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)

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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