Adjustable Robust Optimization for Multi-Period Water Allocation in Droughts Under Uncertainty

Optimal water resource allocation can go some way to overcoming water deficiencies; however, its achievement is complex due to conflicting hierarchies and uncertainties, such as water availability (WA) and water demand (WD). This study develops a robust water withdrawal scheme for drought regions that can balance the trade-offs between the sub-areas and water use participants, ensure sustainable regional system development, and guarantee robust solutions for future uncertainties. A bi-level affinely adjustable robust counterpart (AARC) programming framework was developed in which the regional authority as the leader allocates water to the sub-areas to maximize the intra- and intergenerational equity, and the sub-areas as the followers allocate water to their respective water departments to maximize their economic benefits and minimize water shortages. A case study from Neijiang, China, is given to illustrate the applicability and feasibility of this framework. The novelty of this study is to propose a sustainable bi-level AARC regional water allocation framework which integrates intra- and inter-generational equity of regional water use and priority rules reflected by goal preference programming between water departments under uncertainties of WA and WD simultaneously in water deficient regions.


Introduction
Optimal regional water resource allocation has been found to be an effective approach to mitigating regional water scarcity (Dai et al. 2018), alleviating the contradictions between limited water resources and growing water demand (Tan et al. 2010). Consequently, there have been increased researches into effective water resources allocation, especially in arid and semi-arid areas.
However, water resources planning is always clouded by "cascade" of substantial deep uncertainties indeed (Li et al. 2020), which are introduced by a portfolio of endogenous and exogenous elements, such as sampling uncertainty, human behaviors, environmental processes and hydroclimate change (Herman et al. 2020), with these uncertainties being a greater challenge in regions more prone to drought (Trenberth et al. 2014). Deep uncertainties in water availability (WA) and water demand (WD) affected by environmental or anthropogenic interferences (Li et al. 2020) have posed great challenges for long-term drought emergency problems (Beh et al. 2017).
More recently, many studies have sought to construct optimal water resources management framework under the uncertain WA or WD. Rathnayaka et al. (2017) addressed spatial and temporal variability in WD to obtain sustainable urban water management solutions. Wu et al. (2021) established water and energy resources allocating structure under random WA. Roozbahani et al. (2021) developed an optimal water allocation model which considered variable stream flow. Shan et al. (2021) incorporated complex uncertainties (e.g., precipitation, runoff, evapotranspiration) related to WA into agricultural water resources management model for practical and rational guidance. However, there are few researches focusing on the complex system with WA and WD simultaneously uncertain when making water resources planning, although this is the case in actual states.
Moreover, various techniques have been used in previous studies in response to uncertainties in water resources management, such as, stochastic programming (Beh et al. 2017;Dai et al. 2018), fuzzy logic (Shan et al. 2021) and robust optimization (Mozafari et al. 2020). The first two approaches require that uncertain parameters are subject to perfectly known probability distribution functions or membership functions (Maier et al. 2016). However, the future WA and WD uncertainties for climate change and socio-economic development, which are confronted with distinct future states that may occur (Kwakkelet al. 2010), have no consensus on the potential probability distribution (Trindade et al. 2017). Therefore, methods that set the distribution or membership functions in advance are not appropriate for this situation, which embodies the significance of carrying out the "best policy" that can be immune to data change and guarantee its feasible and near optimal results, notably robust optimization (RO) (Bertsimas et al. 2004). However, previous literatures (e.g., Perelman et al. 2013;Mozafari et al. 2020) using conventional RO are too conservative and the solutions are risk-aversive. To address these concerns, adjustable robust optimization (ARO) proposed by Ben-Tal et al. (2004) divides the decisions into "here and now" which has non-adjustable features and "wait and see", the decisions for which can be made after partial uncertain data known, which are more realistic and less conservatively, and are proved to be below "price of robustness" and little worse than corresponding "ideal" results (Ben-Tal et al. 2004;Kim et al. 2017). Affinely adjustable robust counterpart (AARC) approach where decision variables are formulated as affine functions of the uncertainties, can deal with computational intractability of ARO.
It is noted that optimal water resources allocation is an extraordinary sophisticated system, especially when it comes to contradictory situations under uncertainties of WA and WD in arid and semi-arid regions. Optimum water allocation is a hierarchical structure which needs to consider the conflicts between the upper and lower level (Tu et al. 2015) and should be sustainable considering both the present and the future (Xu et al. 2019). Therefore, it is essential to conceptualize water resources allocation using a hierarchical, multi-period framework under deep uncertainty. However, the complexities and uncertainties associated with the hierarchical resource allocation process and the need to ensure sustainability have not always been fully considered in drought-prone regions.
Consequently, this study proposes a robust bi-level, multi-period allocation policy in water deficient regions under uncertainty. For sustainable water resources planning, maximizing intra-and inter-generational equity is the system objective for the leader (regional authority); the followers (water department) seek for attaining greater economic benefits and fewer water shortages. In addition, the priority rules reflected by goal programming with preferences method have been developed to manage the water department conflicts in drought-prone regions. AARC method which can deal with computational intractable of robust optimization and extended KKT conditions which are able to transform bi-level function to equivalent and solvable single level one, which ensure feasible and less conservative solutions under deep uncertainty. The main contributions of this study are as follows.
• to conceptualize a bi-level, multi-period, sustainable water resources allocation framework for arid and semi-arid regions; • to construct an adjustable robust optimization management model by using affine decision rules that considers the water deficiency uncertainties (WA and WD); • to apply the bi-level, multi-period AARC optimization approach to a case study in a drought-prone region of China.
The remainder of this paper is organized as follows. Problem statement gives the problem statement, Model Formulation details the methodology, Problem-solving approach gives the problem solutions, Case study demonstrates the model in a case study, and Conclusion gives the conclusions.

Conceptual Framework
This paper designs a sustainable regional water resources allocation program which is characterized by a hierarchal structure that includes a dynamic drought decision process. Therefore, a conceptual bi-level regional water allocation framework was constructed, with the regional authority in the upper level, the focus of which is on the ecological environment and intra/intergenerational equity maximization, and the subareas in the lower level, which have income maximization and water shortage minimization as their objective functions. In the lower-level water allocation planning, allocation principles are established to comply with the administrative measures after meeting the minimum ecological water use requirements, with domestic department as the priority, followed by industry and finally agriculture. Due to varying rainfall and possible climate change effects, WA and WD data are usually inexact, vague, and are constantly changing and these uncertainties can have a significant impact on water planning, especially in drought-prone areas. With a specific focus on water deficient regions, this paper employed an AARC method to deal with the WA and WD uncertainties and developed a bi-level multi-period robust water allocation framework. Figure 1 gives the flowchart for the proposed framework that shows the hierarchal structure, the allocation rules and the uncertain environment.

Adjustable Robust Counterpart (ARC)
Robust Optimization is immune to data change, guarantees a feasible, near optimal solution, and does not need to master the true function information. Ben-Tal et al. (2004) introduced Adjustable Robust Counterpart (ARC) methodology, which was found to yield less conservative decisions than classical robust optimization (Zhen et al. 2018) as it encompassed both "here and now" decisions u (that cannot be adjusted to the data) and "wait and see" decisions v (that can be adjusted after uncertain data has been disclosed), that is where the random variables U, V, b are varying in the uncertainty set Z.When V is a certain constant matrix, it is recognized as a fixed recourse that corresponds to a "twostage stochastic programming problem", in which u is the first-stage("here and now") decision and v is the second-stage ("wait and see") decision. This uncertain linear optimization problem which will be addressed next, can be rewritten as Definition. The Adjustable Robust Counterpart (ARC) can be defined as.
Affinely Adjustable Robust Counterpart (AARC). The AARC (Ben-Tal et al.2004) is then introduced to deal with the computational intractability of the ARC, as the linear inequality constraints between the adjustable decisions and random variables can be limited appropriately using affine functions. As in the fixed resource case, the non-variable u can be assumed as given, and U, v, b adopt common assumptions by affinely depending on uncertainty set Z through parameterization mapping v , u ] . The AARC formulation is:

Model Formulation
As shown in Fig. 1, a bi-level multi-period water allocation system for sustainable development in drought-prone regions under uncertainty is considered in this paper. In the upper level, the regional authority as the leader considers the overall socio-economic development when allocating the water to the subareas, with the regional objective function being to maximize the intra-and intergenerational equity under the respective constraints; in the lower-level, each sub-area follower makes water withdrawal decisions based on the consideration to maximize the economic benefits and minimize the water deviation between water demand and water supply. The leader not only considers the upper-decision making variables, but simultaneously takes followers' objective functions and constraints as the upper constraints, and the followers' behaviors have been affected by the upper-level decisions. This section details the mathematical formulations of the bi-level water allocation framework, and the global optimization model of which is shown in Appendix 2 (Eq. (20)). The parameters and decision variables are displayed in Appendix 1.

Upper-Level Objective
In areas where there is a sharp contradiction between water supply and demand, the difference and irrationality of resource allocations are more likely to trigger conflicts. To reduce regional imbalances and promote coordinated development in the whole basin, it is necessary to guarantee water resource allocation equity. The Gini coefficient is an effective tool impartiality, considering population and the gross domestic product (GDP) in each subarea as the criteria. Moreover, for sustainability, equity must consider both spatial and temporal dimensions and focus on individual and intergenerational issues (Kverndokk et al.2014). The spatial equity considerations need to balance the solutions in different sub-areas for the current generation and the temporal equity considerations need to maintain equilibrium between the current and future generations (different periods). The equity functions are: Intra-generational equity: Inter-generational equity: Thus, the Gini coefficient minimization function F equity is described as follows: where ∈ (0, 1) is the weight, u and v are watershed areas u, v ∈ i(i = 1, 2, ...I) , l and q are the periods l, q ∈ t(t = 1, 2, ...T) , and h ki is the Gini coefficient assessment.

Upper-Level Constraints
State transit of stored water: In arid and semiarid areas, the reservoir water storage plays a significant role in providing both spatial and temporal water resources in dry seasons. Therefore, there is a certain functional relationship between the state of the storage water variables and decision-making, that is, the stored water V(t) at the end of period t can be determined by the water storage in the t − 1 period V(t − 1) , the water availability W A t and the allocated water ∑ i x it in period t,as follows: The stored water reservoir capacity is limited within a certain range [V min , V max ] , with its function being to meet several needs, such as power generation, flood control, and the fishing industry. The water storage constraint is: Ecological water requirement constraint: Arid and semi-arid areas have fragile ecological environments and are vulnerable to eco-environmental damage as they have poor resilience and are sensitive to changes in external conditions. To meet this sustainable development requirement, the water allocated to ecosystem e it care must be guaranteed to meet the minimum ecological water demand e min it , as shown in the following constraint: Water availability constraint for sub-area allocation: The volume of water distributed from region to sub-areas, including ecological water e it , and domestic, industrial, agricultural water, cannot exceed the regional water supply: Non-negative constraint: The water withdrawal cannot be negative; therefore, there is the following constraint on the decision variables:

Lower-Level Objective
In the lower level, subarea managers focus more on their own respective benefits and the amount of water shortage, instead of taking a panoramic view of the system. Each subarea provides water to three water users (domestic, industrial, and agricultural users) where b ijt is the economic profit parameter per unit of used water ( CNY∕m 3 ). Therefore, the benefit maximization is expressed as: The objective of maximizing economic benefit: In arid and semi-arid regions, each sub-area manager seeks to reduce their water deficiencies, especially when there is extreme water insufficiency. Thus, water shortage minimization between water demand ∑ j ∑ tW D ijt and water supply ∑ j ∑ t y ijt is the second lowerlevel objective: The objective of minimize water shortage:

Lower-Level Constraints
Water allocation constraint for each sub-area: To make sure there is no water waste or inefficiency, the allocated water y ijt for each water user is constrained within a certain range, that is, between the minimum water requirements L min ijt and the maximum withdrawal targets L max ijt , as follows: Non-negative constraint: As the distributed water cannot be negative, there is the. following constraint on the decision variables:

Allocation Principles
When there is extreme water insufficiency, water distribution among the departments has become the key and difficult problem. As the managers of sub-areas are responsible for making an optimal strategy to maintain daily life and social development, certain water resource allocation priority principles were established as lower-level management tools for drought-prone regions based on China's specific administrative policies. When the ecological environmental water demand is satisfied (Eq. (10)) of the upper constraints), the domestic water demand should be satisfied first, followed by the industrial water demand, with the residual amount being distributed to the agricultural sector. Water scarcity can be mitigated using rational spatial cropping patterns, that is, less allocated water should motivate farmers to adopt more effective irrigation techniques and strategies (Hatem et al. 2020). A goal preference programming method is therefore utilized to realize the lowerlevel distribution principles: where P 0 represents the preferences for the first objective f be i ,P 1 , P 2 , P 3 represent. the water user preferences in the second objective f sh refers to the slack variables. P 1 > P 2 > P 3 represents the priority, that is, the water is first for domestic use, then for industrial use, and finally for agricultural use.

Problem-Solving Approach
The global allocation formulation (Eq. (18)) with uncertain parameters can be re-formulated to computationally tractable equivalent equation by using AARC approach.
Although there is no information regarding the water availability W A t in period t , the actual values from years {1, 2, ...t − 1} can be easily acquired, with x i1 being the "here and now" decisions, and x it being the "wait and see" decisions based on the information from the previous periods I t = {1, 2, ...t − 1} . The AARC methodology assumes that the adjustable variable x it is an affine function of W A r : A r , with the water quantity allocated y ijt depending on data water data water demand W D ijr . y ijt is then mapped with the affine decision rules as follows: , while the future demand W D ijt in the uncertainty set is based on the given parameter perturbations and the nominal demand Therefore, from the uncertainty set, the uncertain AARC problem can be straightforwardly converted (17) min y ijt to a computationally tractable equivalent linear programming equation (Eq. (20)), as demonstrated in Appendix 2. Furthermore, bilevel programming problem can be solved by extensive methodology. Extended Karush-Kuhn-Tucker (KKT) conditions (Lu et al. 2006) are utilized to deal with hierarchical structures with multiple independent followers. The specific steps of transforming the standard bi-level model to the equivalent single-level model are displayed in the Appendix 2.

Case Study
To demonstrate the effectiveness of the proposed robust optimization methodology under uncertainty, this section gives a practical application in Neijiang, China. In this paper, Lingo 18.0 software was utilized to solve the single-level linear programming modelling which is conducted and run on R5-4600H, 2.90 GHz, with 16.00 GB of AMD.

Case study System Overview
Neijiang, a prefecture-level city located on the Tuo River in southeast Sichuan province, China, has a per capita water resource that is usually below 400 m 3 ∕per and water resource utilization above 50%, and has been found to have typical seasonal and year water shortages which is prone to suffer from droughts. There are five subareas in Neijiang, that is, Shizhong, Dongxing, Zizhong, Longchang, and Weiyuan, shown in Fig. 2. There are four main crops (rice, corn, potato, and soybean) in each subarea of Neijiang City with a total of 226 thousand hectares of planting area and 1.3 million tons of crop yield, of which the rice enjoys the largest cultivation land and the highest yield, accounting for almost 35% and 50% respectively in the whole Neijiang City. In these four subareas (Shizhong, Dongxing, Longchang, Weiyuan), the crop with the widest planting area is rice, while corn is extensively cultivated in Weiyuan. And the annual agricultural water consumption is more than 50% of the total water availability in the region. The data and parameters for this problem were extracted from the 2011-2019 Statistical Yearbooks and Water Resources Bulletins from the Regional Bureau of Statistics in Neijiang and its experts opinions. Essential information for the determined parameters and expert decisions are shown in Appendix 3. To meet the allocation principle requirements, the preference weights ( P j ) were diverse and satisfied with P 1 > P 2 > P 3 . The planning period was set to be 3 years and the predicted values being taken as the model reference. In this case, equal importance was given to the intra-and inter-generational equity.

Results and Discussions
This section comprehensively discusses the calculation result and water allocation strategies, and gives a comparative analysis to provide a valuable reference for arid and semiarid regional decision makers under uncertain environments.

Regional Strategies Analysis Under Diverse Uncertainty Levels
We have carried out the regional water allocation strategies of Neijiang city for the proposed methodology under the uncertainties of WA and WD, and conducted four experiments where and respectively equaled to 5%, 10%, 15% and 20% to ascertain the effect of the fluctuations on the performances of the results. The optimal regional objective values are shown in Table 1, each sub-area's objective values are depicted in Fig. 3, and the water withdrawal solutions are illustrated in Fig. 4. (1) Equity, benefit, water shortage analysis At first, we analyzed the objective functions from the perspective of regional authority of the four uncertainty levels. The Gini coefficient ranges from 0 to 1 with 0 being absolute fairness, and the smaller the value, the greater the equity. From the equity results shown in Table 1, it can be seen all the results were between 0 and 0.1, which indicated that the water resources allocations had high inter-and intra-generational equity even though there were fluctuating uncertainty levels, which demonstrated that the AARC water allocation framework is able to make sound decisions for contemporary and future generations even in spite of significant uncertainty. From the regional total economic benefits, it can be seen that the results were sensitive to fluctuations in WA and WD, with the maximum being 4,285,666 CNY at = = 0.1 , 4,052,487 CNY at = = 0.20 higher than the minimum with a 5.8% fluctuation. The regional water shortages were shown to be more vulnerable to indefinite circumstances as they moved from 5,583.80 million m 3 at = = 0.10 to 10,917.36 million m 3 at = = 0.20 with nearly doubled growth, which showed a rapid ascend. Consequently, the regional economic benefits and the amount of water shortages 1 3 presented a worse trend with the increase of uncertainty values, indicating that the uncertainties have a significant impact on water resource management systems that cannot be ignored. Afterwards, optimal objective values were elaborated under diverse uncertainty levels from the perspective of each sub-area's administration. From the results of Fig. 3, we can draw the conclusions that sub-areas' own economic benefit and water shortage are all getting worse when the uncertainty level increases. More specifically, the average value of the benefit reduction of these five sub-areas was 13% where Zizhong had the maximum value of 24%. The added average value of water shortage was 1066.711 million m 3 from = = 0.05 to = = 0.20 , where Weiyuan had the largest added value, that is, 1,462.765 million m 3 . As for the results of five sub-areas, Weiyuan always reached the highest economic benefit values since it possessed the largest benefit coefficient b ijt and greatest GDP G it , bringing more water yield and producing higher economic benefit, while Shizhong got the relatively lowest values due to lower b ijt , G it and less group in population PO it , resulting in worse consequences.
(2) Water allocation strategy analysis As displayed in Table 3 of Appendix C, to illustrate a serious water stress situation, the nominal demand WD * ijt took the average of the previous years and the WA * t reduced over time, with an average of 9,369.123 million m 3 in the first period ( WA * 1 ) and a decrease of 1,000 million m 3 in the following year. The multi-period sub-area and departmental water allocation strategies were examined under several vital uncertainty levels with WA and WD uncertain, as shown in Fig. 4, and the water shortages were based on the maximum possible water loss.
In Fig. 4, when the uncertainty levels of WA and WD increased, regional water allocation amount decreased with maximal deviation 1,120.5031 million m 3 between = = 0.05 and = = 0.20 , which is lower than the maximal fluctuation of water availability in all periods. This indicated that AARC approach is less conservative than the conventional robust optimization with box uncertainty set which constricts the uncertainties on the extreme values simultaneously. In addition, it can also be seen that the water resources allocated all presented a trend of gradual decrease under the four uncertain situations. It is worth nothing that the water allocation strategies in various periods were somewhat similar, especially in the latter two periods ( t = 2 and t = 3 ). In order to obtain high values of intra-generational equity and the especially high inter-generational equity, regional decision makers have a disposition to reduce the deviations among various periods to cope with the increasingly severe conditions by using water conservation measures such as reservoirs and water saving. Therefore, to reduce annual water imbalances and seasonal water, water conservancy projects (e.g., reservoir) are necessary to regulate water storage and water deficiencies.
It is showed in Fig. 4 that the water allocation strategies among the three water departments under various uncertain values, and it's obvious that the proportion of domestic water is the least (almost 20%), followed by industrial water (near 33%), and agricultural water is the largest but not more than 50%. Furthermore, most of these figures were satisfied with water resource allocation principle with priority with less and less water resources, that is, the largest reduction in agricultural water with average value at 626.981 million m 3 , followed by industrial water (623.324 million m 3 ) and domestic water (248.593 million m 3 ). From the perspective of various uncertainty levels, we can see that all the 1 3 allocated amount diminution of water departments are gradually increasing with uncertain values, where domestic figures increased from 171.000 to 333.004 million m 3 , industrial values from 447.195 to 875.403 million m 3 , and agricultural ones from 254.281 to 1074.842million m 3 .
In the five sub-areas, Zizhong acquired the most water allocation volume under the four uncertain conditions due to the largest population and water demand, while Shizhong had the least amount of water allocated when = 0.05, = 0.05 = 0.1, = 0.1 and Weiyuan obtained the least water when = 0.15, = 0.15 and = 0.2, = 0.2 thanks to the fewer population. The water distribution also proved that the system strictly abides by the water shortage policy constraint priority principle from the domestic sector, to the industry sector, and finally to the agricultural sector. For example, in the Dongxing sub-area, when there is a 15% WA and WD uncertainty level, from the first period when there are sufficient water resources to the third water reduction period, the agricultural water sector allocation decreased from 786.448 million m 3 to the minimum of 464.370 million m 3 (by 282.375million m 3 ),while the industrial water consumption reduced by 64.117 million m 3 and there was little change in the domestic sector.

Comparison Discussions
In order to explore the influence of different uncertain parameter on the results, we conducted several experiments under different uncertainty scenarios with various uncertainty levels (from 5%to 20% by 5% rise). There are four scenarios: Scenario 1, in which both the water availability and demand are uncertain; Scenario 2, in which only water availability is indeterministic; Scenario 3, in which only water demand is uncertain; Scenario 4, no uncertain parameters. Table 2  (1) Comparison with scenarios only WA or WD uncertain The results from Table 2 indicated that there were large differences in the economic benefit and water shortage targets as uncertainty values changed; however, the equity was similar regardless of the different scenarios and uncertainty levels. Both Scenarios 2 and 3 emerging the identical tendency, the regional economic benefits are gradually decreasing, while water shortages are increasing along with the rise of uncertainty values, which is similar to the Scenario 1. In addition, Scenario 3 had the largest economic benefit on average at 4,542,303.75 CNY, followed by Scenario 1 at 4,179,622.25 CNY and Scenario 2 at 4,052,756 CNY, which indicated that any variations in water demand have greater impact on the system than the water supply. The result in Scenario 3 was better mainly because dynamic water resource allocations increase the possibility of greater water provision in the region. The benefit in the Scenario 2 was more likely affected by water demand fluctuations as indicated by the higher standard deviation (132,960.6703CNY) than Scenarios 3 (116,486.659CNY) and 1 (97,648.34415CNY). Therefore, it was concluded that water demand management is imperative in improving regional development stability. The lowest regional water shortages were found in Scenario 3, and the highest regional water shortages were in Scenario 2, indicating that water shortages were more vulnerable to fluctuations in the water demand. Interestingly, the result was not as volatile as the one in Scenario 1 when there were double uncertainties, indicating that it is possible that the water resources supply interval uncertainty alleviated the impact of the dynamic demand.

Comparison with Deterministic Programming
To further demonstrate the superiority and effectiveness of the AARC model's response to regional water allocations under dynamic water supply and demand uncertainties, it was compared with conventional deterministic programming (Scenario 4). The deterministic optimization model was formulated with no uncertainty, that is, = 0 and = 0 . In Table 1, the linear programming determined the allocation strategies to be a 0.058 intraand inter-generational equity, 4,559,127 CNY economic benefit, and a 2,884.558 million m 3 water shortage.
The proposed bi-level water allocation framework had preferable spatial and temporal equity in all scenarios with little "price of robustness", and the proposed model had preferable spatial and temporal equity. However, compared with the indeterministic modelling, the economic benefit maximization and water shortage minimization goals results of the certainty case were similar to the scenario with uncertainties of WA and WD and significantly better than the other two scenarios in which there was uncertain demand. Consequently, it was concluded that accurate water demand descriptions are vital for regional water resources planning. Although the water shortages in uncertain environments are greater than the ones in certain environments, the proposed allocation scheme obtained a higher equity and a regional economic benefit close to the certain environment. Therefore, the AARC model, which incorporates uncertainty and robustness, provides more flexible, less conservative solutions, and gives deeper insights in the sub-areas and water user water distribution even in the uncertain environments, which allows decision makers to adjust their uncertainty coefficients to express varying attitudes and diverse future situations.

Management Suggestions
In response to water insufficiency and possibly more serious states, it is necessary to develop water resource allocation frameworks under deep uncertainty that balance supply and demand and promote efficient utilization and water conservation for both the short and long term. Based on above-mentioned results analysis, two policy implications for water resources management in drought regions are proposed.
(1) Regional authority managers need to establish water resources planning frameworks that enable information fluctuate in both water availability and demand. Water resources management is a sophisticated system with unknown water availability and demand before the decision-making and optimal solutions needing to be made that embrace these uncertainties, especially in water deficient region. The results demonstrated 1 3 that the unstable relation between water supply and water demand triggers great contradiction and system loss, and especially, uncertain demand results in greater fluctuations in the overall system development. Moreover, the robust solutions can be immune to data change of supply and demand, and guarantee feasible, near optimal decisions.
(2) Water conservancy facilities (reservoirs) need to be established to ensure emergency water demand, improve water resources storage efficiencies, and reduce water shortages. Reservoirs play an important role in regulating the annual and long-term distribution of water resources, especially when there are long-term water shortages and seasonal droughts. Based on the analysis of the results, the amount of water allocated in the three periods has been narrowed under the increasingly severe situation of water resource, and all figures of inter-and intra-generational equity under uncertain parameters are lower than 0.1, since the reservoir's regulation function on water yield come into play. However, as conservancy project construction inevitably changes the surrounding environment and ecology, corresponding policies and regulations need further study to identify alternative schemes.
(3) Administrative measures are vital by focusing on contingency plans for all sectors and can directly and quickly respond to emergencies in water deficient regions. In the above framework, goal preference programming model was established with priority in different water users, so that in the case of water shortage emergency, the economy can be developed after meeting the minimum living requirements of residents. When there are serious water shortages, the administration can set simple constraints to prioritize the water resources allocation to first to ensure sustainable ecosystem development, and then respectively meet the domestic, industrial production, and agricultural production requirements.

Conclusions
In this paper, a bi-level robust optimization methodology for regional sustainable water resources management was proposed to balance the trade-off among sub-areas and water-use departments and provide regional authority with a practicable and rational water allocation planning under deep uncertainty. In this hierarchal model framework, the upper-level regional authority allocated water to the sub-areas to maximize the equity between the current and future generations, whereas the lower-level sub-areas distributed water to their various water sectors to maximize their own benefits and minimize water shortages. The study closely examined the supply and demand uncertainties and constructed an adjustable robust optimization programming function to provide flexible, less conservative solutions for governments and managers. The affine functions between the water allocated and water availability or demand were defined by using linear rules that transformed the bi-level model into a linear programming problem that was computationally tractable. Priority allocation principles were also established on the lower level through using goal programming with preferences to ensure ecological sustainability and to adhere to administrative policies. The model was then applied to Neijiang, a drought prone area in Sichuan Province, China, to demonstrate the applicability and rationality of the optimization method and elucidate sustainable water allocation solutions. An in-depth water deficiency analysis was conducted by varying the uncertain coefficients, which provides regional authorities and 1 3 sub-area managers with a viable methodology to deal with water supply and demand uncertainty problems in the real world. Regardless of the positive results, there were some research gaps that need to be addressed in future research. For example, the effect of ground water pumping on surface water flows in future years and the complex and interrelated relationship between water supply uncertainty and water demand uncertainty were not considered.

Appendix 1
Notations of the proposed bi-level multi-period water allocation framework: i , water subarea i, i = 1, 2, ...I; j , water department j, j = 1, 2, ...J; t , water resources planning period t, t = 1, 2, ...T; e min it , minimum ecological water demand in subarea i at period t ( m 3 ); b ijt , benefit parameter per unit for water department j in subarea i at period i,(CNY∕m 3 ); L min ijt , minimum water demand of water department j in subarea i at period i ( m 3 ); L max ijt , maximum water demand of water department j in subarea i at period i ( m 3 ); V min , minimum water capacity of stored water ( m 3 ); V max , maximum water capacity of stored water ( m 3 ); WA t , regional available water at period t(m 3 ); WD ijt , water demand for water department j in subarea i at period i(m 3 ); V(0) , regional initial stored water at the beginning ( m 3 ); V(t) , regional stored water at period t at the beginning ( m 3 ); x it , water allocated to subarea i at period t(m 3 ); y ijt , water allocated to water department j in subarea i at period i(m 3 ); e it , water allocated for ecological use in subarea i at period t(m 3 ).

3
Appendix 2 (1) The global optimization of the bi-level water allocation formulation is: x it > 0, e min it , y ijt > 0, ∀i, j, t 1 3 (2) The AARC modelling is as follows: (4) The standard bi-level model with multiple followers is: Let vector u, v, w be the dual variables of the upper and constraints, lower constraints and lower decision variables. The necessary and sufficient condition to guarantee the solutions of Eq. (22) is that there exist vectors u, v, w solve the problem below: B n y n ≤ e min f n (x, y n ) = s n x + r n y n s.t. C n x + D n y n ≤ g n , ∀n x ≥ 0, y n ≥ 0, ∀n B n y n ≤ e C n x + D n y n ≤ g n , ∀n u n B n + v n D n − w n = −r n , ∀n u n (e − Ax − N � n=1 B n y n ) + v n (g n − C n x − D n y n ) + w n y n = 0 x ≥ 0, y n ≥ 0, u n ≥ 0, v n ≥ 0, w n ≥ 0, ∀n 1 3