Interactive Two-Stage Stochastic fuzzy Rough Programming for Water Resources Management

This paper deals with a fuzzy programming approach for treating an interactive two-stage stochastic rough-interval water resource management. The approach has been developed by incorporating an interactive fuzzy resolution method within a rough two-stage stochastic programming framework. The approach can not only tackle dual rough intervals presented as an inexact boundary intervals that exist in the objective function and the leftand right-hand sides of the constraints that are associated with different levels of economic penalties when the promised policy targets are violated. The results indicate that a set of solutions under different feasibility degrees has been generated for planning the water resources allocation. They can help the decision makers to conduct in depth analysis of tradeoffs between economic efficiency and constraint-violation risk, as well as enable them to identify, in an interactive way, a desired compromise between satisfaction degree of the goal and feasibility of the constraints. A management example in terms of rough-intervals water resources allocation has been treated for the sake of applicability of the proposed approach.


Introduction
Recently, the conflict-laden issues of water resources allocation among competing municipal, industrial and agricultural interests have been of increasing concerns.The competition among water users has been intensified due to growing population shifts, shrinking water availabilities, varying natural conditions and deteriorating quality of water resources.The increased water demands and the inadequate water supplies have exacerbated the storage of water resources; this has been considered as a major obstacle to sustainable water resources management.The problem arises when the essential demand cannot be satisfied due to insufficient resources, losses can hardly avoided resulting in variety of adverse impacts on socio-economic development.Therefore, a sound planning water resources allocation is desired to help reduce such losses.However, the water resources management system are complicated with a variety of uncertainties and their interactions when may intensify the conflict-laden issues of water allocation.
To study the above concern problems, a number of optimization techniques ial distributions.In TSP, an initial decision must be made before the realization of random variables (first-stage decision), and then a corrective action can be taken place (second-stage decision).This implies that a second-stage decision can be used to minimize "penalties" that may appear due to any infeasibility.
In general, TS deals with uncertainties expressed as probabilistic distributions and account for economic penalties with recourse against any infeasibility.In many real-world problems' however, the quality and quantity of uncertain/information are often not satisfactory enough to be presented as probabilistic distributions.Even if such distributions for uncertain parameters are available, reflecting them in large-scale optimization models can be extremely challenging.
In comparison, rough-interval mathematical programming (RIMP) is effective in talking of unknown quantities expressed as intervals with known lower and upper bounds.RIMP proves to be an effective approach to deal with the uncertainties in the parameters of the considered problems.Rough-interval two-stage stochastic programming (RITSP) approach for water resources management could not only tockle uncertainties expressed as probabilistic distribution and rough intervals, but also analyze a variety of policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated.
An interactive fuzzy resolution (IFR) method is proposed for solving a two-stage stochastic rough-intervals water resources allocation management problems with fuzzy parameters.IFR permits interactive participations of DMS in all steps of decision process expressing their preference in linguistic terms.A set of solutions could obtained from IFR under different feasibility degrees of the constraints which are meaningful to support in-depth analysis of tradeoffs between economic efficiency and constraint violation risk.IFR enables DMS to identify a compromised solution between two factors in conflict: feasibility of the constraint and satisfaction degree of the goal.
In earlier work in the conflict-laden issues of water resources allocation among competition interests have been of in creasing concerns many researches as Huang  Recently, Jimenez et a;/.(2007) proposed an interactive fuzzy resolution (IFR) method for solving linear programming problems with fuzzy parameters.Also, Wang and Huang (2011) have been proposed an interactive twostage stochastic fuzzy programming (ITSFP) approach for water resource management problem.They develop its approach by incorporating IFR method within an inexact two-stage stochastic programming framework.ITSFP cannot only tackle dual uncertainties presented as a fuzzy boundary intervals that exist in the objective function and the left-and righthand of the constraint, but also permit in-depth analysis of various policy scenarios that are associated with different levels of economic penalties when the promised policy target are violated.This paper deal with a fuzzy programming approach for treating an interactive two-stage stochastic fuzzy (ITSFP) for rough-interval water resources management.The approach has been developed by incorporating an interactive fuzzy resolution method within a rough-interval two-stage stochastic programming (RITSP) framework.ITSFP can not only tackle dual rough-intervals presented as a fuzzy boundary intervals that exist in the objective function and the left-and right-hand/side of the constraints, but also permit in-depth analysis of various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated.A management problem in terms of rough-intervals water resources allocation has been studied for the sake of applicability of the proposed approach.

Preliminaries
In this section, we recall some definitions needed through the paper.

I R 
. We define: 6 2 5 | P a g e J u n e 0 1 , 2 0 1 4   A  can be defined as follows: Remark 1.A triangular fuzzy number (T.F. N) is a special case of a trapezoidal fuzzy number (Tr.F. N.) with 23 aa  .

Problem definition
Consider a problem in which a water manager is response for allocating water to multiple users.The objective of the water resources allocation problem is to maximize the system benefit.Based on local water management polices, a prescribed quantity of water is promised to each user.If the promised water is delivered, it will result in net benefits to the local economy, otherwise, users will have to either obtain water from more expensive sources or curtail their development plans resulting in economic penalties.In such problem, the water flow levels are uncertain (expressed as random variables), while a decision of water allocation target (first-stage decision) must be mate before the realization of random variables, and then a recourse action can be taken after disclosure of a random variables (second-stage decision).Therefore, this 6 2 7 | P a g e J u n e 0 1 , 2 0 1 4 problem under consideration can be formulated as a two-stage stochastic programming (TSP) model as follows (Wang and Huang (2011)): where f is the system benefit ($); i NB is the net benefit to user (i) per 3 m of water allocation 3 ($ / ) m (first-stage revenue parameters); i T is the allocation target for water that is promised to user () E  is the expected value of a random variable; i C is the loss to user () i per 3 m of water not delivered, is the storage of water to user () i when the seasonal flow is 3 () Qm (second-stage decision variables); Q is the total amount of seasonal flows 3 () m (random variable);  is the rate of water loss during transportation; max i T s the maximum allowable allocation amount for user 3 ( )( ) im; m is the total number of water user, I is the water user, e.g., 1 i  for the municipality, 2 i  for the industrial user, 3 i  for the agricultural sector.
The distribution of Q must be approximated by a set of discrete values.Letting Q take values j q with probability ( 1, ..., ) where ij s denotes the amount by which the water-allocation target () i T is not met when the seasonal flow is j q with probability j p .Thus, model (1) van be reformulated as follows: For a difficults may be face the planner to promise a deterministic water-allocation target () i T to a water user, the available flows are uncertain.Thus, we need to define uncertain parameters (rough and interval) of model ( 3).The notion of rough set (Nanda and Majumdar (1992)) given through the following definition: Definition.Let U be a nonempty set and  b a complete sub algebra of the Boolean algebra () PU of subsets of U. the pair ( , ) U  is called a rough universe.Let R be a relation defined as follows: ( , ) The elements of R are called rough sets and the elements of  are called exact sets.
Definition 10.The degree of rough of the set ( , where || || A denotes the length of the interval. In response to the above concerns, rough interval and interval parameters are introduced into model (3) to communicale uncertainties.This leads to an interactive rough interval two-stage stochastic programming (IRITSP)model as follows:

Interactive fuzzy resolution approach
In many real world problems, when the subjective judgment of decision maker (DMS) is influential in the decision making process, the fuzzy set theory is recognized as an effective tool to express their opinions.For example, in RITSP can hardly reflect the ambiguous or vague information from the subjective estimations.Say, DMS estimate that the most possible value for the net benefit to a water user is $90 per 3 m of water allocated, and there is no possibility for it to be lower than $80 or more than $100 per 3 m of water; such a subjective estimation can be expressed as a fuzzy set.
Therefore, an interactive fuzzy resolution (IFR) method is introduced to tackle uncertainties presented as fuzzy sets.
Firstly, consider the following enteractive fuzzy rough interval two-stage stochastic programming (IFRITSP) model: Here, we introduce some concepts and definitions of fuzzy set theory.A fuzzy set () A  in X is characterized by a membership function ( ( )) , where X re[resents a space of points, with an element of X denoted by x (Zadeh According to formulae ( 5) and ( 6), if the fuzzy set () A  is trapezoidal or triangular, its expected interval and expected value can be calculated as follows: The uncertain and / or imprecise nature of many parameters leads to concerns of system feasibility and optimality.It is therefore necessary to answer the following two questions: (1) How to define the feasibility of a decision vector, when the constraints involve fuzzy sets?J u n e 0 1 , 2 0 1 4 For any pair of fuzzy sets ( A  and ) B  , the degree in which A  is smaller than B  can be defined as follows: where  is the degree of the feasibility of a decision vector, and 1   provides a measure of the risk of infeasibility.
According to formula (12), this is equivalent to: The set of decision vectors that are  -feasible denoted by () k  , it is evident that: To address the second question, vectors where 00 ( , ) TS is a better choice (with maximized objective) at least in degree 1 2 . According to formula (12), it is equivalent to: Based in formulae ( 14) and ( 17), vectors is an  -acceptable solution of the initial FRISP model ( 6) if it an optimal solution to the following  -parametric RITSP model: ,   21) can be reformulated as: subject to The second model correspond to the lower bound of the objective-function interval is: subject to , The second model correspond to the lower bound of the objective-function interval is: subject to y are decision a variables.Submodels ( 23) -( 26) are deterministic linear programming problems.The solutions of model 919) under the optimized water-allocation targets are given by the following ( 27) and ( 28):   26).Thus, the optimized rough-water allocation scheme is: The degree of rough of the optimized rough-water allocation, denoted by  , is:

The Algorithm
An interactive algorithm is proposed to solve the  -parametric RITSP problems by using interactive fuzzy resolution (IFR).The IFR (Wang and Huang (2011)) method allows DMS to consider two factors in an interactive way, feasibility of the constraints and satisfaction degree of the goals.A strong desire to obtain a better objective-function value will lead to a lower degree of constraint feasibility; contrarly, a willingness to accept a lower satisfaction degree of the goal will guarantee a higher feasibility of the constraints.Therefore, the DMS have identify a balanced solution between the two objectives in conflict; to improve the objective-function value or to improve the constraint feasibility.
The natural language for establishing a semantic correspondence for different degrees of feasibility is the best Obviously, other scales can be considered depending on the wishes of the DMS.The 11 scales correspond to different risk level of violating the modeling constraints.The higher the scale value, the lower the risk level.In real world problems, the DMS may not willing to admit a high risk of constraints violation.If 0  is the minimum constraint feasibility that the DMS are willing to accept, the feasibility interval of  will be reduce to 0 1   .According to the semantic scale, the discrete values of  can be achieved as follows: In the first step of the IFR method, the space  can be obtained through solving (13)   under each k  .In order to obtain a decision vector that complies with the expectation of the DMS, two conflicting factors should be evaluated: feasibility of the constraints and acceptability of the objective-function value.After obtaining the results under different opt ()  , the DMS are asked to specify a goal () G and its tolerance threshold () G .The foal is expressed by means of a fuzzy goal set () G  whose membership function is: In the second step of the IFR method, the satisfaction degree of the fuzzy goal () G  for each  -acceptable solution, i.e., the membership grade of opt () k f  to () G  can be analyzed through an index: where  denotes the classical integral; S is the support of the fuzzy set opt In the third step of the IFR method, we attempt to identify the balance between feasibility of the constraints and the satisfication degree of the foal in the space of The solution with the highest membership grade will be the final decision for the IFRITSP problem (i.e., model ( 6)) ** opt opt opt ( , ) max {min ( , ( ( )) The detailed solution process can be summarized as follows: Step 1: Formulate the IFRITSP model (6).
Step 2: Reformulate the IFRIRSP model by introducing , where Step 4: Obtain the k  -acceptable solutions through solving the L f   submodel under each k  .
Step 5: Compute the satisfication degree of the fuzzy foal for each  -acceptable solution.
Step 6: Identify a balance (i.e., membership grade of the fuzzy decision) between feasibility of the constraints and satification degree of the foal for each  -acceptable solution.
Step 7: Formulate and solve the L f   submodel by the following the same interactive procedure as that in Step 8: Reach a desired compromise between feasibility of the constraints and the satisfication degree of the goal by considering the solutions from the two submodels.
Step 9: Obtain the k  -feasibility optimal solution: .
Step 10: Obtain the optimal water-allocation scheme: .
Step 11: Repeat steps ( 4)- (10) for obtaining the optimal water allocation scheme: To the two submodels corresponding to U f   .
Step 12: Find the rough degrees of the optimal water allocation scheme: Step 13: Stop.J u n e 0 1 , 2 0 1 4

Case study
The following water resources management problem will be used demonstrate applicability of the developed IFRITSP approach.A water manager is responsible for allocating water from unregulated sector to three users: a municipality, an industry concern and an agricultural sector.These water users need to know how much water they can expect so as to make appropriate decisions on their various activities and investments.If the promised water is delivered a net to the local economy will be generated for each unit water allocated; either the water must be obtained from higher-priced alternatives of the demand must cartailed by reduced industrial and / or agricultural productions, resulting in a reduced system benefit (Wang and Huang (2011)).Table 1 shows the maximum allowable water allocation, the waterallocation target, the fuzzy-rough net benefit to user i per 3 m of water allocated and the loss to user i per 3 m of water not delivered.Moreover, Table 1 presents the fuzzy boundary intervals for the rate of water loss during transportation and the seasonal flows under different probability levels.

Definition 1 .
  denote the set of all closed interval numbers on R. Suppose that[ ,  ]

Definition 3 .
Zadeh (1965)): A fuzzy set A  in XR  is characterized by a membership function () of points, with an element of X denoted by x; A convex fuzzy set A  is a set defined on the real line R with a continuous membership function A   that ca be defined as follows:

Definition 4 .
a continuous and monotonically increasing function on the left-hand side of A  and a continuous and monotonically decreasing function on the right-hand side of A  , respectively.The  -level set of a fuzzy set

sinceA
 is upper semi-continuous, the  -level set of A  defined in (2) forms a closed bounded interval (Wang and Huang (2011)):

Definition 5 .
(Wang and Huang (2011)): The expected interval of a fuzzy set A  , denoted by EI () A  , can be defined as follows: and Huang (2011)): the expected value of a fuzzy set A  , denoted by () EA   , is the half of its expected interval:

Definition 7 .
Let R be the set of real numbers, a fuzzy number a is a mapping: upper semi continuous membership function, 2-a is convex fuzzy set, i.e.,12 ( negativity and technical constraints),

(
1965)); represents the membership grade of x in A  taking value x.Zadeh (1978) define the possibility associated with A  as A   .

( 2 )
How to define the optimality for an objective function with fuzzy coefficients?Several focuses have been developed to answer these questions(Lai and Huang (1994); Rommelfanger and Slovinski (1998); Wang and Huang (2011)).A varity of methods for comparing or ranking fuzzy sets have been reported(Wang and Kerre (1996); Timenez et al., (2007), Wang and Huang (2011)) constructed a fuzzy relationship analysis to compare fuzzy sets: such a method was computationally efficient.
solution if it can be verified that (Wang and Huang (20110): are rough intervals of boundary parameters;  is the feasibility degree of a decision vectors and (1 )   provides a measure about the risk of infeasibility for a decision vectors.According to Wang and Huang (2011), if i T  are consider as uncertain inputs, let the interval variable that are used for identifying an optimized set of target values () i T  in order to support the related policy analysis.Thus, by introducing decision variables () i g model (

T
 are known model (22) can be transformed into four deterministic such model, which are correspond to the lower and upper of the rough objective-function value and correspond to the lower and upper bounds of the objective function interval value.This transformation process is based on an interactive algorithm.For the lower of the rough objective function value, consider the following two submodels.The first model is:

y
are decision a variables.Also, for the upper of the rough objective-function value, consider the following two submodels, the first model correspond to the upper bound of the objective-function interval are those of sub-model (21).Also,


are those of sub-model ( way to reflect the DMS preferences.Therefore, Wang and Huang (2011) and others establish 11 scales, which allowed a sufficient distinction between different levels without being excessive.The 11 scales are as follows: 0.0-Unacceptable solution, 0.1-Practically unacceptable solution, 0.2-Almost unacceptable solution, 0.3-Very unacceptable solution, 0.4-Quite unacceptable solution, 0.5-Neither acceptable solution, 0.6-Quite acceptable solution, 0.7-Very acceptable solution, 0.8-Almost acceptable solution, 0.9 Practically acceptable solution, 1.0-Completely acceptable solution.


-acceptable solutions and the corresponding possibilistic distributions of the objective-function values opt () k f

(
) f  .This an extension of the widely accepted center of gravity defuzzification method, using the goal function () G   as a weighting factor (Jimenez et al., (2007)).

k
-acceptable solutions () O .Now, we consider to build two fuzzy sets: F and S with the following membership functions obtaining a final recommendation, a fuzzy decision () D  is defined by aggregating the two a forenamed fuzzy sets (i.e., D F S     ) (Bellman and Zaeh (1970); Jimênez et al., (2007)): J u n e 0

Step 3 :
Transform the IFRITSP model into four submodels ((20)-(23)), where the first two submodels corresponding to L f   and the second two submodels corresponding to U f   .

Table 2 .
The  -acceptable solutions from the IRISTP model (in 63 10 ) m LU SS   0.