Non radial model of dynamic DEA with the parallel network structure

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701

Int. J. Data Envelopment Analysis (ISSN 2345-458X)

Vol. 1, No. 2, Year 2013 Article ID IJDEA-00126, 12 pages Research Article

Non radial model of dynamic DEA with the parallel

network structure

S.Keikha-Javana, M.Rostamy-Malkhalifehb*

(a) Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran.

(b) Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

Abstract

In this article, Non radial method of dynamic DEA with the parallel network structure is presented and is used for calculation of relative efficiency measures when inputs and outputs do not change equally. In this model, DMU divisions under evaluation have been put together in parallel. But its dynamic structure is assumed in series. Since in real applications there are undesirable inputs and outputs in the proposed model, the assumption of the existence of the intermediate products have been considered. After obtaining period–divisional efficiencies, by considering its weighted arithmetic mean, models are presented for the evaluation of period, divisional and overall efficiency for decision making unit

Keywords: dynamic data envelopment analysis – parallel network – overall efficiency – links and variable carry-overs.

1 Introduction

Data envelopment analysis is a Non parametric method for measuring relative efficiency of decision making units based on multiple inputs and outputs that was invented by Fare and universalized by Charnes et al [2]. One of the drawbacks of this model is the omission of the internal structure of the DMUs. For example, many companies and organizations are comprised of several divisions each one of these division which specific inputs & outputs are linked together and other divisions as well. Also, in real life the activities of such organizations are connected together in several different consecutive. So, for the assessment of the performance of these organizations and companies a model is needed to assess both the period efficiencies and divisional efficiencies and, eventually, the efficiency of overall system.

For the first time in2000, Fare and Grosskopf [5] presented an article under the title of "Network data envelopment analysis" in which the importance of network DEA was emphasized. After that, multiple

*_{Corresponding author, email:[email protected]}

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S.Keikha-Javan, et al /IJDEA Vol. 1, No. 2 (2013) 107-118

models of DEA with network structure were presented (for further studies one can refer to Costelli et al [1] and Chen [3], Cook et al [4] and Lin et al [7]). Also Tone et al [8], developed network DEA according to the SBM model. In this model links and carry-overs between divisions have specific groupings (good link, fixed link). In addition to the structure of desired DMU division, they paid attention to the connections between which this shows the development of network DEA model towards internal structure of the assessed DMUs with the variable links. Ton et al [9], proposed a combinatory model of two models of developed network DEA [8] and dynamic DEA for SBM model [10]. This combinatory model not only enables us in the assessment of overall efficiencies of desired DMU but also is a good guide for further analysis of the period efficiency and divisional efficiency of DMUs.

In this paper the Non radial method of dynamic DEA with parallel network structure has been presented with the assumption of the existence of various links & connections in the structure of the network and dynamic model. Obtaining overall efficiencies, period efficiencies, divisional efficiencies and period-divisional efficiencies in each period of time and in each part of DMUs’ decision making sub-units con be assumed as one of the merits of this method considering the volatile links & connections.

2 Dynamic DEA with parallel network structure

In dynamic DEA with parallel network structure we deal with decision making units n (DMUj, j=1,…, n). Each DMU is divided to q divisions (p=1,…, q) which are placed parallel together. Therefore overall system inputs are divided among all divisions and overall outputs results from the output of all divisions. In this paper their efficiencies and the desired DMU efficiencies in T time period (t=1,…,T) is examined.

The dynamic structure model consists of internal connections that transport intermediate products of t period to t+1 period. In the first period, we don’t have any connection from previous period besides, in the last period of T, we didn't consider any connection for the next period. We grouped these connections into two groups of desirable and undesirable. Desirable carry-overs are treated as outputs (transitional profit, net earned surplus) which we call them as “good” and undesirable carry-overs are treated as inputs (loss carried forward, bad debt, dead stock) which are named “bad” accordingly. So if we consider the number of all dynamic connections in this model as “h”, we will have:

(n-good) + (n-bad) =h

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 

. 1

0

, 1

T

, 1 t 1 1

1 1

1

1 .

(1)

p p

t t t t good t

o r rop dop p

r R d D

t t t t t bad i iop d dop i I d D

q

t t t t t good r rjp d djp

p r R d D i

q T

t d

t p

T

t

q p

E M ax u y z u

v x z

T q

u y z

T q

s t





 

 



 



    

 









 

 



  

 

 _ _

 

 _ _

  

 _ _

 



  



 



 , 1

0

t r

0

u 0 r 1 , , s

0 i 1 , , m

0 1 , ,

0 1 , , n bad

:

0

p p

t t t t t bad t i ijp d djp p

I d D

t i t d t d t

p

p p

v x z u

v

d n good

d u free



 





 

  

   



  

  

 







t ijp

x is input resource i to DMUj for division p in period t.

t rjp

y is output product r from DMUj for division p in period t.

( ,t t 1)good djp

z  is intermediate products d from DMUj at division p from period t to period t+1 with treated as output.

( ,t t 1) bad djp

z  is intermediate products d from DMUj at division p from period t to period t+1 with treated as input.

This model will be able to calculate the overall efficiency of the desired DMU according to sub-unit and dynamic connection after T time period.

3 Calculation of the overall efficiency based on the weighted mean of divisions and periods.

In normal state of DEA, to calculate the efficiency, we divide total weighted outputs to total weighted inputs of the desired DMU. Now that the internal structure DMU is so efficient, to calculate in terms of divisional efficiency & overall efficiency, we use the model of (Zhu et al. 2004) "overall efficiency calculation of decision making unit with network structure by the use of arithmetic mean of the divisional efficiency".

3.1 Period – divisional efficiencies

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S.Keikha-Javan, et al /IJDEA Vol. 1, No. 2 (2013) 107-118         . 1 0 , 1 . 1 0 , 1 . 1 p p p p p p p p

t t t t good t

r rop dop p

t r R d D

op t t t t t bad

i iop d dop

iI dD

t t t t good t

r rjp djp p

r R d D

t t t t t bad

i ijp d djp

iI dD

t d

u y z u M ax

v x z

u y z u v x z

s t                         



t r 0 (2)

u 0 r 1 , , s

0 i 1 , , m

0 1 , ,

0 1 , , n bad

: t i t d t d t p p p v

d n good d u free                

Linear form of model (2) is as follow:

        r . 1 0 , 1

. 1 , 1

0

u

1 (3)

0 .

op

p p

p p p p

t t t t t t good t r rop d dop p r R d D

t t t t t good t t t t t t bad r rjp d djp p i ijp d djp

r R d D i I d D

M ax s

u y z u

v x z

u y z u v x z

t                           



t 0

0 r 1, , s

0 i 1, , m

0 1, ,

0 1, , n bad

: t i t d t d t p p p v

d n good d u free                

Theorem 1: A) Model (3) is always possible. B) 0 t 1

op



 

Proof: A) if assume xt_kop minx_iopt    k i 1, ,m_p and t_k 1_t

kop

v x

 , Also if consider

1,

,

j j

n



 

v_kt 0    k i 1, ,m_p , _dt    0 d 1, ,nbad u_rt 0   r 1, ,s_p ,

0 d 1, , n good

t d

      and u₀t_p 0 , we have:

   

 

, 1 , 1

1 1 1 1

t ,t 1 good t ,t 1 good

t t t t t t

1 1 jp s s jp 1 1 jp n good n good jp , 1

t t t t t t

1 1 jp k kjp m m jp 1 1

1

u y u y β z β z

v x v x v x

p p

t t bad t t bad

t t t y t t t t

op k kop m m op op n bad n bad op

t t bad t

jp n

v x v x v x z z

z



                  

    , 1

0 0 t t bad

t t

badzn bad jp u p



  

(5)

Model (3) is always possible.

B) Due to the previous possible solution and this fact that in each optimum solution at least one of constraints multiplicand (Dual form) is as equality, we have:

   

n good

s m n bad

t , t 1 good t , t 1 bad

t t t t t t

r r p d dop i iop d dop 0

r 1 d 1 i 1 d 1

u y_o β z v x α z ut_p 0

 

 

   

    



   

n good

s m n bad

t , t 1 good t , t 1 bad

t t t t t t

r r p d dop 0 i iop d dop

r 1 d 1 i 1 d 1

u y_o β z ut_p v x α z 1

 

 

   

    



 

n good s

t , t 1 good

t t t

r rop d dop

r 1 d 1

u y β z 1





 

 



We know , 1

0

t t good djp z  

, 0

t rjp

y  , _dt 0, t r

u 0 . Then the sum of positive multi term is always

positive, so  

s ngood

t t t t , t 1 good r rop d dop r 1 d 1

u y β z 0

 

 

 



. We claim  

s n good

t t t t , t 1 good r rop d dop r 1 d 1

u y β z 0

 

 

 



because if we suppose

contradiction  

n good s

t t t t , t 1 good r rop d dop

r 1 d 1

u y β z 0





 

 



then we will have:  

n good s

t t t t , t 1 good r rop d dop

R 1 d 1

u y β z





 

 



. This isn’t

compatible with positive total.

0

t

1

op







.

Definition 1: if *t

1

op





, DMUo is called period-divisional efficient.

By noticing model (2) the period and division efficiency can be defined as convex linear combination.

3.2 Period efficiency

Period efficiency is actually the calculation of overall performance of the desired DMU divisions that can only be evaluated in a specific time period. For this reason it is called period efficiency (the single – period). Calculation of this efficiency is actually the calculation of the desired DMU considering the efficiency of all their divisions. We display it by



_ot . This efficiency can be evaluated by the weighted mean of period – divisional efficiency (t_op ). Which is defined as follows:

 

1

4

q

t p t

o op

p

w









Notice that w pweights shows the share of p division in the efficiency of the

desired period for the unit under evaluation and is

 

, 1

, 1 1

t t t t t bad

i iop d dop

p p

i I dD

p p

q

t t t t t bad i iop d dop i I d D p

p

v x z

w

v x z





 



 







 

 

 



. Due to this

definitionw p, 1

1 q

p p

w



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S.Keikha-Javan, et al /IJDEA Vol. 1, No. 2 (2013) 107-118  





 





 





 





, 1 0 1 , 1 1 , 1 0 1 , 1 1

1 t, p, j . : p p t o p p p p p p q

t t good

t t t t

r rop d dop p

r R d D

p q

t t bad

t t t

i iop d dop

i I d D

p

q

t t good

t t t t

r rjp d djp p

r R d D

p

t t bad

t t t

i ijp d djp

i I q

p d D

u y z u

v x z

s t

u y z u

v x M x z a                              

 



 



  t r 0

u 0 r 1 , , s

0 i 1 , , m

0 1 , ,

0 1 , , n bad

: 5 t i t d t d t p p p v

d n good

d u free                

Model (6) is linear model of from (5).

 





 





   





1 , 1 , 1 0 1

, 1 , 1

0 1

0 t, p, j

. : 1 1 1 p p t p p o q p

p p p p

q

t t good

t t t t

r rop d dop p

r R d D

p

q

t t good t t bad

t t t t t t t

r rjp d djp p i ijp d djp

r R d D i I d D

p

v x z

u y z u

s t

q

u y z u v x

M ax q z                               



 



 



 

t r 0

u 0 r 1, , s

0 i 1, , m

0 1, ,

0 1, , n bad

: 6 t i t d t d t p p p v

d n good d u free                

Theorem 2: A) Model (6) is always possible. B) 0 t 1

o



 

Proof: is similarly to theorem 1 proving.

Definition2: if * t

1 o

  , DMUo is called period efficient.

Corollary 1: *

1 t o

  if and only if



_op*t 1 at least in one of the divisions.

3.3 Divisional efficiency

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Also, if we want to calculate the performance of each one of desired DMU units in a long-time period, we need to calculate divisional efficiency. Calculation this performance is in fact accounted efficiency for each division in a long- time. We show divisional efficiency by _op and we define as the weighted

mean of period-divisional efficiency:

 

1

7

T t t op op

t w

 





, t

w weight show the share t period in the

performance of the desired division for decision making unit and is

( , 1)

1

p p

t t t t t i iop d dop t i I d D

T

t t t t t i iop d dop t i I d D

v x z

w

v x z



  

   

 



 

 

 



 



. Due to

the definition t

w we resulted

1

T t t

w 





.

By considering relation (7), divisional efficiency is defined like following:  





 





 





 





, 1

0 1

, 1 1

, 1

0 1

, 1 1

1 t, p, j

. :

p p

T

t t good

t t t t

r rop d dop p

r R d D

t

op T

t t bad

t t t

i iop d dop

i I d D

t

T

t t good

t t t t

r rjp d djp p

r R d D

t

t t bad

t t t

i ijp d dj

T

t

p

i I d D

M a

u y z u

v x z

s t

u y z u

v x x

z

 



 



 





 





 



 

 

 





 



 

 



 



 

t r

0

u 0 r 1, , s

0 i 1, , m

0 1, ,

0 1, , n bad

:

8

t i

t d

t p

p p

v

d n good d

u free 



  

   

(8)

 





 





   





1

, 1

0 1

, 1 , 1

0 1

0 t, p, j

. : 1

1 1

p p

T

t

p p p p

T

t t good

t t t t

op _{r R} r rop _d _D d dop p

t

T

t t good t t bad

t t t t t t t

r R d D i I d D

t

v x z

u y z u

s t

T

u y M ax

T

z u v x z



 

 





 



 



 

   



 

 



   







 



 



 

t r

0

u 0 r 1, , s

0 i 1, , m

0 1, ,

0 1, , n bad

:

9

t i

t d

t p

p p

v

d n good d

u free 



  

   

Theorem 3: A) this model is always possible. B)

0



_op



1

.

Proof: proving is similar to theorem 1.

Definition3: if*_op 1then DMUo is called divisional efficient.

Corollary 2:*_op 1if and only if



_op*t 1at least in one of the period.

3.4 Overall efficiency

By the use of (2),(5)and(8) models, the overall performance of decision making unit can be written as convex linear combination of parts and periods efficiency and period- divisional efficiency as

model (10).

 

1 1

10

q T

t t o p op

t p

E w 

 





. In this model (wtp) represents the share of p part of t period in the performance of the unit under evaluation which results from the following equation:

 

, 1

1

, 1 1

t t t t t bad i iop d dop

p p

i I d

p t

T

D

p t

p _q

t t bad

t t t

i iop d dop p i I d D

v x z

w

v x z





 





  





  



  

 



 



. According to the definition: 1 1

1

q T

t p t p

w

 





(9)

 





 





 





, 1 0 1 , 1 1 1 , 1 0 1 1 1 . : p p p p p p p p T

t t good

t t t t

r rop d dop p

r R d D

t

o _q

q

p T

t t bad

t t t

i iop d dop

i I d D

t p

q T

t t good

t t t t

r rjp d djp p

r R d D

t p

t t t

i ijp d d

i I d D

u y z u

E

v x z

s t

u y M

z u

v x z

ax                                              



  



 



 





 

t r 0 , 1 1 1

u 0 r 1, , s

0 i 1, , m

0 1, ,

0 1, , n bad

:

1 t, p, j 11

t i t d t d t p q T t p

t t bad jp

p p

v

d n good d u free                           

 

Model (11) can be changed in to model (12).

 





 





    1 1 , 1 0 1 , 1 1

, 1 , 1

0 . : 1 1 1 p p T p p t

p p p p

T

t t good

t t t t

o _{r R} r rop _{d D} d dop p

t

q

t t bad

t t t

i iop d dop

i I d D

p

t t good t t

t t t t t t t

r rjp d djp p i ijp d d

q

p

jp

r R d D i I d D

E u y z u

s t

v x z

T q

u y z u v x

ax T q z M                                         







 

t r 0 1 1

u 0 r 1, , s

0 i 1, , m

0 1, ,

0 1, , n bad

:

0 t, p, j 12

t i t d t d t p q T bad t p p p v

d n good d u free                          

 

Theorem 4: A) This model is always possible. B) *

0



E _o



1

.

Proof: is similar to previous.

Definition4: if *

1 o

E  then DMUo is called overall efficient.

Corollary3: *

1 o

E  if and only if t 1

op

  at least in one of the period and division.

(10)

Proof: suppose * * * *

(u v, , , ) is the optimum solution of model (12). Suppose posterior there exists another possible solution as ( , , ,u v  )such that * * * *

E (_o u v, , , )E u v_o( , , , )  . However

   

* * , 1 , 1

* * , 1

* * , 1 * *

0

1

) )

( ( 1

p p p p

p p

p p p p

t t t t t bad t t t t t bad

i iop d dop i iop d dop

i I d D i I d D

t t t t t t t bad

i i iop d d dop

i I d D

t t t t t good t t t t

r R d D i I d D

v x z v x z

v v x z

u y z u v x z

 

 

 

   



 



   

  

 

   

  





     

   

, 1 , 1 , 1

0

, 1 , 1

* * * *

0

( ) ( ) ( ) ( ) 0

p p p p

t t bad t t t t t good t t t t t t bad

r R d D i I d D

t t t t t t t good t t t t t t t bad

r r rjp d d djp i i ijp d d djp

r R d D i I d D

u y z u v x z

u u y z v v x z

 

   

  

   

 

   

   









     



Since all coefficients must be positive then we have:

* * * *

r ,

, (*)

0 0

t t t t t t t t

r r r r i i i i

t t t t t t t t

d d d d d d d d

u u u u v v v v i

d d

       

 

               

And because t , t t, 1good, t , t t, 1bad rjp djp ijp djp

y z  x z  are constant for both of the solution, then according to (*)

* * * *

E (_o u v, , , )E u v_o( , , , )  This is contradiction byE (_o u v*, *, *, *)E u v_o( , , , )  .

4 A numerical example

We applied this model to a dataset gathered from an insurance company in of exists in Taiwan. (For further studies you may refer to [6]).This company has five evaluation unite each one consists of two parts with an input, an output, a good intermediate product and a bad intermediate product. The performance of the company has been evaluated in two time periods. The data are given in table 1. Table1

Inputs & outputs and intermediate products data.

Yj.t2 Yj.t1

Zj.bad Zj.good

Xj.t2 Xj.t1

DMUj

890062 417620 7004112

4700020 709441

408026 77014798

1967097 670048

7027076 7791109

7074161 Division1

Division2 7

72876 794681 7029070

968147 904712

667980 1888412

4947020 918070

701661 7698002

171777 Division1

Division2 8

46878 448824 1277886

7007800 7174160

097706 41468298

1469469 474797

622222 9966094

7469008 Division1

Division2 4

711447 89741 802106

477980 417294

747680 4710277

479286 760876

74772 907480

707008 Division1

Division2 0

0727 077072 28707

7174762 70710

678489 78094

6101602 70708

992494 77664

8981101 Division1

Division2 7

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Table (2) consists of the performance values of each division of DMUj in each time period and also the performance of each division in overall time period.

Table2

Period–divisional efficiency - divisional efficiency

𝜹

𝒑

𝝆

_𝒑𝟏

𝝆

_𝒑𝟏

DMU

j

7.0000

Division1

Division2

7

026780

7.0000

022980

7.0000

029090

026906

Division1

Division2

8

7.0000

024907

7.0000

Division1

Division2

4

7.0000

021487

7.0000

Division1

Division2

0

7.0000

Division1

Division2

7

Table (3) is also consists of DMU’s under evaluation values. This performance is calculated by the efficiency of each division in each period. Also, each DMU’s overall efficiency value is given in this table.

Table3

Period efficiency - Overall efficiency

𝐄

𝛕

𝟐

𝛕

𝟏

DMU

j

7.0000

7

7.0000

026910

8

7.0000

4

7.0000

0

7.0000

7

5 Conclusion

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and inefficiency of its divisions. But in the usual method, if a unit was inefficiency, we looked for the causes of the desired unit in its sub-units.

Another feature of the presented model is that the same thing can be done for another organization that hasn't any similarity to the surveyed organization by using the parallel network dynamic DEA during the time period and then determined the growth of this organization during the time, eventually compared these two heterogeneous units according to performance growth over the various years. Because simple models of DEA, the basic requirement to compare the decision making units together was their homogeneity. It may also be valuable to investigate the Malmquist index under the Non radial model of dynamic DEA with the parallel network structure model.

References

[1] Castelli L, Pesenti R, Ukovich W. A classification of DEA models when the internal structure of the Decision Making Units is considered, Annals of Operations Research 173(1). (2010). 207-235. [2] Charnes A, Cooper .w .w, Rhodes, E. “Measuring the efficient of making unit”. European journal of operation research 2. (1978). 429-444.

[3] Chen CM. Network DEA: A model with new efficiency measure to incorporate the dynamic effect in production networks, European Journal of Operational Research 194(3) 2009, 687-99.

[4] Cook WD, Liang L, Zhu J. Measuring performance of two-stage network structures by DEA: A review and future perspective, Omega 38 (2010) 423–430.

[5] Fare R, Grosskopf S. Network DEA. Economic planning science (2000), 34, 35-49. [6] Kohkan, T. Network DEA. MSC. Un azad zahedan. (1389).

[7] Li Y, Chen Y, Liang L, Xie J. DEA models for extended two-stage network structures, Omega 40 (2012) 611–618.

[8] Tone K, Tsutsui M. Network DEA: A slacks-based measure approach, European Journal of Operational Research 197(1). (2009). 243-252.

[9] Tone K, Tsutsui M. Dynamic DEA with Network structure: A slacks-based measure approach,

Omega (2013) Accepted manuscript.