On multi-objective linear programming problems with inexact rough interval-fuzzy coefficients

This paper deals with a multi-objective linear programming problem with an inexact rough interval fuzzy coefficients IRFMOLP. This problem is considered by incorporating an inexact rough interval fuzzy number in both the objective function and constrains. The concept of "Rough interval" is introduced in the modeling framework to represent dual-uncertain parameters. A suggested solution procedure is given to obtain rough interval solution for IRFLP(w) problem. Finally,two numerical example is given to clarify the obtained results in this paper.


Introduction
The theory of rough sets proposed by Pawlak (1982) [16], can be regarded as an effective mathematical vehicle for dealing with imprecise and ambiguous data analyses, which can be subsequently applied to pattern recognition, machine learning, and knowledge discovery [[1]- [7]]. The equivalence relation is a key notion in Pawlak's rough set model. All equivalence classes form a partition of a universe of discourse. Using equivalence classes, an arbitrary subset can be approximated by two subsets called the lower approximation and the upper approximation. However, the equivalence relation is a stringent condition that may limit the applications of rough sets in practical problems. Hence, various extensions of Pawlak's rough set were developed from an equivalence relation into a more general mathematical concept, e.g., binary relations by Slowinski (2000) [18]. The theory of rough set deals with approximation of an arbitrary subset of universe by two definable or observable subsets called lower and upper approximation. Dubois and Prade (1990) [5] pointedly that the rough fuzzy set is a special case of the fuzzy rough set. The notion of interval-valued fuzzy set was suggested for the first time by Turksan (1986) [19] and Gorse (1986) [9]. They are applied to the fields of the approximate inference gave Miyamoto (2004) [14] fuzzy sets and  -level sets .Shaocheng (1994) [17] introduced two kinds of linear programming with fuzzy numbers. They are called interval number and fuzzy number linear programming. Guijun and Xiapong (1998) [10] define interval-valued fuzzy number and interval distribution numbers and gave their extended operations. Zhong et al. (1994) [23] study fuzzy random linear programming having fuzzy random variable coefficients and the decision vector of fuzzy random variable. It is generally accepted that these two theories are related, but a listing and complementary, to each other ( Miyamoto (2004) [14], XU (2012) [20] and Yao (1996) [21]). establishes a rough multiple objective programming model for a solid transportation problem. Gong et al., (2008) [8]and Li et al., (2007) [12], introduce an interval-valued fuzzy information system by means of iterating the classical Pawlak's rough set theory with the intervalvalued fuzzy set theory and oles cusses the basic rough set theory for the interval-valued fuzzy information system. There are at least two approaches for the development of the fuzzy rough sets theory: The constructive and axiomatic approaches (Yao (1998) [22]). In the constructive approach, the relation to the universe is the primitive notion. The lower and upper approximation operator are contracted by means of this notion. On the other hand, the axiomatic approach takes the lower and upper approximation operator as primitive notion. In this approaches, a set of axioms is used to characterize approximation operators. Zhang (2012) [24] presents a general framework for the study of interval type-2 rough fuzzy sets by using both constructive and axiomatic approaches. In this paper, multi-objective linear programming problem with an inexact rough interval fuzzy coefficients IRFMOL is introduced.The problem is transformed into the corresponding IRFLP problem using the weighting method, A solution procedure is given to obtain the rough interval solution for the IRFLP(w) problem.
The paper is organized as in the following sections: In section 2, some preliminaries are introduced. In section 3, problem formulation is introduced as specific definition and properties. In section 4, a solution procedure is given to obtain a rough interval solution for the IRFMOLP problem in section 3. In section 5, numerical example is given to clarify the obtained results. Finally, some concluding remarks are reported in section 6.

Preliminaries
In this section, we introduce some notions related to the IRFMOLP problem. The following concepts can be found in [11], [15] and [24]. Number of  -cuts, leading to a series of dual intervals being generated. Among these intervals, the internal has two limits (i.e.

Problem formulation
Consider the rough fuzzy multiobjective linear programming problem (IRFMOLP), as in the following form: Subject to: 0} , are compact rough fuzzy numbers. The above problem can be reformulated by use the weighting method as in the following form: are compact rough fuzzy numbers.

Definition 3 The rough fuzzy vector
which satisfies the conditions in model (2), is said to be a rough fuzzy optimal solution of mode (2) x  is a rough fuzzy vector. then it is said to be a rough fuzzy optimal solution of model 2: for any are rough intervals, furthermore, if M a r c h 23, 2 0 1 5 R x  is a rough fuzzy vector, then it is said to be a rough fuzzy optimal solution of model 2: for any are rough intervals denote them by:

which satisfies the condition in model 1 IRFMOLP, is called a rough fuzzy efficient solution of mode 1 IRFMOLP if and only if there does not exist another
Subject to: 0} , (g) Incorporate the solution of all submodel to the obtain optimal solution as following : After deduce the following programming problems for a upper approximation interval and lower approximation interval It is clear that for any

Basic results
In the following we shall prove some theorems which point out that a rough fuzzy optimal solution of models From eqs (1),(3),(4), (5) and (6), then the second side of eqn (12) From eqs (13) and (14) there followes (11), we also obtain, for any eqs (15) and (16) show that the part of the theorem is correct, too the proof is compleat.  Proof Suppose that U x  is upper approximation fuzzy optimal solution of model U U P (problem ?? ), that is (1) (3) From eqs (1),(3),(4), (5) and (6), then the second side of eqn (18) From eqs (19) and (20) there followes (17), also obtain, for any eqs (21) and (22) show that the part of the theorem is correct, too the proof is compleat.

Conclusion
In this paper, we have obtained an the efficient solutions of multi-objective linear programming problem with inexact rough interval fuzzy coefficients in both objective and constraints. The problem IRFMOLP is transformed into the single objective linear programming problem with an inexact rough interval fuzzy coefficients by using the weighting method IRFLP(w). The new arithmetic operations based convert each an inexact rough fuzzy problem to two problems corresponding to the upper and lower approximation fuzzy set, respectively. Solve all the problems and incorporate the solution of all problems to the obtain optimal solution as follows rough interval.