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.
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].
p
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.
n
p
Thus, the firm chooses its quantity
of production i with the price i
q of the production
to maximize its profit)
ii
(12 )ith in 
p (
F
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
i
q()
ii
g
q.
Additionally, the firm’s cost function is . Thus,
the firm’s profit function is:
iiiii
i()
ii
cq
ith
(i )() ()
i i
F
pqpqqc qpg

 
max ()
ii iii
q
. So, the firm aims
to maximize its profit, that is, ith
)
ii
( )
ii (
F
pqpqpgqc q


i
ith p
p
1
n
i
q


1
min n
pi
Cg




()



. (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.
p
C
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:
()
ii
gq
p
p

ii
q

. (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
nn
1
() (
()
i
i ii
n
ii
i
(
ii
F
pqpqcq
Cgq






()
p 

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:

)
i
ii11
nn
1
( )(
()
ii i
n
ii
i
max
p( )
i i
F
pqpqc q
Cgq






p 

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

1
1
max()( )( )
().
n
ii iiii
pii
n
ii
i
1
n
F
pqpqc qpgq
Cgq



 








(5)
where solves the following problem
i
q
max()( )()
iiiiiii ii
q
F
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:
p
0ii
i
ii
i
i
F
gc
pp
qq

 
q


n
(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 ()
ii
cq
()
ii
g
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
ii
p
p
)
ith
()
ii
cq
(
g
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:
p

1
()
n
ii
i
qgq

i
q
p
p
11
1
nn
iii i
i
ii
ii
nii
iii
qcq gq Fpp
ppqpq
gq
C
gqp



0
i
p

  
 

 







(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).
p
i
q
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
1
p
1
q
pq 10

and 28p

, respectively. Then, the firms’
production conditions are assumed as follows: 11
()gq

, 22 2
2
1
2q()5
g
qq

, 11 1 , . The cost
function of abating the pollution is Cq ,
where
() 3cq q
(
q

)
3
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:
2
111111111
)() ()1023
11
(
F
p q
22
(
p qpgqcqqpq  q
3
2
,
22 222222
)() ()85
F
p qp qpgqcqqpqq

 
p
.
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:
1
pq10 430

 and 2
2
85 30pq

. Thus, we have
17
4
qp  (8)
and
2
85
3
p
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
p
p
22
12
58 5p
1
25 3
8
qqqp

 . And the government’s
objective is formulated as follows:

22
11
2
1
32
11221 2
2
12
()() ()
()
(103)(8)(25)
[10002(25)]
ii iiii
ii
ii
i
Fpqpq cqpgq
Cgq
qqqqpq q
qq










 


According to the Equation (7), the optimal price is
determined by the following equation: p
(
0)
F
pq
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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