Hai Yang and Xiaoning Zhang

1

OPTIMAL TOLL DESIGN IN SECOND-BEST LINK-BASED CONGESTION PRICING

Revised version submitted to TRB on 15 Nov 2002 Total words: 7298 Hai Yang Associate Professor Department of Civil Engineering The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong, P.R. China Tel: 852-2358-7178 Fax: 852-2358-1534 e-mail: [email protected] Xiaoning Zhang (Corresponding author) Graduate Student Department of Civil Engineering The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong, P.R. China Tel: 852-2358-4252 Fax: 852-2358-1534 e-mail: [email protected]

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

2

Abstract: In the traffic assignment literature, it is well known that a marginal-cost toll is charged on each link to drive a user equilibrium flow pattern toward a system optimum in a general network. Although this principle is theoretically reasonable, it is not practically appealing for many reasons. In real life, second-best pricing scheme is more attractive, where only a subset of links is subject to toll charge. Previously most studies in the research area of second-best pricing concern the determination of optimal toll levels on predetermined toll links, whereas very little attention has been so far devoted to the selection of toll locations. This paper deals with the second-best link-based pricing scheme that involves optimal selection of both toll levels and toll locations. Travel cost minimization or social welfare maximization with and without inclusion of implementation cost of toll charge is sought in general networks. Optimization models with mixed (integer and continuous) variables are formulated for determining toll levels and toll locations simultaneously. A binary genetic algorithm is employed to search optimal toll locations dynamically and a simulated annealing method to search optimal toll levels. Keywords: Network equilibrium, congestion pricing, second-best, toll location

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

3

1. INTRODUCTION Road pricing has long been recognized as an efficient way to improve the economic efficiency of the transportation system, and implemented in many metropolises around the world to reduce traffic congestion and pollution. Based on the classical marginal cost pricing principle, Walters (1) examined an efficient system of taxation for a network of highways. Continuing studies of network-based marginal cost pricing were made by Beckmann (2), Dafermos and Sparrow (3), Smith (4), Button (5, 6), Small (7) and Lewis (8). Recently Yang and Huang (9) summarized the marginal pricing principle on the network in the presence of queues and delays. Although the marginal cost-pricing scheme is perfect in theory, it is not practically appealing for many reasons such as the high operating cost and poor public acceptance. Therefore, second-best pricing policies have received more and more attentions in recent years. Previously most studies in the research area of second-best pricing concern the determination of optimal toll levels on predetermined toll links. Yang and Lam (10) presented a bi-level model for optimal toll design problem considering fixed demand, where the upper level program aims to minimize the total network travel time and the lower level program describes the route choice behavior of network users. May and Milne (11) tested and compared various pricing schemes with the Cambridge city network. Recently, Verhoef (12) proposed a mathematical approach to analyze the second-best toll strategies considering elastic demand. Very little attention has been so far devoted to the selection of toll locations. Hearn and Ramana (13) proposed a linear programming approach to determine the minimal number of toll links to achieve a system optimum. But the models are path-based, and the paths with positive flows in system optimum have to be figured out in advance, which is quite difficult for real life networks. Verhoef (12) proposed to determine optimal toll locations based on some sensitivity indicators. By utilizing the mathematical model proposed by Verhoef (12), Shepherd et al (14) investigated the sensitivity of optimal toll charges with respect to pre-specified cordons of different sizes. There is a common point in the existing literature of congestion pricing, that operating cost of toll collection is just ignored. Actually, operating cost is a necessary concern in toll location determination, which cannot be disregarded in real life implementation. This is exactly the reason that the marginal cost pricing cannot be accepted in practice. In fact, operating cost is also very important in the determination of toll location in second-best pricing. Without considering operating cost, we can always achieve system optimum by second-best pricing, such as in reference (13). However, if operating cost is taken into account, system optimum corresponding to first-best pricing case can never be achieved in second-best pricing scheme. There is a tradeoff in determining the number of toll locations, the congestion can be better mitigated with more toll links, but it will raise the operating cost of toll collection. The current paper attempts to balance this tradeoff properly by mathematical programming approach. This paper deals with the second-best link-based pricing scheme that involve optimal selection of both toll levels and toll locations simultaneously using the optimization methods with mixed (integer and continuous) variables. Social welfare maximization with or without inclusion of the implementation cost of toll charge is sought subject to elastic travel demand in general networks. In the following section 2, we investigate how to search the optimal toll levels on given toll links by bi-level mathematical programming. In section 3, a binary genetic algorithm is applied to determine the optimal toll locations and toll levels simultaneously, when

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

4

considering operating cost. In section 4, the minimal number of toll links can be sought to achieve system optimum without considering operating cost. In section 5, an example is given to demonstrate the methods proposed for the second-best link-based pricing scheme. Conclusions are provided in section 6. 2. DETERMINATION OF OPTIMAL TOLL LEVELS ON PREDETERMINED LINKS Let G ( N , A ) be a directed transportation network defined by a set N of nodes and a set A of directed links. Let A ⊂ A be a subset of links (or road segments) subjected to toll charges. Let W denote the set of origin-destination (O-D) pairs, and Rw the set of paths connecting O-D pair w ∈ W . The symbol δwar equals 1 if route r between O-D pair w ∈ W uses link a ∈ A , and 0

otherwise. Let va be the flow on link a ∈ A , and ta ( va ) the travel time on link a ∈ A as a

continuous, convex and strictly increasing function of link flow va . The notation f rw represents the traffic flow on route r ∈ Rw , w ∈ W . Furthermore, the generalized link travel cost (including both travel time and toll charge if any) is given by ca (va , ya ) = ta (va ) + ya if a ∈ A , or simply ca (va , ya ) = ta (va ) otherwise. Let µ w denote the equilibrium minimum travel cost between O-D pair w ∈ W , including both travel time and monetary cost.

Yang and Lam (10) proposed a bi-level programming approach to determine optimal toll pattern on given toll locations considering link capacity constraint. Without considering link capacity constraint, we formulate the following bi-level model for optimal toll design problem with fixed demand. The upper level is to minimize total system travel time, and the lower level is user equilibrium with given O-D demands. The model is given below: Model A: min F1 = ∑ ta (va (y ))va (y )

(1)

y am in ≤ y a ≤ y am ax a ∈ A ,

(2)

y

a∈A

subject to

Here va (y ) , a ∈ A is the solution of the following lower-level problem: va

min v

∑ ∫ c (ω, y )d ω

(3)

f rw = qw , w ∈ W

(4)

a∈ A 0

a

a

subject to

∑

r∈Rw

va =

∑∑

w∈W r∈Rw

w f rw δ ar ,a∈ A

f rw ≥ 0 , r ∈ Rw , w ∈ W

TRB 2003 Annual Meeting CD-ROM

(5) (6)

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

5

The symbol ya represents the toll to be charged on link a ∈ A , and yamax and yamin are the upper and lower bounds of ya . In the above model, demand elasticity is not considered, or in other words the O-D demands qw , w ∈ W are predetermined. Minimization of total travel time can only be considered as objective function of toll design in fixed demand, it cannot be considered in elastic demand. Because in the case of elastic demand, the total travel time can always be minimized to zero by charging an infinite toll on each link. Therefore, both supply and demand have to be considered in toll design when considering elastic demand. Recently, Yang and Zhang (15) proposed a bi-level model to look for optimal toll levels on a given set of toll links considering elastic demand. The upper level program aims to maximize social welfare, and the lower level program is the user equilibrium in terms of generalized travel cost (including monetary toll if any). The model is given below: Model B: max F2 = ∑ y

dw ( y )

∫

w∈W

Dw−1 ( ω) d ω −∑ ta (va (y ))va (y )

(7)

a∈A

0

subject to y am in ≤ y a ≤ y am ax , a ∈ A

(8)

Here va (y ) , a ∈ A and d w ( y ) , w ∈ W solve the following lower-level problem: va

min d,v

dw

∑ ∫ c (ω, y )d ω − ∑ ∫ D a∈A 0

a

a

w∈W 0

−1 w

( ω) d ω

(9)

subject to

∑

r∈Rw

va =

f rw = d w , w ∈ W

∑∑

w∈W r∈Rw

w f rw δ ar ,a∈ A

f rw ≥ 0 , r ∈ Rw , w ∈ W

(10) (11) (12)

where Dw ( µ w ) is the demand between O-D pair w ∈ W as a strictly decreasing and continuous function of O-D travel cost µ w , correspondingly Dw−1 ( d w ) is the inverse of the demand function.

In the above two models, since the equilibrium link flow, va (y ) , a ∈ A and the O-D demand d w ( y ) , w ∈ W generally are non-convex and non-differentiable functions with respect to the toll pattern y , it is very difficult to obtain the global optimum by traditional programming methods. For this kind of problem, the probability based optimization methods have their advantages, because they are able to find the global optimum without the requirement of differentiability. Yang and Zhang (15) employed a simulated annealing method to handle the social and spatial equity issues in congestion pricing. The original procedure is from Aarts and Korst (16)

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

6

and Dekkers and Aarts (17), which can find the global minimum with convergence. This method has also been successfully applied to solve the continuous network design problems (18). Here we also adopt this method to solve the above two bi-level models. To make searching process faster and solution more accurate, random search and local search are used simultaneously. The procedure of the solution method is stated as follows: For the sake of notation clarification, let

{

Ω = y ya

min

≤ ya ≤ ya

max

}

,a∈ A

(13)

denote the feasible set of y with lower and upper bounds. Simulated annealing method

Step 0: Initialization. Given an initial point y ( ) ∈ Ω and parameters 0 < χ0 < 1.0 , 0 < δ < 1.0, 0

0 < t < 1.0 , L0 (integer), m0 and Ts (stop tolerance of temperature). Set k = k1 = 0 .

Step 1: Finding an initial temperature. Uniformly generate m0 random points denoted by z ( ) i

( i = 1,K, m0 ) over the feasible set Ω. For each z (i )

apply equilibrium traffic assignment

to obtain link flow and O-D travel cost associated with z ( ) and then calculate the ) corresponding upper level objective function value F (z ( i ) ) . Let m2 denote the number i

)

)

)

of points z ( ) with F (z ( i ) ) − F (y (0) ) ≥ 0 and ∆F + the average value of those ) ) ) ) F (z ( i ) ) − F (y (0) ) , for which F (z ( i ) ) − F (y (0) ) ≥ 0 . Then the initial temperature T0 is calculated as below i

)

T (0)

 ∆F + if m2 ≤ m0 2;  −1   m2  =  ln  if m2 > m0 2;  m2χ 0 + (1 − χ 0 )(m0 − m2 )  +∞ if m2 = 0.  

(14)

Step 2: Verifying the termination. If T ( ) < Ts , then stop, and y (0) is the optimal toll pattern. Otherwise, go to Step 3. k

Step 3: Checking the termination of a Markov chain. If k1 > L0 N ( N represents the number of decision variables), then go to Step 6. Otherwise, go to Step 4. Step 4: Generation of points. Randomly generate a number denoted by trandom from the interval [0,1]. If trandom > t then apply the method of Hooke-Jeeves (19) from the point y ( 1 ) to find a local solution denoted by x . If trandom ≤ t , then randomly generate a point denoted by x over Ω. k

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

7 )

)

)

)

Step 5: Metropolis’ rule. If F (x) ≤ F (y ( k1 ) ) or if exp(−( F (x) − F (y ( k1 ) )) / T ( k ) ) > random[0,1] )

)

( k +1)

when F (x) > F (y ( k1 ) ) , set y 1 = x , k1 := k1 + 1 and go to Step 3. Otherwise, y ( k1 +1) = y ( k1 ) , k1 := k1 + 1 and go to Step 3.

Step 6. Cooling schedules. Calculate the standard derivation of the values of the objective ) k function F (y ( k1 ) ) ( k1 = 0, , L0 N ) , denoted by σ(T ( ) ) . Set the temperature as

K

T

( k +1)

=T

(k )

 T 1 + 

−1

ln(1 + δ)   , 3σ(T ( k ) ) 

(k )

(15)

k := k + 1 , y (0) = y ( L0 N ) , k1 = 0 and go to Step 2.

The convergence property of the SA algorithm is proved by Dekkers and Aarts (17) mathematically. Note that the original Hooke-Jeeves method is designed for the unconstrained optimization problem. In the problem examined here, there are simple bound constraints, y amin ≤ y a ≤ y amax , a ∈ A only, we can slightly modify the Hooke-Jeeves method to deal with this situation by projecting the trial points onto the region Ω defined by the bound constraints. Furthermore, the objective function evaluation is required in the Hooke-Jeeves method. It means that a user equilibrium traffic assignment is performed at each trial of the local search procedure.

3. A GENETIC ALGORITHM (GA) USED FOR THE DETERMINATION OF OPTIMAL TOLL LOCATIONS From the previous section, we know that given a fixed set of toll links A , an optimal toll pattern y on these links can be found associated with an optimum objective value F . The objective value F is a minimum travel time F1 for fixed demand or a maximum social welfare F2 for elastic demand. So F can be regarded as an implicit function with the toll link set A . In this section, we investigate the problem of the determination of the optimal toll link set A . If the implementation costs of toll charge are simply ignored, without doubt we can always improve the objective F by adding toll links before it reaches system optimum. At least, the F value can be maintained at the same level by setting the tolls on the additional links to be zero and keeping the tolls on the original links unchanged. However, in reality when we add a new toll link in the network, the extra operating cost of this toll station should be taken into account. As a result, there is a tradeoff between optimizing F and minimizing the extra operating cost. As a first step, we can take the total number of toll links as a decision parameter n , which is predetermined by a policy-maker. Based on the results obtained in the first step for various numbers of toll links, we further determine how many toll locations should be selected. To make the problem practicable, a set of candidate toll links M ⊂ A can be predetermined empirically, and only the members in set M can be selected. Generally, the most congested links (bottlenecks) could be chosen as the candidates for avoiding congestion by increasing the perceived travel cost on these links. There are many combinations of choosing n members from the set M . Let l denote the number of members in M , then the total number of combinations is

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

8

( nl ) = n! × (ll!− n )! .

(16)

Sometimes this number could be very large so that it is very hard if not impossible to go through all the combinations. Therefore, we have to find a rational procedure to search the optimal combination efficiently. The genetic algorithm (GA) is able to do this, because it has an efficient procedure of natural selection (20). In the GA method here, a chromosome represents a set of n toll links, and its performance is evaluated by the optimal F value based on this set of toll links. In the process, a number of chromosomes are generated, evaluated and selected through a generation to the next generation, and eventually the chromosome with the best performance is chosen as the optimal toll link set. The length of a chromosome is l , the number of total candidate toll links, a parameter (gene) in the chromosome denotes the state of each candidate link in M . For each candidate m ∈ M , it has only two alternatives, either to be charged or not. Hence a simple binary genetic algorithm is employed, the gene is defined as 1, if candidate m is selected; gm =   0, otherwise.

(17)

Subject to the constraint of the total number of toll links, i.e.

∑g

m∈M

m

=n.

(18)

The candidates m with g m = 1 form the set of toll links A . Therefore, we can encode the parameters (genes) with the simplest binary variables, namely 0-1 integers. For example, the following line (a) represents a chromosome. The length of the chromosome is l = 15 , and the total number of toll links is n = 8 . Assuming the corresponding link numbers of candidates are given in line (b), then the toll link set A = {6,11, 27,39, 45,53, 64,93} . (a)

0

1

1

0

1

0

1

1

1

0

1

0

0

0

1

(b)

1

6

11

18

27

34

39

45

53

59

64

69

73

82

93

The detailed procedure of the binary genetic algorithm is stated as follows:

Step 1: Initial population. Randomly generate an initial population of the toll link combinations. This initial population has N ipop chromosomes and is an N ipop × L matrix filled with random ones and zeros. Different from other genetic algorithm, we have a special constraint (18) here, which means special treatment is required. Firstly, we randomly generate an N ipop × L matrix, in which each element is a random number between 0 and 1. For each row, we round the n largest numbers to 1 and the others to 0, and then this row becomes a chromosome. The following line (c) is an example of the original row randomly generated and line (d) is the resulting chromosome, L = 13 and N = 8 here. (c) (d)

0.42 0.83 0.06 0.28 0.97 0.32 0.59 0.65 0.14 0.74 0.38 0.51 0.79 1

1

0

TRB 2003 Annual Meeting CD-ROM

0

1

0

1

1

0

1

0

1

1

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

9

Step 2: Chromosome evaluation. Each chromosome is translated into a toll links subset A , as shown in line (a) and (b). For each A , we can obtain a minimum total travel time F1* for fixed demand or a maximal social welfare F2* for elastic demand using the models presented in the previous section. The optimal value of − F1* or F2* is taken as the performance (or fitness) index of the corresponding chromosome. Since the process of GA is to find the chromosome with maximal performance index, − F1* instead of F1* is taken as the performance of the chromosome in the case of fixed demand so as to minimize total travel cost.

Step 3: Natural selection. Select part of chromosomes with higher performance indices and discard the rest, according to the principle of “survival of the fittest”. It is somewhat difficult to keep how many chromosomes survive. Keeping too many chromosomes allows the genes with bad performance transferred to next generation. Letting only a few chromosomes survive limits the available seeds giving births to the offspring. The percentage of survived chromosomes is called survival rate. From other researchers’ experience, it is sensible to keep half of them.

Step 4: Crossover. The survival chromosomes are selected to be the parents of the next generation, and pairs are made among them randomly. To give their children chance to perform better than themselves, the mother chromosome and the father chromosome have to exchange genes with each other, and this operation is called a crossover. Subject to the constraint (18) the crossover cannot operate arbitrarily. If a chromosome transfers a gene 1 to its spouse and gets a gene 0 back at one place, he/she has to transfer a gene 0 to his/her spouse and get a gene 1 back at another place. For example, the following line (e) is a chromosome and line (f) is its spouse, if (e) gives its 1st gene “1” to (f) and gets a gene “0” back, and gives its 7th gene “0” to (f) and gets a gene “1” back, the chromosomes (e) and (f) become (g) and (h) respectively. The percentage of exchanged genes is called crossover rate. (e)

1

1

1

0

0

1

0

1

0

0

1

0

1

0

1

(f)

0

0

1

0

1

0

1

1

0

0

1

1

0

1

1

(g)

0

1

1

0

0

1

1

1

0

0

1

0

1

0

1

(h)

1

0

1

0

1

0

0

1

0

0

1

1

0

1

1

Step 5: Mutation. Randomly change some genes for some chromosomes. Constraint (18) works again in mutation, the shifting is in pairs. If a gene in one location shifts from 1 to 0, there should be another gene shifting from 0 to 1. For example, if chromosome (i) shift 7th gene from “1” to “0”

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

10

and shift 10th gene from “0” to “1”, it becomes chromosome (j). The percentage of statechanged genes is called mutation rate. (i)

0

1

0

0

1

0

1

1

0

0

1

1

0

1

1

(j)

0

1

0

0

1

0

0

1

0

1

1

1

0

1

1

Step 6: Verification of stopping criterion. If the convergence criterion has not been reached, go to Step 2 to continue the next generation; otherwise stop. The genetic algorithm method proposed above is able to seek the optimal subset of toll links A , the optimal toll pattern y * and the corresponding F (n)* automatically, for a given number of toll links n . In this sense, F (n)* can be regarded as an implicit function of n . However, up to now the extra operating cost of the toll system has not been taken into account. The problem of finding optimal toll locations considering operating cost is modeled as following. In the case of fixed demand, apart from the total travel time the operating cost is added into the social cost F3 to be minimized. Obviously, the extra operating cost is highly dependent on the total number of toll stations. If we further assume that every toll station has an identical extra operating cost κ , and hence the total operating cost is a linear function of n . Therefore the minimum social cost for n toll links is a function of n as below F3 ( n ) = F1 ( n ) + κn . *

*

(19)

To minimize the social cost, the following model is able to seek the optimal total number of toll links in fixed demand. Model C: min F3 ( n ) = F1 ( n ) + κn

(20)

0≤n≤l

(21)

*

*

n

subject to where l is the number of candidate toll links. Note that there is only one variable in the model, one dimension search method can be applied here to search for the optimal solution of n . In the case of elastic demand, the original social welfare F2 does not include the operating cost, and hence it should be called gross social welfare. If the extra operating cost is taken into account, the net social welfare F4 is obtained by subtracting the operating cost from the gross social welfare. Then the maximal net social welfare for n toll links is given by F4 ( n ) = F2 ( n ) − κn . *

*

(22)

To maximize the net social welfare, the following model is employed to find the optimal total number of toll links considering elastic demand.

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

11

Model D: max F4 ( n ) = F2 ( n ) − κn

(23)

0≤n≤l

(24)

*

*

n

subject to So far, the toll levels and toll locations can be determined by the aforementioned models, provided that the set of candidate toll links and the operating cost of single toll station are given.

4. ACHIEVING SYSTEM OPTIMUM IN SECOND-BEST LINK-BASED PRICING In the first-best pricing scheme, by levying an externality on each link, the user equilibrium solution is driven to a system optimum. In the case of fixed demand, system optimum means minimum travel cost of shipping certain flows to the O-D pairs; in the case of elastic demand, system optimum means maximal social welfare achieved. However, other than marginal cost pricing, there could be many toll patterns that can obtain minimal travel time or maximal social welfare. In the second-best toll schemes, it is meaningful to achieve an optimal flow pattern with a minimal number of toll links. Actually, this can be done by relaxing constraint (18) and modifying the objective functions in the GA algorithm. In the case of fixed demand, it can be modeled as below. Model E: min n + η  F1 (n)* − πTC 

(25)

0≤n≤l

(26)

n

subject to here πTC refers to the minimum total travel time from marginal cost pricing, and η is a big number. The term η  F1 (n)* − πTC  can be regarded as a penalty term. If F1 (n)* < πTC , the penalty term or the second part of the objective function will be very large and hence dominate the objective function value; however, if F1 (n)* = πTC , the whole objective function value becomes simply the number of toll links n . So the model is able to achieve SO firstly and then minimize the number of toll links. In the case of elastic demand, a minimum set of toll link can be chosen to achieve maximal social welfare. The selection of least toll links for SO can be modeled as below. Model F: min n + η π SW − F2 (n)* 

(27)

0≤n≤l

(28)

n

subject to

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

12

here π SW refers to the maximum social welfare obtained by marginal cost pricing. Now at each trial, n is a variable instead of a fixed number. Eventually, only the toll link combination with system optimum and minimum toll link number will be chosen. Note that here the candidate toll links M have to be the whole set of links in the network ( M = A and hence l is the total number of network links), otherwise the system optimum could not be achieved in some cases. In congestion pricing, one obstacle stems from the public objection, because drivers complain that the government implements congestion pricing only for the purpose of revenue. Therefore it is reasonable to make revenue as low as possible. Now we consider how to minimize the toll revenue collected and maintain the flow pattern being system optimum. Since the number of toll locations is not our concern, the toll link set is relaxed to be the whole set of links. In the case of fixed demand, the problem can be modeled as below. Model G: min y

∑ v (y) y a∈ A

a

a

+ η ∑ ta ( va ( y ) ) va ( y ) − πTC  



 a∈A



(29)

subject to y am in ≤ y a ≤ y am ax , a ∈ A ,

(30)

here πTC and η have the same meanings as before. Note that if ya = 0 , a ∈ A , it is not needed to set a toll booth on link a . In the case of elastic demand, the toll design problem can be modeled as following. Model H: min y

∑v (y) y a∈ A

a

a

 + η π SW 

dw ( y )

∑∫

 −  w∈W 

Dw−1 ( ω) d ω −

0

∑ t ( v ( y )) v ( y ) a∈A

a

a

a

   

(31)

subject to y am in ≤ y a ≤ y am ax , a ∈ A .

(32)

In the above two models G and H, the toll locations are given as the whole set of link in the network. Since the toll locations are known, the GA algorithm is not required here, and only the aforementioned simulated annealing method is employed to solve the problems.

5. A NUMERICAL EXAMPLE Example 1 In this example, the toll design problem with fixed demand is demonstrated with the network depicted in Figure 1 and the following data: q1,2 = 5 , q5,6 = 10 ;

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

13

t1 = 20 + v1 , t2 = v2 , t3 = v3 , t4 = 3v4 , t5 = v5 , t6 = 12 + v6 , t7 = 20 + v7 .

The user equilibrium link flows are v1 = 1.444 , v2 = 3.556 , v3 = 3.556 , v4 = 4.778 , v5 = 1.222 , v6 = 1.222 and v7 = 8.778 . At user equilibrium, the total travel time is 395. At system optimum, the link flows are v1 = 3.222 , v2 = 1.778 , v3 = 1.778 , v4 = 3.222 , v5 = 1.444 , v6 = 1.444 and v7 = 8.556 , and the total travel time reduces to 378. Now we apply the genetic algorithm to determine the optimal toll locations and toll levels simultaneously. Since there are only 7 links in the network, all the links are set to be candidate toll links. In the GA procedure, the survive rate, cross rate, mutation rate are set to be 0.3, 0.4 and 0.2 respectively. If we restrict that only one link is subject to toll charge, it is found that link 2 is the best link to place toll station. The optimal toll level is y2 = 8.0 , and the total travel time is reduced to 381. The revenue collected is 14.2. If only two links are permitted to place tolls, link 2 and link 5 are the two best toll links. The optimal toll levels are y2 = 10.0 and y5 = 4.0 , and the total travel time is reduced to the system optimum solution 378. The total revenue is 23.6. This means two toll links are enough to achieve system optimum if the operating cost is simply ignored. If the operating cost is taken into account, it depends on the level of operating cost to consider how to charge tolls. Model C is applied to solve the problem. If the operating cost of a single station is κ < 3 , link 2 and link 5 are charged with the tolls y2 = 10.0 and y5 = 4.0 . If 14 ≥ κ ≥ 3 , only link 2 is charged with y2 = 8.0 . If κ > 14 , no toll can be charged anywhere. Now Model E and Model G are employed to find the optimal toll links to achieve system optimum with minimum number of toll links and minimum amount of revenue. In both cases, still link 2 and links 5 are optimum toll links, and again the optimal toll levels are y2 = 10.0 and y5 = 4.0 . It is confirmed that a system optimum can be achieved with two toll links and with minimum revenue of 23.6.

Example 2 A network in Figure 2 consists of 43 links and 20 nodes, and there is one O-D pair from node 1 to node 20. The demand function is D1,20 = 2500 exp ( −0.02 ×µ1,20 ) .

The performance function of each link takes the following form: 4    va    ta (va ) = t 1.0 + 0.15    .  Ca      0 a

The values of Ca and ta0 are given in Table 1.

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

14

In the city, the value of time is 120HK$/hour. Traffic assignment is conducted according to the user equilibrium principal. As a result, the social welfare is 153,790HK$. Meanwhile, the V/C ratios of the 12 most congested links are greater than 1, these links are 3, 4, 6, 9, 16, 19, 22, 25, 32, 35, 38 and 42. Furthermore, in the case of marginal cost pricing, social welfare under the condition of system optimum is 157,080HK$. Now let us conduct the second-best link-based pricing scheme. The 12 most congested links are viewed as bottlenecks on the network, thus they are taken as the candidate toll links. According to the second-best link-based pricing method proposed in this paper, the genetic algorithm associated with a simulated annealing is used to seek optimal toll locations and toll levels. Totally 10 cases are analyzed when the number of toll links increases from 1 to 10. In each case, the optimal toll links and optimal tolls on them are obtained; in addition, the corresponding maximal social welfare is also calculated. The overall results are given in Table 2. From the table, we can see clearly that we can always increase the social welfare by adding more toll links. When we charge optimal tolls on 10 appropriate links, the social welfare reaches 157,010HK$, which is quite close to the maximal social welfare 157,080HK$. In the above analysis, the implementation costs are ignored. However, as mentioned before, the operating costs spent in toll collection have to be considered. To get net social welfare, the total operating costs spent on all toll stations should be subtracted from the gross social welfare. As long as the unit operating cost is given, there should be an optimal number of toll links, which can be solved mathematically by Model D. It is interesting that Model D can also be interpreted graphically, as shown in Figure 3. A series of points are plotted in the graph, the horizontal axis denotes toll link number and the vertical axis denotes corresponding maximal gross social welfare. If we connect 5 points of A, B, C, D and E with 4 line segments; these 5 points form a convex surface. These 5 extreme points have special meanings: whatever the unit operating cost is, to maximize net social welfare the optimal number of toll links can only be their horizontal axis values, namely 0, 1, 2, 6 or 10. Furthermore, the slopes of the 4 line segments AB, BC, CD and DE are 772, 682, 258 and 184 respectively, which represent the critical unit operating costs for the selection of optimal toll locations. If the unit operating cost of each toll station is less than 184HK$, we can maximize the net social welfare by charging on 10 links. And if the unit operating cost falls in the intervals of (184HK$, 258HK$), (258HK$, 682HK$) or (682HK$, 772HK$), the optimal toll links should be 6, 2 or 1 respectively. However, if the operating cost of one toll station is greater than 772HK$, we cannot impose any toll on any link. Under the first-best marginal-cost pricing scheme, social welfare is 157,080HK$, which gives the upper bound of social welfare for all toll patterns. Actually, this system optimum can also be achieved in the second-best pricing scheme. Now Model F is employed to search the minimal number of toll links to drive the user equilibrium flow pattern to the system optimum. In this case, the set of candidate toll links is extended to the whole set of links on the network, and the constraint of a fixed number of toll links is removed. It is found that at least 14 toll links are required to achieve a system optimum. The toll links and the corresponding optimum tolls are given in Table 3.

6. CONCLUSIONS In this paper, we have examined how to determine the toll levels and toll locations simultaneously for the second-best link-based scheme. Firstly we examined how to determine the

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

15

optimal toll levels on given toll links in the cases of fixed demand and elastic demand. Bi-level models are formulated to minimize total travel cost or to maximize total social welfare while taking into account user’s route choice behavior. A simulated annealing algorithm is applied to solve the models. Then a binary genetic algorithm is employed to determine optimal set of toll links. For different levels of operating cost, the models are able to simultaneously determine both optimal toll links and optimal toll levels. And new models are developed to determine minimum number of toll links or minimum amount of revenue implemented to achieve the system optimum. Finally, numerical examples are illustrated to demonstrate the application of the models.

ACKNOWLEDGMENTS This research described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKUST6203/99E).

REFERENCES 1. Walters, A A (1961) The theory and measurement of private and social cost of highway congestion. Econometrica 29, 676-699. 2. Beckmann, M J (1965) On optimal tolls for highways, tunnels and bridges. In: Vehicular Traffic Science, American Elsevier, New York, 331-341. 3. Dafermos, S C and F T Sparrow (1971) Optimal resource allocation and toll patterns in user-optimized transport network. Journal of Transport Economics and Policy 5, 198200. 4. Smith, M J (1979) The marginal cost pricing of a transportation network. Transportation Research 13, 237-242. 5. Button, K J (Ed.) (1986) Road pricing, a special issue of Transportation Research, Vol. 20A. 6. Button, K J (1993) Transport Economics (2nd Edition). Edward Elgar, England. 7. Small, K A (Ed.) (1992) Congestion pricing. A special issue of Transportation, Vol.19, 287-291. 8. Lewis, N C (1993) Road Pricing: Theory and Practice. Thomas Telford, London. 9. Yang, H and H J Huang (1998) Principle of marginal-cost pricing: how does it work in a general road network? Transportation Research, Vol. 32A, 45-54. 10. Yang, H and W H K Lam (1996) Optimal road tolls under conditions of queueing and congestion. Transportation Research, Vol. 30A, 319-332. 11. May, A D and D S Milne (2000) Effects of alternative road pricing systems on network performance. Transportation Research, Vol.34A, pp. 407-436. 12. Verhoef, E T (2001) Second-best congestion pricing in general networks: algorithms for finding second-best optimal toll levels and toll points. The 9th World Conference on Transportation Research. July 22-27, 2001, Coex, Seoul, Korea.

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

16

13. Hearn, D W and M V Ramana (1998) Solving congestion toll pricing models. In: Equilibrium and Advanced Transportation Modeling (edited by Marcotte, P. and Nguyen, S.) Kluwer Academic Publishers, pp.109-124. 14. Shepherd, S P, A D May, D S Milne (2001) The design of optimal road pricing cordons. The 9th World Conference on Transportation Research. July 22-27, 2001, Seoul, Korea. 15. Yang, H. and Zhang, X. (2002) Multiclass network toll design problem with social and spatial equity constraints. Journal of Transportation Engineering, Vol. 128, 420-428. 16. Aarts, E H L, J Korst (1989) Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing. John Wiley & Sons. 17. Dekkers, A and E Aarts (1991) Global optimization and simulated annealing. Mathematical Programming 50, 367-393. 18. Meng, Q and H Yang (2002) Benefit distribution and equity in road network design. Transportation Research, Vol. 36B, 19-35. 19. Bazaraa, M.S., Sherali, H.D. and Shetty, C.M. (1993) Nonlinear programming: theory and algorithms. John Wiley & Sons, Inc., New York. 20. Haupt, R L and S E Haupt (1998) Practical Genetic Algorithms. John Wiley & Sons, Inc. New York; Chichester; Weinheim; Brisbane; Singapore; Toronto.

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

17

LIST OF TABLES TABLE 1

Input Data for the Network in Example 2

18

TABLE 2

The Optimal Toll Links, Toll levels and Corresponding Maximal Social Welfare for each Given Toll Link Number

19

The Toll Links and Toll Levels to Achieve System Optimum with Minimum Toll Links

20

TABLE 3

LIST OF FIGURES FIGURE 1

Transportation Network Used in Example 1

21

FIGURE 2

Transportation Network Used in Example 2

22

FIGURE 3

The Maximal Social Welfare for a Given Number of Toll Links

23

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Hai Yang and Xiaoning Zhang

18

TABLE 1 Input Data for the Network in Example 2

Link No. 1

ta0 (min) 4.0

2

3.0

3

Link No. 16

ta0 (min) 5.0

1000

17

4.0

5.0

100

18

4

4.0

600

5

3.0

6

Link No. 31

ta0 (min) 3.0

200

32

5.0

300

3.0

600

33

4.0

200

19

5.0

100

34

3.0

600

200

20

4.0

600

35

5.0

100

5.0

100

21

3.0

200

36

4.0

600

7

4.0

700

22

5.0

100

37

3.0

200

8

3.0

400

23

4.0

800

38

5.0

100

9

5.0

300

24

3.0

600

39

3.0

1000

10

4.0

800

25

5.0

300

40

4.0

600

11

3.0

600

26

3.0

600

41

4.0

700

12

5.0

400

27

4.0

600

42

4.0

600

13

3.0

600

28

3.0

800

43

4.0

1000

14

4.0

600

29

5.0

400

15

3.0

600

30

4.0

600

Ca (veh/h) 1000

TRB 2003 Annual Meeting CD-ROM

Ca (veh/h) 300

Ca (veh/h) 800

Paper revised from original submittal.

Determination of optimal toll levels and locations 19 TABLE 2 The Optimal Toll Links, Toll levels and Corresponding Maximal Social Welfare for each Given Toll Link Number Number of Toll Links

Subset of Toll Links

Social welfare (HK$)

Corresponding Tolls on Each Link (HK$)

1

9

3.206

154564

2

9,32

3.048,3.048

155246

3

3,9,32

3.214,3.156,2.984

155502

4

3,9,32,38

3.186,3.116,3.104,3.104

155762

5

3,6,9,32,38

3.230,3.322,3.242,3.128,3.312

156020

6

3,6,9,32,35,38

3.188,3.264,3.212,3.188,3.236,3.248

156276

7

3,4,6,9,32,35,38

3.244,0.206,3.238,3.022,3.212,3.284,3.198

156280

8

3,4,6,16,25,35,38,42

3.312,3.196,3.142,3.200,3.224,3.190,3.312,3.164

156458

9

3,4,6,16,19,25,35,38,42

3.164,3.276,3.260,3.204,3.230,3.204,3.300,3.280,3.242

156734

3,4,6,16,19,22,25,35,38,42 3.192,3.256,3.278,3.210,3.178,3.204,3.220,3.210,3.264,3.232

157010

10

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Determination of optimal toll levels and locations 20 TABLE 3 The Toll Links and Toll Levels to Achieve System Optimum with Minimum Toll Links

Toll link Toll (HK$)

3

4

6

9

16

19

22

25

32

35

38

39

42

43

3.202 3.212 3.202 0.006 3.196 3.210 3.208 3.196 0.006 3.208 6.020 2.884 3.194 2.884

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Determination of optimal toll levels and locations 21

1

1

2

2

3 3

4

6

5 5

4

7

6

FIGURE 1 Transportation Network Used in Example 1

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Determination of optimal toll levels and locations 22

1

1 3

2

6 15

11 28

16

14 16

27 29

40

2 5

7 18

12 31

17

4

6

17 19

30 32

41

3 8

8 21

13 34

18

7

4

10

9

11

12

20

9

22

33 35

42

24

14 37

19

23 25

36 38

43

5 13

10 26

15 39

20

FIGURE 2 Transportation Network Used in Example 2

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.

Determination of optimal toll levels and locations 23

Maximal gross social welfare (10 5HK$)

1.575

E 1.570

D

1.565 1.560 1.555

C

1.550 Max social welfare for given toll link number

B

1.545

Max social welfare in system optimum Convex curve connecting ciritical points

1.540

A 1.535 0

1

2

3 4 5 6 Total number of toll links

7

8

9

10

FIGURE 3 The Maximal Social Welfare for a Given Number of Toll Links

TRB 2003 Annual Meeting CD-ROM

Paper revised from original submittal.