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Water Resources Management

, Volume 27, Issue 7, pp 2125–2136 | Cite as

Waste Load Allocation in Rivers using Fallback Bargaining

Article

Abstract

In this paper, bargaining process between different stakeholders involved in a waste load allocation problem is simulated using the Fallback Bargaining (FB) concept. The paper considers two main parties in a waste load allocation problem. On the one hand, there are wastewater dischargers intending to minimize their treatment costs and on the other hand, there is an environmental protection agency which monitors the river water quality at a checkpoint downstream of the location of dischargers. In this paper, different alternatives which are combinations of dischargers’ treatment scenarios are defined. A water quality simulation model is utilized to estimate the concentration of the water quality indicator along the river based on a selected alternative. If the concentration of water quality indicator in the selected checkpoint violates the water quality standards, a penalty function is used to calculate the amount of penalty assigned to dischargers. The allocated cost to each discharger is computed considering his treatment scenario as well as the penalty allocated to him. Two kinds of Fallback bargaining procedure termed as Unanimity Fallback Bargaining (UFB) and Fallback bargaining with Impasse (FBI), which both aim at minimizing the maximum dissatisfaction of bargainers in a negotiation problem, are utilized for finding a Compromise Set (CS) of alternatives. In this paper, the best alternative (alternatives) among CS members is (are) selected using a social choice theory namely Condorcet winner. The results of these two approaches are compared and the final alternative is selected which shows the initial Tradable Discharge Permits (TDPs) allocated to dischargers. Finally, in order to decrease the total allocated cost to dischargers, initial allocated TDPs are exchanged between them using the Extended Trading Ratio System (ETRS) developed by Mesbah et al. (Environ Model Software 24:238–246, 2009). The applicability and efficiency of the proposed methodology is investigated by applying it to a case study of the Zarjub River in the northern part of Iran.

Keywords

Waste load allocation Fallback bargaining Compromise Set (CS) Tradable Discharge Permits (TDPs) Condorcet winner, the Zarjub River 

1 Introduction

In the past few decades, discharging wastewater into the rivers more than their assimilative capacities has endangered these important ecosystems. Waste Load Allocation (WLA) in rivers which mostly refers to determining the maximum allowable discharge of wastewater into the river by dischargers, usually involves different stakeholders with conflicting interests and priorities. On the one hand, dischargers seek maximum individual satisfaction through minimizing their total costs. On the other hand, Environmental Protection Agency (EPA)’s objective is to minimize the negative environmental impacts of discharging wastewater into rivers. This conflict of interest leads to inevitable disputes between the involved stakeholders.

Game theoretic and also bargaining approaches are among some of the recently used techniques in the literature for resolving the existing conflicts in water resources system management issues including WLA problems. Sadegh and Kerachian (2011) used fuzzy cooperative games for optimally allocation of water resources and associated benefits to water users in a river basin. They showed that these solution concepts are more efficient than the ordinary crisp games by examining them through applying to the Karoon river basin in southern Iran. Nikoo et al. (2012) developed a new methodology for simultaneous allocation of water and waste load in river basins. They incorporated the uncertainties using a nonlinear interval number optimization model. Finally the benefits were reallocated to water users using some solution concepts of the cooperative game theory. Abed-Elmdoust and Kerachian (2012a) utilized fuzzy cooperative games for modeling both inter-basin and intra-basin water allocation. They utilized different methodologies and applied them to three real life scenarios of a water allocation case study in Iran.

Zhang et al. (2012) analyzed different characteristics of water transfer decisions. They used the “satisfaction principle” in a bargaining model with different objectives for inter-basin water transfer. Poorsepahy-Samian et al. (2012) developed a methodology based on game theory for water and discharge permit allocation to agricultural zones in shared rivers and examined it in a case study of the Karoon-Dez river system in Iran.

The Fallback Bargaining Procedure (FBP) is a game theoretic approach which models the bargaining process among stakeholders involved in a negotiation problem using some mathematical rules. In the process of negotiation, the stakeholders compromise their utilities to some degree in order to achieve an agreement. It is expected that FBP minimizes the maximum dissatisfaction of the bargainers (Brams and Kilgour 2001). Application of FBP to conflict resolution in water resources management issues has been very limited.

Brams et al. (2004) proposed a procedure for reaching an agreement on multilateral treaties. The obtained agreement that they termed as minimax outcome was a compromise which minimized the maximum distance to the top preference of any player. They showed that their proposed procedure was not prone to strategizing by states and reduced the states’ incentives to misrepresent their priorities in order to receive better outcomes.

Sheikhmohammady and Madani (2008) and Sheikhmohammady et al. (2010) applied social choice Fallback Bargaining procedures to predict the most possible outcome of the negotiations over the Caspian Sea. Also, Madani et al. (2011) used Fallback bargaining procedures to predict the possible outcomes of the Sacramento-San Joaquin Delta conflict over selecting a new water export strategy.

This paper, perhaps for the first time, intends to simulate negotiations among pollution dischargers in a river system using the Fallback bargaining procedure while there is a penalty function for water quality violations of the standards. Firstly, different alternatives which are the combinations of dischargers’ treatment scenarios are defined. A water quality simulation model is utilized to estimate the concentration of the water quality indicator based on the selected alternative. If the concentration of water quality indicator in the selected checkpoint violates the water quality standards, a penalty function is used to calculate the amount of penalty assigned to that alternative due to violating the water quality standards. The allocated cost to each discharger is estimated considering the selected treatment scenario, the treatment cost function of the discharger and the penalty allocated to him. The Fallback bargaining procedure is utilized for finding a Compromise Set (CS) of alternatives. All members of the CS are of the same priority. Next, the best alternative among CS members is selected using the concept of Condorcet winner (Condorcet 1785). This alternative is considered as the initial Tradable Discharge Permits (TDPs) of dischargers. Finally, in order to decrease the total allocated costs to dischargers, initial allocated TDPs are exchanged between them using the Extended Trading Ratio System (ETRS) developed by Mesbah et al. (2009). To investigate its performance, the methodology is applied to a case study of the Zarjub River in the northern part of Iran. In the following section, the framework of the proposed methodology is presented and its main components are explained.

2 Framework of the Methodology

A flowchart of the proposed methodology is presented in Fig. 1. The main steps of the proposed methodology are explained in the following sections.
Fig. 1

A flowchart of the proposed methodology

2.1 Identifying the Main Characteristics of the System

According to the flowchart, firstly, the main pollution sources and their effluent quantities and qualities are determined. Also, the main water quality indicator as well as the important characteristics of the river such as upstream flow quantity and quality are determined.

2.2 Estimating the Treatment Costs of the Main Dischargers

The total treatment costs of the pollution sources are the construction costs of the wastewater treatment plants as well as the operational costs. In the present paper, the cost estimation is carried out mostly based on the capacity of the domestic wastewater treatment plants. Tsagarakis et al. (2003) estimated the treatment costs of domestic wastewater based on 3 main categories of population sizes. They suggested that the construction cost of a conventional wastewater treatment plant covering a population of 10,000 to 100,000 people would be about $130 per capita and for those covering a population more than 100,000 people, the construction cost would be about $110 per capita. The annual operational costs are considered to be 3 to 5 % of the total construction costs which are in compliance with the operational costs of wastewater treatment plants in Iran.

2.3 Defining the Wastewater Treatment Scenarios of the Main Dischargers

In this paper, the major stakeholders of the system are the wastewater dischargers and an environmental protection agency. The main pollution sources in the study area are considered to be the bargainers taking part in the negotiation process. The bargainers (dischargers) intend to negotiate for their preferred treatment levels. Thus, different levels of treatment are considered to be the treatment scenarios of each bargainer. The alternatives are made of different combinations of the defined scenarios for the bargainers. For example, if there are four bargainers and scenarios 1 to 6 for every bargainer, 1,296 (i.e. 6 × 6 × 6 × 6) alternatives will be made. Alternative 2461, for instance, means that scenarios 2, 4, 6 and 1 are assigned to bargainers 1 to 4, respectively. The role of environmental protection agency in the bargaining process is to put a limit on the bargainer’s wastewater discharges by assigning a penalty cost to each alternative, depending on the amount of water quality violation of the standards at a checkpoint downstream of pollution loads. Thus, the cost associated with every alternative consists of the total treatment cost (according to the treatment scenarios in that alternative) and the amount of penalty allocated to that alternative. The penalty function and how it is calculated will be discussed in the results and discussion section. In this paper, it is assumed that the total penalty cost is equally distributed among dischargers.

2.4 Utilizing the Fallback Bargaining Procedure

Fallback bargaining procedures minimize the maximum dissatisfaction of the bargainers with conflicting objectives involved in a negotiation problem. At first, bargainers rank all alternatives according to their own preferences. It is obvious that each bargainer prefers to select the alternatives which imposes the less cost to that bargainer. Therefore, the ranking is simply based on the costs every discharger has to pay by selecting each alternative. This way, an n × m matrix will be made that its members are the alternatives ranked by the bargainers, n and m are respectively, the number of bargainers and the number of alternatives. After ranking the alternatives and forming the preference matrix, bargainers start the process of bargaining by first selecting their most preferred alternative and then fall-back from their first priorities to the second, third and so on, until they get to a point which all of them agree on. The selected alternative is not necessarily the best choice of the majority of the bargainers.

In this paper, two types of Fallback bargaining procedures namely, Unanimity Fallback Bargaining (UFB) and Fallback Bargaining with Impasse (FBI) are used to find out the outcome of the bargaining process between dischargers. Brams and Kilgour (2001) defined and compared these two procedures. They suggested that UFB selects the alternative(s) receiving unanimous support from all bargainers that is somewhere in the middle of everybody’s ranking. They showed that the selected outcome is a Pareto optimal but not necessarily a unique solution. In the UFB, the outcome is a Compromise Set (CS) consisting of exactly those alternative(s) which maximizes the minimum satisfaction of all bargainers.

FBI is used to consider the possibility for the bargainers to limit their compromise to some level beyond which they prefer to disagree rather than accept a lower-level alternative. This way, bargainers indicate impasse in their preference ranking (denoted by I). The procedure of FBI is exactly the same as UFB, except for once I is reached for a bargainer, the procedure stops for him. Therefore, it is possible that no common agreement is reached before the level descends to every bargainer’s I. Thus, the outcome may be impasse.

If the bargaining process does not reach an impasse, the result of Fallback bargaining is a CS which includes an alternative or a set of alternatives. If the CS matrix consists of only one alternative, this will be the outcome of the bargaining process; otherwise, the final solution of the FBP could be one of the members of the CS. In this paper, the satisfaction of each bargainer (discharger) is quantified based on his treatment cost and Condorcet Social Choice Theory is used for choosing the best alternative among the CS members. The resulted alternative is considered to be the initial Tradable Discharge Permits (TDPs) allocated to dischargers.

2.5 Condorcet Social Choice Theory (Condorcet Winner Concept)

Social choice theory is a theoretical framework for combining individual preferences to provide a collective decision. In this paper, the Condorcet winner concept is used to find out the best alternative among the members of CS. Using this approach, the socially optimum result is selected based on pair-wise comparisons of the alternatives. In general, a Condorcet winner is an alternative that majority prefers it to every other alternative. Therefore, every alternative in the CS is compared to the other ones based on the total cost (the treatment cost and the penalty allocated to dischargers) associated to that alternative. In every pair-wise comparison, there is a winner alternative. This process is repeated for all alternatives until the final alternative which is considered to be the Condorcet winner is chosen. Details and examples about this social choice theory can be found in Brams and Kilgour (2001).

2.6 Trading the TDPs using the Extended Trading Ratio System (ETRS)

In this paper, to minimize the total cost of the system, the initial TDPs obtained from the previous sections are traded between dischargers using the Extended Trading Ratio System (ETRS) developed by Mesbah et al. (2009). In the ETRS, trading Biochemical Oxygen Demand (BOD) discharge permits is conducted while Dissolved Oxygen (DO) is considered as the river water quality indicator. If the river is divided into n different zones each containing a BOD discharger, the transfer coefficient between zones i and j (r ij ) is defined as variation of the concentration of water quality indicator (DO) in zone j (mg/L) as a result of 1 kg increase in BOD load of the discharger in zone i. The amount of BOD discharge permit that two dischargers i and j can trade defines how much BOD (kg) they can trade while the DO concentration in downstream river does not deviate from the water quality standards. This ratio is calculated using the trading ratio between them (t ij ) which is defined as (Mesbah et al. 2010):
$$ \begin{array}{*{20}c} {{t_{ij }}=min\left\{ {\frac{{{r_{ik }}}}{{{r_{jk }}}}} \right\};i<j,} \hfill & {k=\left\{ {j,\ldots,n} \right\}} \hfill \\ \end{array} $$
(1)

3 Case Study

The applicability and efficiency of the proposed methodology is illustrated by applying it to the Zarjub River which stretches from the Talesh Mountains to the Caspian Sea in Northern part of Iran. A 24 km stretch of this river which passes through the Rasht City is considered in this paper. This river is the main source of water supply for agricultural lands in the area. Iran Department of Environment (IDOE) has classified this river as one of the most polluted rivers in Iran. According to water quality measurements, the concentration of some water quality variables such as dissolved oxygen (DO) significantly violates the river water quality standards. Wastewater of 11 domestic pollution sources including the municipal wastewater of the Rasht City is entered into this river (Iran Department of Environment 2005). Figure 2 illustrates a schematic of Zarjub River and the location of the main dischargers and checkpoint. More details about the study area can be found in Niksokhan et al. (2009).
Fig. 2

A schematic of the Zarjub River and the location of dischargers and the checkpoint

4 Results and Discussion

For minimizing the construction and operational cost of wastewater treatment plants, dischargers are grouped together and four main dischargers are formed. Therefore, the river is divided into four reaches, each of which contains one of the aforementioned dischargers (Fig. 2). These dischargers are considered to be the main bargainers of the system. The important characteristics of the dischargers are illustrated in Table 1. According to the proposed methodology, after determining the main characteristics of the system, the assimilative capacity of the river is estimated considering the river water quality standards. Table 2 shows the maximum allowable BOD load per day which can be discharged into the four main reaches of the Zarjub river.
Table 1

Characteristics of the main pollution sources in the study area

Discharger

Wastewater discharge (m3/s)

BOD concentration (mg/L)

1

0.083

100

2

0.414

46

3

0.217

98

4

0.149

180

Table 2

Assimilative capacity of the four reaches on the Zarjub River (kg BOD/day)

Reach 1

Reach 2

Reach 3

Reach 4

432

823

438

1955

In this paper, six treatment scenarios are considered for each discharger. Table 3 shows the BOD concentration in the effluents of dischargers entering into the river according to the six treatment scenarios. Also, the annual operational treatment costs of dischargers are presented in Table 4. Since there exist four dischargers with six treatment scenarios, 1,296 different alternatives are created. Each alternative is a combination of the defined scenarios.
Table 3

The BOD concentration in effluent of each discharger based on different scenarios (mg/L)

Discharger

Treatment scenarios

1

2

3

4

5

6

1

70

58

46

34

22

10

2

32

27

21

16

10

5

3

69

57

45

33

22

10

4

126

104

83

61

40

18

Table 4

Annual operational treatment cost of dischargers in the Zarjub River system (Dollars)

Discharger

Treatment scenarios

1

2

3

4

5

6

1

5200

14500

23800

33150

42450

51800

2

11900

22300

32750

43150

53550

64000

3

13250

24900

36550

48200

59800

71450

4

16750

46850

77000

107100

137250

167350

In this paper, a penalty function is used to assign a penalty cost to the alternatives causing violation of the river water quality standards in the downstream checkpoint (Fig. 2). The penalty function which has been estimated for the Zarjub River by Malakpour-Estalaki et al. (2010) and Abed-Elmdoust and Kerachian (2012b) is as follows:
$$ P=\left\{ {\begin{array}{*{20}c} {ax} \hfill & {0\leq x<m} \hfill \\ {dy+e} \hfill & {x=3.5} \hfill \\ \end{array}} \right. $$
(2)
where, P is the amount of penalty assigned to all dischargers in thousand US Dollars, x and y are respectively the values of the violation of DO and BOD of the water quality standards (mg/L). It is assumed that the penalty cost is apportioned equally among dischargers. In our case study, as proposed by Malakpour-Estalaki et al. (2010) and Abed-Elmdoust and Kerachian (2012b), the values of parameters a, b, c, d, e and m are 1.56, 0, 0, 0.8, −12.73 and 3.5, respectively.

Abed-Elmdoust and Kerachian (2012b) also developed a river water quality simulation model for calculating the DO and BOD concentrations in some control points downstream of dischargers in the Zarjub River. In this paper, for each alternative, the values of the violations of DO and BOD concentrations of water quality standards at the checkpoint is estimated using the mentioned simulation model. The total cost assigned to each alternative is considered to be the total treatment cost of the alternative (the sum of the treatment costs related to the treatment scenarios in that alternative) and also the penalty cost associated with that alternative.

In the next step, the four main dischargers perform their preference rankings over the 1,296 alternatives. For example, discharger 1 prefers alternative 1666 the most, as this imposes the least total cost to this discharger comparing to other 1,295 alternatives. This is because, by selecting this alternative, discharger 1 does not have to treat its wastewater so much while other dischargers treat their effluents almost thoroughly. Therefore, no violation of the river water quality standards occurs. The whole ranking process is performed by dischargers based on the total cost they have to pay by selecting each alternative. The result of this step is a matrix with 4 rows (the number of dischargers or bargainers) and 1,296 columns (the number of alternatives).

Using the concept of UFB, 448 alternatives are selected as the members of the CS. Since the number of CS members is more than one, Condorcet social choice theory is used to select the best alternative. In this case study, the alternative 1465 is selected as the outcome of the bargaining process among dischargers. Table 5 illustrates the characteristics of this alternative.
Table 5

Characteristics of the outcome of Unanimity Fallback Bargaining (UFB) using the Condorcet theory (alternative 1465)

DO in the checkpoint (mg/L)

Annual total cost (USD)

Discharger 1

Discharger 2

Discharger 3

Discharger 4

3.29

28272

66222

94522

160322

In this paper, the FBI process is undertaken considering the maximum cost the dischargers are willing to pay. By choosing the impasse of dischargers less than 90 % of the maximum cost they have to pay (according to their worst alternative), no alternative is found for the CS. When this limit is set to be 90 % of the maximum cost of every discharger, 75 alternatives are chosen as the members of the CS. In this case, the outcome of the FBI is alternative 2665 which is the Condorcet winner among CS members. Table 6 shows the characteristics of this alternative.
Table 6

Characteristics of the outcome of Fallback Bargaining with Impasse (FBI) using the Condorcet theory (alternative 2665)

Discharger

Impasse of bargainers (USD)

Annual total cost (million USD)

DO in the checkpoint (mg/L)

Discharger 1

87633

33246.5

3.5

Discharger 2

95241

82746.5

Discharger 3

92842

90196

Discharger 4

167734

155996

In order to reduce the total cost allocated to dischargers and to provide the incentive for them to take part in a coalition of dischargers, the initial TDPs, which are corresponding to the alternative 1465, are traded among dischargers. This means that dischargers with more efficient treatment facilities can treat more than the amount of treatment suggested by the initial TDPs. On the other hand, the dischargers that has more treatment cost per unit treated wastewater are allowed to treat less than required. Therefore, the total treatment cost of the system will be decreased. In this paper, the ETRS method is used for exchanging the TDPs among dischargers. The violation of the river water quality standards is partially allowed due to the imposing a penalty cost to dischargers. Therefore, the assimilative capacity of the reaches can be estimated based on the maximum BOD load allowed to be entered into it. The transfer coefficients between the location of every discharger and downstream reaches are calculated based on the simulation model. The trading ratios between the four dischargers are obtained using the transfer coefficients. Table 7 shows the transfer coefficients between the reaches. The transfer coefficient a ij is the amount of decrease in DO concentration in point j as a result of 100 kg BOD discharged in point i. Table 8 presents the trading ratio matrix between dischargers obtained using the ETRS method.
Table 7

Transfer coefficients between the reaches (mg/L DO/100 kg BOD)

Reach

Reach

1

2

3

4

1

0.3

0.01

0.01

0.001

2

0

0.12

0.08

0.01

3

0

0

0.058

0.029

4

0

0

0

0.019

Table 8

Trading ratio matrix

Reach

Reach

1

2

3

4

1

1

0.1

0.05

0.07

2

0

1

0.456

0.703

3

0

0

1

1.54

4

0

0

0

1

As suggested by Hung and Shaw (2005), the bilateral TDP exchanges between dischargers are carried out. The results of trading the TDPs are presented in Table 9. In the first round, by investigating the total possible states of trading among dischargers, the best trading state is that discharger 2 sells all of his TDP to discharger 4. The amount of cost reduction based on this trade will be 44,150 Dollars. In the second bilateral trading round, by calculating the reduction costs resulting from mutual trades between dischargers, the maximum benefit (31,150 Dollars) is obtained from trade between discharger 3 (as the seller) and 4 (as the buyer). In the third round, discharger 4 buys discharge permits from discharger 3 to reduce their treatment cost equal to 3,805 Dollars. Therefore, the total cost of the system reduces from 349,500 Dollars to 270,500 Dollars. The overall profit made by trading TDPs is 79,102 Dollars. The calculated total cost of the system after trading is almost equal to the total cost obtained when the objective is minimizing the total cost of the system.
Table 9

Result of trading TDPs among dischargers

Trade round

Reduction in total treatment cost (USD)

Best trading state (seller, buyer)

1st round

44100

(2, 4)

2nd round

31150

(1, 4)

3rd round

3802

(3, 4)

Total

79102

5 Summary and Conclusion

In this paper, perhaps for the first time, waste load allocation to dischargers in a river system was carried out using the Fallback bargaining procedure. A penalty function was also considered for water quality violations of the standards. The costs were allocated to dischargers considering their wastewater treatment scenarios, their treatment cost functions as well as the penalty allocated to them. Then, using the Fallback bargaining procedures, a Compromise Set of alternatives was obtained and the best alternative among CS members, which was selected using the concept of Condorcet winner, was considered as the initial Tradable Discharge Permits (TDPs) of dischargers. Finally, in order to decrease the total allocated costs to dischargers, initial allocated TDPs were exchanged between them using the Extended Trading Ratio System. The applicability and efficiency of the proposed methodology was illustrated through the case study of the Zarjub River in Iran.

In this paper, the dischargers were considered to be of the same importance. In future studies, the outcome of the bargaining can be found by considering the relative weights of the negotiators. Also, every discharger sought for its own benefit individually. The simulation of the negotiation process can also be done by considering different coalitions among the dischargers. In future works, the methodology presented in this paper can be extended to incorporate the existing uncertainties in inputs and parameters of the models.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Faculty of Civil EngineeringK. N. Toosi University of TechnologyTehranIran
  2. 2.Civil EngineeringK. N. Toosi University of TechnologyTehranIran

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