, Y., Kalaba, R., Moore, J., 2005. Arriving on time. Journal of Optimization Theory and Applications 127 (3), 497–513. Frank, H., 1969. Shortest paths in probabilistic graphs. Operations Research 17 (4), 583–599. Friedman, M., Savage, L., 1948. The utility analysis of choices involving risk. Journal of Political Economy 56, 279–304. Fu, L., Rilett, L.R., 1998. Expected shortest paths in dynamic and stochastic traffic networks. Transportation Research Part B 32 (7), 499–516. Hadar, J., Russell, W.R., 1971. Stochastic dominance and diversification. Journal of Economic Theory 3, 288–305. Hall, R.W., 1986. The fastest path through a network with random time-dependent travel time. Transportation Science 20 (3), 182–188. Hanoch, G., Levy, H., 1969. The efficiency analysis of choices involving risk. Review of Economics Studies 36, 335–346. Khani, A., Boyles, S.D., 2015. An exact algorithm for the mean-standard deviation shortest path problem. Transportation Research Part B. http://dx.doi.org/ 10.1016/j.trb.2015.04.002 (in press). Lam, W.H.K., Shao, H., Sumalee, A., 2008. Modeling impacts of adverse weather conditions on a road network with uncertainties in demand and supply. Transportation Research Part B 42, 890–910. Lo, H.K., Luo, X.W., Siu, B.W.Y., 2006. Degradable transport network: travel time budget of travelers with heterogeneous risk aversion. Transportation Research Part B 40 (9), 792–806. Mehrdad Shahabi, S.D.B., Unnikrishnan, Avinash, 2013. An outer approximation algorithm for the robust shortest path problem. Transportation Research Part E 58, 52–66. Miller-Hooks, E.D., Mahmassani, H.S., 2000. Least expected time paths in stochastic, time-varying transportation networks. Transportation Science 34 (2), 198–215. Nie, Y., 2011. Multi-class percentile user equilibrium with flow-dependent stochasticity. Transportation Research Part B 45 (10), 1641–1659. Nie, Y., Wu, X., 2009. Shortest path problem considering on-time arrival probability. Transportation Research Part B 43, 597–613. Nie, Y., Wu, X., Dillenburg, J.F., Nelson, P.C., 2012. Reliable route guidance: a case study from chicago. Transportation Research Part A 46, 403–419. Nie, Y., Wu, X., de Mello, T.H., 2012. Optimal path problems with second-order stochastic dominance constraints. Networks and Spatial Economics 12, 561– 587. Ogryczak, W., Ruszczynski, A., 1999. From stochastic dominance to mean-risk models: Semideviations as risk measures. European Journal of Operational Research 116, 33–50. Shao, H.W.L., Meng, Q., Tam, M.L., 2006. Demand-driven traffic assignment problem based on travel time reliability. Transportation Research Record 1985, 220–230. Tan, Z., Yang, H., Guo, R., 2014. Pareto efficiency of reliability-based traffic equilibria and risk-taking behavior of travelers. Transportation Research Part B 66, 16–31. Toumazis, I., Kwon, C., 2013. Routing hazardous materials on time-dependent networks using conditional value-at-risk. Transportation Research Part C 37, 73–92. Uchida, T., Iida, Y., 1993. Risk assignment: a new traffic assignment model considering risk of travel time variation. In: Daganzo, C. (Ed.), Proceedings of the 12th International Symposium on Transportation and Traffic Theory. Elsevier, Amsterdam, pp. 89–105. Wang, J.Y.T., Ehhrgott, M., Chen, A., 2014. A bi-objective user equilibrium model of travel time reliability in a road network. Transportation Research Part B 66, 4–15.

290

X. Wu / Transportation Research Part B 80 (2015) 275–290

Watling, D., 2006. User equilibrium traffic network assignment with stochastic travel times and late arrival penalty. European Journal of Operational Research 175 (3), 1539–1556. X. Wu, Routing and Traffic Assignment in Stochastic Networks, Ph.D. Thesis, Northwestern University, 2011. Wu, X., 2013. Finding reliable shortest paths in dynamic stochastic networks. Transportation Research Record 2333, 80–90. Wu, X., Nie, Y., 2011. Modeling heterogeneous risk-taking behavior in route choice: a stochastic dominance approach. Transportation Research Part A 45, 896–915. Wu, X., Nie, Y., 2013. Solving the multiclass percentile user equilibrium traffic assignment problem. Transportation Research Record 2334, 75–83. Xing, T., Zhou, X., 2011. Finding the most reliable path with and without link travel time correlation: a lagrangian substitution based approach. Transportation Research Part B 45, 1660–1679.