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

Mohammad Reza Nikoo, Reza Kerachian, Mohammad Hossein Niksokhan

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)


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 %.

Original languageEnglish
Pages (from-to)4539-4552
Number of pages14
JournalWater Resources Management
Issue number15
Publication statusPublished - Oct 2012


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

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Water Science and Technology


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