# Inexact Multistage Stochastic Chance Constrained Programming Model for Water Resources Management under Uncertainties

Authors: Hong Zhang; Minghu Ha; Hongyu Zhao; Jianwei Song
Journal: Scientific Programming
 (2017)

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## Abstract

In order to formulate water allocation schemes under uncertainties in the water resources management systems, an inexact multistage stochastic chance constrained programming (IMSCCP) model is proposed. The model integrates stochastic chance constrained programming, multistage stochastic programming, and inexact stochastic programming within a general optimization framework to handle the uncertainties occurring in both constraints and objective. These uncertainties are expressed as probability distributions, interval with multiply distributed stochastic boundaries, dynamic features of the long-term water allocation plans, and so on. Compared with the existing inexact multistage stochastic programming, the IMSCCP can be used to assess more system risks and handle more complicated uncertainties in water resources management systems. The IMSCCP model is applied to a hypothetical case study of water resources management. In order to construct an approximate solution for the model, a hybrid algorithm, which incorporates stochastic simulation, back propagation neural network, and genetic algorithm, is proposed. The results show that the optimal value represents the maximal net system benefit achieved with a given confidence level under chance constraints, and the solutions provide optimal water allocation schemes to multiple users over a multiperiod planning horizon.


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## Body

## 1. Introduction

With the economic development and population growth, the demands for water resources increase every year. Due to unreasonable exploitation and utilization of water resources, water pollution, and extreme weather conditions, more and more countries and regions are faced with different degrees of water shortage. Making the best use of limited water resources premises the sustainable development of economy and society [1]. In the past few decades, many researchers have applied optimization techniques [2–7] to deal with uncertainties in a number of system components and their interrelationships within water resources systems [8]. Among them, inexact multistage stochastic programming (IMSP) is regarded as a significant method for water resources management. For example, Li et al. [9] proposed an interval-parameter multistage stochastic linear programming model which incorporates inexact optimization and multistage stochastic linear programming to manage water resources. Zhou et al. [10] developed a factorial multistage stochastic programming which is a hybrid methodology of factorial analysis and IMSP to analyze the potential interrelationships among a variety of uncertain parameters and their impacts on system performance for water resources management. Suo et al. [11] proposed an inventory-theory-based inexact multistage stochastic programming for water resources management through introducing an inventory theory into the framework of IMSP to provide reasonable transferring schemes associated with various flow scenarios. IMSP was also combined with other types of uncertainties, such as fuzziness [12–14], to solve the water resources management problems. However, the above models cannot assess the risks in water resources management systems, to which chance constrained programming (CCP) is an effective method to resort to.CCP, pioneered by Charnes and Cooper [15], provides a method to handle uncertainties by specifying a confidence level at which it is desired that the constraint holds [16], which has been applied in several areas [17, 18]. In water resources management, using CCP is conducive to not only making decisions of water allocation but also gaining insights into the tradeoffs between economic objectives and some policy factors for a water resource manager [19]. For instance, Liu et al. [1] provided a factorial multistage stochastic programming with chance constraints to deal with the issues of constraint-violation risks and interactive uncertainties in water resources management systems. However, this kind of CCP models can only be solved in some special cases. Meanwhile, the models should firstly be converted into deterministic equivalents and then be solved by using solution methods of deterministic mathematical programming [16] in the solution process. Complex chance constrained programming models are hard to solve, such as nonlinear models which contain multiply distributed data [20]. In order to overcome the limitations, Liu [21] presented a framework of CCP with the assumption that not only uncertain constraints but also uncertain objective would hold at different confidence levels provided by the decision-maker as an appropriate safety margin and gave a hybrid algorithm to solve the CCP for more cases. It is noticeable that the choice of confidence levels makes the model flexible to more situations. The CCP has been applied to many aspects of system decision-making, such as project scheduling [22], capital budgeting [23], capacitated location-allocation [24], and redundancy-optimization [25]. Up to now, Liu [21]’s CCP has not been applied in the optimization of water resources management although this method is needed because of the complexity of the water resources management systems.In our previous relevant research, [26] extended Liu [21]’s CCP from probability space to Sugeno measure space, in which Sugeno measure was one of representative nonadditive measures. Although the theoretical basis of the CCP was discussed in detail, we did not apply the model to water resources management. Paper [27], which managed water supply risk by constructing a model of water option pricing, did not refer to water resources allocation and assess the risks by using CCP. As an extension of the previous efforts, this study aims to develop an inexact multistage stochastic chance constrained programming (IMSCCP) model, which incorporates the CCP proposed by Liu [21] into the IMSP framework for water resources management. The model can tackle uncertainties in the objective and constraints which present themselves as probability distributions and interval with multiply distributed stochastic boundaries, analyze various policy scenarios when the promised policy targets are violated, and deal with the issue of risks presented as stochastic constraints and stochastic objective with predetermined confidence levels. A hypothetical case study of water resources management within three planning periods is given to demonstrate the applicability of the method. Moreover, a hybrid algorithm incorporating stochastic simulation, back propagation (BP) neural network, and genetic algorithm (GA) is proposed to solve the model. The developed method obtains the results of the model in which the optimal value represents the maximal net system benefit achieved with a given confidence level subject to chance constrains.


## 2. Methodologies

### 2.1. Multistage Inexact Stochastic Programming Model

It is important for a water resources manager to allocate water to multiple users such as municipal, industrial, and agricultural sectors from an unregulated reservoir over a multiperiod planning horizon in an optimized way, which could be formulated as maximizing the expected economic revenue based on the water allocation in the region over the planning horizon [28]. The manager promises a quantity of water to each user. If the promised water is delivered, net benefits will be brought to local economy; otherwise, the deficient water must be obtained from alternative and more expensive sources or the demand must be curtailed, resulting in economic penalties on local economy [29]. Considering the randomness of the water flow and the dynamic feature of the long-term water allocation plans, the problem of water allocation can be formulated as the following scenario-based multistage stochastic programming model for water resources management under uncertainties [9]: (1)max⁡f=∑i=1m∑j=1nNBijTij-∑j=1nE∑i=1mCijDiQjs.t.∑i=1mTij-DiQj≤Qj+εj-1Qj-1,j=1,2,…,nεj-1Qj-1=Qj-1-∑i=1mTij-1-DiQj-1+εj-2Qj-2,j=1,2,…,nTijmax⁡≥Tij≥DiQj≥0,i=1,2,…,m,j=1,2,…,n,where f is the net benefit of the water allocation system ($); NBij is the net benefit when per unit (m3) of water is allocated to user i in period j ($/m3); Tij is the fixed amount of water allocation target promised to user i in period j (m3); Qj is the water flow in period j which is a random variable (m3); DiQj is the amount of water shortage to user i when the seasonal flow is Qj (m3); Cij is the loss when per unit (m3) of water is not allocated to user i in period j ($/m3); E· is the expected value of a random variable; εj-1Qj-1 is the surplus water inflow in period j (m3) according to Qj-1; Tijmax⁡ is the amount of maximum allowable allocation for user i (m3); m is the number of water users; i is the index of water user for i=1,2,3, with i=1 for the municipality, i=2 for the industrial user, and i=3 for the agricultural sector. In the model, DiQj is the decision variable and f is the objective function. It is observed that model (1) reflects nonanticipativity, since a decision must be made in each stage without the knowledge of realizations of random variables in future stages.LetQj take values of qjk with probability pjk for scenarios in each planning period (j), where pjk is the probability of occurrence for scenario k in period j and ∑k=1Kjpjk=1. Then, we have E∑i=1mCiDiQi=∑i=1m∑k=1KjpjkCijDijk,j=1,2,…,n, where Kj is the sum of scenarios in period j; Dijk is the amount of water allocation shortage to user i when the scenario k occurs in period j. Then model (1) can be reformulated as follows [9]:(2)max⁡f=∑i=1m∑j=1nNBijTij-∑i=1m∑j=1n∑k=1KjpjkCijDijks.t∑i=1mTij-Dijk≤Qjk+εj-1k,j=1,2,…,n,k=1,2,…,Kjεj-1k=Qj-1k-∑i=1mTij-1-Dij-1k+εj-2k,j=1,2,…,n,k=1,2,…,KjTijmax⁡≥Tij≥Dijk≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.Considering that the uncertainties exist in variables and coefficients, the fixed values of the parameters, such asTij, NBij, Cij, and Qjk, cannot be determined exactly. Thus, the inexact multistage stochastic programming (IMSP) model, which introduces interval parameters into model (2), is proposed as follows [9]:(3)max⁡f±=∑i=1m∑j=1nNBij±Tij±-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±s.t∑i=1mTij±-Dijk±≤qjk±+εj-1k±,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1±-Dij-1k±+εj-2k±,j=2,3,…,n,k=1,2,…,KjTijmax⁡±≥Tij±≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.


### 2.2. Inexact Multistage Stochastic Chance Constrained Programming Model

Models (1), (2), and (3) do not readily assess the risks, and they only deal with uncertainties in the right hand side such as the water flow Q. It is difficult to handle uncertainties in both the left and right hand sides (i.e., Tij, NBij, and Cij) [1] which are presented as interval with stochastic normal distributed boundaries. In view of the above considerations, Liu et al. [1] combined chance constrained programming (CCP, initiated by Charnes and Cooper [15]) with IMSP to propose the following inexact multistage stochastic programming model with chance constraints to solve problems with the request that chance constraints should hold at least with prescribed levels of probability (i.e., confidence levels):(4)max⁡f±=∑i=1m∑j=1nNBij±Tij±-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±s.tPr∑i=1mTij±-Dijk±≤qjk±+εj-1k±≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1±-Dij-1k±+εj-2k±,j=2,3,…,n,k=1,2,…,KjTijmax⁡±≥Tij±≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.Model (4) can handle uncertainties presented as interval with normally distributed boundaries. However, uncertainties in the water resources management systems have more diverse forms of performance besides normal distribution, which suggests a need for models that can handle uncertainties presented as interval with multiple distributed boundaries. And the confidence levels only in constraints may be not enough to indicate the relationship between the economic objective and the system risk. Moreover, a water resources manager may want to obtain the maximum that the objective function f(x,ξ) achieves with a given confidence level subject to stochastic constraints with other confidence levels, which means that the confidence levels are not only in constraints but also in objective.Liu [21] proposed a framework of nonlinear chance constrained programming with confidence levels occurring in constraints and objective and provided a stochastic simulation based genetic algorithm to solve the CCP. This CCP model can be formulated as follows:(5)max⁡f-s.tPrξ∣fx,ξ≥f-≥αPrξ∣gjx,ξ≤0≥βj,j=1,2,…,n,where x is a decision vector; ξ is a random vector; α and βj are predetermined confidence levels for stochastic objective and stochastic constraint(s), respectively; Pr· denotes the probability of the event in ·. This programming aims to maximize f- that the objective function f(x,ξ) achieves with at least probability of α (max⁡f- is calledα-optimistic value to f(x,ξ)). x is feasible if and only if the probability measure of the set ξ∣gj(x,ξ)≤0 is at least βj for j=1,2,…,n.By incorporating CCP [21] and multistage stochastic programming (model (2)), a multistage stochastic chance constrained programming model for water resources management could be formulated as follows:(6)max⁡f-s.tPr∑i=1m∑j=1nNBijTij-∑i=1m∑j=1n∑k=1KjpjkCijDijk≥f-≥αPr∑i=1mTij-Dijk≤qjk+εj-1k≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k=qj-1k-∑i=1mTij-1-Dij-1+εj-2k,j=2,3,…,n,k=1,2,…,KjTijmax⁡≥Tij≥Dijk≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj,where α and βjk are predetermined confidence levels for stochastic objective and stochastic constraint(s), respectively. This programming aims to obtain the α-optimistic value to the objective function of ∑i=1m∑j=1nNBijTij-∑i=1m∑j=1n∑k=1KjpjkCijDijk. Obviously, model (6) gives predetermined confidence levels for constraints and objective, which indicates more comprehensive risk assessment in the water resources management systems.Considering that the fixed values of the parameters in model (6) cannot be determined exactly, an inexact multistage stochastic chance constrained programming (IMSCCP) model is proposed as follows:(7)max⁡f-±s.tPr∑i=1m∑j=1nNBij±Tij±-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±≥f-±≥αPr∑i=1mTij±-Dijk±≤qjk±+εj-1k±≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1±-Dij-1±+εj-2k±,j=2,3,…,n,k=1,2,…,KjTijmax⁡±≥Tij±≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.LetTij±=Tij-+ΔTijyij, where ΔTij=Tij+-Tij-, i=1,2,…,m, j=1,2,…,n. Model (7) could be replaced by the following form:(8)max⁡f-±s.tPr∑i=1m∑j=1nNBij±Tij-+ΔTijyij-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±≥f-±≥αPr∑i=1mTij-+ΔTijyij-Dijk±≤qjk±+εj-1k±≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1-+ΔTij-1yij-1-Dij-1±+εj-2k±,j=2,3,…,n,k=1,2,…,Kj0≤yij≤1,i=1,2,…,m,j=1,2,…,nTijmax⁡±≥Tij-+ΔTijyij≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.Next, we can change model (8) into two submodels corresponding to the upper and lower bounds for the desired objective function value.Firstly, we formulate submodel (9) corresponding to the upper bound of the objective function value; that is,(9)max⁡f-+s.tPr∑i=1m∑j=1nNBij+Tij-+ΔTijyij-∑i=1m∑j=1n∑k=1KjpjkCij-Dijk-≥f-+≥αPr∑i=1mTij-+ΔTijyij-Dijk-≤qjk++εj-1k+≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k+=qj-1k+-∑i=1mTij-1-+ΔTij-1yij-1-Dij-1-+εj-2k+,j=2,3,…,n,k=1,2,…,Kj0≤yij≤1,i=1,2,…,m,j=1,2,…,nTijmax⁡+≥Tij-+ΔTijyij≥Dijk-≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj,where Dijk- and yij are the decision variables and max⁡f-+ is the α-optimistic value to the net system benefit ∑i=1m∑j=1nNBij+Tij-+ΔTijyij-∑i=1m∑j=1n∑k=1KjpjkCij-Dijk-. In submodel (9), the optimal solutions are denoted by Dijkopt- and yijopt, and the optimal value is denoted by f-opt+.Secondly, submodel (10) corresponding to the lower bound of the objective function value can be formulated as follows: (10)max⁡f--s.tPr∑i=1m∑j=1nNBij-Tij-+ΔTijyijopt-∑i=1m∑j=1n∑k=1KjpjkCij+Dijk+≥f-opt+≥αPr∑i=1mTij-+ΔTijyijopt-Dijk+≤qjk-+εj-1k-≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k-=qj-1k--∑i=1mTij-1-+ΔTij-1yij-1opt-Dij-1++εj-2k-,j=2,3,…,n,k=1,2,…,KjTij-+ΔTijyijopt≥Dijk+≥Dijkopt-,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj,where Dijk+ is the decision variable and max⁡f-- is the α-optimistic value to the net system benefit ∑i=1m∑j=1nNBij-Tij-+ΔTijyijopt-∑i=1m∑j=1n∑k=1KjpjkCij+Dijk+. In submodel (10), the optimal solution is denoted by Dijkopt+ and the optimal value is denote by f-opt-.Thirdly, the real allocation of water to useri when the scenario k occurs in period j can be calculated by Aijkopt±=Tij-+ΔTijyijopt-Dijkopt±.Finally, the optimized interval solution for the decision variableDijkopt±=Dijkopt-,Dijkopt+ and the real interval allocation of water Aijkopt±=Aijkopt-,Aijkopt+ can be obtained, while the optimized interval objective value f-opt±=f-opt-,f-opt+ can be generated through the variation of Dijkopt±.


## 2.1. Multistage Inexact Stochastic Programming Model

It is important for a water resources manager to allocate water to multiple users such as municipal, industrial, and agricultural sectors from an unregulated reservoir over a multiperiod planning horizon in an optimized way, which could be formulated as maximizing the expected economic revenue based on the water allocation in the region over the planning horizon [28]. The manager promises a quantity of water to each user. If the promised water is delivered, net benefits will be brought to local economy; otherwise, the deficient water must be obtained from alternative and more expensive sources or the demand must be curtailed, resulting in economic penalties on local economy [29]. Considering the randomness of the water flow and the dynamic feature of the long-term water allocation plans, the problem of water allocation can be formulated as the following scenario-based multistage stochastic programming model for water resources management under uncertainties [9]: (1)max⁡f=∑i=1m∑j=1nNBijTij-∑j=1nE∑i=1mCijDiQjs.t.∑i=1mTij-DiQj≤Qj+εj-1Qj-1,j=1,2,…,nεj-1Qj-1=Qj-1-∑i=1mTij-1-DiQj-1+εj-2Qj-2,j=1,2,…,nTijmax⁡≥Tij≥DiQj≥0,i=1,2,…,m,j=1,2,…,n,where f is the net benefit of the water allocation system ($); NBij is the net benefit when per unit (m3) of water is allocated to user i in period j ($/m3); Tij is the fixed amount of water allocation target promised to user i in period j (m3); Qj is the water flow in period j which is a random variable (m3); DiQj is the amount of water shortage to user i when the seasonal flow is Qj (m3); Cij is the loss when per unit (m3) of water is not allocated to user i in period j ($/m3); E· is the expected value of a random variable; εj-1Qj-1 is the surplus water inflow in period j (m3) according to Qj-1; Tijmax⁡ is the amount of maximum allowable allocation for user i (m3); m is the number of water users; i is the index of water user for i=1,2,3, with i=1 for the municipality, i=2 for the industrial user, and i=3 for the agricultural sector. In the model, DiQj is the decision variable and f is the objective function. It is observed that model (1) reflects nonanticipativity, since a decision must be made in each stage without the knowledge of realizations of random variables in future stages.LetQj take values of qjk with probability pjk for scenarios in each planning period (j), where pjk is the probability of occurrence for scenario k in period j and ∑k=1Kjpjk=1. Then, we have E∑i=1mCiDiQi=∑i=1m∑k=1KjpjkCijDijk,j=1,2,…,n, where Kj is the sum of scenarios in period j; Dijk is the amount of water allocation shortage to user i when the scenario k occurs in period j. Then model (1) can be reformulated as follows [9]:(2)max⁡f=∑i=1m∑j=1nNBijTij-∑i=1m∑j=1n∑k=1KjpjkCijDijks.t∑i=1mTij-Dijk≤Qjk+εj-1k,j=1,2,…,n,k=1,2,…,Kjεj-1k=Qj-1k-∑i=1mTij-1-Dij-1k+εj-2k,j=1,2,…,n,k=1,2,…,KjTijmax⁡≥Tij≥Dijk≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.Considering that the uncertainties exist in variables and coefficients, the fixed values of the parameters, such asTij, NBij, Cij, and Qjk, cannot be determined exactly. Thus, the inexact multistage stochastic programming (IMSP) model, which introduces interval parameters into model (2), is proposed as follows [9]:(3)max⁡f±=∑i=1m∑j=1nNBij±Tij±-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±s.t∑i=1mTij±-Dijk±≤qjk±+εj-1k±,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1±-Dij-1k±+εj-2k±,j=2,3,…,n,k=1,2,…,KjTijmax⁡±≥Tij±≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.


## 2.2. Inexact Multistage Stochastic Chance Constrained Programming Model

Models (1), (2), and (3) do not readily assess the risks, and they only deal with uncertainties in the right hand side such as the water flow Q. It is difficult to handle uncertainties in both the left and right hand sides (i.e., Tij, NBij, and Cij) [1] which are presented as interval with stochastic normal distributed boundaries. In view of the above considerations, Liu et al. [1] combined chance constrained programming (CCP, initiated by Charnes and Cooper [15]) with IMSP to propose the following inexact multistage stochastic programming model with chance constraints to solve problems with the request that chance constraints should hold at least with prescribed levels of probability (i.e., confidence levels):(4)max⁡f±=∑i=1m∑j=1nNBij±Tij±-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±s.tPr∑i=1mTij±-Dijk±≤qjk±+εj-1k±≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1±-Dij-1k±+εj-2k±,j=2,3,…,n,k=1,2,…,KjTijmax⁡±≥Tij±≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.Model (4) can handle uncertainties presented as interval with normally distributed boundaries. However, uncertainties in the water resources management systems have more diverse forms of performance besides normal distribution, which suggests a need for models that can handle uncertainties presented as interval with multiple distributed boundaries. And the confidence levels only in constraints may be not enough to indicate the relationship between the economic objective and the system risk. Moreover, a water resources manager may want to obtain the maximum that the objective function f(x,ξ) achieves with a given confidence level subject to stochastic constraints with other confidence levels, which means that the confidence levels are not only in constraints but also in objective.Liu [21] proposed a framework of nonlinear chance constrained programming with confidence levels occurring in constraints and objective and provided a stochastic simulation based genetic algorithm to solve the CCP. This CCP model can be formulated as follows:(5)max⁡f-s.tPrξ∣fx,ξ≥f-≥αPrξ∣gjx,ξ≤0≥βj,j=1,2,…,n,where x is a decision vector; ξ is a random vector; α and βj are predetermined confidence levels for stochastic objective and stochastic constraint(s), respectively; Pr· denotes the probability of the event in ·. This programming aims to maximize f- that the objective function f(x,ξ) achieves with at least probability of α (max⁡f- is calledα-optimistic value to f(x,ξ)). x is feasible if and only if the probability measure of the set ξ∣gj(x,ξ)≤0 is at least βj for j=1,2,…,n.By incorporating CCP [21] and multistage stochastic programming (model (2)), a multistage stochastic chance constrained programming model for water resources management could be formulated as follows:(6)max⁡f-s.tPr∑i=1m∑j=1nNBijTij-∑i=1m∑j=1n∑k=1KjpjkCijDijk≥f-≥αPr∑i=1mTij-Dijk≤qjk+εj-1k≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k=qj-1k-∑i=1mTij-1-Dij-1+εj-2k,j=2,3,…,n,k=1,2,…,KjTijmax⁡≥Tij≥Dijk≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj,where α and βjk are predetermined confidence levels for stochastic objective and stochastic constraint(s), respectively. This programming aims to obtain the α-optimistic value to the objective function of ∑i=1m∑j=1nNBijTij-∑i=1m∑j=1n∑k=1KjpjkCijDijk. Obviously, model (6) gives predetermined confidence levels for constraints and objective, which indicates more comprehensive risk assessment in the water resources management systems.Considering that the fixed values of the parameters in model (6) cannot be determined exactly, an inexact multistage stochastic chance constrained programming (IMSCCP) model is proposed as follows:(7)max⁡f-±s.tPr∑i=1m∑j=1nNBij±Tij±-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±≥f-±≥αPr∑i=1mTij±-Dijk±≤qjk±+εj-1k±≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1±-Dij-1±+εj-2k±,j=2,3,…,n,k=1,2,…,KjTijmax⁡±≥Tij±≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.LetTij±=Tij-+ΔTijyij, where ΔTij=Tij+-Tij-, i=1,2,…,m, j=1,2,…,n. Model (7) could be replaced by the following form:(8)max⁡f-±s.tPr∑i=1m∑j=1nNBij±Tij-+ΔTijyij-∑i=1m∑j=1n∑k=1KjpjkCij±Dijk±≥f-±≥αPr∑i=1mTij-+ΔTijyij-Dijk±≤qjk±+εj-1k±≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k±=qj-1k±-∑i=1mTij-1-+ΔTij-1yij-1-Dij-1±+εj-2k±,j=2,3,…,n,k=1,2,…,Kj0≤yij≤1,i=1,2,…,m,j=1,2,…,nTijmax⁡±≥Tij-+ΔTijyij≥Dijk±≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj.Next, we can change model (8) into two submodels corresponding to the upper and lower bounds for the desired objective function value.Firstly, we formulate submodel (9) corresponding to the upper bound of the objective function value; that is,(9)max⁡f-+s.tPr∑i=1m∑j=1nNBij+Tij-+ΔTijyij-∑i=1m∑j=1n∑k=1KjpjkCij-Dijk-≥f-+≥αPr∑i=1mTij-+ΔTijyij-Dijk-≤qjk++εj-1k+≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k+=qj-1k+-∑i=1mTij-1-+ΔTij-1yij-1-Dij-1-+εj-2k+,j=2,3,…,n,k=1,2,…,Kj0≤yij≤1,i=1,2,…,m,j=1,2,…,nTijmax⁡+≥Tij-+ΔTijyij≥Dijk-≥0,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj,where Dijk- and yij are the decision variables and max⁡f-+ is the α-optimistic value to the net system benefit ∑i=1m∑j=1nNBij+Tij-+ΔTijyij-∑i=1m∑j=1n∑k=1KjpjkCij-Dijk-. In submodel (9), the optimal solutions are denoted by Dijkopt- and yijopt, and the optimal value is denoted by f-opt+.Secondly, submodel (10) corresponding to the lower bound of the objective function value can be formulated as follows: (10)max⁡f--s.tPr∑i=1m∑j=1nNBij-Tij-+ΔTijyijopt-∑i=1m∑j=1n∑k=1KjpjkCij+Dijk+≥f-opt+≥αPr∑i=1mTij-+ΔTijyijopt-Dijk+≤qjk-+εj-1k-≥βjk,j=1,2,…,n,k=1,2,…,Kjεj-1k-=qj-1k--∑i=1mTij-1-+ΔTij-1yij-1opt-Dij-1++εj-2k-,j=2,3,…,n,k=1,2,…,KjTij-+ΔTijyijopt≥Dijk+≥Dijkopt-,i=1,2,…,m,j=1,2,…,n,k=1,2,…,Kj,where Dijk+ is the decision variable and max⁡f-- is the α-optimistic value to the net system benefit ∑i=1m∑j=1nNBij-Tij-+ΔTijyijopt-∑i=1m∑j=1n∑k=1KjpjkCij+Dijk+. In submodel (10), the optimal solution is denoted by Dijkopt+ and the optimal value is denote by f-opt-.Thirdly, the real allocation of water to useri when the scenario k occurs in period j can be calculated by Aijkopt±=Tij-+ΔTijyijopt-Dijkopt±.Finally, the optimized interval solution for the decision variableDijkopt±=Dijkopt-,Dijkopt+ and the real interval allocation of water Aijkopt±=Aijkopt-,Aijkopt+ can be obtained, while the optimized interval objective value f-opt±=f-opt-,f-opt+ can be generated through the variation of Dijkopt±.


## 3. Case Study

In this section, the IMSCCP model is applied to the water resources management systems. Just as the statement for the hypothetical problem described in Huang and Loucks [30], a water resources manager shoulders the responsibility of delivering water from an unregulated reservoir to three sectors, that is, municipality, industry, and agriculture, during three periods. All users want to know allocated water amount that they can expect over the three periods. If the allocated water satisfies the demand of the user, per unit water will gain net benefit for the local economy. Otherwise, the user will spend more to obtain water from other reservoirs or curtail their expansion plans, which means per unit water deficiency will gain penalty. Table 1 provides the water allocation targets in the three planning periods. Tables 2 and 3 present the distribution of the stream flows in the three periods. Table 4 shows the related economic data. Obviously, the boundaries of the data in Tables 3 and 4 are random variables satisfying different distribution forms. The objective is to obtain 90%-optimistic value to the net system benefit while the constraints of water availability hold with at least a probability of 95% over the planning horizon.Table 1
Water allocation targets (106m3).
Water allocation targets
Time periods
t
=
1
t
=
2
t
=
3
Municipalityi=1
2,3
2.5,3.5
3,4
Industrial sectori=2
2.5,4
3.5,5.5
4,6
Agriculture sectori=3
3.5,5.5
4,6
4,6.5Table 2
Seasonal flows in the three planning periods (106m3).
Time periods
Seasonal flows
Low flowj=1
Medium flowj=2
High flowj=3
t
=
1
3.5,4.5
6,8
12,15
t
=
2
5,6
7,10
13,17
t
=
3
3.5,4.5
7.5,11
13.5,17.5
Probability
0.2
0.6
0.2Table 3
The distribution forms of the boundaries of seasonal flows in the three planning periods (106m3).
Time periods
Seasonal flows
Low flowj=1
Medium flowj=2
High flowj=3
t
=
1
N
(
3.5,0
.
1
2
)
,
N
(
4.5,0
.
1
2
)
N
(
6,0
.
1
2
)
,
N
(
8,0
.
1
2
)
N
(
12,0
.
1
2
)
,
N
(
15,0
.
1
2
)
t
=
2
N
(
5,0
.
1
2
)
,
N
(
6,0
.
1
2
)
N
(
7,0
.
1
2
)
,
N
(
10,0
.
1
2
)
N
(
13,0
.
1
2
)
,
N
(
17,0
.
1
2
)
t
=
3
N
(
3.5,0
.
1
2
)
,
N
(
4.5,0
.
1
2
)
N
(
7.5,0
.
1
2
)
,
N
(
11,0
.
1
2
)
N
(
13.5,0
.
1
2
)
,
N
(
17.5,0
.
1
2
)
Note. Nμ,σ2 represents a normally distributed random variable with mean μ and standard deviation σ.Table 4
Net benefit and penalty ($/m3).
Time period
t
=
1
t
=
2
t
=
3
Net benefit when water demand is satisfied
Municipality
U
(
90,91
)
,
U
(
110,111
)
U
(
95,96
)
,
U
(
115,116
)
U
(
100,101
)
,
U
(
130,131
)
Industrial sector
N
(
45,0
.
1
2
)
,
N
(
50,0
.
1
2
)
N
(
50,0
.
1
2
)
,
N
(
60,0
.
1
2
)
N
(
55,0
.
1
2
)
,
N
(
70,0
.
1
2
)
Agricultural sector
N
(
30,0
.
1
2
)
,
N
(
35,0
.
1
2
)
N
(
35,0
.
1
2
)
,
N
(
42,0
.
1
2
)
N
(
40,0
.
1
2
)
,
N
(
58,0
.
1
2
)
Penalty when water is not delivered
Municipality
U
(
200,201
)
,
U
(
250,251
)
U
(
220,221
)
,
U
(
275,276
)
U
(
240,241
)
,
U
(
300,301
)
Industrial sector
N
(
60,0
.
1
2
)
,
N
(
85,0
.
1
2
)
N
(
70,0
.
1
2
)
,
N
(
95,0
.
1
2
)
N
(
80,0
.
1
2
)
,
N
(
110,0
.
1
2
)
Agricultural sector
N
(
50,0
.
1
2
)
,
N
(
70,0
.
1
2
)
N
(
55,0
.
1
2
)
,
N
(
75,0
.
1
2
)
N
(
60,0
.
1
2
)
,
N
(
90,0
.
1
2
)
Note. Ua,b represents a uniform distributed random variable and Nμ,σ2 represents a normally distributed random variable.Based on the information shown in Tables1–4, the IMSCCP model for water resources management can be formulated as follows: (11)max⁡f-±s.tPr∑i=13∑j=13NBij±Tij-+ΔTijyij-∑i=13∑k=13p1kCi1±Di1k±-∑i=13∑k=19p2kCi2±Di2k±-∑i=13∑k=127p3kCi3±Di3k±≥f-±≥αPr∑i=13Ti1-+ΔTi1yi1-Di1k±≤q1k±≥β1k,k=1,2,3Pr∑i=13Ti2-+ΔTi2yi2-Di2k±≤q2k±+ε1k±≥β2k,k=1,2,…,9Pr∑i=13Ti3-+ΔTi3yi3-Di3k±≤q2k±+ε2k±≥β3k,k=1,2,…,27ε1k±=q1k±-∑i=13Ti1-+ΔTi1yi1-Di1±,k=1,2,…,9ε2k±=q2k±-∑i=13Ti2-+ΔTi2yi2-Di2±+ε1k±,k=1,2,…,90≤yij≤1,i=1,2,3,j=1,2,3Ti1max⁡±≥Ti1-+ΔTi1yi1≥Di1k±≥0,i=1,2,3,k=1,2,3Ti2max⁡±≥Ti2-+ΔTi2yi2≥Di2k±≥0,i=1,2,3,k=1,2,…,9Ti3max⁡±≥Ti3-+ΔTi3yi3≥Di3k±≥0,i=1,2,3,k=1,2,…,27.Apparently, model (11) is different from the exiting chance constraint programming models for water resources management which transform the original chance constraints into deterministic equivalents via the theories provided by Charnes et al. [31]. In order to solve this model, a hybrid algorithm which consists of stochastic simulation, BP neural network, and GA is proposed as follows.Hybrid Algorithm. In order to solve the model, a hybrid algorithm, which incorporates stochastic simulation, back propagation (BP) neural network, and genetic algorithm (GA), is proposed. At first, model (9) and model (10) should be solved by the hybrid algorithm. In the solution process, stochastic simulation is used to generate input-output data, BP neural network is used to approximate the functions according to the generated input-output data, and GA is used to enhance the optimization process and obtain a solution to the optimization problem. The flowchart figure of the hybrid algorithm is shown in Figure 1.Figure 1
The flowchart figure of hybrid algorithm.Then, the optimized water allocation target and the real allocation of water are achieved. Finally, we obtain the optimal solutions and the optimal value of the model. The concrete step of the algorithm can be summarized as follows.Step 1.
Firstly, give system modeling of model (9). Generate a training set of input-output data for the following uncertain functions:(12)Uijk′yij,Dijk-=max⁡f-+∣Pr∑i=13∑j=13NBij+Tij-+ΔTijyij-∑i=13∑k=13p1kCi1-Di1k--∑i=13∑k=19p2kCi2-Di2k--∑i=13∑k=127p3kCi3-Di3k-≥f-+≥α,Ui1k′′yi1,Di1k-=Pr∑i=13Ti1-+ΔTi1yi1-Di1k-≤q1k+,k=1,2,3,Ui2k′′yi2,Di2k-=Pr∑i=13Ti2-+ΔTi2yi2-Di2k-≤q2k++ε1k+,k=1,2,…,9,Ui3k′′yi3,Di3k-=Pr∑i=13Ti3-+ΔTi3yi3-Di3k-≤q2k++ε2k+,k=1,2,…,27by stochastic simulation (the basic principle could be seen in Liu [21]).
Secondly, construct a BP neural network to approximate the functions according to the generated input-output data and then initialize and train the BP neural network.
Thirdly, initialize a certain number of chromosomes according to the distribution function and check the feasibility of these chromosomes.
Fourthly, calculate the values of the objective function as fitness value by the trained BP neural network.
Fifthly, select the chromosomes by running a standard scheme of the roulette wheel.
Sixthly, update the chromosomes by crossover and mutation operations, test the feasibility of chromosome offspring, and obtain a feasible new population.
Seventhly, choose the chromosome which has the maximal fitness while the function values ofUijk′′(yi1,Dijk-) are greater than βjk.
Eighthly, repeat substep 5 to substep 7 for a given number of cycles.
Ninthly, report the best chromosome. The best solutions obtained in the above way are denoted byDijkopt- and yijopt, and the optimal value is denoted by f-opt+.Step 2.
Based on the optimal solutions obtained by Step1, system modeling of model (10) is given. Generate a training set of input-output data for the following uncertain functions:(13)Gijk′Dijk+=max⁡f--∣Pr∑i=13∑j=13NBij-Tij-+ΔTijyijopt-∑i=13∑k=13p1kCi1+Di1k+-∑i=13∑k=19p2kCi2+Di2k+-∑i=13∑k=127p3kCi3+Di3k+≥f--≥α,Gi1k′′Di1k+=Pr∑i=13Ti1-+ΔTi1yi1opt-Di1k+≤q1k-,k=1,2,3,Gi2k′′Di2k+=Pr∑i=13Ti2-+ΔTi2yi2opt-Di2k+≤q2k-+ε1k-,k=1,2,…,9,Gi3k′′Di3k+=Pr∑i=13Ti3-+ΔTi3yi3opt-Di3k+≤q2k-+ε2k-,k=1,2,…,27by stochastic simulation. The remaining substeps are similar to the substeps in Step 1. Finally, report the best chromosome. The optimal solution obtained in the above way is Dijkopt+ and the optimal value is f-opt-.Step 3.
Achieve the optimized water allocation targetTijopt±=Tij-+ΔTijyijopt and the real allocation of water Aijkopt±=Tijopt±-Dijkopt±.Step 4.
Synthesize the two submodels. The optimal solutions can be summarized asDijkopt±=Dijkopt-,Dijkopt+ and the optimal value can be summarized as f-opt±=f-opt-,f-opt+.


## 4. Results and Discussions

The demands and deficits for water are related to water availability, since the economic benefit will be obtained if the demands are satisfied, while the economic penalties will be generated if the deficits occur. Table1 provides water allocation targets in the three planning periods. Tables 2 and 3 present the information regarding seasonal flows under different probabilities. Tables 1–3 derive from [1]. In the case of insufficient water resources, the water for municipal sector should be delivered preferentially since the highest benefit will be brought when the municipal water demand is satisfied, while the highest penalty will be produced if the promised water is not delivered, followed by the industrial and agricultural sectors which correspond to lower benefits and penalties (see Table 4).The solutions shown in Tables5–7 ensure that the 90% optimistic value to the net system benefit could be obtained subject to the chance constraints with a confidence level of 95%. Therefore, the confidence levels for the net system benefit and the chance constraints during the three planning periods are set to α=0.9 and βjk=0.95 for k=1,2,…,Kj, j=1,2,3. In solution process by using hybrid algorithm, the maximum number of iterations, the learning rate, the momentum term, and the tolerance criterion for the BP neural network are set to be 20000, 0.01, 0.9, and 0.00001, respectively. The population size, the number of generations, the mutation rate, and the crossover rate of GA are set to be 30, 300, 0.1, and 0.7, respectively. The solutions indicate that y11opt=0.644, y21opt=0.743, y31opt=0.250, y12opt=0.449, y22opt=0.281, y31opt=0.182, y13opt=0.212, y23opt=0.211 and y33opt=0.375. Thus, the optimized allocated targets are T11opt±=2.644, T21opt±=3.615, T31opt±=4.000, T12opt±=2.949, T22opt±=4.062, T32opt±=4.363, T13opt±=3.212, T23opt±=4.423, and T33opt±=4.938. These targets would be promised to the three users in the first stage. Obviously, the water manager’s decisions should balance the net benefit and the risk because high net benefit would be brought if the promised water is delivered and high penalties would occur if the demand for water is not achieved.Table 5
Solutions for the first planning period (α=0.9, β=0.95).
Scenario symbol (ijk)
User
Water flow level
Probability (%)
Water targets (106m3)
Water shortage (106m3)
Water allocation (106m3)
111
Municipal
L
20
2.644
1.105,1.174
1.470,1.539
211
Industrial
L
20
3.615
3.342,3.353
0.262,0.273
311
Agricultural
L
20
4.000
3.524,3.769
0.231,0.476
112
Municipal
M
60
2.644
2.116,2.348
0.296,0.528
212
Industrial
M
60
3.615
2.469,2.531
1.084,1.146
312
Agricultural
M
60
4.000
0.889,2.132
1.868,3.111
113
Municipal
H
20
2.644
1.200,1.328
1.316,1.444
213
Industrial
H
20
3.615
2.467,3.159
0.456,1.148
313
Agricultural
H
20
4.000
2.239,3.411
0.589,1.761
y
11
o
p
t
=
0.644, y21opt=0.743, and y31opt=0.250.Table 6
Solutions for the second planning period (α=0.9, β=0.95).
Scenario symbol (ijk)
User
Water flow level
Probability (%)
Associated water flow
Associated probability (%)
Water target (106m3)
Water shortage (106m3)
Water allocation (106m3)
121
Municipal
L
20
L-L
4
2.949
[
0.899,1.683
]
[
1.266,2.050
]
221
Industrial
L
20
L-L
4
4.062
[
0.050,0.764
]
[
3.298,4.012
]
321
Agricultural
L
20
L-L
4
4.363
[
3.032,3.452
]
[
0.911,1.331
]
122
Municipal
M
60
L-M
12
2.949
[
0.369,2.649
]
[
0.300,2.580
]
222
Industrial
M
60
L-M
12
4.062
[
1.641,3.174
]
[
0.888,2.421
]
322
Agricultural
M
60
L-M
12
4.363
[
3.968,4.155
]
[
0.208,0.395
]
123
Municipal
H
20
L-H
4
2.949
[
0.710,1.032
]
[
1.917,2.239
]
223
Industrial
H
20
L-H
4
4.062
[
0.408,1.328
]
[
2.734,3.654
]
323
Agricultural
H
20
L-H
4
4.363
[
1.334,3.588
]
[
0.775,3.029
]
124
Municipal
L
20
M-L
12
2.949
[
2.910,2.931
]
[
0.018,0.039
]
224
Industrial
L
20
M-L
12
4.062
[
1.940,3.328
]
[
0.734,2.122
]
324
Agricultural
L
20
M-L
12
4.363
[
3.400,3.948
]
[
0.415,0.963
]
125
Municipal
M
60
M-M
36
2.949
[
2.941,2.947
]
[
0.002,0.008
]
225
Industrial
M
60
M-M
36
4.062
[
1.566,2.654
]
[
1.408,2.496
]
325
Agricultural
M
60
M-M
36
4.363
[
0.159,3.291
]
[
1.072,4.204
]
126
Municipal
H
20
M-H
12
2.949
[
2.381,2.774
]
[
0.175,0.568
]
226
Industrial
H
20
M-H
12
4.062
[
3.111,3.642
]
[
0.420,0.951
]
326
Agricultural
H
20
M-H
12
4.363
[
1.318,3.148
]
[
1.215,3.045
]
127
Municipal
L
20
H-L
4
2.949
[
2.534,2.669
]
[
0.280,0.415
]
227
Industrial
L
20
H-L
4
4.062
[
1.557,1.709
]
[
2.353,2.505
]
327
Agricultural
L
20
H-L
4
4.363
[
3.310,4.033
]
[
0.330,1.053
]
128
Municipal
M
60
H-M
12
2.949
[
2.755,2.821
]
[
0.128,0.194
]
228
Industrial
M
60
H-M
12
4.062
[
3.088,3.617
]
[
0.445,0.974
]
328
Agricultural
M
60
H-M
12
4.363
[
0.374,0.883
]
[
3.480,3.989
]
129
Municipal
H
20
H-H
4
2.949
[
0.557,1.088
]
[
1.861,2.392
]
229
Industrial
H
20
H-H
4
4.062
[
0.678,2.590
]
[
1.472,3.384
]
329
Agricultural
H
20
H-H
4
4.363
[
0.402,3.609
]
[
0.754,3.961
]
y
12
opt
=
0.449, y22opt=0.281, and y31opt=0.182.Table 7
Solutions for the third planning period (α=0.9, β=0.95).
Scenario symbol (ijk)
User
Water flow level
Probability (%)
Associated water flow
Associated probability (%)
Water targets (106m3)
Water shortage (106m3)
Water allocation (106m3)
131
Municipal
L
20
L-L-L
0.8
3.212
[
2.543,2.696
]
[
0.516,0.669
]
231
Industrial
L
20
L-L-L
0.8
4.423
[
1.114,3.435
]
[
0.988,3.309
]
331
Agricultural
L
20
L-L-L
0.8
4.938
[
4.929,4.935
]
[
0.003,0.009
]
132
Municipal
M
60
L-L-M
2.4
3.212
[
0.410,1.275
]
[
1.937,2.802
]
232
Industrial
M
60
L-L-M
2.4
4.423
[
1.376,3.065
]
[
1.358,3.047
]
332
Agricultural
M
60
L-L-M
2.4
4.938
[
3.665,4.086
]
[
0.852,1.273
]
133
Municipal
H
20
L-L-H
0.8
3.212
[
1.450,1.573
]
[
1.639,1.762
]
233
Industrial
H
20
L-L-H
0.8
4.423
[
0.144,4.207
]
[
0.216,4.279
]
333
Agricultural
H
20
L-L-H
0.8
4.938
[
3.899,4.276
]
[
0.662,1.039
]
134
Municipal
L
20
L-M-L
2.4
3.212
[
2.658,2.876
]
[
0.336,0.554
]
234
Industrial
L
20
L-M-L
2.4
4.423
[
2.895,3.297
]
[
1.126,1.528
]
334
Agricultural
L
20
L-M-L
2.4
4.938
[
1.394,4.583
]
[
0.355,3.544
]
135
Municipal
M
60
L-M-M
7.2
3.212
[
3.171,3.199
]
[
0.013,0.041
]
235
Industrial
M
60
L-M-M
7.2
4.423
[
0.876,3.090
]
[
1.333,3.547
]
335
Agricultural
M
60
L-M-M
7.2
4.938
[
0.782,2.813
]
[
2.125,4.156
]
136
Municipal
H
20
L-M-H
2.4
3.212
[
2.023,2.385
]
[
0.827,1.189
]
236
Industrial
H
20
L-M-H
2.4
4.423
[
2.709,3.584
]
[
0.839,1.714
]
336
Agricultural
H
20
L-M-H
2.4
4.938
[
2.682,3.085
]
[
1.853,2.256
]
137
Municipal
L
20
L-H-L
0.8
3.212
[
2.037,3.022
]
[
0.190,1.175
]
237
Industrial
L
20
L-H-L
0.8
4.423
[
2.642,3.314
]
[
1.109,1.781
]
337
Agricultural
L
20
L-H-L
0.8
4.938
[
1.762,2.322
]
[
2.616,3.176
]
138
Municipal
M
60
L-H-M
2.4
3.212
[
0.929,2.366
]
[
0.846,2.283
]
238
Industrial
M
60
L-H-M
2.4
4.423
[
1.283,1.894
]
[
2.529,3.140
]
338
Agricultural
M
60
L-H-M
2.4
4.938
[
3.712,4.480
]
[
0.458,1.226
]
139
Municipal
H
20
L-H-H
0.8
3.212
[
2.564,3.084
]
[
0.128,0.648
]
239
Industrial
H
20
L-H-H
0.8
4.423
[
4.098,4.263
]
[
0.160,0.325
]
339
Agricultural
H
20
L-H-H
0.8
4.938
[
3.183,4.010
]
[
0.928,1.755
]
1310
Municipal
L
20
M-L-L
2.4
3.212
[
1.780,2.296
]
[
0.916,1.432
]
2310
Industrial
L
20
M-L-L
2.4
4.423
[
1.078,3.080
]
[
1.343,3.345
]
3310
Agricultural
L
20
M-L-L
2.4
4.938
[
4.624,4.856
]
[
0.082,0.314
]
1311
Municipal
M
60
M-L-M
7.2
3.212
[
1.169,3.032
]
[
0.180,2.043
]
2311
Industrial
M
60
M-L-M
7.2
4.423
[
2.145,3.887
]
[
0.536,2.278
]
3311
Agricultural
M
60
M-L-M
7.2
4.938
[
3.619,4.577
]
[
0.361,1.319
]
1312
Municipal
H
20
M-L-H
2.4
3.212
[
1.509,2.804
]
[
0.408,1.703
]
2312
Industrial
H
20
M-L-H
2.4
4.423
[
0.905,2.582
]
[
1.841,3.518
]
3312
Agricultural
H
20
M-L-H
2.4
4.938
[
2.117,2.380
]
[
2.558,2.821
]
1313
Municipal
L
20
M-M-L
7.2
3.212
[
0.628,2.264
]
[
0.948,2.584
]
2313
Industrial
L
20
M-M-L
7.2
4.423
[
2.505,2.865
]
[
1.558,1.918
]
3313
Agricultural
L
20
M-M-L
7.2
4.938
[
4.341,4.492
]
[
0.446,0.597
]
1314
Municipal
M
60
M-M-M
21.6
3.212
[
1.192,3.049
]
[
0.163,2.020
]
2314
Industrial
M
60
M-M-M
21.6
4.423
[
1.062,2.988
]
[
1.435,3.361
]
3314
Agricultural
M
60
M-M-M
21.6
4.938
[
1.389,3.794
]
[
1.144,3.549
]
1315
Municipal
H
20
M-M-H
7.2
3.212
[
1.946,2.103
]
[
1.109,1.266
]
2315
Industrial
H
20
M-M-H
7.2
4.423
[
1.948,3.230
]
[
1.193,2.475
]
3315
Agricultural
H
20
M-M-H
7.2
4.938
[
2.309,2.817
]
[
2.121,2.629
]
1316
Municipal
L
20
M-H-L
2.4
3.212
[
1.733,2.390
]
[
0.822,1.479
]
2316
Industrial
L
20
M-H-L
2.4
4.423
[
1.409,4.280
]
[
0.143,3.014
]
3316
Agricultural
L
20
M-H-L
2.4
4.938
[
4.207,4.583
]
[
0.355,0.731
]
1317
Municipal
M
60
M-H-M
7.2
3.212
[
2.431,2.657
]
[
0.555,0.781
]
2317
Industrial
M
60
M-H-M
7.2
4.423
[
1.526,2.959
]
[
1.464,2.897
]
3317
Agricultural
M
60
M-H-M
7.2
4.938
[
1.660,3.432
]
[
1.506,3.278
]
1318
Municipal
H
20
M-H-H
2.4
3.212
[
0.980,1.552
]
[
1.660,2.232
]
2318
Industrial
H
20
M-H-H
2.4
4.423
[
2.041,3.262
]
[
1.161,2.382
]
3318
Agricultural
H
20
M-H-H
2.4
4.938
[
2.407,3.482
]
[
1.456,2.531
]
1319
Municipal
L
20
H-L-L
0.8
3.212
[
0.145,1.877
]
[
1.335,3.067
]
2319
Industrial
L
20
H-L-L
0.8
4.423
[
1.486,2.048
]
[
2.375,2.937
]
3319
Agricultural
L
20
H-L-L
0.8
4.938
[
3.269,4.521
]
[
0.417,1.669
]
1320
Municipal
M
60
H-L-M
2.4
3.212
[
1.494,2.802
]
[
0.410,1.718
]
2320
Industrial
M
60
H-L-M
2.4
4.423
[
0.358,0.565
]
[
3.858,4.065
]
3320
Agricultural
M
60
H-L-M
2.4
4.938
[
3.166,4.801
]
[
0.137,1.772
]
1321
Municipal
H
20
H-L-H
0.8
3.212
[
1.058,2.266
]
[
0.946,2.154
]
2321
Industrial
H
20
H-L-H
0.8
4.423
[
0.087,4.299
]
[
0.124,4.336
]
3321
Agricultural
H
20
H-L-H
0.8
4.938
[
2.032,4.546
]
[
0.392,2.906
]
1322
Municipal
L
20
H-M-L
2.4
3.212
[
2.553,2.817
]
[
0.395,0.659
]
2322
Industrial
L
20
H-M-L
2.4
4.423
[
4.047,4.256
]
[
0.167,0.376
]
3322
Agricultural
L
20
H-M-L
2.4
4.938
[
0.542,4.078
]
[
0.860,4.396
]
1323
Municipal
M
60
H-M-M
7.2
3.212
[
0.811,2.625
]
[
0.587,2.401
]
2323
Industrial
M
60
H-M-M
7.2
4.423
[
2.370,2.910
]
[
1.513,2.053
]
3323
Agricultural
M
60
H-M-M
7.2
4.938
[
2.874,4.022
]
[
0.916,2.064
]
1324
Municipal
H
20
H-M-H
2.4
3.212
[
3.097,3.119
]
[
0.093,0.115
]
2324
Industrial
H
20
H-M-H
2.4
4.423
[
2.321,3.201
]
[
1.222,2.102
]
3324
Agricultural
H
20
H-M-H
2.4
4.938
[
3.064,3.905
]
[
1.033,1.874
]
1325
Municipal
L
20
H-H-L
0.8
3.212
[
1.332,2.673
]
[
0.539,1.880
]
2325
Industrial
L
20
H-H-L
0.8
4.423
[
0.597,4.261
]
[
0.162,3.826
]
3325
Agricultural
L
20
H-H-L
0.8
4.938
[
1.108,4.456
]
[
0.482,3.830
]
1326
Municipal
M
60
H-H-M
2.4
3.212
[
2.482,2.676
]
[
0.536,0.730
]
2326
Industrial
M
60
H-H-M
2.4
4.423
[
0.553,1.948
]
[
2.475,3.870
]
3326
Agricultural
M
60
H-H-M
2.4
4.938
[
1.731,2.663
]
[
2.275,3.207
]
1327
Municipal
H
20
H-H-H
0.8
3.212
[
0.865,0.866
]
[
2.346,2.347
]
2327
Industrial
H
20
H-H-H
0.8
4.423
[
0.790,4.205
]
[
0.218,3.633
]
3327
Agricultural
H
20
H-H-H
0.8
4.938
[
3.320,3.577
]
[
1.361,1.618
]
y
13
o
p
t
=
0.212, y23opt=0.211, and y33opt=0.375; α-optimistic value to the net system benefit: f-opt±=154.335,235.704.Table5 indicates the optimized solutions under 3 scenarios for the first planning period. For example, D111opt±=1.105,1.174, D112opt±=2.116,2.348, and D113opt±=1.200,1.328 are water shortages for the municipal sector j=1 when the water flow level is low, medium, and high with probability of 20%, 60%, and 20%, respectively. Accordingly, the water allocations are A111opt±=1.470,1.539, A112opt±=0.296,0.528, and A113opt±=1.316,1.444.Table6 indicates the optimized solutions under 9 scenarios for the second planning period. Take water shortages and water allocations of the industrial sector: for example, j=2. D224opt±=1.940,3.328, D225opt±=1.566,2.654, and D226opt±=3.111,3.642 denote water shortages for the industrial sector when the water flow level in the second period is low, medium, and high following the medium flow in the previous period with joint probability of 12%, 36%, and 12%, respectively. Accordingly, the water allocations are A224opt±=0.734,2.122, A225opt±=1.408,2.496, and A226opt±=0.420,0.951.Table7 provides optimized water allocation schemes under 27 scenarios for the third planning period. For example, D1322opt±=2.553,2.817, D2322opt±=4.047,4.256, and D3322opt±=0.542,4.078 indicate the water shortages for industrial, agricultural, and municipal sectors with the corresponding water allocations A1333opt±=0.395,0.659, A2322opt±=0.167,0.376, and A3322opt±=0.860,4.396 respectively, when the water flows are high-medium-low during the entire planning horizon with joint probability of 2.4%.The optimal value (f-opt±=154.335,235.704) represents the 90% optimistic value to the net system benefit subject to the chance constraints with a confidence level of 95%, which provides two extreme values over the planning horizon. Planning for a lower system benefit would be associated with a lower risk of violating the water allocation constraints; conversely, the desire for a higher benefit would correspond to a higher possibility of violating the constraints [9]. When the actual value of each variable fluctuates between its lower and upper bounds, the 90% optimistic values to the net system benefit would change correspondingly between 154.335 and 235.704, which reflects the balance between the system profit and the chance constraints.In the hypothetical case, the water resource is an unregulated reservoir and the uncertainties in the water resources management are expressed as interval with uniform distributed and normally distributed boundaries. However, real-world water resources systems are more complex than the hypothetical case presented, since the water allocated to the users is from regulated reservoir(s), and the representation of the uncertainties in the input may be more diversified. Nevertheless, the hypothetical case still reflects the basic scene and principles and contains main information in the real water resources management systems. Simple as it is, the case is sufficient to study the characteristics and optimization problems of real-world water resources systems. For the real water resources management problems which contain data with multiply distributed (e.g., Gamma, Lognormal) stochastic boundaries, they can also be dealt with through establishing the appropriate IMSCCP model and adding corresponding variables or data in the model. Then the model could be solved by the hybrid algorithm.


## 5. Conclusions

An inexact multistage stochastic chance constrained programming (IMSCCP) model is provided for water resources management, which integrates stochastic CCP proposed by Liu [21], multistage stochastic programming, and inexact stochastic programming. Compared with the existing IMSP with chance constraints, the IMSCCP model proposed in this study contains stochastic variables in the objective function or inexact data with multiply distributed stochastic boundaries. Then the IMSCCP model could be solved by using the hybrid algorithm. After solving the IMSCCP model, the maximum, which is the optimistic value to the net system benefit with a predetermined confidence level, could be obtained subject to the chance constraints with other confidence levels.Considering more real-world situations in the water resources management systems, such as water distribution from regulated reservoir(s), the expansion and development of the reservoir(s), and more uncertainties existing in many system components, further studies can resort to nonlinear programming combined with other uncertain variables such as fuzzy variable andgλ variable to solve the water resources management problems and handle more uncertainties. Moreover, research efforts might also be devoted to the wide application of IMSCCP model in the areas of ecological water requirement system management, waste management planning, electric-power system planning, and so on.



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*Source: 1680813-2017-06-20.xml*