1. travel

A Sample Based Algorithm to Determine Minimum
Robust Cost Path with Correlated Link Travel Times
A. Arun Prakash ∗†and Karthik K. Srinivasan
‡
November 18, 2013
Total word count: 7440 + 1000(3 figures + 1 table) = 8440
Submitted to the TRB 93th Annual Meeting, January 10-14, 2014 for presentation and publication
∗
Corresponding Author
Transportation Engineering Division, Department of Civil Engineering, IIT Madras, Chennai 600036, INDIA. e-mail: [email protected]
‡
Room No: 235, Transportation Engineering Divsion, Department of Civil Engineering, IIT Madras,
Chennai - 600036, INDIA. e-mail: [email protected]
†
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Prakash and Srinivasan
Abstract
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Travel time reliability is an important and a desirable property in route and
departure time choice, specifically for a risk averse traveller. Thus, optimizing for
reliability has been gaining interest in recent past in various fields of transportation, computer science, stochastic optimization etc. The present work addresses
reliability optimization under uncertainty, where travel time distributions are represented by a sample. The weighted mean-standard deviation measure (robust
cost) is adopted as a metric of reliability. The minimum robust cost path problem
with link travel times following a general correlation structure is addressed. A
sampling-based approach which is relatively unused is adopted from the literature
to capture and represent spatial correlations. A novel network transformation and
pruning procedure is proposed which determines exact solution to the problem and
which also circumvents the high dimensionality of the formulations proposed in the
literature. Computational experiments presented demonstrate the efficacy of the
algorithm where it was found to perform well on real world networks. The impact
of sample approximation on the population/true optimal was quantified and was
found to be acceptable.
Keywords: travel time variability, mean-variance, mean-standard deviation, reliability, correlations, most reliable path
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Introduction
Uncertainty in transportation networks can be attributed to both the supply side and
demand side characteristics. Supply side factors include capacity variability, signal controls, incidents, accidents, weather etc, and demand side factors are traveler behaviour,
spatial and temporal demand variation, special events etc. The stochastic interactions
lead to a highly variable system performance and high travel times, especially in peak
periods. To mitigate such impacts, there is growing interest towards optimizing performance in uncertain environment recently. In this regard, this paper addresses the
minimum robust cost path (MRCP) problem with stochastic link travel times having
arbitrary distributions and general correlation structures.
In this study, “robust cost of a path” is a measure of reliability that is defined as
the weighted average of mean and standard deviation of travel times on the path. This
measure is chosen here as the travel times are not restricted to pre-specified commonly
used distributions (e.g. normal, log-normal). Further, the weight for standard deviation
allows a straightforward representation and interpretation of risk aversion to unreliable
travel times . This objective further circumvents the need to evaluate path travel time
distributions using high dimensional multivariate integrals.
The motivation behind this study is threefold. Firstly, it is widely accepted that
users of transportation networks place a high value on reliability while making their
travel decisions [1, 18]. Second, the studies which optimize reliability measures under
uncertainty make restrictive assumptions of link independence or specific correlation
structures. Third, the MRCP problem is difficult to solve for reasons listed below.
Empirical evidence suggests that travel times are significantly correlated across links
and should not be disregarded [13]. Unfortunately, the quantification of travel time
distribution and the estimation of general correlation matrices with empirical data is
non-trivial due to the difficulty in estimating valid correlation structures with empirical
data, and intractability of estimating path travel time distributions analytically. However, the presence of correlations makes the MRCP problem difficult to solve as several
desirable properties such as linearity, link-separability, subpath optimality and subpath
non-dominance (in conventional shortest path problems) do not hold.
Due to these data issues and algorithmic challenges, practical applications for optimizing reliability-based objectives are not yet sufficiently well-developed for realistic
network sizes. In particular, the MRCP problem has not received adequate attention
in the literature. Important exceptions include, [21] which proposes exact solution for
special correlation cases and [24] which addresses the general correlation case but with a
heuristic solution procedure. This study attempts to build upon the advantages of both
the studies and proposes a new exact solution procedure for the general correlation case.
Specifically, this study addresses three objectives: 1) To propose a new subpath
elimination criterion for the MRCP problem. 2) To develop a new and efficient algorithm
to solve the MRCP problem based on the subpath elimination criterion. 3) To investigate
the computational performance of the proposed algorithm on real world networks
To handle correlations, a sampling-based approach with implicit correlation representation is adopted based on the literature [24,25] as as it circumvents correlation quantification issues and leads to increased computational tractability (as explained in Section
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Prakash and Srinivasan
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3.3). However, the solution approaches in this study differ from and contribute to the
literature in the following respects: A new “norm based subpath elimination criterion”
is developed for the robust cost objective to identify and eliminate subpaths that are
sub-optimal. Based on this criterion, a pruning algorithm based on label-correcting is
developed for the MRCP problem. The proposed approach leads to significant enhancements in algorithm performance by applying network transformation techniques (Dial’s
efficiency, Johnson’s transformation, and localization), and their integration with the
sample-based quantification technique. Consequently, the new solution procedure yields
a substantial dimensionality reduction from multiple objectives to a two objective problem and an exact solution is guaranteed. In addition, the quality of the solution from
the sample approximation formulation is also investigated in contrast to prior studies.
The rest of the paper is organized as follows. Section 2 presents a brief review of
related literature and identifies gaps in relation to existing study. The problem definition
and formulation are discussed in Section 3. The solution procedure is presented in Section
4 along with the implementation details. Computational experiments and results are
discussed in Section 5 followed by conclusions in Section 6.
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Literature Review
Travel time reliability has received considerable emphasis in the recent literature. Quantification of travel time variability at network level has various challenges including limited sample size, spare spatial-data availability, separation of sources of variability etc.
Readers are directed to [2] for a more detailed discussion.
In literature, stochastic networks were adopted to appropriately model uncertainty
for network optimization. Various objective functions were chosen for optimization including the least expected travel time (LET), mean-variance measures and probability
measures of reliability. The LET path problem under en-route guidance was addressed
by [6,16,23], where dynamic programming based algorithms with exponential worst case
complexity were implemented. The LET path problem in stochastic and time dependent
networks was address by [9, 10, 17] using stochastic non-dominance criterion along with
label correcting procedures.
Though the LET path problem models decision making under uncertainty, it fails
to take into consideration the users’ risk-averse behavior. Evidence from multiple studies points to the fact that users value reliability considerably [1, 3, 18]. To model this
behaviour [22] introduced a variance constraint into the standard LET problem. The
resulting integer programming problem assuming link independence was solved using a
branch and bound procedure. [19] adopted a similar formulation and proposed a solution procedure applicable for the correlation structures with cycle covariance property
(variance of a path decreases after removal of a cycle), but the performance on large
networks was not tested. Other studies have adopted multi-attribute utility functions
as objective functions including early and late schedule delays [12, 15] and variability as
attributes. [15] showed that this class of problems is NP-hard even for simple quadratic
penalty/disutility functions without link correlations.
Probabilistic measures (on-time arrival reliability) have also been considered as ob4
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Prakash and Srinivasan
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jective function [8, 11, 14, 20]. The need to quantify path travel time distributions and
correlations restrict these approaches to a few common distributions. Readers are referred to [20] for a more detailed review of studies that optimize probabilistic reliability
metrics.
To avoid some of the quantification issues noted above, linear function of mean and
a measure of variability has also been used as a metric of reliability (as in present study)
in the literature. The MRCP problem without correlation can be solved by determining
the non-dominated set in objectives mean and variance by exploiting the subpath nondominance property, which was adopted in [4,5]. But, the significance of correlations and
their effect on optimality has been demonstrated by [20, 21]. [21] proposed an algorithm
and a heuristic for the mean-variance trade-off problem for the case where Cholesky coefficients of the link covariance matrix are positive which is too restrictive and seldom
holds in transportation networks. The problem was reformulated as a multiple-objective
shortest path problem in m dimensions (where m is the number of links) and solved by
determining the non-dominated set. [24] proposed two approximation methods to solve
the MRCP problem with and without link correlations. A sample approximation approach was used to model correlations and a Lagrangian substitution based lower bound
heuristic was proposed for optimization. An average error of over 5.4% and suboptimality
in 25.5% cases was reported in the study.
In summary, compared to the deterministic path optimization models, fewer studies aim to optimize reliability related objectives. Among the few studies that focus
on such stochastic optimization, link travel time are often assumed to be independent
which is restrictive and unrealistic [4] due to data and optimization difficulties.The issues
of distribution quantification, correlation estimation and reliability quantification with
empirical data impose statistical and computational challenges. Unfortunately, the restrictive assumption of link independence or limited correlations can lead to sub-optimal
solutions [20]. Due to these difficulties, practical applications (for reliability-based optimization) on real networks based on empirical data remain elusive. This study tries to
address some of these gaps.
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Problem Definition and Formulation
Context and Scope
The transportation network is represented as a directed graph G(N, A) where, N =
1, 2, . . . n represents the set of nodes and A (|A| = m) represents the set of m directed
arcs/links. The link travel times are assumed to be random variables that follow a multivariate distribution, t ∼ f (µ, Σ) where, µ is the mean travel time vector and Σ is the
link travel time covariance matrix.
The present study addresses a static (time invariant) context, and the minimum
robust cost path is considered to be an a priori path whereby recourse decisions are not
permitted.
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3.2 Minimum Robust Cost Path
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Prakash and Srinivasan
Minimum Robust Cost Path
The mean and variance of travel time of path P represented by µP and σP2 are given by
X
X
X X
µP =
µi and σP2 =
σi2 +
ρij σi σj
(1)
i∈P
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∀i∈P ∀j6=i∈P
i∈P
where µi and σi are mean and standard deviation of travel time on link i, ∀i ∈ A and
ρij represents the correlation coefficient between the travel times of links i, j ∈ A.
The robust cost of path P , RCP , given a weight δ, is defined as a linear combination
of mean and standard deviation of the path travel time.
RCP = δµP + (1 − δ)σP
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(2)
The parameter δ which is between 0 and 1 represents the degree of risk aversion of the
user. A low value of δ represents a highly risk averse user, while a high value represents
a less risk averse user. The objective of the minimum robust cost path problem is to
determine the path P ∗ such that its robust cost RCP ∗ is lower than robust cost RCP of
any other path P from origin r to destination s.
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MRCP:


Min. RCP (x) =δ 
X
µuv xuv  +
(u,v)∈A
0.5

(1 − δ) 
X
2
σuv
xuv +
v:(u,v)∈A
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xuv −
X
v:(v,u)∈A
ρuv−u0 v0 σuv σu0 v0 xuv xu0 v0 
(3a)
u=s
u=t
u ∈ N − {s, t}
(3b)
(u,v)6=(u0 ,v 0 )∈A
(u,v)∈A
X
X
xvu

 1
−1
=

0
where the decision variable xuv = 1 if arc (u, v) belongs to optimal path P and 0
otherwise. The flow conservation constraints (3b) ensures that the series of links carrying
non-zero flows on the optimal solution constitutes a directed path from the source to the
sink.
As noted earlier, subpath optimality and subpath non-dominance properties do not
hold for the MRCP problem. As an illustration, consider the network shown in Figure
1 where the mean and variances of the link travel times are given on the arcs. Let the
correlations between the arcs (1, 3) and (3, 4) be -0.5 and between arcs (2, 3) and (3, 4) be
0.5. Let the weight (δ) be equal to 0.2. Note that for the path 1-3-4 the robust cost is not
additive. Also the path 1-3-4 is the optimal path to node 4 from 1 even-though its subpath
(1-3) is sub-optimal violating the subpath optimality condition. Also, the subpath 1-3 is
dominated on both objectives violating the subpath non-dominance principle.
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3.3 Sample-Approximation Based Reformulation
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Path
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(a) Example Network
Mean
Var.
Robust Cost
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(b) Path/Sub-path Costs
Figure 1: Failure of Subpath Optimality and Subpath non-dominance (δ=0.2, ρ(2,3)(3,4) =
0.5, ρ(1,3)(3,4) = −0.5)
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3.3
Sample-Approximation Based Reformulation
The MRCP problem in equations (3) requires the mean, variance and correlations of link
travel times. However, empirical estimation of correlation structure is data intensive and
non-trivial for reasons noted previously.
To circumvent these difficulties,a sampling based approach is adopted and the objective is accordingly reformulated below
Let tdi represent the travel time realization on P
day d, on the link i measured over
D
−1
D days. The sample mean is given as, µ
ˆi = D
d=1 tdi . Here, the link travel time
distributions are assumed to be stationary, that is all measurements tdi come from the
same multivariate distribution. We estimate the mean and variance of travel time on
path P from the sample by
X
µ
ˆP =
µ
ˆi
(4a)
i∈P
D
D
X
1 X
1 X X
2
2
(tdP − µ
ˆP ) =
tdi −
µ
ˆi
σ
ˆP =
D − 1 d=1
D − 1 d=1 i∈P
i∈P
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Min. RCP (x) = δ
X
i∈A
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(4b)
Note that path travel time realization
on day d, tdP can be expressed as sum of link
P
travel times on that day tdP = i∈P tdi
Using the equations in (4) the robust cost objective in equation (3a) can be reformulated using sample mean and variance instead of the population values as follows:
!
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!2
µ
ˆ i xi
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+ (1 − δ) 
1
D−1
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X
d=1
i∈A
tdi xi −
X
!2 0.5
µ
ˆ i xi 
(5)
i∈A
where x represents the vector of m binary variables xi ∈ {0, 1} which satisfy the flow
constraints of unit flow from source r to sink s. The above expression can be written in
a concise form as follows,
0.5
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T
T T
x C Cx
Min. RCP (x) = δ µ
ˆ x + (1 − δ)
(6)
D−1
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where µ
ˆ represents the vector of sample means of all the links and C is a D × m
matrix of deviations from the mean whose elements are given as cdi = tdi − µ
ˆi . Note
that the variance of a path in (5) and (6) is represented as sum of squares of D separate
link objectives cdP . Thus the problem has been reformulated as a multi-objective (D + 1
including the mean) problem.
Typically multi-day estimates of travel time data are collected by many traffic agencies
using probe vehicles, blue-tooth sources, and historical archives from other ITS sensors.
These data can be be readily used to optimize robust cost based on the reformulated
objective in equation 8 without quantifying distributions or correlations explicitly.
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Proposed Algorithm for determining MRCP
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In this section we provide an overview of the algorithm, discuss subpath elimination
criterion and proof of correctness followed by the implementation details. The pseudo
code of the algorithm is depicted in Figure 2
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Overview of Algorithm
The input to the algorithm is the O-D pair (r, s) for which minimum robust cost path is
sought and the travel time realizations over D days.
The algorithm involves two main procedures namely network transformation to reduce
the dimensionality of the problem and network pruning to find the optimal path.
Network transformation involves two sequential operations: (i) identification of dial
efficient links with link costs as sample mean travel times. This step leads to reduced network dimensionality (∼50%) and acyclic sub-network and (ii) Johnson’s transformation
is applied only to dial efficient links with cost vector ci (defined as difference between
travel time on day d from the sample mean) to obtain reduced cost-vectors wi (defined
in (14)). The advantage of this transformation is two-fold: first the transformed costs
which are non-negative for dial-efficient links unlike the cost vector ci . Second, a super
sink node s∗ and a new link (s, s∗ ) is added to the transformed network and its costs
w(s,s∗ ) are set so that all negative cost elements associated with ci are localized on this
link.
By exploiting this localization of negative costs on to a single arc, a new subpath elimination criterion is proposed for this transformed network with reduced costs (discussed
in detail in Section 4.2.1) and it’s correctness is established. This criterion eliminates
subpaths that satisfy either of the following conditions:
1. Lower bound on variance of any path containing current subpath exceeds a normbased threshold and, lower bound on robust cost of the path is larger than the
corresponding upper bound on optimal path (7a).
2. Lower bound on weighted mean component of robust cost of any path containing
current subpath, exceeds upper bound on robust cost of optimal path (7b).
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4.2 Subpath Elimination Criterion and Proof of Correctness
Prakash and Srinivasan
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Finally, based on this subpath elimination criterion, a new pruning procedure is implemented on the transformed network. This pruning procedure eliminates suboptimal
subpaths using a modified label correcting approach (as described in Section 4.3.2.
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Subpath Elimination Criterion and Proof of Correctness
Description of Subpath Elimination Criterion
In this subsection the norm-based subpath elimination criterion is explained. The elimination criterion is dependent on only two link separable objectives thus reducing the
dimension from D + 1 objectives in equation (5).
Consider an intermediate node u, and a subpath Pr−u , from source r to node u with
two attributes dm (u) and dv (u). Here, dm (u) represents total expected travel time of
the subpath and dv (u) represents the sum of norm-squares (Σi ||wi ||2 , i ∈ Pr−u , obtained
from network transformation) of the transformed cost vectors of the subpath. Let SP m (u)
denote the shortest path label (value) from u to sink s with link costs as µ
ˆi and, SP v (u)
denote the shortest path distance from u to sink s with link costs as ||wi ||2 . Let RCU B
be an upper bound on the optimal robust cost.
Elimination criterion at node u is given by,
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Condition 1
[dv (u) + SP v (u)] ≥ ||β||2 AND
Condition 2
δ [dm (u) + SP m (u)] +
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(1 − δ) v
v
0.5
(d
(u)
+
SP
(u))
−
||β||
> RCU B
(D − 1)0.5
(7a)
OR
Condition 3
δ [dm (u) + SP m (u)] > RCU B
(7b)
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where, −β represents the localized cost vector on the link (s, s∗ ) in the transformed
network.
Condition 1 checks whether the lower bound of variance of the path P with Pr−u as
subpath being greater than the variance on the link (s, s∗ ). Condition 2 evaluates if the
lower bound of robust cost of the path P with Pr−u as subpath is greater than the upper
bound on minimum path robust cost. Condition 3 checks whether the lower bound on
weighted mean component of path P with Pr−u as subpath is greater than the upper
bound on minimum path robust cost. The correctness of this elimination criterion is
established in the next section.
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Proof of Correctness
Proposition 1 establishes that if either conditions 1 and 2 (7a) or condition 3 (7b) holds
at node u for some subpath Pr−u , that subpath cannot be a part of the optimal path
from r to s. Proposition 2 establishes that the optimal robust cost from r to s is one
among the paths that remains after eliminating subpaths that satisfy the elimination
criterion.
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4.2 Subpath Elimination Criterion and Proof of Correctness
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Prakash and Srinivasan
Proposition 1. Any path from source r to an intermediate node u, given by Pr−u , which
satisfies condition (7) cannot be a subpath of the minimum robust cost path from r to
sink s∗ on the transformed network.
Proof. The proposition is proved by demonstrating that LHS of the second inequality in
(7a) and LHS in inequality (7b) are lower bounds on the robust cost of any path P from
source r to sink s∗ which includes Pr−u as a subpath where u 6= s∗ . Consequently, if the
lower bound on robust cost of any path P with Pr−u as a subpath is greater than the
upper bound on optimal path robust cost, then the subpath Pr−u cannot be contained
in the optimal path.
First, the validity of (7a) is demonstrated. Let the first inequality in (7a) be true.
The variance of a path P − s∗ on transformed network from source r to sink s∗ with Pr−v
as subpath from r to node u is given as,
(D − 1)V arP −s∗ =||
X
wi − β||2
i∈P
≥ ||
X
wi ||2 + ||β||2 − 2||
i∈P
X
wi ||||β||
i∈P
!2
||
=
X
wi || − ||β||
(8)
i∈P
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In (8), the inequality follows from norm identity and Cauchy-Schwarz inequality. Note
that all elements of transformed cost-vector are non-negative as a result of transformation
wi 0 (14).
Assuming condition 1 holds and using the fact that Pr−u is the subpath of P − s∗ and
as we have assumed first inequality in (7a) to be true, we can write,
X
X
||
wi ||2 ≥
||wi ||2
(9a)
i∈P
X
||wi ||2 ≥
i∈P
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i∈P
X
||wi ||2 + SP v (u) ≥ ||β||2
(9b)
i∈Pr−u
where SP v (u) denotes the shortest path distance from u to s with link costs as ||wi ||2
. Applying the inequalities (9) onto last expression on (8) we can write.
!2
||
X
wi || − ||β||
i∈P

≥
2
!1/2
X
||wi ||2

 X
− ||β|| ≥ 
||wi ||2 + SP v (u)
i∈P
2
1/2

− ||β||
i∈Pr−u
(10)
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From equations (8) and (10) and given (9) we can write,
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4.3 Implementation of the Algorithm
Prakash and Srinivasan

1/2
 X
(D − 1)V arP −s∗ ≥ 
||wi ||2 + SP v (u)
2

− ||β||
(11)
i∈Pr−u
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2
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Thus we have obtained a lower bound on travel time variance of any path P − s∗ with
Pr−u as its subpath. Similarly the lower bound on mean travel time of P − s∗ can be
obtained as follows,
X
µP −s∗ ≥
µ
ˆi + SP m (u)
(12)
i∈Pr−u
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where SP m (u) denote the shortest path distance from u to s with link costs as µ
ˆi .
Using (12) and (11) we can write

RCP −s∗ ≥ δ 

X
i∈Pr−u
µ
ˆi + SP m (u) +

1/2
(1 − δ)  X
||wi ||2 + SP v (u)

0.5
(D − 1)
i∈P


− ||β||
r−u
(13)
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Thus, the LHS of second inequality in expression (7a) is lower bound on robust cost
of any path P − s∗ with Pr−u as subpath. The LHS of inequality in expression (7b) is a
lower bound on mean travel time of any path P − s∗ with Pr−u as subpath. As variance
of random variable is always non-negative, the LHS of expression (7b) is also a lower
bound on robust cost of any path P − s∗ with Pr−u as subpath.
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Proposition 2. The minimum robust cost path P ∗ between source r and sink s will be
present in the path-set at sink at the termination of pruning procedure.
Proof. This claim is proved by contradiction. Let us assume that the minimum robust
cost path P ∗ is not present in the path-set at the sink s. This implies that anyone of
∗
it’s subpaths, Pr−u
, was discarded as it satisfied the elimination criterion in (7), which
contradicts Proposition 1. Thus, the minimum robust cost path P ∗ will be present in the
path-set at sink after the termination of pruning procedure.
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4.3
Implementation of the Algorithm
The steps of the algorithm presented in Figure 2 are discussed below. An illustrative
example is hosted at webpage http://115.115.108.126/trrhost/mrcp_example.pdf
and is not included here due to space limitations.
11
TRB 2014 Annual Meeting
Paper revised from original submittal.
4.3 Implementation of the Algorithm
Prakash and Srinivasan
Step 0. Inputs:
(i). Network G(N, A), origin r and destination s
(ii). Travel time realizations , tdi where d ∈ D and i ∈ A
Step 1. Network Transformation: To transform cost-vectors to component wise positive
(i). Dial Efficient Links: Perform single sink shortest path algorithm with mean travel times
(ˆ
µi ) as link costs. Determine set of links which are Dial efficient, A0
i ≡ (u, v) ∈ A0 if d(u) − d(v) > 0
where d(u) represent shortest path distance from u to sink s.
(ii). Johnson’s Transformation. Perform Johnson’s transformation for each of day d ∈ D.
For each d ∈ D,
(a). Compute the shortest path labels, ld (v), v ∈ N with cdi = tdi − µ
ˆi , ∀i ∈ A0 as costs.
(b). Compute the transformed costs
wdi = cdi + ld (u) − ld (v), ∀i ≡ (u, v) ∈ A0
βd = −wd,(t,t∗ ) = ld (s) − ld (t)
This results in a transformed network G2 (V, E) where V = N ∪ t∗ and E = A0 ∪ (t, t∗ ).
Step 2. Network Pruning to eliminate subpaths which are sub-optimal
Perform procedure outlined in Figure 3.
Step 3. Finding optimal Robust Cost Path:
Identify the minimum robust cost path from the path (label) set at sink t by calculating the
robust cost of all the paths in it.
Figure 2: MRCP algorithm - Correlated Links
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4.3.1
Network transformation
Network transformations involve two operations 1) eliminating the Dial inefficient links
and 2) transforming the original cost-vectors into reduced costs by applying Johnson’s
transformation.
Identifying Dial efficient links:
In this step, links which do not satisfy a criterion similar to Dial’s criterion used in
STOCH algorithm are eliminated [7]. A weaker condition is employed in the present
study where the links which take the user farther from the given destination in average
time are eliminated. These links are found by solving for single destination shortest path
problem from all sources to sink s with mean travel times from the sample as link costs.
A link (u, v) is dial efficient only if its tail node is closer to the sink than the head node
(i.e. optimal distance labels d(u) − d(v) > 0 ). There are two advantages from this
transformation: the network reduces by 40-50% (number of arcs) and also the resulting
sub-network is acyclic.
Johnson’s Transformation to obtain Reduced Costs:
First, the Dial efficient sub-network above is transformed into a network G1 as follows:
12
TRB 2014 Annual Meeting
Paper revised from original submittal.
4.3 Implementation of the Algorithm
Prakash and Srinivasan
1
1. A new super source node r∗ is added
2
2. Dummy arcs connecting the node r∗ to all nodes are created with zero cost.
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3. Transformed costs ci are defined for the dial efficient links (referred to as real links)
as follows: ci = [c1i , c2i , · · · , cDi ] where cdi = tdi − µ
ˆi represents the difference
between observed link travel time on day d, tdi and sample mean on link i, µ
ˆi over
all days. Note that these costs cdi will be positive on some days and negative on
others. Therefore, the associated path costs and optimal costs on paths with only
real links may either be positive or negative for a given day d.
For the cost vector ci , note that the path from r∗ to any node v in network G1 consists
of real paths (paths containing only dial efficient links) and the virtual path with dummy
arc (r∗ , v). For a given day d, if the minimum path cost on real paths is positive, then
the virtual path (r∗ , v) is optimal as it has a distance label of zero (by construction). On
the other hand, if the minimum path cost on real paths is negative, then the optimal
distance label to node v will be negative. Thus, the optimal distance labels (based on
cost vector ci ) to any node will either be zero or negative.
The Johnson’s transformation is applied on this transformed network G1 to convert
the costs, ci , into reduced costs, wi , which are non-negative. More formally, as per the
Johnson’s transformation, the reduced cost on a given arc i ≡ (u, v) on day d is given as,
wdi = cdi + πd (v) − πd (u)
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where node potential for node u on day d is given as πd (u) = −ld (u) where ld (u) is
the shortest path cost from r∗ to u with cdi as link costs.
The resulting reduced costs on all arcs in G1 can be shown to be non-negative for each
day d. To see this, note that as per the shortest path optimality principle, the optimal
distance labels satisfy the following inequality for all arcs i ≡ (u, v) in network G1
ld (v) ≤ cdi + ld (u)
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(14)
(15)
Substituting πd (u) = −ld (u) and πd (v) = −ld (v) and rearranging (15) yields wdi =
cdi + πd (v) − πd (u) ≥ 0
Next, a new super sink node s∗ is added to the network G1 and a dummy arc (s, s∗ )
between the sink s and super sink s∗ is created. This augmented network is referred to
as G2 (V, E) = G1 ∪ (s, s∗ ).
The reduced cost on this dummy link on day d is set as: wd(s,s∗) = ld (s) − ld (r) and is
denoted as −βd for ease of notation. This term may be either positive or negative. Thus,
the reduced cost transformation ensures that all the negative cost elements are localized
to arc (s, s∗) in network G2 .
Next, it is shown that the path cost on day d from r to s on network G(N, A) based
on cost vector ci , is the same as path cost from r to s∗ on the transformed network G2
with the reduced costs wi .
Consider a path P from r to s in the network G and the corresponding
path P 0 =
P
P ∪ (s, s∗). The original path cost on P for day d is given as cdP = i∈P cdi
The reduced path cost on P 0 for same day is given as,
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TRB 2014 Annual Meeting
Paper revised from original submittal.
4.3 Implementation of the Algorithm
wdP ∪(s,s∗ ) =
X
i∈P
=
X
wdi − βd =
Prakash and Srinivasan
X
[cdi + ld (u) − ld (v)] − βd
i≡(u,v)∈P
cdi + ld (s) − ld (t) − βd =
i∈P
X
cdi
(16)
i∈P
4
Due to this equivalence between path costs on network G from r to s, and reduced
costs on network G2 from r to s∗ , the optimization and subpath elimination criterion are
developed using network G2 and reduced costs wi where all negative cost elements are
localized on to the link (s, s∗ ).
5
4.3.2
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Network Pruning
The transformed network G2 is pruned using a label correcting based approach and
the “subpath elimination criterion” (refer section 4.2.1). The first step in pruning is
precomputation of the following quantities.
First, we obtain SP m (u) and SP v (u), ∀u ∈ V except s∗ . SP m (u) is the minimum
expected travel time from node u to s, and SP v (u) is the shortest path cost from u to s
where, cost on links i ∈ E are the norm-square (||wi ||2 ) of the transformed cost-vectors.
These are obtained by solving single sink shortest path problem using label correcting
procedure, with corresponding costs on the links. Next, the upper bound on optimal
robust cost, RCU B , is computed as the robust cost of least expected travel time (LET)
path.
After precomputation, the network G2 is pruned based on the norm based subpath
elimination criterion given in (7). As shown in Proposition 2 the pruning procedure
returns a set of candidate paths at the sink node s that is guaranteed to contain the
optimal path. A label correcting approach is proposed for pruning which is depicted in
Figure 3 and explained below.
A scan eligible list is maintained consisting of only those nodes whose subpaths are not
discarded by the elimination criterion. The nodes in the scan eligible list are processed
sequentially. Specifically, for each node u in scan-eligible list, each of the outgoing arcs
(u, v) is processed to obtain the labels for mean and variance for connected downstream
v
node v. Specifically dk (u) = [dm
k (u), dk (u)] is maintained for each node u ∈ V . For
v
subpath k, dm
represent the mean travel time and sum of link transformed
k (u) and dk (u) P
cost-vector’s norm-squares ( i∈k ||wi ||2 ) respectively.
Node v is added to the scan-eligible list if atleast one of the new (not stored at v)
subpaths obtained by adding link (u, v) to subpaths at u, do not satisfy the elimination
criteria. The algorithm is terminated when the scan-eligible list is empty. Note that
unlike in the standard label-correcting approaches, multiple labels and predecessors may
be maintained for each node. The optimal path can be shown to be contained in the set
of paths that remain at termination (Proposition 2). Therefore, once the label correcting
algorithm terminates, the optimal path is identified by computing the robust cost of all
non-pruned paths to the destination.
14
TRB 2014 Annual Meeting
Paper revised from original submittal.
4.3 Implementation of the Algorithm
Prakash and Srinivasan
Step 0. Inputs:
(a). Transformed Network G2 (V, E) with source r and sink s
(b). Link attributes which include sample mean of travel times µˆi and transformed cost-vectors
wi ∀i ∈ E
Step 1. Pre-computation..
(a). Compute the shortest path distances
SP m (u) with µ
ˆi as link costs
SP v (u) with ||wi ||2 as link costs
from every node u ∈ V to s, where i ∈ E.
(b). Let RCU B = RCPLET , where PLET is the least expected travel time path from r to s.
Step 2. Initialization.
(a). Set first label of source r to zero for both the objectives
v
dm
1 (r) = d1 (r) = 0
Initialize the set of labels at remaining nodes as null sets
D(u) = ∅ ∀ u ∈ N \ {r}
(b). Add source node r to candidate node list L = {r}
Step 3. Node Selection. IF set L is empty (L = ∅), GOTO Step 5. ELSE Select a node u ∈ L.
Step 4. Node Processing. For all nodes v such that (u, v) ≡ i ∈ E , do the following:
˜
(a). Compute Set D(v)
For every label k of node u,
S
Compute the temporary label d˜k (v) = dk (u) (u, v) to node v obtained by adding arc (u, v)
to label dk (u)
m
ˆi
d˜m
k (v) = dk (u) + µ
v
v
˜
dk (v) = dk (u) + ||wi ||2
(b). Update labels of D(v)
/ D(v) check Subpath Elimination Criterion.
For each acyclic temporary label, d˜k (v) ∈
IF NOT
(7a) OR (7b) for label d˜k (v)
S
THEN set D(v) = D(v) {d˜k (v)}
Step 5. Update List
(a). IF set of labels of node v, D(v) has been modified and node v ∈
/L
L=L∪v
(b). Delete node u from list: L = L\u
(c). Go to Step 2
Step 6. Return. Return the path (label) set D(s) at sink s.
Figure 3: Label Correcting Based Pruning
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TRB 2014 Annual Meeting
Paper revised from original submittal.
4.4 Computational Complexity
1
4.4
Prakash and Srinivasan
Computational Complexity
8
The main steps involve: dial efficient link identification (O[mn]), network transformation
( O[mnD + mD], upper bound computation (O[m + nlogn]), pruning procedure (O[mnκ]
and robust cost evaluation of path set at termination (O[κ])). Here κ is the maximum
size of non-pruned path set size from origin to some non-pruned node. The number of
arcs, nodes, and days in analysis are denoted by m, n and D respectively.
This gives an overall complexity O[mn(D + 1) + mD + (m + nlogn) + mnκ + κ] which
is pseudo-polynomial in pathset size κ.
9
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Computational Experiments
14
Section 5.1 quantifies the performance of the proposed algorithm on real transportation networks followed by analysis of sample approximation on optimal solution quality
(Section 5.2)
The experiments were conducted on a Windows-7 (32-bit) computer with two 2.9
GHz CPUs (Core-2 Duo) and a 3GB RAM.
15
5.1
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Performance on Real World Networks
The performance of the pruning algorithm is studied on real world transportation networks obtained from the website hosted by Bar-Gera [http://www.bgu.ac.il/∼bargera/tntp/].
The link travel time realizations over D days which are assumed to follow a multivariate
shifted lognormal distribution. The univariate shifted lognormal random variable used
for generating these travel times is given by
ti = γi + exp(µi + σi zi )
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(17)
where zi ∼ N (0, 1) is a standard normal random variable and γi represents free-flow
travel time. For the current study, the shifted lognormal parameters for each link on
the network were generated uniformly in the following ranges (65-125 sec/km) for γ,
(20-200 sec/km) for excess mean (E[ti ] − γi ) and (0.05-0.75) for coefficient of variation.
These values were obtained based on data from the Chennai Comprehensive Traffic Study
(CCTS, 2008). The correlation matrix was generated uniformly for each network from
the space of positive definite matrices. Then, the travel time realizations were generated
on all the links for D = 100 days. On each network, the algorithm was applied to find
MRCPs for 25 randomly generated OD pairs.
The performance metrics for analysis include average computation time and average
non-pruned path-set size (at intermediate nodes as well as the destination). The results
from these computational experiments are presented in Table 1(a). The computational
times were found to be less than 5 secs for 98% of OD pairs on moderate to small networks
suggesting acceptable performance for moderately sized ( < 1500 nodes) networks. On
the large network of Austin, the average computational time is around 1.2 minutes which
is still acceptable for a stand alone application.
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TRB 2014 Annual Meeting
Paper revised from original submittal.
5.1 Performance on Real World Networks
Prakash and Srinivasan
(a) Performance on Real World Networks
NETWORK
Name
n
Comp Times(sec)
m
Anahiem
416
914
Chicago Sketch 933 2950
Barcelona
1020 2522
Winnipeg
1052 2836
Chennai
1279 3644
Austin
7388 18961
Avg
90th %
Avg
0.02
0.06
30.51
1.41
0.26
0.64
2.17
2.64
55.32
5.25
152.08 404.64
7.82
53.53
22.40
30.57
88.38
218.71
Std dev
0.01
6.26
0.12
0.67
11.63
72.94
Path-set size at Node
Path-set size at sink
90th %
Avg
12.54 35.29
161.94 138.55
39.94 77.24
67.19 114.44
294.11 180.01
403.48 921.77
45.64
928.12
146.84
307.32
1011.88
3259.40
Std dev
Std dev
90th %
92.11
246.40
3748.07 1207.80
257.06
569.60
732.57 1476.80
3766.06 1804.00
6703.75 13999.20
(b) Quantification of sampling errors (in %) across ODs
Path
lengths
Sample
Size
< 10km
5
10
15
20
25
Mean
1.99
1.56
1.33
1.04
0.87
Stdev
4.28
3.58
2.97
2.24
1.93
Mean
5.05
3.15
2.46
1.73
1.24
Stdev
23.44
18.81
15.62
13.90
12.50
Mean
1.40
1.17
1.34
1.38
1.64
Stdev
2.56
2.31
2.51
2.58
2.72
5
6.16
6.87
2.11
12.58
4.35
4.10
10
15
20
25
4.13
2.82
2.19
1.84
5.31
4.02
3.05
2.75
0.83
0.64
0.79
0.59
8.97
7.14
6.35
5.49
3.32
2.72
2.30
1.90
3.62
3.35
3.01
2.75
5
10
15
20
25
8.15
4.99
3.63
3.03
2.49
6.67
4.66
3.66
3.22
2.69
1.20
0.66
0.51
0.29
0.28
10.22
7.19
5.82
4.99
4.37
3.91
2.94
2.38
2.15
1.85
2.99
2.49
2.16
2.05
1.91
10-20km
>20km
Bias
Var error
Total Error
Table 1: Computational Experiments
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However, the path-set sizes may be higher for a few OD pairs, especially on longer
paths. Network density (arc to node ratio) significantly effects the algorithm’s performance as seen from the results on Chicago, Barcelona and Winnipeg networks. Further
computational experiments suggested a weak influence of network structure on computational times and non-pruned path set size (variables such as assortativity, betweenness
centrality, closeness centrality, and pathsize were found to be influential). Due to space
limitations and the relatively weak influence, these are not reported in the paper.
In contrast, other methods in the literature namely PIND procedure [21] and Lagrangian relaxation [24] do not guarantee the optimal solution. Lagrangian Relaxation
reported an average error of over 5.4% and suboptimality in 25.5% cases though it is
likely to be faster.
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TRB 2014 Annual Meeting
Paper revised from original submittal.
5.2 Analysis of Sample Approximation on Solution Quality
1
5.2
Prakash and Srinivasan
Analysis of Sample Approximation on Solution Quality
34
The travel time distributions being approximated through a sample rises questions about
bias and precision of the solution obtained from sample relative to the population. The
extent of the error is quantified as described below.
A second set of computational experiments are performed on a real network of Chennai, India with 1279 nodes and 3644 links. The empirically observed travel times on 94
links over a period of 100 days (from Nov 2012 to Feb 2013) were used in this set. On
the other links on the networks, independent travel times were generated from link-type
wise marginal distributions obtained from CCTS, 2008 which follow shifted lognormal
distribution.
These travel time realizations (over 100 days) over the complete network are assumed
to constitute the population in the current experiment. Then the sample-based MRCP
algorithm was applied by varying the sample size systematically from 5 to 25 days (in
steps of 5). For each sample size, the MRCP algorithm is applied with 100 draws. This
was done for 15 OD pairs in groups of five selected on basis of their shortest path distances
of less than 10km, between 10-20km and over 20km.
Let Ps∗ and Pp∗ be the optimal paths in the sample and the population respectively. To
quantify the quality of solution with the sample-based objective, the metrics measured
were bias (difference in robust cost between Ps∗ in sample and Pp∗ in sample), varianceerror (difference in robust cost between Pp∗ in sample and Pp∗ in population) and total
error (difference in robust cost between Ps∗ in population and Pp∗ in population).
In Table 1(b) we present the percentage error values averaged over the OD pairs
of similar path length. As expected, as the sample size increases both the mean and
standard deviation of all the errors decrease. The path lengths seems to influence both
the bias and variance components, and hence the total error. The OD pairs with shorter
path lengths have lower sample errors and total errors as the population optimal is more
likely to be found in the sample. The mean total errors across samples for shorter paths
lie in the range 1.17% - 1.64%. In contrast, the errors 1.85% - 4.35% were observed for the
longer paths and may be explained by smaller size of the non-pruned path-set. The OD
pairs with longer paths have lower variance error (0.28% - 1.20%) compared to shorter
paths (1.24% - 4.28%) possibly reflecting a greater tendency to regress towards the mean
in the former case. More importantly, the quality of the solution for the sample-based
robust cost objective is found to be acceptable (1-2%) even with modest sample sizes in
most cases.
35
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Conclusion
This paper presents a sample based algorithm for minimum robust cost path on a network
with link travel time correlations. The problem was formulated as a separable multiobjective problem where a sample based approach was adopted to represent the link
travel time distributions. and reformulate the objective function. The sample based
approach adopted was found to implicitly capture the path correlations thus eliminating
the need to explicitly estimate link travel time correlation matrix which is practically not
18
TRB 2014 Annual Meeting
Paper revised from original submittal.
REFERENCES
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Prakash and Srinivasan
trivial both conceptually and computationally. For this objective, solution procedures
based on sub-path optimality and non-dominance are shown to be inapplicable
A new subpath elimination criterion was developed to eliminate suboptimal subpaths.
This criterion is applied to a suitably transformed network with reduced costs and a pruning algorithm is developed to discard sub-optimal subpaths for the robust cost objective.
The transformations together with the elimination criterion enable the conversion of the
MRCP problem from a D + 1 dimensional problem to a two objective problem leading
to significant computational efficiency gain. The proposed algorithm is shown to yield
an exact solution, and its computational complexity is shown to be pseudo-polynomial
in terms of the number of non-pruned paths. The approach could also be extended with
repetitive application to multi-day and quasi dynamic scenarios
The empirical findings from computational experiments performed led to following
insights,
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small to medium (< 1500 nodes) sized networks with computational time being
less than 5 sec for 98% OD pairs. On large networks, the computational times are
higher, particularly for OD pairs with substantial spatial separation. In such cases,
the number of non-pruned paths increases considerably.
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less than 5 % for various sample sizes. The OD pairs with shorter path lengths have
smaller bias than those with longer path lengths. In contrast, the variance component of such O-D pairs was found to be larger. The total error was found to increase
with increasing path lengths, and may be potentially improved by applying the algorithm repeatedly based on real-time traffic state updates and re-optimization
28
The directions for future research include the transformation of costs-vectors into
positive components while negative cycles exist. Extending this present algorithm to
time-dependent networks, online routing, and non-stationary distributions offer interesting and valuable scope for further investigations.
29
Acknowledgments
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This research is supported in part by the Center of Excellence in Urban Transport funded
by Ministry of Urban Development, Government of India. This support is gratefully
acknowledged. The authors would also like to thank Ravi Seshadri for his inputs in the
discussions on ideas in the present study.
34
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TRB 2014 Annual Meeting
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TRB 2014 Annual Meeting
Paper revised from original submittal.