A Review of Ranking Models in Data Envelopment Analysis

In the course of improving various abilities of data envelopment analysis (DEA) models, many investigations have been carried out for ranking decision-making units (DMUs). This is an important issue both in theory and practice. There exist a variety of papers which apply different ranking methods to a real data set. Here the ranking methods are divided into seven groups. As each of the existingmethods can be viewed fromdifferent aspects, it is possible that somewhat these groups have an overlappingwith the others. The first group conducts the evaluation by a cross-efficiency matrix where the units are selfand peer-evaluated. In the second one, the ranking units are based on the optimal weights obtained from multiplier model of DEA technique. In the third group, superefficiency methods are dealt with which are based on the idea of excluding the unit under evaluation and analyzing the changes of frontier.The fourth group involvesmethods based on benchmarking, which adopts the idea of being a useful target for the inefficient units.The fourth group uses themultivariate statistical techniques, usually applied after conducting the DEA classification.The fifth research area ranks inefficient units through proportional measures of inefficiency. The sixth approach involves multiple-criteria decision methodologies with the DEA technique. In the last group, some different methods of ranking units are mentioned.


Introduction
Data envelopment analysis (DEA) first proposed by Charmes et all [8], in evaluating the efficiency of an educational center in USA, and developed by Banker et all [4].In evaluating the relative efficiency of each Decision making units (DMUs) by DEA, we obtain scores between zero and one.In this way, usually more than one unit may be efficient in the DEA models and their scores are 1.Also, pay attention that the number of efficient units in the Variable return to scale (VRS) models is not less than the Constant return to scale (CRS) models.Therefore, the researchers proposed some methods to difference these efficient units.This concept has named Ranking efficient units in the DEA.There are lots of ranking methods and each of them has special quality and property to rank efficient units.Charnes et all [5], counted the number of times that an efficient DMU play the role of benchmark unit for others, and used this norm to rank these units.Since, finding the reference set of a DMU is not easy, their model is not a suitable method.Charnes et all [6], proposed another method to find the benchmark DMUs.They changed the rate of outputs and evaluate the change of efficiency score.However, they didn't distinguish how they can do it.Sexton et all [18], suggested the cross efficiency method.In this model they used the weights which they had obtained them by solving each of n-linear problems.They evaluate the efficiency of each DMU, n times and store these data in a matrix.Each row of this matrix contains the cross efficiency score of DMUs.They compute the average of these rows and store the result as a ranking measure.It seams this is a good method, but it may have some difficulties.The biggest problem appears when the DEA models have alternative solutions.Finally, pay attention that there are some techniques and strategies in DEA which they affect on ranking.For example, Thompson et al [20], used the assurance regions.In their technique, the number of efficient DMUs may be decrease.But it isn't a suitable method because finding suitable weights isn't easy.Adler et al [2], proposed another method to difference between DMUs.In their model they decrease the number of inputs and outputs by component analysis.Therefore, the number of efficient DMUs is decreased.But in general this model couldn't use for a complete ranking.In this paper the most number of ranking models have been discussed and in some sense they are compared.In 2nd section we review DEA and its basic models.In 3rd section we discuss about ranking models and their weakness and forte.In section 4 real data is used for illustrative propose, and finally section 5 is conclusion.

Review of DEA models
Consider n, DMUs with m inputs and s outputs.The input and output vectors of DMU j (j = 1, . . ., n) are X j = (x 1j , . . ., x mj ) t , Y j = (y 1j , . . ., ysj) t which X j ≥ 0, X j = 0, Y j ≥ 0, Y j = 0.By using the non-empty, constant return to scale, convexity and possibility postulates, the production possibility set (PPS) is made as follows: The envelopement form of CCR model in the input oriented case is as follows: and its dual, the multiplier form of CCR model in the input oriented case is: Theorem 1.All of these definitions about efficiency are equivalent in T c (the production possibility set of CCR model): ( By adding the constrains 1λ = 1,1λ ≥ 1,1λ ≤ 0 the BCC, BCC − CCR,CCR − BCC models obtained respectively.Cooper et al [9], introduced the additive model to analyze the efficiency of DMUs.In this model they evaluate the maximum distance between DMUs and the frontier.Their proposed model is as follows; In this model the evaluated DMU is efficient iff z 8 = 0.There is an important point that the additive model can not give a number such as an efficiency score for DMUs.After that Cooper et al [9], suggested the additive model with constant return to scale by omitting the constraint 1λ = 1.Tone [21,22], proposed the following model and named it slack based measure (SBM) model: Clearly, problem (4) is a rational problem.By charnes-cooper transformation [6] the following linear programming problem is obtained: Let (τ * , t * , γ * , W − * , W + * ) be an optimal solution of (5), then the optimal solution of SBM model is: In this model DMU o is efficient if and only if ρ * = 1.This condition is equivalent to S − * = 0, S + * = 0.

Ranking models
According to the previous introduction, ranking is used to difference between efficient units.In this section we are going to review ranking models in the Data Envelopment Analysis.

AP model
Anderson and Peterson [3], proposed supper efficiency model.They omitted the efficient DMU from the PPS (T c )c and ran CCR model for another units to rank it.Their proposed model is: and its dual is: For efficient units θ * ≥ 1 and for inefficient units 0 < θ * ≤ 1.This model has tow difficulties: (1) AP model may be infeasible for special data in input oriented case.
(2) This model is unstable for some DMUs which one of their data components is near to zero.
Finally, it is important to note that these difficulties are only in input oriented models, and there aren't any difficulties in the output oriented models.

MAJ model
To solve the important difficulties of AP model, Mehrabian et al [16], proposed another motel for ranking efficient units.Their proposed model is: Theorem2: The necessary and coefficient conditions for feasibility of MAJ model is that in evaluating of DMU o , or y ro = 0, r = 1, . . ., s or there exists DMU j , j = o such that y rj = 0.

Revised MAJ model
According to the theorem2, the MAJ model may be infeasible in some cases.to solve this problem, Saati et al [17], proposed the following model: However,decreasing the inputs and increasing the output are considered together respectively.Theorem3: problem 9 always is feasible.

The slack adjusted DEA model
In the CCR model, there are some difficulties to introduce ε.To solve these difficulties, Sueyoshi et al [19], proposed the slack adjusted DEA model (SA-DEA), and used it to rank efficient units.Their model for ranking is: where R − i = max j x ij , R + r = max j y rj , for every i, j.SA − DEA has the same difficulties of AP model.We denote δ * as rhe optimal value of problem (9).To rank the extreme efficient units, Sueyoshi used the adjusted index number (AIN): where E is the set of efficient units.This index number belongs to interval [1,2].For details see [19].

The gradient line method
Jahanshahloo et al [14], used the gradient line to rank extreme efficient units .In this way they defined the plan P 0 which contains the point (X o , Y o, and the set S o as follows: The corresponding gradient equation to DMU o in P 0 with center (0, 0) is: where: Later they made this model: Let (U * t , V * t ) be the optimal solution.(U * t , −V * t ) is the gradient of hyperplane which supports on T c (T c is obtained by omitting the evaluated DMU in T c ).The intersection of T c and the supported hyperplane is: , and it is an efficient face in T c .clearly, problem ( 13) is always feasible.
Theorem4: in problem ( 14) Proof.See [14].Now, again consider the set S 0 corresponding to DMU o in (α, β space.Intersection of S 0 and efficient surface of T c is a half line where its equation is: where V * t Xo U * t Yo is the slope of (15).To rank DMU o the length of connecting arc DMU o with intersection point of line (15) and previous ellipse is calculated in (α, β) space.This intersection is as follows: Now, the length of connecting arc DMU o to the point corresponding to α * , β * in (α, β) plan calculated as follows: Note that this model has no suggestion for ranking non extreme efficient units.

Ranking by common set of weights
The DEA models evaluate the units in the best situations.For example, the multiplier form of CCR model solves n linear problem to evaluate n DMUs, and for each DMU evaluates a set of weights.Now, can we evaluate all DMUs with a set of weight?Hosseinzadehlotfi et al. [10], proposed common set of weight (CSW) model which evaluates all DMUs with a set of weight.They based it on this fact that the feasible region of n problems are the same.Their suggested model for ranking extreme efficient units is: By special transformation [10], the above goal fractional programming turning to the following non linear problem: The following ratio is used as a measure to rank efficient units after solving the above problem: where E is the set of efficient units.The higher score is the higher rank.

Ranking by l 1 norm
Jhanshahloo et al. [13], used l 1 norm for ranking extreme efficient units.Consider the following model to obtain the ranking score of j=1,j =o λ j y rj ≥ y r , r = 1, . . ., s, where Γ o c (X, Y ) is the distance of (X o , Y o from (X, Y ) BY l 1 norm in constant return to scale model.Clearly, problem 17 is a non linear problem.To change it to a linear once, consider the following set: See [13].Now let (X * , Y * , λ * ) be the optimal solution of (20), then by the last theorem, to change (20) as a linear problem we should add where α = s r=1 y ro − m i=1 x io is a constant value.This problem is always feasible and bounded.The objective value of the last problem is used for ranking efficient units.

Ranking by using the concept of advantage model
Jahanshahloo et al. [15], are tried with due to price ratio constraint that is exerted to the advantage model. 2 The same price ratio constraint are added to the multiplier form of AP model (and MAJ model).The modified AP model is as follows: DMU o is efficient in this model if and only if A 0 ≥ 1.For more details see [15].Because of control weight may be the number of CCR efficient units is decreased in this model.

Ranking by using SBM model
This model were proposed by Tone, [21,22].Let T c be the new PPS obtained by omitting the evaluated DMU o and . By definition of T c this distance is not less than 1.Therefore the super efficiency score of DMU o is defined as the optimal solution of the following problem: In this model δ * ≥ 1 if and only if (X o , Y o ) ∈ T c .One of the most important property of this problem is the unit invariance property of δ.By charnes-cooper transformation, the following linear problem is obtained: clearly we can remove the last restriction by t ≥ 0.

Ranking by using the tchebycheff norm
Jahanshahloo et al. [12], ranked extreme efficient units by L∞ norm.Let A = (X o , Y o ) be the evaluated DMU o and A be a member of T c .The proposed model is: By the definition of L ∞ we obtain the following linear problem:

Ranking by mont carlo method
Up to now, proposed model could not rank non extreme efficient units.Jahanshahloo et al. [11], proposed a new model by using the mont carlo method, but it is not a DEA model.Definition: The region of exclusive domination (RED)for an efficient DMU is a subregion of PPS where this DMU dominates all point of PPS and non of the other (observed) DMUs dominates these points, and it it is illustrates in figure 1.

RED(B)
Figure 1: RED The measure of RED has been used as a criterion for ranking all efficient units.Generally, calculating this measure is analytically very hard and in some cases is impossible.The authors used mont carlo method to calculate the one dimensional integral: b a f(x)dx where the function f(x) is bounded and 0 ≤ f(x) ≤ c, a ≤ x ≤ b.Algorithm of this method is as follows: (1) Generate a sequence (Uj) 2N j=1 of 2N random numbers that are uniformly distribute on(0,1).
(2) Arrange the random numbers into N pairs (U 1 , U 1 ), ..., (U N , U N ) in any fashion such that each number is used exactly once.
(3) Compute f(X i ) where Count the number of N H which f(X i ) > cU i (5) Estimate the integral I by: θ Now, consider DMU o as an efficient one.We use last technique to find the measure of this area which is dominated exclusively by this DMU.This area is generally bounded, but for some DMU it may be unbounded.The authors bounded the whole region for each DMU.
Let W + = (x + 1 , ..., x + m , y + 1 , ..., y + r ) = (x + , y + ) where x i + = min j {x ij } and y + r = max j {y rj }, be an ideal unit and let W − = (x − 1 , ..., x − m , y − 1 , ..., y − r ) = (x − , y − ) where y − r = 0 for each r and x − i may be achieved in the following way: x ij W − andW + assure us that the RED for each DMU is bounded.With this assumption the measure of whole region is i for each i and y + r ≥ 0 fore each r.By generating random numbers W = (x 1 , ..., x m , y 1 , ..., y m ) is constructed where By counting all hits (above conditions) we should estimate the measure of RED by using the formula: N where N H is the number of vectors W = (X, Y ) and N is the number of generated vectors.

Illustrative example
In this section we are going to compare the foregoing models.Table2: ranking of efficient units Table of data included 23 units with 5 inputs and 2 outputs: