model for dealing with dual-role factors in dea: extensions gongbing bi,jingjing ding,liang...

MODEL FOR DEALING WITH DUAL-ROLE FACTORS IN

DEA: EXTENSIONS

GONGBING BI,JINGJING DING,LIANG LIANG,JIE WU

Presenter : Gongbing Bi

School of ManagementUniversity of Science and Technology of China

Contents

• Introduction

• Cook’s dual-role model and extension

• Extended dual-role DEA model

• Conclusions

Introduction(1/2)

• In traditional application of DEA, all factors involved can be clearly classified as inputs or outputs. However, Beasley (1990, 1995) first addressed the factors that could be treated as both inputs and outputs. Factors having the property of both input and output are referred to as dual-role factors.

Introduction(2/2)

• Examples for dual-role factors

(1) The number of nurse trainees on staff

(2) Graduate students

(3) Research income

(4) etc.

Introduction(3/3)

• Beasley’s papers (1990, 1995), the author proposed a new framework to deal with dual-role factors in DEA. However, Cook et al. (2006) showed that the Beasley’s methodology might cause unreasonable evaluation results. In addition, the inconsistency in the logic to deal with flexible factors in input and output sides was explored. To correct the apparent flaws, Cook et al. (2006) provided new ways to handle flexible factors in DEA. In their paper, flexible factor was treated as both input and output simultaneously just as how Beasley did. However, Cook et al. recommended treating the flexible factor as being nondiscretionary in the input side when input-oriented DEA model was used.

Cook’s dual-role model and modification(1/4)

• Cook’s dual-role model

1 1

1 1

1 1

1 1

(1 )

. . 1, 1,2,...,

(1 )

{0,1}, , , 0, , ,

s L

r ro l l lor l

m L

i io l l loi l

s L

r rj l l ljr l

m L

i ij l l lji l

l r i l

y d wMax

v x d w

y d ws t j n

v x d w

d l v r i l


• Flaw in the method

If the DMU under evaluation is CCR efficient , then the classification of dual-role factor will depend on optimizing algorithm that we use to solve the previous model


• Property 1 if DMUo is characterized as efficient by CCR model when dual-role factors are excluded from consideration, DMUo is indifferent towards the classification of dual-role factor ,in other words, the efficiency of DMUo doesn’t change under any classification of dual-role factors.

• This is why the algorithm dependent classification could happen


• modification According to property 1, it is proposed that (1) DMUs are evaluated while dual-role factors are exclud

ed by CCR model before cook’s model is applied. At this step, the DMUs which are indifferent towards the classification of dual-role factors are known.

(2) the model is solved for each DMU that is not indifferent to the classification.

(3) the number of DMUs which have optimal solution d=0 is counted and the majority rule is applied to determine the status of each dual-role factor

Extended DEA model(1/11)

• Suppose there are n DMUs in a reference set. Each DMU has s outputs, denote Yk=(y1k,y2k,…,ysk)T , m inputs, denote Xk=(x1k,x2

k,…,xmk)T and L dual-role factors, denote W

k=(w1k,w2k,…,wLk)T.


• If dual-role factors are classified as inputs, the production possibility set having constant returns-to-scale characteristic

• If dual-role factors are classified as outputs, the CRS production possibility set

11 1 1

{( , , ) | , , , 0}n n n

Ti i i i i j

i i i

T X Y W X X Y Y W W

21 1 1

{( , , ) | , , , 0}n n n

Ti i i i i j

i i i

T X Y W X X Y Y W W


• Dual-role are considered as both inputs and outputs simultaneously, the production possibility set ( )1 2T T T

1 1 1

{( , , ) | , , , 0}n n n

Ti i i i i j

i i i

T X Y W X X Y Y W W


• The definition of production probability set T is also based on the following facts which are consistent with the property of dual-role factors as both inputs and outputs simultaneously:

i) If some dual-role factor of DMUo under evaluation is considered as relatively excess when compared with efficient frontier (such dual-role factor has input characteristic). This means that DMUo will gain an advantage if the dual-role factor is considered as output.

ii) If some dual-role factor of DMUo under evaluation is considered as relatively in short when compared with efficient frontier (such dual-role factor has output characteristic). This means that DMUo will gain an advantage if the dual-role factor is considered as input.


Based on the production

possibility set of dual-

role factors, a model that

deals with dual-role

factors that act as both

inputs and outputs is

constructed as model (1).

1

1

1

s.t. 1,...,

1,..., (1)

1,...

0

n

i ij oji

n

i ij oji

n

i il oli

i

Min

x x j m

y y j s

w w l L


• We proceed to find the relation between model (1) and CCR model in which all factors have clear classification. This is formulated in terms of the following result:

• Theorem 1 if DMUo achieves a score by model (1), then there exist at least one classification of dual-role factors under which the CCR efficiency is 1.


• Proof

Case 1: when dual-role factors are excluded, DMUo is evaluated as efficienct by CCR model.

According to Property 1, it is easy to know that Theorem 1 holds since the efficiency score of DMUo under any classification of dual-role factors is 1


• Case 2: when dual-role factors are excluded (model (2)), DMUo is evaluated as inefficienct by CCR model.

• Lemma 1 All feasible solutions to model (2) with constitute a convex set C.

• Lemma 2 If we set ,the CCR efficiency is less than 1, where I is the index set of dual-role factors that are classified as input, O is that classified as output, and O and indicate the partition of dual-role factors that satisfy the following condition:

1

,I I O O M

1

1

1

.

, {1,..., }.

n

i il oli

n

i il oli

n

i il oli

w w l I

w w l O

w w l M

I O M I O M D L


where is any point belonging to C • For each partition i.e. I and O are disjointed and the

union gives D={1,…,L}, we define sub set of convex set C as follows where i takes on values in .

• Lemma 3 If DMUo is evaluated as inefficient by CCR model, then Ci is nonempty for any .

• Lemma 4 If is nonempty for any , then there exists a point such that

1 2( , ,..., , )Tn

{1,...,2 }L

1 21 1

{ ( , ,..., , ) | . }n n

i n i il ol i il oli i

C x C w w l I w w l O

{1,...,2 }Li

iC {1,...,2 }Li1 2( , ,..., , )Tnx C

1

1...2n

Li il ol

i

w w l


• Proof of Case 2: Now we prove Theorem 1 by contradiction. Assume that Theorem 1 were not true in Case 2 . By Lemma 3, is nonempty for any. According to Lemma 4, there exists a point x that belongs to C such that .This contradicts the assumption that the optimal value to model (1) is 1. So Theorem 1 holds.

iC

1

1...2n

Li il ol

i

w w l

Extended DEA model(11/11)• The converse of previous the

roem need not be ture.• Example Consider a set of DMUs sho

wn in Table 1. Table 2 shows the results of all DMUs under different classification of dual-role factors and model (1). From the results of DMU4, we know that the unit is characterized as efficient if dual-role factor is classified as input. However, if the dual-role factor is treated as both input and output simultaneously the efficiency score by model (1) is 0.8.

Conclusions• Beasley (1990, 1995) first addressed the factors that could be treated as both

inputs and outputs. As pointed out by Cook et al. (2007), dual-role factors have become an important and very much under-researched topic.

• We first address the problem of uncertainty with Cook’s method. A property is introduced to provide the decision maker with more information on which DMU is in fact indifferent towards the classification of dual facts. An extended DEA model is proposed to incorporate dual-role factors in DEA literature when dual-role factors can be treated as inputs and outputs simultaneously. We study dual-role factor that serves as input and output simultaneously from production possibility set’s perspective, which is quite different from the existing methods. In addition, the relation between the proposed model with the traditional CCR model are explored in this paper.

• Dual-role factors, by its nature, have the property that for any DMUo evaluated the values of them should be equal to the counterparts of the efficient DMU that serves as benchmarking. One may think how we can improve the amount of dual-role factors aiming at improving the overall efficiency. However, we believe that properties such as returns-to-scale and congestion of DMU should be determined by some extended models incorporating the property of dual-role factors and the efficiency of each DMU concerning dual-role factors then can be studied from those perspectives. This is a direction for future research.

Thank you

model for dealing with dual-role factors in dea: extensions gongbing bi,jingjing ding,liang...

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