In this paper, in order to evaluate the performance of a DMU in Production Possible Set (PPS) with Variable Return to Scale (VRS) technology, we provide models to obtain non negative weights for inputs for outputs and a nonnegative scalar corresponding to inputs and a nonnegative scalar corresponding to outputs which for the weights and scalars, the number of which DMUs for each one its virtual output plus the scalar corresponding to inputs does not exceed (is less than, if any) its virtual input plus the scalar corresponding to inputs be maximum, provided that for DMU under evaluation, the virtual output plus the scalar corresponding to inputs does not exceed (is less than, if any) the virtual input plus the scalar corresponding to inputs and the virtual input will be positive. We call these weights and scalars the relatively best weight in input-oriented (the relatively strongest weight in input-oriented, if any) for the DMU under evaluation, and if all the weights be positive we call them the best weight in input-oriented (the strongest weight in input-oriented, if any) for the DMU under evaluation. Also, we define input-oriented efficiency and input-oriented strictly efficiency (input-oriented strongly efficiency), respectively, as ratio the number of which DMUs for each one per the relatively best weight in input-oriented and the best weight in input-oriented (the relatively strongest weight in input-oriented), its virtual input plus the scalar related to inputs does not exceed (is less) its virtual input plus the scalar related to outputs, to the total DMUs. Similarly we define the relatively best weight in output-oriented (the relatively strongest weight in input-oriented, if any), the best weight in output-oriented (the strongest weight in output-oriented, if any), output-oriented efficiency and output-oriented strictly efficiency (output-oriented strongly efficiency). The relatively best weight in input-oriented (the relatively strongest weight in input-oriented) indicates normal vector of a superface in the PPS with VRS assumption that the DMU under evaluation is on the superface and the maximum number of which DMUs their performance are no worse than (is better than) the DMU under evaluation separate from the rest of DMUs, with the constraint that the virtual input be positive. Accordingly, we can interpret the rest of the definitions of non-negative weights for inputs and for outputs and nonnegative scalars related to inputs and outputs. In this paper, we present the relationship between these definitions of efficiency with efficiency in the DEA models with VRS assumption.
The input-oriented BCC Banker et. al
On the other word, we are going to obtain non negative weights for the inputs and the outputs of DMU under evaluation, a nonnegative scalar related to inputs and a nonnegative scalar related to outputs which for the weights and scalars, the number of which DMUs for each one its income plus subsidization does not exceed (is less than, if any) its cost plus setup fee be maximum, provided that for DMU under evaluation its income plus subsidization will be equal to its cost plus setup fee and cost resulting from the inputs of the DMU will be positive. We call these weights and scalars, the relatively best weight in input-oriented (the relatively strongest weight in input-oriented, if any) for the DMU under evaluation, and if all of these weights be positive, we call them the best weight in input-oriented (the strongest weight in input-oriented, if any) for the DMU under evaluation. Also, we define input-oriented efficiency and input-oriented strictly efficiency (input-oriented strongly efficiency), respectively, as ratio the number of which DMUs for each one per the relatively best weight in input-oriented and the best weight in input-oriented (the relatively strongest weight in input-oriented), its virtual input plus the scalar corresponding to inputs does not exceed (is less) its virtual input plus the scalar corresponding to outputs, to the total DMUs. Similarly, the relatively best weight in output-oriented (the relatively strongest weight in input-oriented, if any), the best weight in output-oriented (the strongest weight in output-oriented, if any), output-oriented efficiency and output-oriented strictly efficiency (output-oriented strongly efficiency) are defined. Also we are going to obtain non negative weights for the inputs and the outputs of DMU under evaluation, a nonnegative scalar corresponding to inputs and a nonnegative scalar related to outputs which for the weights and scalars, the number of which DMUs for each one its income plus subsidization does not exceed (is less than, if any) its cost plus setup fee be maximum, provided that for DMU under evaluation its income plus subsidization will be equal to its cost plus setup fee and both cost resulting from the inputs and income resulting from outputs of the DMU will be positive. We call these weights and scalars the relatively best weight (the relatively strongest weight, if any) for the DMU under evaluation. If all the weights be positive, we call them the best weight (the strongest weight, if any) for the DMU under evaluation. Also, we define efficiency and strictly efficiency (strongly efficiency, if any), respectively, as ratio the number of which DMUs for each one per the relatively best weight in input-oriented and the best weight in input-oriented (the relatively strongest weight, if any), its virtual input plus the scalar related to inputs does not exceed (is less, if any) its virtual input plus the scalar corresponding to outputs, to the total DMUs. The relatively best weight in input-oriented (the relatively strongest weight in input-oriented) indicates normal vector of a superface in the PPS with Variable Return to Scale (VRS) assumption that the DMU under evaluation is on the superface and the maximum number of which DMUs their performance are no worse than (is better than) the DMU under evaluation separate from the rest of DMUs, with the constraint that the virtual input be positive
Suppose we have n ≥ 2 peer observed DMUs,{DMUj : j = 1,2, … , n } which produce multiple outputs yrj , (r = 1, … , s), by utilizing multiple inputs xij , (i = 1, … , m). The input and output vectors of DMUj are denoted by xj and yj, respectively, and we assume that xjand yj are semipositive, i.e. , xj ≥ 0, xj ≠ 0 and yj ≥ 0 , yj ≠ 0 for i = 1, … , n. We use by (xj, yj) to descript DMUj, and specially use (xo, yo)(oϵ{1,2, … , n}) as the DMU under evaluation. Throughout this paper, vectors will be denoted by bold letters.
The production set of the BCC model is defined as a set of semi-positive (x, y) as follows
Pv = {(x, y)| x ≥ ∑n j=1 λjxj & y ≤ ∑n j=1 λjyj & ∑n j=1 λj = 1 }, where (λ1, … , λn) is a semi-positive in ℝn. The input-oriented BCC model evaluates the efficiency of each by solving the following linear program:
θ∗ = minθ,
∑ λjxj ≤ θxo,
∑ λjyj ≥ yo, (1) s. t.
∑ λj = 1,
λj ≥ 0, j = 1, … , n, where θ is a scaler. Because xj and yj are semipositive for j = 1,2, … , n , θ∗ > 0. Also since (θ, λ = (λ1, … , λn)) is a feasible solution to Model (1), where θ = 1, λj = 0(j ≠ o), λ = 1, then θ∗ ≤ 1. Thus 0 < θ∗ ≤ 1. θ∗ represents the input-oriented BCC-efficiency value of DMUo.
Definition 1. (input-oriented BCC-efficient). The performance of DMUo is the input-oriented BCC-efficient if and only if θ∗ = 1.
The dual problem of Model (1) is expressed as:
z∗ = max ut yo + uo, vtxo = 1, t t (2) s. t. u yj + uo ≤ v xj, j = 1,2, … , n, u ≥ o, v ≥ o, where vϵℝm and uϵℝs are row vectors and represent dual variables corresponding to Eq. (1) and Eq. (2), respectively. From strong duality theorem θ∗ = z∗ , thus 0 < z∗ ≤ 1.
The two-phase process for BCC model evaluates the efficiency of DMUo by solving the following linear program:
min θ − ε(∑ si− + ∑ sr +),
i=1 r=1
∑ λjxij + si− = θxio , i = 1, … , m,
n (3) ∑ λjyrj − sr + = yro , r = 1, … , s, s. t. j=1
∑ λj = 1,
λj ≥ 0, si− ≥ 0, sr + ≥ 0, for all j, for all i, for all r, where ε > 0 is the non-Archimedian element.
Definition 2. (BCC-efficient). The performance of DMUo is BCC-efficient if only if an optimal solution (θ∗, λ∗, s∗−, s∗+) of the two-phase Model (4) satisfies θ∗ = 1, s−∗ = 0, s+∗ = 0.
The dual multiplier form of program Model (4) is expressed as:
max ut yo + uo,, vtxo = 1,, t t (4) s. t. u yj + uo ≤ v xj , for all j, u ≥ 1ε, v ≥ 1ε.
By Definition 2 and by strong duality theorem, the performance of DMUo is BCC-efficient if only if an optimal solution (u∗, v∗) of Model (4) satisfies u∗t yo + uo = 1.
Definition 3. (Reference set) reference set of DMUo denoted by Eo is defined as:
Eo = {DMUj|jϵ{1, … , n}& λj∗ > 0 in some optimal solution (θ∗, λ∗, s−∗, s+∗) of (3)}.
Theorem 1. The DMUs in Eo are BCC-efficient.
Proof: see
Definition 4. (extreme BCC-efficient) DMUo is extreme BCC-efficient if only if Eo = {DMUo}.
Theorem 2. If DMUo be extreme BCC-efficient, then DMUo is BCC-efficient Proof: see
Theorem 3. DMUo is extreme BCC-efficient iff has an optimal objective function valve of one.
min θ − ε ∑ λj,
∑ λjxij + si− = θxio , i = 1, … , m,
+ (5) ∑ λjyrj − sr = yro , r = 1, … , s, s. t. j=1
∑ λj = 1,
λj ≥ 0, si− ≥ 0, sr + ≥ 0 for all j, for all i, for all r, Proof: let DMUo not be extreme BCC-efficient. Then, there exists an optimal solution (θ∗, λ∗, s∗−, s∗+) of Model (2) such that a λj∗ > 0(j ≠ o). Also, since (θ, λ = (λ1, … , λn)) is a feasible solution to Model (4), where θ = 1, λj = 0(j ≠ o), λo = 1, thus θ∗ ≤ 1. Therefore θ∗ − ε ∑j≠o λ∗j < 1. Let the solution objective function valve of Model (2) and (4) is less one, and let (θ ̃, λ ̃) is an optimal solution of the model, then either θ ̃ < 1 or (θ ̃ = 1 and ∑j≠o λ ̃j > 0). If θ ̃ < 1, DMUo isn't extreme BCC-efficient. If θ ̃ = 1 and ∑j≠o λ ̃j > 0, then either (s ̃−, s ̃+) ≠ (0,0) or (s ̃−, s ̃+) = (0,0), where s ̃− = θ ̃, xo − ∑n j=1 λ ̃jxj, and s ̃+ = ∑n j=1 λ ̃jyj − yo. If (s ̃−, s ̃+) ≠ (0,0), since(θ ̃, λ ̃, s ̃−, s ̃+) is a feasible solution of Model (2), thus DMUo isn't BCC-efficient, therefore DMUo isn't extreme BCC-efficient. If (s ̃−, s ̃+) = (0,0), then either (θ ̃, λ ̃, s ̃−, s ̃+) is an optimal solution of Model (2) or isn't. If (θ ̃, λ ̃, s ̃−, s ̃+) be an optimal solution of Model (2), since ∑n j=1 λ ̃j > 0, thus DMUo isn't extreme BCC-efficient. If (θ ̃, λ ̃, s ̃−, s ̃+) not be an optimal solution of Model (2), then there exists an optimal solution(θ∗, λ∗, s∗−, s∗+) of Model (2) such that θ∗ = 1 and (s∗−, s∗+) ≠ (0,0), thus DMUo isn't extreme BCC-efficient.
with Variable Return to Scale Technology
Definition 5. Let Λv ⊂ ℝm+s+1 be Λ = {(u, v, uo, vo)|uϵℝs & ϵℝm & uoϵℝ≥0& voϵℝ≥0& (u, v) ≥ (0,0)}.
We define a map fv: Λv → ℕ⋃{0} by fv(u, v, uo, vo) = |{DMUj|jϵ{1,2, … , n}&vtxj + vo ≥ utyj + uo}|,
where Λ defined by Eq. (6).
Definition 6. Let (u ̅, v ̅, u ̅o, v ̅o)ϵΛv. We say (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in input-oriented (in output-oriented) in Λv for DMUo if
v ̅txo + vo = u ̅tyo + uo& v ̅txo > 0 (u ̅tyo > 0),
and
for all (u, v, uo, vo) ((u, v, uo, vo)ϵΛc & vtxo > 0 (utyo > 0) & vtxo + vo = utyo + uo, ⟹ fv(u ̅, v ̅, u ̅o, v ̅o) ≥ fv(u, v, uo, vo)).
Definition 7. Let (u ̅, v ̅, u ̅o, v ̅o)ϵΛv. We say (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in Λv for DMUo if v ̅txo + vo = u ̅tyo + uo & v ̅txo > 0 & u ̅tyo > 0.
And
for all (u, v, uo, vo)((u, v, uo, vo)ϵΛc & vtxo + vo = utyo + uo& v ̅txo > 0 & u ̅tyo > 0 ⟹ fv(u ̅, v ̅, u ̅o, v ̅o) ≥ fv(u, v, uo, vo)).
Definition 8. Let (u ̅, v ̅, u ̅o, v ̅o)ϵΛv. We say (u ̅, v ̅, u ̅o, v ̅o) is best weight in Λv for DMUo if
v ̅txo + vo = u ̅tyo + uo& (u ̅, v ̅) > (0,0).
Definition 9. Let (u ̅, v ̅, u ̅o, v ̅o)ϵΛv. We say (u, v, uo, vo) is relatively strongest weight in input-oriented (in output-oriented) in Λv for DMUo if
v ̅txo + vo = u ̅tyo + uo& v ̅txo > 0 (u ̅tyo > 0) & gv(u ̅, v ̅, u ̅o, v ̅o) ≥ 1,
and for all (u, v, uo, vo)((u, v, uo, vo)ϵΛv & vtxo > 0 (utyo > 0) & vtxo + vo = utyo + uo ⟹ gv(u ̅, v ̅, u ̅o, v ̅o) ≥ gv(u, v, uo, vo)).
Definition 10. Let (u ̅, v ̅, u ̅o, v ̅o)ϵΛc. We say (u ̅, v ̅, u ̅o, v ̅o) is relatively strongest weight in Λv for DMUo if
v ̅txo + vo = u ̅tyo + uo& v ̅txo > 0 & u ̅tyo > 0 & gv(u ̅, v ̅, u ̅o, v ̅o) ≥ 1,
and
for all (u, v, uo, vo)((u, v, uo, vo)ϵΛv & vtxo > 0 utyo > 0 & vtxo + vo = utyo + uo ⟹ gv(u ̅, v ̅, u ̅o, v ̅o) ≥ gv(u, v, uo, vo)).
Remark 1. Since xj and yj are semipositive, it follows ∑s r=1 yro > 0, ∑m i=1 xio > 0. Now if ∑s r=1 yro = ∑m i=1 xio, by taking vt = (1, … ,1)ϵℝm, ut = (1, … ,1)ϵℝs, uo = 0 and vo = 0, if ∑s r=1 yro > ∑m i=1 xio, by taking α =
∑s r=1 yro t s t m s
( ⁄∑m x ), u = α(1, … ,1)ϵℝ , v = α(1, … ,1)ϵℝ , uo = 0 and vo = 0, and finally if ∑r=1 yro <
i=1 io
m ∑m i=1 xio t s t m
∑i=1 xio, by taking β = ( ⁄∑s y ), u = β(1, … ,1)ϵℝ , v = β(1, … ,1)ϵℝ uo = 0 and vo = 0, we
r=1 ro
have vtxo + vo = utyo + uo, vtxo > 0 utyo > 0, u ≥ 1ε, v ≥ 1ε.
This shows that there is not any relatively strongest weight in input-oriented (in output-oriented) in Λv for DMUo if
for all (u, v, uo, vo)((u, v, uo, vo)ϵΛv & vtxo > 0 (utyo > 0) & vtxo + vo = utyo + uo ⟹ gv(u, v, uo, vo) = 0).
And there is not any relatively strongest weight in Λc for DMUo if for all (u, v, uo, vo)((u, v, uo, vo)ϵΛv & vtxo > 0 utyo > 0 & vtxo + vo = utyo + uo ⟹ gv(u, v, uo, vo) = 0).
Also there is not any strongest weight in Λv for DMUo if
for all (u, v, uo, vo)((u, v, uo, vo)ϵΛc & (u, v) > (0,0) & vtxo + vo = utyo + uo ⟹ gv(u, v, uo, vo) = 0).
Definition 10 (input-oriented (output-oriented) -efficiency). If (u ̅, v ̅, u ̅o, v ̅o) be relatively best weight in input-oriented (output-oriented) in Λc for DMUo , then input-oriented (output-oriented) Λv-efficiency of
fv(u ̅,̅ v,u ̅o,̅ vo)
DMUo = n Definition 11 (input-oriented (output-oriented) -efficient). DMUo is said to be in input-oriented (in output-oriented) Λv-efficient if input-oriented (output-oriented) Λc-efficiency of DMUo = 1.
Definition 12 ( -efficiency). If (u ̅, v ̅, u ̅o, v ̅o) be relatively best weight in Λc for DMUo, then efficiency
fv(u ̅,̅ v,u ̅o,̅ vo)
Λv = n .
Definition 13 ( -efficient). DMUo is said to be Λv-efficient if Λc-efficiency = 1.
Definition 14 (strictly -efficiency). If (u ̅, v ̅, u ̅o, v ̅o) be best weight in Λv, for DMUo then strictly Λv-
fv(u ̅,v ̅,u ̅o,v ̅o)
efficiency = n .
Definition 15 (strictly -efficient). DMUo is said to be strictly Λv-efficient if strictly Λv-efficiency of DMUo = 1.
Definition 16 (input-oriented (output-oriented) strongly -efficiency). If there is a (u ̅, v ̅, u ̅o, v ̅o) relatively strongest weight in input-oriented (output-oriented) in Λv for DMUo, then input-oriented (output-
gv(u ̅,v ̅,u ̅o,v ̅o)
oriented) strongly Λv-efficiency = n−1 .
Definition 17. If there is not any relatively strongest weight in input-oriented (output-oriented) in Λv for DMUo, then input-oriented (output-oriented) strongly Λc-efficiency = 0.
Definition 18 (input-oriented (output-oriented) strongly -efficient). DMUo is said to be input-oriented (output-oriented) strongly Λv-efficient if input-oriented (output-oriented)strongly Λv-efficiency of DMUo = 1. Definition 19 (strongly -efficiency). If there is a (u ̅, v ̅, u ̅o, v ̅o) relatively strongest weight in Λv for DMUo
gv(u ̅,v ̅,u ̅o,v ̅o)
Then strongly Λv-efficiency = n−1 .
Definition 20. If there is not any relatively strongest weight in Λv for DMUo, then strongly Λv-efficiency = 0. Definition 21 (strongly -efficient). DMUo is said to be strongly Λv-efficient if strongly Λv-efficiency of DMUo=1.
Proposition 1. Let (u ̅, v ̅, u ̅o, v ̅o) be relatively best weight in input-oriented (output-oriented) in Λv for DMUo, and let (u ̅, v ̅) > (0,0). Then (u ̅, v ̅, u ̅o, v ̅o) is best weight in Λv for DMUo.
Proof: if (u ̅, v ̅, u ̅o, v ̅o) not be best weight in Λv for DMUo, then by Definition 7, there is some (u ̃, v ̃, u ̃o, v ̃o)ϵΛv such that (u ̃, v ̃) > (0,0), v ̃txo + v ̃o = u ̃tyo + u ̃o and fv(u ̅, v ̅, u ̅o, v ̅o) < fv(u ̃, v ̃, u ̃o, v ̃o). On the other hand, since xj and yj are semipositive, v ̃txo + v ̃o > 0 (u ̃tyo + u ̃o > 0). Therefore, noting that (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo,.fv(u ̅, v ̅, u ̅o, v ̅o) ≥ fv(u ̃, v ̃, u ̃o, v ̃o) which is contradiction with this fact that fv(u ̅, v ̅, u ̅o, v ̅o) < fv(u ̃, v ̃, u ̃o, v ̃o). Thus (u ̅, v ̅, u ̅o, v ̅o) is best weight in Λv for DMUo.
Proposition 2. If (u ̅, v ̅, u ̅o, v ̅o) be relatively best weight in Λv for DMUo, and if (u ̅, v ̅) > (0,0). Then (u ̅, v ̅, u ̅o, v ̅o) is best weight in Λv for DMUo.
Proof: similar to the proof of Proposition 1.
Proposition 3. If there is a (u ̅, v ̅, u ̅o, v ̅o) relatively strongest weight in input-oriented (output-oriented) in Λv for DMUo, and if u ̅tyo > 0 (v ̅txo > 0 ). Then (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in Λc for DMUo.
Proof: if (u ̅, v ̅, u ̅o, v ̅o) be relatively strongest weight in input-oriented (output-oriented) in Λc for DMUo, and if u ̅tyo > 0 (v ̅txo > 0 ), then
for all (u, v, uo, vo)((u, v, uo, vo)ϵΛv & vtxo > 0 (utyo > 0) & vtxo + vo = utyo + uo ⟹ gv(u ̅, v ̅, u ̅o, v ̅o) ≥ gv(u, v, uo, vo)). (8) Now we assume (u ̃, v ̃, u ̃o, v ̃o)ϵΛv be relatively best weight in Λv for DMUo, then v ̃txo > 0, u ̃tyo > 0, v ̃txo + v ̃o = u ̃tyo + u ̃o,
and fv(u ̅, v ̅, u ̅o, v ̅o) ≤ fv(u ̃, v ̃, u ̃o, v ̃o).
Also, by Eq. (8), we have gv(u ̅, v ̅, u ̅o, v ̅o) ≥ gv(u ̃, v ̃, u ̃o, v ̃o).
(10) On the other hand, by Definition 1 and Definition 2, we have fv(u ̅, v ̅, u ̅o, v ̅o) ≥ gv(u ̅, v ̅, u ̅o, v ̅o),
and
fv(u ̃, v ̃, u ̃o, v ̃o) ≥ gv(u ̃, v ̃, u ̃o, v ̃o). (11) By Eqs. (1)-(4), we have fv(u ̅, v ̅, u ̅o, v ̅o) = fv(u ̃, v ̃, u ̃o, v ̃o) = gv(u ̅, v ̅, u ̅o, v ̅o) = gv(u ̃, v ̃, u ̃o, v ̃o).
Thus (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in Λc for DMUo.
Proposition 4. If there is a (u ̅, v ̅, u ̅o, v ̅o) relatively strongest weight in input-oriented (output-oriented) in Λv for DMUo, and if u ̅tyo > 0 (v ̅txo > 0 ). Then (u ̅, v ̅, u ̅o, v ̅o) is relatively strongest weight in Λv for DMUo. Proof: similar to the proof of Proposition 3.
Proposition 5. If there is a (u ̅, v ̅, u ̅o, v ̅o) relatively strongest weight in input-oriented (output-oriented) in Λv for DMUo, and if (u ̅, v ̅) > (0,0), Then (u ̅, v ̅, u ̅o, v ̅o) is strongest weight in Λv for DMUo.
Proof: Similar to the proof of Proposition 3.
Proposition 6. If there is a (u ̅, v ̅, u ̅o, v ̅o) relatively strongest weight in Λc for DMUo, and if (u ̅, v ̅) > (0,0), Then (u ̅, v ̅, u ̅o, v ̅o) is strongest weight in Λv for DMUo.
Proof: Similar to the proof of Proposition 3.
Theorem 1. Let (u ̅, v ̅, u ̅o, v ̅o) be relatively best weight in input-oriented (output-oriented) in Λc for DMUo, let p = fv(u ̅, v ̅, u ̅o, v ̅o) let
{DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̅txj + v ̅o ≥ u ̅tyj + u ̅o}, (12)
and let t̅ = (t̅1, … , t̅n) with 0, jϵ{j1, … , jp}, 1, jϵ{1, … , n} − {j , … , j }. (13)
Then (u ̅, v ̅, u ̅o, v ̅o, t̅) is an optimal solution of the following model:
vtxo + vo − (utyo + uo) = 0, vtxo ≥ ε (utyo ≥ ε), s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 0 , for all j ( Μ ≫ 0), u ≥ 0, v ≥ 0, uo ≥ 0, vo ≥ 0, tjϵ{0,1}.
Conversely, if (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution of Model (4), then (u ̃, v ̃, u ̃o, v ̃o) is a relatively best weight in input-oriented (output-oriented) in Λc for DMUo.
Proof: since (u ̅, v ̅, u ̅o, v ̅o) is a relatively best weight in input-oriented (output-oriented) in Λc for DMUo, then, by Eqs. (5) and (6), (u ̅, v ̅, u ̅o, v ̅o, t) ̅ is a feasible solution of Model (4). On the other hand, since (u ̃, v ̃, u ̃o, v ̃o, t̃) is an optimal solution for Model (4), we have n − ∑n j=1 t̃j ≥ n − ∑n j=1 t̅j, therefore fv(u ̃, v ̃, u ̃o, v ̃o) ≥ fv(u ̅, v ̅, u ̅o, v ̅o). Also fc(u ̃, v ̃, u ̃o, v ̃o) ≤ fc(u ̅, v ̅, u ̅o, v ̅o), since (u ̅, v ̅) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo.
Hence, fv(u ̃, v ̃, u ̃o, v ̃o) = fv(u ̅, v ̅, u ̅o, v ̅o) = n − ∑n j=1 t̅j = n − ∑n j=1 t̃j.
Thus (u ̅, v ̅, u ̅o, v ̅o, t) ̅ is an optimal solution to Model (4), and (u ̃, v ̃, u ̃o, v ̃o) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo.
Theorem 2. Let (u ̅, v ̅, u ̅o, v ̅o, t) ̅ be an optimal solution of the following model:
vtxo + vo − (utyo + uo) = 0, vtxo = 1 (utyo = 1), (14) s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 0 , for all j ( Μ ≫ 0), u ≥ 0, v ≥ 0, uo ≥ 0, vo ≥ 0, tjϵ{0,1}.
Then (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo.
Proof: let (u ̃, v ̃, u ̃o, v ̃o) be relatively best weight in input-oriented (output-oriented) in Λc for DMUo, let p = fc(u ̃, v ̃, u ̃o, v ̃o), let {DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̃txj + v ̃o ≥ u ̃tyj + u ̃o},
Then
v ̃txo + v ̃o = u ̃tyo + u ̃o, v ̃txo > 0 (u ̃tyo > 0).
So that by taking k = v ̃txo(k = u ̃tyo), u ̂ = u ̃⁄k, v ̂ = v ̃⁄k, u ̂o = u ̃o⁄k, and v ̂o = v ̃o⁄k, we have v ̃txo > 0 (u ̃tyo > 0), (u ̂, v ̂) ≥ (0,0), v ̂txo + v ̂o = ( u ̂tyo + u ̂o), v ̂txji + v ̂o − (u ̂tyj + u ̂o) ≥ 0, i = 1, … , p. (15)
Thus (u ̂, v ̂, u ̂o, v ̂o, t̂) is a feasible solution for Model (15), where t̂ = (t̂1, … , t̂n) with
0 jϵ{j1, … , jp}.
t̂j = { , 1 jϵ{1, … , n} − {j1, … , jp}
Therefore, sine (u ̅, v ̅, u ̅o, v ̅o, t) ̅ is an optimal solution for Model (5), n − ∑n j=1 t̂j ≤ n − ∑n j=1 t̅j. Hence fc(u ̂, v ̂, u ̂o, v ̂o) ≤ fc(u ̅, v ̅, u ̅o, v ̅o). (16) Also by Model (15)
fc(u ̃, v ̃, u ̃o, v ̃o) ≤ fv(u ̂, v ̂, u ̂o, v ̂o). (17) On the other hand, since (u ̃, v ̃, u ̃o, v ̃o) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo and v ̃txo + v ̃o = u ̃tyo + u ̃o, v ̃txo > 0 (u ̃tyo > 0).
We have
fc(u ̃, v ̃, u ̃o, v ̃o) ≥ fc(u ̅, v ̅, u ̅o, v ̅o). (18) Thus, by Models (16)-(18), fc(u ̃, v ̃, u ̃o, v ̃o) = fc(u ̅, v ̅, u ̅o, v ̅o),
Therefore (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo.
Corollary 1. Let (u ̅, v ̅, u ̅o, v ̅o) be an optimal solution of Model (5), then input-oriented (output-oriented) Λv-efficiency of
f (u ̅,̅ v,u ̅ ,̅ v ) n−∑n j=1 t̅j
DMUo = c n o o = n .
Proof: Theorem 2.
Corollary 2. DMUo is input-oriented (output-oriented) Λv-efficient if only if the optimal objective function value of Model (5) is zero.
Proof: Theorem 2.
Corollary 3. If there is some optimal solution(u ̅, v ̅, u ̅o, v ̅o) for Model (5) such that v ̅txo > 0 (u ̅tyo > 0), then (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in Λc for DMUo.
Proof: Proposition 1 and Theorem 2.
Corollary 4. If there is some optimal solution(u ̅, v ̅, u ̅o, v ̅o) for Model (5) such that (u ̅, v ̅) > (0,0), then (u ̅, v ̅, u ̅o, v ̅o) is best weight in Λc for DMUo.
Proof: Proposition 2 and Theorem 2.
Corollary 5. If there is some optimal solution(u ̅, v ̅, t̅) for Model (2) such that (u ̅, v ̅) > (0,0), and ∑n j=1 t̅j = 0, then DMUo is strictly Λc-efficient.
Proof: Proposition 1 and Theorem 2.
Theorem 3. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution of the following model:
vtxo + vo − (utyo + uo) = 0, vtxo ≥ ε, utyo ≥ ε, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 0 , for all j ( Μ ≫ 0), u ≥ 0, v ≥ 0, uo ≥ 0, vo ≥ 0, tjϵ{0,1}.
Then (u ̃, v ̃, u ̃o, v ̃o) is a relatively best weight in Λc for DMUo Conversely, let (u ̅, v ̅, u ̅o, v ̅o) be relatively best weight in Λc for DMUo, Then (u ̅, v ̅, u ̅o, v ̅o, t) ̅ , where t̅ is defined by Models (5) and (6) is an optimal solution of Model (3).
Proof: similar to the proof of Theorem 1.
Theorem 4. Let (u ̅, v ̅, u ̅o, v ̅o, t̅) be an optimal solution of the following model:
vtxo + vo − (utyo + uo) = 0, vtxo ≥ 1, utyo ≥ 1, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 0 , for all j ( Μ ≫ 0), u ≥ 0, v ≥ 0, uo ≥ 0, vo ≥ 0, tjϵ{0,1}.
Then (u ̅, v ̅, u ̅o, v ̅o) is relatively best weight in input-oriented (output-oriented) in Λc for DMUo.
Proof: similar to the proof of Theorem 2.
Corollary 6. Let (u ̅, v ̅, u ̅o, v ̅o) be an optimal solution of Model (4), then Λv-efficiency of
f (̅,̅,u ̅ ,̅ v ) n−∑ = t̅
DMUo = c n o o = n .
Proof: Theorem 4.
Corollary 7. DMUo is Λv-efficient if only if the optimal objective function value of Model (4) is zero.
Proof: Theorem 4.
Corollary 8. If there is some optimal solution(u ̅, v ̅, u ̅o, v ̅o) for Model (4) such that (u ̅, v ̅) > (0,0), then (u ̅, v ̅, u ̅o, v ̅o) is best weight in Λc for DMUo.
Proof: Proposition 2 and Theorem 4.
Corollary 9. If there is some optimal solution(u ̅, v ̅, t̅) for Model (5) such that (u ̅, v ̅) > (0,0), and ∑n j=1 t̅j = 0, then DMUo is strictly Λc-efficient.
Proof: Proposition 1 and Theorem 4.
Theorem 5. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution of the following model:
vtxo + vo − (utyo + uo) = 0, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 0 , for all j ( Μ ≫ 0), u ≥ 1ε, v ≥ 1ε, uo ≥ 0, vo ≥ 0, tjϵ{0,1}.
Then (u ̃, v ̃, u ̃o, v ̃o) is a relatively best weight in Λv for DMUo.
Conversely let (u ̅, v ̅, u ̅o, v ̅o) be best weight in Λc for DMUo. Then (u ̅, v ̅, u ̅o, v ̅o, t̅), where t̅ is defined by Models (5) and (6), is an optimal solution of Model (5).
Proof: by Remark 1 there is a (u, v, uo, vo)ϵΛv such that vtxo + vo = utyo + uo, u ≥ 1ε, v ≥ 1ε.
Thus (u, v, uo, vo, t), where t is defined by Models (5) and (6), is a solution feasible for Model (6).
Let (u ̅, v ̅, u ̅o, v ̅o) be best weight in Λv for DMUo, let p = fc(u ̅, v ̅, u ̅o, v ̅o), and let {DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̅txo ≥ u ̅tyo},
Then (u ̅, v ̅) > (0,0), v ̅txo + v ̅o = u ̅tyo + u ̅o, v ̅txji + v ̅o − (u ̅tyji + u ̅o) ≥ 0, i = 1, … , p.
Thus by taking k = min {min{u ̅r} , min{v ̅i}} we have (u ̅, v ̅) ≥ k(1,1), v ̅txo + v ̅o = u ̅tyo + u ̅o, v ̅txji + v ̅o − (u ̅tyji + u ̅o) ≥ 0, i = 1, … , p.
Thus (u ̅, v ̅, u ̅o, v ̅o, t) ̅ , where t̅ is defined by Model (6) is a feasible solution for Model (5), and since (u ̃, v ̃, u ̃o, v ̃o, t̃) is an optimal solution of the model, therefore n − ∑n j=1 t̃j ≥ n − ∑n j=1 t̅j. Hence fc(u ̃, v ̃, u ̃o, v ̃o, t̃) ≥ fc(u ̅, v ̅, u ̅o, v ̅o),
Also fc(u ̃, v ̃, u ̃o, v ̃o, t̃) ≤ fc(u ̅, v ̅, u ̅o, v ̅o),
Since (u ̅, v ̅, u ̅o, v ̅o) is a best weight in Λc for DMUo. Thus fc(u ̃, v ̃, u ̃o, v ̃o, t̃) = fc(u ̅, v ̅, u ̅o, v ̅o) = n − ∑n j=1 t̅j = n − ∑n j=1 t̃j.
It follows that also (u ̃, v ̃, u ̃o, v ̃o, t̃) is best weight in Λc for DMUo, and (u ̅, v ̅, u ̅o, v ̅o) is an optimal solution of Model (5).
Theorem 6. Let (u ̅, v ̅, u ̅o, v ̅o, t̅) be an optimal solution of the following model:
vtxo + vo − (utyo + uo) = 0, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 0 , for all j ( Μ ≫ 0), u ≥ 1, v ≥ 1, uo ≥ 0, vo ≥ 0, tjϵ{0,1}.
Then (u ̅, v ̅, u ̅o, v ̅o) is best weight in input-oriented (output-oriented) in Λc for DMUo.
Proof: similar to the proof of Theorem 5.
Corollary 10. Let (u ̅, v ̅, u ̅o, v ̅o) be an optimal solution of Model (6), then strictly Λv-efficiency of
f (̅,̅,u ̅ ,̅ v ) n−∑ = t̅
DMUo = v n o o = n .
Proof: Theorem 6.
Corollary 11. DMUo is strictly Λ -efficient if only if the optimal objective function value of Model (6) be zero. Proof: Theorem 6.
Theorem 7. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution for the following model:
vtxo + vo − (utyo + uo) = 0 , vtxo ≥ ε (utyo ≥ ε), s. t. vtxj + vo − (utyj + uo) + Μtj ≥ ε, j ≠ o ( Μ ≫ 0), u ≥ 0, v ≥ 0, tjϵ{0,1}, j ≠ o,.
where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n). Then if ∑j≠o t̃j < n − 1, (u ̃, v ̃, u ̃o, v ̃o) is a relatively strongest weight in input-oriented (output-oriented) in Λc for DMUo. But if ∑j≠o t̃j = n − 1, there is not any relatively strongest weight in input-oriented (output-oriented)in Λc for DMUo. Conversely if there is not any relatively strongest weight in input-oriented (output-oriented) in Λc for DMUo, then the optimal objective function value of Model (7) is equal to n − 1. But if there exists a (u ̅, v ̅, u ̅o, v ̅o)ϵΛv such that (u ̅, v ̅, u ̅o, v ̅o) is relatively strongest weight in input-oriented (output-oriented) in Λv for DMUo, then by taking p = gv(u ̅, v ̅, u ̅o, v ̅o), {DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n} & v ̅txj + v ̅o ≥ u ̅tyj + u ̅o}. (19)
And t̅ = (t̅1, … , t̅o−1, t̅o+1, … , t̅n).
With 0, jϵ{j1, … , jp}, t̅j = { 1, jϵ{1, … , n} − {j , … , j , o}. (20)
(u ̅, v ̅, u ̅o, v ̅o, t) ̅ is an optimal solution for Model (8) and ∑j≠o t̅j < n − 1.
Proof: if there is a (u ̅, v ̅, u ̅o, v ̅o)ϵΛv such that (u ̅, v ̅, u ̅o, v ̅o) is relatively strongest weight in input-oriented (output-oriented) in Λc for DMUo, then gv(u ̅, v ̅, u ̅o, v ̅o) ≥ 1 and (u ̅, v ̅, u ̅o, v ̅o, t) ̅ , where t̅ is defined by Models (11) and (12), is a feasible solution for Model (7). Thus, since (u ̃, v ̃, u ̃o, v ̃o, t̃) is an optimal solution for Model (7), we have∑j≠o t̃j ≤ ∑j≠o t̅j. Therefore, gv(u ̃, v ̃, u ̃o, v ̃o) ≥ gv(u ̅, v ̅, u ̅o, v ̅o) ≥ 1.
Thus the optimal objective function value of Model (7) is less n − 1. On the other hand, gc(u ̃, v ̃, u ̃o, v ̃o) ≤ gc(u ̅, v ̅, u ̅o, v ̅o), since (u ̅, v ̅, u ̅o, v ̅o) is relatively strongest weight in Λc for DMUo. Consequently, gc(u ̃, v ̃, u ̃o, v ̃o) = gc(u ̅, v ̅, u ̅o, v ̅o). Thus (u ̅, v ̅, u ̅o, v ̅o, t) ̅ is an optimal solution for Model (7), and (u ̃, v ̃, u ̃o, v ̃o) is a relatively strongest weight in input-oriented (output-oriented in Λc for DMUo. Also it easy to show, if there is not any relatively strongest weight in input-oriented (output-oriented in Λc for DMUo, then the optimal objective function value of Model (7) is equal to n − 1.
Theorem 8. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution for the following model,
vtxo + vo − (utyo + uo) = 0 , vtxo ≥ 1 (utyo ≥ 1), s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 1, j ≠ o ( Μ ≫ 0), u ≥ 0, v ≥ 0, tjϵ{0,1}, j ≠ o,
where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n). Then if ∑j≠o t̃j < n − 1, (u ̃, v ̃, u ̃o, v ̃o) is a relatively strongest weight in input-oriented (output-oriented) in Λc for DMUo. But if ∑j≠o t̃j = n − 1, there is not any relatively strongest weight in input-oriented (output-oriented)in Λc for DMUo.
Proof: similar to the proof of Theorem 7.
Corollary 12. let (u ̅, v ̅, u ̅o, v ̅o) be an optimal solution of Model (8), then input-oriented (output-oriented) Λv-efficiency of,
gv(̅,̅,u ̅o,̅ vo) (n−1)−∑j≠o tj
DMUo = n−1 = n−1 .
Proof: Theorem 8.
Corollary 13. DMUo is input-oriented (output-oriented) Λv-efficient if only if the optimal objective function value of Model (6) is zero.
Proof: Theorem 8.
Theorem 9. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution for the following model:
vtxo + vo − (utyo + uo) = 0 , vtxo ≥ ε , utyo ≥ ε, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ ε, j ≠ o ( Μ ≫ 0), u ≥ 0, v ≥ 0, tjϵ{0,1}, j ≠ o,
where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n). Then if ∑j≠o t̃j < n − 1, (u ̃, v ̃, u ̃o, v ̃o) is a relatively strongest weight in Λc for DMUo. But if ∑j≠o t̃j = n − 1, there is not any relatively strongest weight in Λc for DMUo. Conversely if there is not any relatively strongest weight in Λc for DMUo, then the optimal objective function value of Model (9) is equal to n − 1. But if there exists a (u ̅, v ̅, u ̅o, v ̅o)ϵΛv such that (u ̅, v ̅, u ̅o, v ̅o) is relatively strongest weight in Λv for DMUo, then (u ̅, v ̅, u ̅o, v ̅o, t̅), where t̅ is defined by Models (10) and (11), is an optimal solution for Model (9) and ∑j≠o t̅j < n − 1.
Proof: similar to the proof of Theorem 7.
Theorem 10. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution for the following model,
vtxo + vo − (utyo + uo) = 0 , vtxo ≥ 1 , utyo ≥ 1, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 1, j ≠ o ( Μ ≫ 0), u ≥ 0, v ≥ 0, tjϵ{0,1}, j ≠ o.
where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n). Then if ∑j≠o t̃j < n − 1, (u ̃, v ̃, u ̃o, v ̃o) is a relatively strongest weight in Λc for DMUo. But if ∑j≠o t̃j = n − 1, there is not any relatively strongest weight in Λc for DMUo.
Proof: similar to the proof of Theorem 7.
Corollary 14. let (u ̅, v ̅, u ̅o, v ̅o) be an optimal solution of Model (10), then strongly Λv-efficiency of,
gv(̅,̅,u ̅o,̅ vo) (n−1)−∑j≠o tj
DMUo = n−1 = n−1 .
Proof: Theorem 10.
Corollary 14. DMUo is strongly Λv-efficient if only if the optimal objective function value of Model (10) be zero.
Theorem 11. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution for the following model:
vtxo + vo − (utyo + uo) = 0, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ ε, j ≠ o ( Μ ≫ 0), u ≥ 1ε, v ≥ 1ε, tjϵ{0,1}, j ≠ o,
where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n). Then if ∑j≠o t̃j < n − 1, (u ̃, v ̃, u ̃o, v ̃o) is a strongest weight) in Λc for DMUo. But if ∑j≠o t̃j = n − 1, there is not any in Λc for DMUo. Conversely if there is not any strongest weight in Λc for DMUo, then the optimal objective function value of Model (11) is equal to n − 1. But if there exists a (u ̅, v ̅, u ̅o, v ̅o)ϵΛv such that (u ̅, v ̅, u ̅o, v ̅o) is strongest weight in Λv for DMUo, then (u ̅, v ̅, u ̅o, v ̅o, t̅), where t̅ is defined by Models (11) and (12), is an optimal solution for Model (11) and ∑j≠o t̅j < n − 1.
Proof: similar to the proof of Theorem 7.
Theorem 12. Let (u ̃, v ̃, u ̃o, v ̃o, t̃) be an optimal solution for the following model,
vtxo + vo − (utyo + uo) = 0, s. t. vtxj + vo − (utyj + uo) + Μtj ≥ 1, j ≠ o ( Μ ≫ 0), u ≥ 1, v ≥ 1, tjϵ{0,1}, j ≠ o,
where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n). Then if ∑j≠o t̃j < n − 1, (u ̃, v ̃, u ̃o, v ̃o) is strongest weight in Λc for DMUo. But if ∑j≠o t̃j = n − 1, there is not any strongest weight in Λc for DMUo.
Proo: similar to the proof of Theorem 7.
Theorem 13. DMUo is input oriented (output oriented) Λc-efficient if only if DMUo is input oriented BCC efficient.
Proof: let DMUo be input oriented Λc-efficient, then, by Corollary 2, the optimal objective function value of Model (5) is zero. Thus letting (u ̅, v ̅, u ̅o, v ̅o, t̅) be an optimal solution for Model (5), we have u ̅ ≥ 0, v ̅ ≥ 0, v ̅txo + v ̅o = u ̅tyo + u ̅o, v ̅txo = 1(u ̅tyo = 1), v ̅txj + v ̅o − (u ̅tyj + u ̅o) ≥ 0 , j = 1, . . , n.
Hence (u ̅, v ̅, w ̅ o), where w ̅o = u ̅o − v ̅o, is an optimal solution for Model (2). therefore DMUo is input oriented (output oriented) BCC - efficient. Conversely, let DMUo is input oriented (output oriented) BCC - efficient, and let (u ̃, v ̃, u ̃o, v ̃o) is an optimal solution of Model (2), then u ̃ ≥ 0, v ̃ ≥ 0, v ̃txo = u ̃tyo + w ̃o, v ̅txo = 1(u ̃tyo = 1), v ̃txj − (u ̃tyj + w ̃o) ≥ 0 , j = 1, . . , n,
Thus (u ̃, v ̃, u ̃o, v ̃o, t̃), where w ̃o w ̃o ≥ 0, u ̅o = { 0 w ̃o < 0, 0 w ̃o ≥ 0, v ̅o = { −w ̃o w ̃o < 0.
and t̅ = 0ϵℝn, is a feasible solution for Model (2). Therefore the optimal objective function value of Model (5) is zero. Hence, by Corollary 2, DMUo is input oriented (output oriented) Λc-efficient.
Theorem 14. If DMUo be Λc- efficient, then DMUo is both input oriented BCC - efficient. and output oriented BCC – efficient.
Proof: let DMUo is Λc- efficient. Then, by Corollary 6, the optimal objective function value Model (7) is zero, thus letting (u ̅, v ̅, u ̅o, v ̅o, t̅) be an optimal solution for the model, we have u ̅ ≥ 0, v ̅ ≥ 0, v ̅txo + v ̅o = u ̅tyo + u ̅o, v ̅txo ≥ 1, u ̅tyo ≥ 1, v ̅txj + v ̅o − (u ̅tyj + u ̅o) ≥ 0 , j = 1, . . , n.
So that by taking k = v ̅txo, u ̃ = (u ̅⁄k),and v ̃ = (v ̅⁄k), we have u ̃ ≥ 0, v ̃ ≥ 0, v ̃txo + v ̃o = u ̃tyo + u ̃o, v ̃txo = 1, v ̃txj + v ̃o − (u ̃tyj + u ̃o) ≥ 0 , j = 1, . . , n.
Thus (u ̃, v ̃, w ̃o), where w ̃ o = u ̃o − v ̃o, is a feasible solution for Model (4), and since v ̃txo = 1 it follows that the optimal objective function value of Model (4) is equal to one. Hence DMUo is input oriented BCC - efficient. Similarly, we can show that DMUo is output oriented BCC – efficient.
Theorem 15. DMUo is strictly Λv- efficient if only if DMUo is BCC - efficient.
Proof: let DMUo is strictly Λv- efficient. Then, by Corollary 6, the optimal objective function value Model (7) is zero, thus letting (u ̅, v ̅, u ̅o, v ̅o, t) ̅ be an optimal solution for the model, we have u ̅ ≥ 1, v ̅ ≥ 1, v ̅txo + v ̅o = u ̅tyo + u ̅o, v ̅txj + v ̅o − (u ̅tyj + u ̅o) ≥ 0 , j = 1, . . , n.
Thus (u ̅, v ̅, w ̅o), where w ̅ o = u ̅o − v ̅o, is a feasible solution for the following model min vtxo − (utyo + w ̅o), vtxj − (utyj + w ̅ o) ≥ 0 for all j, s. t.
u ≥ 1, v ≥ 1,
which for the feasible solution the objective function value of the model is equal to zero. Therefore the optimal objective function value Model (9) is zero. Therefore, by strong duality theorem, the optimal objective function value of the following model,
max ∑ si− + ∑ sr +,
i=1 r=1
∑ λjxij + si− = xio for all i,
s. t. n ∑ λjyij − sr + = yro for all r,
λj ≥ 0, si− ≥ 0, sr + ≥ 0 for all j, for all i , for all r,
which is dual form of Model (9), equal to one. Consequently DMUo is BCC – efficient. It is easy to show that if DMUo is BCC – efficient, then DMUo is strictly Λv- efficient.
Theorem 16. DMUo is input oriented strongly Λc- efficient if only if DMUo is extreme BCC - efficient. Proof: let DMUo be input oriented strongly Λc- efficient. Then, by Corollary 3.8, the optimal objective function value Model (8) is zero, thus letting (u ̅, v ̅, u ̅o, v ̅o, t) ̅ be an optimal solution for the model, where t̅ = (t̅1, … , t̅o−1, t̅o+1, … , t̅n) = 0ϵℝn−1.
we have u ̅ ≥ 0, v ̅ ≥ 0, u ̅o ≥ 0, v ̅o ≥ 0, v ̅txo ≥ 1, v ̅txo + v ̅o = u ̅tyo + u ̅o, v ̅txj + v ̅o − (u ̅tyj + u ̅o) ≥ 1 , j ≠ o.
Thus (u ̅, v ̅, w ̅o), where w ̅ o = u ̅o − v ̅o, is a feasible solution for the following model min vtxo − (utyo + wo), vtxo − (utyo + wo) ≥ 0, vtxj − (utyj + wo) ≥ 1 j ≠ o, s. t.
vtxo ≥ 1, u ≥ 0, v ≥ 0, wois free.
Therefore, by strong duality theorem, the optimal objective function value of the following model
max θ + ε ∑ λj,
∑ λjxj + θxo ≤ xo,
∑ λjyj ≥ yo, s. t.
∑ λj = 1,
λj ≥ 0, for all j,
which is dual form of Model (9), equal to one. Hence, by Theorem 2, DMUo is extreme BCC-efficient. Conversely, if DMUo is extreme BCC-efficient, by Theorem 2 and Corollary 8, DMUo is input oriented strongly Λ - efficient.
In Section 3, we provided models to obtain non negative weights for inputs, nonnegative weights for outputs, a nonnegative scalar corresponding to inputs and a nonnegative scalar corresponding to outputs which for the weights and scalars, the number of which DMUs for each one its virtual output in addition to the scalar corresponding to inputs does not exceed (is less than, if any) its virtual input in addition to the scalar corresponding to inputs be maximum, provided that for DMU under evaluation, the virtual output in addition to the scalar corresponding to inputs does not exceed (is less than, if any) the virtual input in addition to the scalar corresponding to inputs and the virtual input will be positive. We called these weights and scalars, the relatively best weight in input-oriented (the relatively strongest weight in input-oriented, if any) for the DMU under evaluation, and if all of the weights be positive we called them best weight in input-oriented (the strongest weight in input-oriented, if any) for the DMU under evaluation. The relatively best weight in input-oriented (the relatively strongest weight in input-oriented) indicates normal vector of a superface in the PPS with VRS assumption that the DMU under evaluation is on the superface and the maximum number of which DMUs their performance are no worse than (is better than) the DMU under evaluation separate from the rest of DMUs, with the constraint that the virtual input be positive. Accordingly, we can interpret the rest of the definitions of non-negative weights for inputs and for outputs and nonnegative scalars related to inputs and outputs. Also in this paper, we presented the relationship between these definitions of efficiency with efficiency in the DEA models with VRS assumption. These normal vectors can be applied as a criterion for efficiency analysis and ranking of a set of peer DMUs with interval scale data. Specially, the relatively strongest weight in input-oriented (in output-oriented), both indicate extreme CCR-efficiency and provide a performance measure DMUo with interval scale inputs and/or outputs. Also the relatively strongest weight and the strongest weight can be applied for ranking extreme CCR-efficient DMUs and BCC-inefficient DMUs.
The authors declare that they have no conflict of interest regarding the publication of this paper.
No datasets were generated or analyzed during the current study. All necessary theoretical models and proofs are included within the article.
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.