#### Abstract

, the set of Pareto efficient (weak efficient) points of a set with respect to a cone in , is expressed as a differencebetweentwo sets and ( and ). Using the new representation, the properties of are proved more easily than before. When or is in the form of union, intersection, sum, or difference of two sets or two cones, respectively, the properties of 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.

#### 1. Introduction and Preliminaries

Multiobjective programming is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints (see [1–3]). 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 by solving the linear programming [4–6]. 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 [7–9]. There is a multiobjective linear programming, corresponding to a DEA model, such that a is (weak) DEA efficient if and only if (associating with the ) 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 in , is called a Pareto efficient (weak efficient) point of , if there is no with such that . 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 , , , 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 , , , 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.

Let , , , and be sets in :

#### 2. 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 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.

Proposition 4. *Assume that , , and are nonempty sets and , , and are cones in , . Then*(i), *if *,(ii),(iii),(iv), *for any *, *is a convex cone*.

* Proof. * For convenience, let , and let . Since , we have . And then

By Theorem 3, it is sufficient to show that
for all , where , and . Then , otherwise , a contradiction. Similarly . Hence

Since and is a convex cone, we have , ,

On the other hand, For any , , , . If , then , a contradiction. Hence, , , that is, .

A spacial case of (ii) in Proposition 4 is that if is convex and is a cone.

Besides Proposition 4, we state the following properties.

Corollary 5. *
Consider the following:*

Proposition 6. *Let , , and be sets in . Then
*

*Proof. *It is obvious that and . Hence

Let , and let in Proposition 6. We have Corollary 7.

Corollary 7. *If and 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.

Lemma 8. *If is a convex cone, then
*

*Proof. *If , the result obviously holds.

Otherwise, for all , if , such that , where .

If , it contradicts that . Hence . such that , so , which still contradicts that . In consequence, .

A more accurate relationship between the two efficient sets in Lemma 8 is described in the following proposition.

Proposition 9. *If is a convex cone, then
*

*Proof. * obtained by Lemma 8.

Note that , . So .

Reciprocally, for all , , . If , with such that , in contradiction to . Therefore,

Usually , is what Example 10 shows.

*Example 10. *
Consider the following:

About the efficient point set of differences between two sets and , 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 , 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.

Proposition 13. *
Consider the following:*

*Proof. *
We have

*Remark 14. *It is obvious that if and , then

Proposition 15. *
Consider the following:*

*Proof. *Notice that .

*Remark 16. *(i) Usually the equality does not hold in Proposition 15. Let , . Then

(ii) for Proposition 15, if , then
(iii) since , Proposition 11 can be obtained by Proposition 15.

#### 3. Efficiency of Four Most Representative DEA Models

For each of DEA models, Charnes et al. and Wei et al. (e.g., see [5, 7, 9]) establish an associative multiobjective linear programming model and prove that a is DEA efficient if and only if (associating with the ) is a Pareto efficient solution of the multiobjective linear programming problem (for the details about DEA efficient of DMUs in DEA model, see [4, 5] and the references therein). This beautiful conclusion provides the multiobjective linear programming as an efficient tool to solve DEA problems. So the key point now is to find all efficient solutions of the multiobjective linear programming problems. This section offers a simple way to do this.

By the results of Section 2, in the following, we investigate the multiobjective linear programming problems corresponding to the four DEA models and develop the structures of the efficient point sets and the efficient solution sets of these programmings. Based on these, the relationships between the DEA efficiency of the four DEA models are obtained.

Denote to be the input vector for the decision making unit (), and , to be the output vector for the decision making unit, for . For convenience the notations , for are given. Let , and let . The orderings in and are defined by and , respectively. , , and . The production possibility set , is based on postulate sets which are presented with a brief explanation (see [4, 5]). The four most representative models are, briefly, , and , which correspond to different production possibility sets , , , and , respectively, [4–10]

Obviously, the following equalities and inclusions hold:

For distinguishing the DEA efficiency of DMUs in DEA models, according to the equivalence of DEA efficiency in DEA models and Pareto efficiency in multiobjective linear programming obtained by Charnes et al. and Wei et al., we introduce the multiobjective linear programming problem corresponding to DEA models in the following. For the details about DEA models, see [7, 8].

Consider multiobjective linear programming problem: For , , , or , the four multiobjective linear programming problems , , , and correspond to the four DEA models respectively.

*Definition 17. * is called an Pareto efficient (weak efficient) solution of , if there is no , such that

Since defines the ordering in , the previous inequality can be written as
When is a convex pointed cone, all conclusions in Section 2 hold for multiobjective programming problem . Theorem 18 provides the equivalency of DEA efficiency of DMUs and the Pareto efficiency of the points corresponding to the DMUs. Consequently, the structures of the sets of efficient solutions to DEA models are obtained by solving the multiobjective programmings. Then the relationships between the DEA efficiencies of DMUs in DEA models are deduced.

Theorem 18 (see [5]). *A is DEA efficient if and only if is a Pareto efficient solution of .*

For convenience, we denote by . In the following, we investigate the structures of the efficient point sets and the efficient solution sets of the four vector optimization problems. The relations (), (), (), and () imply:

As an example, consider the multiobjective linear programming problem . Let denote the efficient point set, and let denote the efficient solution set. Note that By Definitions 2 and 17, Since we have Similar argument is applied to other three multiobjective linear programming problems. Therefore the following two theorems hold.

Theorem 19. *
Consider the following:*

Theorem 20. *
Consider the following:*

Inclusions , and Theorem 20 imply Theorem 21.

Theorem 21. *
Consider the following:*

Theorem 21 shows the relationships between the solutions of the four vector optimization problems. Therefore, by Theorem 18, the relationships between DEA efficiencies of DMUs in DEA models are as the following: (i) if DMU_{0} is DEA efficient, then it is DEA efficient. If is DEA efficient, then it is DEA efficient;(ii) if DMU_{0} is DEA efficient, then it is DEA efficient. If is DEA efficient, then it is DEA efficient.

By Proposition 13, along with the equality and , we obtain following Theorem 22.

Theorem 22. *
Consider the following:*

By Theorem 22, we conclude that if , then and ; by , , and , we declaim that if is DEA efficient, then it is also DEA efficient and DEA efficient.

Proposition 15 and infer Theorem 23.

Theorem 23. *
Consider the following:*

Consider that , and , if , then . Similarly, . Therefore, the following consequences are obtained:(i) if DMU_{0} is DEA efficient, then it is also DEA efficient; (ii) if DMU_{0} is DEA efficient, then it is also DEA efficient.

#### 4. 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, ; 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.

#### Acknowledgments

This research was supported in part by the Zhejiang Natural Science Foundation of China Grant Y6110054, the Shanxi Scholarship Council of China Research Grant 2010087.