In this section, we use the data recorded in
Data set.
This table is shown in the PDF version of the article.
By using the data set from
Example results.
This table is shown in the PDF version of the article.
In this paper, in order to evaluate the performance of a Decision Making Unit (DMU) in a Production Possible Set (PPS) with Constant-to-Scale (CRS) technology, we provide models to obtain nonnegative weights for inputs and outputs which for the weights, the number of which DMUs for each one its virtual output does not exceed (is less than, if any) its virtual input be maximum, provided that for DMU under evaluation, the virtual output will be equal to the virtual input and the virtual input will be positive. We call these weights the relatively best weight (the relatively strongest weight, if any) for the DMU under evaluation, and if all the weights are 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), respectively, as the ratio of the number of DMUs for each one for the relatively best weight and the best weight (the relatively strongest weight); its virtual input does not exceed (is less) its virtual input, to the total DMUs. The relatively best weight in input-oriented (the relatively strongest weight, if any) indicates the normal vector of a surface in the PPS with CRS assumption that the DMU under evaluation is on the surface and the maximum number of which DMUs their performance is 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, it can be interpreted the rest of the definitions of non-negative weights for inputs and outputs based on separation hyperplanes. Also, in this paper, we present the relationship between these definitions of efficiency with efficiency in the DEA models with constant returns to scale assumption.
The CCR Charnes et al.
In other words, the derived CCR multiplier form obtains nonnegative weights for the inputs and the outputs of the DMU under evaluation by maximizing the income resulting from the outputs of the DMU, provided that the income resulting from the outputs of each DMU does not exceed the cost resulting from the inputs, and the cost resulting from the inputs of the DMU under evaluation be positive. Thus, the derived CCR multiplier form evaluates DMU under evaluation in the best conditions.
In this paper, with respect to one's inspiration from the derived CCR multiplier form, to evaluate the performance of a DMU in comparison with a set of DMUs, we obtain nonnegative weights for inputs and outputs which for the weights, the number of which DMUs for each one its virtual output does not exceed (is less than, if any) its virtual input be maximum, provided that for DMU under evaluation, the virtual output will be equal to the virtual input and the virtual input will be positive.
In other words, we are going to obtain negative weights for the inputs and the outputs of DMU under evaluation per weighs, the number of which DMUs for each one its income does not exceed (is less than, if any) its cost be maximum, provided that for DMU under evaluation, its income will be equal to its cost and cost resulting from the inputs of the DMU will be positive
Also, we define efficiency and strict efficiency (strong efficiency), respectively, as the ratio of the number of DMUs for the relatively best weight and the best weight (the relatively strongest weight); those virtual inputs do not exceed (is less) those virtual input, to the total DMUs
Suppose we have n ≥ 2 peer observed DMUs,{DMUj: j = 1,2, … , n } which produce multiple outputsyrj, (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 xj and 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 Pc of the CCR model
Pc = {(x, y)|x ≥ ∑ λjxj&y ≤ ∑ λjyj&λj ≥ 0 j = 1, … , n},
j=1 j=1
where (λ1, … , λn) is a semipositive in ℝn.The input-oriented CCR model evaluates the efficiency of each DMUo by solving the following linear program:
θ∗ = min θ , ∑ λjxj ≤ θxo(a), n (1) s. t.
∑ λjyj ≥ yo (b), 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), λo = 1, then θ∗ ≤ 1. Thus 0 < θ∗ ≤ 1.θ∗represents the input-oriented CCR-efficiency value of DMUo.
Definition 1. (Radial efficiency): The performance of DMUo is radial efficient if and only if θ∗ = 1.
The dual problem of Model (1) is expressed as:
z∗ = max ut yo, vtxo = 1, t t (2) s. t. u yj ≤ v xj, j = 1,2, … , n.
u ≥ o, v ≥ o, where vϵℝm and uϵℝs are row vectors and represent dual variables corresponding to (a) and (b), respectively.From strong duality theorem θ∗ = z∗ , thus 0 < z∗ ≤ 1
The two-phase process for the CCR 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(c), j=1 (3) s. t. n ∑ λjyrj − sr + = yro , r = 1, … , s(d), λj ≥ 0, si− ≥ 0, sr + ≥ 0, for all j, for all i, for all r, where ε > 0 is the non-Archimedian element. The presence of a non-Archimedean element, ε, in the function of Model (2) effectively allows the minimization over θ to preempt the optimization involving the slacks s− = (s1 −, … , sm − ), s+ = (s1 +, … , ss +).
Definition 2. (CCR-efficient). The performance of DMUo is CCR-efficient if only if an optimal solution ((θ∗, λ∗, s∗−, s∗+)of the two-phase Model (2) satisfies θ∗ = 1, s−∗ = 0, s+∗ = 0.
The dual multiplier form of the program Model (2) is expressed as:
max ut yo, vtxo = 1, t t (4) s. t. u yj ≤ v xj , for all j, u ≥ 1ε, v ≥ 1ε.
optimal solution (θ∗, λ∗, s∗−, s∗+) of Model (2) such that θ∗ = 1 and (s∗−, s∗+) ≠ (0,0), thus DMUo is not extreme CCR-efficient.
Definition 6. Let Λc ⊂ ℝm+s be Λc = {(u, v)|uϵℝs&vϵℝm&(u, v) ≥ (0,0)}. (6) We define a map fc:Λc → ℕ⋃{0}.
By fc(u, v) = |{DMUj|jϵ{1,2, … , n}&vtxj ≥ utyj}|, (7) where symbol | . | is the cardinality of sets.
Definition 7. We define a map gc:Λc → ℕ⋃{0}.
By gc(u, v) = |{DMUj|jϵ{1,2, … , n}&vtxj > utyj}|, (8) where Λc is defined by Model (6).
Definition 8. Let (uo, vo)ϵΛc. We say (uo, vo) is a relatively best weight in Λc for DMUoif vo t xo = ut oyo&ut oyo > 0,
for all (u, v)((u, v)ϵΛc&utyo > 0 &vtxo = utyo ⟹ fc(uo, vo) ≥ fc(u, v)).
Definition 9. Let (uo, vo)ϵΛc. We say (uo, vo) is the best weight in Λcfor DMUoif vo t xo = ut oyo&(uo, vo) > (0,0),
for all (u, v)((u, v)ϵΛc&(u, v) > (0,0)&vtxo = utyo ⟹ fc(uo, vo) ≥ fc(u, v)).
Definition 10. Let (uo, vo)ϵΛc. We say (uo, vo) is relatively strongest weight in Λc for DMUo if vo t xo = ut oyo&ut oyo > 0 &gc(uo, vo) ≥ 1,
for all (u, v)((u, v)ϵΛc&utyo > 0 &vtxo = utyo ⟹ gc(uo, vo) ≥ gc(u, v)).
Definition 11. Let (uo, vo)ϵΛc. We say (uo, vo) is the strongest weight in Λc for DMUoif vo t xo = ut oyo&(uo, vo) > (0,0) &gc(uo, vo) ≥ 1, for all (u, v)((u, v)ϵΛc&(u, v) > (0,0)&vtxo = utyo ⟹ gc(uo, vo) ≥ gc(u, v)).
Remark 1. Since xj and yj are semi-positive, it follows
∑s r=1 yro > 0, ∑m i=1 xio > 0.
Now if ∑s r=1 yro = ∑m i=1 xio, then, by taking vt = (1, … ,1)ϵℝm, and ut = (1, … ,1)ϵℝs, we have vtxo − utyo, utyo > 0, u ≥ 1ε, v ≥ 1ε.
If ∑s r=1 yro > ∑m i=1 xio, then, by taking ∑s r=1 yro t s α = ( ⁄∑m x ),u = α(1, … ,1)ϵℝ ,
i=1 io
and, vt = α(1, … ,1)ϵℝm,
vtxo − utyo, utyo > 0, u ≥ 1ε, v ≥ 1ε.
Finally if ∑s r=1 yro < ∑m i=1 xio then, by taking ∑m i=1 xio β = ( ⁄∑s y ),
r=1 ro
ut = β(1, … ,1)ϵℝs,
vt = β(1, … ,1)ϵℝm,
vtxo − utyo, utyo > 0, u ≥ 1ε, v ≥ 1ε.
This shows that there is not any relatively strongest weight in Λc for DMUo if for all (u, v)((u, v)ϵΛc&utyo > 0 &vtxo = utyo ⟹ gc(u, v) = 0).
Also, there is no strongest weight in Λc for DMUo if for all (u, v)((u, v)ϵΛc&(u, v) > (0,0)&vtxo = utyo ⟹ gc(u, v) = 0).
Definition 12. (Λc-efficiency) If (uo, vo) be relatively best weight in Λc for DMUo, then Λc-efficiency of
fc(uo,vo)
DMUo= n Definition 13. (Λc-efficient) DMUo is said to be Λc-efficient if Λc-efficiency of DMUo = 1.
Definition 14. (Strictly Λc-efficiency) If (uo, vo) be the best weight in Λc, for DMUo then strictly
fc(uo,vo)
Λc- efficiency = n .
Definition 15. (Strictly Λc-efficient) DMUo is said to be strictly Λc-efficient if strictly Λc-efficiency of DMUo = 1.
Definition 16. (Strongly Λc-efficiency) If there is a (uo, vo) relatively strongest weight in Λc for DMUo. Then, strongly
gc(uo,vo)
Λc-efficiency = .
Definition 17. If there is not any relatively strongest weight in Λc for DMUo, then strongly Λc-efficiency =0. Definition 18. (Strongly Λc-efficient) DMUo is said to be strongly Λc-efficient if strongly Λc-efficiency of DMUo = 1.
Proposition 1. Let (u ̅, v ̅) be relatively best weight in Λcfor DMUo, and let (u ̅, v ̅) > (0,0). Then (u ̅, v ̅) is the best weight in Λc for DMUo.
Proof: If (u ̅, v ̅) not be the best weight in Λc for DMUo, then by Definition 4, there is some (u ̃, v ̃)ϵΛc such that (u ̃, v ̃) > (0,0), v ̃txo = u ̃tyo,
fc(u ̅, v ̅) < fc(u ̃, v ̃), Thus, since xj and yj are semi-positive, u ̃tyo(= v ̃txo) > 0. Therefore, noting that (u ̅, v ̅) is relatively best weight in Λc for DMUo,. fc(u ̅, v ̅) ≥ fc(u ̃, v ̃) which is in contradiction with the fact that fc(u ̅, v ̅) < fc(u ̃, v ̃). Thus (u ̅, v ̅) is the best weight in Λc for DMUo.
Proposition 2. Let (uo, vo) be relatively strongest weight in Λc for DMUo, and let (uo, vo) > (0,0). Then (uo, vo) is the strongest weight in Λc for DMUo.
Proof: Similar to the proof of Theorem 2.
Theorem 5. Let (u ̅, v ̅) be relatively best weight in Λc for DMUo, let p = fc(u ̅, v ̅)
{DMUj1, … , DMUjp} = {DMUj│jϵ{1, … , n} &̅ vtxo ≥ u ̅tyo}, (a1).
Letand let t̅ = (t̅1, … , t̅n) with 0 jϵ{j1, … , jp}, t̅j = { (a2) 1 jϵ{1, … , n} − {j1, … , jp}.
Then (u ̅, v ̅, t) ̅ is an optimal solution for the following model
vtx − uty = 0, uty ≥ ε, (9)
o o o
s. t. vtxj − utyj + Μtj ≥ 0, for all j, ( Μ ≫ 0) u ≥ 0, v ≥ 0, tjϵ{0,1}.
If (u ̃, v ̃, t̃) be an optimal solution of Model (9), then (u ̃, v ̃) is a relatively best weight in Λc for DMUo.
Proof: Since (u ̅, v ̅) is a relatively best weight in Λc for DMUo, then, by (a1) and (a2), (u ̅, v ̅, t) ̅ is a feasible solution to Model (9). Also, since (u ̃, v ̃, t̃) is an optimal solution for Model (9), we have
n − ∑ t̃j ≥ n − ∑ t̅j.
j=1 j=1
Therefore fc(u ̃, v ̃) ≥ fc(u ̅, v ̅),
fc(u ̃, v ̃) ≤ fc(u ̅, v ̅).
Since (u ̅, v ̅) is relatively best weight in Λc for DMUo. Hence, fc(u ̃, v ̃) = fc(u ̅, v ̅) = n − ∑n j=1 t̅j = n − ∑n j=1 t̃j.
Thus, (u ̅, v ̅, t) ̅ is an optimal solution to Model (9), and (u ̃, v ̃) is relatively best weight in Λc for DMUo.
Theorem 6. Let (u ̅, v ̅, t) ̅ be an optimal solution for the following model
vtx = 1, uty = 1, (10) s. t. vtxj − utyj + Μtj ≥ 0, for all j, ( Μ ≫ 0) u ≥ 0, v ≥ 0, tjϵ{0,1}.
Then, (u ̅, v ̅) is relatively best weight in Λc for DMUo.
Proof: Let (u ̃, v ̃) be relatively best weight in Λc for DMUo, let p = fc(u ̃, v ̃) with
{DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̃txo ≥ u ̃tyo}, Then v ̃txo = u ̃tyo, u ̃tyo > 0,
v ̃txji − u ̃tyji ≥ 0, i = 1, … , p.
So that, by taking k = u ̃tyo, u ̂ = u ̃⁄k, and v ̂ = v ̃⁄k, we have (u ̂, v ̂) ≥ (0,0), v ̂txo = u ̂tyo = 1, v ̂txji − u ̂tyji ≥ 0, i = 1, … , p. (a3).
Thus (u ̂, v ̂, t̂) is a feasible solution for Model (10), where t̂ = (t̂1, … , t̂n) with 0 jϵ{j1, … , jp}, t̂j = { 1 jϵ{1, … , n} − {j1, … , jp}.
Therefore, (u ̅, v ̅, t) ̅ is an optimal solution for Eq. (10), n − ∑n j=1 t̂j ≤ n − ∑n j=1 t̅j).
Hence fc(u ̂, v ̂) ≤ fc(u ̅, v ̅) (a4), by (a4),p ≤ fc(u ̂, v ̂).
Thus, since (u ̃, v ̃) is relatively best weight in Λc for DMUo, p = fc(u ̃, v ̃) = fc(u ̂, v ̂). (a5).
Also, since (u ̃, v ̃) is relatively best weight in Λc for DMUo, we have fc(u ̃, v ̃) ≥ fc(u ̅, v ̅), then, by (a3), (a4), and (a5), fc(u ̃, v ̃) = fc(u ̅, v ̅), Thus (u ̅, v ̅) is relatively best weight in Λc for DMUo.
Corollary 1. Let (u ̅, v ̅, t) ̅ be an optimal solution of Model (10), then
f (u ̅,̅ v) n−∑n j=1 t̅j
Λc-efficiency of DMUo= c n = n .
Proof: Theorem (6).
Corollary 2. DMUo is Λc-efficient if and only if the optimal objective function value of Model (10) is zero. Proof: Theorem (6).
Corollary 3. If there is some optimal solution (u ̅, v ̅, t̅) for Model (10) such that (u ̅, v ̅) > (0,0), then (u ̅, v ̅) is the best weight in Λc for DMUo.
Proof: Proposition 1 and Theorem (7).
Corollary 4. If there is some optimal solution (u ̅, v ̅, t) ̅ for Model (10) such that (u ̅, v ̅) > (0,0), and ∑n j=1 t̅j = 0, then DMUo is strictly Λc-efficient.
Proof: Proposition 1and Theorem (7).
Theorem 7. Let (u ̃, v ̃, t̃) be an optimal solution for the following model
min ∑ tj, vtx − uty = 0, (11) s. t. vtxj − utyj + Μtj ≥ 0, for all j, ( Μ ≫ 0) u ≥ 1ε, v ≥ 1ε, tjϵ{0,1}.
Then (u ̃, v ̃) is the best weight in Λc for DMUo.
Proof: First, let us show Model (11) is feasible. Since x j and yj are semi-positive, it follows that ∑s r=1 yro > 0, ∑m i=1 xio > 0.
Now, if ∑s r=1 yro = ∑m i=1 xio, then, by taking vt = (1, … ,1)ϵℝm, and ut = (1, … ,1)ϵℝs, we have vtxo − utyo, u ≥ 1ε, v ≥ 1ε.
If ∑s r=1 yro > ∑m i=1 xio, Then, by taking ∑s r=1 yro t s α = ( ⁄∑m x ), u = α(1, … ,1)ϵℝ ,
i=1 io
vt = α(1, … ,1)ϵℝm,
vtxo − utyo, u ≥ 1ε, v ≥ 1ε.
If ∑ yro < ∑ xio,
r=1 i=1
Then, by taking ∑m i=1 xio t s β = ( ⁄∑s y ), u = β(1, … ,1)ϵℝ ,
r=1 ro
vt = β(1, … ,1)ϵℝm,
vtxo − utyo, u ≥ 1ε, v ≥ 1ε.
Thus (u, v, t), where t is defined by (a1) and (a2), is a feasible solution for Model (11).
Let (u ̅, v ̅) be the best weight in Λc for DMUo, let p = fc(u ̅, v ̅), and let {DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̅txo ≥ u ̅tyo}, Then (u ̅, v ̅) ≥ (0,0), v ̅txo = u ̅tyo, v ̅txji − u ̅tyji ≥ 0, i = 1, … , p.
Taking k = min {u ̅tyo, min{u ̅r} , min{v ̅i}},
(u ̅, v ̅) ≥ k(1,1), v ̅txo = u ̅tyo, u ̅tyo ≥ k(> 0), v ̅txji − u ̅tyji ≥ 0, i = 1, … , p.
Thus (u ̅, v ̅, t) ̅ , where t̅ is defined by (a2), is a feasible solution for Model (11), and since (u ̃, v ̃, t̃) is an optimal solution of Model (11); therefore, n − ∑n j=1 t̃j ≥ n − ∑n j=1 t̅j.
Hence fc(u ̃, v ̃) ≥ fc(u ̅, v ̅). Also fc(u ̃, v ̃) ≤ fc(u ̅, v ̅), since (u ̅, v ̅) is the best weight in Λc for DMUo. Thus fc(u ̃, v ̃) = fc(u ̅, v ̅) = n − ∑n j=1 t̅j = n − ∑n j=1 t̃j.
Then it follows that also (u ̃, v ̃) is the best weight in Λc for DMUo, and (u ̅, v ̅, t̅) is an optimal solution of Model (11).
Theorem 8. Let (u ̃, v ̃, t̃) be an optimal solution for the following model
vtxo − utyo = 0, (12) s. t. vtxj − utyj + Μtj ≥ 0, for all j, ( Μ ≫ 0) u ≥ 1, v ≥ 1, tjϵ{0,1}.
Then (u ̃, v ̃) is the best weight in Λc for DMUo.
Proof: Similar to the proof of Theorem 5.
Corollary 5. Let (u ̅, v ̅, t) ̅ be an optimal solution of Model (12), then strictly
f (u ̅,̅ v) n−∑n j=1 t̅j
Λc-efficiency of DMUo= c n = n .
Proof: Follows from Theorem (8).
Corollary 6. DMUo is strictly Λc-efficient if only if Model (12) has an optimal objective function value of zero. Proof: Follows from Theorem (8).
Theorem 9. Let (u ̃, v ̃, t̃) be an optimal solution for the following model
vtxo − utyo = 0, vtxo ≥ ε, (13) s. t. vtxj − utyj + Μtj ≥ ε, j ≠ o, (Μ ≫ 0), u ≥ 0, v ≥ 0, tjϵ{0,1}, j ≠ o, where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n).
If ∑j≠o t̃j < n − 1, then (u ̃, v ̃) is a relatively strongest weight in Λc for DMUo, but if ∑j≠o t̃j = n − 1, then 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 (13) is equal to n − 1, but if there exists a (u ̅, v ̅)ϵΛc such that (u ̅, v ̅) is relatively strongest weight in Λc for DMUo, then by taking {DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̅txj > u ̅tyj} (a6).
With p = gc(u ̅, v ̅), and t̅ = (t̅1, … , t̅o−1, t̅o+1, … , t̅n).
With 0 jϵ{j1, … , jp}, t̅j = { ( a7) 1 jϵ{1, … , n} − {j1, … , jp, o}.
(u ̅, v ̅, t) ̅ is an optimal solution for Model (13) and ∑j≠o t̅j < n − 1.
Proof: Let there be a (u ̅, v ̅)ϵΛc such that (u ̅, v ̅) is relatively strongest weight in Λc for DMUo, then gc(u ̅, v ̅) ≥ 1, so by taking that p = gc(u ̅, v ̅) with {DMUj1, … , DMUjp} = {DMUj|jϵ{1, … , n}&v ̅txj > u ̅tyj},
u ̅ ≥ 0, ̅ v ≥ 0, v ̅txo = u ̅tyo , v ̅txo > 0, v ̅txji − u ̅tyi > 0 , i = 1, . . , p.
So by taking k = min {u ̅tyo, min{v ̅txji − u ̅tyi}}, We have k > 0 and u ̅ ≥ 0, v ̅ ≥ 0, v ̅txo = u ̅tyo , v ̅txo ≥ k, v ̅txji − u ̅tyi ≥ k , i = 1, . . , p.
Thus (u ̅, v ̅, t) ̅ is a feasible solution for Model (13) where t̅ is defined by (a6) and (a4), therefore, the optimal objective function value of Model (13) is less n − 1. Also, since (u ̃, v ̃, t̃) is an optimal solution for the Model (13), n − 1 − ∑j≠o t̃j ≥ n − 1 − ∑j≠o t̅j.
It follows gc(u ̃, v ̃) ≥ gc(u ̅, v ̅). On the other hand, gc(u ̃, v ̃) ≤ gc(u ̅, v ̅), since (u ̅, v ̅) is relatively strongest weight in Λc for DMUo. Consequently, gc(u ̃, v ̃) = gc(u ̅, v ̅). Thus (u ̅, v ̅, t) ̅ is an optimal solution for Model (13), and (u ̃, v ̃) is a relatively strongest weight in Λc for DMUo. Also, it is easy to show if there is not any relatively strongest weight in Λc for DMUo, then the optimal objective function value of Model (13) is equal to n − 1. Theorem 10. Let (u ̃, v ̃, t̃) be an optimal solution for the following model
vtxo − utyo = 0, vtxo ≥ 1, (14) s. t. vtxj − utyj + Μtj ≥ 1, j ≠ o, (Μ ≫ 0), u ≥ 0, v ≥ 0, tjϵ{0,1}, j ≠ o, Where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n).
If ∑j≠o t̃j < n − 1, then (u ̃, v ̃) is a relatively strongest weight in Λc for DMUo, but if ∑j≠o t̃j = n − 1, then there is not any relatively strongest weight in Λc for DMUo.
Proof: Similar to the proof of Theorem 10.
Corollary 7. Let (u ̃, v ̃, t̃) be an optimal solution for Model (11), then strictly
g (u ̃,̃ v) n−∑n j=1 t ̃j
Λc-efficiency of DMUo= c n−1 = n−1 .
Proof: Theorem (10).
Corollary 8. DMUo is strongly Λc-efficiency if only if Model (11) has an optimal objective function value of zero.
Proof: Theorem (10).
Theorem 11. Let (u ̃, v ̃, t̃) be an optimal solution for the following model
vtxo − utyo = 0, (15) s. t. vtxj − utyj + Μtj ≥ ε, j ≠ o, (Μ ≫ 0), u ≥ 1ε, v ≥ 1ε, tjϵ{0,1}, j ≠ o, where t̃ = (t̃1, … , t̃o−1, t̃o+1, … , t̃n).
Thus (u ̃, v ̃) is a feasible solution for Model (4), and since u ̃tyo = 1, it follows that the optimal objective function value of Model (9) is equal to one. Hence, by Corollary 6, DMUo is CCR–efficient.
Conversely, let DMUo be CCR–efficient, then, by Theorem 1; there exists some optimal solution (u ̂, v ̂) for Model (4) such that u ̂ > 0, v ̂ > 0 v ̂txo = u ̂tyo = 1, v ̂txj − u ̂tyj ≥ 0 , j = 1, . . , n.
Thus, by letting k = min {min{v ̂ } , min{u ̂ }},u ̿ = (u ̂⁄ ), i i r r k and v ̿ = (v ̂⁄ ). We have k > 0 and u ̿ ≥ 1, v ̿ ≥ 1, v ̿txo = u ̿tyo, v ̿txj − u ̿tyj ≥ 0 , j = 1, . . , n.
Therefore (u ̿, v ̿, t) ̿ , where t̿ = 0ϵℝn, is a feasible solution for Model (7); hence, the optimal objective function value of the model is zero. Consequently, by Corollary 6, DMUo is strictly Λc- efficient.
Theorem 15. DMUo is strongly Λc- efficient if only if DMUo is extreme CCR-efficient.
Proof: Let DMUo be strongly Λc- efficient. Then, by Corollary 8, the optimal objective function value Model (8) is zero, thus letting (u ̅, v ̅, t) ̅ be an optimal solution for the model where t̅ = (t̅1, … , t̅o−1, t̅o+1, … , t̅n) = 0ϵℝn−1,
u ̅ ≥ 0, v ̅ ≥ 0, v ̅txo − u ̅tyo = 0, v ̅txo ≥ 1, v ̅txj − u ̅tyj ≥ 1 , j = 1, . . , n.
Therefore, by taking k = v ̅tx (= u ̅ty ), u ̃ = (u ̅⁄ ), v ̃ = (v ̅⁄ ), and α = (1⁄ ), o o k k k
u ̃ ≥ 0, v ̃ ≥ 0, v ̃txo = u ̃tyo = 1, v ̃txj − u ̃tyj ≥ α(> 0), j ≠ o.
Thus (u ̃, v ̃) is an optimal feasible solution for the following model max utyo, vtxo = 1, vtxo − utyo ≥ 0, (16) s. t. t t v xj − u yj ≥ ε , j ≠ o, u ≥ 0, v ≥ 0.
Therefore, by the strong duality theorem, the optimal objective function value of Model (5), which is the dual form of Model (9), is equal to one. Hence, by Theorem 2, DMUo is extreme CCR-efficient. Conversely, if DMUo is extreme CCR-efficient, then, by Theorem 2 and Corollary 8, DMUo is strongly Λc- efficient.
Theorem 16. Let DMUo be strongly Λc- efficient, then there exists some (u ̅, v ̅)ϵΛc such that (u ̅, v ̅) is also a relatively strongest weight in Λc for DMUo and a strongest weight in Λc for DMUo.
Proof: Suppose DMUo is strongly Λc- efficient, then, by Theorem 14, DMUo is extreme CCR-efficient. Thus, by Theorem 4, Theorem 3, and Definition 2, it follows that the optimal objective function value of the following model is one
min θ − ε(∑ si− + ∑ sr +) – ε2 ∑ λj ,
i=1 r=1 j≠o
∑ λjxij + si− = θxio , i = 1, … , m , j=1 (17) s. t. n ∑ λjyrj − sr + = yro , r = 1, … , s, λj ≥ 0, si− ≥ 0, sr + ≥ 0. for all j, for all i, for all r, Therefore, by strong duality theorem, the optimal objective function value of the following model, which is the dual form of Model (10), is equal to one max utyo, vtxo = 1, vtxo − utyo ≥ 0, (18) s. t. t t v xj − u yj ≥ ε , j ≠ o, u ≥ 1ε, v ≥ 1ε.
Hence letting (u ̅, v ̅) be an optimal solution for Model (11), we have u ̅ ≥ 1ε, v ̅ ≥ 1ε, u ̅tyo = v ̅txo = 1, v ̅txj − u ̅tyj ≥ ε , j ≠ o.
Since (u ̅, v ̅, t) ̅ where t̅ = 0ϵℝn−1 is an optimal solution also for Model (8) and for Model (11), it follows (u ̅, v ̅) is also a relatively strongest weight in Λc for DMUo and a strongest weight in Λc for DMUo.
In this section, we use the data recorded in
Data set.
This table is shown in the PDF version of the article.
By using the data set from
Example results.
This table is shown in the PDF version of the article.
In Section 3, we provided models to obtain nonnegative weights for inputs and outputs for the weights, the number of which DMUs for each one its virtual output does not exceed (is less than, if any) its virtual input be maximum, provided that for DMU under evaluation, the virtual output will be equal to the virtual input and the virtual input will be positive. We called these weights the relatively best weight (the relatively strongest weight, if any) for the DMU under evaluation, and if all the weights were positive, we called them the best weight (the strongest weight, if any) for the DMU under evaluation. The relatively best weight (the relatively strongest weight, if any) indicates the normal vector of a surface in the PPS with returns to scale constant assumption that the DMU under evaluation is on the surface and the maximum number of which DMUs their performance is no worse than (is better than) the DMU under evaluation separate from the rest of DMUs. Also in this paper, we presented the relationship between these definitions of efficiency with efficiency in the DEA models with constant returns to scale assumption. The normal vectors can be applied as a criterion for efficiency analysis and ranking of a set of peer DMUs with interval scale data. Especially the relatively strongest weight, both indicate extreme CCR-efficiency and provide a performance measure DMUo with interval scale inputs and/or outputs. Also the strongest weight can be applied for ranking extreme CCR-efficient DMUs and CCR-inefficient DMUs.
We acknowledge the support and constructive feedback received during the course of this research.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Data are available from the corresponding author upon reasonable request.