An Optimization Model for Water Management under the Dual Constraints of Water Pollution and Water Scarcity in the Fenhe River Basin, North China

Sustainable watershed development suffers from severe challenges, such as water pollution and water scarcity. Based on an analysis of water quality and water utilization in the Fenhe River Basin, an inexact two-stage stochastic programming model with downside-risk aversion was built for optimal water resource allocations for the four primary water use sectors (industry, domestic use, agriculture, and the environment) in the Fenhe River Basin. The model aims to maximize the comprehensive watershed benefits, including water benefits, water costs, water treatment costs, and downside risks. The constraints are water quality, available water resources, and sectoral demands in different hydrological scenarios. The results show that pollutant emissions decrease as risk-aversion levels increase and show the opposite trend in the midstream and downstream areas. The increase in water resource allocation for agriculture and reduction in ecological water indicate that agriculture suffered the greatest water shortage and risk. Improving water recycling and coordinating the transferred water resources increases the comprehensive benefits and reduces sectoral risks. The model effectively manages rational water allocations under dual constraints and provides support for coordinating socio-economic development and environmental protection in the river basin.


Introduction
Water resources play a central role in human survival, ecosystem security, and socioeconomic development [1][2][3]. However, with the rapid population expansion and economic development, water shortages and water pollution have become severe problems worldwide [4][5][6]. Excessive water resource exploitation, inefficient utilization, and increased pollutant emissions have exacerbated the deterioration of water quality and water scarcity [7,8]. Water pollution is the most pressing issue in China, and water scarcity has become a critical constraint for regional socio-economic development [9,10]. High-quality and sustainable development based on improving water quality and rational water resource utilization is a developmental priority in China. Therefore, optimal regional and sectoral water resource allocation for coordinated development between the social economy and the environment are important.
Previous studies reveal that optimal allocation of water resources is one of the most effective resolutions for addressing water pollution, water shortages, and rising water demand [11][12][13][14][15][16]. Moreover, appropriate policy interventions, such as adjusting the industrial structure and technological progress, are necessary to achieve harmonious economic and societal development [17]. Additionally, there are many uncertainties in the water environment system, including stochasticity of available water resources caused by climate change [18][19][20][21], uncertainties regarding plans and policies, and the complexity of

Study Area
The Fenhe River (110°30′ E-113°32′ E, 35°20′ N-39°00′ N) is the second largest branch of the Yellow River and the largest river in Shanxi, with a total length of 694 km and a watershed area of approximately 39,471 km 2 [40]. The Fenhe River Basin has a semi-humid climate, the multi-year (1956-2010) average rainfall of the entire basin is 504.8 mm (with a ten-year decreasing trend), and the water surface evaporation is 900-1200 mm. The total available water resources (from 1956-2010) of the Fenhe River Basin are 2.656 billion m 3 [41].
The Fenhe River Basin is highly urbanized and agriculturally developed. The utilization of surface water development is over 70%, and the average utilization of groundwater development reaches 85% [42]. However, the Fenhe River Basin is a relatively severely water-deprived region with a water resource per capita of 378 m 3 -18% of the national average [40]. The utilization rate of water resources has long exceeded 70%, leading to severe water conflicts between the social economy and the environment [42]. Moreover, due to over-exploitation, uneven allocation, and low utilization efficiency of water resources, excessive pollutant discharge has caused severe environmental problems in the Fenhe River.
In this study, the scope of the Fenhe River Basin was determined by analysis of data from a Digital Elevation Model in theShanxi province, using the hydrological analysis module of ARCGIS. The river basin is divided into the upstream, midstream, and downstream areas and a total of 16 water environment control units with the corresponding water quality sections ( Figure 2).

Study Area
The Fenhe River (110 • 30 E-113 • 32 E, 35 • 20 N-39 • 00 N) is the second largest branch of the Yellow River and the largest river in Shanxi, with a total length of 694 km and a watershed area of approximately 39,471 km 2 [40]. The Fenhe River Basin has a semi-humid climate, the multi-year (1956-2010) average rainfall of the entire basin is 504.8 mm (with a ten-year decreasing trend), and the water surface evaporation is 900-1200 mm. The total available water resources (from 1956-2010) of the Fenhe River Basin are 2.656 billion m 3 [41].
The Fenhe River Basin is highly urbanized and agriculturally developed. The utilization of surface water development is over 70%, and the average utilization of groundwater development reaches 85% [42]. However, the Fenhe River Basin is a relatively severely waterdeprived region with a water resource per capita of 378 m 3 -18% of the national average [40]. The utilization rate of water resources has long exceeded 70%, leading to severe water conflicts between the social economy and the environment [42]. Moreover, due to over-exploitation, uneven allocation, and low utilization efficiency of water resources, excessive pollutant discharge has caused severe environmental problems in the Fenhe River.
In this study, the scope of the Fenhe River Basin was determined by analysis of data from a Digital Elevation Model in theShanxi province, using the hydrological analysis module of ARCGIS. The river basin is divided into the upstream, midstream, and downstream areas and a total of 16 water environment control units with the corresponding water quality sections ( Figure 2).

Model Development
This study considers long-run programming of 15 years and three periods (i.e., 2021-2025, 2026-2030, and 2031-2035). Three hydrological scenarios (low, medium, and high) reflect different water environment carrying capacities and water resources. The ITSDP model for integrating water benefits and downside-risk control under dual constraints of water shortage and pollution in the Fenhe River Basin was formulated as follows: where f ± is the comprehensive benefit of the basin (10 6 million CNY) over the programming periods.
1. Water utilization benefits: where i denotes the control unit; k denotes the water sectors ( k = 1 for industry, k = 2 for domestic, k = 3 for agriculture, and k = 4 for the environment); t denotes different periods; t L denotes the length of period t, and the values are fixed at 5 years; ikt UNB ± represents the sectoral water-use benefit (10 4 CNY/10 4 m 3 ); ikt IAW ± represents the pre-allocation of water resources for each sector (10 4 m 3 /year); and jkt RW ± represents reused water resources for each sector (10 4 m 3 /year).

Model Development
This study considers long-run programming of 15 years and three periods (i.e., 2021-2025, 2026-2030, and 2031-2035). Three hydrological scenarios (low, medium, and high) reflect different water environment carrying capacities and water resources. The ITSDP model for integrating water benefits and downside-risk control under dual constraints of water shortage and pollution in the Fenhe River Basin was formulated as follows: where f ± is the comprehensive benefit of the basin (10 6 million CNY) over the programming periods.

1.
Water utilization benefits: where i denotes the control unit; k denotes the water sectors (k = 1 for industry, k = 2 for domestic, k = 3 for agriculture, and k = 4 for the environment); t denotes different periods; L t denotes the length of period t, and the values are fixed at 5 years; UNB ± ikt represents the sectoral water-use benefit (10 4 CNY/10 4 m 3 ); represents the pre-allocation of water resources for each sector (10 4 m 3 /year); and RW ± jkt represents reused water resources for each sector (10 4 m 3 /year).

2.
Water shortage penalty: where h denotes hydrological scenarios, p h denotes the occurrence probability of scenario h, PNB ± jkt represents the reduction of net benefit per unit of water resource not delivered (10 4 RMB/10 4 m 3 ), and DW ± ikth represents the allocation deficit of water resources for each sector (10 4 m 3 /year).

3.
Cost of water supply: where CW ± jkt represents the cost of the water supplies (10 4 CNY/10 4 m 3 ) and CRW ± jkt is the cost of reused water (10 4 CNY/10 4 m 3 ).

5.
Cost of ecological water: where GW ± ith is the ecological water (i.e., ecological water is to purify excessive pollutant emission) and CEW ± it is the cost of ecological water. 6.
Downside risk constraints: where ω represents the risk control level and DRisk ± ikt is the sectoral downside risk. Constraints:

2.
Water sector demand constraints: where WD ± minikt represents the minimum water resources requirement and WD ± maxikt represents the maximum water resources requirement (10 4 m 3 /year).

3.
Regional wastewater treatment capacity constraints: where α ikt is the wastewater emission coefficient and ATW ± ikt represents the wastewater treatment capacity (10 4 tons).

4.
Regional wastewater reuse capacity constraints: where ξ jkt is the wastewater reuse rate.

5.
Water environment carrying capacity constraints: where r is the controlled water pollutant (r = 1 for chemical oxygen demand (COD), r = 2 for ammonia nitrogen (NH 4 -N), r = 3 for total phosphorus (TP)), EC ± ikrt represents the concentration of pollutants after wastewater treatment (mg/L), IDR krt represents the river load ratio of different pollutants, and ALD ± irth is the environmental capacity (tons) of different pollutants.

6.
Downside risk where PRW ± ikth represents the actual benefit, Ω ± ikt represents the expected regional benefit, Delta ± ikth represents the positive deviation from the expected benefit, and DRisk ± ikt represents the downside risk.
The objective is to maximize the comprehensive benefits of the river basin, including the benefits of water resource sectors, water shortage penalties, and the cost of water supply, wastewater treatment, and wastewater reclamation. The constraints are for the relationships between decision values and water quality requirements, including available water resources, environmental water carrying capacity, and downside risks.
Using an interactive algorithm, the ITSDP model can be transformed into two deterministic sub-models that correspond to the lower and upper bounds of the desired objective function value. The  Table 1 lists the available water resources in the upper, middle, and lower reaches of the Fenhe River, including surface water, groundwater, and transferred water from the Yellow River. These were calculated based on regional water resource planning and management policies.

System Benefits and Risks
In this study, the optimal results were obtained without the downside-risk control constraints for a ω value fixed at 0. Further, values of 5, 15, 30, and 50 were selected to reflect increasing risk control levels and obtain the corresponding optimal results of water resource allocation and pollutant emissions. In this section, we compare and analyze the differences in the system benefits and risks of the Fenhe River Basin under different risk levels. Figure 3 shows the system benefits of water resource utilization and corresponding risks for different ω values. The figure shows a similar change in benefits and risks. × 10 6 million for ω of 0, 5, 15, 30, and 50, respectively. Both benefits and risks decrease slightly as the ω values increase, indicating that certain water resources transfer to lower-benefit sectors to meet the stronger risk-control requirement. As the risk control level increases, the model will optimize the water resource allocations for units and sectors based on factors such as water efficiencies, pollutant emission intensities, and water environment carrying capacities.

System Benefits and Risks
In this study, the optimal results were obtained without the downside-risk control constraints for a ω value fixed at 0. Further, values of 5, 15, 30, and 50 were selected to reflect increasing risk control levels and obtain the corresponding optimal results of water resource allocation and pollutant emissions. In this section, we compare and analyze the differences in the system benefits and risks of the Fenhe River Basin under different risk levels. Figure 3 shows the system benefits of water resource utilization and corresponding risks for different ω values. The figure shows a similar change in benefits and risks. The fer to lower-benefit sectors to meet the stronger risk-control requirement. As the risk control level increases, the model will optimize the water resource allocations for units and sectors based on factors such as water efficiencies, pollutant emission intensities, and water environment carrying capacities.  These risks correspond to the lower bound of water resources and environmental carrying capacity, and more water resources are allocated to sectors with high benefits and low pollutant emission intensities.   risks are zero, indicating that the system's benefits and risks are balanced. Compared to the other areas, the upstream area has relatively sufficient water resources and environmental carrying capacity. The developed ITSDP model can optimize water resource allocation strategies to meet the highest system benefit under different risk control requirements. Furthermore, the supply of ecological water resources to purify water quality reduces the amount of water available for production and increases risks. The strong water quality constraint plays a decisive role in the optimal allocation of water resources in this study.  Compared to the other areas, the upstream area has relatively sufficient water resources and environmental carrying capacity. The developed ITSDP model can optimize water resource allocation strategies to meet the highest system benefit under different risk control requirements. Furthermore, the supply of ecological water resources to purify water quality reduces the amount of water available for production and increases risks. The strong water quality constraint plays a decisive role in the optimal allocation of water resources in this study.

Water Resource Allocation and Pollutant Emissions
In this section, we comprehensively analyzed the pollutant emissions and water resource allocation strategies of the Fenhe River Basin under different risk control levels, to study the key constraints of the sustainable development of the basin. Figures 5-7 show the emissions of the main pollutants (COD, NH 4 -H, and TP) during the study periods. These figures indicate that emissions of the three pollutants gradually increase for scenarios 1, 2, and 3, which are influenced by the increasing water resources and environmental carrying capacities. For example, in period 1, for ω values of 15, the amounts of TP emission Figure 7  , and 50. These differences are closely related to regional industrial structures, water resource allocation strategies for sectors, and various pollutant emission intensities. Additionally, Figures 5-7 show more significant changes in period 3, and datasets such as the available water resources for scenario 2 are closer to regional policies and plans; thus, the following analysis would be carried out for period 3, scenario 2.                 Considering the downward trend of sectoral risks and pollutant emissions, the decrease in total water consumption demonstrates that benefits and costs are the main factors affecting resource allocation strategies for agriculture in the upstream area. In the midstream area, water resource allocations show differences across the four sectors. Water consumption of the industry remains unchanged as ω values increase due to relatively high sectoral benefits. Water consumption of the domestic and agriculture sectors shows opposite tendencies.  [11,226.32, 12,351.39], [11,296.26, 12,040.38], [10,971.21, 11,503.76], and [10,769.79, 11,360.53] × 10 4 m 3 for ω values of 0, 5, 15, 30, and 50, and these show a gradual downward trend. Oppositely, as ω values increase, more water resources are allocated to agriculture in order to balance sectoral benefits. The same tendency appears in the downstream area for the two sectors. However, water resource allocation strategies for the industry show obvious fluctuation; the amounts of fresh water are [11,962.2, 17,436.72], [11,682.12, 19, for ω values of 0, 5, 15, 30, and 50. Considering the downward trend of industrial risks in this area, it indicates that the developed ITSDP model optimizes water resource allocations based on different regional industrial structures and benefits. ing various changes due to regional and sectoral differences in pollutant emission intensities. In the midstream area, it shows fluctuation, and the amounts of ecological water are [51,044. In the upstream and downstream areas, the amounts of ecological water show a downward trend as ω values increase; the developed model optimizes water resource allocation and reduces pollutant quantities entering the river in these areas.

Policy Scenarios Analysis
Risks and ecological water consumption indicate that water resources are insufficient for coordinating watershed socio-economic development and ensuring environmental quality in the Fenhe River basin. In this section, we set up two policy intervention scenarios to study the potential effect of appropriate policy interventions for further optimization of water resource allocation. To reflect the severe water shortage conditions, we consider a low water resource level (h = 1) and a ω value of 15 as the baseline scenario (S1), and another two policy scenarios are set: (1) S2: The water reuse rate gradually increases 10% over time; (2) S3: Coordinating the transferred water resources in the whole river basin for optimal regulation.
The risks in the three scenarios (Table 4)  The results indicate that improving the water reuse rate (S2) and coordinating the water resources of the whole river basin (S3) could increase the benefits and control regional risks, and S3 shows a more significant effect. The results can support decision makers to Additionally, ecological water consumption in this study reflects the gaps between pollutant emissions and environmental carrying capacity for different values, showing various changes due to regional and sectoral differences in pollutant emission intensities. In the midstream area, it shows fluctuation, and the amounts of ecological water are [51,044. 31 In the upstream and downstream areas, the amounts of ecological water show a downward trend as values increase; the developed model optimizes water resource allocation and reduces pollutant quantities entering the river in these areas.

Policy Scenarios Analysis
Risks and ecological water consumption indicate that water resources are insufficient for coordinating watershed socio-economic development and ensuring environmental quality in the Fenhe River basin. In this section, we set up two policy intervention scenarios to study the potential effect of appropriate policy interventions for further optimization of water resource allocation. To reflect the severe water shortage conditions, we consider a low water resource level (h = 1) and a ω value of 15 as the baseline scenario (S1), and another two policy scenarios are set: (1) S2: The water reuse rate gradually increases 10% over time; (2) S3: Coordinating the transferred water resources in the whole river basin for optimal regulation.
The risks in the three scenarios ( The results indicate that improving the water reuse rate (S2) and coordinating the water resources of the whole river basin (S3) could increase the benefits and control regional risks, and S3 shows a more significant effect. The results can support decision makers to formulate water management requirements and coordinate the optimal allocation of water resources in the entire river basin.  Figure 10 shows the deficits in water resource pre-allocations in different scenarios. The results show a regional difference. For example, in period 1, the water deficits of agriculture in S2 decrease from [3753.41, 8668.13] × 10 4 m 3 (S1) to [3539.54, 8535.12] × 10 4 m 3 , and in S3, they increase to [7255.75, 9467.58] × 10 4 m 3 . In the midstream and downstream areas, the results show different changes. The agricultural scale in the upstream area is relatively below, and S3 regulates the transferred water resources in the whole river basin, which leads to more water resources being allocated to other areas of higher benefits. formulate water management requirements and coordinate the optimal allocation of water resources in the entire river basin.  Figure 10 shows the deficits in water resource pre-allocations in different scenarios. The results show a regional difference. For example, in period 1, the water deficits of agriculture in S2 decrease from [3753.41, 8668.13] × 10 4 m 3 (S1) to [3539.54, 8535.12] × 10 4 m 3 , and in S3, they increase to [7255.75, 9467.58] × 10 4 m 3 . In the midstream and downstream areas, the results show different changes. The agricultural scale in the upstream area is relatively below, and S3 regulates the transferred water resources in the whole river basin, which leads to more water resources being allocated to other areas of higher benefits.  60,000 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2S3 S1S2 S3 70,000 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2S3 S1S2 S3 70,000 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2S3 S1S2 S3   (S3), respectively. It indicates that more water resources are allocated to the production sectors with higher benefits over the river basin. Water shortages mainly appear in the downstream areas, according to requirements of water resources and environment protection. The results show that water recycling is a critical factor for addressing water shortage and reducing pollutant emissions, and appropriate policy interventions would further optimize water resource allocation and effectively alleviate water shortage and water pollution in the Fenhe River Basin.  (S3), respectively. It indicates that more water resources are allocated to the production sectors with higher benefits over the river basin. Water shortages mainly appear in the downstream areas, according to requirements of water resources and environment protection. The results show that water recycling is a critical factor for addressing water shortage and reducing pollutant emissions, and appropriate policy interventions would further optimize water resource allocation and effectively alleviate water shortage and water pollution in the Fenhe River Basin.

Conclusions
In this study, an improved ITSDP model was built for optimal water resource allocation and pollutant emissions under dual constraints and uncertainties in the Fenhe River Basin, a highly urbanized, densely populated, typically water-deprived area with high degrees of contamination. The proposed model simultaneously addresses the uncertainties presented as interval values and probability distributions by integrating the IPP and TSP methods. The introduction of the downside risk aversion method effectively avoids possible risks caused by uneven water resource allocations. By solving the ITSDP model, optimal water resource allocation and pollutant emissions for the primary water use sectors were calculated for the programming periods under different scenarios. Additionally, we obtained the amounts of ecological water supplies required for purifying excessive pollutant emissions to identify areas with significant environmental water problems. These results, such as pollutant emissions and ecological water supplies, subject to strong water quality constraints, could provide a basis for regional emission permit systems. Furthermore, the optimal results under different risk levels and policy scenarios, namely S2 and S3, could reflect the subjective will of decision-makers for regional socio-economic development, environment protection, and possible changes of regional planning and policies.
The aim of this study was to use the ITSDP model to develop an effective approach to determine and optimize water resource allocations. The coordination of water quality

Conclusions
In this study, an improved ITSDP model was built for optimal water resource allocation and pollutant emissions under dual constraints and uncertainties in the Fenhe River Basin, a highly urbanized, densely populated, typically water-deprived area with high degrees of contamination. The proposed model simultaneously addresses the uncertainties presented as interval values and probability distributions by integrating the IPP and TSP methods. The introduction of the downside risk aversion method effectively avoids possible risks caused by uneven water resource allocations. By solving the ITSDP model, optimal water resource allocation and pollutant emissions for the primary water use sectors were calculated for the programming periods under different scenarios. Additionally, we obtained the amounts of ecological water supplies required for purifying excessive pollutant emissions to identify areas with significant environmental water problems. These results, such as pollutant emissions and ecological water supplies, subject to strong water quality constraints, could provide a basis for regional emission permit systems. Furthermore, the optimal results under different risk levels and policy scenarios, namely S2 and S3, could reflect the subjective will of decision-makers for regional socio-economic development, environment protection, and possible changes of regional planning and policies.
The aim of this study was to use the ITSDP model to develop an effective approach to determine and optimize water resource allocations. The coordination of water quality protection and socio-economic development could support the establishment of water-based industrial structures and layouts. The results suggest that this approach is applicable and effective for the optimization of water resource allocation and water quality management in the Fenhe River Basin and could also be applied in other contexts or water-stressed areas. However, this model does not consider the gross ecosystem production from wa-ter resources utilizations, which has been a significant indicator of regional sustainable development. Further studies are needed to address these limitations.

Conflicts of Interest:
The authors declare no conflict of interest.