Allocation of allowable pollution load by using the bankruptcy solution rules

Rivers are potential cases to raise conflicts related to water quality because they usually have several stakeholders. In many cases, the upstream users' actions, such as considerable withdrawals or adding pollution load to the river flow in forms of domestic, industrial (sanitary), and agricultural waste or sewer, can threaten downstream stakeholders' benefits. Pollution may have only financial and economic consequences, but it can sometimes threaten the environment and human health. In any case, if a simple conflict is not solved, it can be turned into a serious social conflict. Therefore, the development of methods to solve water quality conflicts between stakeholders of river systems can be regarded as a very important issue in the area of water resources management.

Bankruptcy solution methods are a type of method that can be used to solve asset allocation conflicts. The objective of these rules is to distribute a limited asset over several agents when the asset is not enough to satisfy their claims (Herrero & Villar 2001). Several methods have been developed so far to solve the bankruptcy problem in which their difference is usually related to their different definitions of equity.

In recent years, bankruptcy solution methods have attracted considerable attention in the area of water resources management and their performance in water resources conflict resolution has been assessed in several studies. Some important issues related to water resources management, such as allocation of limited amount of available water to multiple stakeholders, can be illustrated as a bankruptcy problem.

Kampas & White (2003) presented some solutions based on the bankruptcy solution rules to make various allocation permissions for a small watershed in the southwest of England in order to control agricultural pollution. The results showed different effects of the different rules on the solution of benefit distribution problem between the stakeholders based on the equity interpretation. In addition, the study played a positive role in the quantification of equity for the allocation permission.

Kucukmehmetoglu & Guldmann (2004) developed a linear programming model to allocate required water to uses including irrigation, urban consumption, and on-stream hydroelectricity production in the three riparian countries of the rivers Euphrates and Tigris (i.e., Turkey, Syria, and Iraq), by maximization of the aggregate net benefits from water uses considering water-conveyance costs. The constraints including water-conservation balances and maximum and minimum water consumption were regarded and the model was used to evaluate the economic consequences of different cooperation and non-cooperation strategies possible by the riparian countries. Cooperative game-theory concepts were adopted to recognize stable water allocations that were beneficial for all three countries to cooperate. According to the results, there was an allocation of the total benefits, under various scenarios of future energy prices and agricultural productivities, that is able to make this global cooperation interesting for all three countries.

Sheikhmohamady & Madani (2008) utilized three bankruptcy solution rules including constrained equal awards (CEA) rule, proportional (PRO) rule, and adjusted proportional (APRO) rule to solve the conflict between five coastal countries of the Caspian Sea about the use of oil and gas resources. According to the results, the rule CEA outperformed the other two rules in terms of social choice rules.

Madani (2010) studied the use of game theory for conflict resolution in water resources management through non-collaborative games. The capabilities of the methods for understanding and solving real-world water resources conflict were evaluated considering the dynamic structure of water resources problems.

Mahjouri & Ardestani (2010) and Mahjouri & Bizhani-manzar (2013) suggested river quality management policies by using conflict resolution methods and optimization models. The results showed the positive role of cooperative strategies to maximize net benefit of surface water resources when both quality and quantity aspects of the water resource are regarded.

Ansink & Weikard (2012) transformed a shared-river problem into a bankruptcy problem and then applied bankruptcy solution rules to solve the problem. They used PRO, CEA, constrained equal losses (CEL), and Talmud (TAL) rules and suggested that the rules can be suitable tools to solve the studied problem.

Madani & Zarezadeh (2012) reviewed performances of the bankruptcy solution rules including PRO, CEA, CEL, TAL, and Piniles' rules to solve a hypothetical conflict problem about a groundwater resource use.

Also, Zarezadeh et al. (2012) applied four bankruptcy solution rules including PRO, APRO, CEA, and CEL to the problem of conflicts between eight provinces located in Sefidrud basin in Iran to present a fair allocation structure for some climatic and development scenarios. They concluded that the structure based on CEL provides the most acceptable allocation solutions.

Nikoo et al. (2013) presented a stochastic dynamic programming model to optimize allocation of water and waste load in Dez reservoir–river system considering the existing uncertainties in reservoir inflow, waste loads, and water demands. They used some soft computing techniques such as regression tree induction, fuzzy K-nearest neighbor, Bayesian network, support vector regression, and adaptive neuro-fuzzy inference system to increase model computational ability and speed.

Liu et al. (2014) developed an integrated optimization model to allocate water quantity and waste load in Northwest Pearl River Delta in China. The maximization of the economic benefits and the minimization of water shortages and maximization of waste load were regarded as objectives of the model. They applied a second-generation non-dominated genetic algorithm to solve the multi-objective allocation model.

In addition, Madani et al. (2014) proposed a new approach based on bankruptcy solution to solve inter-province river conflicts where the stakeholders' total demand exceeds the existing resources. They presented four optimization models based on the rules PRO, APRO, CEA, and CEL.

Mianabadi et al. (2014) proposed a new approach based on the bankruptcy solution by considering agents' contributions in total assets besides their claims. They assessed its performance in a case study on the River Euphrates and a hypothetical case study. The assessment showed the superiority of the proposed approach compared to the other methods.

Furthermore, Mianabadi et al. (2015) proposed a weighted bankruptcy solution to solve the problem of demands exceeding resources. They described the special features of the proposed rule and applied the method to the water allocation problem of Tigris River, which is a shared river between the countries Turkey, Syria, and Iraq, as a case study. The results indicated that the proposed rule can facilitate the negotiation about water allocation in the transboundary river basins.

Sechi & Zucca (2015) consider higher user's tension to pay the cost as the higher priority of equal requested volume and introduced a method for water resources allocation in a complicated system under water scarcity conditions by using bankruptcy solution rules. They applied the rules PRO, APRO, CEA, CEL, and TAL to a simple water system and a complicated and multi-purpose water system in Italy. APRO and TAL rules were identified as the appropriate options in terms of balancing the existing resources according to the results.

Arjoon et al. (2016) proposed a method for welfare distribution in transboundary river basins based on stakeholders' contribution aimed at maximizing economic benefits of the water use and fair sharing of the benefits. They evaluated the distribution of total benefits based on the bankruptcy solution rules by applying the proposed method to the basin of the River Nile.

Also, Degefu & He (2016) studied the application of the bankruptcy solution rules to water resources conflict in the transboundary basin of the River Nile. They considered the relative value of each riparian country as its water contribution in forming a large collaborative coalition and proposed a method to calculate the water contribution of riparian countries of a bankrupt river basin. In addition, they recommended weighting the riparian stakeholders' water claims and deficits (shortages) by considering their relative vulnerability against water deficit (shortage).

Oftadeh et al. (2016) proposed a new bankruptcy solution method and compared its performance with that of PRO rule in Zarrinerud sub-basin of Urmia Lake basin located in the northwest of Iran. The performances of the methods were evaluated based on their ability in decreasing the water allocated to irrigated agriculture and allocating it to the environmental demand of Urmia Lake, but none of them compensate for the deficit completely.

Degefu et al. (2017) combined bankruptcy theory with the concept of asymmetric bargaining solution in order to develop a water allocation mechanism for transboundary river basins under water scarcity conditions, focusing on the River Nile basin.

Bozorg-Haddad et al. (2018) studied water allocation to the urban-industrial, agricultural, and environmental uses downstream of Zarrinerud dam in the northwest of Iran. The water allocation was considered as a bankruptcy problem in which the allocation was implemented by using the bankruptcy rules PRO, APRO, CEA, and CEL. Finally, CEA was identified as the best rule for water allocation based on the performance criteria and bankruptcy allocation sustainability index.

Moridi (2019) developed a framework for river water quality conflict resolution by applying the bankruptcy solution rules to the allocation of pollution discharge permits. The pollution point sources were considered as the beneficiaries which release their wastewater to the river with minimum treatment costs. Four well-known bankruptcy rules were used to form objective functions for an optimization–simulation river water quality model which utilized the model QUAL2Kw as the river water quality simulation model with a particle swarm optimization (PSO) model.

Wickramage et al. (2020) used five methods based on bankruptcy solution rules to solve the conflicts related to water resources shortage between the upstream and downstream states of Missouri River in the US. They evaluated the performances of applied rules and identified an adjusted CEA rule and PRO rule as the most suitable choices.

Janjua & Hassan (2020a, 2020b) studied the performances of various bankruptcy solution rules for water allocation between riparian provinces of the Indus River basin in Pakistan. They proposed some new methods based on bankruptcy solution rules for surface water and groundwater allocation. The researchers concluded that bankruptcy solution rules are able to address mismatches between supply and demand in shared river basins.

Farjoudi et al. (2021) presented a probabilistic water quality management model based on bankruptcy rules to solve the conflicts between the Environmental Protection Agency and polluters in river systems. Dissolved oxygen was considered as the water quality factor and the bankruptcy rules were used to allocate wastewater cooperatively and improve the water quality at a control point. Also, a simulation–optimization model consisting of the river water quality model QUAL2Kw and PSO was utilized to optimize the pollution load allocation. The assessment of the results related to the deterministic and probabilistic models showed that the suggested methodology may decrease the waste load and increase dissolved oxygen considerably. In addition, the application of the probabilistic model let pollution sources release pollution loads more than the deterministic model. Among the evaluated bankruptcy rules, the TAL rule provided a more appropriate performance with higher dissolved oxygen and waste load criteria.

Although there are several studies related to the application of the bankruptcy solution rules to water resources conflicts, the use of bankruptcy rules to solve the conflicts related to river water quality has not been studied considerably so far. Therefore, the main objective of the current study is to evaluate the capability of bankruptcy solution rules to solve river water quality conflicts. In this study, performances of four bankruptcy solution rules including PRO, CEA, CEL, and TAL rules are evaluated by applying them to a case study in Karun River basin located in the southwest of Iran. Also, in the current study, the objective is to allocate equitable and optimum pollution loads to pollution sources in a way that the water quality (electrical conductivity (EC)) does not exceed an allowable value in a control point of the downstream of the river. Equitable allocation of pollution loads relates to the perceived fairness in the distribution of loads between polluters and tends to be defined by socio-economic and political factors (e.g., whether small-scale polluters should be allowed to pollute more than large-scale polluters, or vice versa); while optimal allocation of pollution loads relate to the strategy that reduces the total pollution load without regarding fairness. One of the novel aspects of the current study is to consider the amount of each flow discharge in addition to the pollution concentration of each pollution source. Considering the magnitude of each flow discharge makes it possible to have a clearer image of the share of each pollution source in polluting the river water and support the identification of both equitable and optimal pollution allocation loads.

To study a water quality conflict in a river, first, it is required to determine a reach of the river and a point at the end of the reach to assess the water quality. The reach includes a main flow and some tributaries which join the main flow over the reach. Both main flow and tributaries can be considered as pollutant resources or inflows. If pollution load exceeds an allowable value in the endpoint, there is a need to decrease the pollution load of inflows so that pollution concentration in the endpoint decreases to a value lower than the allowable value.

The problem of determining the allowable pollution load of each pollutant inflow can be regarded as an allocation problem with a limited resource or asset and multiple agents where the sum of the claims is greater than the total asset. Thus, it is a bankruptcy problem and it is possible to use bankruptcy solution rules to solve it.

In the current study, four bankruptcy solution rules including PRO, CEA, CEL, and TAL rules were used to solve a water quality conflict within a reach of Karun River in Iran.

To apply TAL rule to a river water quality conflict, first, the values of pollution loads of all inflows must be reduced to their initial values. After the reduction, if the pollution load in the control point of the river is greater than the allowable value ⁠, then CEA rule is used to allocate allowable pollution loads to the inflows constrained that the new pollution load for each inflow i does not exceed half of its initial value ⁠. Also, the constraint of Equation (3) must be considered. On the other hand, if the pollution load in the control point of the river is lower than the allowable value ⁠, then CEL rule is used to allocate allowable pollution loads to the inflows constrained that the new pollution load for each inflow i is not determined lower than half of its initial value ⁠. The constraint of Equation (3) must be applied.

There are several physical, chemical, and biological factors effective on pollution load in the control point. Thus, after reducing the allowable pollution loads of the inflows over the river reach, it is necessary to determine the pollution load of the control point by river water quality modeling and it is not possible to calculate it only by adding the new pollution loads of inflows to each other.

To apply bankruptcy-based models and assess their performance in solving river water quality conflicts, a reach of Karun River in the southwest of Iran, from downstream of Gotvand storage dam to upstream of the city Ahvaz was chosen as a case study (Figure 1). Karun River flow, released from Gotvand dam, was considered as the main streamflow and three river flows, two urban and industrial wastewater flows, and seven agricultural drainage flows were considered as inflows. The control point was located upstream of the city of Ahvaz.

Since salinity has been the most conflicting issue about the water quality of Karun River, the river water quality models were formed in terms of electrical conductivity (EC) which is the main water quality parameter to evaluate the water salinity. Also, focusing on one water quality parameter is useful to evaluate the performance of models more clearly.

According to the results related to the assessment of EC variation in the selected reach of the river, the models were formed based on the mass-balance process because the processes such as diffusion did not show a considerable effect on the water quality in terms of EC. Discharge and EC values can be seen in Table 1.

The water quality optimization models were developed by R 4.0.2 (R Core Team 2020) programming language and environment.

In the first step, PRO rule was applied to the case study for the three EC thresholds. For each threshold, the optimization model (Equations (11) and (12)) was applied to the river reach and the EC value was calculated for each flow by the river water quality model. As seen in Table 2, for the three EC thresholds 1,000, 1,500, and 2,000 μmhos/cm in the river control point, the proportion of allowable pollution load or EC of the flows to their initial values based on PRO rules are equal to 0.47, 0.71, and 0.94, respectively.

In the second step, CEA rule was applied to the case study for the three EC thresholds. As observed in Table 3, by applying the model based on the CEA rule, the flows or pollution sources where their pollution loads are not very large (i.e., the sources 2, 3, 5, 6, 7, 8, 10, 11, 12, 13) do not have to decrease their initial values of EC, but the pollution load or flow EC of the large pollution sources (i.e., the sources 1, 4, 9) must be reduced. From this point of view, CEA rule can be considered as a rule that is desirable for small pollution sources and allows them to continue their activity keeping their current polluting conditions. On the other hand, the large pollution sources do not take advantage of CEA rule application.

In the third step, the CEL rule was applied to the case study for the three EC thresholds. As observed in Table 4, the use of the CEL-based model results in the highest benefits for the greatest pollution sources (i.e., the sources 1, 4, 9), because they can always keep a part of their current pollution load or flow EC, whereas the small pollution sources (i.e., the sources 2, 3, 5, 6, 7, 8, 10, 11, 12, 13) have to reduce their pollution load (flow EC) to 0 in several cases. Therefore, it can be concluded that the CEL rule may be more appropriate for the large pollution sources than small sources.

In the fourth step, TAL rule was applied to the case study for the three EC thresholds. Since the EC value in the control point calculated by the river water quality model exceeded the allowable threshold of 1,000 μmhos/cm, to apply TAL rule, CEA rule was applied to half of the initial values of the pollution loads in order to reduce the values of pollution loads. On the other hand, for the allowable EC threshold values 1,500 and 2,000 μmhos/cm in the control point, the EC value in the control point calculated by the river water quality model did not exceed the thresholds. Thus, CEL rule was used after reducing the pollution loads of polluting flows to half of their initial values in order to increase the values of pollution loads (in comparison with half of the pollution loads).

As presented in Table 5, by using CEA rule for the EC threshold of 1,000 μmhos/cm, the smaller pollution sources are allowed to import half of their initial pollution loads to the river, while the larger sources, such as sources 1, 9, and 4, were not allowed to exceed the CEA value which is lower than half of pollution loads. For the EC thresholds 1,500 and 2,000 μmhos/cm, by applying the CEL rule, more pollution sources are allowed to import pollution loads that are greater than half of their initial pollution loads. In these cases, the greater sources need to lower decreases in their pollution loads. In general, it seems that the application of TAL rule is more desirable for the greater pollution sources in comparison with PRO and CEA rules, while it provides lower benefit for them compared to CEL rule.

The results of applying the four models based on the bankruptcy rules can be compared in Figures 2–4 for the EC thresholds 1,000, 1,500, and 2,000 μmhos/cm, respectively. According to the figures, the most beneficiary choice for the large pollution sources (e.g., the sources 1, 4, and 9) is the model based on the CEL rule, whereas the most desirable choice for the small sources (e.g., the sources 2, 3, 5, 6, 7, 8, 11, 12, and 13) is the model based on CEA rule.

The results related to the application of the models based on PRO and TAL rules often can be observed between those related to the models based on CEA and CEL rules. The application of the PRO-based model always results in an identical relative (proportional) decrease in the pollution loads of the pollution sources.

It is clear that the allowable pollution loads of the pollution sources increase by increasing the allowable EC threshold in the river control point. Also, the results provided by the models get closer to each other as the allowable EC threshold increases in the river control point in a way that the results provided by the models based on the rules PRO and CEA are very similar to each other for the EC threshold 2,000 μmhos/cm.

The differences between the results provided by the models based on CEL and TAL rules with those of the other two models in the cases related to the small pollution sources are because of their small discharges of these sources that cause a considerable decrease in their EC values which, in some cases, reduce the EC value of a flow to 0. It should be noted that for the EC thresholds 1,500 and 2,000 μmhos/cm, the TAL-based model uses CEL rule after decreasing the pollution loads to half of their initial values and, thus, the results related to the models based on CEL and TAL rules are more similar to each other for these two EC thresholds.

Figures 5–7 show the water quality variation in terms of EC along the river reach for the allowable EC thresholds 1,000, 1,500, and 2,000 μmhos/cm in the control point, respectively. As seen in the figures, among the results provided by the four models, the greatest EC variation range over the river reach is related to the CEA-based model. Also, the CEL-based model showed the smallest range of EC variation for different EC thresholds in the river control point. The EC variation over the river reach by applying the models based on PRO and TAL rules provides the results between those related to the models based on the CEA and CEL rules. For the threshold 2,000 μmhos/cm, the results provided based on CEL and TAL rules are very similar to each other because in the application of the TAL-based model the CEL rule is utilized after reducing the pollution loads of the pollution sources to half of their initial values for this allowable EC threshold in the river control point.

The pollution sources 2 and 3 do not play a key role in changing the river water quality because of their small discharge values, which result in the small pollution load values whether their flow EC values are large or not. On the other hand, the pollution sources such as sources 4 and 9 change the river water quality considerably because of their large pollution loads (EC multiplied by discharge). The important factor about source 4 with very high EC value and source 9 is the great discharge value.

Bankruptcy solution methods have been used in several studies during the last decade to solve the conflicts related to water resources. However, most of the conducted studies focused on quantitative water resource conflicts. In the current study, four bankruptcy rules including PRO rule, CEA rule, CEL rule, and TAL rule were used to develop optimization models in order to solve the problem of allocating allowable pollution loads to pollution sources of the river reach. The models were applied to a river reach of Karun River in the southwest of Iran as a case study. The electrical conductivity was chosen as the main river water quality variable in the water quality modeling because water salinity has been one of the most challenging issues related to the water quality of Karun River in recent years. The conclusions of this study can be summarized as below:

1. CEA is the most appropriate rule for small pollution sources because, in many cases, it allows them to keep their polluting conditions while it makes large pollution sources decrease their pollution loads considerably.

2. CEL is the most desirable rule for large pollution sources because it may let them decrease only a relatively small part of their initial pollution loads whereas it may determine the allowable pollution loads of small pollution sources equal to zero (100% decrease).

3. PRO and TAL rules usually provide results between those provided by CEA and CEL and, thus, they may be the most probable solution for reaching an agreement between small and large pollution sources' stakeholders.

4. The results related to the application of TAL rule can get closer to those related to CEA or CEL depending on the value of allowable pollution load in the river control point.

The performance assessment of bankruptcy solution methods in the cases including more water quality variables can make the advantages and limitations of these methods clearer in future study. In addition, some recently developed bankruptcy solution rules can be used in future studies on solving water quality conflicts.