https://orcid.org/0009-0006-3410-6331
Abstract
The uncertainty of socioeconomic development and climate change poses challenges to the sustainable management of water resources in Taiyuan, China. The study proposes a type-2 fuzzy chance-constrained ordered multi-objective fractional programming (T2FCC-ORMOFP) model to address the allocation of water under uncertainties. The model incorporates type-2 fuzzy programming and chance-constrained programming into an overall framework to handle multiple uncertainties under multi-level and multi-objective conflicts, while using fractional programming to reflect the marginal benefits of the system. It prioritizes the principles of optimal, fair, and stable distribution by the upper-level governing authorities, while considering the social and economic environmental benefits of lower-level decision-making. The optimal water allocation scenarios were obtained under different hydrological guarantee rates, violation levels of water supply constraints, and net economic benefits of type-2 fuzzy numbers for 2030. Additionally, groundwater is replaced by reclaimed water in varying proportions for flexible water supply. The results show that the water demand in Taiyuan cannot be ignored, and the risk of water shortage is more sensitive for the agricultural and industrial sectors. The recycled water substitution strategy can optimize the water supply structure and improve social and economic benefits (9–12%). Also T2FCC-ORMOFP can improve overall efficiency (20%) compared with a single-level model.
HIGHLIGHTS
The model takes into account the upper-level criteria of optimality, equity, and stability, along with the lower-level considerations of social and economic environmental benefits.
It tackles the intricacies stemming from the design of type-2 fuzzy- chance-constrained ordered multi-objective fractional programming.
Utilizing reclaimed water as an alternative can effectively mitigate the overexploitation of groundwater resources in Taiyuan.
INTRODUCTION
The rapid development of economic society and the impact of climate change have intensified the disparity between water supply and demand, turning water shortage into an urgent concern. In recent years, the optimal allocation of water resources has garnered widespread attention as a potentially effective method to enhance the efficiency of water resources allocation and alleviate water shortages (Hou et al. 2022, 2023; Wang et al. 2023). This optimal allocation involves multiple subjects, objects, parameters, and objectives, resulting in structural complexity. Mathematical programming methods are capable of identifying the allocation of scarce water resources to meet the requirements of various uses (Dalcin & Marques 2020; Wang & Xue 2021; Wei et al. 2023).
However, traditional optimization methods often concentrate on modeled inputs and outputs, neglecting the optimization of system efficiency as an output/input ratio. To address this, fractional programming (FP), as advocated by Nie et al. (2016), provides an unbiased measurement and is advantageous for addressing ratio optimization problems. Guo et al. (2014) proposed a FP method that supports water resources management, addressing ratio optimization issues in water resources systems. Additionally, the system of water resources allocation often encompasses multiple conflicting objectives, escalating the complexity of decision-making. According to Li et al. (2019), multi-objective programming (MOP) is deemed an effective approach to tackle this complexity. The intricacies of water resources allocation systems are also evident in the distinct requirements and priorities of decision-makers at varying levels (Yue et al. 2020). For example, higher-level authorities balance conflicts between optimality, equity, and stability in allocating scarce water resources, while regional administrations focus on maximizing individual economic efficiency and complying with environmental regulations. These diverse requirements pose challenges for practical water resources allocation. However, few research works have fully considered the distinct configuration requirements of the upper and lower levels concerning optimization metrics such as optimality, stability, equity, efficiency, and sustainability of the solutions. Therefore, this study designs an ordered model (OR) as a leader–follower framework, aiming to prioritize the leader's requirements. Simultaneously, it integrates MOP and FP into the OR model to address challenges posed by inter-level decision-making requirements, conflicting objectives, and ratio optimization problems.
Additionally, there is a significant amount of uncertainty in water resources planning systems, especially concerning policies and measures that impact water resources management (Koltsaklis et al. 2015). In the past, extensive research has been conducted to tackle such uncertainties (Li et al. 2020a; Yue et al. 2022; Nematian 2023). Among them, interval parameter programming proposed by Kumar & Bhurjee (2022) can handle uncertainties that fluctuate within a certain range, which cannot be quantified as membership or distribution functions. It is challenging to address uncertainties represented by random variables, so stochastic programming (SP) was introduced to solve problems involving input data represented as probabilities (Xu et al. 2019). However, it is difficult to incorporate other forms of uncertainty represented as probability distributions into the optimization framework. By contrast, the FP has advantages in solving decision-making problems with imprecise information (Wang et al. 2020; Li et al. 2022b). Conventional fuzzy programming can handle decision-making problems where each fuzzy number in a fuzzy set is defined by a membership function. However, in water resources planning systems, the membership functions of fuzzy sets are also fuzzy, meaning they cannot be represented as crisp values. Type-2 fuzzy sets (T2FS) can address such high levels of uncertainty. Some research has already applied T2FS in areas such as production programming, transportation cost, and fuzzy logic to describe their high degrees of uncertainty (Jana & Jana 2020; Roy & Maiti 2020; Wang et al. 2022). The variability of water availability is sensitive to climatic factors, and the stochastic nature of climate change results in a similar stochastic nature of water availability variability. Li et al. (2020a, 2020b) addressed the parameter uncertainty caused by randomness through chance-constrained programming (CCP). However, research simultaneously considering multiple uncertainties within the leader–follower multi-objective FP framework remains limited. Therefore, this study attempts to incorporate T2FS, CCP, FP, and MOP into the OR model to address the complexity and uncertainty issues in water resources allocation systems.
Consequently, this study aims to develop a new method (T2FCC-ORMOFP) for planning of water management systems under multiple uncertainties. The key contributions of this method include the following: (1) balancing decision-making conflicts among multiple subjects and objectives in a leader-centric bi-level (BL) model; (2) improving the descriptive accuracy of highly uncertain parameters in objective functions and constraints, effectively addressing multiple uncertainties represented by type-2 fuzzy parameters and random parameters; and (3) providing a sustainable water resources management scheme for replacing groundwater with reclaimed water. Subsequently, using the water resources management system in Taiyuan as an example, the effectiveness of the T2FCC-ORMOFP model is verified. The research plans under different scenarios can offer forward-looking guidance for industrial development and water supply structure optimization in Taiyuan.
METHODOLOGY
The framework of the T2FCC-ORMOFP model for the study.
Ordered model
and 
are the upper-level and lower-level decision variables; then (
, 
) and (
, 
) are the objective function and constraint set of the upper and lower levels, respectively.The constructed OR model considers the preferences of different decision-makers at the upper and lower levels. However, the model faces challenges in handling the numerous uncertainties present in the water resources system. To address the stochastic and fuzzy uncertainties in the water resources system, CCP and T2FS are introduced in the proposed OR model.
The ordered model incorporating chance-constrained programming and type-2 fuzzy sets
refers to the vectors of T2FS; and 
, 
are random variables with a probability distribution function 
. The variable 
represents the probability that the event 
is established, where p is the pre-given probability level of violating the constraints.
, 
, and given the cumulative distribution function of 
, 
, and the probability of violating constraint i, the rest of the symbols are the same as above.
can be expressed as 
per Liu & Liu (2010), where 
, 
, and 
are real values and 
, 
are two parameters characterizing the spreads of primary membership grades of T2TFS. Due to the fuzzy membership function of a type-2 fuzzy number, the computational complexity is very high in practical applications. To avoid this, Qin et al. (2011) developed the CV-based methods of reduction for the type-2 fuzzy variable, and the optimistic CV, the pessimistic CV, and the CV of 
, are defined as Equation(3a)–(3c):
Type-2 fuzzy chance-constrained ordered multi-objective fractional program
and 
are constants. The rest of the symbols are the same as mentioned previously. According to Rakhshan et al. (2016), Equations (5a)–(5g) can be converted into linear versions as Equations (6a)–(6i), as follows:
is the new decision variable; the rest of the symbols are the same as mentioned previously.Development of T2FCC-ORMOFP model
A T2FCC-ORMOFP model was developed for planning the water resources system of Taiyuan City for 2030. The system involves 10 administrative districts, four water sources (surface water, groundwater, external water transfer, and reclaimed water), and five water-use sectors (domestic, agricultural, industrial, service, and ecological sectors). The developed model is designed with two levels of decision-makers: the upper level comprises authorities responsible for the water resources system in Taiyuan, focusing on achieving optimality, fairness, and stability in the water distribution scheme for each administrative district of Taiyuan. The lower level includes decision-makers representing the water-use sectors in each administrative district, primarily concerned with their socioeconomic benefits.
Upper-level model
For the upper-level decision-makers, the allocation of water should take into consideration the optimality, equity, and stability of the allocation solution in the subareas.
Objective functions:
- (1)Optimality objective. The percentage between the least squares solution and the ideal solution is used as the optimization target for social optimality. The least squares method aims to minimize the sum of the squares of deviations between the ideal and actual solutions, so that the total dissatisfaction among different stakeholders is minimized. To account for bias toward higher claims in water resource allocation, Qin et al. (2020) proposed to measure dissatisfaction in the form of percentage deviation from the ideal solution. Hence, the least squares solution-based optimality objective can be expressed as Equation (7a):(7a)
- (2)Equity objective. The equitable sharing of the consumed water amount is quantified by the Gini coefficient, signifying equal access to water and the benefits of water use. The Gini coefficient, proposed by Gini (1921), is a commonly used measure of inequality in the distribution of income or wealth. It has also been applied to assess distributional equity in various disciplines (Wüstemann et al. 2017; Chen et al. 2022; Duan et al. 2022), with notable results, particularly in the field of water resources management (Zhou et al. 2015; Mehdi et al. 2020; Lou et al. 2023). Computed based on the Lorenz curve diagram and the equivalence line, the Gini coefficient is calculated via the relative mean difference divided by the average (Cullis & Koppen 2007). The equity objective, based on the Gini coefficient, can be defined as Equation (7b):(7b)
- (3)Stability objective. The power index method is commonly applied to identify solution stability in decision-making research (Teasley & McKinney 2011). It describes the willingness of an agent to cooperate and calculates the ratio of its loss when it leaves the cooperative coalition to the sum of the cooperative coalition (Madani & Hooshyar 2014). Widely used in water resources management (Read et al. 2014), the stability objective based on the power index can be defined as Equations (7c) and (7d):(7c)
(7d)
Subject to:
- (1)Water availability constraint. The total water allocation cannot exceed the resources available.(7e)
- (2)Water demand constraints. The amount of the water allocation should not exceed its expectation, even though it should be greater than its minimum water requirement.(7f)
Lower-level model
At the lower-level, economic benefits and environmental sustainability are considered.
Objective functions:
- (1)Socioeconomic benefits objective. Economic benefits are a necessary criterion for water allocation (Hu et al. 2016), and the maximum net economic benefit of the minimum regional water shortage is adopted as the socioeconomic benefit target. The socioeconomic benefits objective based on water scarcity and net economic benefits can be defined as Equation (8a):(8a)
- (2)Environmental objective: The environmental objective is to minimize emissions of pollutants from sectors to reflect the sustainability of the solution. A representative pollutant indicator, chemical oxygen demand (COD), was chosen to describe the level of contamination (Wang et al. 2019; Li et al. 2022a, 2022b). Therefore, the COD-based environmental objectives can be defined as Equation (8b):(8b)
Subject to:
- (1)Regional water availability constraint. The total water in the subareas allocated to the different sectors cannot exceed the sum of the actual water received.(8c)
- (2)Water source availability constraint. The supply of each water source cannot exceed its available water supply.(8d)
- (3)Water demand constraints. The water allocated to each sector should fall between the minimum demand and the normal demand.(8e)
- (4)Non-negative constraints.(8f)
Model solution
Upper-level model:
(i = 1,2,3) denote the maximum estimate for target 
(i = 1,2,3), 
(i = 1,2,3) denote the minimum estimate for target
, and 
denote the satisfaction level for the target, as follows:
, 
, and 
for 
, 
, and 
, are obtained, then the normalized single-objective programming is as Equation (12).
The quantum particle swarm optimization (QPSO) algorithm, using a linearly decreasing contraction–expansion coefficient strategy based on the quantum 
potential well with 1 → 0.5 is adopted to iteratively optimize Equation (12) for solving (Chen et al. 2023). The flowchart of the QPSO algorithm is shown in Figure SI 1.
Lower-level model:
According to Equations (6a)–(6i), the lower multi-objective FP model is transformed into a single-objective two-times programming model:
and 
are the target weight factors for 
and 
, respectively.Equation (14) represents the quadratic programming model, which balances multiple objectives using weights and can be efficiently solved by the Gurobi 11.0 solver.
The specific solution process for solving the T2FCC-ORMOFP model can be summarized as follows:
Step 1: Identify and collect uncertain variables and their associated probability distribution information.
Step 2: Formulate a T2FCC-ORMOFP model.
Step 3: Convert T2TFS into a chosen deterministic type via defuzzification based on the CV method, and transform the stochastic constraint into deterministic constraint through the CCP model.
Step 4: Transform the multi-objective programming of the upper and lower levels into single-objective programming separately.
Step 5: Obtain the optimal solution for the upper-level objectives and decision variables.
Step 6: The decision variables obtained from the upper-level model are substituted into the constraints of the lower-level model to solve the lower-level model.
Step 7: Obtain the optimal water allocation plan.
CASE STUDY
Study area
The study area.
To address the water scarcity problem, Taiyuan's water supply system relies mainly on surface water, groundwater, and reclaimed water. Surface water is supplemented through inter-basin water transfer projects, and both surface water and external water transfers are the primary water sources for all users. Groundwater resources are also crucial for drinking water, but overexploitation is a significant issue. The introduction of the Yellow River water into Taiyuan in 2003 provided favorable conditions for controlling and managing groundwater overexploitation. However, with rapid economic and population growth, the demand for water has significantly increased. In recent years, Taiyuan has increased utilization of reclaimed water to improve water resources efficiency, alleviate water scarcity, reduce pollution, and improve the ecological environment. Developing an optimal management method to meet the growing water demand from different regions with multiple water sources and water-use units is of great significance. Additionally, the presence of various uncertain parameters in the water distribution system makes problem-solving complex. Therefore, it is necessary to establish an optimization model under uncertain conditions to support the sustainable management of water resources.
Parameter determination
The data for the T2FCC-ORMOFP model mainly include hydrological data, socioeconomic data, and so on. The water supply of Taiyuan City in the planning year (2030) was predicted based on the water resources bulletin of Taiyuan (2006–2018) combined with the topic of water resources allocation in the whole region of Shanxi Province. To gain a better understanding of the water resources carrying capacity in Taiyuan City during dry and extra dry years, and to identify effective management methods, this study focuses on hydrological years with hydrological guarantee rates of 75 and 95% for water resources allocation analysis.
The availability of surface water quantity in Taiyuan City is predominantly influenced by climate change, exhibiting stochastic characteristics that can be described through a probability distribution. In this study, five 
levels (i.e., 0.01, 0.05, 0.10, 0.15, and 0.20) for water supply are considered, which implies that the water supply should be satisfied with probabilities of 0.99, 0.95, 0.90, 0.85, and 0.80. The different 
levels help to investigate the risk of violating the water supply constraints and generate the required water supply solutions. Table 1 shows the predicted values of water supply in Taiyuan for various 
values with 75 and 95% hydrological guarantee rates for the planning year. The estimation of maximum water demand for different hydrological guarantee rates in the planning year utilized the quota method. This involved referencing the Taiyuan National Economic and Social Development Statistical Bulletin (2014–2018) and the Taiyuan Urban Overall Development Strategic Planning Study. The calculation of minimum water demand was determined by multiplying the maximum water demand by appropriate coefficients: 0.95 for domestic, service, and ecological water requirements; 0.9 for industrial water requirements; and 0.8 for agricultural water requirements (Yue et al. 2020). And Table SI 2 shows the maximum and minimum water demand projections for Taiyuan City with 75 and 95% hydrological guarantee rates for the planning year. To better understand the relationship between water resources and water-use sectors, Table SI 1 provides the water supply relationship coefficients based on water quality and demand per water user. In addition, Table SI 3 illustrates the COD concentrations and sewage discharge coefficients.
Projected water supply for different hydrological guarantee rates in Taiyuan City for 2030 (106 m3)
| pi | |||||
|---|---|---|---|---|---|
| 0.01 | 0.05 | 0.1 | 0.15 | 0.2 | |
| Hydrological guarantee rate | |||||
| 75% | 1,065.4 | 1,075.6 | 1,081.0 | 1,084.7 | 1,087.6 |
| 95% | 1,056.1 | 1,062.9 | 1,066.5 | 1,069.0 | 1,070.9 |
| pi | |||||
|---|---|---|---|---|---|
| 0.01 | 0.05 | 0.1 | 0.15 | 0.2 | |
| Hydrological guarantee rate | |||||
| 75% | 1,065.4 | 1,075.6 | 1,081.0 | 1,084.7 | 1,087.6 |
| 95% | 1,056.1 | 1,062.9 | 1,066.5 | 1,069.0 | 1,070.9 |
The net benefit coefficient of water use is a crucial parameter for calculating the economic efficiency of water distribution, playing a significant role in guiding the rational allocation of water resources. However, obtaining the water-use benefit coefficients and cost coefficients is often challenging, as they rely on statistical yearbooks, survey reports, and previous studies that involve human judgment, leading to a high degree of uncertainty. To address this issue, T2Fs can be employed as a potential expression of multi uncertainty for these parameters. The defuzzification method reported by Cheng (2004) can be used to construct T2Fs. In this study, the net water-use benefit factor is expressed as T2Fs, which is provided in Table 2. These parameters are essential references for water resources management and water supply planning in Taiyuan City. They can contribute to developing sustainable water management strategies that ensure sufficient water supply for all water-use sectors, while considering the impact of climate change.
2030 sector-wise water-use net benefit coefficients (Yuan/m3)
| District | Domestic | Agriculture | Industry | Services | Ecology |
|---|---|---|---|---|---|
| Xiaodian district | (727,757,832;θl,θr) | (22,39,56; θl,θr) | (658,702,776;θl,θr) | (3922,4227,4684; θl,θr) | (692,722,797;θl,θr) |
| Yingze district | (677,707,782;θl,θr) | (41,73,105; θl,θr) | (610,653,725;θl,θr) | (8027,9524,11,420;θl,θr) | (642,672,747;θl,θr) |
| Xinghualin district | (394,428,480;θl,θr) | (97,147,196; θl,θr) | (360,379,418;θl,θr) | (7827,9524,11,220;θl,θr) | (359,393,445;θl,θr) |
| Jiancaoping district | (528,562,613;θl,θr) | (25,33,57; θl,θr) | (467,511,584;θl,θr) | (4989,5287,5735; θl,θr) | (493,527,578;θl,θr) |
| Wanbailin district | (576,610,661;θl,θr) | (3,6,12; θl,θr) | (515,559,632;θl,θr) | (6368,8065,9761; θl,θr) | (541,575,626;θl,θr) |
| Jinyuan district | (262,282,321;θl,θr) | (14,31,38; θl,θr) | (180,230,279;θl,θr) | (637,850,1125; θl,θr) | (227,247,286;θl,θr) |
| Qingxu county | (330,349,388;θl,θr) | (6,22,39; θl,θr) | (247,297,346;θl,θr) | (1428,1757,2251; θl,θr) | (295,314,353;θl,θr) |
| Yangqu county | (757,787,862;θl,θr) | (118,151,183;θl,θr) | (689,733,807;θl,θr) | (1716,2083,2632; θl,θr) | (722,752,827;θl,θr) |
| Loufan county | (297,316,355;θl,θr) | (36,53,85; θl,θr) | (214,264,313;θl,θr) | (1507,1837,2133; θl,θr) | (262,281,320;θl,θr) |
| Gujiao city | (178,198,236;θl,θr) | (36,56,94; θl,θr) | (95,145,194; θl,θr) | (1734,2150,2605; θl,θr) | (143,163,201;θl,θr) |
| District | Domestic | Agriculture | Industry | Services | Ecology |
|---|---|---|---|---|---|
| Xiaodian district | (727,757,832;θl,θr) | (22,39,56; θl,θr) | (658,702,776;θl,θr) | (3922,4227,4684; θl,θr) | (692,722,797;θl,θr) |
| Yingze district | (677,707,782;θl,θr) | (41,73,105; θl,θr) | (610,653,725;θl,θr) | (8027,9524,11,420;θl,θr) | (642,672,747;θl,θr) |
| Xinghualin district | (394,428,480;θl,θr) | (97,147,196; θl,θr) | (360,379,418;θl,θr) | (7827,9524,11,220;θl,θr) | (359,393,445;θl,θr) |
| Jiancaoping district | (528,562,613;θl,θr) | (25,33,57; θl,θr) | (467,511,584;θl,θr) | (4989,5287,5735; θl,θr) | (493,527,578;θl,θr) |
| Wanbailin district | (576,610,661;θl,θr) | (3,6,12; θl,θr) | (515,559,632;θl,θr) | (6368,8065,9761; θl,θr) | (541,575,626;θl,θr) |
| Jinyuan district | (262,282,321;θl,θr) | (14,31,38; θl,θr) | (180,230,279;θl,θr) | (637,850,1125; θl,θr) | (227,247,286;θl,θr) |
| Qingxu county | (330,349,388;θl,θr) | (6,22,39; θl,θr) | (247,297,346;θl,θr) | (1428,1757,2251; θl,θr) | (295,314,353;θl,θr) |
| Yangqu county | (757,787,862;θl,θr) | (118,151,183;θl,θr) | (689,733,807;θl,θr) | (1716,2083,2632; θl,θr) | (722,752,827;θl,θr) |
| Loufan county | (297,316,355;θl,θr) | (36,53,85; θl,θr) | (214,264,313;θl,θr) | (1507,1837,2133; θl,θr) | (262,281,320;θl,θr) |
| Gujiao city | (178,198,236;θl,θr) | (36,56,94; θl,θr) | (95,145,194; θl,θr) | (1734,2150,2605; θl,θr) | (143,163,201;θl,θr) |
Optimized allocation results
Water allocation
, 0.05, 0.10, 0.15, 0.20), and three sets of economic type-2 fuzzy parameter crisp values (i.e., (
) = (0.9, 0.1), (0.5, 0.5), (0.1, 0.9)) are involved in the decision-making process. The optimal water allocation results for subregions under uncertain conditions are shown in Figure 3(a). It can be seen that Xiaodian District has the highest water supply, followed by Qingxu County, with average water supplies of 244.9 × 106 m3 and 203 × 106 m3, respectively. Loufan County has the lowest average distribution at 38.4 × 106 m3, which is 12.3% of the average distribution in Xiaodian District. A higher water resources allocation implies a greater risk of water shortage and requires more resources to meet the demand. The x-axis represents the risk of violating water supply constraints 
, ranging from 0.01 to 0.2. The total water allocation increases with higher 
values. Qingxu County shows a significant increase in allocation, with an average increase of 4.7%. This indicates that Qingxu County tends to allocate more water resources to cope with higher uncertainty risks. At a hydrological guarantee rate of 75%, the water allocation in Qingxu County is significantly higher than at 95%, with an average increase of 3.5%. This suggests that the government tends to allocate more resources under lower guarantee rate conditions to address greater uncertainty risks. Figure 3(b) shows the optimal water allocation schemes for different water users in various scenarios. In the future, industrial water usage will dominate in Taiyuan City, with agricultural and service sector water usage accounting for average shares of 22 and 21%, respectively. This aligns with Taiyuan's emphasis on industrial strength and diversified industry development. As 
increases, water distribution in the industrial and agricultural sectors exhibits an upward trend, while other sectors remain relatively stable. This highlights the sensitivity of the industrial and agricultural sectors to incremental water resources allocation and to the higher risk of water supply failure they face. 
Water allocation with different hydrological guarantee rates under different scenarios: (a) water allocation by region and (b) water allocation for different users.
Water allocation with different hydrological guarantee rates under different scenarios: (a) water allocation by region and (b) water allocation for different users.
Optimization of objectives
, and constraint violation probability 
scenarios at two hydrological guarantee rates. The constraint violation probability level 
is negatively correlated with the optimality and stability values of the upper-level objectives, while it is positively correlated with the equity value. This implies that as 
increases, the optimality and stability of the allocation of water resources in the region tend to decrease, while equity tends to increase. At a 75% hydrological guarantee rate, the best allocation scheme demonstrates lower optimality and stability compared with the scheme at a 95% hydrological guarantee rate. However, smaller values indicate better performance in terms of optimality, stability, and equity. As 
increases, the regional allocation scheme becomes more stable and optimal but less equitable. This indicates that increasing the quantity of water in situations of water shortage results in a more stable and superior allocation scheme, but it doesn't necessarily ensure a more equitable distribution among different regions. It's important to note that the differences in interregional equity are minimal (less than 2%) under conditions of water shortage. Therefore, prioritizing the optimality and stability of the allocation scheme are crucial when making decisions regarding regional water allocation. 
Optimal objectives under different scenarios: (a) 75% hydrological guarantee rate; (b) 95% hydrological guarantee rate; (c) the socioeconomic benefits optimal objective under different scenarios (yuan/m3); and (d) the optimal pollutant emissions under different scenarios 103(m3).
Optimal objectives under different scenarios: (a) 75% hydrological guarantee rate; (b) 95% hydrological guarantee rate; (c) the socioeconomic benefits optimal objective under different scenarios (yuan/m3); and (d) the optimal pollutant emissions under different scenarios 103(m3).
The lower-level model incorporates socioeconomic and environmental considerations, where the socioeconomic objective aims at maximizing economic benefits and minimizing water deficit (Figure 4(c)), and the environmental objective focuses on minimizing pollutant discharge (Figure 4(d)). Figure 4(c) shows the average optimal socioeconomic objective values for different sets of spreads (
) at various 
values and hydrological guarantee rates, along with the gap between the benefit values of the spreads and the average value. The average optimal socioeconomic target values increase with 
, and the benefits at a 75% hydrological guarantee rate are 1.67–2.29 times higher than those at a 95% hydrological guarantee rate for the corresponding 
values. This indicates that higher probability of water supply constraint violation leads to greater socioeconomic benefits. Additionally, at the corresponding 
as 
decreases and 
increases, the corresponding socioeconomic benefits increase. This suggests that socioeconomic benefits are more sensitive to changes in economic parameters. Figure 4(d) demonstrates that pollutant emissions are higher at a 75% hydrological guarantee rate compared with a 95% hydrological guarantee rate, approximately 1.01 times higher. An increase in 
results in increased pollutant emissions, indicating a trade-off between allocating more water for higher economic efficiency and less water to avoid pollutant discharges.
Furthermore, allocation scenarios and models' upper and lower target values are influenced by varying hydrological guarantee rates, the probability of violating water supply constraints 
, and type-2 fuzzy spreads 
of economic parameters. Clear regularities are observed in the effects of hydrological guarantee rates and 
values, while the impacts of spreads 
on target values are relatively minor and lack a distinct regularity, except for socioeconomic objectives of the lower-level. This indicates that multiple uncertainties can influence water allocation patterns in subareas. Among these uncertainties, variations in 
and hydrological guarantee rates directly affect water allocation to each sector, influencing the scheme and target values for both upper-level decision-makers and lower-level followers. Conversely, the spread 
primarily reflects changes in net economic benefit parameters, directly affecting socioeconomic benefits. Table SI 4 displays the crisp values of type-2 fuzzy numbers representing the net economic benefits for the domestic water sector in Xiaodian District under different spread values. The maximum net economic benefits, 769.04 yuan/m3, occur when 
is set to (0.1, 0.9). Conversely, the minimum net economic benefits, 767.03 yuan/m3, are observed with (0.9, 0.1), reflecting changes of less than 0.3%. Therefore, the variation in the spread 
will influence the allocation scheme and target value, but its impact is minimal. It's worth noting that while the spread has a minor effect on the optimal allocation scheme and target value, the critical value approach used in this study provides more of an indicative scheme.
DISCUSSION
Optimization of water resources
Percentage of regional water deficit and percentage of incremental water allocation under different scenarios: (a) 75% hydrological guarantee rate and (b) 95% hydrological guarantee rate.
Percentage of regional water deficit and percentage of incremental water allocation under different scenarios: (a) 75% hydrological guarantee rate and (b) 95% hydrological guarantee rate.
increases, notable changes in water satisfaction occur in the agriculture and industry sectors. Notably, at a 75% guarantee rate, these sectors show higher satisfaction than at 95%, indicating their sensitivity to water scarcity. Decreased water supply disproportionately affects these sectors owing to their higher consumption and shortage probabilities. Particularly, at a 95% hydrological guarantee rate and 
, water supply tension may occur, posing challenges to the operation and development of the industrial and agricultural sectors. Therefore, in the future development of Taiyuan City, it is crucial to strengthen attention to the water demand of the industrial and agricultural sectors and implement corresponding measures such as improving water-use efficiency, promoting water-saving technologies, and optimizing water-use structures to ensure their normal operation and sustainable development. 
Satisfaction of individual water users in different scenarios: (a) 75% hydrological guarantee rate: and (b) 95% hydrological guarantee rate.
Satisfaction of individual water users in different scenarios: (a) 75% hydrological guarantee rate: and (b) 95% hydrological guarantee rate.
Reclaimed water replacement strategies
To ensure sustainable groundwater use in Taiyuan City, addressing overexploitation and its negative consequences on socioeconomic development and the environment is crucial. One effective strategy is to allocate reclaimed water to replace groundwater. Three scenarios are considered: (1) the baseline scenario (BAS) with the projected water supply in 2030 with a hydrological guarantee rate of 75%, 
= (0.5, 0.5), and 
; (2) S1 involves a Low Proportion Replacement, strictly reducing groundwater extraction and requiring a 30% replacement of groundwater with reclaimed water according to the BAS; and (3) S2, High Proportion Replacement, intensifies efforts to control groundwater extraction, mandating a 50% replacement of groundwater with recycled water.
Water allocation schemes with different reclaimed water replacement ratios (106m3). (a) BAS Scenario, (b) S1 Scenario and (c) S2 Scenario.
Water allocation schemes with different reclaimed water replacement ratios (106m3). (a) BAS Scenario, (b) S1 Scenario and (c) S2 Scenario.
Contrasting T2FCC-ORMOFP and T2FCC-MOFP.
Additionally, Table 3 indicates that increasing the replacement ratio of reclaimed water can enhance socioeconomic benefits without causing a rise in pollutant emissions. When the replacement ratio of reclaimed water is increased to 30% (S1) from non-replacement (BAS), socioeconomic benefits see an approximate 9% increase. Furthermore, with a replacement ratio increased to 50% (S2), the benefits rise by about 21% compared with BAS, while pollutant emissions remain relatively stable. This indicates a gradual increase in socioeconomic benefits with an increment in the replacement ratio of reclaimed water. The results can be attributed to the combined effect of water supply parameters and net economic benefits. Externally transferring water and surface water contribute more to the economic benefits in the services sector compared with groundwater. Similarly, reclaimed water contributes more to the economic benefits in the industrial sector compared with surface water. Thus, adjusting the water distribution structure can enhance economic benefits, while pollutant emissions remain unaffected since the overall water consumption by each sector remains unchanged. The results demonstrate that the replacement of reclaimed water can alleviate groundwater overexploitation, optimize the water distribution structure, and improve social and economic benefits, achieving efficient resource utilization.
Socioeconomic benefits and pollutant emissions of different reclaimed water replacement ratios
| pi | BAS | S1 | S2 | |||
|---|---|---|---|---|---|---|
| Socioeconomic benefits (yuan/m3) | Pollutant emissions (m3) | Socioeconomic benefits (yuan/m3) | Pollutant emissions (m3) | Socioeconomic benefits (yuan/m3) | Pollutant emissions (m3) | |
| 0.01 | 639.6 | 158,051.1 | 697 | 158,041.8 | 775.1 | 158,045.5 |
| 0.05 | 777.9 | 158,619.4 | 850.9 | 158,788.3 | 943.1 | 158,786 |
| 0.1 | 879.5 | 159,024.4 | 961 | 158,923.1 | 1,065.2 | 159,017.1 |
| 0.15 | 964.1 | 159,292.4 | 1,053.2 | 159,128.8 | 1,168.1 | 159,306.8 |
| 0.2 | 1,043.9 | 159,457.9 | 1,141 | 159,406.4 | 1,264.9 | 159,388.8 |
| pi | BAS | S1 | S2 | |||
|---|---|---|---|---|---|---|
| Socioeconomic benefits (yuan/m3) | Pollutant emissions (m3) | Socioeconomic benefits (yuan/m3) | Pollutant emissions (m3) | Socioeconomic benefits (yuan/m3) | Pollutant emissions (m3) | |
| 0.01 | 639.6 | 158,051.1 | 697 | 158,041.8 | 775.1 | 158,045.5 |
| 0.05 | 777.9 | 158,619.4 | 850.9 | 158,788.3 | 943.1 | 158,786 |
| 0.1 | 879.5 | 159,024.4 | 961 | 158,923.1 | 1,065.2 | 159,017.1 |
| 0.15 | 964.1 | 159,292.4 | 1,053.2 | 159,128.8 | 1,168.1 | 159,306.8 |
| 0.2 | 1,043.9 | 159,457.9 | 1,141 | 159,406.4 | 1,264.9 | 159,388.8 |
Comparison with the single-layer multi-objective model
Figure 8 illustrates the performance comparison between T2FCC-ORMOFP and the single-layer multi-objective model (T2FCC-MOFP) under a hydrological guarantee rate of 75% and 
set to (0.1, 0.9). The results indicate that the T2FCC-ORMOFP model prioritizes optimality, fairness, and stability, reflecting the upper-level decision-makers' goal management. By contrast, the T2FCC-MOFP model emphasizes higher socioeconomic benefits and lower pollutant emissions of the lower-level decision-making. To standardize and balance the impact of each feature on the model, a standardized method is applied to normalize the five performance items. This eliminates the size difference between different features and prevents certain features with excessively large values from disproportionately affecting the model. The comparison of the combined benefit values between the two models reveals that the scenarios generated by the T2FCC-ORMOFP model yield approximately 1.2 times higher combined benefits than those obtained by the T2FCC-MOFP model. This suggests the T2FCC-ORMOFP model, by prioritizing upper-level goals while considering lower-level needs, enhances overall benefits. However, it's crucial to consider specific problem characteristics and practical requirements when applying the T2FCC-ORMOFP model, as different metrics and datasets may yield different results.
CONCLUSIONS
The T2FCC-ORMOFP model is developed to address water resources system programming under uncertain conditions. This model integrates BL, T2FS, and CCP techniques to handle the interests between decision-makers at different levels and to address uncertainties using probability distributions, membership functions, and fuzzy sets. Additionally, the model evaluates the risk of violating system constraints under uncertainty. Therefore, the T2FCC-ORMOFP model can address issues related to multiple levels, stakeholders, users, objectives, and uncertainties.
By applying this model to Taiyuan City's water resources management system, optimal water resources allocation schemes for subregions and water-consuming sectors are obtained under different scenarios, taking into account different water endowment conditions, supply constraint violation risks, and net economic benefits represented by type-2 fuzzy numbers. These solutions prioritize the principles of optimization, fairness, and stability as advocated by higher-level supervisory authorities, while also taking into account the social, economic, and environmental benefits emphasized by the lower-level management agencies. The goal is to ensure the efficient utilization and sustainable development of the water supply system. In this study, a water supply structure involving groundwater substitution with recycled water was proposed, enhancing local water management flexibility. Compared with single-level models, the decision strategies derived from the T2FCC-ORMOFP model are more comprehensive and effective. The research findings show the following:
- (1)
There are differences in water resources allocation among administrative regions in Taiyuan City, with Qingxu County facing greater uncertainty and water shortage risks;
- (2)
The industrial and agricultural water-consuming sectors in Taiyuan City face higher water shortage and supply violation risks, requiring attention;
- (3)
There are minimal differences (<2%) in equity among regional allocation plans, and priority should be given to optimality and stability, balancing positive socioeconomic benefits from increasing water supply with negative environmental impacts;
- (4)
The reclaimed water replacement strategy can alleviate the current water resources situation in Taiyuan City, optimize the water supply structure, improve socioeconomic benefits, and promote sustainable economic and social development.
- (5)
In addition, compared with the T2FCC-MOFP model, the T2FCC-ORMOFP model is able to take into account the interests and goals of different levels while improving overall efficiency by 20%.
In future research, this model can be applied to other water-deficient areas and other management issues, such as environmental and energy management. Additionally, interval parameter programming and more robust optimization methods should be considered to handle multiple uncertainties and expand the boundaries of the research system to achieve efficient resource utilization and allocation.
ACKNOWLEDGMENTS
All authors are very grateful to the editor and anonymous reviewers for their valuable comments. 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), and the Natural Science Foundation of Shanxi Province, China (No. 202203021221050).
DATA AVAILABILITY STATEMENT
Data cannot be made publicly available; readers should contact the corresponding author for details.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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