On two-stage fuzzy random programming for water resources management

In this paper, a two-stage fuzzy random programming for a management problem in terms of water resources allocation having fuzzy random variable coefficients and decision vector of random variables is studied. The first results show the fact that a fuzzy pseudorandom optimal solution of a two-stage fuzzy random programming may be resolved into a two of pseudorandom optimal solutions of relative two-stage random programmings. The subsequent results present a method that a fuzzy pseudorandom optimal solution of two-stage fuzzy random programming is structured by a two of pseudorandom optimal solutions of a relative two two-stage random programmings. A numerical example was given to clarify the obtained results.


INTRODUCTION
Stochastic programming deals with situations where some or all the parameters of a mathematical programming problem are described by stochastic variables rather than by deterministic quantities.Several models have been presented in the field of stochastic programming (Stancu-Minasian et al., 1976).Contini (1978) developed an algorithm for stochastic goal programmings when the random variables are normally distributed with known means and variances.He transformed the stochastic problem into an equivalent deterministic quadratic programming problem, where the objective functions consisted of maximizing the probability of a vector of goals lying in the confidence region of a predefined size.Teghem et al. (1986) and Leclercq (1982) have presented interactive methods in stochastic programming.Two major approach to stochastic programming as recognized by Goicoechea et al. (1982) and Kambo (1984) are: 1. Chance constrained programming, 2. Two-stage programming.
In fuzzy decision making problems, the concept of a maximizing decision was proposed by Bellman and Zadeh (1990).Fuzzy linear programming problem with coefficients was formulated by Negoita (1970) and called robust programming.Dubois and Prade (1982) investigated linear fuzzy constraints.Tanaka and Asai (1984) also proposed a formulation of fuzzy linear programming with fuzzy coefficients.Wang and Wang (1985) built another theory of fuzzy programming on the basis of "Satisfaction degree" which has been used in engineering design.Wang and Qiao (1993) put forward some model linear programming with fuzzy random variable coefficients, and studied the solution and distribution problems of a fuzzy random programming.Over the past decades, the conflict-laden issues of water resources allocation among competing municipal, industrial and agricultural interests have been of increasing concerns (Huang and Chang, 2003;Wang et al., 2003).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 (Li and Huang, 2008).
The increased water demands and the inadequate water supplies have exacerbated the shortage of water resources; this has been considered as a major obstacle to sustainable water resources management.When the essential demand cannot be satisfied due to in sufficient resources, losses can hardly be avoided resulting in a variety of a diverse impact on Socio-economic development (Lu et al., 2010).Wang and Adams (1986) proposed a two-stage optimization framework of planning reservoir operations, inflows were modeled as a periodic Markov processes.Eiger and Shamir (1991) developed a model for an optimal multi-period operation within a multi-reservoir system.Ferreto et al. (1998) examined a long-term by drothermal scheduling of multi-reservoir systems using a two-stage dynamic programming approach.
In many real-world problems, however, the quality and quantity of uncertain information is often not satisfactory enough to be presented as probabilities distribution.Even if such distributions for uncertain parameters are available, reflecting them in large-scale optimization models can be extremely challenging (Huang and Loucks, 2000).Wang and Huang (2011) have also developed an interactive two-stage stochastic fuzzy programming approach through incorporating an interactive fuzzy resolution method within an inexact two-stage stochastic programming framework.This paper will further be devoted to a model proposed in Wang and Huang (2011).Under a certain a condition we discuss the solution method of a fuzzy two-stage stochastic programming, which has fuzzy random variable coefficients and the decision vector of random variables.Ammar and Khalifa (2014) introduced a fuzzy programming approach for treating an interactive twostage stochastic rough interval water resource management.

PRELIMINARIES
In order to discuss our problem conveniently, we shall state some necessary results on interval arithmetic, fuzzy numbers and fuzzy random variables.Moore (1979) introduced the concept of closed interval numbers.
denote the set of all closed interval numbers on R.
, where T is an index set, we define In particular, for [ , ] ( ), denote the set of all bounded closed fuzzy numbers (that is, compact on R, for any 2. For any .
The order relation " "  on 0 ( ) F R is defined by f g  if and only if and only if f g (a) , where T is an index set.
Definition 5: Let ( , , ) M N P be a probability measure space.
A mapping MN (Wang and Zhang, 1992).Denote the set of all fuzzy random variables on ( , )  MN by Moreover, let  be an algebraic operation on 0 ( ) F R .
The algebraic operation  on ( ) F R M may be defined by for any w M  , and ab  if and only if any wM  .

STATEMENT OF PROBLEM
A typical two-stochastic programming model for water resources management is (Wang and Huang (2011) as follows: max   Based on the findings of Huang and Loucks (2000), the problem (1) can be reformulated as follows: 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 .
In this paper, we study the problem (2) with fuzzy random variable coefficients and decision vector of random variables as: Model 1: (two-stage fuzzy random programming) Find ( , ) ST to max The solution method of Model 1 will be discussed under the condition , and .Model 1 may be rewritten as follows:

Definition 6:
The ** ( , ) ST which satisfy the condition in Model 2, is called a fuzzy random optimal solution of Model 2.
Observe that Where, , for any Corresponding to Model 2, we structure the following programming problems: For any given ( 0, 1]   Models 2.1-2.2 are linear programming problems with random variable coefficients; we may use the simple method discussed in Wang and Qiao (1993) to find random optimal solutions of the two models.

S T G
 , then ( , )   , ( , ) for any It follows that from Definitions 2 and 5: , and That is to say, ( , ) , ( , ) is a random optimal solution of Model 2, then for any , we have: (1) is a random optimal solution of Model 2.2, (2) is a random optimal solution of Model 2.1, (3) . It follows by Theorem 1, and Definition 1, and observing the conditions  , this leads to (3) of the Theorem 2 is true.

Numerical example
Consider the Model 1 with the following economic data illustrated in

Conclusions
In this paper, a two-stage fuzzy random programming for water resources management having fuzzy random variable coefficients and decision vector of random variables has been presented.The results have shown that the fuzzy random optimal solution of the problem under consideration has relative two-stage random programmings.
Water-allocation target constraints) negativity and technical constraints)Where f = system benefit ($); by using Lemma 1, we can see that