Water Resources Management

, Volume 26, Issue 15, pp 4539–4552 | Cite as

Equitable Waste Load Allocation in Rivers Using Fuzzy Bi-matrix Games

  • Mohammad Reza NikooEmail author
  • Reza Kerachian
  • Mohammad Hossein Niksokhan


This paper presents a new game theoretic methodology for equitable waste load allocation in rivers utilizing fuzzy bi-matrix games, Non-dominated Sorting Genetic Algorithms II (NSGA-II), cooperative game theory, Bayesian Networks (BNs) and Probabilistic Support Vector Machines (PSVMs). In this methodology, at first, a trade-off curve between objectives, which are average treatment level of dischargers and fuzzy risk of low water quality, is obtained using NSGA-II. Then, the best non-dominated solution is selected using a non-zero-sum bi-matrix game with fuzzy goals. In the next step, to have an equitable waste load allocation, some possible coalitions among dischargers are formed and treatment costs are reallocated to discharges and side payments are calculated. To develop probabilistic rules for real-time waste load allocation, the proposed model is applied considering several scenarios of pollution loads and the results are used for training and testing BNs and PSVMs. The applicability and efficiency of the methodology are examined in a real-world case study of the Zarjub River in the northern part of Iran. The results show that the average relative errors of the proposed rules in estimating the treatment levels of dischargers are less than 5 %.


Waste load allocation Fuzzy bi-matrix game Water quality Bayesian Network (BN) Support Vector Machine (SVM) 


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Mohammad Reza Nikoo
    • 1
    Email author
  • Reza Kerachian
    • 2
  • Mohammad Hossein Niksokhan
    • 3
  1. 1.Department of Civil Engineering, East Tehran Branch, Islamic Azad UniversityTehranIran
  2. 2.School of Civil Engineering and Center of Excellence for Engineering and Management of Civil Infrastructures, College of EngineeringUniversity of TehranTehranIran
  3. 3.Faculty of EnvironmentUniversity of TehranTehranIran

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