An Application of Bilevel Programming Problem in Optimal Pollution Emission Price

doi:10.4236/jssm.2011.43039

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Journal of Service Science and Management, 2011, 4, 334-338

doi:10.4236/jssm.2011.43039 Published Online September 2011 (http://www.SciRP.org/journal/jssm)

An Application of Bilevel Programming Problem

in Optimal Pollution Emission Price

Guang-Min Wang1,2, Lin-Mao Ma1, Lan-Lan Li1

1School of Economics and Management, China University of Geosciences, Wuhan, China; 2Hubei Province Key Laboratory of

Systems Science in Metallurgic a l Proce ss, Wuhan University of Science and Technology, Wuhan, China.

Email: [email protected]

Received February 25th, 2011; revised April 6th, 2011; accepted April 24th, 2011.

ABSTRACT

Charging for the pollu tion is one of the ways to enhance the environm ental quality. The appropriate price of the po llu-

tion emission is the most important question of the research on how to charge for the pollution. So, by constructing a

bilevel programming model, we provide a novel way for solving the problem of charging for the pollution. In our model,

the government (or the social regulation) chooses the optimal price of the pollution emission with consideration to

firms’ response to the price. And the firms choose their optim al quantities of the production to maximize their profits at

the given price of the pollution emission. Finally, a simple example is illustrated to demonstrate the feasibility of the

proposed model.

Keywords: Bilevel Programming, Pollution Emission, Price Control Problem

1. Introduction

Rapid economic development and population growth in

China have left a legacy of widespread environmental

pollution in the last two decades [1,2]. So, the research

on environmental pollution is very important to enhance

the environmental quality [3,4]. Now, three basic ways,

such as regulation, Pigovian tax and transaction of

emission permits, were used to abate the environmental

pollution in developed countries. Because the firms’

marginal costs are less than the social marginal cost, the

firms will emit excess pollution, which shows that the

firms will have negative extern ality to the environmental

quality. For effectively dealing with the externality,

which can not be solved by the market, the government

regulation is adopted by prescribing the maximal quantity

of the pollution emission, which is an administrative

meaning to abate the environmental pollution. While the

asymmetry information makes it hard to reach the ideal

goal. So, Pigovian tax is adopted by imposing tax on the

pollution to make the externality cost internal and give

the firms an incentive to decrease the quantities of the

pollution emission, which is an economic meaning to

abate the environmental pollution. Based on the idea of

making the environmental externality cost internal,

transaction of emission permits corrects the distortion of

the market’s price resulted from the exposure of Pigovian

tax to effectively abate the environmental pollution by

use of definition of initial emission rights and the

allocation market of initial emission rights as well as the

trading market of emission rights. Now, China is

experiencing an unprecedented discharge of pollutants

within a relatively short time compared with developed

nations, in which discharges were spread over a century

or more [1,5]. Thus, the research on how to charge for

pollution is one of the most important work to enhance

the environmental quality. Among various factors, the

price of the emission permits must be the first place,

because it influences not only the environmental quality

but also the allocation of natural resources and supply

and demand of commodity. Thus, the scientific and

reasonable price is the key to perform system of charging

for pollution successfully [6].

Many authors have attempted to use techniques in the

pollution abatement problem [7,8]. Additionally, a

serious shortcoming of these optimization models is that

complete information on th e production and damage cost

functions of every firm is assumed to be known.

Although, each firm may know its own production cost

functions, there is no reason to believe that this informa-

tion will be readily available to the central authority [9].

Furthermore, Amouzegar and Jacobsen have conceptuali-

zed the problem in terms of a multilevel frame work [10].

An Application of Bilevel Programming Problem in Optimal Pollution Emission Price335

Later, Amouzegar and Moshirvaziri presented two opti-

mization models for hazardous waste capacity planning

and treatment facility locations by investigating the

complex behavior of firms in the presence of central

planning decisions and price signals which can best be

captured by a bilevel programming model [9].

In this paper, we propose a bilevel programming

model different from the above bilevel models to abate

the environmental pollution, in which the government(or

the social government) chooses the price of the pollution

emission to maximize the social profits by considering

the firms’ response to the price, and then the firms

maximize their profits by choosing the optimal quantity

of the production at the given price. Our model aims to

discuss not only the scientific an d reasonable price of the

pollution emission to maximize the social profits but also

the firms’ choosing the optimal quantity of the production

at the given price to maximize their profits. The remai-

ning of the paper is organized as follows: Section 2

presents a bilevel programming model to determine the

price of emission permits; Section 3 gives the algorithm

for this bilevel programming model; a computational

example is presented in Section 4 to demonstrate the

feasibility of the model; finally, a conclusion and future

work are given in Section 5.

2. The Bilevel Programming Model

If the government (or the social regulation) chooses the

price of the pollution emission, then each firm will

be in response to the price, and then, the government will

adjust repeatedly the price according to the response of

the firms until the government obtains the optimal price

of pollution emission to maximize the social profits

while each firm gain its maximizing profit at the given

price. It can be seen that this process is the decision

problem with hierarchical structure and the bilevel pro-

gramming problem is a useful tool to solve this kind of

problem [11].

Next, we will give some assumption before construct-

ing the model. Supposing that there are firms to pro-

duce different productions and emit the same pollution.

For simplicity, each firm only produces one kind of pro-

duction. And the quantity of the pollution is only deter-

mined by the quantity of the production. Then, the charge

of the per unit pollution emission is the same for differ-

ent firms, and let denote the charge of the per unit

pollution emission.

Thus, the firm chooses its quantity

of production i with the price i

q of the production

to maximize its profit)

(12 )ith in 

p (

pq as the pr ice of pollution

emission is set by the government. And let the quantity

of the pollution be the function of the quantity of produc-

tion , that is, the quantity of the pollution is

q()

Additionally, the firm’s cost function is . Thus,

the firm’s profit function is:

iiiii

i()

ith

)() ()

i i

pqpqqc qpg



 

max ()

ii iii

. So, the firm aims

to maximize its profit, that is, ith

)

( )

ii (

pqpqpgqc q





ith p





min n









()







. (1)

where is the price of the production manufactured

by the firm, and i is fixed because the firms are

all the price acceptors in the competitive mar ket.

Thus, at the given price of the pollution emission,

the total of the pollution is . Obviously, the

marginal cost of abating th e pollu tion is decreasing as th e

quantity of the pollution, that is, abating pollution is

economy of scale. Thus, we consider the situation that

the pollutions are all abated by the government. Hu

Zhenpeng et al. [12] determined the optimal price by

minimizing th e cost of abating the pollution, namely, the

government (or the social regulation) chooses to

minimize his objective function formulated as follows:

()





. (2)

where



, which is the increasing function, is the cost

function of abating the pollution. However, only to con-

sider minimizing the cost of abating the pollution is not

complete, because the pollution is the logical result of

manufacturing the production for better our lives. Thus,

we should consider maximizing the social profits at the

same time consider minimizing the cost of abating the

pollution. Hence, we treat the cost of abating the pollu-

tion as the cost of manufacturing social production. So,

the government's objective function is formulated as fol-

lows:



) )

ii11

() (

()

i ii

(

pqpqcq

Cgq













()





g q













(3)

where C



, which is the increasing function, is the cost

function of abating the pollution. Thus, the government

aims to maximize his objective function formulated as

follows:



)

ii11

( )(

()

ii i

max

p( )

i i

pqpqc q

Cgq

















g q













(4)

Hence, we can propose the programming model for-

mulated as follows:

An Application of Bilevel Programming Problem in Optimal Pollution Emission Price

336



max()( )( )

().

ii iiii

pii

pqpqc qpgq

Cgq





 

















(5)

where solves the following problem

max()( )()

iiiiiii ii

pqpqpgqc q .

where . Obviously, the model is a bilevel

programming problem. Next, we will discuss the algo-

rithm for this model.

12i n

3. The Proposed Algorithm for the Model

Although Bracken and McGill [13] gave the original

formulation for bilevel programming in 1973, the prob-

lem started receiving the attention motivated by the game

theory [14] till the early eighties. And many authors stud-

ied bilevel programming intensively and contributed

themselves into those fields [15-18]. However, the bilevel

programmin g is neither contin uous anywhere nor convex

even if the objective functions of the upper level and

lower level and the constraints are all linear because the

objective function of the upper level, which, generally

speaking, is neither linear nor differentiable, is decided

by the solution function of the lower level problem. Bard

proved that the bilevel linear programming is a NP-Hard

problem [19] and even it is a NP-Hard problem to search

for the locally optimal solution of the bilevel linear pro-

gramming [20]. So, it is greatly difficult to solve the

bilevel programming for its non-convexity and non-con-

tinuity. When , the problem (5) is similar with the

price control problem, which has been researched by

some authors with hypothesis that there only one solution

to the lower level programming for fixed the upper level

decision variable [21-24]. Recently, Yibing Lv et al. dis-

cussed a class of weak price control problems with non-

unique lower level solutions and study the existence of

solution via a penalty method [25]. In this paper, we dis-

cuss the situation that there are firms based on

above referenc es.

1n

(2)n

After the government chooses the price of the

pollution mission, the firms choose their quantity of the

production to maximize their profits, and the optimal

quantity of production is determined by the following

equation:

0ii



 

q



(6)

where . From the Equation (6), we can see

that the firm’s optimal quantity of production is

determined by not only the firm’s cost function

and polluting function

1 2i 

ith ()

()

q but also the price of

the pollution emission. In fact, it is the function of the

price of the pollution emission because the

firm’s cost function and polluting function

)

ith

()

(

q are all changeless because its production condi-

tions are changeless in a relatively short period.

Thus, at the given price of the pollution emission,

the total of the pollution is , where , the

response to , is determined by the Equation (6). So the

optimal price of the pollution emission is determined

by the following equation:



()

qgq



i

iii i

nii

iii

qcq gq

Fpp

ppqpq

gqp









 



 



 















(7)

The optimal price can be obtained by solving the

Equation (7) with the Equation (6), and then the optimal

quantities of production is computed according to

the Equation (6).

4. Experiment

In this section, we will illustrative a simple example to

demonstrate the feasibility of our model.

Example 4.1. There are two firms to produce different

production and emit the same pollution while the gov-

ernment chooses the price of the pollution emission.

Supposing the two firms’ quantity of the production are

and 2 and the prices of the productions are



and 28p



, respectively. Then, the firms’

production conditions are assumed as follows: 11

()gq



, 22 2

2q()5



, 11 1

, . The cost

function of abating the pollution is Cq ,

where

() 3cq q

(



)

22 2

()cqq

( )10002q

112 2

()q gqg



.

According to the assumption in the example (4.1), we

can easily get the two firms’ profit function formulated

as follows:

111111111

)() ()1023

(

p q

(

p qpgqcqqpq  q

22 222222

)() ()85

p qp qpgqcqqpqq



 

According to the Equation (6), at the given price

of the pollution emission, the firms’ optimal quantities of

the production are determined by the following equations:

pq10 430



 and 2

85 30pq



. Thus, we have

 (8)

and

q

. (9)

An Application of Bilevel Programming Problem in Optimal Pollution Emission Price337

which show that the firms’ optimal quantities of the pro-

duction are involved with the price of the pollution

emission , and increas e when decreases. This accords

with the real situations. So, the total of the pollution is

58 5p

25 3

qqqp



 . And the government’s

objective is formulated as follows:



11221 2

()() ()

()

(103)(8)(25)

[10002(25)]

ii iiii

Fpqpq cqpgq

Cgq

qqqqpq q





















 





According to the Equation (7), the optimal price is

determined by the following equation: p

(



. (10)

Thus, the optimal price 0 9280p



 is obtained by

solving the Equation (10) with the Equations (8) and (9).

Following, the optimal quantities 1 and

2 of the two firms’ are obtained by the Equa-

tions (8) and (9).

1886q

0 611q

From the simple example, we aim to reveal how the

government (or the social government) chooses the op-

timal price of the pollution emission to maximize the

social profits by considering the firms’ response to the

price, and how the firms determine the optimal quantity

of the production at the given price to maximize their

profits.

5. Conclusions and Future Work

In this paper a bilevel programming problem is proposed

to determine the optimal price of the pollution emission,

which is a novel way to discuss this problem. And an

example is solved to illustrate the feasibility of the model,

which can provide some consultations for the deci-

sion-makers. In the future, there are more researches to

do, such as considering that there are more than one kind

of production and pollution emissions and so on, so that

more real problems are solved to ab ate the env ironmental

pollution.

6. Acknowledgements

The authors would like to thank the anonymous editors

and reviewers for their useful comments and suggestions.

And the work is supported by the Social Science Foun-

dation of Ministry of Edu cation (No. 10YJC630233) and

Hubei Province Key Laboratory of Systems Science in

Metallurgical Process (Wuhan University of Science and

Technology) (No. B201003).

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