Type-2 fuzzy chance-constrained linear fractional programming model for a water resource management system: A case study of Taiyuan city, China

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 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.

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.

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.

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 ⁠.

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 2–4 provide comprehensive details.

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)

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., p_i = 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., (θ_l, θ_r) = ((0.9,0.1), (0.5,0.5), (0.1,0.9)).

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 (p_i) 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.

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.