AN EVALUATION OF VICS’S BENEFIT BY USINGMULTIPLE USER STOCHASTIC EQUILIBRIUM MODEL
Shoshi Mizokami, Professor, Dr. Eng.
Department of Civil and Environmental Engineering, Kumamoto University,
2-39-1 Kurokami, Kumamoto, 860-8555; JAPAN,
Tel: +81-96-342-3541; Fax: +81-96-342-3507
smizo@gpo.kumamoto-u.ac.jp
http://www.civil.kumamoto-u.ac.jp/keikaku/

SUMMARY
This paper aims to develop a mathematical model that is able to forecast the traffic flow on thetransportation network and evaluate the system benefits. This model is basically based on theordinary multiple user network equilibrium model, but we can formulate the other equivalentmathematical optimization problem using an appropriate definition on VICS users sharemodeling. By using this model, we can measure the effectiveness that is consistent with thedemand-modeling framework. Furthermore, we investigate the sensitivity not only of traffic
flow but also of the share of VICS users by several numerical analyses. As a result, we canfind the interest results that VICS information sometimes lessens the total travel time and thatthe social total cost can ameliorate even if we charge fee for VICS information.

INTRODUCTION
In Japan, development of ITS (Intelligent Transportation System) has already commenced asa means of drastically transforming traffic systems. Using advanced information andtelecommunication networks as well as control technologies, ITS seeks organic integrationand management of users, roads, vehicles and other transport means. ITS is expected to help
resolve traffic congestion, accidents, and other road traffic problems, as well as improve theamenity and convenience for users. ITS consists of nine development areas. One of them is
Vehicle Information and Communication Systems, so called VICS. This is thee mostadvanced area that promptly provides the latest necessary road traffic information to drivers
via car navigation systems. The world's first VICS service started in Japan in April 1996. Thissystem is composed of VICS-Unit, Beacons and VICS center. The most distinctive pointcompared with the ordinal navigation system is the two-way communication system. Thissystem aims to improve the efficiency of its service so as to meet driver’s requests, cut down
travel costs by shortening the time of transportation, ensure safety by accurate information on
the flow of road traffic, and preserve the environment by streamlining traffic. Furthermore, itaims to contribute to social and economic development. This system help drivers choose
appropriate routes for smooth and comfortable driving, so that VICS is generally believed to1

be efficient in alleviating congestion and enhancing the performance of traffic networks.
However, lately VICS is about to become Dinosaur. The reasons are as follows;(1) First, there is the difference in the type of beacons and contents of information that eachagency adapts, so information is incompatible to each other. We cannot see the traffic
conditions of expressway on main trunk roads.
(2) Second, the development of only devices and systems like VICS-Unit and Navigation
system was conducted earlier than surveys on the relation between driver’s behavior andtraffic information. If this kind of surveys will not be conducted, we cannot decide when
and where and what kind of traffic information will be provided.
(3) In addition, from the social and economic points of view, it seems that the provision of
traffic information to drivers does not always have social benefit.
(4) So, we have to take some tools to evaluate the traffic information systems.
A fundamental requirement for VICS applications is the development of route choice
behavior models with or without travel time information and the evaluation of the system andindividual driver benefit. For the latter subject, we should study the travel demand forecastingunder the provision of traffic information, and evaluation method of traffic information
systems in the real world transportation networks. Recently there has been substantial
development in modeling and evaluating VICS, however, it has not been sufficient. Then, we
propose a normative mathematical model with which we forecast multiple route choice
behavior equilibrium flow pattern with endogenous demand of VICS (Vehicle Information
and Communication System) users. This model is basically based on the multiple user
network equilibrium model provided by Yang (1998), but we can formulate the otherequivalent optimization-programming problem using an alternative and appropriate definition
on VICS users demand modeling. By using this model, we can measure the social benefit thatis consistent with the demand-modeling framework. Furthermore, we obtain a very interesting
paradox that the VICS information might not always produce social efficiency to thetransportation. Furthermore, we find the useful results that total cost of the system issometimes lower in the case when the VICS information cost is charged to drivers than in thecase when the VICS information cost is free of charge.

NETWORK EQUILIBRIUM MODELBASED ON MULTIPLE ROUTE CHOICE BEHAVIOR
Consider a network G=(N, A), where N is the set of nodes and A is the set of links in the road
network. Let R and S be the set of origin and destination nodes. r ∈ R and s ∈ S are origin
and destination node, respectively. We assume that the travel cost for each link, a ∈ A , is afunction of the flow, x a , on the link. This is described by an increase and strictly convex
function of link flow, t a = t a ( x a ) .
Suppose there are two types of drivers; one who uses and the other one who doesn’t use thetravel time information proposed by VICS on the common road network. When VICS canprovide travel cost information to drivers, it is assumed that their perceptions of the travel cost
improve and become more concentrated around the real travel costs. The drivers who use the
VICS information will receive complete information on the minimum cost route, so they are2

able to find the minimum travel cost route in a deterministic user-optimal manner. Whereas
the drivers who do not use the VICS information will only have incomplete information onroute cost, and may choose alternatives which may not be in their best interest, and hence may
take higher travel cost routes in a stochastic manner. Then, we can consider that drivers are
assumed to have different perceptions of travel cost, so we describe the multiple traffic flowwhich results from these two route choice manners by mixed user equilibrium (UE) andstochastic user equilibrium (SUE) principle assignment. The difference between bothbehavioral principles is distinguished by the degree of the variance of travel cost perceptionsof drivers. This multiple logit based route choice behavior model can be formulated as thefollowing equivalent mathematical optimization program, which was proposed by Yang
(1998):
xa

Min: Z (x, f 1 , f 2 ) = ∑ ∫ t a (ω ) + ∑
a∈ A

s.t.

0

∑f

g =1, 2

rs
g ,k

1

∑∑ f
θ

rs
g ,k

ln( f grs,k / q rsg )(1)

g rs k∈K= q rsg , ∀r ∈ R, s ∈ S , g = 1,2

(2)

k∈K

x a = ∑ ∑ ∑ f grs,k δ ars,k , a ∈ A(3)

f grs,k ≥ 0, ∀k ∈ K rsg , r ∈ R, s ∈ S , g = 1,2(4)

rs k∈K rs g =1, 2

where g(=1,2) appears the driver’s group whose level of perceptions is different. f grs,k is theroute flow by driver’s group g on route k ∈ K rsg . δ ars,k is the link-route incident matrix whichelement is δ ars,k =1 if route k between OD pair rs uses link a, 0 otherwise. q rsg is the demandof driver’s group g between OD pair rs. This model results in two classes of SUE flow
distribution with different degree of travel cost variability: one (g=1) is associated with thedrivers who don’t use the VICS information with a variability parameter θ 1 of thelogit-based route choice model and the other one (g=2) is associated with the drivers who use
the VICS information with a variability parameter θ 2 . In this case, the constraintθ 1 < θ 2 → ∞ should be kept between them.
It is clear that the optimization program (1)-(4) will lead to the multiple logit-based stochastic
user equilibrium assignment models, because we can get the following logit-based route
choice probability as the first-order optimality conditions of this program:Prsg,k = exp(−θ g c krs )∑ exp(−θ c ) ∀k ∈ K , r ∈ R, s ∈ S , g = 1,2
g

rs
k

g
rs

(5)

k∈K rswhere c krs = ∑ t a ( x a )δ ars,k , being the actual travel cost on the k’th route. It is easy to conduct
its proof, so we will neglect the details.

DEFINITION OF VICS USERS SHARE MODEL
We define the VICS users share model. We assume that the share of VICS users, shown by3

g=2, is endogenous variable. As the same manner that Yang considers, this demand can be
determined by the difference of the benefit, S rs1 − S rs2 , generated by the travel cost
information system. It is natural that the share of VICS users assumes to be described by thefollowing logit type model:
Pr[2 | rs ] =1
1 + exp[α + β ( S rs1 − S rs2 )],r ∈ R, s ∈ S(6)where α , β are parameters that should be estimated. We will show two methods to define
the benefit derived from VICS.

Definition By The Travel Cost Saving
As the benefit, Yang used the average travel cost shown in the following equation:
S rsg = ∑ Prsg, k (c rs (f 1 , f 2 )) ⋅ c krs( g = 1,2)

(7)

k ∈K rs

In Eq.(7), we need to calculate the route cost. However, it is difficult to eliminate all routes in
the large-scale road network. We will propose an effective method which is able to calculate
the expected value of travel cost by using link flow by rs and g, x ag,rs , obtained from the Dial’s
backward procedure as follows:
S rsg = ∑∑ x ag, rs ⋅t a (∑ ∑ x ag,rs ) / q rsg
a∈ A rs( g = 1,2)

(8)

rs g =1, 2

Definition By The Improvement Of Uncertainty
As the index of benefit, we will introduce the following uncertainty function which means theuncertainty level of travel cost between OD. For logit-based route choice model as shown in
Eq. (5), the uncertainty function is 1
S rsg = min c krs − −ln ∑ exp(−θ g c krs )
k∈K rs θ g k∈K rs{ }( g = 1,2)

(9)This definition is consistent with the VICS share and route choice demand models which are
based on the random utility theory, so it should be much useful for the evaluation of VICS.

EQUIVALENT MATHEMATICAL OPTIMIZATION MODEL
When we use the demand function with the assumption of benefit (8), the equilibrium level of
VICS user share becomes an endogenous variable to be determined as same as multiple
behavior equilibrium flow. However, we have been unable to introduce an appropriate
stochastic user equilibrium model that is formulated by mathematical program for the above4

multiple behavior equilibrium model with elastic VICS user share demand. Yang proposed a
general parametric nonlinear programming problem and investigated its existence, uniquenessand stability. Furthermore, he proposed a revised solution algorism, so called the mixed
equilibrium assignment algorithm (MEAA), based on the standard method of successive
average (MSA) where Dial’s algorithm is used to find a decent direction of the objectivefunction without path enumeration. When we calculate the benefit by Eq. (8), this algorithm ismuch useful because we can obtain link flow by rs and g, x ag,rs by using Dial’s algorithm.
In the case that we use Eq. (9) as the VICS share demand function, we can find the equivalentmathematical optimization program as follows:
Min: Z (x, f1 , f 2 , q 1 , q 2 )= ∑ ∫ t a (ω )dω + ∑
xa

a∈ A

1

∑∑f

g =1, 2 θ g rs k∈K

0

rs
g ,k

ln( f grs,k / q rsg )

1

2
q rs

rs

−

rs

s.t.

0

∑ q = q , ∀r ∈ R, s ∈ Sgrsω∑ ∫ (ln q − ω + α )dω
β

(10)

rs

rs

(11)

g =1, 2

∑f

rs
g ,k

= q rsg , ∀r ∈ R, s ∈ S , g = 1,2(12)k∈K rsx a = ∑ ∑ ∑ f grs, k δ ars,k , a ∈ A(13)

q rsg ≥ 0, ∀r ∈ R, s ∈ S , g = 1,2(14)

f grs,k ≥ 0, ∀k ∈ K rs , r ∈ R, s ∈ S , g = 1,2(15)

rs k∈K rs g =1, 2

where q rs is the total demand between OD pair rs and is given. In this model, not only pathflow but also the demand of VICS users, q rsg , becomes unknown variable. This is one of thestochastic user equilibrium models with elastic modal choice demand. It is clear that thesolution of this model is satisfied with Eqs. (5), (6) and (9) because the first order conditionsof this model are satisfied with these all equations, so we skip the detailed proof.

SOLUTION ALGORITHM
We can apply the ordinal solution algorithm performing MEAA (the Mixed equilibrium
assignment Algorithm) within the updating process of VICS user demand. We show theprocedure to solve the above optimization model:
Step 0: Set an initial value for the VICS user demand, q rs2 ( n ) , and thus
q 1rs( n ) = q rs − q rs2( n ) , r ∈ R, s ∈ S .
Set n:=0.
Step 1: Conduct MEAA.
Step 1-0: Perform an all-or-nothing loading for q rs2 ( n ) and a stochastic network

loading for q 1rs( n ) based on a set of initial free-flow travel time t a(k ) . Find
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sets of link flows of both groups, and x a(k ) = ∑ x g(k,a) . Set k:=0
g =1, 2

Step 1-1: Update link cost t a = t a (x a ) . For the updated link cost, perform the Dial’s
Forward Pass process and find a set of link weight of both groups, {Wg ,a }.
(k )

(k )

{ } by performing an

Step 1-2: Find the set of auxiliary link flow y a(k ) = y g(k, a)

all-or-nothing loading for q rs2 ( n ) and a stochastic network loading for q 1rs( n )based on a current set of link travel cost t a(k ) .
Step 1-3: Update the new flow pattern of both groups by setting
1
x g(k,a+1) = x g(k,a) + ( y g(k,a) − x g(k,a) )kStep 1-4: If some convergence conditions on x g(k,a+1) are satisfied, then go to Step 2.
Otherwise, let k:=k+1 , and go to Step 1-1.
Step 2:Calculate S rsg (n ) , r ∈ R, s ∈ S , g=1,2 by Eqs.(8), (9), and the set of auxiliary
VICS users demand by1
q rs2(n ) = q rs ⋅, q 1rs(n ) = q rs − q rs2(n ) , r ∈ R, s ∈ S
1( n )
2(n )1 + exp[α + β ( S rs − S rs )]Step 3: If

q rs2(n +1) − q rs2(n ) ≤ ε

for all r ∈ R, s ∈ S , where ε

is a predetermined

tolerance value, then stop. Otherwise let n:=n+1 and go to Step 1.

NUMERICAL ANALYSIS
Here are some numerical examples that will be presented to investigate how the degree of theperceived travel cost variability of drivers, and congestion level on the network and so on
affect the economic benefit of VICS and share of VICS users. The model network, that is thesame as Yang’s one, is shown in Figure 1. As the link performance function, we use the BPR
type function which parameters are shown in Figure 1 as well. It is assumed that there is only
one OD pair from node-1 to node-12 in the base case. The sensitivity parameters of VICS

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users share function are assumed to be α = 1.75 , β = 0.50 . In only the case that we define
the user’s benefit by improvement of driver’s uncertainty, we should use. β = −0.50 . Thosevalues are the same as Yang’s ones.
We will show some basic attributes of this model by following some numerical analyses. Firstis the sensitivity of VICS users share against the change of the total number of trips and theshort distance trip ratio in total trips. The short distance trip is the trip between node-1 andnode-7. A smaller value of θ 1 implies a higher degree of drivers’ perceived travel costvariability, and thus the VICS information might produce more utility to the VICS users. Of
course, we assume that the value of θ 2 is infinite. However, when we use the average travel
time as the definition of a VICS benefit, the VICS users share decreases as the total number oftrips increases. On the other hand, when we use the uncertainty function value as a VICS
benefit, there is the opposite tendency. The second is the sensitivity of the VICS users shareagainst the short distance trips ratio in the total number of trips. As the short distance trips
from node-1 to node-7 increases, VICS users ratio increase in the case that we use the average
travel cost as the VICS benefit. In the case that the uncertainty function is used as thedefinition of VICS benefit, there is the opposite tendency. The VICS might produce more
benefit to the longer distance trips, so the uncertainty value seems to be appropriate as theindex that we defined in the VICS benefit. From these results of analyses, it is good to say

7that the second definition regarding the user’s share model has better attributes than the firstone.
Figures 2 and Figure 3 show the change of two kinds of indices that evaluate the efficiencyof this whole transportation system. First index is the total travel time and the other is the sumof uncertainty value. These indices are shown against the total number of trips under threedifferent equilibrium assignment methods such that all trips are assigned by deterministic user
equilibrium (UE) and stochastic user equilibrium (SUE) and multiple user stochastic
equilibrium (MUSE) model. These values of UE and MUSE are standardized by the value ofSUE. These values increase monotonously as the total number of trips increases. For the sumof uncertainty value shown in Figure 5, the reasonable result such that UE>MUSE>SUE is
kept. On the other hand, the total travel time by UE and MUSE is bigger than by SUE when
the total number of trips is many in Figure 4. This result means a paradox, that the VICS
information might not always produce social efficiency to the transportation system in case ofheavy traffic congestion when we evaluate the social efficiency by used total travel time.

When we assume the cost that the drivers need to get the VICS information, the uncertainty
function of VICS uses is written as follows;
S rs2 = −1θ2ln ∑ exp(−θ 2 c krs ) − C info

(16)

k∈K rswhere C info is the VICS information getting cost. This unit is changed into the unit ofuncertainty value. Figure 5 shows the change of the total cost of this transportation systemagainst the cost of the VICS information under the total number of trips. The values areshown by the ratios to the value in the case of free VICS information cost. From this figure,
we find that there is a maximum point of total cost, which is the worst cost of the VICS
information from a viewpoint of total travel cost for the transportation system. Furthermore,when the total number of trips is 1,600, the total cost of the system is lower than when the
VICS information cost is free of charge.

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CONCLUSIONS AND FURTHER EXTENSIONS
We conclude our studies and show some further research subjects;(1) We have proposed a multiple users stochastic equilibrium model with elastic VICS users
demand. The VICS users demand is defined by the travel time saving and travel cost
accuracy improving of driver. The latter model is formulated by a mathematicalprogramming model based on the drivers’ route choice and VICS information usage
choice behavior.
(2) Through some numerical analyses, there are some different characteristics between themin the change of equilibrium VICS users share against the traffic congestion level.
(3) We obtain a very interesting paradox that the VICS information might not always produce
social efficiency to the transportation.
(4) Furthermore, we find the useful results that total cost of the system is sometimes lower inthe case when the VICS information cost is charged to drivers than in the case when the
VICS information cost is free of charge.
Now, we attempt to expand our model and numerical sensitivity analysis into the inter-modaltransportation system, because the advanced navigation systems like a VICS intend to provide
the route guidance traffic information not only for cars but also for public transportation inorder to use the inter-modality.

REFERENCES(1) Yang, H. (1998) Multiple Equilibrium Behaviors and Advanced Traveler Information
Systems with Endogenous Market Penetration. Transportation Research 32B/3, 205-218.
(2) Catoni, S. and Pallottino, S. (1991) Traffic Equilibrium Paradoxes. Transportation Science
25/3, 240-244.
(3) Harker, P. T. (1988) Multiple Equilibrium Behaviors on Network. Transportation Science
22/1, 39-46.
(4) Shoshi, M. (2002) An Evaluation Method of VICS Using Multiple User Stochastic
Equilibrium Model. Journal of Infrastructure Planning and Management(in Japan) 709/
Ⅳ-56, 105-115.

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