Multiobjective programming is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints (see [

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 [

Now let us recall the definition of efficiency and the representation of the set of efficient points deduced by the definition in vector optimization.

Given a nonempty set

In [

Since

the structures of the production possibility sets and the relationships between these sets (

the equivalence of Pareto efficiency in multiobjective linear programming and DEA efficiency in DEA model (Theorem

the particularity of structures of the set of solutions to the multiobjective linear programmings corresponding to DEA models (detailed in Section

For the four most representative DEA models

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

This paper is organized as follows. Section

The following notations are used in the paper.

Let

In this section, a new representation of

Given a nonempty set

Clearly, the following result holds.

Equations (

Definition

Assume that

For convenience, let

By Theorem

Since

On the other hand, For any

A spacial case of (ii) in Proposition

Besides Proposition

Consider the following:

Let

It is obvious that

Let

If

In the following, we investigate some new properties of the efficient set when

If

If

Otherwise, for all

If

A more accurate relationship between the two efficient sets in Lemma

If

Note that

Reciprocally, for all

Usually

Consider the following:

About the efficient point set of differences between two sets

Consider the following:

We have

In the previously mentioned proposition, if

Propositions

Consider the following:

We have

It is obvious that if

Consider the following:

Notice that

(i) Usually the equality does not hold in Proposition

(ii) for Proposition

For each of DEA models, Charnes et al. and Wei et al. (e.g., see [

By the results of Section

Denote

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 [

Consider multiobjective linear programming problem:

Since

A

For convenience, we denote by

As an example, consider the multiobjective linear programming problem

Consider the following:

Consider the following:

Inclusions

Consider the following:

Theorem

if DMU_{0} is

if DMU_{0} is

By Proposition

Consider the following:

By Theorem

Proposition

Consider the following:

Consider that

if DMU_{0} is

if DMU_{0} is

In this paper, Definition

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.