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
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Dam1
0.025 0.007 7.5 0.08 1419 354Salavat-Abad2
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Sewage4
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0.0050.006100.0513118.653.4Sou Junction70.005 0.003 100.045 1293 3.25 5.4 Darvishan
Junction8
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Fig. 4. Removal rates of cBOD at different discharge units under two different game theories
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Fig. 5. Construction, maintenance, and operation costs of cBOD at discharge units under two different game
theories QR�_=�@G' �'��� Y2$ % �� ]����\[� (�@/� �� �$�* �'�
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��-9�=�@G' �'��� Y2$ % �� ]����\[� (�@/� �� �$�* �'�z�5i��Y��� ����� Table 3. Construction, maintenance, and operation costs at different discharge units for two different game theory
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
��-9�=H @b� ��' ]�% +��� �5*��3 ��&(�@/� �� �'����� z�5i�������'Table 4. Fine paid by cBOD discharge units in various game theory approaches
Number of discharger
Game theory method 1 2 3 4 5 6 7 8Total cost (billion Rial)
Nash bargaining 1.11 0.44 0.11 0.00 0.00 0.00 2.15 0.08 3.89 Stackelberg 3.18 2.73 0.82 0.00 7.93 0.00 16.94 0.27 31.86
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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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Fig. 7. cBOD fine unit value under two different game theories compared with with Iran EPA value
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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
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