A New Representation of Efficient Point Sets and Its Applications in DEA

E(M, K) (E w (M, K)), the set of Pareto efficient (weak efficient) points of a set M with respect to a cone K in R, is expressed as a differencebetween two sets M and M + K\{0} (M and M + intK). Using the new representation, the properties of E(M, K) are proved more easily than before. When M or K is in the form of union, intersection, sum, or difference of two sets or two cones, respectively, the properties of E(M, K) are considered. Most of the properties are proved by the binary operations of sets, which is a new method in the multiobjective optimization. Then these properties are used to solve some types of multiobjective linear programming problems corresponding to Data Envelopment Analysis (DEA) models. The structures of the DEA efficient solution sets of four most representative DEA models are developed. Further more, the relationships between efficiencies of the four DEA models are deduced.


Introduction and Preliminaries
Multiobjective programming is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints (see [1][2][3]).Data Envelopment Analysis (DEA) is a nonparametric method in operations research and economics for the estimation of production frontiers.It is used to empirically measure productive efficiency of decision maker units DMU 1 , . . ., DMU  by solving the linear programming [4][5][6].Charnes et al. and Wei et al. establish the equivalence of (weak) DEA efficient solutions in DEA model and (weak) Pareto solutions of multiobjective linear programming [7][8][9].There is a multiobjective linear programming, corresponding to a DEA model, such that a DMU  0 (1 ≤  0 ≤ ) is (weak) DEA efficient if and only if (  0 ,   0 ) (associating with the DMU  0 ) is a (weak) Pareto efficient solution of the multiobjective linear programming whose feasible region is the production possibility set (see [10]).
In this paper, we propose a new representation for the set of Pareto efficient (weak efficient) points.With the help of the new representation, not only the properties of the set of Pareto efficient (weak efficient) points which are given in [1,3,11] can be proved more simply, but also more new properties can be obtained.It is these new properties that reveal the relationships between the set of solutions and different multiobjective linear programmings which correspond to different DEA models.Further, the relationships between efficiencies of DMUs in different DEA models are obtained by a new way.Wei et al. [10] develop a famous method to translate production possibility sets in the intersection form and in the sum form and find all DEA efficient DMUs.For each of the four most representative DEA models, we offers a simple way to get all DEA efficient DMUs by the binary operations of sets.Now let us recall the definition of efficiency and the representation of the set of efficient points deduced by the definition in vector optimization.

Definition 1 ([1]
, efficiency, weak efficiency).Given a nonempty set  and a cone  with int  ̸ = 0 in   ,  0 ∈  is called a Pareto efficient (weak efficient) point of , if there is no  ∈  with  ̸ =  0 such that  0 ∈  +  ( 0 ∈  + int ).The set of all Pareto efficient (weak efficient) points of  is denoted by (, ) (  (, )).(, ) (  (, )) is called the efficient (weak efficient) point set of .By Definition 1, we have that In [2] when  =   + , the set is also described as follows: Since  =   + is a convex pointed cone, (1) and (3) are equivalent.So are (2) and (4).In Section 2, we give a new representation of the efficient (weak efficient) point set, which is expressed as the difference of two sets.The idea of the new representation is motivated by the following facts in the area of DEA: (i) the structures of the production possibility sets and the relationships between these sets (5); (ii) the equivalence of Pareto efficiency in multiobjective linear programming and DEA efficiency in DEA model (Theorem 18); (iii) the particularity of structures of the set of solutions to the multiobjective linear programmings corresponding to DEA models (detailed in Section 3).
For the four most representative DEA models  2 ,  2 , , and  (for the details about the models, see [4,5]), each of the DEA models associates with a production possibility set which is also the feasible set of the multiobjective linear programming corresponding to this DEA model.The production possibility sets are denoted by   2  ,   2 ,   , and,   , respectively.The following relations hold (the structures of these production possibility sets are presented in Section 3): It is the specialty of the relations of the production possibility sets and the equivalence of DEA efficiency and Pareto efficiency that motivate us to propose a new representation of (weak) efficient point set.Using the new representation,we obtain some new properties of efficient point set (, ), when the set  or  is in form at union, intersection, sum, or difference of two sets.By these properties, it is easier to get the relationship between the DEA efficiency of the four DEA models than before.
This paper is organized as follows.Section 2 introduces the new representation of the efficient (weak efficient) point set of a set, discusses some new properties of efficient point set.Using the new expression of the set (, ) (  (, )), most of these properties are proved by the binary operations of sets.The multiobjective linear programming problems corresponding to the four DEA models are studied in Section 3. The structures of the efficient point sets and the efficient solution sets of the multiobjective linear programming problems are developed, and then the relationships between DEA efficiencies of DMUs in four DEA models are revealed.Section 4 is devoted to the conclusion.
The following notations are used in the paper.

Some Properties of the Efficient Point Set
In this section, a new representation of (, ) (  (, )) is presented.Then we prove that it is equivalent to the original ones when  is a cone.Lastly we focus on the properties of (, ), when  or  is in the form of the union, intersection, sum, or difference of two sets.Most of the proofs are completed by the binary operations of sets, which is a new method in multiobjective optimization.
Definition 2. Given a nonempty set  and a cone  with int  ̸ = 0 in   , the efficient (weak efficient) point set of  with respect to  is defined by Clearly, the following result holds.
Theorem 3. Equations (1) and (7) (( 2) and (8)) are equivalent, that is, Definition 2 gives a new representation of (, ) that is, the efficient point set of a set  is the difference between two sets.Using this new representation, we prove the properties of (, ).Proposition 4 comes from Luc [1], Papageorgiou [3], and Guerraggio et al. [11].We give an easier proof of this proposition.

Corollary 7.
If  1 and  2 are cones, then In the following, we investigate some new properties of the efficient set when  is the union, intersection, sum, or difference of two sets.
Otherwise, for all A more accurate relationship between the two efficient sets in Lemma 8 is described in the following proposition.
Reciprocally, for all  ∈ ( Usually ( Example 10.Consider the following: About the efficient point set of differences between two sets  1 and  2 , Proposition 11 gives the conclusion, without requiring  to be a convex but a cone.Proposition 11.Consider the following: Proof.We have Remark 12.In the previously mentioned proposition, if ( 1 ∩  2 ) +  0 = ( 1 +  0 ) ∩ ( 2 +  0 ), the equation holds.
Propositions 13 and 15 present the properties of the efficient point sets; when  is the union or intersection of two sets, respectively,  is not required to be a convex.

Conclusion
In this paper, Definition 2 presents a new representation to (, ).Then some new properties of (, ) are deduced by the new representation (Propositions 6 to 15 and their corollaries).Most of the properties are proved by mean of the binary operations of sets, which is a new method in multiobjective optimization.These conclusions are used to deal with the multiobjective linear programming problems corresponding to the four most representative DEA models.We investigate the structures of the efficient solution set of the four DEA models (Theorems 19 and 20) and deduce the relationships between DEA efficient solution sets of the four DEA models (Theorems 21 and 22).For each of the four DEA models, by Theorem 20,  =  0 \(+ 10 ); that is, all the DEA efficient DMUs are obtained by the binary operations of sets.Therefore, this is a simple way to get all DEA efficient DMUs theoretically.By using the new representation, it may be able to discuss the effects upon DEA efficiency of DMUs when the number of the DMUs changes in DEA models.