Modeling of the Market Penetration of Advanced Traveler Information Systems Using a Mixed Network Equilibrium Model

Copyright © IFAC Transportation Systems Chania, Greece, 1997

MODELING OF THE MARKET PENETRATION OF ADVANCED TRA VELER INFORMATION SYSTEMS USING A MIXED NETWORK EQUILIBRIUM MODEL Hai Yang

Department of Civil & Structural Engineering, The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong Abstract. We consider a specific Advanced Traveler Infonnation System (ATIS) whose objective is to reduce drivers' time uncertainty with recurrent network congestion through provision of traffic infonnation to drivers. Suppose drivers who are equipped with an ATIS will receive complete infonnation, and hence be able to find the minimum travel time routes in a user-optimal manner, while drivers who are not equipped with an ATIS will have only incomplete infonnation, and hence may take longer travel time routes in a stochastic manner. We propose a convex programming model and an algorithm to solve this mixed behavior equilibrium problem for any given level of market penetration of ATIS . Furthennore, suppose that the infonnation benefit received by a driver who buys an ATIS is measured as the travel time saving (stochastic mean travel time minus deterministic minimum travel time in a mixed behavior equilibrium), and the market penetration of ATIS is detennined by a continuous increasing function of the infonnation benefit, then we have a variable, mixed behavior equilibrium model with endogenous market penetration in an ATIS environment. We establish the existence, uniqueness and stability of this perfonnance and demand equilibrium of ATIS, and propose an iterative procedure to calculate the ATIS market penetration and the resulting equilibrium network flow pattern. Keywords: Route choice, ATIS, user equilibrium, market penetration, nonlinear programming. Civil Engineering

achieved by ATIS under varied level, amount, and extent of information, number of drivers who received information, and level of congestion.

1. INTRODUCTION Recently there has been substantial development in modeling and evaluating Advanced Traveler Information Systems (ATIS). These systems are generally believed to be efficient in alleviating congestion and enhancing the performance of traffic networks, but there are diminishing marginal returns as more information is provided (Koutsopoulos and Lotan, 1990). A fundamental requirement for ATIS applications is the development of route choice behavior models in the presence or absence of information and the evaluation of the system and individual driver benefit, thereby detennining the actual advantages of ATIS implementations.

An important factor that controls the benefits of ATIS is the level of market penetration, defined as the percentage of road users equipped with the system. Most studies have focused on the investigation of the benefits by assuming varied levels of market penetration. It was found that the benefits to the system, the equipped and unequipped drivers strongly depend on the level of market penetration (Emmerink et al., 1995). In addition, a high level of market penetration could possibly lead to overreaction and a deterioration of the network wide performance (Mahmassani and Jayakrishnan, 1991 ; Ben-Akiva et aI., 1991).

Clearly, drivers with and without an ATIS will have different behaviors of route choice because they have different perception of travel time. Three routing criteria: user-equilibrium (UE), systemoptimum (SO) and stochastic user-equilibrium (SUE) for both guided and unguided drivers in an ATIS environment have been frequently used by a number of authors in their investigation on the system benefit of ATIS. Most research efforts have been focused on the total travel time reduction

There is little question that ATIS affects both the equipped and the non-equipped drivers. ATIS can reduce the travel time for the equipped drivers by eliminating uncertainty, and in addition, makes them better off than the non-equipped drivers. It is this private gain that governs the market potential and economic viability of these systems. Therefore, it becomes an important issue to model the market equilibrium of ATIS. From this perspective,

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Emmerink et al. (1994) provided a framework for analyzing the market potential of these systems from an economic point of view. However, the demand relationship between equipped and unequipped drivers has not yet been incorporated into the ATIS modeling explicitly.

We now consider route choice behavior of two specific classes of drivers in the network. The first class of drivers is equipped with ATIS, and thus has complete information about the road traffic condition. This class is assumed to make route choice decision in a deterministic user-optimum manner. Namely, ifJ;w > 0 then c; = cw' and

In this paper, we propose a mixed equilibrium assignment model for determining the market equilibrium of the ATIS. Here we consider a specific information system (ATIS) whose objective is to reduce variations of travel time perceptions of drivers with recurrent network congestion through provision of travel information. Suppose drivers who are equipped with an ATIS will receive complete information, and hence be able to find the minimum travel time routes in a user-optimal manner, whereas drivers who are not equipped with an ATIS will have only incomplete information, and may select alternative which may not be in their best interest, and hence may take longer travel time routes in a stochastic manner. We propose a convex programming formulation for this mixed behavior equilibrium model over a common network. Furthermore, suppose that the information benefit received by a driver who buys an ATIS is measured as the travel time saving (stochastic mean travel time minus deterministic minimum travel time in a mixed behavior equilibrium), and the market penetration of ATIS is determined by a continuous increasing function of the information benefit, we thus lead up to a variable, mixed behavior equilibrium model with endogenous market penetration in an ATIS environment. We establish the existence, uniqueness and stability of this performance and demand equilibrium of ATIS, and propose an iterative procedure to calculate the ATIS market penetration and the resulting equilibrium network flow pattern.

r

(2) if = 0 then c; ~ c.., where C w is the minimum travel time between 0-0

pair we W, j,'" is the flow on route re R associated with the equipped drivers: (3) J;w =dw, w e W

l..

rER.,

where d.., is the demand of equipped drivers between 0-0 pair we W. The second class of drivers is not equipped with ATIS, and is assumed to choose routes in a stochastic user-optimum manner due to their perception errors of route travel time and diversity of preferences. Suppose the probability of this class of drivers choosing the r-th route, p;, is specified by a logit formula: ~w

p,

where d w is the demand of unequipped drivers

r

dw+dw=d.."weW.

0:,=1

(6)

2.2. Model Fonnulation

For given demands of the two groups of drivers, d w ' d..,. we W, the mixed behavior equilibrium model can be formulated as the following equivalent minimization program: min F(v,C,f,a,d) =

R

l..

aeA

r

t.(x)d.x

+.!. l.. 2',i,"lnJ,'" e WEW,.Jt.

(7)

(1)

subject to l..J;w =dw' we W

aeA

where

(5)

between 0-0 pair we W, and is the flow of unequipped drivers on route re R. Suppose that the total travel demand, d w ' we W over the network is given. Obviously,

Therefore, the travel time on a route, r E between 0-0 pair W E W is given by

W

(4)

,

J,w = p;d.." re Rw' we W,

on that link. This is described by an increasing and strictly convex function of link flow, t a = t a (V a ) .

WE

W

where the positive value of parameter e is related to the variation of travel time perceptions of drivers. This variability parameter measures the sensitivity of route choices to travel cost. Therefore, the logitbased SUE assignment will result in the following route flow:

Consider a network G=(N,A) , where N is the set of nodes and A is the set of links in the network. Let W be the set of all 0-0 pairs in the network and Rw be the set of routes between 0-0 pair WE W. Oue to traffic congestion, we assume that the travel time for each link, a E A, is a function of the flow, V a ,

Rw '

w,we

k

2.1. Multiple Route Choice Behavior

rE

R

kER ..

2. FORMULATION OF THE MODEL

C; = L/a (va )0:"

=""'"exp(-ec;) (e ..,),re exp - c

if route r between 0-0 pair w uses

"'R,.

link a, and 0 otherwise.

1370

(8)

(9) (10)

d.. +d.. =d.. , WEW f." ~ 0, rE R.. , WE W ~ 0, r ER,

.

f., A

..

W E

On the other hand, for the route flow associated with the equipped drivers, we have the following optimality conditions:

(11)

W

if!," > 0, then'dL(f,f,A)/'dj," = 0;

(12)

if!," = 0, then 'dL(f, f, A)/'d!," > 0.

where link flow is defined by v.

=I I(tr. + J,.. ~:, WEW

Since

(13)

a EA.

'dL(f,f,A)

reil..

=l+w

CJ),

We now show that the minimization program (7)(13) will lead to the logit-based stochastic user equilibrium assignment model (4) and (5) for unequipped drivers, and the deterministic userequilibrium assignment model (2) for equipped drivers. Substituting the constraints (8), (9) directly to the objective function (7) and to eqn.(10), we can view the problem as a minimization problem with respect to route flows only, subject to the constraint

If,'" + IJ," = d.. ,

WE

W

°then c; = A.. { if!," = °then c" ~ A , , .

er

w'

rER ,w E W.

(20)

(21)

..

Finally we mention the existence and uniqueness of the solutions without proof. Assume that the demand, d", is positive for each WE W, and the travel cost function, t. (v. ), is positive and continuous for each a EA. Then there exists an optimal solution to the minimization program (7)(12), which satisfies the aforementioned optimality conditions, and hence is the SUE flows for the unequipped driver and UE flows for equipped drivers, respectively. Furthermore, from the strict convexity of the second term of the objective function (7), the stochastic equilibrium route flows, J,"', rE R", W E W, associated with the unequipped drivers are unique. Further assume that the travel cost function, t. (v.), is strictly increasing, then the

~A.. (~J," + ,~tr. - d.. J(15)

where A is the Lagrange multiplier associated with constraint (14). First we consider the route flow associated with the unequipped vehicles. Equating the partial derivatives L(f, f, A)/ai to zero will give the following optimality conditions for this program:

i(lnJ," +1)+ ~t.(v.)o: -A .. = 0, (16)

eqUilibrium link flows, v.' for all a E A are also unique. Note that the deterministic equilibrium route flows, j,", rE R", W E W, associated with the equipped drivers are, however, not unique.

We note that 'dUJ,f,A)/'df does not exist for

J," = 0,

= .. - A w

In other words, the UE conditions are satisfied for the route flows associated with the equipped drivers. The minimum route travel cost or O-D travel cost is given by c'" =A.. , W E W.

(14)

rE R.. , WE W.

aJUar

aeA

we have if j ,.. >

Denote the partial Lagrangean of problem (7)-(12) as

A) =F(f,f) -

= £..J ~ t"(v ,5:" -A

rER .. , WE W,

and the nonnegative constraints (11) and (12).

L(f,f,

(19)

the solution of these conditions is only

J," > 0, r E R.. , W E W. In fact, it can be shown that the path flows J,", rE R.. , WE W must valid for

2.3. Market Penetration

be strictly positive at any optimal solution point. We now consider modeling of the market penetration of A TIS. Suppose that the market penetration of the system is an endogenous variable, and can solely be determined by the benefit derived from the information system. In other words, drivers will decide to equip a driver information system based on the travel time savings generated by the system. When not equipped with an ATIS, drivers may take non-optimal routes due to incomplete information. With a logit-based SUE assignment shown in eqn.(4), the average travel cost of an unequipped driver, between O-D pair W E W is equal to

We now focus on the routes belonging to a specific t (v ,5:" O-D pair WE W. Note that c", = ~ ~aEA a IV ar , IJ

being the actual route travel cost, we have

J," = exp[-S(C; -

A.. )}

(17)

rE R...

Solving eqns.(17) and (9), we can easily obtain

A..

p,

J,"

exp(-Sc")

d..

'"'" exp(-Sc

=-A- = ~

"

n

rE R , WE W,

.

(18)

which means that the drivers without an information system will choose their routes in accordance with the logit-based route choice probability.

c. ,

1371

(22) When equipped with an ATIS, drivers are able to find and choose the minimum cost route from their origin to destination. Therefore, the travel time saving generated by the information system can be measured by the difference between the average route travel cost and the minimum route travel cost: (23)

mini?lize F(v,d)

(25)

subject to v E Sed) where F: RI"I x D lwl ~ R ,

(26)

Dlwl

={d: 0 ~ d

w

~ ~C W E

W}

lwl

and S: D ~ 'P(RI"I) is a point-to-set mapping or constraint mapping defined by the constraints (8)(13). Problem (25)-(26) is a minimization problem with respect to v for any given value of d.

Note that Sw ' WE W is always positive that means that buying an information system will always benefit individual driver if we omit the cost for purchasing the information system itself.

Now we define function cp: D lwl ~ [-00,+00] and point-to-set mapping <1>: D lwl ~ 'P(RI"I) : cp(d) = inf{F(v,d)lv E S(d)}

Now we can assume that the number of drivers to buy the information system is determined by the following exponential function: d = w

(d) =

dw

=

w

1+exp[ex + ~(cw - cw ) ] '

WE

W

(28)

where cp(d) and (d) are refereed to as optimal value function and optimal set mapping.

l+exp(ex+~sJ

d

{v E RIAIIF(v,d) = cp(d), vE Sed)}

(27)

Continuity Theorem: If the separable link cost function , c.(v.), a EA , in the transportation network G(N,A) are continuous and strictly monotone, then the solution of link flow, v, to the problem (7)-(13) is a continuous function of demand d (or cl =d - d).

(24)

where ex, ~ are parameters which are assumed to be common for all O-D pair WE W, these parameters could be calibrated once data are available. This function is hereinafter refereed to as the market penetration function . Note that any other factors such as the cost of the system could be incorporated into the market penetration function as exogenous variables.

Proof. The proof can be inferred from the results of stability and sensitivity analysis of nonlinear programming problem (see, e.g., Hogan, 1973). Because the constraint mapping S: Dlwl ~ 'P(RI"I)

is defined by a set of linear inequalities and equalities and thus is continuous at any d' E D lwl ,

3, EXISTENCE, UNIQUENESS AND STABILITY

and the objective function F: RI"I x D lwl ~ R is

With the assumption of eqn.(24), the equilibrium level of market penetration now becomes an endogenous variable to be determined from the mixed behavior equilibrium model and the traveler information system. We now investigate the existence, uniqueness and stability of the demandsupply equilibrium of the information system. Note that we have been unable to find a nice mathematical program for the above mixed behavior equilibrium model (7)-(13) with endogenous market penetration (24), and hence we are unable to establish the equilibrium mathematical properties through mathematical programming analysis. Nevertheless, a variational inequality formulation of the problem could be considered, and hence its mathematical properties can be also established based on the fundamental theory of variational inequality.

continuous at S( d' ) x {d' } , furthermore the optimal solution of link flow, v(d'), to the problem (7)(13) is unique at d =d' , therefore the optimal set mapping (d) is continuous. Due to the uniqueness of the link flow solution, we can conclude that the mixed behavior equilibrium link flow pattern, v, depends continuously upon the demand d (or cl = d - d). Because the equilibrium link flow, v(d) , is a continuous function of demand d, and due to the continuity of the link cost function c. (v.) , a EA, we can easily infer that both the equilibrium minimum O-D travel cost c w (d) and the equilibrium average O-D travel cost cw(d) between any O-D pair WE W are also continuous functions of demand d (or cl = d - d).

We now regard d (or cl =d - d) as a vector of parameters and rewrite problem (7)-(13) as a general parameteric non linear programming problem:

Existence Theorem: If the separable link cost function , c. (v.) , a EA, in the transportation network G(N,A) are continuous and strictly

1372

monotone, then there is at least one solution of the equilibrium market penetration, d, to eqn.(24).

made. Akamatsu (1995) derived a dual formulation of SUE problem and developed an entropy decomposition method, where the route choice entropy is decomposed and expressed in a more convenient form in terms of link and node variables only. Therefore, an equivalent optlITuzation program for the logit-based SUE assignment involving link-based variables only can be formulated, which allows for the full application of the convex combination method.

Proof. The equilibrium condition of market penetration (24) can be expressed as d=r(d). (29) where ro.(d) =

1+ aexp

~(~o. r Co. (d) - Co. (d)

(30)

According to the continuity theorem, the mapping r :D 1w1 c R 1w1 ~ R1w1 defined by eqn.(29) is a continuous mapping on the compact convex set Dlwl. Also, for any de D 1w1 , clearly r(d) E D !W!

We note that all these methods could be adapted to solve our mixed behavior equilibrium model (7)(12) for any given market penetration. In application of these algorithms, the descent direction at each iteration could be found be solving an all-or-nothing assignment for the UE flow and a stochastic network loading problem for the SUE flow based on the current set of link travel times. The mixed equilibrium assignment algorithm (MEAA) can be described below. MEAA: Step 0: Initialization. Perform an all-or-nothing assignment for do.' W E Wand a stochastic

so that rD 1w1 c D lwl. We can thus apply Brouwer's fixed point theorem to conclude that r has at least one fixed point in D lwl . Consequently, existence of the eqUilibrium solution of network flow and market penetration is guaranteed. Note that there might be multiple equilibrium points of the market penetration because the minimum route travel cost c .. (d) and the average route travel cost Co. (d), and hence the travel time saving generated by the information system do not necessarily vary monotonically with the demand d (or d) . Furthermore, the equilibrium point might be either stable or unstable. Fig.l illustrates these possible situations, where at full market penetration traffic flows in the road network becomes the standard user equilibrium. A stable equilibrium corresponds to the point of intersection of the benefit curve and penetration curve, in the sense that the system will converge to this level of market penetration, and the system will return to this level for small deviations in market penetration. An unstable equilibrium corresponds to the point, in the sense that any arbitrarily small deviation from it may cause that the market penetration and traffic flow move to a different equilibrium state.

network loading for do.'

WE

W based on a

set of initial free-flow travel time t~ . This generates a set of link flows v~~) , a EA. Set

k:=O. Step 1: Update. Update link cost t~t) = t.(v~t» ), a E A. Step 2: Direction finding. Perform an a11-ornothing assignment for do. and a stochastic

network loading for do.' based on the current set of link travel time

t~t), a E A.

This yields an auxiliary link flow pattern u~t), a E

A.

Step 5: Move. Find the new flow pattern by setting v(t.1) =V(k) + a ( k) (U(k) - V(k) \ a E A. a

a

0"

Q

Step 6: Convergence criterion. If convergence is attained, then stop. Otherwise let k:=k+ 1 and go to Step 1. In this algorithm, the step size a ( k) could be determined a priori. For example the following simplest move-size sequences could be applied:

4. A SOLUTION ALGORITHM

Recently there has been substantial development of the various solution methods for the logit-based SUE assignment. Powell and Sheffi (1982) developed the method of successive average (MY A) where Dial's algorithm (Dial, 1971) is used to find a descent direction of the objective function without path enumeration. Chen and Alfa (1991) proposed the partial convex combination method proposed where a supoptimal rather than a fixed step size is searched using a linear search method to minimize the integral network cost function or the first term of the objective function (7). Bell et al. (1993) developed the iterative balancing and convex combination method where some improvements over the partial convex combination method are

a(t )

=){.

Alternatively, the step size could be determined by partially minimizing the following term of the objective function: minimize Z(a) = OSaSl

L Jor .

vel)+a(,i.l:)_vC1»)

OEA.

..

t (x)d.x.

"

This one-dimensional search could be extended to the whole objective function (7) if the entropy decomposition method (Akimatsu, 1995) or the iterative balancing method (Bell et al., 1993) is employed.

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Emrnerink, R. H. M., Nijkamp, P., Rietveld, P. and Axhausen, K. W. (1994) The economics of motorist information systems revisited. Transport Review 14, 363-388. Emrnerink, R. H. M., Axhausen, K. W ., Nijkamp, P. and Rietveld, P. (1995) The potential of information provision in a simulated road transport network with non-recurrent congestion. Transportation Research 3C, 293309. Hogan, W. W . (1973) Point-to-set maps in mathematical programming. SIAM Review 15, 591-603. Koutsopoulos, H. N. and Lotan, T. (1990) Motorist information systems and recurrent traffic congestion: Sensitivity analysis of expected results. Transportation Research Record 1281, 145-158. Mahmassani, H. and Jayakrishnan, R. (1991) System performance and user response under real-time information in a congested traffic corridor. Transportation Research 25A, 293307. Powell, W. B. and Sheffi, Y. (1982) The convergence of equilibrium algorithms with predetermined step sizes. Transportation Science 16, 45-55.

The procedure for determination of the market penetration is straightforward according to the convergence pattern shown in Figure 1: Step 0: Determine an initial value of market penetration, d~'), and thus

d~')

=c( - d~'), w E W.

Set n:=O.

Step 1: Perform MEAA to obtain

<.) and c:') ,

W E

W.

Step 2: Update d(.·I) = d.. .. 1 + exp[a + ~(c~') d("1) w

=d

w

_d(··I) W

WE t

- c~'»)]

,

W.

Step 3: If Id~"I) - d~')1 < E for all W

E W,

where

E is

a

predetermined tolerance, then stop. Otherwise let n:=n+ 1 and go to Step 1. CONCLUSIONS We have proposed a variable multiple behavior equilibrium model with endogenous market penetration in an ATIS environment. The market share of A TIS is determined from the private travel time saving derived from the information system. Our numerical example demonstrates that the proposed algorithm has generally a fast convergence toward the stable equilibrium point of the demand and supply of the information system. Both the degree of travel time uncertainty and the level of total travel demand affect the resultant equilibrium market penetration, the effect of the former is significant, whereas the later is relatively small.

Non-equipped Saving ----------------------------------

IT.E

Equipped

REFERENCES Akamatsu, T. (1995) Decomposition of path choice entropy in general transport networks. Transportation Science (submitted). Bell, M.G.H., Lam, W.H.K., Ploss, G. and Inaudi, D. (1993) Stochastic user equilibrium assignment and iterative balancing. Proceedings of the 12th International Symposium on the Theory of Traffic Flow and Transportation, Berkeley, 427-439. Ben-Akiva, M., De Palma, A. and Kaysi, I. (1991) Dynamic network models and driver information systems. Transportation Research 25A,251-266. Chen, M. and Alfa, A. S. (1991) Algorithms for solving Fisk's stochastic traffic assignment model, Transportation Research 25B, 405412. Dial, R. B. (1971) A probabilistic multipath traffic assignment algorithm which obviates path enumeration. Transportation Research 5, 83-111.

Market Penetration (%) Figure l(a)

(100%)

Penetration Curve Benefit

Market Penetration (%) Figure l(b)

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(100%)