New radial FNDEA-R models are proposed with flexible measures using the input/output ratio or vice versa in the input and output orientation under constant returns to scale in general two-stage network to classify flexible measures to achieve the highest relative efficiency of the DMU under evaluation. A general two-stage network structure is pictured in
General two-stage network.
This figure is shown in the PDF version of the article.
If there are L1 flexible measures to stage one wl j, (l1 = 1, … , L1) and L2 flexible measures to the second stage wl j, (l2 = 1, … , L2) of which the statues of input or output are not given, some DMUs can employ these measures as inputs while other DMUs can be employed as outputs.
A new radial FNDEA model is presented using envelopment form of CCR in the input orientation in general two-stage network DEA to assess the relative efficiency and categorize flexible measures.
(1) (2)
In this model, M is a big positive number. For each l1 and l2, binary variables d and d ∈ {0,1} to the first
l1 l2
and second stage are used for identifying that factors l1 and l2 are used as inputs to the stages one and two,
(1) (2) (1) (2)
d = 0 and d = 0 or outputs of stages one and two d = 1 and d = 1.
l1 l2 l1 l2
Definition 3. DMUo is called a FNDEA efficient unit if and only if θFNDEA = 1 .
The technology set TFNDEA is defined as follows:
1 1 1
∑ λ w ≤ w or
n n j j 1 1 ′ ′ j=1
∑ λj xj ≤ x; ∑ λj yj ≥ y ; n n ;
j=1 j=1 1 1 2 1
∑ λj wj ≥ ∑ λj wj 1 ′ ′ 2 (j=1 j=1 ) TFNDEA = (x, w , y , z, x , w , y) n .
| 2 2 2 ∑ λ w ≤ w or
n n n n j j 1 2 2 ′ ′ 2 j=1 1 2
∑ λj zj ≥ ∑ λj zj ; ∑ λj xj ≤ x ; ∑ λj yj ≥ y; n ; λj ≥ 0, λj ≥ 0, for all j |j=1 j=1 j=1 j=1 2 2 2 ∑ λj wj ≥ w { ( j=1 ) }
Definition 4. Model (3) is feasible and has units-invariant.
Here new radial FNDEA-R models are presented in the presence of flexible measures under constant returns to scale in the output and input orientation depending on input/output ratios and vice versa where each flexible measure functions as input for some DMUs and output for the others to achieve a maximum relative efficiency of the DMU under evaluation in general two-stage network.
A general two-stage network structure is illustrated in
x x1 xm x1 xm x1 xm
{ = ( , … , , , … , , … , , … , )} with ý = (ý1, … , ý H), x = (x1, … , xm).
ý ý 1 ý 1 ý 2 ý 2 ý H ý H
1 1 1 1 1 1 1
w w wL w wL w wL 1 1 1 1 1 1 1
{ = ( , … , , , … , , … , , … , )} with ý = (ý1, … , ý H), w =
ý ý 1 ý 1 ý 2 ý 2 ý H ý H
(w1, … , wL ).
x x1 xm x1 xm x1 xm 1 1 1
{ = ( 1 , … , 1 , 1 , … , 1 , … , 1 , … , 1 )} with w = (w1, … , wL ), x =
ẃ1 w w w w w w 1
1 1 2 2 L1 L1
(x1, … , xm).
1 1 1 1 1 1 1
w w wL w wL w wL 1 1 1 1 1 1 1
{ = ( , … , , , … , , … , , … , )} with y = (y1, … , ys), w =
y y1 y1 y2 y2 ys ys
(w1, … , wL ).
x́ x́ 1 x́ T x1 x́ T x́ 1 x́ T
{ = ( , … , , , … , , … , , … , )} with y = (y1, … , ys), x́ = (x́ 1, … , x́ T).
y y1 y1 y2 y2 ys ys
2 2 2 2 2 2 2
w w wL w wL w wL 1 2 1 2 1 2 2
{ = ( , … , , , … , , … , , … , )} withy = (y1, … , ys), w =
y y1 y1 y2 y2 ys ys
(w1 , … , wL ).
x́ x́ 1 x́ T x1 x́ T x́ 1 x́ T 2 2 2
{ = ( 2 , … , 2 , 2 , … , 2 , … , 2 , … , 2 )} with w = (w1 , … , wL ), x́ =
w2 w w w w w w 2
1 1 2 2 L2 L2
(x́ 1, … , x́ T).
x x1 xm x1 xm x1 xm
{ = ( , … , , , … , , … , , … , )} with z = (z1, … , zK), x = (x1, … , xm).
z z1 z1 z2 z2 zK zK
z z1 zK z1 zK z1 zK
{ = ( , … , , , … , , … , , … , )} with y = (y1, … , ys), z = (z1, … , zK).
y y1 y1 y2 y2 ys ys
These ratios are defined. A new radial FNDEA-R-I model is presented under constant returns to scale in the input orientation using the input/output ratios for classifying flexible measures to achieve maximum relative efficiency of the DMU in two-stage network.
(1) (2)
In this model, M is a big positive number. Binary variables d and d ∈ {0,1} in the first and second stage
l1 l2
are introduced for each l1 and l2 to determine that factors l1 and l2 are inputs for the first stage and second
(1) (2) (1) (2)
stage, d = 0 and d = 0 or outputs for the first stage and second stage d = 1 and d = 1.
l1 l2 l1 l2
Definition 5. DMUo is called a FNDEA-R-I efficient unit if and only if θFNDEA−R−I = 1.
Theorem 1. Model (4) is feasible.
1 2 1 2 1 2
Proof: suppose dl = 0, for all l1,dl = 1, for all l2 and λo = 1, λo = 1, λj = 0, λj = 0; for all j, j ≠ o, θ = 1. So, this is a feasible solution for the Model (4).
The technology set TFNDEA−R−I is defined as follows:
∑ λj ( ′ ) ≤ ( ) or n y y′ | x x j
1 j j=1
∑ λj ( ′ ) ≤ ( ) ; n n ; y y′ 1 1 j xj x wj w
j=1 1 2
1 ′ ′ 2 | ∑ λj ( 1) ≤ ( ), ∑ λj ( ) ≤ ( ) (x, w , y , z, x , w , y) w w1 y y 1 ′ ( j=1 j=1 ) TFNDEA−R−I = x ≻ 0, w ≻ 0, y ≻ 0 n ′ 2 wj w , z ≻ 0, x ≻ 0, w ≻ 0, y ≻ 0 | 2 ∑ λj ( ) ≤ ( ) or n n n ′ y y x ′ j xj x zj z j x j=1
1 2 2 1 2
∑ λj ( ) ≤ ( ) ; ∑ λj ( ) ≤ ( ) ; ∑ λj ( ) ≤ ( ) ; n ; λj ≥ 0, λj ≥ 0; for all j z z y y y y ′ ′ | j j j xj x
j=1 j=1 j=1 2
∑ λj ( 2) ≥ ( ) w w2 { ( j ) } Theorem 2. Model (4) has units-invariant.
Proof: we let,
Then we rewrite Model (4) as follows:
Thus, we remove αi, β , δh, γ , , σr, πk from constraints so Model (4) has units-invariant.
l1 t 2
If the ratio of outputs to inputs is given, we can use FNDEA-R model in the output orientation. To evaluate DMUo, we present a new radial FNDEA-R-O model under constant returns to scale in the output orientation using the output/ input ratio for classifying flexible measures for maximizing the relative efficiency of the DMU evaluated in two-stage network as follows:
Definition 6. DMUo is called a FNDEA-R-O efficient unit if and only if φFNDEA−R−O = 1. The efficiency corresponding to DMUo is defined as ∗ .
φFNDEA−R−O
The technology set TFNDEA−R−O is defined as follows:
TFNDEA−R−O =
yj y
∑j=1 λj ( 1) ≤ ( )or
′ ′ w w1 yj y j
| ∑j=1 λj ( ) ≥ ( ) ; ( 1 ) ,
x x w w1 y y 1 ′ ′ 2 j n 1 j n 2 j
(x, w , y , z, x , w , y) ∑ λ ( ) ≥ ( ), ∑ λ ( ) ≥ ( )
j=1 j j=1 j 1 1 xj x wj w
x ≻ 0, w ≻ 0, y ≻ 0, z ≻ 0, |
n 2 yj y
′ 2 ∑ λ ( ) ≤ ( )or x ≻ 0, w ≻ 0, y ≻ 0 j=1 j 2 2
z z y y y y wj w n 1 j n 2 j n 2 j 1 2
|∑j=1 λj ( ) ≥ ( ) ; ∑j=1 λj ( ) ≥ ( ) ; ∑j=1 λj ( ′) ≥ ( ) ; ( 2 ) , λj ≥ 0, λj ≥ 0; for all j
x x z z x x′ w 2 j j j n 2 j w
∑j=1 λj ( ′ ) ≥ ( )
x x′
{ j }
Theorem 3. Model (5) is feasible.
1 2 1 2 1 2
Proof: suppose dl = 0, for all l1,dl = 1, for all l2 and λo = 1, λo = 1, λj = 0, λj = 0; for all j, j ≠ o, φ = 1.
So, this is a feasible solution for the Model (5).
Definition 7. Model (5) has units-invariant.