An Extended Aggregated Ratio Analysis in DEA

Liang Liang*

School of Management

University of Science and Technology of China

He Fei, An HuiProvince, P.R. China 230026

Email:

Jie Wu

School of Management

University of Science and Technology of China

He Fei, An HuiProvince, P.R. China 230026

Email:

[005-0135]

Submitted to POMS International Conference - Shanghai 2006

May 2006

An Extended Aggregated Ratio Analysis in DEA

Liang Liang, Jie Wu

Abstract:We propose an extended aggregated ratio analysis model that is similarly extended from CCR to C2GS2model in DEA. This extended model also offers an insight into frontier analysis. Whether a DMU is on the frontier or efficient frontier can be informed by proposed extended model. Several results developed in the paper are coincident with that in the literature.

Key words: Data envelopment analysis (DEA); C2GS2model; Ratio; Efficient frontier

1. Introduction

Ratio analysis is a popular technology to evaluate the performance in economy. The ratio is used to evaluate the relation among variables. In order to realizehome the performance of DMU from multi-perspectives, such as yield rate, liquidity and quality of asset, we select these variables. The number of ratio lies on the aims of analysis. Because they are easily understood, ratio analysis is used widely in some fields, like financial investment, credit card and insurance. But there are some disadvantages in using. First, only by observation of one aspect of behavior of DMU, orobtaining one numeral by combination of multi-perspectives, the result isn’t satisfying；Second, the ratio obtained by financial descriptive data is very large, may puzzle persons and make contradictions among ratio numbers. So, ratio analysis factually uses some limited principles to evaluate the efficiency of unit. Due to incomplete consideration of process with multi-inputs and multi-outputs, and difficult to find the optimal in peer set, classical ratio analysis usually is ineffective in application to evaluate the efficiency of unit.

To overcome above limitations of ratio analysis, D. Wu et. al.(2005）presentedan aggregated ratio analysis model, and proved that the model proposed is equivalent to CCR model, that is, to any given DMU, the efficiency of aggregated ratio model is the same asthe efficiency of CCR model, and the best DMU in aggregated ratio model is on the DEA frontier.

In this paper, we propose an extended aggregated ratio analysis model that is similarly extended from CCR to C2GS2 model in DEA.This extended model also offers an insight into frontier analysis. Whether a DMU is on the frontier or efficient frontier can be informed by proposed extended model.The rest of this paper unfolds as follows. Section 2 presents the extended aggregated ratio analysis model. The frontier analysis is introduced in section3. and finally concluding remarks are made in section 4.

2. Extended aggregated ratio analysis model

Let the observed input and output vectors of DMUjbe

respectively.

Then the mathematical programming problem of the C2GS2”ratio-form” DEA model is stated as:

(1)

and the C2GS2 Envelopment Form is:

（2）

Definition 1: DMU0 is DEA efficient if and only if there exists an optimal solution of（1）and an optimal solution of (2) such that .

Now we introduce aggregated ratio analysis modelwhich can also be used to evaluate DMUs.

For any given DMUj, the output-input ratio vector is defined as ,we introduce the following aggregated ratio model for DMU0:

(3)

Definition 2: DMU0 is ratio efficient if there exists an optimal solution and with such that.

Theorem 1: DMU0 is ratio efficient if and only if it is DEA efficient.

Proof: We only need to show that the optimal objective value of (1) is equal to the optimal objective value of (3), i.e. .

First, we would like to show.

Let and be the optimal solution of (3) and be its optimal objective value, therefore,

(4)

and (5).

Equation (4) can be rewritten as :

(6)

By multiplying .

On both numerator and denominator of (6), we have:

(7)

Let

Then Eq.(7) can be written as (8)

Similarly (9)

（8）and (9) indicate is a feasible solution to the C2GS2 model in (1) and thus .

Now let us prove that.

Suppose that is an optimal solution of the C2GS2 model in (1) and is its optimal objective value.

Then (10) and (11)

Multiplying (10) by and summarize on , after simple arrangement we then have:

(12)

Where

Eq.(12) can be written as (13)

Let

Then (13) can be written as (14)

Divide both sides of (14) by, we have:

(15) similarly,

Therefore is a feasible solution of model (3) and thus.

Then

3. The frontier analysis

In this section, we will introduce the character of the frontier.

Theorem 2: If there exist weights combination of and such that, then DMU0 is on the frontier. Furthermore, if , then DMU0 is on efficient frontier.

Proof: For , we denote .Let we have

and

It means that and satisfies the first constraint of ratio model (3)

Since , DMU0 is on the frontier.

Now if ,we know that and .

Then is an optimal solution of (3) with . DMU0 is efficient.

The theorem above indicates that DMU with the highest aggregated ratio of output divided by input is on the frontier,it also proves the possibility of finding DMU on the frontier using different combinations of input/output ratio.

4. Conclusions

In this paper, we propose an extended aggregated ratio analysis model that is similarly extended from CCR to C2GS2 model in DEA.This extended model also offers an insight into frontier analysis. Whether a DMU is on the frontier or efficient frontier can be informed by proposed extended model.

References

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