Zhu J., Cook W.D. (Eds.) Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis

45

bounded data for

DMU

o

(and other DMUs). That is, we calculate the

following linear program (Zhu, 2001),

oj

dataexactBOryy

BOryyy

dataexactBIixx

BIixxx

j

ro

oj

rjj

roroo

oj

rj

j

io

oj

ijj

io

o

io

o

oj

ijj

oo

≠≥

∉≥

∈≥+

∉≤

∈≤+

=

∑

≠

0

); (

;

); (

;

~

tosubject

~

min

~

*

λ

λλ

λ

θλλ

θθ

(20)

Consider DMU1 in Table 3-1, we have

.5,4,3,2,0

(Revenue) 200060090012001000

)(Judgement

~

6.01 8.0 1 9.0

(Cost) 100200200150150

osubject t

~

min

5432

15432

5432

1

=≥

≥+++

≤+++

j

λ

λλλλ

θλλλλ

λλλλ

θ

Now, suppose the lower bound of judgment input for DMU1 can be

increased by

σ

, and the upper bounds of judgment input for other DMUs

can be decreased by

σ

′

. Based on Zhu (2001), if

σ

′

=

*

1

~

θ

= 25/9, then

DMU1 remains efficient (

*

1

π

=

*

1

θ

= 1). Thus, the efficiency stability region

for judgment input is larger than the range of [0.6, 0.7] for DMU1. When

DMU1 is under evaluation, any DMU1’s judgment input value within the

range of [0.6, 0.7] is an optimal solution such that

*

1

π

=

*

1

θ

= 1 remains true.

Table 3-3 reports the optimal values of judgment input by using model (20).

As a result, the (multiple) optimal solutions on the bounded input can be

retrieved for the variable-alternation algorithm.

We next convert the ordinal data into a set of exact data using three

DMUs, namely, DMU4, DMU3 and DMU5, in Table 3-1. We have (i)

DMU4 has the lowest rank. We use “1” as the satisfaction output value for

all DMUs; (ii) DMU3 has the highest rank. We use “1” for the satisfaction

output value for DMU3 and use 0 for other DMUs; and (iii) DMU5 is ranked

Chen & Zhu, Interval and Ordinal Data

46

third. We use “1” for the satisfaction output value for DMUs 2, 4 and 5 and

use “0” for other DMUs.

Table 3-3. Alternative Optimal Solutions

DMU Under Evaluation

DMU1 DMU2 DMU3 DMU4 DMU5

DMU1 [0.6, 0.7] 0.7 0.7 0.7 0.7

DMU2 [0.8, 0.9] 0.8 [0.8, 0.9] [0.8, 0.9] [0.8, 0.9]

DMU4 [0.7, 0.8] [0.7, 0.8] [0.7, 0.8] 0.7 [0.7, 0.8]

Table 3-4. Converting Ordinal Data Into Exact Data

DMU under evaluation

Satisfaction DMU1 DMU2 DMU3 DMU4 DMU5

y

21

1 1 0 1 1

y

22

0 1 0 1 0

y

23

1 1 1 1 1

y

24

0 0 0 1 0

y

25

0 1 0 1 1

Table 3-5. MIDEA Results

Efficiency score slack

Referent DMU

DMU

*

o

θ

Revenue

Cost

*

j

λ

1 1 0

0

*

1

λ

= 1

2 0.875 1000

31.25

*

1

λ

= 1

3 1 0

0

*

3

λ

= 1

4 1 1100

100

*

1

λ

= 1

5 0.7 1400

40

*

1

λ

= 1

Table 3-4 reports the set of exact data for the satisfaction output across all

five DMUs when a specific DMU is under evaluation. Table 3-5 reports the

results from model (13) when we use the exact data from Table 3-4. It can be

seen that both MIDEA approaches yield the identical efficiency scores. The

standard DEA approach indicates that DMUs 2, 4 and 5 have non-zero slack

values.

3.4 Application

This section applies this standard DEA-based MIDEA approach to the 33

telephone offices in Kim, Park and Park (1999). Table 3-6 reports the data

with man power (

1

x ), operating costs (

2

x ) and number of telephone lines

(

3

x ) as the three inputs and local revenues (

1

y ), long distance revenues

(

2

y ), international revenues (

3

y ), operation/maintenance level (

4

y ) and

customer satisfaction (

5

y ) as the five outputs. Note that

4

y and

5

y are

ordinal data. With respect to Kim, Park and Park (1999), the current paper

Chapter 3

47

reports

4

y differently with “1” for the worst and “33” for the best (“5” for

the best in

5

y ), since larger output values are preferred in DEA.

Table 3-6. Data for Telephone Offices

DMU

1

x

2

x

3

x

1

y

2

y

3

y

4

y

a

5

y

b

1 239 7.03 158 47.1 16.67 34.04 28 2

2 261 3.94 163 37.47 14.11 19.97 26 3

3 170 2.1 90 20.7 6.8 12.64 19 3

4 290 4.54 201 41.82 11.07 6.27 23 4

5 200 3.99 140 33.44 9.81 6.49 30 2

6 283 4.65 214 42.43 11.34 5.16 21 4

7 286 6.54 197 47.03 14.62 13.04 9 2

8 375 6.22 314 55.48 16.39 7.31 14 1

9 301 4.82 257 49.2 16.15 6.33 8 3

10 333 6.87 235 47.12 13.86 6.51 6 2

11 346 6.46 244 49.43 15.88 8.87 18 2

12 175 2.06 112 20.43 4.95 1.67 32 5

13 217 4.11 131 29.41 11.39 4.38 33 2

14 441 7.71 214 61.2 25.59 33.01 16 3

15 204 3.64 163 32.27 9.57 3.65 15 4

16 216 3.24 154 32.81 11.46 9.02 25 2

17 347 5.65 301 59.01 17.82 8.19 29 1

18 288 4.66 212 42.27 14.52 7.33 24 4

19 185 3.37 178 32.95 9.46 2.91 7 2

20 242 5.12 270 65.06 24.57 20.72 17 1

21 234 2.52 126 31.55 8.55 7.27 27 2

22 204 4.24 174 32.47 11.15 2.95 22 3

23 356 7.95 299 66.04 22.25 14.91 13 2

24 292 4.52 236 49.97 14.77 6.35 12 3

25 141 5.21 63 21.48 9.76 16.26 11 2

26 220 6.09 179 47.94 17.25 22.09 31 2

27 298 3.44 225 42.35 11.14 4.25 4 2

28 261 4.3 213 41.7 11.13 4.68 20 5

29 216 3.86 156 31.57 11.89 10.48 3 3

30 171 2.45 150 24.09 9.08 2.6 10 5

31 123 1.72 61 11.97 4.78 2.95 5 1

32 89 0.88 42 6.4 3.18 1.48 2 5

33 109 1.35 57 10.57 3.43 2 1 4

Source: Kim et al. (1999).

a

Ordinal ranks (33 = the best; 1 = the worst).

b

Five ordinal scales (5 = the best;

1 = the worst).

Chen & Zhu, Interval and Ordinal Data

48

Table 3-7. Exact Data When DMU29 is under Evaluation

Weak ordinal relations Strong ordinal relations Ratio bounded data

DMU

4

y

5

y

4

y

5

y

4

y

5

y

1 33 0 5.42743 0 5.42743 0.45662

2 33 33 4.74053 1 4.74053 1

3 33 33 2.95216 1 2.95216 1

4 33 33 3.86968 1.68000 3.86968 1.68000

5 33 0 6.21387 0 6.21387 0.45662

6 33 33 3.37993 1.68000 3.37993 1.68000

7 33 0 1.50073 0 1.50073 0.45662

8 33 0 2.10485 0 2.10485 0.20850

9 33 33 1.40255 1 1.40255 1

10 33 0 1.22504 0 1.22504 0.45662

11 33 0 2.75903 0 2.75903 0.45662

12 33 33 7.11426 2.82240 7.11426 2.82240

13 33 0 7.61226 0 7.61226 0.45662

14 33 33 2.40985 1 2.40985 1

15 33 33 2.25219 1.68000 2.25219 1.68000

16 33 0 4.43040 0 4.43040 0.45662

17 33 0 5.80735 0 5.80735 0.20850

18 33 33 4.14056 1.68000 4.14056 1.68000

19 33 0 1.31080 0 1.31080 0.45662

20 33 0 2.57853 0 2.57853 0.20850

21 33 0 5.07237 0 5.07237 0.45662

22 33 33 3.61653 1 3.61653 1

23 33 0 1.96715 0 1.96715 0.45662

24 33 33 1.83846 1 1.83846 1

25 33 0 1.71819 0 1.71819 0.45662

26 33 0 6.64884 0 6.64884 0.45662

27 33 0 1.07000 0 1.07000 0.45662

28 33 33 3.15882 2.82240 3.15882 2.82240

29 33 33 1 1 1 1

30 33 33 1.60578 2.82240 1.60578 2.82240

31 33 0 1.14490 0 1.14490 0.20850

32 0 33 0 2.82240 0.90909 2.82240

33 0 33 0 1.68000 0.82645 1.68000

Chapter 3

49

Table 3-8. Efficiency Scores without Weight Restrictions

Weak ordinal relations Strong ordinal relations Ratio bounded data

DMU efficiency RTS efficiency RTS efficiency RTS

1 1 CRS 1 CRS 1 CRS

2 1 CRS 1 CRS 1 CRS

3 1 CRS 1 CRS 1 CRS

4 0.94322 DRS 0.88097 DRS 0.86641 DRS

5 1 DRS 0.99714 IRS 0.98523 IRS

6 0.91747 DRS 0.85080 DRS 0.83543 DRS

7 0.87524 DRS 0.87167 DRS 0.86893 DRS

8 0.72117 DRS 0.72117 DRS 0.72117 DRS

9 0.88812 DRS 0.83096 DRS 0.82752 DRS

10 0.76098 DRS 0.75611 DRS 0.75260 DRS

11 0.78351 DRS 0.77986 DRS 0.77699 DRS

12 1 CRS 1 CRS 1 CRS

13 1 CRS 1 CRS 1 CRS

14 1 CRS 1 CRS 1 CRS

15 1 CRS 0.85782 DRS 0.84854 DRS

16 1 CRS 0.96872 IRS 0.94574 IRS

17 0.91257 DRS 0.88730 DRS 0.86063 DRS

18 0.94468 DRS 0.85697 DRS 0.84235 DRS

19 1 CRS 0.79177 IRS 0.78686 IRS

20 1 CRS 1 CRS 1 CRS

21 1 CRS 1 CRS 1 CRS

22 1 CRS 0.80056 IRS 0.78029 IRS

23 0.86482 DRS 0.85685 DRS 0.85363 DRS

24 0.95990 DRS 0.90019 DRS 0.89546 DRS

25 1 CRS 1 CRS 1 CRS

26 1 CRS 1 CRS 1 CRS

27 0.97963 DRS 0.97963 DRS 0.97858 DRS

28 0.97363 DRS 0.90326 DRS 0.88385 DRS

29 0.95283 DRS 0.81847 IRS 0.81728 IRS

30 1 CRS 0.93254 DRS 0.92583 DRS

31 1 CRS 0.79833 IRS 0.79414 IRS

32 1 CRS 1 CRS 1 CRS

33 1 CRS 0.88997 IRS 0.88288 IRS

We set M equal to 33 for the two outputs in weak ordinal relations. Based

upon (18) and (19) and Theorem 1, we can have a set of exact data when a

telephone office is under evaluation. (The second and third columns of

Chen & Zhu, Interval and Ordinal Data

50

Table 3-7 report a set of exact data on

4

y and

5

y when DMU29 is under

evaluation.)

We use model (13) to analyze the efficiency, because in addition to the

efficiency scores, it provides (i) the efficient projections; (ii) benchmarks

with magnitudes; and (iii) information on RTS classification

6

. The second

column of Table 3-8 records the efficiency scores under the assumption of

weak rankings. It can be seen that our approach yields the same efficient

DMUs and larger efficiency scores for inefficient DMUs compared to those

in Kim, Park and Park (1999) where so called strict rankings (strong ordinal

relations) are assumed

7

3.5 Converting the strong ordinal data and ratio

bounded data into a set of exact data

Recall that

rk

y -

1, −kr

y ≥

χ

and

ik

x -

1, −ki

x ≥

χ

are not valid forms to

represent strict rankings under model (10). We propose the following correct

and valid modifications when

k

DMU is under evaluation

1, +kr

y ≥

r

χ

kr

y

,

(

r

χ

> 1) and

ik

x ≥

i

η

1, −ki

x (

i

η

> 1)

(21)

Since (21) is units invariant, it allows scale transformations.

1, +kr

y ≥

r

χ

kr

y

,

and

ik

x ≥

i

η

1, −ki

x

in (10) are equivalent to

1, +kr

Y

≥

r

χ

kr

Y

,

and

ik

X ≥

i

η

1, −ki

X in (11) respectively, if the scale-transformation and

variable-alternation based approach is used. Note that (21) may allow all

data equal to zero. However, the proposed method of finding exact data does

not allow such cases to occur.

Based upon the discussion on converting weak ordinal data into a set of

exact data, we can set

rk

y = 1 and

ik

x = 1, and further we have

8

⎪

⎩

⎪

⎨

⎧

=

−

)11(

)1(

, …, k-j = for DMU x

, …, nj = k+U for DM y

j

kj

iij

j

kj

rrj

η

χ

when

r

χ

and

i

η

are given.

Furthermore, parts of (8) and (9) actually represent strong ordinal

relations when

rj

L =

r

χ

or

ij

H

1

=

i

η

.

6

Because of the possible multiple optimal solutions, method described in Zhu and Shen

(1995) is used. (See Seiford and Zhu (1999) for a detailed discussion on RTS estimation.)

7

This indicates that optimal solution is not achieved in Kim, Park and Park (1999). As noted

in Zhu (2003a;b), this is due to the fact that incorrect forms of ordinal relations are used in

Kim, Park and Park (1999).

8

We can easily select a set of exact data for

rj

y (j = 1, …, k) and

ij

x (j = k+1,…, n). For

example, we can set these

rj

y very close to zero and meanwhile (22) is satisfied.

Chapter 3

51

In Kim, Park and Park (1999),

4

L = 1.07 for the forth output and

5

L =

8.04 for the fifth output. Thus, if only strong ordinal relations are assumed,

we have

1, +kr

y ≥

r

L

kr

y

,

or

r

kr

L

y

≥

+

,

1,

(r = 4, 5)

(22)

When

k

DMU is under evaluation, we let

rk

y = 1 and we have a set of

exact data consisting of (i)

rj

y =

kj

r

L

−

(r = 4, 5; j = k+1, …, n) and (ii)

rj

y =

j

ε

≈ 0 such that

1, +jr

y ≥

r

L

rj

y (j = 1, …, k-1). The fourth and fifth

column of Table 3-2 present a set of exact data on

4

y and

5

y when DMU29

is under evaluation and strong ordinal relations in (22) are imposed.

The fourth and fifth columns in Table 3-8 report the efficiency scores and

the RTS classification when (correct and valid) strong ordinal rankings are

assumed. It can be seen that 8 efficient DMUs under weak ordinal relations

become inefficient under strong ordinal relations.

Moreover, note that if we assume strong ordinal relations as in (22), too

much flexibility may still be allowed in

rj

y =

j

ε

≈

0 (j = 1, …, k-1).

Therefore, we introduce

kr

y

,

<

r

U

1, −kr

y

or

r

kr

U

y

≤

−1,

,

(r = 4, 5)

(23)

to further restrict the values on

rj

y (j = 1, …, k-1).

Ratio bounded data (22) and (23) can also be converted into a set of exact

data via the following two steps.

Step 1: Set

o

rj

y = 1 and

o

ij

x = 1

Step 2: We have bounded data for other DMUs:

rj

L <

rj

y <

rj

U and

ij

G <

ij

x

<

ij

H

(j ≠

o

j ) which can further be converted into exact data.

Step 1 is valid is because there are no other constraints on

o

rj

y and

o

ij

x .

However, if

o

rj

y

and

o

ij

x

can take values within certain ranges as given in

(3), we have two cases associated with step 1. (Case 1) If

o

j

DMU is under

evaluation, we set

o

rj

y =

o

rj

y and

o

ij

x =

o

ij

x . (Case 2) If

o

j

DMU is not

under evaluation, we set

o

rj

y

=

o

rj

y and

o

ij

x

=

o

ij

x

.

Now, when

k

DMU is under evaluation, if we set

rk

y = 1, then we have

rj

y =

jk

r

U

−

1

(j = 1, …, k-1) where r = 4 and 5 and

4

U = 1.10 and

5

U =

23.04. The last two columns of Table 3-7 report a set of exact data on

4

y

and

5

y when DMU29 is under evaluation and when ratio scale bounds are

introduced.

When ratio bounded data are imposed, the efficient DMUs stay the same

and the efficiency scores for inefficient DMUs only slightly drop (see the

Chen & Zhu, Interval and Ordinal Data

52

sixth column in Table 3-8) compared to those under strong ordinal relations.

Note that the objective of model (10) is maximization and better efficiency

ratings are obtained in the current study compared to those in Kim, Park and

Park (1999)

9

.

Turning to the RTS classifications in Table 3-8, note that when weak

ordinal relations are replaced by strong ones, the RTS classification of some

DMUs changes. However, the RTS classification stays the same when strong

ordinal relations are replaced by ratio bounded data. This may be due to the

fact that ratio bounded can be viewed as strong ordinal relations.

4. TREATMENT OF WEIGHT RESTRICTIONS

The above discussion and the proposed method assume that weight

restrictions related to imprecise data in forms (4)-(9) are not present. The

following theorem shows that adding weight restrictions related to imprecise

outputs and inputs in forms (4)-(9) does not change the efficiency ratings.

Thus, these particular weight restrictions are redundant and can be removed

before solving the model (10). As a result, the standard DEA model based

approach can be used.

Theorem 2: Suppose the fth input and the dth output are imprecise data

given by (4)-(9), and

*

o

π

is the optimal value to (10), then

*

o

π

remains unchanged if the following weight restrictions related

to the fth input and dth output are added into model (10)

f

r

f

RODOrf

β

μ

α

≤

∩∉

≤

),(

(f ∈DO or RO) (24)

d

i

d

B

RIDIif

A ≤

∩∉

≤

ω

),(

(d ∈DI or RI) (25)

where f(

•) is a function on

r

μ

(

i

ω

) related to exact outputs

(inputs).

[Proof]: See Zhu (2003a).

10

Theorem 2 indicates that the optimal value to model (10) is equal to the

optimal value to the following model (model (10) with weight restrictions

(24) and (25))

9

This implies that Kim, Park and Park (1999) did not provide the optimized results.

10

This theorem is true because (4)-(9) are units invariant.

Chapter 3

53

0,

),(

)(

1

0 ..

max

1

11

1

≥

∈≤

∩∉

≤

∈≤

∩∉

≤

Θ∈

=

≤−

=

+

−

=

==

=

∑

∑∑

∑

ir

d

i

d

f

r

f

rrj

iij

io

m

i

ij

m

i

irj

s

r

ro

s

r

ro

RIorDIdB

RIDIif

A

ROorDOf

RODOrf

y

x

xyts

y

ωμ

ω

β

μ

α

ω

ωμ

μπ

(26)

In other words, the same efficiency ratings can be obtained by either

solving model (10) or model (25). If one obtains the efficiency ratings under

model (10), the same efficiency ratings are obtained for model (26). The

method developed in the previous sections can be applied to solve model

(10) without affecting the efficiency ratings. As a result, model (26) is

solved indirectly.

Although we cannot set exact data for partial data in model (26), Theorem

2 provides an alternative where the efficiency ratings under (26) are obtained

via setting exact data in model (10) and solving model (10). i.e., the

proposed approach is not affected by the presence of weight restrictions,

since solving model (10) with proposed approach is equivalent to solving

model (26) directly

11

. To obtain the efficiency ratings under model (26), we

only need to solve model (10) via setting exact data in model (10), because

the weight restrictions represented by (24) and (25) are redundant.

Note that Kim, Park and Park (1999) incorporate the six sets of weight

restrictions. Of which two are given as

11

Note that the following two cases are different. Case I: Setting the variables by exact data in

model (10) and then finding the efficiency ratings. (This provides the same efficiency

ratings as those obtained from solving model (26) directly.) Case II: Setting the variables

by a set of exact data in model (26) and finding the efficiency scores under model (26)

(This leads to a different problem.) Case I represents the objective of the current study or

the objective of solving model (26).

Chen & Zhu, Interval and Ordinal Data

54

3.5 <

4

321

μ

++

< 8 and 7 <

5

321

μ

+

< 18

Based upon Theorem 2, these two restrictions are redundant constraints

in (1)

12

. Therefore, we only incorporate the remaining three sets of weight

restrictions specified in Kim, Park and Park (1999)

0.8 <

2

1

v

< 3,

2

3

v

< 1, 1 <

1

2

μ

< 2, 1 <

1

3

μ

< 2

The second and third columns of Table 3-9 report the efficiency scores

under ratio bounded data and weak ordinal relations, respectively, when the

above weight restrictions are incorporated in model (10) or (12). In this case,

model (12) is used. Only seven DMUs, namely DMUs 1, 12, 13, 20, 25, 26,

and 32, are efficient with ratio bounded data, and 13 with weak ordinal

relations.

We next use this data set to very Theorem 2. Table 3-10 provides two

sets of optimal solutions from model (12) when DMU2 is under evaluation.

The second and third columns give the exact data on

4

y and

5

y for all 33

DMUs, respectively. We have

*

1

μ

=

*

3

μ

= 0.006116,

*

2

μ

= 0.012233,

*

4

μ

=

0.188621 and

*

5

μ

= 0.069929 with

*

2

π

= 0.78247. In this case,

*

4

*

3

*

2

*

1

)(

μ

μμμ

++

=

0.129705 and

*

5

*

3

*

2

*

1

)(

μ

μμμ

++

= 0.349855.

Now, we let

4

γ

= 4 •

)(

*

3

*

2

*

1

*

4

μμμ

μ

++

= 30.83921 and

5

γ

= 8

•

)(

*

3

*

2

*

1

*

5

μμμ

μ

++

=

22.86662. We next multiply the two previous outputs by 30.83921 and

22.86662, respectively and obtain a new set of exact data on

4

y and

5

y (see

the fourth and fifth columns in Table 3-10). From model (11), we have

*

1

~

μ

=

*

3

~

μ

=

*

4

~

μ

= 0.006116,

*

2

~

μ

= 0.012233, and

*

5

~

μ

= 0.003058 with

*

2

π

=

0.78247. In this case,

*

4

*

3

*

2

*

1

~

~~~

μ

μμμ

++

= 4 and

*

5

*

3

*

2

*

1

~

~~~

μ

μμμ

++

= 8. This numerical

example indicates that the weight restrictions related to the ranking data are

redundant in model (12) (or model (10)) and do not affect the proposed

approach.

12

In this case, f(•) =

∑

=

3

1r

r

μ

.

Chapter 3

Zhu J., Cook W.D. (Eds.) Modeling Data Irregularities and Structural Complexities in Data Envelopment Analysis

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