A New Procedure of Two Stage Data Envelopment Analysis Model

under Strict Positivity Restriction

M. D. Nasution

, M. R. Syahputra

, H. Mawengkang

, A. A. Kamil

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Sumatera Utara, Padang Bulan

20155, Medan, Indonesia

Telkom University, West Java, Bandung, Indonesia

Keywords: Positivity Restriction, Two Stage DEA.

Abstract: Data Envelopment Analysis (DEA) is a mathematical non-parametric approach for measuring relative

efficiency of homogenous decision making units (DMUs) performing. This approach will evaluates the

efficiency score of entities. The efficiency is defined as the maximum of the ratio of the sum of its weight

output to the sum of its weight inputs. The objective value is subject to the conditions that are corresponding

to ratios for each DMU be less than or equal to one. Strict positivity of the weights in the theoretical and the

computational result is an important condition to identify whether the DMU

is efficient or not. One method

that can be used to achieve this condition was considering a positive lower bound on its weights, known as a

non-Archimedean infinitesimal, 𝜀. In fact, it is very hard to find a set of positive weights among all the

alternative solutions of multiplier model. This paper show that a new procedure two-stage approach can solve

the decision-making problems that are modelled on the DEA-CCR model under strict positivity restriction.

1 INTRODUCTION

In determining the performance of an organization and

increasing productivity, the efficiency level must be

measured. In general, efficiency is expressed in the

form of a comparison between input (input) and output

(output). But in a company there may be different

input and output entities, in aspects of resources,

activities, environmental factors. So in general

measurement of efficiency is difficult to use. So to be

able to measure the level of efficiency with different

input and output entities can be done using Data

Envelopment Analysis (DEA) (Charnes et al., 1978).

Charnes et al (1979) proposed the model as a

fractional programming problem. After that, the

model was transformed as a simple linear

programming problem with a objective function and

some criteria. DEA's main objective is to determine

efficient conditions based on existing problem

scenarios. In this case the efficiency can be

interpreted as the maximum ratio of the weighted

output to the weighted input with the constraints

corresponding to each DMU.

Based on the basic concept of the CCR model

found by Charnes et al., (1978), known as the DEA

CCR, that the unit shows performance the best is with

one efficiency score. This shows that the score it is

part of the production boundary that cannot be

compared to the boundary area. Further techniques

that combine principles the basic DEA is known as

"Super Efficiency Analysis" introduced by Andersen

and Peterson (1993).

In his paper, Thompson et al (1993) discussed

several ways to eliminate zero weight in the DEA

problem. Various methods have been carried out,

including modifying the DEA model as carried out by

Charnes et al (1997). In his paper, Charnes et al

(1979) added a positivity requirement, using the

parameter 𝜀. This method is the right way to do it, but

this method has complex limitations and complexities

because we don't know the right value for 𝜀.

By this situation, Yao (2003) and Amin &

Toloo

(2004) conducted related research and found the

right number for 𝜀.

Cooper et al (2001) in his paper discuss about a

method that solved zero weight problem in DEA.

proposed two-stage method. This procedure is for

selecting non zero weights from the alternative

optimal solution of the multiplier model in a DEA.

Saen (2010) said that it is very hard to find a set of

positive weight among all the alternative solutions of

multiplier models.

Nasution, M., Syahputra, M., Mawengkang, H. and Kamil, A.

A New Procedure of Two Stage Data Envelopment Analysis Model under Strict Positivity Restriction.

DOI: 10.5220/0010095310311035

In Proceedings of the International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches (ICOSTEERR 2018) - Research in Industry 4.0, pages

1031-1035

ISBN: 978-989-758-449-7

1031

2 BASIC DEA-CCR MODEL

In this study the author uses the DEA-CCR model as

the basis for the model that will later be developed.

The basic DEA-CCR model in (1) is formed for

evaluating the efficiency of DMU

(Charnes et al,

1978). Suppose there are n DMU

𝐷𝑀𝑈



, 𝑗 

1,2,…,𝑛 , that will be evaluate the efficiency values.

Each of DMU consumes the amounts 𝑥



𝑥



of 𝑚

inputs 𝑖1,2,…,𝑚 , and will produce the amounts

𝑦



𝑦



 of 𝑠 outputs 𝑟1,2,…,𝑠 𝑥



0



,𝑥







,𝑦



0



,𝑦



0



. Then the DEA-CCR model is

defined as follows

max𝜃  𝑢







𝑦



𝑠.𝑡. 𝑣







𝑥



 1 1

𝑢







𝑦



𝑣







𝑥



0 𝑗1,2,…,𝑛

𝑣



 0 𝑖  1,2,…,𝑚

𝑢



 0 𝑟  1,2,…,𝑠

Where 𝑥



𝑥



 and 𝑦



𝑥



 are inputs and output

respectively. Meanwhile the weights of 𝑖-th input and

𝑟-th output are indicated by 𝑣



and 𝑢



respectively.

Completion of the model (1) will get the optimal

value for multipliers. Therefore, the model (1) is often

referred to as the multipliers form of the CCR

problem.

3 AN IMPROVED DEA-CCR

MODEL

The issue of strict positivity is important in the DEA.

Although there are many alternative optimal

solutions, it is still difficult to determine the level of

efficiency of each DMU. Therefore, Charnes et al

(1979) modifies the model (1) as follows:

max𝜃  𝑢







𝑦



𝑠.𝑡. 𝑣







𝑥



 1 2

𝑢







𝑦



𝑣







𝑥



0 𝑗1,2,…,𝑛

𝑣



 𝜀 𝑖  1,2,…,𝑚

𝑢



 𝜀 𝑟  1,2,…,𝑠

where 𝜀0 is an infinitecimal element that smaller

than any positive real number.

4 AN IMPROVED FORMULA OF

TWO STAGE DEA

The first step we must take to develop the DEA-CCR

model is to complete the model (1) in the first stage.

If the value 𝜃



∗

1, then DMU

is said to be CCR-

inefficient. If the model (1) has obtained its efficiency

value, then the next step is to solving the following

model (3) in the second stage.

max 𝛿

𝑠.𝑡. 𝑣







𝑥



 1 3

𝑢







𝑦



𝑣







𝑥



 0

𝑢







𝑦



𝑣







𝑥



 0 𝑗  𝑜

𝑣



 𝛿  0 ∀𝑖

𝑢



 𝛿  0 ∀𝑟

𝑣



,𝑢



,𝛿  0 ∀𝑖,𝑟

After solving the model (3) in the second stage, the

next step is to check the optimal solution. If the value

of 𝛿

∗

0, then we get



𝑢

∗

,𝑣

∗



0



. If that so, the

DMU

is called to be efficient.

In model (1) and (3), we replace the constrain

∑

𝑣



𝑥







1 with

∑

𝑣



𝑥







𝐾, 𝐾 is an arbitrary

nonnegative number to improve the recent procedure

of two stage DEA. Therefore, we rewrite the models

(1) and (3) respectively as follows:

maxΘ



𝑢







𝑦



𝑠.𝑡. 𝑣







𝑥



 𝐾 4

𝑢







𝑦



𝑣







𝑥



0 𝑗1,2,…,𝑛

𝑣



 0 𝑖  1,2,…,𝑚

𝑢



 0 𝑟  1,2,…,𝑠

maxΔ

𝑠.𝑡. 𝑣







𝑥



 1 5

𝑈







𝑦



𝑉







𝑥



 0

ICOSTEERR 2018 - International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches

1032

𝑈







𝑦



𝑉







𝑥



 0 𝑗  𝑜

𝑣



 Δ  0 ∀𝑖

𝑢



 Δ  0 ∀𝑟

𝑉



,𝑈



,𝛿  0 ∀𝑖,𝑟

In this paper a simple example will be given to

seeing the proposed DEA model application. In

addition, there will also be case example from bank

performance. As a tool, we use LINDO for solving

and making analysis of the models.

5 NUMERICAL EXAMPLE

As a simple numerical example, we evaluate 7 DMU

with two inputs and two outputs as shown in Table 1.

First, applying stage I to evaluate each DMUs. We

have four DMU which efficiency score is 1 as shown

in Table 2.

Table 1: Input and output of 7 bank

DMU In

Out

DMU

19 3 5 4

DMU

6 4 2 4

DMU

7 2 2 2

DMU

3 3 5 5

DMU

1 5 3 3

DMU

9 2 2 6

DMU

3 4 2 3

We evaluate the data on Table 1 using LINDO. By

means of 𝛿

∗

of the DMU

and DMU

is greater than

zero, it means that both of them are efficient.

Table 2: The result of numerical example using LINDO

DMU

Stage I Stage II

𝜃



∗

𝑣



∗

𝑣



∗

𝑢



∗

𝑢



∗

𝛿

∗

𝑣



∗

𝑣



∗

𝑢



∗

𝑢



∗

DMU

1.0000 0.0000 0.5000 0.2500 0.0000 0.0000 0.0000 0.5000 0.2500 0.0000

DMU

0.4710 0.0588 0.1961 0.0000 0.1569

DMU

0.5000 0.0000 1.0000 0.3750 0.1250

DMU

1.0000 0.2500 0.0000 0.2500 0.0000 0.1250 0.1875 0.1250 0.1250 0.1250

DMU

1.0000 0.5000 0.0000 0.0000 0.5000 0.0000 0.5000 0.0000 0.5000 0.0000

DMU

1.0000 0.0000 1.0000 0.0000 0.2000 0.0625 0.2500 0.3750 0.6250 0.1875

DMU

0.5000 0.2500 0.0000 0.0000 0.2500

Table 2 shows the results of stages I and II obtained

using LINDO. From Table 2 it can be seen that the

efficient DMUs are DMU

and DMU

. This is due to

the optimal value of DMU

and DMU

which are

𝛿

∗

 0.1250 ; 𝛿

∗

 0.0625.

As another example, data from 50 banks was

provided. There are three inputs and 3 outputs. This

problem is solved by an improved two-stage DEA.

The optimal values of the first stage and the objective

function of the second stage are showed by the last

two columns of Table 3. There are eight DMU

whose

optimal value 𝛿

∗

0.

Table 3: Input and output of the 50 DMU

DMU

uts Out

uts Sta

e 1 Sta

e II

Empl. Cost Debt. Deposits Income Loan

𝜃



∗

𝛿

∗

DMU

32 161 446,869 551,768 2,068 1,209,876 1.0000 0.000005448

DMU

19 2,026 22,345 87,365 2,848 103,573 1.0000 0.000017493

DMU

14 1,456 12,830 50,206 2,755 208,456 0.8943

DMU

5 4,566 145 77,436 1,554 12,789 1.0000 0.000096789

DMU

18 1,324 21,567 24,794 1,638 45,790 0.6360

DMU

18 1,562 25,689 25,894 1,448 44,567 0.7862

DMU

16 1,468 54,243 95,804 1,578 80,942 0.6453

DMU

17 1,884 39,453 25,266 1,895 35,790 0.8543

DMU

9 1,636 12,456 28,885 1,572 55,782 1.0000 0.000036918

DMU

13 1,993 7,623 34,226 1,277 209,765 0.8764

DMU

8 1,934 34,562 87,990 1,445 45,674 1.0000 0.000032445

DMU

11 1,279 2,487 77,567 2,051 77,833 0.4325

DMU

17 2,426 11,453 45,698 2,745 50,975 0.5543

DMU

14 1,236 10,934 78,965 2,774 120,987 0.5547

A New Procedure of Two Stage Data Envelopment Analysis Model under Strict Positivity Restriction

1033

DMU

14 2,011 22,176 88,784 2,341 35,678 0.5722

DMU

7 2,894 26,832 33,489 1,090 58,542 0.3974

DMU

12 1,500 8,643 56,779 1,462 556,709 0.3894

DMU

9 1,475 3,411 69,055 1,572 450,097 0.4490

DMU

5 1,290 1,421 67,784 1,635 169,005 0.5768

DMU

6 2,094 3,744 92,675 1,725 33,789 0.5947

DMU

6 2,068 5,321 38,000 1,613 87,734 0.3462

DMU

8 2,848 31,589 65,470 2,025 56,733 0.5231

DMU

9 2,755 4,215 34,226 1,486 34,098 0.5279

DMU

8 1,554 65,782 87,990 3,566 66,990 0.4469

DMU

7 1,638 20,021 77,567 2,324 59,032 0.5103

DMU

9 1,448 25,072 95,804 4,572 133,456 0.3974

DMU

7 1,578 14,081 25,266 1,498 12,500 0.5478

DMU

7 1,895 16,702 28,885 1,874 31,567 0.4580

DMU

7 1,572 6,574 34,226 1,536 51,578 0.4356

DMU

6 1,277 5,432 87,990 1,984 76,890 0.5569

DMU

7 1,445 7,331 77,567 1,935 34,590 0.5021 0.000034526

DMU

7 2,051 2,361 45,698 1,289 98,004 0.4592

DMU

8 2,745 2,093 78,965 2,426 95,709 0.3678

DMU

9 2,774 2,100 88,784 1,236 39,056 0.5946 0.000033468

DMU

5 2,341 1,946 33,489 2,011 34,781 0.5793

DMU

7 1,090 1,421 56,779 2,894 72,890 0.5198

DMU

8 1,462 3,744 37,586 1,500 39,357 0.5356 0.000005549

DMU

6 1,572 5,321 77,895 1,475 55,490 0.5782

DMU

5 1,635 31,589 76,880 1,290 33,789 0.5583

DMU

9 1,725 4,215 34,556 2,094 87,734 0.5932

DMU

5 1,545 65,782 67,032 1,678 56,733 0.5435

DMU

6 1,792 25,689 87,004 1,568 65,470 0.5321

DMU

6 1,227 54,243 79,034 3,899 34,226 0.5271

DMU

7 1,967 39,453 66,503 1,257 87,990 0.4367

DMU

5 1,215 12,456 80,933 1,065 77,567 0.3561

DMU

5 1,157 7,623 79,335 1,803 95,804 0.5519

DMU

6 1,592 34,562 44,897 2,560 56,903 1.0000

DMU

5 1,278 2,487 76,449 1,774 103,466 0.4706

DMU

6 1,373 11,453 77,803 1,356 12,890 0.5335

DMU

4 1,298 10,934 69,067 2,508 33,390 1.0000

Table 4: The strictly positive weights of the efficient DMU

DMU

uts Out

uts Sta

e II

𝑣



∗

𝑣



∗

𝑣



∗

𝑢



∗

𝑢



∗

𝑢



∗

𝛿

∗

DMU

0.00000544 0.00007144 0.00002824 0.0000544 0.0000544 0.0000544 0.0000544

DMU

0.00001749 0.00037749 0.00001049 0.0001749 0.0003669 0.0001749 0.0001749

DMU

0.00009678 0.00009678 0.00078578 0.0002578 0.0009678 0.0009678 0.0009678

DMU

0.00003691 0.00003691 0.00014991 0.0003691 0.0003691 0.0003691 0.0003691

DMU

0.00003244 0.00003244 0.00093744 0.0003244 0.0003244 0.0003244 0.0003244

DMU

0.00003452 0.00003452 0.00067452 0.0003452 0.0003452 0.0003452 0.0003452

DMU

0.000033468 0.004733468 0.000031569 0.00033468 0.00033468 0.00033468 0.0003346

DMU

0.000005549 0.000391549 0.000101549 0.00005549 0.00005549 0.00005549 0.0000554

DMU with optimal value will then determine its

strictly positive weight value as shown in table 4.

From table 3 it can be seen that there are 8 DMUs that

have optimal values in stage 2, they are DMU

DMU

, DMU

and

DMU

. The eight efficient DMUs will then be

determined its strictly positive weight as shown by

Table 4.

ICOSTEERR 2018 - International Conference of Science, Technology, Engineering, Environmental and Ramiﬁcation Researches

1034

6 CONCLUSIONS

In this paper, in order to achieving strictly positive of

multipliers, we have to eliminate the role of non-

Archimedean (ε), in the DEA models. The model

used in this study is the multiplier form of the DEA-

CCR model. By considering that all weights on its

constraints are non-negative number.

In the first stage, we solved a new CCR model

to specifying the CCR-efficient DMUs using LINDO.

At this stage we get an efficient DMUs. In the second

stage we will evaluate the efficient DMU that we get

in the first stage to get the strictly positive value for

their inputs and outputs.

On the other hand, from the computational test

result using LINDO, we have to pay attention to gain

the accuracy of computations result. This method is

able to provide better efficiency results for cases of

positive strictly constraints. This will help decision

makers in making decisions on issues with scenarios

that correspond to the proposed model.

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A New Procedure of Two Stage Data Envelopment Analysis Model under Strict Positivity Restriction

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