1.1 Research background

In to optimize the redistribution of water resources and improve the efficiency of water use, the Chinese government actively encourages the development of water rights trading through market mechanisms. In 2012, the Chinese government proposed to actively conduct trials of trading water rights and pollution emission rights. In 2017, the government clearly proposed to "accelerate the construction of water rights markets, promote the confirmation and entry transactions of water resources use rights". Furthermore, the following requirement was stipulated: "the market needs to play a decisive role in resource allocation [1]".

According to national requirements, the Ministry of Water Resources and other relevant ministries actively promote transactions of water rights. In 2014, the Ministry of Water Resources issued the "Notice on Carrying out Water Rights Pilot Work", and initiated pilot work in Ningxia, Jiangxi, and five other provinces. These pilot projects provided experience and references for the national promotion of the construction of an appropriate water rights system. In 2016, the Ministry of Water Resources issued “the Interim Measures for the Administration of Water Rights Transactions”, which clarified the main forms of water rights transactions. These include regional water rights transactions, water withdrawal rights transactions, and water rights transactions of irrigation water users. In 2016, the Chinese water rights exchange was put into operation. In 2018, nine departments, such as the National Development and Reform Commission and the Ministry of Finance, jointly issued the "Action Plan for Establishing a Market-oriented and Diversified Ecological Protection Compensation Mechanism". This action plan proposed to encourage and guide water rights trading as well as to improve existing water rights trading platforms. According to statistical data, since 2016, more than 500 water rights transactions have been processed through the China water rights exchange. This means that Chinese water rights trading has entered a good stage of development.

The scientific calculation of the amount of tradable water rights is one of the core requirements of water rights trading. Together, the amount of tradable water rights, transaction actors, water rights transaction period, and standard water measurement constitute the four basic elements that affect water rights transaction prices [2]. The amount of tradable water rights refers to the amount of water resources, the transfer of which needs to be clarified in the water rights transaction agreement. For the transferor, under the premise of satisfying the transferor’s own production, living, and ecological water use, the amount of tradable water rights represents the maximum amount of water rights that can be sold through the water rights trading market. For the transferee, the amount of tradable water rights is the maximum amount of water rights purchased through the water rights trading market to compensate for the gap in water demand for production, life, and ecology.

The determination of the amount of tradable water rights must consider the natural and social capacities of both parties of the transaction, especially the transferor’s water consumption of the social and economic development and ecological safety. If the amount of tradable water rights to be traded between both parties is too large, it is very likely that a shortage of water resources will affect the social and economic development of the transferor in the future. Moreover, in serious cases, such a situation may cause disputes, and even damaged the durability and stability of the water rights transaction market. Therefore, studying the quantitative method of the scientific measurement of tradable water rights is not only conducive to determining the scale of water rights traded by both parties, but also to the sustainable development of the water rights trading market.

1.2 Research question

How to scientifically measure the amount of tradable water rights is a difficult issue that must be overcome for the development of water rights trading. At present, the method of using the transferor’s water saving potential for measuring the amount of tradable water rights has been generally accepted by the academic community [3]. This method offers the advantages of clear logical thinking and good interpretability. However, the application of this method faces two difficulties: the first difficulty is the involvement of many links and parameters in the measurement of the water-saving potential, which restricts the accurate measurement by many objective and subjective factors. Furthermore, the calculation is often too work intensive; the second difficulty is that while the water-saving potential is calculated, the proportion of water-saving potential that can be used for water rights transactions varies between different regions. The purpose of water rights trading is the improvement of the utilization efficiency of water resources. An unreasonable proportion of water-saving potential used for water right transaction can easily cause calculation errors, and may cause conflicts between the transferor and the transferee in the execution of the transaction agreement.

The basic premise of water rights trading is a difference in the marginal benefits of different entities that use water resources. The process of water rights trading is actually a process in which water use rights are transferred from the party with lower marginal revenue to the party with higher marginal revenue.

In general, the amount of tradable water rights agreed upon in the water rights transaction agreement mainly depends on the amount of tradable water rights the transferor can transfer. In the water rights transaction process, the legal rights of the transferee must be protected. Furthermore, the transferor’s water shortage risk cannot be increased. Therefore, measuring the amount of tradable water rights from the perspective of controlling the risk of water shortages the transferor faces is more conducive to safeguarding the rights and interests of both parties of the transaction, and can avoid potential risks caused by the transaction process. The present investigation showed that since China’s water right trading market is not yet complete, disputes between both parties of the execution of the transaction agreement often occur because of problems associated with the transaction volume and the transaction price. A case of water rights trading between DongYang and YiWu, Yiwu City (the transferee) yielded the permanent use rights of 50 million m³ of water resources per year from DongYang City (the transferor) for a one-time payment of 200 million CNY. In recent years, a number of people in DongYang, i.e., the transferor, have begun to believe that water rights trading has damaged their long-term interests, which has caused conflicts between both sides and led to the emergence of hidden dangers of social instability. Therefore, studying the measurement method of tradable water rights from the perspective of avoiding the water risk of the transferor has both theoretical and practical significance.

1.3 Literature review

How to determine the amount of tradable water rights is one of the key technical issues of the water rights trading mechanism. In recent years, with the development of a number of water rights trading practices, scholars have begun to conduct research on the tradable water rights from multiple angles: (1) Foreign scholars mainly studied the analysis methods of the amount of tradable water rights based on restrictive factors. Marino and Kemper [4] suggested that the establishment of water rights transaction must have sufficient information resources of the amount of tradable water rights and necessary infrastructure for measuring and transferring the traded water resources. Matthews et al. [5] showed that water rights trading for the transfer of agricultural irrigation water to industrial water will change river runoff and aggravate droughts during dry periods. Dellapenna [6] suggested that water rights trading may impact the ecological environment, economic development, and the lives of residents at water rights transfer sites. Therefore, Georgia stated that the amount of tradable water for cross-basin water rights trading must be the remaining water volume after meeting the water demand of the river basin. Guo et al. [7] jointly analyzed water rights trading and water conservation management and found that they are highly compatible and related. Moreover, it was also found that the combination of water rights trading and water conservation management can promote the implementation of water conservation management contracts. To accurately quantify the comprehensive value of water resources, based on market and administration, Danyang Di et al [8] proposed a water rights transaction game for optimizing the water transaction volume and transaction price for each region of the Yellow River Basin. Shen [9] proposed the concept of standard water, established a standard water quantity measurement model, converted the exchanged water quantity into a standard water quantity, and used DongYang and Yiwu as case studies to calculate the water rights transaction amount. The results help to ensure the fairness and permanence of the water rights transaction process. (2) Chinese scholars have also conducted theoretical and quantitative research on the amount of tradable water. Wu [10] established a two-stage water rights transaction equilibrium price calculation model based on shadow price theory. In their model, the first stage is the unilateral measurement stage, which constructs a shadow price model from both the transferor and the transferee. The second stage is the bilateral coupling measurement stage, which aims to maximize the overall benefit of the national economy, and uses the differential game equilibrium theory to determine the equilibrium water price. Hu et al. [11] proposed to use actual water users as subjects of water rights, and to change the water withdrawal system to a water consumption right system; moreover, they clearly defined water rights as water consumption rights, thus clarifying the ownership of property rights. Li et al. [12] considered that the amount of water rights allowed to be traded and how to calculate the volume of water taken in actual transactions is one of the key technical issues of the water rights trading mechanism. Under the premise of a given water use efficiency, by analyzing the industry’s advanced water-saving technologies (both in China and internationally), comparing existing and planned water-saving indicators, and analyzing the water-saving potential of specific regions and industries, the theoretical tradable water volume can be obtained. Zhao et al. [13] used the maximum overall economic and social benefits of a river basin as objective function, and applied the control of total water consumption and water loss as constraints to construct a water withdrawal right transaction model. By taking the Shaying River Basin as example, this model was used to optimize the second water rights transaction scheme. Tan [14] considered the difference in water guarantee rates between agriculture and industry, as well as the encroachment of industrial water on agricultural water; then, the amount of tradable water was calculated and converted from agricultural water to industrial water. Liu et al. [15] constructed a water rights transaction decision-making model, selected water rights transaction behaviors, made transaction decisions, and analyzed the potential water rights transaction volume in the study area. Their research assumed that decision-making behavior is constrained by the maximization of economic benefits of the main water body, and is also affected by the cost of water saving and the benefits obtained following the transaction. The research results provide a basis for the construction of the regional water rights market and the determination of the tradable water volume in the regional water market. Su [16] summarized the implementation of water rights conversion pilot projects in Ningxia and the Inner Mongolia Autonomous Region of the Yellow River Basin, and evaluated it from the four aspects of society, economy, ecology, and theory. Finally, the characteristics, problems, and prospects of the Yellow River water rights conversion scheme ware identified.

The above analysis showed that existing research on the amount of tradable water rights can be divided into two categories: ① With regard to qualitative research, existing research mainly focused on the control of the trading market, as well as the design of the trading mechanism and ownership management. ② With regard to quantitative research, foreign scholars studied methods for measuring the amount of tradable water rights mainly from the remaining water after meeting the water demand of the river basin; domestic scholars mainly studied this amount from the water saving potential.

Research on how to measure the amount of tradable water rights based on the change of water shortage risk has not yet been reported. Changes in the water supply and water demand of an area may change the water shortage risks of this area, which would then affect the socio-economic and ecological security of that area. This means that water rights transactions will inevitably cause changes in the transferor’s water supply to ensure that own needs are met. Consequently, the amount of tradable water rights should be adjusted according to the water shortage risk of the transferor.

1.4 Research idea

This paper proposes a new method for estimating the amount of tradable water rights. The basic principle is to measure the amount of tradable water rights under the premise of controlling the risk that the transferor experiences water shortage. In theory, this approach compensates for the deficiency of existing calculation methods such as large amounts of required calculations and calculation errors. In practice, this approach is more conducive to safeguard the rights and interests of both parties and to avoid potential risks associated with the water supply in the transaction process.

The steps of this model are summarized in the following:

1. Step 1: The risk evaluation index of water shortage of the transferor is constructed. Here, the combined method is used to measure the weights of different indicators. A fuzzy comprehensive evaluation model is used to evaluate the risk categories of water resources shortage.
2. Step 2: The mid-term and long-term water shortage risk thresholds of the transferor are set. According to the water inflow frequencies of 50%, 75%, and 90%, the evaluation index value of the planning year in class A and class B is predicted by different methods. The simulated annealing algorithm is used to calculate the theoretical value of tradable water rights.
3. Step 3: Combined with the case study of Helan County in Ningxia Autonomous Region, China, the recommended value of tradable water rights of this County is obtained by comprehensive analysis.
4. Step 4: Discussion and analysis of the results. To verify the robustness of the model, various methods were used to calculate the weights of evaluation indexes and to compare the calculation results of tradable water rights. To verify the feasibility of the proposed method, the results are compared with the obtained conclusions when the water-saving potential is used.

A basic flow chart of the proposed method is shown in Fig 1.

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Fig 1. Basic flow chart of the proposed method.

https://doi.org/10.1371/journal.pone.0254428.g001

The innovation of this work is reflected in to following: A new method for measuring tradable water rights is proposed from the perspective of assessing the water shortage risk of the transferor. Specifically, the water shortage risk evaluation index of the transferor is constructed from the following four factors: natural and social conditions of water resources of the transferor, water supply capacity, water demand degree, and regional water ecological environment change trend. The fuzzy comprehensive evaluation method is used to construct the risk classification evaluation model of water resource shortage. A tradable water rights measurement method is proposed based on a simulated annealing algorithm.

The remainder of this paper is organized as follows: Section 2 introduces the study area. Section 3 introduces the methods. Section 4 analyzes the results. Section 5 discusses the findings and Section 6 concludes the study.

3.1. Evaluation model of transferor’s water shortage risk

3.1.1. Establishing the transferor’s water shortage risk evaluation index system.

According to Malthus’s Absolute Resource Scarcity Theory and Ricardo’s Relative Resource Scarcity Theory [18], the scarcity of water resources can be classified into absolute scarcity and relative scarcity. The natural endowment of water resources is the main source of its “absolute” scarcity. Because of the rapid socio-economic development over recent years, water resources are increasingly "relatively" scarce. The two main reasons for their relative scarcity are: first, the imbalance between supply and demand, and second, with continuously increasing utilization of water resources, the discharge of sewage also increases, which affects the quantity of available high-quality water resources and causes the scarcity of water resources.

Based on the above analysis, and related literature [19–24] about the water shortage risk evaluation index system, the evaluation index system of transferor’s water shortage risk was divided into the following four aspects: natural and social conditions of water resources, supply of water resources, demand for water resources, and the ecological water environment, as shown in Table 1.

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Table 1. Index system of transferor’s water shortage risk.

https://doi.org/10.1371/journal.pone.0254428.t001

Of the factors listed in Table 1, after the transaction of water rights, the transferor’s "water resources natural and social conditions" and "water ecological environment" factors will generally not change because of the amount of traded water rights. The indicators "water supply per capita" and "water supply rate" under the "water resources supply" category will be directly and negatively affected by the trading of water rights. The "surface water supply ratio" and "ground water supply ratio" will not show obvious changes. Furthermore, the water rights transaction is bound to encourage the transferor to conduct positive water-saving policies, and to restrain water resource demand indicators. Based on the above analysis, this paper focuses on the impact of tradable water rights on transferor’s "water supply per capita" and "water supply rate" indicators. For the prediction of relevant indicators the influence of tradable water rights on water resource demand will be appropriately adjusted by parameters.

3.2. Method for measuring tradable water rights of the transferor

3.2.2. Determination of the tradable water rights based on the simulated annealing algorithm model.

The simulated annealing algorithm is a stochastic optimization algorithm based on the Monte Carlo iteration strategy. The random solution generated by the simulated annealing algorithm starts from a higher initial temperature, and finds the global optimal solution in the solution space with the continuous decrease of temperature [25]. This paper designed 16 indexes to measure the water resources shortage risk. Among them, the indexes of "water supply per capita" and "water supply rate" change with changing tradable water rights, which then influences the risk level of water resource shortages. In this paper, the measurement of tradable water rights represents an attempt to investigate the change of the water resources shortage risk level through the change of tradable water rights. It satisfies the mechanism of the simulated annealing algorithm based on metropolis criterion to explore the target. Moreover, in the process of using the simulated annealing algorithm to calculate tradable water rights, this paper designs the situation of "increasing" or "decreasing" the tradable water rights, to avoid the algorithm falling into local optimization. The algorithm includes four core steps, and the specific process is shown in Fig 2.

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Fig 2. Flow chart of the simulated annealing algorithm for calculating tradable water rights.

https://doi.org/10.1371/journal.pone.0254428.g002

1. Step 1: "Start". Data is imported and the model is initialized. Formulas (6)–(8) are used to determine the index weight (a₁ in Fig 2) and predict the index value in the case of no water right trading in a year and a certain frequency of water (a₂ in Fig 2). Setting the parameters of the simulated annealing algorithm (b₁ in Fig 2), where the initial temperature is T_start and the end temperature is T_end. The cooling equation is as follows: (10) Setting θ ← T_start, and the initial value of cycle count n ← 1. An initial value of tradable water right is randomly generated, and is denoted as y⁽¹⁾(b₂ in Fig 2), and is used as current optimal solution.
2. Step 2: "Judgment", judging whether the new plan can be accepted. For the current tradable water rights y⁽ⁿ⁾, the affected index values can be calculated (c₁ in Fig 2). Then, the impacts of the change of tradable water rights on the "water supply per capita" and the "water supply rate" are analyzed. The specific formulas are shown in the following: (11) (12)
   Where, y⁽ⁿ⁾ represents the current (nth round) tradable water rights assignment, P_supply represents the total population of the region, and W_total represents the total regional water resources.
   Using Formulas (3)–(5), the membership degree can be determined according to the standard value of the evaluation grade (c₂ in Fig 2). Formula (2) is used to calculate the transferor’s water resources shortage risk level (c₃ in Fig 2), and diagnose it by using the discriminant criteria set in Formula (9): if the condition (9) is met, the new scheme will be "accepted"; if the condition (9) is not met, the new scheme will be "rejected".
3. Step 3: "Adjustment", this paper adjusts and optimizes the scheme through iteration. According to the diagnosis results, if the new scheme is "accepted", the iterative process will be automatically transferred to the "increase area" of tradable water rights; however, if the new scheme is "rejected", the iterative process will be automatically transferred to the "reduction area" of tradable water rights. The order is n ← n + 1, and a new scheme can be obtained by "increasing" (e₁ in Fig 2) or "reducing" (e₂ in Fig 2) the currently tradable water rights. The adjustment formula of the new scheme is shown as Formula (13): (13) where, Δ represents the increment or the reduction, and is predetermined according to the allowable deviation of the tradable water rights.
   The order is θ ← T(n) (f in Fig 2). Judging the relationship between θ and T_end, if θ ≥ T_end, Step 2 and Step 3 are repeated to accept the next round of diagnosis, if θ < T_end, then go to Step 4 and output the current scheme as optimal scheme.
4. Step 4: "Output", outputting the scheme by taking the average value of two rounds as the theoretical value of tradable water rights (h in Fig 2): (14)

3.2.4. Index prediction.

To calculate the risk level of water shortage of the transferor in a planning year (i.e., the kth year), it is necessary to predict the evaluation index of the planning year, and to analyze the change of the index value according to the three types of water inflow frequencies. Moreover, to improve the accuracy of the prediction, this paper divides the evaluation indicators into two categories based on data availability. These are indicators A and B. Category A indicators can be directly obtained (or their development trend can be estimated) based on the annual average value, as well as relevant national and local planning reports. For category B indicators, it is generally impossible to directly obtain forecast data through planning reports; therefore, reasonable methods should be adopted for their prediction. Moreover, as the specific prediction method of category B index is related to the case, the specific prediction model is proposed based on the sample data and the characteristics of the case in this paper. The specific division of category A and B indicators is shown in Table 2.

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Table 2. Prediction methods of different evaluation indexes.

https://doi.org/10.1371/journal.pone.0254428.t002

4.1. Data Sources, index prediction, and parameter determination

1. Relevant parameters.
   The inflow frequencies were set to 50%, 75%, and 90%. The current year was 2018. Since the water rights trading period of Helan County is basically set for 20–25 years and is implemented from 2020, two planning years are set: 2030 (medium-term) and 2040 (long-term).
2. Sample data sources.
   Since the Chinese government proposed to implement the strictest water resource management system in 2011, to make the sample data comparable, this paper focuses on the analysis of basic data from 2012 to 2018. ① Data reflecting the natural endowment of water resources uses the average value of many years, and is mainly obtained from local water resources bulletins or water resources statistical bulletins. These dates contain total water resources, surface water, underground water, precipitation, evaporation, among other relevant factors. ② Social and economic data are obtained from governmental and departmental work reports or socio-economic statistical yearbooks, and mainly include total population, land area, GDP, industrial added value, and irrigation area. ③ Other water resources and water environmental data are mainly obtained from or are referred to relevant policy documents, planning texts, and research results of Helan County. Specific data sources mainly refer to the "Ningxia Water Resources Allocation Guarantee Plan (2016–2020)", the "Ningxia Agricultural Irrigation Water Quota" (2014), the "Ningxia Current Status of Agriculture in 2018 Report on the Results of Measurement and Analysis of the Effective Utilization Coefficient of Irrigation Water”, the “Outline of the 13th Five-Year Plan for National Economic and Social Development of Helan County", the "Thirteenth Five-Year Plan for Water Conservancy Development of Helan County" (2016), and the "Thirteenth Five-Year Plan for Agricultural Industry Development in Helan County" (2016). These are the data sources of industrial sewage discharge, domestic sewage discharge, water function area compliance rate, water supply, water consumption, effective utilization coefficient of farmland irrigation water, water consumption of 10⁴ Yuan industrial added value, and water consumption rate.
3. Prediction of relevant indicators.
   According to statistical data, the average water consumption of Helan County from 2012 to 2018 was 433.8 million m³, among which, 399.9 million m³ of water originated from the Yellow River, accounting for 92.21%; and 33.8 million m³ originated from groundwater, accounting for 7.79%. Among this consumption, agricultural and ecological water withdrawals reached up to 411.8 million m³, accounting for 94.9% of the total water consumption. Consequently, the runoff of the Yellow River trunk stream was the main water source of Helan County. Considering regional factors, the measured runoff data from 1980 to 2018 of Lanzhou hydrological station in the mainstream of the Yellow River were selected as the main basis for adjusting the change of water supply under different inflow frequencies.
   The related prediction process of category A indicators are shown in Table 3.
   Category B indicators need to build a model for prediction. Specific ideas and prediction models are listed in Table 4.
   Through the above model or method, the predict results for each indicator were obtained, as shown in Table 5.
4. The evaluation index corresponds to the evaluation standard value range.
   In reference to relevant national standards, prior literature, and in combination with the "vast land and few people" characteristics of Helan County, the evaluation index corresponding to the evaluation standard value range was determined, as shown in Table 6.
5. Determination of index weight. The combined weight is determined according to Formulas (6)–(8), as shown in Table 6.
6. Determination of Δ. Based on the analysis of existing water rights trading cases in Ningxia Autonomous Region, the allowable deviation of tradable water rights was set as 500,000 m³; therefore, Δ = 1 million m³.

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Table 3. Forecast description of Class A indicators.

https://doi.org/10.1371/journal.pone.0254428.t003

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Table 4. Prediction description and prediction model or method of category B indicators.

https://doi.org/10.1371/journal.pone.0254428.t004

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Fig 3. Changes of each indicator from 2012 to 2018.

(a) Change of total water consumption. (b) Change of utilization coefficient of farmland irrigation water. (c) Change of water consumption rate. (d) Change of industrial water consumption. (e) Change of domestic water consumption.

https://doi.org/10.1371/journal.pone.0254428.g003

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Table 5. Actual value in 2018 and forecast value of indicators in 2030 and 2040 with different inflow frequencies.

https://doi.org/10.1371/journal.pone.0254428.t005

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Table 6. Index weights and corresponding evaluation standard value ranges.

https://doi.org/10.1371/journal.pone.0254428.t006

4.2. Analyzing the risk evolution trend of the shortage of water resources in Helan County without water rights trading

1. Water shortage risk in 2018
   According to the data in Table 2, the grading interval shown in Table 3, and Formulas (3)–(5), r_ij can be obtained as follows:
   According to the weight of Table 3, and Formula (2), B = (0.389,0.179,0.176,0.082,0.174). This means that the evaluation grade is v₁.
2. Water shortage risk without water rights trading in 2030
   Similarly, the water shortage risk assessment grade in 2030 without water right trading would be as follows: Under a water inflow frequency of 50%, B = (0.370, 0.237, 0.168, 0.050, 0.174), the evaluation grade is v₁; under a water inflow frequency of 75%, B = (0.317, 0.273, 0.185, 0.051, 0.174), the evaluation level is v₁; under a water inflow frequency of 90%, B = (0.253, 0.337, 0.185, 0.051, 0.174), the evaluation level is v₂.
3. Water shortage risk without water rights trading in 2040
   Similarly, the water shortage risk assessment grade in 2040 without water right trading would be as follows: Under a water inflow frequency of 50%, B = (0.366, 0.251, 0.154, 0.055, 0.174), the evaluation grade is v₁; under a water inflow frequency of 75%, B = (0.301, 0.269, 0.201, 0.055, 0.174), the evaluation level is v₁; under a water inflow frequency of 90%, B = (0.239, 0.331, 0.201, 0.055, 0.174), the evaluation level is v₂.
4. Evolution trend analysis
   The evaluation result for 2018 is compared with the predicted results for 2030 and 2040 under the water inflow frequency of 50%, and a radar chart is drawn, as shown in Fig 4. Fig 4 shows that the risk grade of water resource shortages in 2018, 2030, and 2040 all maintained the v₁ level, but a trend of "gravity shift backward" exists among the five evaluation levels. This means that the value of v₁ tends to decrease, while the value of v₂ tends to increase.

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Fig 4. Comparison of evaluation grades for three years without water right trading.

https://doi.org/10.1371/journal.pone.0254428.g004

4.3. Calculation of tradable water rights in 2030 based on risk analysis of water shortage

Based on the impact water rights trading exerts on the risk of water shortage in Helan County in 2030, using MATLAB (version 7.01), the simulated annealing algorithm and the theoretical values of the amount of tradable water rights under the water inflow frequency of 50%, 75%, and 90% were calculated. The process of increasing and decreasing is calculated by equivalent method, and Δ = 1 million m³.

1. Under a water inflow frequency of 50%, the corresponding index value is obtained by Formulas (11) and (12), and then based on the process shown in Fig 1, the amount of tradable water rights is calculated. Finally, after iteration, based on Formula (14), the theoretical value of the amount of tradable water rights was obtained as 75 million m³.
2. Similarly, under a water inflow frequency of 75%, after iteration, the theoretical value of the amount of tradable water rights is 15 million m³.
3. Under a water inflow frequency of 90%, since the water resource supply quantity has exceeded the bottom line of water demand, Helan County cannot provide the required amount of tradable water rights anymore. This means that Helan County, as the transferor, has a theoretical value of the amount of tradable water rights of 0.

4.4. Calculation of the amount of tradable water rights in 2040 based on risk analysis of water shortage

Similarly, it can be calculated that:

1. Under a water inflow frequency of 50%, the corresponding index value can be obtained by Formulas (11) and (12). Then, based on the process shown in Fig 1, the amount of tradable water rights can be calculated. Finally, after iteration, based on Formula (14), the theoretical value of the amount of tradable water rights is obtained as 65 million m³.
2. Under a water inflow frequency of 75%, after iteration, the theoretical value of the amount of tradable water rights is 17 million m³.
3. Under a water inflow frequency of 90%, similarly, Helan County also cannot provide the required amount of tradable water rights. This means that Helan County, the transferor, has a theoretical value of the amount of tradable water rights of 0.

The theoretical values of the amount of tradable water rights under three scenarios in the two planning years are compared and analyzed, as shown in Fig 5.

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Fig 5. Comparison of theoretical values of the amounts of tradable water rights in different planning years and scenarios.

https://doi.org/10.1371/journal.pone.0254428.g005

4.5 Determination of the recommended value of tradable water rights

According to the above analysis, the following conclusions can be presented. During the implementation of water right trade in Helan County, to maintain a "low" level of risk in 2030, the theoretical values of tradable water rights are 75 million m³ and 15 million m³, under 50% and 75% water frequencies, respectively. There no tradable water rights exist under a 90% frequency. Similarly, to ensure that the risk classification in 2040 maintains "low" or "lower" levels, the theoretical value is 75 million m³ under 50% frequency and 17 million m³ under 65% frequency. Tradable water shortage exists at a frequency of 95%.

1. Helan County needs to control the water shortage crisis that is simultaneously caused by water right trading in 2030 and 2040. Therefore, the intersection of the calculation results of the two planning years should be implements with relatively strict standards.
2. The theoretical values of tradable water rights in Helan County vary greatly under different water frequencies. It is therefore necessary to not only consider the adverse effects induced by natural factors, but also the driving factors of completed transactions to both society and economy. Therefore, given the existence of episodes of extreme situations under 90% frequency, it is more reasonable and realistic to apply calculation results with 50% and 75% water frequencies as the main basis to determine the amount of tradable water rights.

In summary, the amount of tradable water rights in Helan County should be controlled at [15 million m³, 65 million m³], and the recommended value is 40 million m³ using the mean value method.

5.1 Comparison with other weight calculation methods

The subjective and objective combination method was adopted to obtain the weight of water shortage risk assessment indicators; then, the tradable water rights were calculated. To test the possible effects of differently weighted calculation methods on the tradable water rights measurement results, the subjective weight measurement methods CRITIC and CRITIC-M, as well as the objective weight measurement methods LBWA and FUCOM [28–30] were used to calculate the weights. Then, the corresponding tradable water rights were calculated, and the changes in the results of tradable water rights were analyzed.

5.1.1 Four other types of weight calculation methods.

1. The CRITIC method. To improve the accuracy of the calculation, the evaluation index values of five different counties located in the upper reaches of the Yellow River were collected. Helan County was used as the case study, and six schemes were established for analysis and calculation. The following steps were taken: ① Constructing the initial decision matrix X = [ξ_ij]_m×n, where ξ_ij represents the attribute value of index j of scheme i; ② Normalizing the index value. To maximize criteria, was used; to minimize criteria, where was used; ③ Calculating the standard deviation σ_j; ④ Constructing the matrix L = [l_jk]_n×n, containing the coefficients of linear correlation of vectors ξ_j and ξ_k; ⑤ is used to calculate the weights, where and W_j = σ_j⋅φ_j.
2. The CRITIC-M method. The data of the same six schemes are also used for analysis and calculation in this case. The following steps are taken: ① Constructing the initial decision matrix X = [ξ_ij]_m×n, where ξ_ij represents the attribute value of index j of scheme i; ② To normalize the index value, and maximize criteria, where is used; to minimize criteria, where is used. ③ Calculating the standard deviation σ_j; ④Formula is used to calculate the weights, where .
3. The LBWA method. Ten experts were invited to form a panel of experts. These experts mainly came from the China Water Rights Exchange, local water resources trading center, university professors, and the water administration department. The following steps were taken: ① The most important criterion was determined from the set of criteria S = {C₁, C₂, …, C₁₆}. In the defined problem, criterion C₁₀ was selected as the most important/influential criterion; ② criteria are grouped by their levels of significance. In accordance with the preferences of decision makers, the criteria are grouped in the following subsets/levels: S₁ = {C₁₀, C₁, C₂, C₃, C₆, C₇}, S₂ = {C₁₁, C₁₂, C₁₃, C₁₄, C₁₅, C₁₆}, S₃ = {C₄, C₅}, and S₃ = {C₈, C₉}; ③ The maximum value of the scale is defined for comparing the criteria; let r = 6. Based on the preferences of decision makers, the following relationships can be defined: Level S₁, I₁₀ = 0, I₆ = I₇ = 1, I₁ = I₂ = 2, I₆ = 3; Level S₂, I₁₁ = I₁₂ = I₁₃ = I₁₄ = I₁₅ = 1, I₁₆ = 2; Level S₃, I₄ = I₅ = 1; Level S₄, I₈ = I₉ = 4. ④ Let r₀ = 9 and calculate the corresponding value f(C_i); ⑤ the weight w_i is calculated.
4. The FUCOM method. The 10 experts mentioned above were also employed for this method. The following steps were taken: ① The decision-makers ranked the criteria as: C₁₀ > C₃ > C₂ > C₁ > C₇ > C₆ > C₁₁ > C₁₂ > C₁₃ > C₁₄ > C₁₅ > C₁₆ > C₄ > C₅ > C₈ > C₉; ② Based on the decision-makers’ preferences, the comparative priorities of the ranked criteria were determined and the vector of the comparative priorities of the evaluation criteria was obtained as listed in Table 7.

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Table 7. Vector of the comparative priorities of the evaluation criteria.

https://doi.org/10.1371/journal.pone.0254428.t007

③ According to and , the relationship between index weights should be calculated; ④ the optimization model was constructed and solved to obtain the index weight

The weights calculated by the four methods are shown in Table 8:

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Table 8. Weights calculated by the four methods.

https://doi.org/10.1371/journal.pone.0254428.t008

5.1.2 Comparative analysis of water rights trading volumes under different weights.

In accordance with the inflow frequency, 50%, 75%, and 90% of water rights were assumed, and the weights were calculated (as shown in Table 8) and were used according to the process shown in Fig 1. The theoretical value of tradable water rights in 2030 and 2040 was calculated by using Formula (14), as shown in Table 9. For the convenience of comparison, the theoretical value of water right trading volume, calculated by combination weight above, is listed in Table 9.

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Table 9. Theoretical values of water right trading volume under four weights (10⁴ m³).

https://doi.org/10.1371/journal.pone.0254428.t009

1. The calculation results of 2030 show that: ① At 50% incoming water frequency, the result calculated by the CRITIC method is 78 million m³, which implies an increase of 3 million m³ over the original result, representing an increase of 4%. The results obtained by the other three methods are all lower than the original results, where the LBWA and FUCOM methods are both reduced by 5 million m³, representing a reduction ratio of 6.7%, and the CRITIC-M method is reduced by 1 million m³, representing a reduction ratio of less than 2%. ② At 75% water inflow frequency, the result calculated by the CRITIC method is 17 million m³, which is 2 million m³ more than the original result, representing an increase of 13.3%. The results obtained by the other three methods are not different compared with the original results. ③ Under a water inflow frequency of 90%, the calculation results do not change.
2. The calculation results of 2040 show that: ① The results calculated by CRITIC and LBWA at 50% incoming water frequency did not change compared with the original results. The results calculated by CRITIC-M and FUCOM are both 55 million m³, implying a decrease by 10 million m³, with a reduction ratio of 15.4%. ② In case of an incoming water frequency of 75%, the results calculated by the LBWA method did not change compared with the original results. The results measured by CRITIC, CRITIC-M, and FUCOM methods are reduced by 1 million m³, 2 million m³, and 5 million m³, respectively, compared with the original results. The associated reduction rates are 6.7%, 13.3%, and 33.3%, respectively. ③ Under a water inflow frequency of 90%, the calculation results did not change.

In general, the theoretical value of the water right trading volume calculated by these four weight calculation methods does not significantly deviate from the original results except in specific cases. This shows that the model proposed in this paper has strong robustness. To control the measurement error, this paper suggests that in addition to the FUCOM method, the other three weight calculation methods CRITIC, CRITIC-M, and LBWA can also be selectively used in practical applications.

5.2 Comparative analysis of tradable water rights based on the water-saving potential

According to the “Water Rights Trade Indicator Analysis Report in Helan Country of China”, the amount of water-saving in the status year and the water-saving potential in the planning year of Helan County can be calculated as follows:

1. In the current year, the annual average water-saving amount from 2012 to 2019 was 13.6839 million m³, where 9.45 million m³ originate from governmental investment in slab-lined canals and efficient water-saving irrigation projects; 423 million m³ originate from water users’ investment in water-saving irrigation projects.
2. For the planning year of 2045, the comprehensive water-saving potential is 95.46 million m³ based on the right index. The approach of subentry calculating the total potential is 63.6302 million m³, 56.6187 million m³ of which originates from the government’s water-saving reform engineering on irrigated areas, and 7011500 m³ originates from water users’ efficient water-saving irrigation projects. In Helan County, the water-saving potential mostly originates from agriculture, and the key measurement index data include the reduction of rice planting areas and an increased efficiency of water-saving in irrigated areas. The rice planting area in Helan County for the planning year will be controlled within 129500 mu, which represents a decrease by 57500 mu compared with the current situation; the proportion of the efficient water-saving irrigation area will increase by 40%, i.e., to 235800 mu.

From the perspective of controlling the water shortage risk of transferors, the amount of tradable water rights of Helan County is [15 million m³, 365 million m³]. The upper end of this interval is very close to the water-saving potential of 63.6302 million m³, as calculated by sub items; however, it differs from the value 95.46 million m³, which was obtained based on the right index. The main reason is that under the right index allocation, the right confirmation quantity holds a margin of the actual water supply in the current year. Only part of the water-saving potential can be used for water rights transactions; consequently, the suggested value of the tradable water rights is 40 million m³, retaining approximately 2/3 of the water saving potential of 63.6302 million m³. This proportion is in line with mainstream beliefs, and further verifies the feasibility to control transferors’ risk of water shortage. These results show that the method proposed in this paper forms a good complementary and corroborative relationship with that of using the water-saving potential of the transferor to calculate the amount of tradable water rights.