Equitable setting target by using DEA

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Equitable setting target by using DEA

Kavous Soleimani Damane1*, Reza Kargar2, Mohadeseh Rozpeykar3

(1) Islamic Azad University, Jiroft Branch, Jiroft, 7861736343, Kerman, Iran (2) Islamic Azad University, Gom Branch, Gom, Iran

(3) Mathematic teacher in Jiroft, Jiroft, Iran

Copyright 2015 © Kavous Soleimani Damane, Reza Kargar and Mohadeseh Rozpeykar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

One application of DEA to apply fixed costs to input sourcear or setting target in out put that is done by decision maker for units. At first, Mr Cook & Kress in troduced DEA as an approach for allocating fixed costs, many people introduced such models [3]. For completing this issue, one model for equitable allocation of costs or sorce was introduced by Wade D. Cook & Joe Zhu [6]. In this proposal obtaining a model for fixed equtable setting target in units was proposed in order to efficiency before & after the setting target nevar changed.

Keywords: DEA-equitable setting target, efficiency, equitable allocation. 1 Introduction

DEA is an un parametric way that define propotional evaluation of decision maker units with multiple inputs & out puts, this issue began in (1978 [1]) by Edward Rhodes on forals essay that for first time introduced an unparametric way. Results of this research by helping of charnez & coopor generalized C.C.R & it led to B.C.C & then anther basic models suchas additive models, S.B.M, conical vatio model,…

In part, varios articles for DEA were written, one of the most important application & models for decision maker units is cost allocation & fixed setting target (DMUS).

For cost allocating, one constraint was added as in put, one constaint was added as an out put in fixed setting target. At first DMU was evaluated and its efficiency were calculated. Models are presented in such way that their efficiency never changed befor & after allocating & setting target.

For first time, Kress & cook proposed amodel for allocating, also Dr. Jahanshahloo & Hosseinzadeh in their essay obtained a constraint in which allocations remain without any changes, so this model was used in many articles and essays & many searchers were done in this field [3-5].

Joe Zhu & Wade. D. Cook, presented a model. This model was not applicable in some allocations. This issue was proved by ruiyue line & then a complete model for allocation was presented. By using of ruyue

Available online at www.ispacs.com/dea

Volume 2015, Issue 1, Year 2015 Article ID: dea-00071, 7 Pages doi:10.5899/2015/dea-00071

Research Article

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lins model, we want to have a model with fixed output allocation or in other words fixed setting target for dicision marker units and at last we introduce a model for setting target [6].

2 C.C.R model

Suppose n decision making unit were evaluated & each of them cost m in put to produce s out put. For

decision making units j 1,...,n. input cost

X

_j



(

x

₁_j

,...,

x

_mj

)

to produce

Y

_j



(

y

₁_j

,...

y

_sj

)

and also

u

_r

in equivalent weight with r out put & v_i is equivalent weight with j input. DEA efficiency defined by ratio of sum of inputs & out puts.

C.C.R envelopment model by

o j

DMU

is as follow

(2.1)

Evaluable

DMU

_jo is workable when



1

o j



& equivalent lent slacks with constraints in optimal solution

were zero with out this, it is not workable. C.C.R with in put nature is as follow.

(2.2)

Envelopment model CCR with output nature is for obtaining the measurement of units that are evaluated

by

o j

DMU

is as follow

1

min

. .

1,..., .

0 1, 2,..., .

o

o o

o j

n

j ij j ij j

n

j rj rj j

j

s t

x

i

m

y

r

s

j

n

























1

1 1

1

max

,

. .

0 ,

1

0 1, 2,

,

0 1, 2,

,

o

o o

o

s

r rj r

s m

r rj i ij j

r i

m

i ij i

r

i

u y

s t

u y

v x

u

r

s

v

i

m



 





















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1

max

. .

1,..., .

0 1, 2,..., .

o

o o

j

n

j ij ij

j

n

j rj j rj

j

s t

x

i

m

y

r

s

j

n

























(2.3)

With are zero. Solution

. nd equivalent scales in optimal a

1 

o j



will be efficient if jo

DMU

Evaluable

out this, it is not workable multiplicative model. C.C.R with out put nature is as follow.

(2.4)

3 Setting target model

Suppose fixed P, common expectations of n, was DMU. If

p

_j was expected amount from

DMU

_j. So

we have the following relationship between �_�

(3.5)

at first, before set tins target with model 1 we calculate the DMU efficiency amount. We suppose the

efficiency amount o j



is

o j

DMU

. Now if we suppose

p

_j as

DMU

_j output,



s



1 

output for each DMU, so we can explain the following model that is CCR model with input nature as follow

(3.6)

1

1 1

1

min

. .

0 1, 2,..., ,

1

0 1, 2,..., ,

0 1, 2,..., .

o

o m

i ij i

s m

r rj i ij

r i

s r rj r

i r

v x

s t

u y

v x

j

n

u y

v

i

m

u

r

s



 





















1

n j j

p

P





1

min

. .

1,..., .

o

o o

o j

n

j ij j ij j

n

j rj rj j

n

s t

x

i

m

y

r

s

p























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If o j



~ was the calculated efficiency of this model. By comparing models (2.1) and (3.6), we can state the

following relationship   ~ 1

o

o j

j



. If

o j

DMU

was not efficient, it mean according to defined cons taint

in

o j

DMU

,~ 1

o j



remain un efficient because above setting target condition protect optimal condition &

the amount of objective function remain without changes. We suppose that  o j



is optimal solution of model (2.1), so we define N, f as follow

} Is Efficient or Boundary j

DMU

|

j

=}

F

{

Boundary Is not Efficient or

j

DMU

|

j

=}

N

According to dominate condition for DEA, we have the following relationship



j

jj

j

_o



F

o







,

0 



o j

DMU

We have For each

(3.7)

We can present the following model for obtaining the maximum amount of setting target for DMU with cons taints (3.5) and (3.7) and un negative variable

(3.8)

In order to present setting target between unefficient DMU, we can obtain the following model by adding ,

and replace above model with it 7)

3. to relation ship (

) (s_t tN

un negative variable an

1 *

max . .

,

, ,

, 0,

o j

t

j j t t

j F n

j j

t l

t N

j t

p s t

p s p t N p P

s S p s











  

 

 



(3.9) t

j j t

j F

p p t N







 



1

max

. .

,

0,

o j

t

p t

j F n

j j

j

p

s t

p

t

N

p

P



 









 



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, that is calculated as follow



N t

t

s

is optimal amount of l

s

In which

(3.10)

Above models are possible & for their demonstrating, refer to (3.6) we suppose the optimed amount of model (3.9) as

o j

p

_max it is clear that it is the expect maximum amont for

o j

DMU

. It is ideal that we can

set target maximum amount of o j

p

_max for each

o j

DMU

in one system, but it is not possible. Equitable setting target were proportional with their capacities, it mean:

o j

j j

j

p

o o

,

max





But it is difficult, so we try to obtain amount as these ratio have the minimum of differences, so the following model can be proposal,

max min

max max min max

*

1 *

min

. .

/

,

1,..., ,

/

,

1,..., ,

,

0.

j j

t

j j t t

j F n

j j

t l

t N

j t

P

s t

P

p

j

n

P

p

j

n

p

s

p

t

N

p

P

s

S

r

s



















 











(3.11)

,

min max P P 

minimized the differences between min & max ratio for all DMU if we can obtain

min max P P 

From the solution of model (3.11) it mean that the amount of objective function is zero, then

for each j,jo

o o

j j j

j

p

max max



*

1

min . .

, ,

,

, 0.

l t

t N

t

j j t t

j F n

j j

j t

S s

s t

p s p t N p P

p s







  



 



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4 Solution algorithm (sum up of setting target)

1) We can obtain the efficiency amount of 

o j



for each

DMU

jo with the solution of model number (2.1)

2) Distinguish sets N, f according to above definitions

3) Calculate amount of

s

l _{with model number10}

4) Obtain the maximum setting target capacity

DMU

joor

p

maxjo with model number (3.9) 5) Calculate amount of equitable setting target

 

n

p

₁

,



by solving model number (3.11)

5 examples

Suppose 12 following DMU with 2 in puts & out puts.

DMU1 2 DMU1 1 DMU1 0 DMU 9 DMU 8 DMU 7 DMU 6 DMU 5 DMU 4 DMU 3 DMU 2 DMU 1 DMU 38 53 50 30 31 33 55 22 27 25 19 20 Input1 284 306 268 244 206 235 255 158 168 160 131 151 Input2 250 260 250 190 152 220 230 94 180 160 150 100 Output 1 120 147 100 100 80 88 90 66 72 55 50 90 Output 2 0.9582 0.9551 0.8706 0.960 4 0.796 3 0.902 0 0.834 8 0.763 5 1 0.882 7 1 1 *



We can obtain allocations related to above DMU with setting target

P

1



150 ,

P

2



110

12

p

11

p

10 p 9 p 8 p 7 p 6 p 5 p 4

p

3 p 2

p

1

p

15.6897 19.2793 17.4009 12.3557 11.0830 12.9440 15.5877 7.2189 12.5286 9.2417 8.3352 8.3352 1

P

11.5058 14.1581 12.7606 9.0608 8.1276 9.4923 11.4310 5.2939 9.1877 6.7773 6.1125 6.1125 2

P

6 results

This article showed we can suppose equitable allocations as same as equitable setting target for decision making units. In this essay was assumed as a fixed scale. It mean that we can obtain results with varied

scale by using C.C.R models but adding





n j j 1

1 

to above models & with B.C.C model.

References

[1] A. Charnes, W. W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (6) (1978) 429-444.

http://dx.doi.org/10.1016/0377-2217(78)90138-8

[2] R. D. Banker, A. Charnes, W. W. Cooper, Some methods for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (9) (1984) 1078-1092.

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[3] W. Cook, M. Kress, Characterzing an equitable allocation of shared costs: a DEA approach, European Journal of Operational Research, 119 (1999) 625-661.

http://dx.doi.org/10.1016/S0377-2217(98)00337-3

[4] G. R. Jahanshahloo, F. Hosseinzadeh lotfi, N. Shoja, M. Sanei, An alternative approach for equitable allocation of shared costs by using DEA, Applied mathematics and Computation, 153 (2004) 267-274.

http://dx.doi.org/10.1016/S0096-3003(03)00631-3

[5] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Moradi, A DEA approach for fair allocation of common revenue, Applied Mathematics and Computation, 160 (2005) 719-724.

http://dx.doi.org/10.1016/j.amc.2003.11.027

[6] W. D. Cook, J. Zhu, Allocation of shared costs among decision making units: a DEA approach, Computeres & Operations Research, 32 (2005) 2171-2178.

http://dx.doi.org/10.1016/j.cor.2004.02.007

[7] Ruiyue Lin, Allocating fixed costs or resources and setting targets via data envelopment analysis, Applied Mathematics and Computation, 217 (2011) 6349-6358.