Continuous network design with emission pricing

as a bi-level optimization problem

Tom V. Mathew1 and Sushant Sharma2

1 Assistant Professor, Department of Civil Engineering, Indian Institute of Technology

Bombay, India; PH: (9122) 2257 7349; FAX: (9122) 2257 7302; email: vmtom@iitb.ac.in2 Research Scholar, Department of Civil Engineering, Indian Institute of Technology

Bombay, India; PH: (9122) 2576 7349; email:sushantsharma@iitb.ac.in

1. Abstract

Traffic network design problem attempts to find the optimal network expansion poli-

cies under budget constraints. This can be formulated as a bi-level optimization prob-

lem: the upper level determines the optimal link capacity expansion vector and the

lower level determines the link flows subject to user equilibrium conditions. However,

in the context of environmental concerns, drivers route choice includes travel time as

well as emission pricing. This study is an attempt to solve network design problem

when the user is environment cautious. The problem is formulated as a bi-level con-

tinuous network design problem with the upper level problem determines the optimal

link capacity expansions subject to user travel behavior. This behavior is represented

in the lower level using the classical Wardropian user equilibrium principles. The up-

per level problem is an example of system optimum assignment and can be solved

using any efficient optimization algorithms. Genetic Algorithm is used because of its

modeling simplicity. The upper level will give a trial capacity expansion vector and

will be translated into new network capacities. This then invokes the lower level with

these new link capacities and the output is a vector of link flows which are passed to

upper level. The upper level then computes the objective function and GA operators

are applied to get a new capacity expansion vector and the process is repeated till con-

vergence. The model is first applied to an example network and the optimum results

are shown. Finally the model is applied to a large case study network and the results

are presented.

2. Introduction

The foremost option that strikes transportation engineers and planners, alike, while

considering the growth in traffic demand is to expand the capacity of existing congested

links or build new links. In such cases, selecting new links and adding capacity to

existing links becomes an interesting problem. The planner has to make decision while

considering, on the one hand, minimization of total system cost under limited budget

and, on the other hand, behavior of the road users. This problem is stated as network

design problem (NDP). The objective of NDP is to achieve a system optimal solution

by choosing optimal decision variables in terms of link improvements. The decision

taken by the planner affects the route choice behavior of road users and need to be

considered while assigning traffic to the network. Network design models concerned

with adding indivisible facilities (for example a lane addition) are said to be discrete

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UsuarioResaltado

UsuarioNota adhesivay el nivel inferior determina el flujo de enlace sujeto a condiciones de equilibrio usuario. Sin embargo, en el contexto de las preocupaciones ambientales, la eleccin de ruta del conductor incluye el tiempo de viaje, as como los precios de emisin. Este estudio es un intento de resolver el problema de diseo de la red cuando el usuario es cauteloso del medio ambiente.

NDP, whereas those dealing with divisible capacity enhancements (for example road

widening) are said to be continuous NDP. Among the several route choice models, the

most popular one is based on Wardrops first principle and can be formulated as an

equivalent minimization problem.

This equilibrium network design problem can be formulated as bi-level program-

ming problem. At the upper level, the total system travel time, subjected to the con-

struction cost available for the link capacity expansion is minimized. At the lower

level, the user equilibrium flow is determined using Frank Wolfe algorithm. Although,

there are few attempts to implement these models, a case study illustrating the appli-

cation of the model to a large network is absent. Therefore, this study is an attempt to

find optimal capacity expansion for a large city network

3. Literature Review

The first discrete NDP was developed by LeBlanc (1975) and later it was extended

to the continuous version (LeBlanc and Abdulaal, 1979). Hook-Jeeves pattern search

algorithm (HJ Algorithm) was used for solving these models. Another kind of network

design problem was proposed by Yan and Lam (1996) for optimizing road tolls under

condition of queuing and congestion. Optimization of reserve capacity of whole sig-

nal controlled network as bi-level programming problem in network design was first

time attempted by Wong (1997). Meng et al. (2001) have proposed an equivalent sin-

gle level continuously differentiable optimization model for the conventional bi-level

continuous network design problem. Ziyou and Yifan (2002) combined the concept

of reserve capacity with continuous equilibrium network problem. Chen and Yang

(2004) came out with two models that considered spatial equity and demand uncer-

tainty in the network design problem. Both models were solved by a simulation based

genetic algorithm. Meng et al. (2004) did a comprehensive study of static transporta-

tion network optimization problems with stochastic user equilibrium constraints.

Genetic algorithm (GA) based approach for optimizing toll and maximizing reserve

capacity was attempted by Yin (2000). They summarized the advantages of using

GA based approach as efficient and simple. Ceylan and Bell (2004) used GA-based

approach for optimizing traffic signal, while including drivers routing. The major

drawback of GA-based model is the expensive fitness evaluation in each generation for

the large route network design problem, which is combinatorial in nature (Aggarwal

and Mathew, 2004).

To consider the environmental parameters in network design problems various

studies have been carried out with traffic assignment like a multi-objective decision

model with system optimum conditions (Tzeng and Chen, 1993). A new kind of as-

signment called system equitable traffic assignment was attempted taking generalized

environmental cost function (Bendek and Rilett, 1998). Nagurney (2000) considered a

multi-criteria traffic network model with emission terms in objective function.

4. Model Formulation

Determining the optimal link capacity expansions is formulated as a bi-level continu-

ous network design problem with the upper level problem minimizes the system travel

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cost subject to users travel behavior. This behavior is represented in the lower level

using the Wardropian users equilibrium principles. However, the link cost function is

modified to represent both the link travel time as well as the environmental concerns.

The upper level problem is an example of system optimum assignment and can be

solved using any efficient algorithms. Although the link cost function is modified, it is

still convex and is therefore solved using Frank-Wolfe algorithm.

The following notation has been used for continuous NDP formulation: A is the

set of links in the network, is the set of OD pairs, q is the vector of fixed OD pair

demands, qrs q is the demand from node r to node s , R is the set of paths between

OD pair r and s, f the vector of path flows, f = [f rsk ], x the vector equilibrium link

flows, x = [xa], y is the vector of link capacity expansions, y = [ya], B the allocated

Budget for expansion, conversion factor from emission to travel cost, ea the emission

at link a, the link-path incidence matrix, = [rsa,k]. The bi-level problem can bemathematically represented as below:

Upper Level

Minimize Zy =

a xata(xa, ya) (1)

subject toa aya B; ya 0 : a A

Lower Level

Minimize Zx =

a

xa0

ttwa (xa) + cea(xa) dx (2)

subject tok f

rsk = qrs : k K; r, s q;

xa =

r

s

k

rsa,kf

rsk : r, s q; a A; k K

f rsk 0 : r, s q; k K

xa 0 : a A

5. Solution Approach

A flowchart of the solution approach is given in Fig. 1. The algorithm starts in the

upper level by reading all the inputs like network details, demand matrix, budget, link

expansion cost functions, travel time function, and emission cost functions. The upper

level algorithm will give a trial capacity expansion vector and will be translated into

new network capacities. The lower level algorithm is then invoked with these new link

capacities. In this level, the demand matrix is assigned into the network considering

both travel time function and emission cost function. The output of this model is a

vector of link flows which is passed to the upper level. The upper level then computes

the objective function which is the system travel time from link flows and assigns

penalty if the budget constraint is violated. This objective function value is given

to the upper level algorithm which supplies a new capacity expansion vector and the

process is repeated till convergence.

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UsuarioResaltado

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The upper level problem is solved using genetic algorithmwhich is an iterative pro-

cedure that borrows the ideas of natural selection and can easily solve many complex

problems.

Y

X

Y

Traffic Assignment

(Lower Level)

Flows

Cost of Construction

Converged

Yes No

Stop

Link Capacity

Minimize System

Network Data

Investment Function

Convergence Criteria

OD Matrix

Emission Function

User Equilibrium

Cost

Upper Level

GA

Operations

Travel Time Function

Figure 1: Flowchart representing solution approach

The emission function has been incorporated along with the travel time by convert-

ing that into equivalent cost. It is supposed that the user is being charged for producing

the emissions in the network. The amount of pollution on link a is given by

ea = xaefda (3)

where xa is the traffic flow on link a, ea is pollution by vehicles on link a in grams (gm),

ef is emission factor (gm/km), da is length of link a (km). The emission generated is

multiplied by c which is a factor for converting emission into cost.

6. Motivating example

1

2

4

3

k = 30

k=20

k =20

x4

x3x1

k = 20+z1 x2 x5

d14 = 50

d34 = 80

= 4

= .15

k = 30+z2 tf = 1

= 0:004 0:3

t0 = tf

h1 + (x

k)i+ Ea

Figure 2: Bi-level example network

Pune

Figure 3: Pune city network

To appreciate the working of the bi-level problem, an example network having 4

nodes and 5 links (Fig. 2) is solved. The two OD pairs involved are q34 and q14 and

has 50 and 80 trips respectively. The budget is taken as 10 units. The link expansion

vector is initialized to 0. User equilibrium is performed to get the required link flows

x1*, x2*, x3*, x4*, x5*. Now these flows are input to upper level from where we get

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UsuarioResaltado

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UsuarioNota adhesivaLa funcin de emisin se ha incorporado junto con el tiempo de viaje conviertiendo eso en costo equivalente. Se supone que el usuario est siendo acusado por producir las emisiones en la red. La cantidad de contaminacin en el enlace a est dada por

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Table 1: Example network results: link flow values, capacity expansion vectors, and

total system travel time (TSTT) for six iterations

No x0* x1* x2* x3* x4* UE TSTT y1* y2* SO

1 42.92 37.96 54.76 11.84 76.68 405.47 846.01 4.56 5.43 785.93

2 40.93 39.81 56.93 16.00 74.27 391.06 764.72 4.68 5.31 764.71

3 40.88 39.85 56.87 15.99 74.33 391.05 764.62 4.72 5.28 764.62

4 40.87 39.86 56.85 15.97 74.35 391.06 764.66 4.73 5.27 764.66

5 40.86 39.88 56.85 15.99 74.35 391.05 764.57 4.73 5.27 764.57

6 40.86 39.88 56.85 15.99 74.35 391.05 764.57 4.73 5.27 764.57

a new set of link expansion vectors y1*, y2* which minimizes the system travel time.

Same values of y1*, y2* were used as input for new run of User equilibrium. This

iteration is repeated until the total system travel time from lower level and upper level

converges. The results are tabulated in Table 1. The convergence of the solution (y1,

y2) to the optimum value illustrates the working of the model.

7. Case Study

The network of Pune city was considered for the case study. This city has an area of

138 square kilometers. The city is divided into 97 zones out of which 85 are internal

12 are external zones. There are 273 road nodes 1131 road links, including walking

and non walking links. Fig. 3 shows the network of the city.

Various traffic flow characteristics like , , free flow speed and running speed

for all the links were found after surveying. Link characteristics were collected and

trip table was generated for the whole network after validating the OD counts. The

emission factor for all the vehicle has been supposed as 25gm/Km. The emission which

stays in and causes pollution is supposed to be 30% of whole pollution generated. Thisis multiplied by a factor which gives the external cost of emission in environment

which is 0.004 Rs/gm. In order to study the network expansion, a budget of Rs. 600

crores and a maximum expansion of 100% is considered.

8. Conclusion

Finding optimal network capacity expansion where the user is environment conscious

for a transport network is attempted in this study. The problem is formulated as a

continuous bi-level optimization problem: the upper level solves the problem of finding

the optimal network expansion values and the lower level represents the user travel

behavior. The former is solved using GA while Frank-Wolfe algorithm is used in

the later case. An example network is used to illustrate the working of the model.

A large case study network is solved which demonstrates the ability of the solution

methodology to handle large problem. Further study is needed to incorporate a better

representation of environmental concerns.

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