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Chapter 16

DATA ENVELOPMENT ANALYSIS WITH

MISSING DATA

A Reliable Solution Method

Chiang Kao

and Shiang-Tai Liu

Department of Industrial and Information Management,

ational Cheng Kung University,

Tainan 701, Taiwan, Republic of China, ckao@mail.ncku.edu.tw

Graduate School of Business and Management, Vanung University, Chung-Li, Tao-Yuan

320, Taiwan, Republic of China, stliu@vnu.edu.tw

Abstract: In data envelopment analysis (DEA), the input and output data from all of the

decision making units (DMUs) to be compared are required. If, for any reason,

some data are missing, then the associated DMU must be eliminated to make

the approach applicable. This study proposes a fuzzy set approach to deal with

missing values. The value of a DMU in an input (or output) which is missing

is represented by a triangular fuzzy number constructed from the values of

other DMUs in that input (or output). A fuzzy DEA model is then used to

calculate the efficiencies, which are usually also fuzzy numbers. We use a

problem with complete data to investigate the effect of this approach when

1%, 2%, and 5% of the values are missing. While the conventional DMU-

deletion method will overestimate the efficiencies of the remaining DMUs, the

fuzzy set approach produces results which are very close to those calculated

from complete data. The average error in estimating the true efficiency is less

than 0.3%. Most importantly, the fuzzy set approach is able to calculate the

efficiencies of all DMUs, including those with some values missing.

Key words: Data envelopment analysis, efficiency, missing data, fuzzy set

1. INTRODUCTION

Since the pioneering work of Charnes et al. (1978), data envelopment

analysis (DEA) has been widely studied from both the theoretical and

292

practical points of view. Different models have been developed to measure

the efficiency of a group of decision making units (DMUs) which utilize the

same inputs to produce the same outputs under different conditions.

Applications for different types of organizations have also been reported

(Seiford 1996, 1997, Cooper et al. 2000).

The basic idea of DEA is to allow each decision making unit to use

different virtual multipliers, the most favorable, in calculating its relative

efficiency expressed as the ratio of aggregated output to aggregated input. In

selecting the multipliers, it is required that the ratio of aggregated output to

aggregated input calculated from the multipliers selected by the DMU

concerned should not exceed 1.0 for all DMUs. Let Xij, j=1,…, s and Yik,

k=1,…, t denote the jth input and kth output, respectively, of DMU i, i=1,…,

n. Banker et al. (1984) develop the following mathematical program to

calculate the efficiency of DMU r:

)( .max

∑∑

rjjrk

XvvYuE

niXvvYu

ijjik

,...,1 ,1)( s.t.

=≤+

∑∑

(1)

,0,

vvu

>ε≥ unrestricted in sign,

where u

and v

are the multipliers associated with output k and input j,

respectively, to be determined from (1) and ε is a small non-Archimedean

number (Charnes et al. 1979, Charnes and Cooper 1984) imposed to avoid

DMU r from assigning zero weight to unfavorable factors. E

is the relative

efficiency of DMU r, where E

=1 indicates efficiency and E

<1 inefficiency.

Model (1) is a linear fractional program which can be solved by

transforming to a linear program (Charnes and Cooper 1962) and utilizing

any linear programming solver. The underlying assumption of this model is

variable returns to scale. If v

is set to zero, then the model boils down to one

under the assumption of constant returns to scale

As indicated by its name, data envelopment analysis is based on data. To

calculate the relative efficiency, data from all DMUs are required. If any

observation of a DMU in the group is missing, then this DMU must be

deleted from the group in order to calculate the efficiency of all other

DMUs. A consequence of this is overestimation of the efficiency of some

DMUs, because the number of DMUs for comparison is decreased. As more

DMUs are deleted due to lack of data, the resulting efficiencies will be

biased high to a larger extent. Therefore, it is desirable to keep those DMUs

with some observations missing in the group by making some amendments.

The problem of missing data has been widely discussed in statistical

analysis (Allison 2002, Rubin 2004, Schafer 1997). There are formulas for

Chapter 16

293

estimating missing values in randomized blocks, Latin squares, etc.

(Snedecor and Cochran 1967). In general applications, a probability

distribution such as triangular or beta is usually assumed for the data which

are missing (Law and Kelton 1991). The studies of Simar and Wilson (1998,

2000) use this idea, although they are not developed for dealing with missing

data. To the knowledge of the authors, the only journal article which

proposes a methodology for handling missing data in DEA is the one by Kao

and Liu (2000b). Instead of assuming a probability distribution for missing

values, a membership function of fuzzy set theory is assumed. The fuzzy set

approach allows the derivation of the membership function of the efficiency

from the membership functions of the input and output data (Kao and Liu

2000a). With the membership functions of the efficiencies, generalized

means can be calculated and rankings of the DMUs can be subsequently

made. This approach has been successfully applied by Kao and Liu (2000b)

to measure the efficiency of 24 university libraries in Taiwan, where 3 out of

144 observations were missing. Since the data in that study were really

missing, whether the calculated efficiencies are correct or not cannot be

verified. In this study we select a problem with complete data from the DEA

literature, randomly delete some data to make it incomplete, and modify the

approach of Kao and Liu (2000b) to calculate the efficiency. The results

from the incomplete data sets are compared with those from the complete

data set to investigate the reliability of the fuzzy set approach.

In the following, the fuzzy set approach of Kao and Liu (2000b) for

handling missing data is first briefly reviewed. Secondly, different deletion

rates are applied to the complete data set of a problem to calculate the

efficiency from the incomplete data sets by using a modified approach of

Kao and Liu (2000b). The efficiencies and ranks calculated from incomplete

data are then compared with those calculated from the complete data.

Finally, conclusion is made based on the comparison.

2. THE FUZZY SET APPROACH

When there are data in a group of DMUs which, for any reason, are

missing, their values must be estimated so that the DEA approach can be

applied. The simplest way is to find the most likely value to represent the

missing data. However, how to find a representative value is a problem. In

statistical analysis it is usually to assume that the missing data follows a

probability distribution. Unfortunately, this approach involves sophisticated

mathematical derivation and cumbersome simulation analysis, and none of

the existing studies in DEA has successfully accomplished this task.

Alternatively, Kao and Liu (2000b) tackle this problem by assuming the

Kao & Liu, Data Envelopment Analysis with Missing Data

294

missing data to be fuzzy numbers and then applying the fuzzy DEA

approach of Kao and Liu (2000a).

)

(

.max

∑∑

rjjrk

XvvYuE

,...,1 ,1)

(

s.t.

niXvvYu

ijjik

=≤+

∑∑

(2)

,0,

vvu

>ε≥ unrestricted in sign.

Since the missing data

and

are fuzzy numbers, the

efficiency

calculated from them must be fuzzy as well. The characteristic

of a fuzzy number is described by its membership function. Let

and

μ denote the membership functions of

and

, respectively.

According to Zadeh’s extension principle (Yager 1986, Zadeh 1978), the

membership function for

can be expressed as:

)}(|,, ),( ),(.{min.sup)(

yxEzkjiyxz

rik

ikijr

∀

, (3)

where E

(x,y) is the efficiency of DMU r calculated from Model (1).

The membership function can be viewed either vertically or

horizontally. Vertically,

)(

μ shows the possibility of occurrence for the

value z, while horizontally, at a level α in the range of 0 and 1, it shows the

range of values whose possibility of occurrence is greater than or equal to α.

In fuzzy set terminology it is called α-level set, or α-cut, which is expressed

as:

])(,)[()(

ijij

XXX

ααα

= (4a)

},)(|)

({.[min

αμ

≥∈=

ijijx

xXSx

}])(|)

({max.

αμ

≥∈

ijijx

xXSx

])(,)[()(

ikik

YYY

ααα

= (4b)

},)(|)

({.[min

αμ

≥∈=

ikiky

yYSy

}])(|)

({max.

αμ

≥∈

ikiky

yYSy

where

)

(

XS and )

(

YS are the supports of

and

, respectively.

The membership function

can be constructed by deriving its α-cut at

different α values. According to (3),

)(

μ is the minimum of )(

Chapter 16

295

and

)(

, ∀ i, j, k. We need ,)( ,)(

≥

≥μ

ikij

and at

least one

)(or )(

ikij

equal to α, kji ,,

∀

, such that

)( yx,

Ez = to satisfy α=μ )(

. Furthermore, all α-cuts form a nested

structure with respect to α (Zimmermann 1996); viz.,

[

)(

] ⊆ [

)(

] and [

)(

] ⊆

[

)(

], for

≤α<α<

. To find the membership function

μ , it suffices to find the lower and upper bounds of the α-cut of

which, based on (3), can be solved as:

.min)(

)()(

kji

YyY

XxX

ikik

ijij

∀

≤≤

αα

⎪

⎩

⎪

⎨

⎧

>ε≥

=≤+

∑∑

sign. in edunrestrict ,0 ,

,,...,1 ,1)( s.t.

)/( max.

vvu

nixvvyu

xvvyuE

ijjik

rjjrk

(5a)

.max)(

)()(

kji

YyY

XxX

ikik

ijij

∀

≤≤

αα

⎪

⎩

⎪

⎨

⎧

>ε≥

=≤+

∑∑

sign. in edunrestrict ,0 ,

,,...,1 ,1)( s.t.

)/( max.

vvu

nixvvyu

xvvyuE

ijjik

rjjrk

(5b)

In measuring the relative efficiency of DMU

, its smallest value is

derived by setting the output level of this DMU and the input levels of all

other DMUs to their lowest possible values and setting the input level of

this DMU and the output levels of all other DMUs to their highest possible

values.Conversely, to find the highest relative efficiency of a DMU, one

will set the output level of this DMU and the input levels of all other DMUs

to their highest possible values and set the input level of this DMU and the

output levels of all other DMUs to their lowest possible values. Therefore,

the two-level mathematical model (5) can be simplified to the following

conventional one level model (Kao 2006)

See also Chapter 3.

Kao & Liu, Data Envelopment Analysis with Missing Data

296

))(()( .max)(

∑∑

αα

rjj

XvvYuE

()

1))(( s.t.

≤

∑

rjj

XvvYu

(6a)

riniXvvYu

ijj

≠=≤

∑

,,...,1 ,1))(()(

αα

,0,

vvu

>ε≥ unrestricted in sign.

))(()( .max)(

∑∑

αα

rjj

XvvYuE

1))(()( s.t.

≤

∑

rjj

XvvYu

αα

(6b)

riniXvvYu

ijj

≠=≤

∑

,,...,1 ,1))(()(

αα

,0,

vvu

>ε≥ unrestricted in sign.

The α-cut of

is obtained from (6) as ])(,)[()(

EEE

ααα

= and

the membership function of

is constructed from

)(

E at different α

values.

The most difficult part of this approach is the determination of the

membership function. Kao and Liu (2000b) use a triangular function to

represent the missing value, where the three vertices of the function are

represented by the smallest, largest, and median observations of the factor

corresponding to the missing value. As a result, different DMUs with

missing values occurring for the same factor have the same membership

function, which, intuitively, is not appropriate. In this study, we change the

left, right, and top vertices of the triangular function to the values

corresponding to the smallest, largest, and median ranks which have

appeared in other inputs (or outputs) for the DMU with missing values.

Consider a case of four inputs (outputs will be treated the same way), and the

value of the fourth input of a DMU is missing. Suppose the ranks for the

values of this DMU in the first three inputs are 3, 7, and 8. Then the values

corresponding to ranks 3, 8, and 7 of the fourth input are used as the left,

right, and top vertices, respectively, of the membership function. If,

excluding the inputs corresponding to the missing values, there are two

inputs left, then the ranks of the values in these two inputs will be used for

determining the left and right vertices, and the average rank for determining

the top vertex. If there is only one input left, then its rank will be used for

determining all three vertices. In this case, the fuzzy number degenerates to a

deterministic value. Finally, if there is no input left, then this DMU should

Chapter 16

297

be deleted from the group for comparison. Alternatively, one can use the

ranks in outputs for substitution.

Notably, when some data are missing, not only will the DMUs with

values missing have fuzzy efficiencies, but also other DMUs with complete

data. It is also possible for DMUs with some values missing to have

deterministic efficiencies. In this case the efficiency must be 1. All these are

dependent on the nature of the frontier facets, whether they are deterministic

or fuzzy.

3. A CASE ANALYSIS

Kao and Liu (2000b) applied the fuzzy set approach to measure the

efficiency of 24 university libraries in Taiwan where three values were

missing. Although the efficiencies have been calculated, there is no way to

verify whether the results are correct or not, and thus is no way to prove

whether this approach can produce correct results. For this reason, this study

uses a case with complete data and creates the condition of incomplete data

by deleting a number of observations to investigate how reliable the fuzzy

set approach is.

Table 16-1. Input and output data of the forest reorganization problem

Factor 1

2 3 4 5 6 7

DMU Input 1 Input 2 Input 3 Input 4 Output 1 Output 2 Output 3

1 67.55 82.83 44.37 60.85 26.04 85.00 23.95

2 85.78 123.98 55.13 108.46 43.51 173.93 6.45

3 80.33 104.65 53.30 79.06 27.28 132.49 42.67

4 205.92 183.49 144.16 59.66 14.09 196.29 16.15

5 51.28 117.51 32.07 84.50 16.20 144.99 0.00

6 82.09 104.94 46.51 127.28 44.87 108.53 0.00

7 123.02 82.44 87.35 98.80 43.33 125.84 404.69

8 71.77 88.16 69.19 123.14 44.83 74.54 6.14

9 61.95 99.77 33.00 86.37 45.43 79.60 1252.62

10 25.83 105.80 9.51 227.20 19.40 120.09 0.00

11 27.87 107.60 14.00 146.43 25.47 131.79 0.00

12 72.60 132.73 44.67 173.48 5.55 135.65 24.13

13 84.83 104.28 159.12 171.11 11.53 110.22 49.09

14 202.21 187.74 149.39 93.65 44.97 184.77 0.00

15 66.65 104.18 257.09 13.65 139.74 115.96 0.00

16 51.62 11.23 49.22 33.52 40.49 14.89 3166.71

17 36.05 193.32 59.52 8.23 46.88 190.77 822.92

The case to be investigated is the forest reorganization problem by Kao

and Yang (1992). That problem is considered by Seiford (1996) as one of the

most novel applications in DEA. Its data set is composed of 17 DMUs,

where each DMU uses four inputs to produce three outputs, with a total of

Kao & Liu, Data Envelopment Analysis with Missing Data

298

119 observations. Table 16-1 shows the data set of this problem. The

missing rates considered are 1%, 2%, and 5%. For a total of 119

observations, these rates correspond to 1, 2, and 6 observations. The

observations to be deleted are selected randomly. In order to make a

consistent analysis, the observations deleted in the case with the smaller

missing rate are also deleted in the case with the larger missing rate. This

type of experimental design provides a more accurate picture of the effect of

the missing rate on efficiency estimation.

The six observations selected randomly, as underlined in Table 16-1, are

located at (2,4), (3,6), (8,1), (9,1), (13,6), and (15,2). For the case of 1%

missing rate, we assume the observation at (2,4), i.e., the fourth input of the

second DMU, is missing. The values of DMU 2 in inputs 1, 2, and 3 are

ranked tenth, thirteenth, and fourteenth, respectively, in the corresponding

inputs. Therefore, the values corresponding to ranks 10, 13, and 14 in the

fourth input, which occur at DMUs 7, 6, and 11, respectively, are chosen as

the left, top, and right vertices of the triangular function, and the function

constructed is (98.80, 127.28, 146.43). By applying Model (6), one obtains

the α-cut of the fuzzy efficiency

at a specific α value, ])(,)[(

αα

The average of the central values of all α-cuts serves as an estimate of the

deterministic efficiency of DMU r. This average value can also be used for

ranking. The experience of Chen and Klein (1997) is that three or four cuts

are sufficient to obtain a good estimation. In this study we calculate the α-

cut at five α values: 0, 0.25, 0.5, 0.75, and 1.

When the observation at (2,4) is missing, with its value replaced by the

fuzzy number (98.80, 127.28, 146.43), there are five DMUs, viz., 1, 3, 6, 8,

and 12, whose efficiencies become fuzzy. In Table 16-2, the first pair of

columns, with the heading “0% missing,” shows the efficiencies and their

corresponding ranks calculated from complete data. There are ten efficient

DMUs and their ranks are not specified. The second pair of columns shows

the estimated efficiencies and their corresponding ranks when the value at

(2,4) is assumed missing. The total absolute difference between the original

efficiency and the estimated efficiency of the 17 DMUs, as shown at the

bottom of Table 16-2, is 0.0251. Its average is 0.0015. The absolute

difference of each DMU divided by the original efficiency is the error in

estimating the true efficiency. As shown at the bottom of Table 16-2, the

average error for the 17 DMUs is only 0.1597%, indicating an accurate

estimation. In addition to efficiency scores, their ranks are also very close to

the original ones. There are only two DMUs whose ranks are different from

the original case, with a total difference of 2 and an average difference of

0.1176.

For the case of 2% missing rate, we assume the observation at (3,6), in

addition to that at (2,4), is missing. Following the same procedure, we obtain

the triangular function (120.09, 131.79, 135.65) for the missing value at

Chapter 16

299

(3,6). This triangular function is constructed from the values of DMUs 10,

11, and 12, corresponding to ranks 8, 10, and 12, respectively, in the second

output. When the values at (2,4) and (3,6) are assumed fuzzy, there are six

DMUs, viz., 1, 3, 6, 8, 12, and 13, whose efficiencies are fuzzy numbers.

Note that although DMU 2 has a fuzzy observation at the fourth input, it has

a deterministic efficiency of 1. The estimates of the efficiencies and their

corresponding ranks are shown in the third pair of columns of Table 16-2.

Interestingly, the results are better than the case of 1% missing rate. The total

absolute difference between the original efficiencies and the estimated

efficiencies of the 17 DMUs is only 0.0088, with an average difference as

small as 0.0005. From the absolute difference of each DMU, we obtain

0.0598% as the average error in estimating the true efficiency. The rankings

are exactly the same as that of the original case. The reason is quite

straightforward. As more values are missing, the resulting efficiency of each

DMU is fuzzier. However, this does not imply that the generalized mean of

the fuzzy efficiency is farther away from the true efficiency.

Table 16-2. Results of the fuzzy set approach

0% missing 1% missing 2% missing 5% missing

DMU Eff. Rank Eff. Rank Eff. Rank Eff. Rank

1 0.8283 14 0.8340 14 0.8340 14 0.8359 14

2 1.0000 1.0000 1.0000 1.0000

3 0.9581 12 0.9773 11 0.9609 12 0.9718 11

4 1.0000 1.0000 1.0000 1.0000

5 1.0000 1.0000 1.0000 1.0000

6 0.9572 13 0.9573 13 0.9573 13 0.9582 13

7 1.0000 1.0000 1.0000 1.0000

8 0.7941 16 0.7941 16 0.7941 16 0.7993 16

9 1.0000 1.0000 1.0000 1.0000

10 1.0000 1.0000 1.0000 1.0000

11 1.0000 1.0000 1.0000 1.0000

12 0.8172 15 0.8173 15 0.8173 15 0.8173 15

13 0.7441 17 0.7442 17 0.7442 17 0.7610 17

14 0.9644 11 0.9644 12 0.9644 11 0.9644 12

15 1.0000 1.0000 1.0000 1.0000

16 1.0000 1.0000 1.0000 1.0000

17 1.0000 1.0000 1.0000 1.0000

Total absolute difference 0.0251 2 0.0088 0 0.0445 2

Average difference 0.0015 0.1176 0.0005 0 0.0026 0.1176

Average error 0.1597% 0.0598% 0.2786%

Finally, for the case of 5% missing rate, we assume the observations at

(8,1), (9,1), (13,6), and (15,2), in addition to those at (2,4) and (3,6), are

missing. The triangular fuzzy numbers associated with these four values are

Kao & Liu, Data Envelopment Analysis with Missing Data

300

(51.28, 82.09, 82.09), (51.28, 51.62, 67.55), (74.54, 120.09, 144.99), and

(82.44, 104.28, 193.32). Following the same procedure for missing rates of

1% and 2%, we obtain the efficiency estimates and their corresponding ranks

as shown in the last pair of columns of Table 16-2. This time there are

eleven DMUs whose efficiencies are fuzzy numbers. However, the total

absolute difference between the original efficiencies and the estimates is not

much, a value of 0.0445. The average difference for each DMU is only

0.0026 and the average error in estimating the true efficiency is 0.2786%.

Only two DMUs have ranks that are different from those calculated from the

original data.

In sum, the errors in estimating the true efficiency for three missing rates

are 0.1597%, 0.0598%, and 0.2786%. Despite the mild increasing trend, they

indicate a very accurate estimation.

4. A COMPARISON

Conventionally, when some values are missing, the corresponding DMUs

are deleted from the group for comparison. In the case of the 1% missing

rate, where the value at (2,4) is assumed missing, DMU 2 will be deleted,

and the remaining 16 DMUs will apply Model (1) to calculate their

efficiencies. Similarly, in the case of the 2% missing rate, DMUs 2 and 3

will be deleted. Finally, for the case of the 5% missing rate, DMUs 2, 3, 8, 9,

13, and 15 will be deleted, leaving 11 DMUs to calculate their efficiencies.

The first part of Table 16-3 shows the original efficiencies of the 17

DMUs and the relative ranks of the DMUs of three cases: 1%, 2%, and 5%

missing rates. The entries with a dash indicate that the corresponding DMUs

have been deleted from the group. Empty entries correspond to efficient

DMUs, so there is no need to show their ranks. The second part of Table 16-3

shows the relative efficiencies of 16 DMUs and their corresponding ranks

when DMU 2 is deleted. As expected, when a DMU is deleted, the new

efficiencies of the remaining 16 DMUs are greater than or equal to their

original counterparts. The total difference and average difference are 0.2063

and 0.0129, respectively, where the average is approximately nine times

larger than that of the fuzzy set approach. Recall that the average from the

fuzzy set approach is 0.0015. The error in estimation measured from the

absolute difference of each DMU is 1.5388%, which is approximately ten

times larger. Meanwhile, the total difference and average difference in ranks

are 6 and 0.375, respectively. Compared with the average difference of the

fuzzy set approach, the ratio is 3.19.

For the case of 2% missing rate, where DMUs 2 and 3 are deleted, the

results in Table 16-3 show that the efficiencies are the same as the case of

the 1% missing rate. This is because the additional DMU being deleted, viz.,

Chapter 16