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dx.doi.org/10.22093/wwj.2017.58676.2231 118 Journal of Water and Wastewater Vol. 29, No. 4, 2018 Leader- Follower and Nash Bargaining Game Theory Models for Optimum Waste Load Allocation, Gheshlagh River as Case Study M. Saadatpour 1 , H. Khoshkam 2 Assist Prof, School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran (Corresponding Author) msaadatpour@iustacir 2 MSc Graduated Student of Civil and Environmental Engineering, College of Environment, Karaj, Alborz, Iran (Received July 31, 2016 Accepted Jan. 1, 2017) To cite this article : Saadatpour, M., Khoshkam, H., 2018, “Leader-follower and nash bargaining game theory models for optimum waste load allocation, gheshlagh river as case study.” Journal of Water and Wastewater, 29 (4), 118-131. Doi: 10.22093/wwj. 2017.58676.2231 (In Persian) Abstract Waste load allocation (WLA) is one of the most important elements when evaluating in water quality management problems. Due to multiple and sometimes conflict objectives in WLA problems, a set of Pareto optimal solutions is derived with evolutionary algorithms in which one of these Pareto fronts could be influenced by conflicts. In this research study, simulation-optimization approach was applied by QUAL2Kw simulation model and particle swarm optimization (PSO) as optimization algorithm to assign cBOD point source pollutions for specific location along Gheshlagh River. To reduce the conflicts between beneficiaries for the optimum operation of water resources in river, the level leader-follower and the Nash bargaining game theory models were applied. The results showed that the construction, maintenance and operation costs of the treatment plants for leader-follower and Nash bargaining game theories were about 192 and 293 billion Rial, respectively. The penalties for violating the environmental regulations set by the Iranian environmental protection agency (EPA) for the above theory models were found to be about 32 and 3.9 billion Rial, respectively. Furthermore, the estimated penalty tariff for each overdischarge of allowed cBOD under Stackelberg and Nash bargaining game theories were about 10.8 and 3 Rial, per environmental violation, respectively. The estimated penalty tariff in Stackelberg game is extremely close to current Iran’s EPA penalty tariff. Keywords: Particle Swarm Optimization Algorithm, Leader-Follower Game, Nash Bargaining Game, Optimum Waste Load Allocation, QUAL2Kw Model.
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Page 1: dx.doi.org/10.22093/wwj.2017.58676.2231 118 Leader ... · dx.doi.org/10.22093/wwj.2017.58676.2231 118 Journal of Water and Wastewater Vol. 29, No. 4, 2018 ˘ˇ ˆ Leader- Follower

dx.doi.org/10.22093/wwj.2017.58676.2231 118

����� ��� �� Journal of Water and Wastewater ��� �� ���� ����� ����� Vol. 29, No. 4, 2018

Leader- Follower and Nash Bargaining Game Theory

Models for Optimum Waste Load Allocation, Gheshlagh River as Case Study

M. Saadatpour1, H. Khoshkam2

1.Assist.Prof.,SchoolofCivilEngineering,IranUniversityofScienceandTechnology,Tehran,Iran(CorrespondingAuthor) [email protected]

2. MScGraduatedStudentofCivilandEnvironmentalEngineering,CollegeofEnvironment,Karaj,Alborz,Iran

(Received July 31, 2016 Accepted Jan. 1, 2017)

To cite this article : Saadatpour, M., Khoshkam, H., 2018, “Leader-follower and nash bargaining game theory models for optimum waste load

allocation, gheshlagh river as case study.” Journal of Water and Wastewater, 29 (4), 118-131. Doi: 10.22093/wwj. 2017.58676.2231 (In Persian)

Abstract Waste load allocation (WLA) is one of the most important elements when evaluating in water quality management problems. Due to multiple and sometimes conflict objectives in WLA problems, a set of Pareto optimal solutions is derived with evolutionary algorithms in which one of these Pareto fronts could be influenced by conflicts. In this research study, simulation-optimization approach was applied by QUAL2Kw simulation model and particle swarm optimization (PSO) as optimization algorithm to assign cBOD point source pollutions for specific location along Gheshlagh River. To reduce the conflicts between beneficiaries for the optimum operation of water resources in river, the level leader-follower and the Nash bargaining game theory models were applied. The results showed that the construction, maintenance and operation costs of the treatment plants for leader-follower and Nash bargaining game theories were about 192 and 293 billion Rial, respectively. The penalties for violating the environmental regulations set by the Iranian environmental protection agency (EPA) for the above theory models were found to be about 32 and 3.9 billion Rial, respectively. Furthermore, the estimated penalty tariff for each overdischarge of allowed cBOD under Stackelberg and Nash bargaining game theories were about 10.8 and 3 Rial, per environmental violation, respectively. The estimated penalty tariff in Stackelberg game is extremely close to current Iran’s EPA penalty tariff.

Keywords: Particle Swarm Optimization Algorithm, Leader-Follower Game, Nash Bargaining Game, Optimum Waste Load Allocation, QUAL2Kw Model.

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Discharge Loading rate

(g/s)Flow

Rate(m3/s)cBOD (mg/L)

Distance from Downstream

(km) Reach

Number Name of Discharger Unit

20.80.120041 1Nanleh Sewage9.130.0180033 3Industrial Park2.570.00390028 3Sanandaj Livestock Slaughter0.20.0045025 4Fajr Concrete Foundation

136.631.497.9423 4Effluent of Wastewater Treatment Plant 0.140.0034021 5Asphalt Production and Recycle

61.640.032054.7919 5Landfill Leachate1.660.002698.6314 6Par Chicken Slaughter

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Side Slope Bed Slope

River’s bed Width

(m)

Manning Roughness Coefficent

Elevation Above Sea

Level (m)

Distance from Downstream

(Km) Length (Km)

Reach Name

Reach Number

0.030.00340.09147949.45Headwater00.02 0.003 5.5 0.09 1447 39 10.45 Gheshlagh

Dam1

0.025 0.007 7.5 0.08 1419 354Salavat-Abad2

0.020.005100.06139028.566.35Baharan30.015 0.005 100.055 1362 22.95 5.7 Sanandaj

Sewage4

0.01 0.001 12.5 0.055 1357 17.55 5.4 Landfill Leachate5

0.01 0.005 12.5 0.05 1332 12.05 5.5 Par Chicken Slaughter6

0.0050.006100.0513118.653.4Sou Junction70.005 0.003 100.045 1293 3.25 5.4 Darvishan

Junction8

0.01 0.021 10 0.05 1226 03.25 Gavroud Junction9

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Fig. 4. Removal rates of cBOD at different discharge units under two different game theories

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methods Number of discharger

Game theory method 1 2 3 4 5 6 7 8 Total cost

(billion Rial)

Nash bargaining 3.99 6.03 3.22 0.09 188.33 0.03 90.79 0.91 293.39 Stackelberg 14.36 2.08 1.37 0.17 123.82 0.03 48.09 1.65 191.57

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Number of discharger

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Fig. 6. Fines paid by violating cBOD discharge load at different discharge unit under two different game

theories QR�{=H @b� ��' ]�% +��� �5*��3 ��&(�@/� �� �'�

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�5$ :��� � .���& �� nB� 6nW�] ��� H n, �n5�� nN$ �� �n��������ni5$� �� B� �� |����5��$�n@���ny�\] �����n� +n���?��+��G$ ��(��� �� .�����n$�j n$���n� DRn$ ��n��%�% 6

�� ���G�'���� ��' ]�% D��2�b�&n� H n@�n� a� n] �n� n@'�\[� ��/�����n$�* �n'����n� �n�����n�� >Cn]�cBOD �5n)�� ��$ �� nn]�G' % nn@�nn@ �nn'��5nn)�� �� nnY2$ D�� nn]� �nn����%

HY� �����\[� �����$�* �� �'� �� �� $����� <�Uh�n& 6n�?5� L5����i� ����cBOD n� �nN�Z �n���� RJK��n��"n� � ���n5 $�,@T�' .�� �� <��� <�& �� B� Dn5�� �n�������n� +n���

?� �y�\]�+��� ��z����n, �n� �n$�� ��Ln�� <�n1 �n�

�' nn]�% +nn��� Uh�nn& �nn*��3 6nn?� �� nn�����b�� DH nn@nn <�' ]�%�5�@W�%Y,�$�G' 6�?5� G�@�'�n5��� $�n, .�n����� ����� D|����5�� ���n?� �ny�\] �����n� RB$ 6�� +�n�

�� �' ]�% �� F�� ] ���� L�� D�'� ���@7�b��n� zn�i5� H @� Uh�& <5�� N$ ���� �&�� ���� D5)�5)�R'�� �j�� .���

G'�nn@ �nn'�HnnY� D�� nn]� ���nn���� nnY2$ %���\nn[����nn$�* % �nn'G'�@�'�$ z�i� Uh�&��� G! ���n@7 n��' ����n&�� ���n� DU5n�H��� .���

Fig. 7. cBOD fine unit value under two different game theories compared with with Iran EPA value

QR��=(�@/� �� ��& ���� ��B� % ���� z�5i� ��'

�$�@� +�, �� ���B�

��nnB�G' �nn@ �nn'�HnnY� �nn��� H nn, �nn*��3 ���nn����\[����$�* 5*��3 Uh�& % �'�n?� �ny�\] �����n� �n��+��n�

' +����' ]�% �� (�b�/\� �� H @��� (�b� ]�% 'H @�� �i)� +�?� ������ .��, ���i� ��� ��h�& E��7� �� ���

�� R��JK�n��� n5�� �� "n�cBOD % ����nN$ RnB$ E�n�7� �n� D�� ��* ��5@S E% &) ����3J.(@T�'��' ���� D<�H n, L��

+�?� �y�\] ������ +��� �'�/%� �� ( ' �� G�$ ����

��-9 m=�@G' ���B� �'��' ]�% �5*��3���� +�?� ������ ���� % H @b� Table 5. Fine paid for over discharging cBOD at each discharge unit and the total income gained by Iran EPA under

two different game theories Number of discharger

Game theory method

Costs paid by Every cBOD Discharger Unit (billion Rial) Environmental Protection Agency’s income (billion Rial)

1 2 3 4 5 6 7 8

Nash bargaining 5.09 6.47 3.33 0.09 188.33 0.03 92.93 0.99 3.89 Stackelberg 17.53 4.81 2.19 0.17 131.75 0.03 65.03 1.92 31.86

0

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1 2 3 4 5 6 7 8

Fine

Paie

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arge

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OD

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No. of cBOD Discharge Unit

Nash Bargaining Stackelberg

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Nash Bargaining Stackelberg Iran EPA

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Uni

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