advertisement

International Journal of Applied Operational Research Vol. 1, No. 2, pp. 65-73, Autumn 2011 Journal homepage: www.ijorlu.ir

F. Rezai Balf , R. Shahverdi

**Received:**

May 17, 2011 ;

**Accepted:**

September 4, 2011

**Abstract **

In data envelopment analysis, anyone can do classification decision units with efficiency scores. It will be interesting if a method for classification of DMUs without regarding to efficiency score is obtained. So in this paper, the classification of Decision Making Units (DMUs) is done according to the additive model without being solved for obtaining scores efficiency. This is because it is known that the additive model is the simplest non-radial model in DEA. In fact, the classification of DMUs to a set of efficient, weak efficient, and inefficient units, based upon feasibility concept is done here. Especially, the models and theorems for this aim are presented.

**Keywords **

Data Envelopment Analysis, Classification, Decision Making Units.

**1 Introduction **

Data envelopment analysis (DEA), as originally proposed by Charnes, Cooper and Rhodes [1] is a nonparametric frontier estimation methodology for evaluating the relative efficiencies. This methodology is nonparametric in a sense, because it does not require an assumption about a functional form of the efficient frontier; and therefore, there is no parameter estimation. DEA clusters the DMUs as “efficient” or “inefficient” depending on their relative geometric location with respect to an efficient frontier. The unit on the frontier is called efficient; otherwise, it is called inefficient. One major problem with a radial measure of technical efficiency is that it doesn't reflect all identifiable potential for increasing outputs and reducing inputs. In economy, the concept of efficiency is intimately related to the idea of Pareto optimality. A DMU is Pareto optimal only when it is not possible to improve any input or output without worsening some other input or output. In other words, when positive output and input slacks are presented at the optimal solution of CCR or BCC [2], radial DEA models, the corresponding radial projection of an observed input-output combination does not meet the criterion of Pareto optimality [3] and it should not be qualified as an efficient point. The Pareto efficient units are divided into two groups: extreme efficient and non-extreme efficient. However, in this paper we consider one of several non-radial models of DEA, that is, the additive model, [4], as it does yield a projection to the efficient subset of efficiency frontier. The suggested approach is based on replacing the concept of feasibility instead of solving the DEA models.

*** Corresponding Author. (**

**) E-mail:**

[email protected] (F. Rezai Balf)

**F. Rezai Balf, R. Shahverdi **

Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.

66 F. Rezai Balf, R. Shahverdi / IJAOR Vol. 1, No. 2, 65-73, Autumn 2011 (Serial #2) The study is arranged as follows: In section 2 several DEA basic models are given. In section 3 we introduce the additive model based on DEA. In section 4, we are determined to distinguish between the resulting classification of DMUs as efficient, weak efficient and inefficient based upon feasibility. Section 5 illustrates an example and section 6 deals with the conclusion.

**2 The DEA basic models **

This section presents some basic definitions and concepts readable and useful for other sections. They won't be discussed in details. Suppose that the producer uses the input vector

*x*

*R*

*m*

to produce the output vector

*y*

*R*

*s*

; i.e., all data are assumed to be nonnegative, but at least one component of every input and output vector is positive. A pair of such semi positive input

*x*

and output

*y*

, are called the decision making units. The Production Possibility Set (PPS) is represented as

*T*

*R m s*

input

*x*

can produceoutput

*y*

} .

*T*

The production possibility set

*T*

are given as follows: is generally close and convex. Additional properties set 1.

2.

3.

The observed DMU belongs to

*T*

; ( (

*x j*

,

*y j*

)

*T*

If ( , )

*T*

, then For any ( , )

*T*

( , )

*T*

,

*t*

0.

, any semi-positive ( , ) with ,

*x j*

1,...,

*n x*

and

*y*

).

*y*

Arranging the data set in matrices defined by

*T c*

*T X*

*x*

,

*X*

*T Y*

(

*x j*

) , and

*y*

,

*Y*

0} . (

*y j*

) ,

*j*

1,...

*n*

, the PPS Where

*R n*

is a semi-positive vector. Adding the constraint

*T e*

1 , with

*e T*

can be (1,1,...,1)

*T*

, to

*T c*

is equivalent to omitting the postulate 2, then

*T v*

is defined as follows:

*T v*

*T X*

*x*

,

*T Y*

*T*

1, 0}

*T c*

and

*T v*

are called the production possibility sets of CCR model [1] and BCC model [2] in DEA to gather with constant return to scale and variable return to scale respectively. To evaluate the efficiency of

*DMU o*

(

*o*

, the input-oriented model (1) and the output oriented model (2) are applied. Min s.t.

*T X*

x , o

*T Y*

*y o*

, .

(1) and

A Method for DMUs Classification in DEA 67 Max s.t.

*T T Y X*

x , o

*y o*

, .

The above mentioned models are called envelopment form

*CCR*

model if { (2) 0} and envelopment form

*BCC*

model if { 0,

*e T*

1} .

**3 DEA additive model **

Consider the additive model with input-output crisp data. This model was introduced by [3]: Max

*w*

*T e s*

_

*T e s*

s.t.

*j n*

1

*j x j*

*s*

x , o

*j n*

1

*j y j s*

0,

*s*

*y o*

,

*s*

0, .

(3) or equivalently we have: Max s.t.

*w*

*i m*

1

*s i*

*r s*

1

*s r*

*j n*

1

*j x ij*

*s i*

*x io*

,

*i*

1,...,

*m*

,

*j n*

1

*j y rj*

*s r*

*y ro*

,

*r*

(4)

*s i*

,

*s r*

.

0,

*i*

1,..., In the literature of data envelopment analysis different models are proposed to determine efficient units, weak efficient units and inefficient units. Here, we like to take a deeper look at the additive model in order to classify the DMUs with it. The classical DEA models rely on the assumption that inputs have to be minimized and outputs have to be maximized. Following the efficient subset (ET), weak efficient (WET) and inefficient (INET) of the production possibility set

*T*

E

*T*

*T x*

can be defined respectively as follow:

*y*

*T*

} . WE

*T*

INE

*T*

*T*

*T*

E

*T*

(

*x*

*x*

,

*y*

, ((

*x*

*y*

)

*x*

,

*y*

*y*

)

*T*

}

*T*

) }

68 F. Rezai Balf, R. Shahverdi / IJAOR Vol. 1, No. 2, 65-73, Autumn 2011 (Serial #2) To explain the underpinnings of the above definitions, we can use the Fig. 1, which represents the efficient and inefficient Sets, in Farrell model with two inputs and one output.

*x*

2

*F A B C D E x*

1

**Fig. 1**

*O*

In Fig. 1, the symbol

*O*

point out (0, 0,1) and the Set , , } is called efficient units,

*E*

is weak efficient unit and

*F*

is inefficient unit. Therefore, it is obvious that according to the above definitions, the below properties can be resulted. E

*T*

E

*T*

WE

*T*

WE

*T*

INE

*T*

INE

*T*

*PPS*

The symbol is applied for empty set. The above properties express the union and intersection of set efficient, weak efficient and inefficient. The union of them is production possibility set and their intersection is the empty set. In the following sections, we are willing to separate DMUs into three groups,

*ET*

,

*WET*

, and

*INET*

with our focus on the additive model. Thus, one can classify DMUs to measure the efficiency for DMUs. If

*DMU o*

be a unit under evaluation by model (4), then it is an efficient unit when

*w*

unit when

*w*

* 0, for some

*i*

0 or for some

*r*

0 * 0 and weak efficient , and finally it is an inefficient unit if

*w*

* 0, for all values of

*i*

,

*s i*

0 and for all values of

*r*

0 . It is assumed that the obtained

*w*

* represents the total sum of inefficiencies (total sum of slacks) or

*DMU o*

[4].

**4 Classification of DMUs in the additive model **

In the earlier part of this paper, we mentioned how to classify DMUs with respect to efficiency score in DEA additive model. It will be interesting if we attain a method for classifying DMUs without regarding to efficiency score. Our focus is based on feasibility additive model. This work is done by revising the additive model. To start, we modify the model (4) as model (5), with replacing

*s*

*e*

,

*s*

*e*

, instead of

*s*

0,

*s*

0 . The term is a non-Archimedean infinitesimal positive number which is smaller than any positive real number.

A Method for DMUs Classification in DEA 69 Max

*T e s*

_

*T e s*

s.t.

*j n*

1

*j x j*

*s*

x , o

*s*

*y o*

, (5)

*j n*

1

*j y j s*

*e*

, .

Now by replacing value (

*m*

*s*

)

*s*

*e*

, with

*s*

,

*s*

, in model (5), and by removing the constant of objective function, we obtain the model (6) as follows: Max

*T e s*

_

*T e s*

s.t.

*j n*

1

*j x j*

*s*

x o

*e*

,

*j n*

1

*j y j*

*s*

*y o*

*e*

,

*s*

0,

*s*

0, .

(6) Using the model (5) or (6), we determine the set of efficient DMUs and also inefficient DMUs. This subject will relate to Theorem 1.

**Theorem 1. Proof. **

Suppose that the model (6) has feasible solution then Suppose (

*s*

,

*s*

*DMU o*

be a feasible solution model (6). INE

*T*

.

Then we obtain:

*j n*

1

*j x j*

*s*

x o

*e*

and

*j n*

1

*j y j*

*s*

*y o*

equivalently, we obtain:

*j n*

1

*j x j*

(

*s*

*e*

) x o and

*j n*

1

*j y j*

*e*

or (

*s*

*e*

)

*y o*

. Hence, it implies:

*j n*

1

*j x ij*

*x io*

,

*i*

1,...,

*m*

, and

*j n*

1

*j y rj*

In other words, we have: (

*j n*

1

*j x j*

,

*j n*

1

*j y j*

)

*y ro*

, (

*x o*

,

*y o*

) , that is,

*DMU o r*

1,...,

*s*

. INE

*T*

, because

*DMU o*

is dominated by (

*j n*

1

*j x j*

,

*j n*

1

*j y j*

) .

1

**Corollary 1. **

Suppose that the model (6) is not feasible then

*DMU o*

E

*T*

WE

*T*

. 1 We say

*DMU p*

:(

*x p*

,

*y p*

) dominates

*DMU q*

:(

*x q*

,

*y q*

) if and only if

*x p*

least one component.

*x q*

and

*y p*

*y q*

and inequality be strict in at

70 F. Rezai Balf, R. Shahverdi / IJAOR Vol. 1, No. 2, 65-73, Autumn 2011 (Serial #2) As demonstrated in Theorem 1, we can only distinguish between efficient and inefficient DMUs, with respect to the feasibility of model (6). Now, we show a model and two theorems for the separation of efficient units of weak efficient units. Considering the following model (7): Max

*w*

0 s.t.

*j n*

1

*j x ij*

*x io*

,

*i*

1,...,

*m*

,

*y ro*

,

*r*

(7)

*j n*

1

*j y rj*

.

**Theorem 2. **

Suppose (

*x o*

,

*y o*

)

*T*

infeasible.

**Proof. **

Suppose (

*x o*

,

*y o*

) . A unit (

*x o*

,

*y o*

) WE

*T*

if and only if the model (7) is according the weak efficiency definition, it is obvious that there is

*PPS*

, such that

*x*

*j n*

1

*j x j*

,

*y*

*j n*

1

*j y j*

and

*j n*

1

*j x uj*

*x uo*

,

*u*

{1,...,

*m*

}, or

*j n*

1

*j y vj*

*y vo*

,

*v*

. It implies that is dominated by (

*x o*

,

*y o*

) . Hence, the model (7) is infeasible. Conversely, suppose that the model (7) is infeasible, since (

*x o*

,

*y o*

)

*T*

it means that there is

*u*

{1,...,

*m*

}, or

*v*

{1,..., } such that

*j n*

1

*j x uj*

it implies assertion, that is, (

*x o*

,

*y o*

) WE

*T*

.

**Theorem 3. **

Suppose (

*x o*

,

*y o*

)

*T*

. A unit (

*x o*

,

*y o*

) E

*T*

*x uo*

, or

*j n*

1

*j y vj*

*y vo*

, and if and only if model (7) is feasible.

**Proof. **

With contradiction, suppose that the model (7) is not feasible. Then according to Theorem 2, (

*x o*

,

*y o*

) which is a contract. Also (

*x o*

,

*y o*

*T*

, then with respect to (

*x o*

,

*y o*

)

*T*

, we find

*DMU o*

WE

*T*

, and also according to Theorem 2, the model (7) is infeasible. It is also a contract. The proof is complete. In order to recognize that

*DMU o*

E

*T*

or

*DMU o*

WE

*T*

, we apply the model (7) with respect to feasibility concept. So far, the DMUs have been divided in two groups, efficient and weak efficient. Now, for classification of efficient DMUs, at first we present the reference set definition.

**Definition 1.**

For

*DMU o E O*

{

*j*

*j*

* , its reference set,

*E o*

, is defined by 0 in some optimal solution of model (3)} . All the references of a

*DMU o*

are efficient units. A

*DMU o*

is dominated by one of them or their some combination. Therefore, we have several results as follows: 1.

*DMU o*

is inefficient; then it is not a member of reference set

*E o*

, [5]. 2.

*DMU o*

is extreme efficient, then it is only reference for itself [1].

A Method for DMUs Classification in DEA 71 3.

*DMU o*

is non-extreme efficient, then there exists at least one member of reference set

*E o*

, except itself. Hence, by utilizing the foregoing definition and results, we present the following model for recognization and classification of extreme efficient and non-extreme efficient based on the additive model. Max

*w*

*T e s*

_

*T e s*

s.t.

*j n*

1

*j x j*

*s*

x , o

*j n*

1

*o*

*j*

0,

*y s*

0,

*j*

*s*

*s*

*y o*

, 0, .

(8)

**Theorem 4. **

Suppose

*DMU o*

E

*T*

. The model (8) has feasible solution if and only if

*DMU o*

be non-extreme efficient.

**Proof. **

Suppose

*DMU o*

(8) so that

*o*

E

*T*

and (

*s*

0 . Then we have 0,

*s*

*n*

*j*

1

*j o*

*j x j n*

*j*

1

*j o*

*j y j*

x , o

*y o*

.

, be a feasible solution of model (9) This means that

*DMU o*

is non-extreme efficient; because there is at least one other unit in

*E o*

, other than itself. Conversely, suppose that

*DMU o*

E

*T*

and

*DMU o*

be non-extreme efficient, then with respect to its definition, in addition to it, there exists at least one other unit in

*E o*

. This follows that for , the equation (9) holds. Therefore,

*o*

0 .

**Theorem 5. **

Suppose

*DMU o*

E

*T*

. The model (8) has not feasible solution if and only if

*DMU o*

be extreme efficient.

**Proof. **

By contradiction

**, **

suppose that

*DMU o*

E

*T*

and the model (8) has feasible solution. Then by Theorem 4, we obtain Conversely, if

*DMU o*

E

*T*

and

*DMU o*

as non-extreme efficient and this is a contract.

*DMU o*

isn’t an extreme efficient, it will be non-extreme efficient. Therefore, according to Theorem 4, the model (8) has a feasible solution, also it is a contract. Thus the proof is complete. So in this paper, we have used some theorems which helps us to recognize and classify DMUs based on feasibility concept.

72 F. Rezai Balf, R. Shahverdi / IJAOR Vol. 1, No. 2, 65-73, Autumn 2011 (Serial #2)

**5 An example **

Consider 10 samples of DMUs with two inputs and two outputs which are shown at Table 1.

**Table 1.**

The set of units with two inputs and two outputs DMU 1 2 3 4 5 6 7 8 9 10 Input 1 81 85 56.7 91 216 58 112.2 293.2 186.6 143.4 Input 2 87.6 12.8 55.2 78.8 72 25.6 8.8 52 0.1 105.2 Output 1 5191 3629 3302 3379 5368 1674 2350 6315 2865 7689 Output 2 205 0 0 8 639 0 0 414 0 66 Applying the model (6) to { 0} with 0.1

, we find that DMUs 1, 2, 5, 8 and 9 are efficient, but the other DMUs are inefficient. In other words, the model (6) is infeasible when we evaluate DMUs 1, 2 , 5, 8 and 9, on the other hand the model (6) be feasible for DMUs 3, 4, 6, 7, and 10 with optimal objective value 311.27, 2164.44, 1229.05, 948.76 and 509.28, respectively. Therefore, we have: Efficient DMUs ={DMU , DMU , DMU , DMU , DMU } 1 2 5 8 9 Now, using the model (7), to distinguish the extreme efficient unit of non-extreme efficient unit, we obtain that all DMUs are extreme efficient, because the model (7) is infeasible, when we evaluate every one of them with it.

**6 Conclusions **

In this paper we introduced some of the models and theorems in order to identify and classify DMUs regarding to feasibility based upon the DEA revised additive model. At first, we divided DMUs in two bundles of efficient and inefficient, and then the efficient DMUs are classified to two groups; that is, extreme-efficient and non-extreme efficient. In this study, we solved several models to identify and classify DMUs, being worthwhile and significant, when one can only use the feasibility property for models. Finally, we recommended to readers to focus on obtaining and classifying DMUs without solving its corresponding models, for sample refer to [6], such that, the efficient DMUs are ranked in it without solving the model.

**References**

1.

2.

Charnes, A., Cooper, W. W., Rhodes, E., (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429-444. Banker, R. D., Charnes, A., Cooper, W. W., (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), 1078-1092.

A Method for DMUs Classification in DEA 73 3.

4.

5.

6.

Charnes, A., Cooper, W. W., Golany, B., Seiford, L., Stutz, J., (1985)b. Foundations of data envelopment analysis for Parato-Koopmans efficient empirical production functions. Journal of Econometrics, 30, 91 107. Bardhan, I., Bowlin, W. F., Cooper, W. W., Sueyoshi, T., (1996). Models for efficiency dominance in data envelopment analysis. Part I: Additive models and MED measures. Journal of the operations Research Society of Japan, 39, 322-332. Cooper, W. W., Li, S., Seiford, L. M., Tone, K., (1999). Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, Kluwer Academic Publisher, Norwell, Mass. Jahanshahloo, G. R., Hosseinzadeh, F., Zhiani Rezai, H., Rezai Balf, F., (2007). Finding strong defining hyperplanes of Production Possibility Set. European Journal of Operational Research, 177(1), 42-54.