Uncertainties arising from extreme climate events and human activities pose a challenge to the efficient allocation of water resources. In this study, a type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP) is developed to support the water resource management system under uncertainty by incorporating type-2 fuzzy sets, chance-constrained programming, and fractional programming into a comprehensive multi-objective optimization framework. The model enables the trade-off between economic, social, and environmental sustainability and provides water supply solutions associated with different levels of fuzzy uncertainty and risk of violating constraints. The T2F-CCLFP model is applied to Taiyuan, Shanxi Province, China, to support its water resource management. Results reveal that: (i) the industrial structure is transitioning toward diverse industries from energy and heavy industry dominance; (ii) external water transfer will be the major water-supply sources for the city in the future, accounting for 55 and 50% of the total water supply in 2025 and 2030, respectively; (iii) the water-supply security of the city is enhanced by provoking the utilization of reclaimed water (the annual growth rate is 13.9%). The results are helpful for managers in adjusting the current industry structure, enhancing water supply security, and contributing to the sustainable development of socio-economic and water systems.

  • Complexities due to multi-objective functions, fractional non-linearity, chance-constraint, and type-2 fuzzy uncertainties are solved.

  • Economic efficiency, fairness, and sustainability of the water-resource allocation system are balanced.

  • T2F-CCLFP has superior ability to express uncertainty, compared with CCLFP.

In recent decades, the demand for water has experienced a significant increase due to factors such as population growth, rapid industrialization, and accelerated urbanization. In addition, the availability of future water resources is increasingly uncertain as a result of extreme climate events and human activities (Hasti & Ahmad 2021). This situation has the potential to worsen conflicts related to water supply and demand, as well as impede socio-economic sustainability (Gunasekara et al. 2014). To address this issue and alleviate the supply–demand imbalance, effective water resource management measures must be implemented. One of the primary strategies for addressing this issue involves the optimization of water resource allocation (Xu et al. 2019). Optimization of water resource allocation, as a non-engineering measure, offers advantages such as low cost and ease of implementation (Li et al. 2022). It can also strike a balance between economic development and environmental protection, ensuring fair distribution of water resources among different sectors.

Numerous optimization techniques have been previously developed to address the challenge of efficiently and environmentally managing the imbalance between water resource supply and demand (Shen & Speed 2009; Divakar et al. 2011; Fu et al. 2019). Hu et al. (2016) established a multi-objective double-layer model of water resources in the basin that can not only reflect the principles of fair and stable water distribution by basin authorities, but also ensure the maximization of economic benefits of each division, and the model is applied to the Qujiang River Basin in China, which proves the rationality of the model. Tian et al. (2019) introduced a water allocation method that aims to address conflicts arising from diverse objectives and competing regions. By leveraging Sperner's lemma, the proposed method facilitates the attainment of Nash equilibrium in the distribution process. This approach proves to be highly effective in managing water resources and exhibits promising potential for application in other river basins. Li et al. (2021) constructed a theoretical framework of water resources allocation based on ecological priority, and established a mathematical model of water resources allocation based on the maximization of economic, social and ecological benefits, which realized the rational allocation of water resources under the ecological priority mode. The aforementioned studies primarily focus on water resource efficiency, equity, and ecological protection. However, it is important to note that they may not fully consider the complexities and obstacles related to uncertainty factors in water supply and demand. These uncertainties can pose difficulties in effectively solving the problem of water resource allocation. To overcome these challenges, it is essential to incorporate uncertainty factors into water resource management strategies.

The uncertainty and complexity of the water distribution process arise primarily from uncertainty factors resulting from the random nature of natural events, the ambiguity of objectives and constraints, and errors in parameter estimation (Regulwar & Gurav 2011; Yue et al. 2020), especially in analyses and studies of hydrological characteristics, hydraulic construction, and water demand and supply. To deal with these uncertainties, mathematical methodologies such as interval parameter programming, fuzzy programming (FP), and stochastic programming are adopted (Gu et al. 2013; Ren et al. 2017; Li et al. 2019; Zhang et al. 2019). In the context of water resources systems, various parameters, including reservoir inflow (river flow), rainfall, and evapotranspiration, demonstrate stochastic properties. This characteristic has led to the widespread use of stochastic planning in water resources management, particularly the chance-constrained programming (CCP) method. The CCP method utilizes probability density functions (PDFs) to represent random parameters, providing valuable information on the level of violation risk between the objective function and constraints efficiently (Xiao et al. 2021). Although the CCP method effectively handles the probability distribution on the right side of constraints, it fails to consider any ambiguity present in the constraints (Guo & Huang 2009).

In the context of water resource management systems, numerous parameters demonstrate fuzzy characteristics. These include the volume of water consumed as well as the associated benefits and costs (Yue et al. 2020). FP can effectively capture ambiguity in resource allocation. The FP method utilizes fuzzy sets to incorporate membership concepts and address uncertainty in events. In practice, however, accurately deriving a fuzzy set membership function in the presence of multiple uncertainties can be challenging (Jana et al. 2017). Zadeh (1975) introduced type-2 fuzzy sets (T2FS) as an extension of ordinary fuzzy sets (Qin et al. 2011). This extension allows for an additional degree of design freedom by providing a membership degree that ranges from 0 to 1. The T2FS has been found to improve the handling of fuzzy uncertainty information in a more effective manner compared to traditional fuzzy sets (Suo et al. 2017). However, same as ordinary fuzzy sets, T2FS cannot reflect the risk of violating uncertainty constraints, which may lead to information loss and reduced decision robustness if such uncertain information is not adequately handled. Therefore, combining CCP and T2FS to solve multiple uncertainty problems in realistic applications can effectively deal with fuzzy sets and probability distributions (Cheng et al. 2018).

In the context of water management uncertainty, it becomes imperative to address the optimal problem of marginal benefits associated with optimizing the ratio of physical or economic quantities within the existing system. Thus, the linear fractional programming (LFP) can better accommodate the decision-making problems encountered in real-world scenarios (Emam 2013), and is designed to maximize output per unit of water input. This approach aligns with the principle of water resources utilization efficiency and is also consistent with the principles of sustainable development (Ren et al. 2016). In order to optimize the industrialization and structure of water resources allocation, it is important to balance the efficiency of water resources utilization with the objectives of promoting social stability and achieving sustainable development.

Hence, in order to tackle the intricacies and uncertainties associated with water resource management systems, it is suggested that incorporating T2FS, CCPs, and LFPs into a multi-objective planning framework for sustainable water resource management is both imperative and significant. However, this integration is seldom explored in the current literature. Therefore, this paper develops a water resource management model based on type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP). The T2F-CCLFP model incorporates objectives such as efficiency, equity, and sustainability. The proposed model was applied to Taiyuan city, situated in Shanxi Province, a region characterized by semi-humid conditions. This paper presents the primary contributions as follows: (1) The integration of fractional planning as a means to achieve benefit objectives in multi-objective planning, enabling the development of more sustainable water allocation schemes by simultaneously considering economic development, social equity, and environmental protection; (2) Uncertainties are effectively addressed in this approach through the utilization of probability and likelihood distributions. Additionally, the method is capable of handling multiple uncertainties by employing T2FS. This tool offers decision-makers a range of optimal solutions that are determined by varying degrees of fuzziness and violations. The model facilitates the determination of water resource allocation outcomes among various water sources and water users. The analysis also uncovers the connections between water supply sources and competing users, offering valuable insights for the modification of industrial and water supply frameworks in Taiyuan City. The developed model provides prospective guidance for efficient water resource management under multiple uncertainties.

The coordination of social, economic and ecological benefits is essential in the allocation of water resources to different sectors. This necessitates the incorporation of three key criteria, namely fairness, efficiency, and sustainability, into the water resources allocation model. Meanwhile, a water resources management model of T2F-CCLFP was developed to solve the large amount of uncertain information in the water resources allocation system (as shown in Figure 1). The framework comprises four primary components: (1) CCP to handle stochastic uncertainty; (2) T2FS to address multiple fuzzy uncertainties; (3) establishment of a multi-objective fractional programming model to reflect water distribution sustainability, equity, and efficiency; and (4) development and application of a T2F-CCLFP model with multiple water sources and multiple water users in Taiyuan City by incorporating CCP and T2FS into component (3).
Figure 1

Framework of the T2F-CCPFP model for the study.

Figure 1

Framework of the T2F-CCPFP model for the study.

Close modal

Type-2 trapezoidal fuzzy programming

The T2FS is a proposed as an extension of type-1 fuzzy set (T1FS). It incorporates the concept of possibility theory to quantify the degree of uncertainty associated with a fuzzy set. The true values of T2FS are represented as ordinary fuzzy sets within the unit interval, indicating fuzzy truth values (Dutta & Jana 2017). Compared to T1FS, T2FS are useful in situations where more uncertainty needs to be dealt with, which makes them very attractive in many practical problems (Taká 2014). Liu et al. (2014) extended the concept of type-2 trapezoidal fuzzy variables (T2TrFV) based on the type-2 triangular fuzzy variables (T2TFV) proposed by Liu & Liu (2010), the flexibility of trapezoidal fuzzy variables is better than that of triangular fuzzy variables (Zhang et al. 2014).

If is denoted as type-2 trapezoidal fuzzy variable, then , where are real values, , are two parameters that characterize the degree of uncertainty of taking the value x, they are used to represent the spreads of primary possibilities of type-2 trapezoidal fuzzy variable.

Then, for , the secondary possibility distribution function of is in the form:
(1)
The complexity associated with directly addressing the ambiguity of T2FS in practical applications is significantly high. Hence, it is imperative to convert the outputs of T2FS to T1FS by employing reduction techniques, followed by the process of defuzzification in order to obtain a crisp output. Qin et al. (2011) proposed a critical value (CV)-based reduction method for a type-2 fuzzy variable. Let be a T2FV with secondary possibility distribution function , the method is to introduce the CVs as representing values for T1FS , i.e., (optimistic CV), (pessimistic CV), (neutral CV). These CVs are defined as Equations (2a)–(2c):
(2a)
(2b)
(2c)
To obtain crisp values in practical applications, Dutta & Jana (2017) deduced the expectation of the corresponding T2TrFV reduction based on the three CVs of T2TrFV for defuzzification, as shown in Equations (3a)–(3c):
(3a)
(3b)
(3c)
where , , and represent the reductions of the T2TrFV , obtained by the optimistic, pessimistic, and neutral CV reduction methods, respectively.

T2FS can handle fuzzy information in objective functions and constraints but cannot solve the problem of uncertain planning with random variables, however, chance-constrained planning can compensate for this deficiency.

Type-2 fuzzy chance-constrained programming

CCP, one of the branches of stochastic optimization techniques, was initially introduced by Charnes & Cooper (1959). The capability of the CCP to effectively address the uncertainty related to random parameters and quantify the risk of constraint violation is evident. Incorporating the T2FS within the CCP framework, a type-2 fuzzy chance-constrained programming (T2F-CCP) model can be formulated as Equations (4a)–(4e):
(4a)
subject to:
(4b)
(4c)
(4d)
where X is the decision variable; , , are coefficients in the model: is the parameter in the right hand of constraint i; represents the probability that the event is established, is the pre-given probability level of violating the constraints, and is the confidence level of the constraint condition i; is the vector of T2TrFS.
According to Suo et al. (2017), when are deterministic and are random, Equation (4b) becomes linear one:
(4e)
where , given the cumulative distribution function of , and the probability of violating constraint i.

T2F-CCP can deal with multiple fuzzy information and can also reflect components of uncertainty expressed as probability distributions with unknown correlations. Water resource systems must take into account optimization concerns, including system efficiency. Fractional planning is a reliable method for evaluating system efficiency and accurately representing the complexities of the real-world.

Type-2 fuzzy-chance constrained linear fractional programming

LFP is a mathematical framework that extends the principles of linear programming (LP) to address multi-objective optimization problems. It offers a quantitative assessment of system efficiency (Guo et al. 2013). Through incorporating T2TrFV and CCP into the framework of LFP, a T2F-CCLFP model can be formulated. Thus, it is:
(5a)
subject to:
(5b)
(5c)
(5d)
where and are scalar constants; D is a coefficient in the model; the rest of the symbols are the same as above.
According to Zhou et al. (2015), Equations (5a)–(5d) can be converted into a linear version as Equations (6a)–(6e):
(6a)
subject to:
(6b)
(6c)
(6d)
(6e)
where is the new decision variable, and the original variable X equal to .

The specific solution of the integrated the T2F-CCLFP model can be summarized as:

Step 1: Formulate a T2F-CCLFP model.

Step 2: Convert T2TrFV into a chosen deterministic type via defuzzification based on the CV method.

Step 3: Transform the stochastic constraint into deterministic constraint through CCP model.

Step 4: Transform the LFP model into its corresponding linear version.

Step 5: Solve the transformed model and get the corresponding optimal solution.

Step 6: Repeat Steps 3–5 under different values of , and .

As shown in Figure 2, Taiyuan (N: 37°27′–38°25′; E: 111°30′–113°09′) is the capital city of Shanxi Province, China. The city of Taiyuan is situated in the Taiyuan Basin, which is located in the northern-central region of Shanxi. The city is governed by six districts, three counties, and one county-level city. As of the end of 2022, the city has a population of 5.435 million and covers a total area of 6,988 km2. Taiyuan is a typical water-deficient city with uneven inter-annual precipitation, relatively limited rainfall (the annual rainfall is approximately 468.4 mm) and large evapotranspiration (the average annual evapotranspiration is around 1,057.4 mm). The climate of Taiyuan city is characterized as a continental monsoon climate. The average annual temperature is recorded at 9.5 °C, with the highest temperature occurring during the summer at 23.5 °C, and the lowest temperature observed in winter at −6.8 °C.
Figure 2

The study areas.

Figure 2

The study areas.

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In recent decades, Taiyuan has undergone rapid economic development and sustained population growth, leading to a significant increase in water resource demand. According to the Shanxi Water Resources Bulletin, water consumption in Taiyuan city has grown at an annual rate of 6.5%, rising from 428.39 × 106 m3 in 2012 to 764.28 × 106 m3 in 2021. The city of Taiyuan is confronted with a significant shortage of local water resources, posing a considerable challenge to the city's sustainable development. Consequently, the procurement of water from external sources and its subsequent transfer to Taiyuan is necessary to fulfill the demand. The projects known as the ‘Yellow River Diversion’ have played a crucial role in mitigating the imbalance between water supply and demand. However, the reliance on external water supply also poses the risk of insufficient water availability. Therefore, ensuring water supply security is of utmost importance for the development of the city.

The city's current water supply strategy involves the utilization of surface water, groundwater, and external water transfer to fulfill its growing water requirements. Nevertheless, the utilization of reclaimed water maintains a consistent level of low usage. In order to tackle this issue, the government has implemented a range of measures aimed at promoting the integrated utilization of reclaimed water alongside other water resources. Thus, achieving a coordinated allocation of multiple water sources is an effective approach to mitigate the imbalance between water resource supply and demand in the city. In addition, the water resources allocation system faces numerous uncertainties that can impact the optimization process and decision-making. For example, the optimization of water resources system is often subject to some subjective judgments, which will affect the relevant parameters in the calculation of water demand. As a result, the estimated water demand is typically highly uncertain and can be represented using type-2 fuzzy numbers. Climate change has a significant impact on the surface water supply. In normal years, the availability of surface water is greater when compared to dry years. The availability of surface water is a stochastic phenomenon influenced by climate conditions and can be mathematically represented using a probability distribution function.

The annual water demand projections for Taiyuan city can be estimated by utilizing historical data from the Taiyuan Water Resources Bulletin, which covers the period from 2006 to 2018.

To estimate the annual water demand projections for Taiyuan city, historical data from the Taiyuan Water Resources Bulletin spanning from 2006 to 2018 can be utilized. Table 1 shows the maximum and minimum water demands of Taiyuan city in 2025 and 2030 under the extremely dry year, dry year and normal year scenarios, which are highly uncertain and are expressed by type-2 fuzzy numbers. The minimum water demand is determined by multiplying the maximum water demand by specific coefficients. The coefficients used to calculate the minimum values are as follows: 0.95 for domestic water, services industry, and ecological water; 0.9 for industrial water; and 0.8 for agricultural water, are used to obtain the minimum values (Yue et al. 2020). In this study, five levels of surface water supply (i.e., 0.01, 0.05, 0.10, 0.15, and 0.20) in different planning years were considered. It also assists managers in gaining insight into the trade-off between demand uncertainty and the risk of violating constraints. The socio-economic parameters, such as the water use coefficient, benefit coefficient, and water cost, were primarily sourced from statistical yearbooks, government reports, and prior research studies. Tables 24 provide comprehensive details.

Table 1

The maximum and minimum water demands of Taiyuan city in 2025 and 2030 under different hydrological years (unit: 106 m3)

Hydrological yearsThe maximum water demand
The minimum water demand
2025203020252030
Normal year (817.5,871.3,889.6,1035.9;θl,θr(787.5,960.7,986.1,1153.4;θl,θr(733.2,741.2,757.5,888.4;θl,θr(708.9,827.4,864.9,1102.1;θl,θr
Dry year (850.8,906.9,926.1,1078.7;θl,θr(815.9,995.4,101.9,1195.1;θl,θr(759.9,769.7,786.7,922.6;θl,θr(731.7,853.9,892.6,1135.4;θl,θr
Extremely dry year (855.1,911.6,930.8,1084.4;θl,θr(819.9,100.1,102.4,1200.9;θl,θr(763.4,773.4,790.5,927.1;θl,θr(734.8,857.8,896.5,1140.1;θl,θr
Hydrological yearsThe maximum water demand
The minimum water demand
2025203020252030
Normal year (817.5,871.3,889.6,1035.9;θl,θr(787.5,960.7,986.1,1153.4;θl,θr(733.2,741.2,757.5,888.4;θl,θr(708.9,827.4,864.9,1102.1;θl,θr
Dry year (850.8,906.9,926.1,1078.7;θl,θr(815.9,995.4,101.9,1195.1;θl,θr(759.9,769.7,786.7,922.6;θl,θr(731.7,853.9,892.6,1135.4;θl,θr
Extremely dry year (855.1,911.6,930.8,1084.4;θl,θr(819.9,100.1,102.4,1200.9;θl,θr(763.4,773.4,790.5,927.1;θl,θr(734.8,857.8,896.5,1140.1;θl,θr
Table 2

The order coefficient of water supply

Water supply sourceDomesticAgricultureIndustryServicesEcology
Surface water 0.5 0.3 0.2 0.33 0.17 
Ground water 0.33 0.1 0.1 0.17 
Reclaim water 0.2 0.3 0.5 
External water 0.17 0.4 0.4 0.5 0.33 
Water supply sourceDomesticAgricultureIndustryServicesEcology
Surface water 0.5 0.3 0.2 0.33 0.17 
Ground water 0.33 0.1 0.1 0.17 
Reclaim water 0.2 0.3 0.5 
External water 0.17 0.4 0.4 0.5 0.33 
Table 3

The cost coefficient of the water sector in 2025 and 2030 (unit: yuan/m3)

District2025
2030
DomesticAgricultureIndustryServicesEcologyDomesticAgricultureIndustryServicesEcology
Xiaodian district 2.8 0.6 4.8 50 2.8 0.8 80 
Yingze district 2.8 0.6 4.8 50 2.8 0.8 80 
Xinghualin district 2.8 0.6 4.8 50 2.8 0.8 80 
Jiancaoping district 2.8 0.6 4.8 50 2.8 0.8 80 
Wanbailin district 2.8 0.6 4.8 50 2.8 0.8 80 
Jinyuan district 2.8 0.6 4.8 50 2.8 0.8 80 
Qingxu county 2.8 0.6 4.8 50 2.8 0.8 80 
Yangqu county 2.8 0.6 4.8 50 2.8 0.8 80 
Loufan county 2.8 0.6 4.8 50 2.8 0.8 80 
Gujiao city 2.8 0.6 4.8 50 2.8 0.8 80 
District2025
2030
DomesticAgricultureIndustryServicesEcologyDomesticAgricultureIndustryServicesEcology
Xiaodian district 2.8 0.6 4.8 50 2.8 0.8 80 
Yingze district 2.8 0.6 4.8 50 2.8 0.8 80 
Xinghualin district 2.8 0.6 4.8 50 2.8 0.8 80 
Jiancaoping district 2.8 0.6 4.8 50 2.8 0.8 80 
Wanbailin district 2.8 0.6 4.8 50 2.8 0.8 80 
Jinyuan district 2.8 0.6 4.8 50 2.8 0.8 80 
Qingxu county 2.8 0.6 4.8 50 2.8 0.8 80 
Yangqu county 2.8 0.6 4.8 50 2.8 0.8 80 
Loufan county 2.8 0.6 4.8 50 2.8 0.8 80 
Gujiao city 2.8 0.6 4.8 50 2.8 0.8 80 
Table 4

The benefit coefficient of the water sector in 2025 and 2030 (unit: yuan/m3)

District2025
2030
DomesticAgricultureIndustryServicesEcologyDomesticAgricultureIndustryServicesEcology
Xiaodian district 634.8 39.2 584.8 4,000.0 599.8 749.3 39.2 699.3 4,201.7 714.3 
Yingze district 611.8 73.1 561.8 9,090.9 576.8 699.4 73.1 649.4 9,523.8 664.4 
Xinghualin district 361.5 144.4 311.5 9,090.9 326.5 424.5 144.4 374.5 9,523.8 389.5 
Jiancaoping district 490.5 32.1 440.5 5,000.0 455.5 557.6 32.1 507.6 5,263.2 522.6 
Wanbailin district 535.4 5.4 485.4 7,692.3 500.4 605.6 5.4 555.6 8,064.5 570.6 
Jinyuan district 209.2 31.1 159.2 800.0 174.2 277.3 31.1 227.3 840.3 242.3 
Qingxu county 300.6 22.0 250.6 1,666.7 265.6 345.0 22.0 295.0 1,751.3 310.0 
Yangqu district 691.0 150.6 641.0 1,960.8 656.0 779.9 150.6 729.9 2,057.6 744.9 
Loufan district 220.1 50.0 170.1 1,754.4 185.1 311.8 50.0 261.8 1,841.6 276.8 
Gujiao city 134.5 51.1 84.5 2,040.8 99.5 193.1 51.1 143.1 2,141.3 158.1 
District2025
2030
DomesticAgricultureIndustryServicesEcologyDomesticAgricultureIndustryServicesEcology
Xiaodian district 634.8 39.2 584.8 4,000.0 599.8 749.3 39.2 699.3 4,201.7 714.3 
Yingze district 611.8 73.1 561.8 9,090.9 576.8 699.4 73.1 649.4 9,523.8 664.4 
Xinghualin district 361.5 144.4 311.5 9,090.9 326.5 424.5 144.4 374.5 9,523.8 389.5 
Jiancaoping district 490.5 32.1 440.5 5,000.0 455.5 557.6 32.1 507.6 5,263.2 522.6 
Wanbailin district 535.4 5.4 485.4 7,692.3 500.4 605.6 5.4 555.6 8,064.5 570.6 
Jinyuan district 209.2 31.1 159.2 800.0 174.2 277.3 31.1 227.3 840.3 242.3 
Qingxu county 300.6 22.0 250.6 1,666.7 265.6 345.0 22.0 295.0 1,751.3 310.0 
Yangqu district 691.0 150.6 641.0 1,960.8 656.0 779.9 150.6 729.9 2,057.6 744.9 
Loufan district 220.1 50.0 170.1 1,754.4 185.1 311.8 50.0 261.8 1,841.6 276.8 
Gujiao city 134.5 51.1 84.5 2,040.8 99.5 193.1 51.1 143.1 2,141.3 158.1 

Based on the proposed T2F-CCLFP method, considering the conflicts among fairness, efficiency, and sustainable development principles in water resource systems, the study problem can be formulated as follows:

Objective function:

  • (1)
    Social objective: The primary aim is to reduce the Gini coefficient, a measure of income inequality, in order to promote social equity. To achieve an equitable distribution of water resources across different zones, it is crucial to prioritize fairness during the water distribution process. The Gini coefficient, typically used to measure income inequality, has more recently been applied to assess land and water use inequalities (Cullis & Koppen 2007). Therefore, according to the definition of the Gini coefficient, the fairness of water allocation can be assessed by the fairness of per capita water allocation in sub-regions. The smaller the water shortage difference in each sub-district, the lower the value of the Gini coefficient. The Gini coefficient is a numerical measure that ranges from 0 to 1. A lower value indicates a more equal distribution. The equation for calculating the Gini coefficient for water allocation is represented as (7a)–(7f):
    (7a)
    where represents the equity objective, and respectively are the water supply quantity (m3) supplied by the water source i of sub-area m and sub-area to the water user j; and are the population of sub-area m and sub-area , and is the total number of water sources, water users and sub-areas respectively, where .
  • (2)
    Economic objective: The primary goal in this context is to optimize the ratio between the total economic benefit value (calculated by multiplying the net economic benefit and water consumption of each sub-area) and the actual available water supply. This can be mathematically represented as:
    (7b)
    where represents the efficiency objective; is the water supply benefit coefficient (yuan/m3) generated by the water source i to the water user j in sub-area m; is the water supply cost coefficient (yuan/m3) generated by water source i to water user j in sub-area m; is the order coefficient of water supply reflects the order of water supply from source i to water user j.
  • (3)
    Environmental objective: To achieve sustainable environmental development, it is necessary to consider the impact of wastewater generated by water-using sectors on the environment while simultaneously meeting the water demand of social and economic systems. The evaluation of environmental sustainability typically involves minimizing the emissions of regional pollutants. However, quantitatively expressing the concept of environmental sustainable development is challenging due to the diverse components of pollutant emissions. According to Wang et al. (2019) and Li et al. (2022), the chemical oxygen demand (COD) is used as the evaluation index for pollutants in this paper. Therefore, the environmental goal of water allocation is to minimize COD emissions in water-using sectors.
    (7c)
    where represents the sustainability objective, is the pollutant content (mg/L) in the pollutant emissions by different water users in each sub-area, and is the pollutant emissions coefficient of different water use sectors.

Model constraints:

  • (1)
    Water availability constraints: the allocation amount cannot exceed the total amount of available water:
    (7d)
    where is the maximum available water supply in the study area (m3).
  • (2)
    Water supply constraints: the sum of the allocated water from any water source (surface water, groundwater, external water, and reclaimed water) should not exceed its corresponding maximum water supply:
    (7e)
    (7f)
    where is the maximum storage capacity of surface water (m3), p is the probability of violating the constraint conditions, is the maximum water supply of the water source, and , other symbols are the same as above.
  • (3)
    Water demand constraints: The allocated water for each water use sector should range between the maximal and minimal demand:
    (7g)
    where represents the lower limit of water demand (m3) of water user j, and represents the upper limit of water demand (m3) of water user j, both of which are type-2 fuzzy numbers.
  • (4)
    Non-negative constraints:
    (7f)

Combining the above objective functions and constraints, the T2F-CCLFP water resources management model can be established as:
(8a)
(8b)

In this study, the decision-making process involved two planning years, three hydrological years, four types of water resources, five consumption sectors, and five levels of violation risk characterizing the water supply constraints (i.e., pi = 0.01, 0.05, 0.10, 0.15, and 0.20). Additionally, there were three sets of crisp values representing the type-2 fuzzy parameters of water demand, namely (i.e., (θl, θr) = ((0.9,0.1), (0.5,0.5), (0.1,0.9)).

Results and discussion of optimal water allocation

Solution schemes

By solving developed T2F-CCLFP model in the optimization method, optimal water allocation results among different water use sectors in different subareas during different periods under different hydrological years were obtained. Figure 3 shows the optimal water allocation for different water users among different scenarios. The results indicate that the water allocations for Domestic, Industry, and Services sectors are projected to increase by 2030 compared to 2025. In contrast, Agriculture sector water allocation is forecasted to decrease during the same period, while the level of allocation for Ecology remains relatively stable. Furthermore, the average annual growth rate of water distribution in the service sector in 2030 is anticipated to surpass that of the industrial sector. Take and in normal years as an example, the average agricultural water consumption would be 193.1 × 106 m3 in 2025 and 167.2 × 106 m3 in 2030, with an annual average decrease rate of 2.7%. Comparatively, the average annual growth rate of water distribution in the industrial sector would be 2.88%, while the service sector's average annual growth rate would be 5.26%, nearly double that of the industrial sector. Such development is mainly due to the improvement of agricultural water-saving appliances, which have improved the efficiency of agricultural water usage and subsequently reduced agricultural water allocation. The substantial growth rate in water distribution for the service sector reflects the adjustment of Taiyuan's industrial structure, transitioning from energy and heavy industries to the service industry. Consequently, the corresponding water distribution to production departments will also change to align with the evolving industrial structure. Besides, in normal years, and both have effects on the water-requirement schemes, with the decrease of and the increase of , the water consumption of various sectors are gradually increasing. In the year of 2025, the average total water consumption would be 902 × 106 m3 when (, ), and 904 × 106 m3 when (, ). However, in dry years and extremely dry years, there is no obvious regularity in the effects of and , due to the amount of water allocation in the dry years would be dependent on the available water supply.
Figure 3

The optimal water allocation for different water users among different scenarios.

Figure 3

The optimal water allocation for different water users among different scenarios.

Close modal
Figure 4 presents the solution of water supply obtained from the T2F-CCLFP model. It is evident that both the overall trend of water allocation amount and the characteristics of each water source are depicted. The results show that external water transfer serves as the primary source of water supply for the city, with the proportion of external water transfer projected to be 55% in 2025 and 50% in 2030. More specifically, the allocation of surface water exhibits a declining trend from normal years to extremely dry years due to reduced natural inflow. However, the total water volume is higher in dry years compared to normal years, primarily attributed to the increased external water supply during dry years. Furthermore, in comparison to the baseline level in a reference year (with groundwater supply amounting to 230.46 × 106 m3), the groundwater supply is projected to decrease by 73.89 × 106 m3 in 2025 and by 113.40 × 106 m3 in 2030. In addition, the supply of reclaimed water is expected to show a substantial increase of 124.77 × 106 m3 in 2030 compared to 2025, reflecting an annual growth rate of 13.9%. Results indicate that the water supply structure in both 2025 and 2030 would be optimized, especially in 2030. Furthermore, as the level decreases, there is a reduction in surface water supply while groundwater supply remains relatively stable. To meet water demand requirements when surface water supply is inadequate, priority is given to the utilization of external water and reclaimed water, thus avoiding excessive exploitation of groundwater resources.
Figure 4

The amount of water supply under different scenarios.

Figure 4

The amount of water supply under different scenarios.

Close modal
Figure 5 shows the differences in industrial development and water use structure of Taiyuan districts under different hydrological years in 2025 and 2030. Among them, Yangqu and Qingxu are dominated by agriculture, with agricultural water use accounting for 47–59 and 40–54% of the total water use, respectively. Meanwhile, Wanbailin, Jianqaoping, Xiaodian, Loufan and Gujiao are dominated by industrial water use, accounting for 46–48, 57–61, 49–51, 46–49, and 62–65 of the total water use, respectively. In contrast, living water use accounts for a larger share in Yingze and Xinghualing, accounting for 43–45 and 48–50% of the total water use, respectively. Jinyuan District has a balanced share of water use in agriculture, industry and services, all around 25%, the water use structure of the administrative districts reflects, to some extent, their industrial characteristics and development priorities. In addition, the study shows that the water consumption of the service sector in most counties and districts will increase in 2030 compared with 2025, which is in line with the overall development pattern of Taiyuan's industrial structure.
Figure 5

The water structure of each district in Taiyuan city under different hydrological flows in 2025 and 2030: (a) normal year (2025), (b) normal year (2030), (c) dry year (2025), (d) dry year (2030), (e) extremely dry year (2025), and (f) extremely dry year (2030).

Figure 5

The water structure of each district in Taiyuan city under different hydrological flows in 2025 and 2030: (a) normal year (2025), (b) normal year (2030), (c) dry year (2025), (d) dry year (2030), (e) extremely dry year (2025), and (f) extremely dry year (2030).

Close modal

Socio-economic-ecological benefits

Figure 6 presents the comparison of economic efficiency values for optimal distribution results of 2025 and 2030 under various scenarios. The findings highlight that, irrespective of the hydrological years, the economic efficiency of water allocation in 2030 surpasses that of 2025. Additionally, it can be observed that the economic benefits are slightly higher in dry and extreme dry years than in normal years due to the increase in agricultural water allocation in these years compared to normal years, resulting in higher economic benefits. For example, the average economic efficiency of dry years in 2025 would be 1.11 times that of normal years. However, in extreme dry years, the annual water inflow is lower than in dry years, and there is a severe shortage of water supply resulting in an inability to meet the water needs. As a result, the economic benefits are lower in extreme dry years than in dry years.
Figure 6

Economic efficiency under different hydrological years in each scenario.

Figure 6

Economic efficiency under different hydrological years in each scenario.

Close modal
Equity needs to be considered in the water distribution process. The Gini coefficients in 2025 and 2030 under different scenarios are presented in Figure 7. The results show that the Gini coefficient would be mainly affected by hydrological flow. Taking the year of 2025 as an example, the Gini coefficients of normal years would be the lowest about 0.281, while in extremely dry years would be around 0.31. Additionally, the Gini coefficients for dry years would be slightly lower than that of the extremely dry years. This is because in normal years, where water supply is relatively sufficient, the distribution of water among administrative district would be more even; in dry and extremely dry years, as water demand increases and supply decreases, the uniformity of water distribution decreases. However, the Gini coefficient for per capita water distribution remains within a reasonable range. Furthermore, under the same hydrological flow and corresponding scenarios, the Gini coefficient in 2030 would be slightly smaller than that in 2025, implying a more uniform distribution of water resources among administrative divisions in 2030.
Figure 7

Gini coefficients under different hydrological years in each scenario.

Figure 7

Gini coefficients under different hydrological years in each scenario.

Close modal
Figure 8 illustrates pollutant emissions in various scenarios under different hydrological years in 2025 and 2030 to assess the environmental sustainability of the configuration scheme. It is observed that the pollutant emissions in 2030 would be more than that in 2025. In addition, the results show that although water distribution would be minimized during normal years, pollutant emissions would be highest during such periods, followed by dry years and extremely dry years. Due to the increased demand for water for agriculture in dry years and extreme dry years, the allocation of water to industry and services would relatively reduce to meet the minimum water needs of the agricultural sector. Table 5 also reveals that, apart from the ecological sector, the concentrations of pollutant emissions from other sectors are 5–6 times greater than those from the agricultural sector. Consequently, the reduction of water allocation in these sectors would significantly decrease overall pollutant emissions. From Figures 68, under the corresponding hydrological scenario, the general trend of economic benefits from water distribution is negatively correlated with the trend of per capita Gini coefficient, while positively correlated with the overall trend of pollutant emissions.
Table 5

Pollutant emissions of different water sectors

Pollutant emissionsDomesticAgricultureIndustryServices
Pollutant emission coefficient 0.8 0.1 0.6 0.8 
COD emission concentration (mg/L) 300 50 230 300 
Pollutant emissionsDomesticAgricultureIndustryServices
Pollutant emission coefficient 0.8 0.1 0.6 0.8 
COD emission concentration (mg/L) 300 50 230 300 
Figure 8

Pollutant emissions under different hydrological years in each scenario.

Figure 8

Pollutant emissions under different hydrological years in each scenario.

Close modal

From Figures 6 and 8, due to sufficient water supply in normal years, the variation of and that characterizes water demand uncertainty degree has an impact on economic efficiency and pollutant emissions. The economic efficiency and pollutant emissions of water allocation both have an uptrend with the increase of and the decrease of , respectively, while in dry years and extremely dry years, as the water supply cannot meet the water demand, the variation of and has no obvious regularity influence on the economic efficiency and pollutant emissions. Figure 7 illustrates the irregular fluctuation of the Gini coefficient with the change of and . The economic efficiency, Gini coefficient, and pollutant emissions are all related to the water supply constraint-violation (pi) level in dry years and extremely dry years. Figures 6 and 8 show the overall trend of economic efficiency and pollutant emissions would increase with the raising of , while the Gini coefficient in Figure 7 shows the opposite trend. The three objective values fluctuate with the values of and , , which indicates that dual uncertainties would affect the city's water-allocation pattern. Furthermore, it should be noted that while the spreads had minimal impact on the optimal targets, the CV approach used in this study offers more indicative schemes rather than a range of alternative solutions.

Comparison between the CCLFP model and the T2F-CCLFP model

Figure 9 presents the economic-environmental-social indicators obtained from T2F-CCLFP and CCLFP models. In most scenarios, T2F-CCLFP is expected to have slightly higher economic efficiency than CCLFP, as well as lower levels of pollutant emissions and Gini coefficients. For example, in normal years, when , the economic efficiency, pollutant emissions and Gini coefficient of T2F-CCLFP would be 688 yuan/m3, 118,729 m3, and 0.2809, respectively, while for CCLFP, they would be 679 yuan/m3, 121,859 m3 and 0.2810. It indicates that the T2F-CCLFP model would achieve higher economic efficiency, more uniform water distribution per capita, and lower pollutant (COD) emissions. Compared to the singular format of CCLFP, the T2F-CCLFP framework is more effective in incorporating all relevant uncertain information into the decision-making process. Simplifying a problem may lead to unreliable or misleading solutions. The loss of an added degree of design freedom can be disadvantageous in water resources systems marked by a high degree of input data uncertainty, which may provide unreliable decision support for managers. T2F-CCLFP offers the ability to address heightened levels of uncertainty relating to CCLFP due to its additional degree of freedom. Therefore, in contrast to CCLFP, T2F-CCLFP is more advanced in its ability to reflect uncertainty.
Figure 9

Comparison of economic–environmental–social objects from T2F-CCLFP and CCLFP.

Figure 9

Comparison of economic–environmental–social objects from T2F-CCLFP and CCLFP.

Close modal

The real-world case study demonstrated that the T2F-CCLFP model is an effective tool for sustainable water allocation in Taiyuan city, particularly under conditions of uncertainty. The model takes into account the trade-off between efficiency, equity, and environmental sustainability, enabling more practical and reasonable decision-making. Additionally, the model addresses the stochastic nature of water supply, represented by probability distributions, as well as the dual uncertainty of water demand characterized by two types of fuzzy numbers, accurately capturing its randomness and fuzziness. Overall, the recommended configuration scheme in this model reflects a balance between economic benefits, equitable water allocation, and environmental requirements. Pursuing higher economic benefits alone may pose risks of water scarcity and hinder the fulfillment of environmental sustainability requirements. On the other hand, accepting lower economic benefits ensures water supply security and meets environmental protection goals. Therefore, the model provides a range of indicative scenarios based on water supply risk constraints and water demand ambiguities rather than final solutions to complex water management challenges.

In this study, the T2F-CCLFP model has been developed for supporting refined management of water resources in Taiyuan city under uncertainties. T2F-CCLFP is capable of addressing ratio optimization issues and effectively dealing with parameter uncertainties expressed as T2FS and probabilistic distribution. By incorporating type-2 fuzzy programming, CCP, and fraction programming into a framework, a T2F-CCLFP-based water resources planning model is formulated for Taiyuan. The model considers multiple water sources, multiple users, multiple treatment technologies, and various hydrological flows. With the help of the T2F-CCLFP method, solutions of water supply and allocation associated with different fuzzy degrees of freedom and different constraint-violation risks have been obtained, while meeting the requirements of equity, efficiency and sustainable development. Results obtained suggest that: (a) compared with 2025, the average annual growth rate of industrial water consumption in Taiyuan city will decrease in 2030 (2.88%), and the average annual growth rate of water consumption in the service industry will increase(5.26%), indicating that industrial development model of Taiyuan city is adjusted from energy and heavy industry-based to the common development of multiple industries; (b) the water supply structure has been optimized in both 2025 and 2030, ground water consumption is cut down (which would decrease by 73.89 × 106 m3 and 113.40 × 106 m3 compared to 2018), and the city's water supply structure is mainly based on external water transfer, while increasing the amount of water supply of reclaimed water, forming a water supply structure of joint water supply of multiple water sources, and improving the emergency guarantee capacity of water resources; (c) the industrial water use structure of Taiyuan's 10 administrative districts has distinct focuses, which partly reflect their industrial characteristics and development priorities. In general, water consumption in the service sector has witnessed growth across all districts, aligning with the overall development pattern of Taiyuan's industrial structure; (d) the general trend of economic benefits of water distribution is negatively correlated with the trend of per capita water consumption Gini coefficient, while positively correlated with the overall trend of pollutant emissions. The results are beneficial to decision makers to adjust the city's current industrial structure as well as to enhance the city's water-supply security.

In addition, T2F-CCLFP is not only capable of handling uncertainty expressed as T2FS but also effectively addresses uncertainty in terms of probabilistic distribution. T2F-CCLFP enables decision-makers to make trade-offs among constraint violation risk, equity, economic efficiency and sustainability. The developed approach can be applied to other similar regions encountering water scarcity issues. However, there are also potential limitations and extensions that should be addressed in future research. For example, mine water resources were not considered in this study based on local status of water utilization in the city rich in mineral resources. Other water resources such as mine water or rain water could be further considered in order to improve the applicability of the T2F-CCLFP model. Moreover, when economic parameters and available water resources are expressed as intervals, interval LP should be integrated into the T2F-CCLFP framework for better accounting for more complex uncertainties.

This research was supported by the National Key Research and Development Program of China (No. 2019YFC0408601), the National Natural Science Foundation of China (No. 52279020), the Special Fund for Science and Technology Innovation Teams of Shanxi Province (No. 202204051002027), the Natural Science Foundation of Shanxi Province, China (No. 202203021221050).

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.

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