Interval Network DEA 1. Introduction

Applied Mathematical Sciences, Vol. 1, 2007, no. 15, 697 - 710

Interval Network DEA F. Hosseinzadeh Lotfi1 Department of Mathematics, Science & Research Branch Islamic Azad University, Tehran, Iran G. Tohidi Department of Mathematics, Islamic Azad University Tehran-Center Branch, Tehran, Iran T. Soleimany Department of Mathematics, Science & Research Branch Islamic Azad University, Tehran, Iran Abstract Basic Data Envelopment Analysis (DEA) Models measure the perfect performance of a unit . Indeed Decision Making units (DMU) or production processes are seen as a black box and the parts of the DMU arent included in evaluation. But in interval Network DEA we are allowed to consider the evaluation of changes occurred within the process. In this paper, we’ve developed Network DEA models on interval inputs and outputs.

1. Introduction The problem of measuring efficiency is one of the main problems which is also important for economical experts. Farrell [1], for the first time, proposed Non-parametric methods to measure the efficiency of DMUs . He made a set called Production Possibility Set (PPS) , using observations and undeniable principles ruling on the science of the set, and he called its frontier, that is a linear piswise a production function. Then charnes, cooper and Rhodes [2] extended Farell’s work , and proposed CCR model. After, BCC Model was put forward by Banker, charnes and cooper [3]. These two models caused DEA 1

Corresponding author,Email: hosseinzadeh− [email protected]

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branch rapidly improve from operation Research . In many practical examples, DMUs or production processes, themselves include subunits that are related to each other through networks. It means that a subunits output may be an input for another subunit, which ultimately, these actions and reactions made ways to ultimate output production of system. The paper authorized by Fare and Grosskopf [4] is a basis for this work. In this present paper, there will be, in the first section, are view on CCR model and Network DEA models. Then in the section.3, we will develop Network DEA Models on interval inputs and outputs. Finally, in the last section, we will have conclusions.

2. Network DEA In this section, firstly, we have a review on CCR model then we examine forms and models for Network DEA. Suppose n, DMU exist. Each DMU has, N, inputs and M outputs. In fact , DMUj the inputs and outputs . (Xj , Yj ) = (xj1 , · · · , xjN , yj1, · · · , yjM ), j = 1, · · · , n. PPS of CCR model isn defined as follows: n Tc = {(X, Y ) | X ≥



j=1

λ j Xj , Y ≤



j=1

λj Yj , λj ≥ 0, j = 1, · · · , n}

The CCR model in input oriented on Tc is as follws: Min θ s.t

n 

j=1 n  j=1

λj xji ≤ θxpi , i = 1, · · · , N, (1) λj yjr ≥ ypr ,

λj ≥ 0,

r = 1, · · · , M, j = 1, · · · , n,

Now, we review Network DEA models that are extracted from fare and Grosskopf: A) Assume that there are two production processes p1 and p2 , each producing an output vectory 1 andy 2, respectively. Moreover assume that the two processes use the same source of input x . inparticalar the allocation of x to p1 andp2 .

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Interval network DEA

y1

y2

P1

P2

x1

x2

6 6     6 6

Fig.1.Resource constraint:x1 + x2 ≤ x The model of fig . 1 can be formalized as: β(ˆ x11 , xˆ12 , · · · , xˆ1N , xˆ22 , · · · , xˆ2N ) = {(y 1 + y 2 ) = yr1

≤

n  j=1

j=1

1 λ1j yjr ,r

= 1, · · · , M,

n  j=1

λ1j x1j1

≤

(yr1 + yr2 ) :

r=1

x11 ,

λ1j x1ji ≤ xˆ1i , i = 2, · · · , N, λ1j ≥ 0, j = 1, · · · , n

yr2 ≤ n 

n 

M 

n  j=1

2 λ2j yjr , r = 1, · · · , M,

n  j=1

(2)

λ2j x2j1 ≤ x21 ,

λ2j x2ji ≤ xˆ2i , i = 2, · · · , N, λ2j ≥ 0, j = 1, · · · , n,

j=1 x11 +

x21 ≤ xˆ11 }.

For simplicity, we assume that only the first input can be allocated between the two processes. B)The full Network model, illustrated in fig .2. includes three subprocesses 1, 2 and 3. To these, a source 0 and sink 4 are added . The sourece gives the Network exogenous input x , which are allocated to the subprocesses.

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4 1y

1 y Q 1 *  ?  0x  4 s Q  3 x 3y - 4 0 -y ` x 0 3    Q  2 Q 33  s 6 4 0x 2  2y 2y 

1

Fig.2. the network technology The model of fig. 2 can be formolized as: β(x) = {y = (41 y,42 y,43 y) : 4 3 yr n  j=1 n  j=1 n  j=1

≤

n 

j=1 3 λ3j 0 xji

4

λ3j 3 yjr , r = 1, · · · , M 3 , ≤30 xi , i = 1, · · · , N,

3

λ3j 1 yjr ≤31 yr , r = 1, · · · , M 1 , 3

λ3j 2 yjr ≤32 yr , i = 1, · · · , M 2 ,

λ3j ≥ 0, j = 1, · · · , n, (31 yr +41 yr ) ≤ n  j=1

n 

j=1

λ1j (31 yjr +41 yjr ), r = 1, · · · , M 1 ,

(3)

1

λ1j 0 xji ≤10 xi , i = 1, · · · , N,

λ1j ≥ 0, j = 1, · · · , n, (32 yr +42 yr ) ≤ n  j=1

2 λ2j 0 xji

n 

j=1

λ2j (32 yjr +42 yjr ), r = 1, · · · , M 2 ,

≤20 xi , i = 1, · · · , N,

λ2j ≥ 0, j = 1, · · · , n, 1 2 3 0 xi +0 xi +0 xi ≤ xi , i = 1, · · · , N}. C )The two production processes are denoted bypt andpt+1 . Each uses timespecific inputs xt and xt+1 to produce time-specific final outputsf y t and f y t+1 (fig. 3) . In addition pt produces intermediate output that are used as input inpt+1 .

701

Interval network DEA f t

y

f t+1

xt

xt+1

y

6 6   h t−1 y P t - h yt -P t+1 -h y t+1   6 6

Fig.3. the dynamic technology

the model of Fig .3 can be formalized as: β(xt , xt+1 ,h y t−1 ) = {(f y t , (f y t+1 +h y t+1 )) : (f yrt +h yrt ) ≤ n  j=1 n  j=1

n 

j=1

t t λtj (f yjr +h yjr ), r = 1, · · · , M,

λtj xtji ≤ xti , i = 1, · · · , N, h

t−1 λtj yjr ≤h yrt−1 , r = 1, · · · , M,

(4)

λtj ≥ 0, j = 1, · · · , n, (f yrt+1 n 

+

h

yrt+1 )

t+1 λt+1 j xji

≤

≤

n 

j=1

xt+1 i ,i

f t+1 h t+1 λt+1 j ( yjr + yjr ), r = 1, · · · , M,

= 1, · · · , N,

j=1 n t+1 h t yjr ≤h yrt , r = j=1 λj ≥ 0, j = 1, · · · , n}. λt+1 j

1, · · · , M,

3. Interval Network DEA In this section, our goal is to develop the previous section models on interval data. L U Suppose that: xji ∈ [xLji , xUji ] and yjr ∈ [yjr , yjr ],j = 1, · · · , n; i = 1, · · · , N, r = 1, · · · , M. The model of Fig. 2 can be formolized as:

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Min θ n 4 34 3 s.t 3 ypr ≤ j=1 λj 3 yjr , r = 1, · · · , M , n 

3

i = 1, · · · , N,

3

r = 1, · · · , M 1 ,

λ3j 0 xji ≤30 xpi ,

j=1 n 

λ3j 1 yjr ≤31 ypr ,

j=1 n 

3

λ3j 2 yjr ≤32 ypr ,

j=1

3 1 ypr

≤

4 1 ypr

≤

n 

1

n  13

λj 1 yjr ,

j=1 n 14 j=1 λj 1 yjr ,

λ1j 0 xji ≤10 xpi ,

(5)

r = 1, · · · , M 2 , r = 1, · · · , M 1 , r = 1, · · · , M 1 ,

i = 1, · · · , N,

j=1

3 2 ypr

≤

4 2 ypr

≤

n 

2

n  j=1 n  j=1

3

λ2j 2 yjr , r = 1, · · · , M 2 , 4

λ2j 2 yjr , r = 1, · · · , M 2 ,

λ2j 0 xji ≤20 xpi ,

i = 1, · · · , N,

xpi +30 xpi ≤ θxpi , 0, λ2j ≥ 0, λ3j ≥ 0,

i = 1, · · · , N, j = 1, · · · , n,

j=1 1 2 0 xpi +0 λ1j ≥

Assume that the feasible region of the problem (5) is called S1 . Now suppose thatDMUp has the highest possible input and the lowest possible output, and the other of DMUs have the lowest possible inputs; and the highest possible outputs i.e: (XpU , YpL ) and (XjL , YjU ) j = 1, · · · , n; j = p The problem (6) can be formolized as:

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Interval network DEA

θL = Min θ n 4 L 34 U 34 L 3 s.t 3 ypr ≤ j=1,j=p λj 3 yjr + λp 3 ypr , r = 1, · · · , M , n 

3

λ3j 0 xLji + λ3p 30 xUpi ≤30 xUpi ,

i = 1, · · · , N,

j=1,j=p n 

L U U λ3j 31 yjr + λ3p 1 ypr ≤31 ypr ,

3

r = 1, · · · , M 1 ,

j=1,j=p n 

3

r = 1, · · · , M 2 ,

L U U λ3j 32 yjr + λ3p 2 ypr ≤32 ypr ,

j=1,j=p 3 L 1 ypr ≤ 4 L 1 ypr ≤ n  j=1,j=p 3 L 2 ypr ≤ 4 L 2 ypr ≤ n 

n

13 U j=1,j=p λj 1 yjr 14 U j=1,j=p λj 1 yjr

n 1

3

L + λ1p 1 ypr , r = 1, · · · , M 1 , 4 L + λ1p 1 ypr , r = 1, · · · , M 1 ,

1

λ1j 0 xLji + λ1p 0 xUpi ≤10 xUpi , n

23 U j=1,j=p λj 2 yjr n 24 U j=1,j=p λj 2 yjr 2

(6)

i = 1, · · · , N,

3

L + λ2p 2 ypr , r = 1, · · · , M 2 , 4 L + λ2p 2 ypr , r = 1, · · · , M 2 ,

2

λ2j 0 xLji + λ2p 0 xUpi ≤20 xUpi ,

j=1,j=p 1 L 2 L 3 L U 0 xpi +0 xpi +0 xpi ≤ θxpi , λ1j ≥ 0, λ2j ≥ 0, λ3j ≥ 0,

i = 1, · · · , N, i = 1, · · · , N, j = 1, · · · , n,

Assume that the feasible region of the problem (6) is called S2 . Now suppose that DMUp has the lowest possible input and the highest possible output, and the other of DMUs have the highest possible inputs; and the lowest possible outputs, i.e: (XpL , YpU ) and (XjU , YjL ) j = 1, · · · , n; j = p.The problem (7)can be formolized as:

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θU = Min θ n 4 U 34 L 34 U 3 s.t j=1,j=p λj 3 yjr + λp 3 ypr , r = 1, · · · , M , 3 ypr ≤ n 

3

j=1,j=p n 

λ3j 0 xUji + λ3p 30 xLpi ≤ 30 xLpi ,

i = 1, · · · , N,

U L L λ3j 31 yjr + λ3p 1 ypr ≤31 ypr ,

3

r = 1, · · · , M 1 ,

3

r = 1, · · · , M 2 ,

j=1,j=p n 

U L L λ3j 32 yjr + λ3p 2 ypr ≤32 ypr ,

j=1,j=p 3 U 1 ypr ≤ 4 U 1 ypr ≤ n  j=1,j=p 3 U 2 ypr ≤ 4 U 2 ypr ≤ n 

n

13 L j=1,j=p λj 1 yjr n 14 L j=1,j=p λj 1 yjr 1

3

U + λ1p 1 ypr , r = 1, · · · , M 1 , 4 U + λ1p 1 ypr , r = 1, · · · , M 1 ,

1

λ1j 0 xUji + λ1p 0 xLpi ≤10 xLpi , n

23 L j=1,j=p λj 2 yjr n 24 L j=1,j=p λj 2 yjr 2

(7)

i = 1, · · · , N,

3

U + λ2p 2 ypr , r = 1, · · · , M 2 , 4 U + λ2p 2 ypr , r = 1, · · · , M 2 ,

2

λ2j 0 xUji + λ2p 0 xLpi ≤20 xLpi ,

j=1,j=p 1 U 2 U 3 U L 0 xpi +0 xpi +0 xpi ≤ θxpi , 1 2 3 λj ≥ 0, λj ≥ 0, λj ≥ 0,

i = 1, · · · , N, i = 1, · · · , N, j = 1, · · · , n,

Assume that the feasible region of the problem (7) is called S3 . ∗ ∗ Theorem 1:Ifθ∗ ,θL ,θU are the optimal values of the problems (5) , (6) and (7) L U respectively and for each i , j and r,xji ∈ [xLji , xUji ] and yjr ∈ [yjr , yjr ],then, L∗ ∗ U∗ θ ≤θ ≤θ . Proof:since all the three problems; (5) , (6) and (7) are minimized, and those ∗ ∗ three have objective function θ , thus in order to proof θL ≤ θ∗ ≤ θU , we should just enough proof that: S3 ⊆ S1 ⊆ S2 . First we proof that S3 ⊆ S1 .Assume (λ, θ) ∈ S3 so:   U L ≤ nj=1,j=p λ3j 43 yjr ≤ nj=1,j=p λ3j 43 yjr (1 − λ3p )43 ypr ≤ (1 − λ3p )43 ypr  ⇒ nj=1 λ3j 43 yjr ≥43 ypr , r = 1, · · · , M 3 , (a) n

 λ3j 30 xji ≤ nj=1,j=p λ3j 30 xUji ≤ (1 − j=1,j = p  ⇒ nj=1 λ3j 30 xji ≤ 30 xpi , i = 1, · · · , N, (b)

n

33 j=1,j=p λj 1 yjr  3 ⇒ nj=1 λ3j 1 yjr

n

33 j=1,j=p λj 2 yjr  3 ⇒ nj=1 λ3j 2 yjr

λ3p )30 xLpi ≤ (1 − λ3p )30 xpi



U L ≤ nj=1,j=p λ3j 31 yjr ≤ (1 − λ3p )31 ypr ≤ (1 − λ3p )31 ypr ≤31 ypr , r = 1, · · · , M 1 , (c)



U L ≤ nj=1,j=p λ3j 32 yjr ≤ (1 − λ3p )32 ypr ≤ (1 − λ3p )32 ypr ≤32 ypr , r = 1, · · · , M 2 , (d)

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Interval network DEA n

13 j=1,j=p λj 1 yjr  3 ⇒ nj=1 λ1j 1 yjr

n

14 j=1,j=p λj 1 yjr  4 ⇒ nj=1 λ1j 1 yjr



L U ≥ nj=1,j=p λ1j 31 yjr ≥ (1 − λ1p )31 ypr ≥ (1 − λ1p )31 ypr ≥31 ypr , r = 1, · · · , M 1 , (e)



L U ≥ nj=1,j=p λ1j 41 yjr ≥ (1 − λ1p )41 ypr ≥ (1 − λ1p )41 ypr ≥41 ypr , r = 1, · · · , M 1 , (f)

n

 λ1 1 x ≤ nj=1,j=p λ1j 10 xUji ≤ (1 − j=1,j n=p j101 ji ⇒ j=1 λj 0 xji ≤ 10 xpi , i = 1, · · · , N, (g)

n

23 j=1,j=p λj 2 yjr  3 ⇒ nj=1 λ2j 2 yjr

n

24 j=1,j=p λj 2 yjr  4 ⇒ nj=1 λ2j 2 yjr

λ1p )10 xLpi ≤ (1 − λ1p )10 xpi



L U ≥ nj=1,j=p λ2j 32 yjr ≥ (1 − λ2p )32 ypr ≥ (1 − λ2p )32 ypr ≥32 ypr , r = 1, · · · , M 2 , (h)



L U ≥ nj=1,j=p λ2j 42 yjr ≥ (1 − λ2p )42 ypr ≥ (1 − λ2p )42 ypr ≥42 ypr , r = 1, · · · , M 2 , (i)

n

 λ2 2 x ≤ nj=1,j=p λ2j 20 xUji ≤ (1 − j=1,j n=p j202 ji ⇒ j=1 λj 0 xji ≤ 20 xpi , i = 1, · · · , N, (j)

1 2 3 1 0 xpi +0 xpi +0 xpi ≤0 ⇒10 xpi +20 xpi +30 xpi

λ2p )20 xLpi ≤ (1 − λ2p )20 xpi

xUpi +20 xUpi +30 xUpi ≤ θxLpi ≤ θxpi ≤ θxpi , i = 1, · · · , N, (k)

Thus, we conclued from (a) to (k) that: (λ, θ) ∈ S1 then S3 ⊆ S1 . so the ∗ amount of objective function wont be better i.e,θ∗ ≤ θU , (*) Now we proof that S1 ⊆ S2 . Assume (λ, θ) ∈ S1 so: n

n 34 U 34 3 4 j=1,j=p λj 3 yjr ≥ j=1,j=p λj 3 yjr ≥ (1 − λp )3 ypr ≥ (1 − n U L L ⇒ j=1,j=p λ3j 43 yjr + λ3p 43 ypr ≥ 43 ypr , r = 1, · · · , M 3 , (l)

n

n 33 L 33 3 3 j=1,j=p λj 0 xji ≤ j=1,j=p λj 0 xji ≤ (1 − λp )0 xpi ≤ (1 n ⇒ j=1,j=p λ3j 30 xLji + λ3p 30 xUpi ≤ 30 xUpi , i = 1, · · · , N, (m)

− λ3p )30 xUpi

n

 L λ3j 31 yjr ≤ nj=1,j=p λ3j 31 yjr ≤ (1 − λ3p )31 ypr ≤ (1 − j=1,j = p  L U U ⇒ nj=1,j=p λ3j 31 yjr + λ3p 31 ypr ≤ 31 ypr , r = 1, · · · , M 1 , (n)

n

 L λ3j 32 yjr ≤ nj=1,j=p λ3j 32 yjr ≤ (1 − λ3p )32 ypr ≤ (1 − j=1,j = p  L U U ⇒ nj=1,j=p λ3j 32 yjr + λ3p 32 ypr ≤ 32 ypr , r = 1, · · · , M 2 , (o)

n

n 13 U 13 1 3 j=1,j=p λj 1 yjr ≥ j=1,j=p λj 1 yjr ≥ (1 − λp )1 ypr ≥ (1 − n U L L ⇒ j=1,j=p λ1j 31 yjr + λ1p 31 ypr ≥ 31 ypr , r = 1, · · · , M 1 , (p)

n

14 U j=1,j=p λj 1 yjr

≥

n

14 j=1,j=p λj 1 yjr

L λ3p )43 ypr

U λ3p )31 ypr

U λ3p )32 ypr

L λ1p )31 ypr

L ≥ (1 − λ1p )41 ypr ≥ (1 − λ1p )41 ypr

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F. Hosseinzadeh Lotfi et al n

14 U j=1,j=p λj 1 yjr

L L + λ1p 41 ypr ≥ 41 ypr , r = 1, · · · , M 1 , (q)

n

n 11 L 11 1 1 j=1,j=p λj 0 xji ≤ j=1,j=p λj 0 xji ≤ (1 − λp )0 xpi ≤ (1 n ⇒ j=1,j=p λ1j 10 xLji + λ1p 10 xUpi ≤ 10 xUpi , i = 1, · · · , N, (r)

− λ1p )10 xUpi

n

n 23 U 23 2 3 j=1,j=p λj 2 yjr ≥ j=1,j=p λj 2 yjr ≥ (1 − λp )2 ypr ≥ (1 − n U L L ⇒ j=1,j=p λ2j 32 yjr + λ2p 32 ypr ≥ 32 ypr , r = 1, · · · , M 2 , (s)

n

n 24 U 24 2 4 j=1,j=p λj 2 yjr ≥ j=1,j=p λj 2 yjr ≥ (1 − λp )2 ypr ≥ (1 − n U L L ⇒ j=1,j=p λ2j 42 yjr + λ2p 42 ypr ≥ 42 ypr , r = 1, · · · , M 2 , (t)

n

n 22 L 22 2 2 j=1,j=p λj 0 xji ≤ j=1,j=p λj 0 xji ≤ (1 − λp )0 xpi ≤ (1 n ⇒ j=1,j=p λ2j 20 xLji + λ2p 20 xUpi ≤ 20 xUpi , i = 1, · · · , N, (u)

1 L 2 L 3 L 1 0 xpi +0 xpi +0 xpi ≤0 ⇒10 xLpi +20 xLpi +30 xLpi

L λ2p )32 ypr

L λ2p )42 ypr

− λ2p )20 xUpi

xpi +20 xpi +30 xpi ≤ θxpi ≤ θxUpi ≤ θxUpi , i = 1, · · · , N, (v)

Thus, we conclued from (l) to (v) that: (λ, θ) ∈ S2 then S1 ⊆ S2 . so the ∗ amount of objective function wont be better i.e,θL ≤ θ∗ , (**). ∗ ∗ we conclued from (*) and (**)that, θ∗ ∈ [θL , θU the model of fig 1 can be formolized as: Min θ s.t

1 ypr n  j=1 n  j=1

≤

j=1

λ1j x1j1

j=1 n 

1 λ1j yjr , r = 1, · · · , M,

≤ x1p1 ,

λ1j x1ji ≤ θˆ x1pi , i = 2, · · · , N,

2 ypr ≤ n 

n 

n  j=1

2 λ2j yjr , r = 1, · · · , M,

(8)

λ2j x2j1 ≤ x2p1 , λ2j x2ji ≤ θˆ x2pi , i = 2, · · · , N,

j=1 x1p1 . x1p1 + x2p1 ≤ θˆ λ1j ≥ 0, λ2j ≥ 0, j =

1, · · · , n

Now suppose that DMUp has the highest possible input and the lowest possible output, and the other of DMUs have the lowest possible inputs and the highest possible outputs.The problem (9) can be formolized as:

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Interval network DEA

θL = Min θ L 1 ypr

s.t

j=1,j=p

n  j=1,j=p n  j=1,j=p L 2 ypr

n 

≤

λ1j L x1j1 + λ1p U x1p1 ≤ U x1p1 , λ1j L x1ji + λ1p L x1pi ≤ θU xˆ1pi , i = 2, · · · , N, n 

≤

j=1,j=p

n  j=1,j=p n 

1 1 λ1j U yjr + λ1p L ypr , r = 1, · · · , M,

2 2 λ2j U yjr + λ2p L ypr , r = 1, · · · , M,

(9)

λ2j L x2j1 + λ2p U x2p1 ≤ U x2p1 , λ2j L x2ji + λ2p L x2pi ≤ θU xˆ2pi , i = 2, · · · , N,

j=1,j=p L 1 xp1 +L x2p1 ≤ θU xˆ1p1 . λ1j ≥ 0, λ2j ≥ 0, j = 1, · · · , n

Now suppose that DMUp has the lowest possible input and the highest possible output, and the other of DMUs have the highest possible inputs and the lowest possible outputs the problem (10) can be formolized as: θU = Min θ U 1 ypr

s.t

j=1,j=p

n  j=1,j=p n  j=1,j=p U 2 ypr

n 

≤

λ1j U x1j1 + λ1p L x1p1 ≤ L x1p1 , λ1j U x1ji + λ1p U x1pi ≤ θL xˆ1pi , i = 2, · · · , N, n 

≤

n  j=1,j=p n 

1 1 λ1j L yjr + λ1p U ypr , r = 1, · · · , M,

j=1,j=p

2 2 λ2j L yjr + λ2p U ypr , r = 1, · · · , M,

(10)

λ2j U x2j1 + λ2p L x2p1 ≤ L x2p1 , λ2j U x2ji + λ2p U x2pi ≤ θL xˆ2pi , i = 2, · · · , N,

j=1,j=p U 1 xp1 +U

x2p1 ≤ θL xˆ1p1 .

λ1j ≥ 0, λ2j ≥ 0, j = 1, · · · , n ∗

∗

Theorem 2:Ifθ∗ ,θL ,θU are the optimal values of the problems (8) ,(9)and(10) ∗ L U , yjr ],then, θL respectively and for each i , j and r,xji ∈ [xLji , xUji ] and yjr ∈ [yjr ∗ ≤ θ∗ ≤ θU .

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Proof:The proof of this is much like the proof theorem (1) . the model of fig 3. can be formolized as: Min θ s.t

f t ypr

≤

h t ypr

≤

n  j=1 n  j=1

n  j=1 n  j=1

h

t λtj yjr , r = 1, · · · , M,

λtj xtji ≤ θxtpi , i = 1, · · · , N, h

t−1 t−1 λtj yjr ≤ θh ypr , r = 1, · · · , M,

f t+1 ypr

≤

h t+1 ypr

≤

n 

f

t λtj yjr , r = 1, · · · , M,

n  j=1 n  j=1

t+1 λt+1 j xji

(11)

f

t+1 λt+1 yjr , r = 1, · · · , M, j h

t+1 λt+1 yjr , r = 1, · · · , M, j

≤ θxt+1 pi , i = 1, · · · , N,

j=1 n t+1 h t t yjr ≤h ypr , r = 1, · · · , M, j=1 λj t+1 t λj ≥ 0, λj ≥ 0, j = 1, · · · , n,

Now suppose that DMUp has the highest possible input and the lowest possible output, and the other of DMUs have the lowest possible inputs and the highest possible outputs the problem (12) can be formolized as: θL = Min θ s.t

fL t ypr hL

n 

≤

t ypr ≤

n  j=1,j=p n  j=1,j=p

j=1,j=p n  j=1,j=p

t + λtp ypr , r = 1, · · · , M,

λtj

hU t yjr

t + λtp ypr , r = 1, · · · , M,

hL

t−1 λtj yjr + λtp

≤

hL t+1 ypr

≤

j=1,j=p

fU t yjr

fL

hL

λtj L xtji + λtp U xtpi ≤ θU xtpi , i = 1, · · · , N,

f L t+1 ypr

n 

λtj

n  j=1,j=p n  j=1,j=p L

λt+1 j

hU t−1 ypr

t−1 ≤ θh ypr , r = 1, · · · , M,

f U t+1 yjr

t+1 + λt+1 ypr , r = 1, · · · , M, p

hU

U

fL

hL

t+1 t+1 λt+1 yjr + λt+1 ypr , r = 1, · · · , M, j p

t+1 U t+1 λt+1 xt+1 xpi ≤ θU xt+1 j ji + λp pi , i = 1, · · · , N,

(12)

709

Interval network DEA n 

hL

t λt+1 yjr + λt+1 j p

j=1,j=p λtj ≥ 0, λt+1 j

hU t ypr

U

t ≤h ypr , r = 1, · · · , M,

≥ 0, j = 1, · · · , n,

Now suppose that DMUp has the lowest possible input and the highest possible output, and the other of DMUs have the highest possible inputs and the lowest possible outputs. The problem (13) can be formolized as: θU = Min θ s.t

fU t ypr

≤

hU t ypr

≤

n  j=1,j=p n  j=1,j=p

n  j=1,j=p n  j=1,j=p

fU t ypr , r

= 1, · · · , M,

hL

hU t ypr , r

= 1, · · · , M,

t λtj yjr + λtp

λtj U xtji + λtp L xtpi ≤ θL xtpi , i = 1, · · · , N, λtj

f U t+1 ypr

≤

hU t+1 ypr

≤

n 

fL

t λtj yjr + λtp

hU t−1 yjr

hL

n  j=1,j=p n  j=1,j=p

fL

fU

hL

hU t+1 ypr , r

t+1 λt+1 yjr + λt+1 j p t+1 λt+1 yjr + λt+1 j p

(13)

t+1 ypr , r = 1, · · · , M,

= 1, · · · , M,

U

t+1 L t+1 λt+1 xt+1 xpi ≤ θL xt+1 j ji + λp pi , i = 1, · · · , N,

j=1,j=p U n t+1 h t yjr j=1,j=p λj t+1 t λj ≥ 0, λj ≥ 0, j ∗

L

t−1 t−1 + λtp ypr ≤ θh ypr , r = 1, · · · , M,

hL

L

t t + λt+1 ypr ≤h ypr , r = 1, · · · , M, p = 1, · · · , n,

∗

Theorem 3:Ifθ∗ ,θL ,θU are the optimal values of the problems (11) ,(12)and(13) L U , yjr ],then, respectively and for each i , j and r,xji ∈ [xLji , xUji ] and yjr ∈ [yjr L∗ ∗ U∗ θ ≤θ ≤θ . Proof:The proof of this is much like the proof theorem (1) .

4. conclusion In this paper we have extended the work of Fare an Grosskopf ( 2000 ) to obtain a theoretically justified method for efficiency indexes of DMUs with interval inputs and outputs .The precise data are the special case of interval data. So

710

F. Hosseinzadeh Lotfi et al

the proposed method in the general case of the previous method about network DEA with precise data . on the other hand, we are allowed to consider all changes Occurred within the prossess of sub- DMUs when the efficiency of a DMU is evaluated . Based on the proved theorems, the efficiency value of DMUs with interval data in network DEA lies in a interval .

References [1] Farrell M.J. The measurement of productive Efficiency. Journal of the Royal statistical society, 1957. [2] Charnes A, Cooper W.W, Rhodes E. Measuring the efficiency of decision making units . European journal of operational Research, 1978 . [3] Banker R.D, Charnes A, Cooper W.W. Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis.management Sicience, 1984. [4] Fare R, Grosskopf S. Network DEA . Socio-Economic Planning Sciences, 2000. Received: August 12, 2006