INTRODUCTION
Data Envelopment Analysis (DEA) was initially proposed by Charnes
et al. (1978) (the CCR model) and was improved by other scholars,
especially Banker et al. (1984) (the BCC model)
to evaluate the performance of a homogeneous group of operating decisionmaking
units (DMUs), such as schools, hospitals, or sales outlets, which use a set
of resources, referred to as inputs and transform them into a set of outcomes,
referred to as outputs. For evaluating the performance of DMUs, DEA successfully
divides them into two categories: efficient DMUs and inefficient DMUs. DEA does
this by assigning a relative efficiency score to each DMU such that the DMUs
in the efficient category have identical relative efficiencies equal to one
and the rest have the relative efficiency between zero and one (Cooper
et al., 2000). DEA does not provide more information about the efficient
DMUs; nevertheless, it is not appropriate to claim that they have identical
performance in actual practice. However, there are many methods to rank the
efficient DMUs (Adler et al., 2002; Andersen
and Petersen, 1993; Li et al., 2007; Jahanshahloo
et al., 2008; Liu and Peng, 2008). As regards
for the inefficient DMUs, the majority of the methods presented in the different
studies have not attempted to evaluate the performance of the inefficient DMUs
beyond the relative efficiency scores attained from the standard DEA models.
Nevertheless, there are some methods for ranking the inefficient DMUs, (Bardhan
et al., 1996; Torgersen et al., 1996;
Baek and Lee, 2009; Jahanshahloo
and Afzalinejad, 2006), which suggest methods different from the standard
DEA models for evaluating the performance of inefficient DMUs. One concept,
derived in Bardhan et al. (1996), attempts to
rank the inefficient DMUs using a Measure of Inefficiency Dominance (MID) according
to their average proportional inefficiency in all inputs and outputs. Moreover,
Torgersen et al. (1996) ranked the inefficient
DMUs by counting the number of DMUs that need to be removed from the analysis
before they are considered efficient. However, a complete ranking cannot be
assured since many DMUs may receive the same ranking score. In the method suggested
by Jahanshahloo and Afzalinejad (2006), a fullinefficient
frontier having been defined, DMUs are ranked based on their distance from this
frontier. The method presented by Baek and Lee (2009)
proposes the use of the LeastDistance Measure in order to evaluate the performances.
By considering these methods and all methods in the literature, the distance
between the inefficient DMUs and the efficient frontier of the Production Possibility
Set (PPS) is a very important criterion for evaluating the performance of the
inefficient DMUs. On the other hand, since the efficient frontier of the PPS
contains the efficient DMUs, the less the distance of an inefficient DMU from
the efficient frontier, the easier it is to remove its inefficiency by less
variations in its inputs and outputs. Hence, the closer the DMU under assessment
to the efficient frontier, the better the performance of this DMU. But the majority
of the methods do not obtain the distance of the inefficient DMUs from the efficient
frontier exactly for evaluating the performance of the DMUs. For example, the
CCR and BCC models approximate the distance of the inefficient DMUs from the
efficient frontier of the PPS in a manner similar to the methods used for evaluating
the performance of the inefficient DMUs. However, since these models do not
obtain the distance of an inefficient DMU from the efficient frontier accurately,
there exists some difficulties for evaluating the performances in practice.
There are several methods to determine the minimum distance of an inefficient
DMU from the efficient frontier (Cherchye and van Puyenbroeck,
2001; Cooper et al., 2000; Frei
and Harker, 1999; Gonzalez and Alvarez, 2001; Lozano
and Villa, 2005; Silva Portela et al., 2003;
Baek and Lee, 2009), but some of them have some difficulties
when used in practice. The procedure by Gonzalez and Alvarez
(2001) cannot always guarantee to reach the efficient projection that is
actually the closest to the assessed DMU; Frei and Harker
(1999) proposed an algorithm to obtain all the efficient facets that do
not always completely describe the efficient frontier and the approach in Cooper
et al. (2000) might lead to a projection that is not an efficient
point. The approaches by Cherchye and Van Puyenbroeck (2001)
and Silva Portela et al. (2003) do find the solution
that is wanted. Nevertheless, they need in a previous stage to identify all
the efficient facets of the PPS frontier in order to determine the closest efficient
projection. The method presented by Lozano and Villa (2005)
finds the sequence of targets ended in the efficient frontier and the efficient
projection found in the final stage is generally closer to the original unit
than the onestep projection is, but it is not necessarily the closest efficient
projection to the original unit. The recently presented method by Baek
and Lee (2009) proposes the use of the LeastDistance Measure in order to
obtain the shortest projection from the evaluated DMU to the efficient frontier,
thus allowing an inefficient DMU to find the easiest way to improve its efficiency.
In this paper, first, a method is developed using a branchandbound algorithm
that is able to determine the minimum distance of the inefficient DMUs from
the efficient frontier (by norms  . _{1} and  . _{∝}
and then, using these distances, a score is assigned to each inefficient DMU
for evaluating the performance of inefficient DMUs such that the less the score
of an inefficient DMU, the better its performance. The proposed method has the
following advantages:
• 
As is known, obtaining the minimum distance of an inefficient
DMU from the efficient frontier is a very important subject in DEA. This
method finds these distances by an interesting branchandbound algorithm
and hence proposes the use of these distances in order to obtain the shortest
projection from the evaluated DMU to the efficient frontier, thus allowing
an inefficient DMU to find the easiest way to improve its efficiency 
• 
Owing to obtaining the minimum distance of the inefficient
DMUs from the efficient frontier exactly and using these distances for evaluation,
the method proposed in this paper is more realistic for evaluating the performance
of the inefficient DMUs compared to other methods and also it is able to
remove the existing difficulties in some models, say, the CCR and BCC models 
DATA ENVELOPMENT ANALYSIS
Consider a set of n DMUs, Ω = {DMU_{1}, DMU_{2}, ...,
DMU_{n}}, where, each DMU_{j} (jεJ = {1,2, ..., n}) produces
s outputs y_{rj} (r = 1,2, ..., s) using m inputs x_{ij} (i
= 1,2, ..., m). Define x_{j} = (x_{1j}, x_{2j},...,
x_{mj})^{t}
and y_{j} = (y_{1j}, y_{2j},..., y_{sj})^{t}
as the input and output vectors of DMU_{j}, respectively, all components
of vectors x_{j} and y_{j} for all DMUs being nonnegative and
each DMU having at least one strictly positive input and output, that is: x_{j}≥0,
x_{j} ≠ 0 and y_{j}≥0, y_{j} ≠ 0; j ε
J. Also, X = [x_{1}, x_{2},...,x_{n}] and Y = [y_{1},
y_{2},...,y_{n}] are mxn and sxn matrices of inputs and outputs,
respectively. The production possibility set (PPS) T is represented as:
The definition of the PPS depends on the postulates selected by the manager.
Charnes et al. (1978) deduced the following PPS,
considering some postulates. This set is denoted by T_{c}, which signifies
the prevalence of constant returns to scale.
Banker et al. (1984) introduced the following
PPS by considering some postulates, which is obtained by adding the constraint
to
T_{C}, where,1_{n} is a vector in ú^{n} with
all components equal to one and "." shows the dot or inner product of the two
vectors. This set is denoted by T_{v}, which signifies the prevalence
of variable returns to scale. .
Definition 1: (x_{o}, y_{o}) εT is called an efficient unit in T if there is no other (x, y) εT such that (x,y)≤(x_{o},y_{o}) and strict inequality holds in at least one component.
By the above definition, (x_{o}, y_{o})εT is an efficient unit if it is not possible to improve any of its inputs or outputs in T without worsening some other inputs or outputs. The set of all efficient units is a subset of the PPS, which is called the efficient frontier and is denoted by ∂^{E} (T). That is:
If (x_{o},y_{o})εT is not an efficient unit, then it is an inefficient unit. This means that it is possible to improve the inputs or outputs of this DMU in the PPS without worsening some other inputs or outputs.
Now, we introduce the input and outputoriented CCR and BCC models for assessing the relative efficiency of (x_{o},y_{o}) by the assumption of constant and variable returns to scale and in order to compare these models with our method for evaluating the performance of the inefficient DMUs:
• 
The inputoriented BCC model corresponding to (x_{o},y_{o})
ε T_{v} 
• 
The outputoriented BCC model corresponding to (x_{o},y_{o})
ε T_{v} 
Also, by replacing T_{c} with T_{v} in the constraints of Models 1 and 2, the input and outputoriented CCR models corresponding to (x_{o}, y_{o}) ε T_{c} are obtained, respectively, whose optimal values are represented by θ_{c}(x_{o}, y_{o}) and φ_{c}(x_{o}, y_{o}), respectively. Using these models, the performance of the inefficient DMUs can be evaluated according to the scores that are obtained at the end of the analysis. For the inputoriented CCR/BCC model corresponding to (x_{o},y_{o}), the more θ_{c}(x_{o}, y_{o})/θ_{v}(x_{o}, y_{o}), the better the performance of (x_{o}, y_{o}). But for the outputoriented CCR/BCC model corresponding to (x_{o}, y_{o}), the less φ_{c}(x_{o}, y_{o})/φ_{v} (x_{o}, y_{o}), the better the performance of (x_{o}, y_{o}).
DEA was introduced to provide a ranking for all DMUs. For efficient DMUs, there
are many methods to do this (Adler et al., 2002;
Andersen and Petersen, 1993; Li et
al., 2007; Jahanshahloo et al., 2008;
Liu and Peng, 2008). For inefficient DMUs, the minimum
distance between DMU_{o} and the boundary of the PPS is approximated.
Since the PPS frontier contains the efficient DMUs, the less the minimum distance,
the more the efficiency. For estimating the minimum distance between DMU_{o}
and the boundary of the PPS, DEA models try to decrease/increase the inputs/outputs
of DMU_{o} as much as possible. This means that the inputs/outputs of
DMU_{o} are decreased/increased until a DMU on the PPS frontier is obtained
and this DMU is called the efficient pattern of DMU_{o}. The distance
between DMU_{o} and its efficient pattern is considered as an inefficiency
score. To achieve efficiency, it is suggested that DMU_{o} decrease/increase
its inputs/outputs until the efficient pattern is achieved. Hence, the closer
the efficient pattern to DMU_{o}, the easier it is to remove the inefficiency
of DMU_{o} by less variations in its inputs and outputs. There are many
models in DEA to obtain the efficient pattern for the DMU under assessment,
which can be divided into two classes.
• 
Radial models: In these models (for example, the CCR
and BCC models), the inputs/outputs of the DMU under assessment are decreased/increased
proportionally until we reach the PPS frontier 
• 
Nonradial models: In these models (for example, the
Additive model), the inputs/outputs of DMU_{o} are decreased/increased
such that the weighted sum of the variations is maximized, in which case
the DMU reaches the PPS frontier 
The following example shows the drawbacks of the radial and nonradial models
to determine the efficiency score and efficient pattern.
Example 1 (Motivating example): Consider the inputoriented CCR model,
one of the famous radial models, as follows:
θ_{c}(x_{o}, y_{o}) =
min{θ:(θx_{o}, y_{o})εT_{c}} 
For the above model, (θ_{c}(x_{o},y_{o})x_{o},y_{o})
is an efficient pattern and its inefficiency score is:
where, w_{i} = 1/x_{io} if x_{io} ≠ 0 and w_{i} = 0 if x_{io} = 0, i = 1,2,.., m. There are two drawbacks for Model CCR:
• 
The pattern is not necessarily efficient unit and could be
weakly efficient unit 

Fig. 1: 
The production possibility set in the illustrative example 
• 
Even if the pattern is efficient unit, it might not necessarily
be the closest efficient unit to the DMU under assessment 
Figure 1 demonstrates the intersection of the PPS T_{c}
for DMU_{A} to DMU_{H} and the hyperplane y = 1. Consider DMU_{o}
:= DMU_{F} as the DMU under assessment. For this DMU, θ_{c}(DMU_{F})
= 1. So, its inefficiency score is zero and it is considered as its own efficient
pattern, though it is not an efficient unit. Now, let DMU_{o} := DMU_{B}
be the DMU under assessment. An efficient pattern for this DMU is
and its inefficiency score is 1θ_{c}(DMU_{B}) = 7/19. However, it can be shown that the closest efficient units to this DMU are f(7, 5/4, 1) and h(28/5, 8/5, 1) by the ._{1} distance and ._{∝} distance, respectively. Now, there are three efficient patterns for DMU_{B}, which are obtained by the CCR model and the ._{1} and ._{∝} methods. It is evident that DMU_{B} can achieve efficiency more easily (by less variations in its inputs) if it chooses the efficient patterns obtained by the ._{1} and ._{∝} methods. In fact, if DMU_{B} chooses the efficient pattern of the CCR model, then its inputs should be decreased by (49/19, 21/19). In this case, the sum of the variation and the maximum variation to attain efficiency are 70/19 and 49/19, respectively. If the efficient pattern by the ._{1} method is chosen, then the inputs should be decreased by (0, 7/4) and the sum of the variations and the maximum variations are 7/4 and 7/4, respectively. If the efficient pattern by the ._{∝} method is selected, then the inputs should be decreased by (7/5, 7/5) and the sum of the variations and the maximum variation are 14/5 and 7/5, respectively. Therefore, if the efficient patterns obtained by the ._{1} and ._{∝} methods are chosen, then efficiency can be achieved by less variations in its inputs and outputs. In this study, we are going to provide a method which is capable of determining the closest efficient patterns by the ._{1} and ._{∝} methods for the DMU under assessment. In what follows, it will be shown that the abovementioned difficulty of radial models is also present in nonradial models and is even worse.
Consider the Additive model, one of the well known nonradial models, as follows:
In the Additive model, the efficient pattern for DMU_{o} is (x_{o}s¯*, y_{o}+s^{+}*), (“*" denotes optimality) and the inefficiency score is:
In all optimal solutions of the Additive model, the efficient pattern is efficient and hence Additive Model can be rewritten as follows:
The above model reveals that the Additive model chooses the farthest efficient unit (by the ._{1} distance) among all efficient units dominating DMU_{o}. Thus, not only the efficient pattern of the Additive model is not the closest efficient unit, it is indeed the farthest efficient unit among those dominating DMU_{o}. For example, consider DMU_{B} as the DMU under assessment in Fig. 1 again. The Additive model has proposed (3,3,1) as an efficient pattern and its inefficiency score is 4. So, DMU_{B} should decrease its inputs by (4,0) to achieve efficiency. In this case, the sum of the variations and the maximum variation to attain efficiency is 4. Therefore, for this DMU the Additive efficient pattern is worse than the efficient patterns proposed by the CCR model and the ._{1} and ._{∝} methods. Table 1 and 2 show the efficient patterns and the inefficiencies of all DMUs.
Note that we consider the PPS T_{v} to continue our discussion, but we can have a similar argument for T_{c}.
Table 1: 
Inefficiency scores and efficient pattern by the CCR and
additive models 

Table 2: 
Inefficiency scores and efficient pattern by the ._{1}
and ._{∝} methods 

THE BRANCHANDBOUND ALGORITHM FOR EVALUATING THE PERFORMANCE OF INEFFICIENT DMUS AND OBTAINING THE CLOSEST EFFICIENT TARGET
Consider a set of DMUs Ω with the input and output vectors as given in the previous section. In this section, using the branchandbound algorithm, we obtain the minimum distance between the inefficient DMUs and the efficient frontier of T_{v}.
Assume that E is the indices' set of the efficient DMUs in Ω and IE = Ω\E and also consider DMU_{o} = (x_{o}, y_{o}) as the observed inefficient DMU under assessment, that is: oεIE.
Considering an arbitrary norm, say ., the minimum distance of DMU_{o} from ∂^{E}(T_{v}) can be obtained as:
For the sake of linearization of the problems that will be solved by the branchandbound algorithm, we consider ._{∝} and ._{1} in Problem Eq. 3. Considering ._{1}, this problem leads to the following problem:
Let (x,y) = (x_{o}s¯, y_{o}+s^{+}) while s¯ εú^{m} and s^{+} εú^{s}, then Problem Eq. 4 is transformed to the following problem:
Note that ∂^{E}(T_{v}) is not usually a convex set and hence Problem Eq. 5 is not a convex problem. Also, it is very difficult to determine ∂^{E}(T_{v}) exactly; therefore, solving Problem Eq. 5 is not simple. But the following theorem shows that Model Eq. 5 is equivalent to a solvable problem.
Theorem 1: The optimal value of Problem Eq. 5 is equal
to the optimal value of Problem Eq. 6. That is: .
Proof: Since, ∂^{E}(T_{v}) is a nonempty and closed
set and the objective function of Problem Eq. 5 is nonegative
and continuous, this problem has a finite optimal value and its optimal solution
belongs to ∂^{E}(T_{v}). Suppose that (s¯*, s^{+}*)
is an optimal solution of Problem Eq. 5. This means that (x_{o}s¯*,
y_{o}+s^{+}*) ε ∂^{E}(T_{v}). Hence,
the optimal value of Model Eq. 7 must be zero.
Then, the KarushKuhnTucker Conditions in linear programming imply that there
is
such that:
Relation Eq. 8 shows that
is a feasible solution for Problem Eq. 6. Therefore,
Conversely, suppose that
is an optimal solution of Problem Eq. 6. Then the KarushKuhnTucker
Conditions in linear programming imply that the optimal value of Model Eq.
7 must be zero. Thus (x_{o}s¯*,y_{o}+s^{+}*)
ε∂^{E}(T_{v}). Then (s¯*, s^{+}*) is
a feasible solution for Problem Eq. 5. Thus:
And this completes the proof.
The objective function of Problem Eq. 6 is nonlinear owing to the presence of the absolute function, but the following theorem shows that it can be linearized.
Theorem 2: Assume that we perform the following change of variables in Problem Eq. 6:
then Problem Eq. 6 is equivalent to the following problem:
Proof: To prove this theorem, it is sufficient to show that in all optimal solutions of Problem Eq. 6, we have:
The above relation holds if
and
for each i and r. By contradiction, assume that for an optimal solution of Problem
Eq. 6 there is an index i such that If
we replace and
by
and
respectively and the other values of variables remain unchanged, then a new
feasible solution is obtained with a smaller objective value, which is a contradiction
to the optimality condition.
Now, we extend the above argument for ._{∞}. Suppose that in Problem Eq. 3, ._{∞} is chosen as the distance function. Then, Problem Eq. 3 leads to the following problem:
If we let (x, y) = (x_{o}s¯,y_{o}+s^{+}) while s¯εú^{m} and s^{+} ε ú^{s}, then Problem Eq. 10 yields the following problem:
The following theorem shows that Model Eq. 11 is equivalent to a nonlinear problem that is solvable. The proof of this theorem is similar to that of Theorem 1 and hence is omitted.
Theorem 3: The optimal value of Problem Eq. 11 is
equal to the optimal value of Problem Eq. 12. That is:
Proof: The proof of this theorem is similar to that of Theorem 1.
Although, the objective function of Problem Eq. 12 is nonlinear
owing to the presence of absolute functions, using the change of variable Problem
Eq. 12 can be converted to Problem Eq. 13
with the linear objective function:
Hence, for obtaining the minimum distance of DMU_{o} from ∂^{E}(T_{v})
by ._{1} and ._{∞}, we must solve Problems Eq.
9 and 13, respectively. But notice that these models
are nonlinear programming problems because of the existence of the constraint:
Now, in the following we present a branchandbound algorithm that solves Problem Eq. 9 by solving some linear programming problems. It is clear that solving Problem Eq. 13 is similar to Problem Eq. 9.
THE BRANCHANDBOUND ALGORITHM FOR SOLVING PROBLEM EQ. 9
Problem 9 provides the theoretical foundation for our new algorithm. To elaborate on the following algorithm, it must be stated that the algorithm for solving Problem 9 considers a branchandbound tree corresponding to Problem 9. This tree contains some nodes and edges and one linear programming problem has been assigned to each node and one variable has been assigned to each edge. To solve Problem 9, this algorithm moves along the edges and considers some nodes and solves the linear programming problems corresponding to them.
Also, the algorithm uses the leftmost node searching method for traveling to all the nodes of the tree. All nodes of the tree have been coded by vector k and each path in the branchandbound tree is determined by this vector and its dimension is the current depth of the tree. The components of vector k are 0 or 1, in which 0 demonstrates moving along the left edge and 1 demonstrates moving along the right edge. Further moving along some paths is stopped if the conditions presented in Steps 1,2,3,4 and 8 hold. By these conditions, it is not necessary to travel to all nodes. In other words, Problem 9 is solved by traveling to only some nodes and solving their linear programming problems, which greatly reduces the computational complexity.
Scalar w_{k} demonstrates whether or not Node k has been visited. That
is to say, if w_{k} = 0, then the left and right edges of Node k have
not been visited and if w_{k} = 1, then the left edge of Node k has
been visited but the right edge has not. If w_{k} = 2, then the left
and right edges of Node k have been visited. W_{k} is the set of remaining
indices E that can be added to or
and
can develop the depth of the tree. Also, in each node, sets of indices and
are
used for imposing conditions:
on Problem (9) without condition Eq. 14.
We consider the branchandbound algorithm for solving Problem (9) as follows.
Step 0: Input
(notice that k is a vector) and then go to Step 1.
Step 1: Solve the following problem:
If it is feasible, then go to Step 2, else go to Step 5.
Step 2: If then
go to Step 3, else go to Step 5.
Step 3: If (
"*" shows the optimality) then set:
and go to Step 8, else go to Step 4.
Step 4: If W_{k} = Ø, then go to Step 5, else go to Step 9.
Step 5: Remove the last component of vector k and save the obtained
vector in .
If
then go to step 7, else set
and go to step 6.
Step 6: If k =(0) and w_{k} = 2, then go to step 10, else go to Step 5.
Step 7: First set and
then set w_{k} = 2. Then, add one component 1 to the righthand side
of the components of vector k such that the dimension of this vector is increased
by 1. Save this obtained vector in .
Then select Also
set: ,
At last set k = and
then go to Step 1.
Step 8: If k = (0) then go to Step 10, else go to Step 5.

Fig. 2: 
The flowchart of the branchand bound algorithm for solving
problem 9 
Step 9: First set w_{k} = 1 and then add one component 0 to
the righthand side of the components of vector k, such that the dimension of
this vector is increased by 1 and save this obtained vector in .
Then select Also,
set: ,
At
last set and
then go to Step 1.
Step 10: Print and
F*.
The flowchart of this algorithm is represented as Fig. 2.
TWO NUMERICAL EXAMPLES
Here, we present two examples to explain the method presented in this study. The first example indicates how to obtain the minimum distance of the inefficient DMUs from the efficient frontier by the branchandbound algorithm. Also, this example demonstrates the weakness of the input and outputoriented BCC models in comparison with our method for evaluating the performance of the inefficient DMUs. We present this simple example to demonstrate the weakness of two input and outputoriented BCC models in comparison with our method contemporaneously. Also, since the PPS can be shown as a simple graphic in this example, it is possible to check geometrically whether the minimum distances obtained by our method are correct or not. Also, this example is related to the variable returns to scale. But the second example is related to the constant returns to scale and this example compares the results of evaluating the performance of the inefficient DMUs obtained by the our method and the two input and outputoriented CCR models.
Example 2: Consider set Ω = {DMU_{A}, DMU_{B}, DMU_{C}, DMU_{D}, DMU_{E}, DMU_{F}, DMU_{G}, DMU_{H}, DMU_{I}, DMU_{J}}, with one input and one output for each DMU as shown in Table 3 as the set of the observed DMUs. Also, Table 3 demonstrates the efficiency status of DMUs and Fig. 3 demonstrates the PPS T_{v} produced by these DMUs.
Considering Fig. 3, the points on the line segments and
compose the efficient frontier of T_{v}. Now, we explain how to obtain
the minimum distance by ._{1} of DMUs from the efficient frontier
by the branchandbound algorithm. Obtaining the minimum distance of other inefficient
DMUs from ∂^{E}(T_{v}) is similar to this. To obtain the
minimum distance of DMU_{I} from ∂^{E}(T_{v}),
Steps 0 and 1 imply that first, we must solve the following problem:
Table 3: 
Data and the efficiency status 


Fig. 3: 
The production possibility set T_{v} 
After solving Problem Eq. 16, which is corresponding to the root node, the optimal
value
= 7.25 and the optimal solution:

Fig. 4: 
The tree corresponding to DMU_{I} 
are obtained. Since, the optimal solution of Problem (16) does not satisfy
condition (14), we move to Node 2 as shown in Fig. 4 and by
imposing constraint λ_{A} = 0 on Problem (16), solve the following
problem:
By which the optimal value
and the optimal solution:
are obtained. But since the optimal solution of Problem Eq.
17 satisfies condition (14) and
then we give the new value to
F*, that is, we set F* = 8. Also, further moving along the left and right edges
of Node 2 is stopped. Thus, we return to Node 1 and from Node 1 move to Node
3. The problem corresponding to Node 3 is the following problem which must be
solved:
After solving Problem Eq. 18, the optimal value =
7.25 and the optimal solution:
are obtained. Since, the optimal solution of Problem (18) does not satisfy
condition (14) and also since
then further moving from this node is not stopped. Therefore, we go to Node
4. By solving the problem corresponding to Node 4,
and
are obtained. Then, we move to Node 5 and solve the following problem:
Problem Eq. 19 has the optimal value and
the optimal solution:
Since, Node 5 is the final node, then further moving from this node is impossible
and since ,
we do not vary the value F*, which is F* = 8. Now, we move to Node 6, whose
corresponding problem is Problem Eq. 20 and is infeasible:
Next, we go to Node 7 and solve its corresponding problem. But since the optimal
solution and optimal value of the problem corresponding to this node are similar
to those of Nodes 3 and 4, hence we move to the new Node 8.
Table 4: 
The results of evaluation by our method 

Table 5: 
The results of evaluation by the BCC model 

By solving the problem corresponding to this node, the optimal value and the
optimal solution of this problem are obtained as follows.
In reality, since this solution satisfies condition (14) and and
the problem related to the next node (Node 9) is infeasible, therefore the minimum
distance of DMU_{l} from the efficient frontier is 7.5. If we
similarly obtain the minimum distances of the other inefficient DMUs from the
efficient frontier by ._{1} and ._{∝} and also
obtain the optimal value of the input and outputoriented BCC models for the
inefficient DMUs, then Table 4 and 5 are
obtained.
As shown in Table 5, the inputoriented BCC model evaluates the performances of DMU_{E} and DMU_{F} identically, while these DMUs have equal inputs and DMU_{F} has greater output than DMU_{E}. Therefore, DMU_{F} must have a better performance than DMU_{E}. But considering Table 4, our method evaluates their performances correctly by both ._{1} and ._{∞} and gives a better rank to DMU_{F} than DMU_{E}. Also, the outputoriented BCC model ranks DMU_{G} and DMU_{H} identically, while these DMUs have equal outputs and the input of DMU_{G} is less than the input of DMU_{E}. However, our method evaluates their performance correctly and gives a better rank to DMU_{G} than DMU_{H}.
Example 3: Consider the DMUs in Table 6 which records
behavior intended to serve as a basis for evaluating the performance of the
inefficient hospitals in terms of two inputs, number of doctors and number of
nurses and two outputs identified as number of outpatients and inpatients (each
in units of 100 persons/month).
Table 6: 
Data of the observed DMUs 

Table 7: 
The results of evaluation of the performances 

Also, Table 6 demonstrates the efficiency status of these
DMUs. The results of evaluation using our method and the input and outputoriented
CCR models are exhibited in Table 7.
CONCLUSION
In this study, a method has been presented for evaluating the performance of the inefficient DMUs, which is based upon the minimum distance of these DMUs from the efficient frontier. This method is different and more realistic compared to the other methods of evaluation of the performances and it is able to remove the existing difficulties in some standard DEA models, say, the CCR and BCC models. We proposed a methodology based on the branchandbound algorithm that allows us to obtain the closest targets and the best efficient patterns for the inefficient DMUs by ._{∞} and ._{1}. These norms are considered for the sake of the linear property and computational advantages. However, other norms and criteria could be regarded as well. The general argument behind this idea is that the closest targets and best efficient patterns suggest directions of improvement for the inputs and outputs of the inefficient units that can help them to achieve efficiency with the least effort.