An Interval-Parameter Fuzzy Linear Programming with Stochastic Vertices Model for Water Resources Management under Uncertainty

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Introduction
Water shortage has become serious issue in process of urbanization as well as socioeconomic development gradually, especially in metropolis, where water resources are limited.Therefore, effective allocation of water resources to various users is important.There are many uncertain factors in practical management and decision making of complex water resources system, which might result in significant difficulties in optimizing water resources allocation.Conventional deterministic optimization methods have difficulties in reflecting these uncertainties [1].Many researchers have tried to tackle these uncertain problems through fuzzy programming, interval programming, and stochastic programming [2][3][4][5][6][7][8][9][10].In many real-world problems, several types of uncertainties may exist together in a complex system.Therefore, hybrid uncertainty methods have been desired for solving the problem with several types of uncertainties.Based on inexact chance-constrained programming (ICCP) method, Huang [11] proposed a hybrid inexact-stochastic water management model, which improves upon the existing inexact and stochastic programming approaches by allowing both distribution information in right hand and uncertainties in left hand or coefficients of objective.Hybrid uncertainties, including interval and stochastic distribution information in parameters and coefficients, can be directly communicated into the optimization process through representing the uncertain parameters or coefficients as fuzzy sets and random variables [3,9,10,[12][13][14][15].In these hybrid uncertainty approaches, each coefficient or parameter has only one kind of uncertainty, and the stochastic distribution information is treated with the discrete way.

Mathematical Problems in Engineering
Due to the complexity of the real world, highly uncertain information may exist.Such boundaries of interval parameters of the optimization model are also uncertain.Nie et al. [16] proposed an interval parameter fuzzy-robust programming (IFRP) model through introducing the concept of fuzzy boundary interval.The parameters of the IFRP model were represented as interval numbers with fuzzy uncertain boundary, and the uncertainties were directly communicated into the optimization process and resulting solution.In this way, the robustness of the optimization process and solution can be enhanced.Considering the interval uncertain of boundaries of interval parameters, Liu and Huang [17] proposed a dual interval two-stage restrictedrecourse programming (DITRP) method for flood-diversion planning.Liu et al. [18] established a dual-interval linear programming (DILP) model by introducing ILP approach into the existing interval-parameter linear programming framework.Some of coefficients and parameters in the DILP model were represented as interval-parameter with interval vertices.The DILP approach improved the ILP method by allowing dual uncertainties (presented as dual intervals) to be incorporated into the optimization process.For the fuzzy feature of boundaries of interval parameters, Li et al. [19] proposed a dual-interval vertex (DIV) method by incorporating the vertex method within an intervalparameter programming framework, and a fuzzy vertex analysis approach was proposed for solving the DIV model.The DIV approach can tackle uncertainties expressed as dual intervals that exist in both objective function and left-hand and right-hand sides of the constraints.However, these above methods hardly deal with the dual uncertainties including stochastic distribution attributes.Considering the stochastic attribute of boundaries of interval parameters in objective functions and constraints, Han et al. [20] proposed an interval linear programming with stochastic vertices (ILPSV) method to tackle dual uncertainties, which were presented as interval parameter with stochastic vertices problem.The fuzzy attribute of objective and constraints is not for concern in the ILPSV model.Therefore, one potential approach for better accounting for integrated uncertainties of parameters of model is to incorporate the stochastic distribution within a general fuzzy linear programming framework.This leads to an interval-parameter fuzzy linear programming with stochastic vertices method under dual uncertainty.
The objective of this paper is to propose an intervalparameter fuzzy linear programming model with stochastic vertices by coupling inexact fuzzy linear programming (IFLP) and stochastic vertices method.Highly uncertain information for the lower and upper bounds of interval parameters that exist in optimization model due to the complexity of the real world can be effectively handled through allowing the stochastic boundary of interval parameter to be incorporated into the optimization processes.In addition, the dual uncertainty concept (being stochastic boundaries of interval) is presented when the available information is highly uncertain for boundaries of interval parameter of objective functions and constraints.A hybrid intelligent algorithm based on Liu [21,22] has been proposed for solving the developed model.The developed IFLPSV model is then applied to allocation of multisource water to multiple users in Beijing city of China in 2020, where water resources shortage is a challenging issue.

Methodology
2.1.Dual Uncertain Linear Programming Model.In many practical problems, the lower and upper bounds of some interval parameters in a water resources management system can rarely be acquired as deterministic.Instead, they can only be expressed by interval, fuzzy, or stochastic numbers.For a system with such dual uncertainty, an interval-parameter linear programming with stochastic vertices is generated as follows: where  1(a).The interval number of dual uncertainty linear programming is shown in Figure 1(b), and the vertices may possess a certain distribution function, such as formal distribution.The -axis in Figure 1 does not represent any physical meaning since the figure is used to illustrate an interval number or an interval number with randomly distributed lower and upper bounds.
The models (2a), (2b), (2c), and (2d) are an interval parameter linear programming with stochastic vertices, and it possesses the randomly distributed attribute of stochastic vertices and interval attribute of interval parameter; therefore, it is a dual uncertain optimization model.The traditional solution algorithm for interval-parameter linear programming (ILP) is not applicable to solve the models (2a), (2b), (2c) and (2d) due to stochastic variables existence.

Illustrative Example
The following illustrative example can be formulated to demonstrate the applicability of the proposed method: where,  1 ,   1, no units for all results since it is only an illustrative example).
Therefore, the solutions of problem models (7a), (7b), (7c), (7d), and (7e) are  Water shortage problem is severe, and there has appeared four times water crisis in urbanization process of Beijing city.The per capita water resource in Beijing city is less than 300 cubic meters, which is one-thirtieth of the world average.Beijing has become one of the most water-scarce megacities in the world.Water resource has become a key factor which limits urbanization process and socioeconomic development in Beijing city.
In order to meet water requirements of Northwest and North China, the South-to-North Water Diversion Projects have been developed early.The Middle Route of South-to-North Water Diversion Project will transfer water from the Danjiangkou Reservoir on the Hanjiang river, a tributary of the Yangtze river, to Beijing city through opening channel.The Middle Route Project will be completed in 2014.The total amount of transferred water to Beijing city ranges from 1.0 to 1.4 billion m 3 per year, which will be another important part of water resources for Beijing city in the future.

IFLPSV Optimization Model for Water Resources Allocation in Beijing
City.Reasonable allocation of urban water resources is usually a multiobjective problem.In this study, the objective functions include net economic benefit maximization and greenbelt irrigation area maximization as shown in formulas (10a) and (10b).The objective functions are subject to ten kinds of constraints, including water delivery capacity limit from river or lake as formula (10c), Guanting reservoir water supply capacity limit as formula (10d), Miyun reservoir water supply capacity limit as formula (10e), groundwater supply capacity limit as formula (10f), reused water supply capacity limit as formula (10g), South-to-North Water Transfer supply capacity limit as formula (10h), the lowest requirement limit of domestic water consumption as formula (10i), the lowest requirement limit of industry water consumption as formula (10j), the lowest requirement limit of agriculture water consumption as formula (10k), the lowest requirement limit of greenbelt water irrigation as formula (10l), nonnegativity constraint is as formula (10m): where  1 ,  2 are the interval objective function of the net economic benefit maximization and greenbelt irrigation area maximization, respectively,   ( = 1, 2, 3, . . ., ;  = 1, 2, 3, . . ., ) denotes the interval amount of water source  supply to  water user,  = 6 is the total number of water source (lift or deliver water from river and lake, Guanting reservoir, Miyun reservoir, groundwater, reused water, and South-to-North Water Transfer);  = 4 is the total number of water users (domestic, industry, agriculture and environment),   is the benefit coefficient of the water source  supply to  user, which belongs an interval number,  is the interval coefficient of greenbelt irrigation water consumption per acre,  deliv is the maximization water deliver capacity from rivers and lakes, which is an interval number,  reser1 is the water supply capacity of the Guanting reservoir, which is an interval number,  reser2 is the water supply capacity of the Miyun reservoir, which is an interval number,  ground is the groundwater supply capacity, which is an interval number,  reuse is the reused water supply capacity, which is an interval range,  trans is the available water amount from the South-to-North Water Diversion Project, which is an interval number,  domes is the lowest amount of domestic water requirement, which belongs an interval range, the reused water for domestic is small, it is ignored,  indus is the lowest amount of industry water requirement, which is an interval number,  agri is the lowest amount of agriculture water requirement, depending on weather and rainfall.The value of agriculture water requirement is an interval number with normal distribution boundaries, (12.05× 10 8 , 10000) m 3 and (12.01× 10 8 , 10000) m 3 respectively;  env is the lowest amount of greenbelt irrigation water requirement, which is an interval number.In this study, the greenbelt irrigation water requirement includes river water supplement and urban roadway watering.
The capacity of water source supply under different flow frequencies is shown in Table 3.
Scenario 1 denotes the normal flow year, in which frequency of flow is 50%.The total amount of water resource is  The results show that domestic has higher priority than industry on water consumption under water shortage situation.The amount of water consumption of agriculture and environment reach its least requirement in Beijing city.Therefore, the amounts of agricultural and environmental water consumption are almost the same under three scenarios.In the dry year or extraordinary dry year, the water crisis might be relieved by reducing industrial water consumption.
Figure 7 shows the allocation results of different water source supplies to different water users in Beijing city under three scenarios.Domestic water consumption of Beijing city is mainly from groundwater, reservoirs, and South-to-North Middle Route Project in 2020.Although the total amount of the domestic water consumption is the same under three scenarios, the amounts of water in Guanting and Miyun reservoirs (reservoir1 and reservoir 2 in Figure 2) for the domestic sector evidently decreased with rainfall reduction.Moreover, groundwater is the primary and reliable water source for the domestic sector.The allocated groundwater for domestic sector is raised from [738.For example, the reused water amount for environment is remarkably raised from [303.9, 323.9] million m 3 in scenario 1 to [348.9, 368.9] million m 3 in scenario 2 and to [400.8, 420.8] million m 3 in scenario 3. The results show that transfer water from South-to-North Middle Route Project has significant impact on water supply structure in Beijing city.It will mitigate the water shortage issue of social and economy development in Beijing city in some extent, particularly in dry year and extraordinary dry year.

Conclusion
An interval-parameter fuzzy linear programming with stochastic vertices (IFLPSV) has been developed for water resources allocation under dual uncertainty.The developed IFLPSV model improves upon the existing IFLP method by allowing the uncertain boundaries of interval parameter to be incorporated into the optimization processes.A hybrid intelligent algorithm based on genetic algorithm and artificial neural network was used to the developed model.The developed IFLPSV considers fuzzy, interval, and stochastic attributes of parameters and coefficients of objective functions and constraints.The application results indicated that it is effective for regional water resources allocation and planning under dual uncertainty.IFLPSV may provide more satisfactory solutions for an optimization problem under dual uncertainty.The developed IFLPSV model has been applied to multisource water allocation among multiple users in Beijing city in 2020, where water resources shortage is a challenging issue.The results indicate that transfering water from South-to-North Middle Route Project has an important impact on water supply structure in Beijing city, particularly in dry year and extraordinary dry year.The developed model and solution algorithm can be extended to other water resources system and environment management, where bounds of interval parameters or coefficients are dual uncertainty.
In this study, only two objectives including economic and greenbelt irrigation area maximization have been considered when carrying out water allocation in Beijing city.In the future, other objectives, such as social and environmental objective, might also be involved into water resources allocation in Beijing city.Moreover, interactive method for solving multi-objective optimization model is supposed to be incorporated into the methodology, so that the decision makers can make compromise among conflict objectives.
There are many uncertain factors in practical management and decision making of water resources system.The interval programming is an effective method to tackle uncertainties expressed as interval values with known lower and upper bounds.However, due to high complexity of water resources system, sometimes it is hard to determine the exact values of both the lower and upper bounds.They might be with characteristics of interval, fuzzy, or random.Therefore, in this paper, the dual uncertainties are defined as interval or fuzzy numbers with lower and upper bounds with interval, fuzzy, or random characteristics.Although the bounds of interval parameters might be randomly distributed, the elements within the interval are regarded as certain numbers.In future research, physical meaning of the stochastic attribute of upper and lower bounds of interval parameters is suggested to in-depth investigation.Further discussion should be undertaken to answer whats the difference between interval number with randomly distributed bounds when the elements within the interval are simply considered as certain numbers and interval parameter as simply a random parameter with its own distribution.

Figure 1 :
Figure 1: Interval number with certain and uncertain vertices.

Figure 2 :
Figure 2: Flow chart of solving the IFLPSV model by hybrid intelligent algorithm.

Figure 5 :
Figure 5: Location illustration of main water resources in the study area.

Table 1 .
4078, 1.5333] and  2 = [0.6014,0.6080], and interval objective value is  = [30.9548,42.6913].In the work by Li et al. [19], dual uncertain parameters were interval values, and the DIV method was employed.The results of illustrative examples (6a), (6b), (6c), and (6d) by ILP, DIV, ILPSV, and IFLPSV method are shown in The ILP method can tackle uncertainties expressed as interval values with known lower and upper bounds, but the distribution functions of uncertain parameters are unknown.The result of objective value from ILP is [[29.438,32.150], [42.172, 45.784]].The solutions might compose fuzzy information due to the parameter's large boundary range.The DIV can tackle the uncertainties presented as interval parameter with fuzzy vertices by incorporating the vertex method within an interval parameter programming.The result of objective value from DIV is [[30.099,31.455],[43.053, 44.895]].Some feasible solutions might be missed by DIV method due to using the discrete vertex way, and the DIV method is unable to deal with the vertex presented as certain stochastic distribution function problem.The upper bound (42.6913) of objective value from IFLPSV is smaller than the interval [43.053, 44.895] of upper bound of DIV method.The reason is that normal distribution of upper and lower bounds of the interval parameter has more information than that of DIV model.The ILPSV method can tackle the interval-parameter linear programming with stochastic vertices.The result of objective value from IFLPSV is [30.9548,42.6913].The upper and lower bounds of the objective value from IFLPSV have less uncertainty than ILPSV since parameters with membership function have less interval parameters of ILP with interval bounds.Therefore, the uncertainty degree of the solution from the IFLPSV is also less.The developed IFLPSV model is an integrated optimization model under dual uncertainty, which considers fuzzy, interval, and stochastic attributes of parameters and coefficients of optimization model.

Table 4 :
Table 4shows the results of water allocation to water users under three scenarios.The total amount of allocated water is [6184, 6285] million m 3 in Results of water resources allocation among water users under three scenarios.

Table 4 and
Figure 6, the industry and domestic are main water utilization sectors in Beijing city.For example, in scenario 1, the amount of water allocation to industry is[2146, 2146]million cubic meters, which accounts for [34.1%, 34.7%] of total allocated water.The amount of allocated water to domestic is [1634, 1634] million cubic meters, which accounts for [26.0%, 26.4%] of total allocated water.In scenario 2, the amount of water allocation to industry is declined to [1712, 1712] million cubic meters, which accounts for [29.3%, 29.8%] of total allocated water.The amount of allocated water to domestic is [1634, 1634] million cubic meters, which accounts for [27.9%, 28.4%] of total allocated water.In scenario 3, the industrial water consumption is declined to [1361, 1361] million cubic meters, which only accounts for [24.7%, 25.2%] of total allocated water.The amount of allocated water to domestic is [1632, 1634] million cubic meters, which accounts for [29.7%, 30.2%] of total allocated water.Although the amount of domestic water consumption is almost not changed, its proportion is more than industrial in scenario 3. The proportion of domestic water consumption increases from [26.0%, 26.4%] in scenario 1 to [29.7%, 30.2%] in scenario 3.
2, 738.2] million m 3 in scenario 1 to [764.3, 764.3] million m 3 in scenario 2 and to [808.2, 808.2] million m 3 in scenario 3.Meanwhile, the water amount for the domestic sector from South-to-North Middle Route Project is raised from [325.7, 325.7] million m 3 in scenario 1 to [401.8, 401.8] million m 3 in scenario 2 and to [495.7, 495.7] million m 3 in scenario 3. The South-to-North Middle Route Project gradually becomes rainfall reduction in dry year and extraordinary dry year, the exploitation amount of groundwater and South-to-North Middle Route Project water is raised to compensate water shortage of the agriculture sector.The allocated groundwater for agriculture is raised from [586.7, 599.7] million m 3 in scenario 1 to [620.8, 633.8] million m 3 in scenario 2 and to [679.9, 686.7] million m 3 in scenario 3. The amount of South-to-North Middle Route Project water for agriculture is also raised from [187.2, 187.2] million m 3 in scenario 1 to [271.3, 271.3] million m 3 in scenario 2 and to [374.2, 374.2] million m 3 in scenario 3. The environment mainly uses reservoirs water, groundwater, reused water, and South-to-North Middle Route Project water.The water amount of Guanting and Miyun reservoirs for environment is declined with rainfall reduction.Contrarily, the groundwater, reused water, and transfer water for environment are all raised.