Ranking efficient decision making units in data envelopment analysis based on reference frontier share

Accepted Manuscript

Ranking efficient decision making units in data envelopment analysis based on reference frontier share M.J. Rezaeiani, A.A. Foroughi PII: DOI: Reference:

S0377-2217(17)30605-7 10.1016/j.ejor.2017.06.064 EOR 14546

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

27 June 2016 24 June 2017 27 June 2017

Please cite this article as: M.J. Rezaeiani, A.A. Foroughi, Ranking efficient decision making units in data envelopment analysis based on reference frontier share, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.064

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ACCEPTED MANUSCRIPT Highlights • The concept of the reference frontier share is introduced. • A model is proposed to measure the reference frontier share. • A measure is provided for ranking efficient units. • The new approach is compared with other existing approaches.

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• The new approach can rank the extreme and non-extreme efficient units.

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Ranking efficient decision making units in data envelopment analysis based on reference frontier share M.J. Rezaeiani, A.A. Foroughi∗ Department of Mathematics, University of Qom, Qom 3716146611, Iran

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Abstract Data envelopment analysis (DEA) is a powerful technique for performance evaluation of decision making units (DMUs). Ranking efficient DMUs based on a rational analysis is an issue that yet needs further research. The impact of each efficient DMU in evaluation of inefficient DMUs can be considered as additional information to discriminating among efficient DMUs. The concept of reference frontier share is introduced in which the share of each efficient DMU in construction

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of the reference frontier for evaluating inefficient DMUs is considered. For this purpose a model for measuring the reference frontier share of each efficient DMU associated with each inefficient one is proposed and then a total measure is provided based on which the ranking is made. The new approach has the capability for ranking extreme and non-extreme efficient DMUs. Further, it has no problem in dealing with negative data. These facts are verified by theorems, discussions and numerical examples.

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Keywords: Data envelopment analysis, Ranking, Reference frontier share

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1. Introduction

Data envelopment analysis (DEA), introduced by Charnes et al. [1], is a mathematical programming technique for efficiency analysis and performance evaluation of decision making

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units (DMUs). Each decision making unit uses multiple inputs to produce multiple outputs. The basic models of DEA such as CCR [1], BCC [2], additive [3] and SBM [4] divide the set of

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DMUs into two subsets, efficient and inefficient. One of the main objectives that is followed in performance evaluation is discriminating among efficient DMUs to provide a complete ranking of DMUs. Several approaches for getting more information about DMUs and therefore carrying

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out an enhanced ranking of DMUs have been proposed. Among them we can refer to weights restrictions [5–12], cross efficiency [13–20], common weights [21–26], super-efficiency [27–29], multiple criteria DEA model [30–33], inverted frontier [34–36], reference-based approaches [37– 40] and etc.

Weights restrictions can help to prevent the assignment of unreasonable weights, and therefore to reduce the number of efficient DMUs that become efficient by using unreasonable weights. This helps to increase the discrimination power of DEA and improve the ranking of DMUs. On the other hand, this approach needs expert opinions and other information, while this informa∗

Corresponding author, Tel: +982532103054 Email addresses: [email protected] (M.J. Rezaeiani), [email protected] (A.A. Foroughi)

Preprint submitted to Elsevier

July 1, 2017

ACCEPTED MANUSCRIPT tion often is out of reach or earning this information is a difficult task. This point is noted by Doyle and Green [14]. In cross efficiency approach, each DMU is evaluated by favorable weights of other DMUs and then its efficiency score is determined by averaging the efficiency scores follow from these evaluations. The main problem with this approach is the existence of alternative optimal weights. Various forms of secondary objectives for choosing among alternative optimal weights were proposed in [14–17, 19]. However, each of these secondary objectives yields different results and selecting one of them needs preference information. Apart from this, cross efficiency analysis

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has not received enough attention with variable returns to scale (VRS) assumption. The input oriented VRS model (BCC) can generate negative efficiencies in cross efficiency table. Wu et al. [41] added some constraints to the BCC model to avoid negative cross efficiency. Soares de Mello et al. [42] studied the consequences of the modified model and noted that the proposed modifications change the original frontier and define new efficiency scores. Lim and Zhu [43] proposed that the VRS cross efficiency evaluation be done using a series of CCR models under the translated Cartesian coordinate system.

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The common weights approach was proposed in order to set efficiency evaluation in a common framework. The original DEA models evaluate each DMU with its favorable weights while it seems more reasonable to evaluate all DMUs with a common set of weights. Kao and Hung [24], using compromise solution, proposed an approach for finding a common set of weights. More comprehensive work in context of common weights is that of Liu and Peng [23]. However, the problem of multiple optimal weights may affect the applicability of their work. Ramezani-

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Tarkhorani et al. [44] reported this problem and proposed an approach for dealing with this problem. Jahanshahloo et al. [25] suggested an approach for providing two common sets of

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weights using ideal and special lines based on the work of Liu and Peng [23]. Sun et al. [26] proposed some models for generating two common sets of weights using ideal and anti-ideal DMUs. Carrillo and Jorge [45] proposed a common weights approach based on compromise

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programming. They used the minimum distance from the ideal point of aggregated input and output level to produce a set of common weights. Some other works that we can refer to are those of Karsak and Ahiska [46, 47] and Foroughi [48]. Nonetheless, almost all of the work on

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common weights approach is for constant returns to scale assumption. Super-efficiency is an exclusion approach for further discrimination and ranking of DMUs. It excludes the DMU under evaluation from the sample and then evaluates this DMU with

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respect to the new production possibility set created by other DMUs. It has been extended for all returns to scale assumptions. However, it may result in infeasibility of envelopment models or unboundedness of multiplier models [49, 50]. These facts restrict the applicability of the super-efficiency models, especially in the case of variable returns to scale. There have been several attempts for dealing with this problem e.g. [51–61]. Even if we ignore the infeasibility or unboundedness problem, super-efficiency models cannot rank non-extreme efficient DMUs in cases where there is more than one such DMU. Another issue that creates some doubt about the application of the super-efficiency approach is the findings of Banker and Chang [62]. They conducted a simulation study to evaluate the performance of super-efficiency approach in ranking of efficient DMUs and outlier detection. Their findings showed that the super-efficiency

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ACCEPTED MANUSCRIPT approach performs satisfactorily in outlier detection rather than in ranking efficient DMUs. Multiple criteria DEA (MCDEA) model was proposed by Li and Reeves [30]. They attached two different criteria to the traditional DEA objective as a way to limit the freedom of DMUs in choosing the weights that are unfavorable to other DMUs. This can reduce the number of efficient DMUs and hence improve the ranking. However, in this method there is no guarantee to provide a complete ranking. Further, choosing a solution from available solutions needs preference information. Some goal programming formulations of the MCDEA model have been proposed. One of the formulations is a bi-objective weighted model of Ghasemi et al. [31]. A

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more analytic work for presenting a valid goal programming formulation for MCDEA model has been implemented by Rubem et al. [33]. In addition to reporting some inconsistencies of the previous models, they proposed a new weighted goal programming model to solve the MCDEA model. Chaves et al. [32], considering two objectives at the same time in MCDEA model, presented dual formulations to provide benchmarks for inefficient DMUs.

The inverted frontier approach was introduced aiming at considering both efficient (best practice) and anti-efficient (worst practice) frontier. Entani et al. [34] defined a type of in-

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terval efficiency. Takamura and Tone [63] used the inverted frontier to evaluate positives and negatives of the candidate sites in the problem of site selection for government agencies of Japan. Paradi et al. [64] applied the worst practice to credit risk evaluation. Amirteimoori [35] defined full-efficiency and full-inefficiency concepts using inverted frontier idea, and, to provide full discrimination among efficient DMUs, applied a slack-based version of super-efficiency. Shen et al. [36] provided three composite DEA indicators using efficient and anti-efficient fron-

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tiers, which can improve the discrimination power of DEA and ranking of efficient DMUs. However, in cases that some DMUs lie on both efficient and anti-efficient frontier, further dis-

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crimination requires an auxiliary tool such as super-efficiency. Reference-based approaches consider the importance of efficient DMUs as references for other DMUs to provide a criterion for discriminating among efficient DMUs. Charnes et al. [37]

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considered the number of times an efficient DMU can be in reference groups of inefficient DMUs as a factor for discriminating among efficient DMUs. Torgersen et al. [38] emphasized on total level of improvements proposed to inefficient DMUs by each efficient DMU as a measure of

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importance of efficient DMUs. Zhu [65] applied a similar measure to performance evaluation of the Fortune 500 companies which indicates the role of each efficient company in evaluating inefficient companies. Lu and Lo [66] introduced the concept of interactive benchmark to

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evaluate each DMU as a benchmark for other DMUs. Jahanshahloo et al. [39] proposed an exclusion approach called changing the reference set. It measures the changing in efficiency scores of other DMUs when the DMU under evaluation is excluded from the reference set. The DMU which removing it from the reference set makes a greater change in efficiency scores of other DMUs will get a better rank. This approach was proposed under the constant returns to scale assumption and its applicability for variable returns to scale assumption was not tested. Chen and Deng [40] developed this approach with some modifications for variable returns to scale assumption. Some other researchers have extended different research area related to selecting the best DMU. Among them we can refer to Foroughi [67, 68], Wang and Jiang [69], Toloo [70, 71] and

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ACCEPTED MANUSCRIPT Lam [72]. The work of Foroughi [68] also presented a generalized approach for ranking with all returns to scale assumptions. The key point that we want to use here for further evaluation and increasing discrimination among efficient DMUs is the share of each efficient DMU in construction of the reference frontier for inefficient DMUs. We extend the concept of the reference set to reference frontier. The reference frontier of an inefficient DMU is the set of all efficient points of the production possibility set that dominate it. The targets proposed by a DEA model to an inefficient DMU are members of the reference frontier of it. In fact, we will show that each point of the reference frontier of an

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inefficient DMU can be a target for it. The efficient DMUs can be discriminated based on their role in construction of the reference frontiers for inefficient DMUs. Another aspect of this idea is to consider the role of each efficient DMU in providing the targets for inefficient DMUs that can help to improve their efficiency. Based on this idea we propose models to provide a measure for comparing efficient DMUs and ranking. The new approach does not have any problem such as infeasibility or unboundedness and it is expected that can rank all extreme and non-extreme efficient DMUs.

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The rest of the paper is organized as follows: section 2 briefly presents some preliminaries. The new approach is presented in section 3. Some numerical examples are given in section 4. Section 5 concludes the paper. 2. Preliminaries

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Consider a sample of n DMUs, each of which uses m inputs to produce s outputs. Let the ith input and the rth output of DMU j be xij and yrj , respectively. Let the input and output vectors of DMU j be xj = (x1j , . . . , xmj ) and yj = (y1j , . . . , yrj ), respectively. We

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consider the production possibility set under the variable returns to scale assumption as defined by Banker et al. [2]. One of the valuable features of the DEA models is providing reference sets for inefficient DMUs and determining alternative targets that show the road map to improving

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their efficiency. There are some models to provide alternative targets, e.g., in Thanassoulis and Dyson [73], Zhu [74], and Estellita Lins et al. [75]. As noted by Estellita Lins et al. [75], radial

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projections often lead to inefficient portions of the frontier. The additive models of DEA allow us to search in efficient frontier to access all possible targets. Therefore, we use the additive models.

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The simple additive model, proposed by Charnes et al. [3], is as follows:

max

s.t.

m P

i=1 n P

j=1 n P

j=1 n P

s− i +

s P

r=1

s+ r

λj xij + s− i = xio

i = 1, . . . , m

λj yrj − s+ r = yro r = 1, . . . , s

λj = 1 j=1 + s− i ≥ 0 , sr λj ≥ 0

≥0

(1)

i = 1, . . . , m, r = 1, . . . , s j = 1, . . . , n 5

ACCEPTED MANUSCRIPT DMU o is efficient if and only if in the optimal solution of model (1) we have s−∗ = 0 (i = i 1, . . . , m) and s+∗ r = 0 (r = 1, . . . , s). We denote the set of efficient and inefficient DMUs by E and IE , respectively. It is proved that in optimal solution of this model for each inefficient DMU j we have λ∗j = 0. The set of DMUs with positive lambda in an optimal solution of this model is called the reference set of DMU o that is a subset of the set of efficient DMUs. Note that model (1) may have alternative optimal solutions and therefore the reference set of a DMU need not to be unique. The target for DMU o with respect to an optimal solution λ∗ , which also is called the P  n n P projection of DMU o onto the efficient frontier, is defined as (ˆ xo , y ˆo ) = λ∗j xj , λ∗j yj . j=1

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j=1

This is a point on the efficient frontier to which the performance of DMU o is compared. The dual problem of model (1) is as follows:

s.t.

s P

r=1 m P i=1

vi xio − vi xij −

vi ≥ 1

s P

r=1 s P

r=1

ur yro + u0 ur yrj + u0 ≥ 0 j = 1, . . . , n i = 1, . . . , m

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min

ur ≥ 1

(2)

r = 1, . . . , s

u0 f ree

DMU o is efficient if and only if the optimal objective value of model (2) is zero. Note that, by the complementary slackness condition, if for some j in an optimal solution of model (1) we

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have λ∗j > 0, then the constraint associated with j in optimality of model (2) must be binding, m s P P i.e. vi∗ xij − u∗r yrj + u∗0 = 0. This means that DMU j lies on one of the efficient facets of r=1

i=1

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the production possibility set that constitute the efficient frontier.

Lovell and Pastor [76] introduced the general weighted additive model below:

s.t.

m P

i=1 n P

wi− s− i + λj xij +

j=1 n P

r=1 s− i =

i = 1, . . . , m

λj yrj − s+ r = yro

r = 1, . . . , s

λj = 1 j=1 + s− i ≥ 0 , sr

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wr+ s+ r xio

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j=1 n P

s P

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max

λj ≥ 0

≥0

(3)

i = 1, . . . , m, r = 1, . . . , s j = 1, . . . , n

In this model, wi− (i = 1, . . . , m) and wr+ (r = 1, . . . , s) are nonnegative constant weights determined by user. The proposed targets for inefficient DMUs are dependent on these weights. Moreover, each efficient point dominating an inefficient DMU can be a target for it with respect to a weighted additive model with appropriate weights. This result is stated by the following theorem. For convenience, the set of efficient points dominating DMU o is denoted by No . Theorem 1. Let (¯ x, y ¯) ∈ No . There are suitable weights wi− (i = 1, . . . , m) and wr+ (r = 6

ACCEPTED MANUSCRIPT 1, . . . , s) so that one of the proposed targets for DMU o by model (3) is (¯ x, y ¯). Proof. Since (¯ x, y ¯) ∈ No , (¯ x, y ¯) is lying on one of the efficient facets of the production possibility

set. Hence, the point (¯ x, y ¯) can be represented as a convex combination of theoefficient DMUs P n P ˆ  ˆ ˆ lying on this facet. Let (¯ x, y ¯) = λj xj , λj yj and L = j ∈ E|λj > 0 . Consider the j∈E

j∈E

following model:

s.t.

s P

r=1 m P i=1 m P i=1

vi xio − vi xij − vi xij −

vi ≥ 1

s P

r=1 s P

r=1 s P r=1

ur yro + u0 ur yrj + u0 = 0 j ∈ L ur yrj + u0 ≥ 0 j ∈ /L i = 1, . . . , m

ur ≥ 1

r = 1, . . . , s

(4)

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u0 f ree

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min

Let vˆi (i = 1, . . . , m), u ˆr (r = 1, . . . , s) and u ˆ0 be an optimal solution to this model. Obviously, this model is equivalent to the following model in terms of the optimal objective function value:

r=1 m P i=1 m P i=1

vi xio − vi xij − vi xij −

vi ≥ vˆi

ur ≥ u ˆr

r=1 s P

r=1 s P r=1

ur yro + u0

ur yrj + u0 = 0 j ∈ L ur yrj + u0 ≥ 0 j ∈ /L

(5)

i = 1, . . . , m r = 1, . . . , s

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u0 f ree

s P

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s.t.

s P

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min

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Now consider the following model:

min

r=1 m P

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s.t.

s P

i=1

vi xio −

vi xij −

vi ≥ vˆi

s P

r=1 s P

r=1

ur yro + u0 ur yrj + u0 ≥ 0 j = 1, . . . , n

ur ≥ u ˆr

i = 1, . . . , m r = 1, . . . , s

u0 f ree

The dual to model (6) is as follows:

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(6)

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s.t.

m P

i=1 n P

j=1 n P

j=1 n P

vˆi s− i +

s P

r=1 λj xij + s− i

u ˆr s+ r = xio

i = 1, . . . , m

λj yrj − s+ r = yro r = 1, . . . , s

λj = 1 j=1 + s− i ≥ 0, sr λj ≥ 0

≥0

(7)

i = 1, . . . , m, r = 1, . . . , s j = 1, . . . , n

Now a feasible solution for model (6) is as follows: vi = vˆi (i = 1, . . . , m), ur = u ˆr (r = 1, . . . , s) and u0 = u ˆ0 .

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Also a feasible solution for model (7) is as follows: ˆ j (j ∈ L), λj = λ

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max

λj = 0 (j ∈ / L),

s− ˆi (i = 1, . . . , m) and i = xio − x

ˆr − yro (r = 1, . . . , s). s+ r =y

These feasible solution pairs of models (6) and (7) satisfy complementary slackness condition.

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Hence, both are optimal for their respective models. Therefore, wi− = vˆi (i = 1, . . . , m) and wr+ = u ˆr (r = 1, . . . , s) is a suitable choice for weights to get (¯ x, y ¯) as a target for DMU o.

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This theorem motivates us to work with reference frontiers rather than reference sets. We define reference frontier as below.

Definition 1. (Reference frontier). The set of efficient points dominating inefficient DMU o is

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called the reference frontier of DMU o, i.e., the reference frontier of DMU o is No . The members of the reference sets of a DMU are used to evaluate it and the presence

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of a DMU in the reference sets of other DMUs shows the importance of that DMU. In fact, this can be considered as a criterion for further evaluation of efficient DMUs. This matter is considered by some authors. Charnes et al. [37] considered the number of times an efficient

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DMU can be in the reference sets of inefficient DMUs as a factor for discriminating among efficient DMUs. Torgersen et al. [38] suggested a method for ranking efficient DMUs by their importance as benchmarks for inefficient DMUs. They calculated the level of improvement in each specific output created by an efficient DMU when is a member of the reference set of other DMUs. Using the aggregation of this improvement levels they constructed an index referred to as the reference-share of each efficient DMU in terms of each specific output. By comparing these reference-shares, they provide a ranking for efficient DMUs. Zhu [65] also extended this approach and used it for performance ranking of Fortune 500 companies. Lu and Lo [66] proposed an interactive benchmark approach in which each DMU is evaluated with respect to any other DMUs as its benchmark. Then, by averaging the results, a score for

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ACCEPTED MANUSCRIPT each DMU is obtained and ranking is made. Jahanshahloo et al. [39] considered the influence of efficient DMUs in efficiency evaluation of inefficient DMUs in a different manner. They proposed to measure the impact of excluding each efficient DMU from the sample on the efficiency scores of inefficient DMUs. The extension of their work by Chen and Deng [40] also follows the same objective. Based on the previous discussions, there is sufficient motivation for considering the importance of DMUs in performance evaluation of other DMUs as a criterion for discriminating among efficient DMUs. The next section presents our new approach in which the share of each efficient

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DMU in construction of the efficient frontier has the key role in discriminating among efficient DMUs. 3. The new approach

Let o ∈ E and p ∈ IE . The reference frontier of DMU p, Np , includes all possible efficient

targets for DMU p. Each element of Np can be represented as a convex combination of the

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elements of E. The share of an efficient DMU o in Np is defined as the maximum value of the coefficient associated with DMU o in all convex combinations representing the elements of Np . In other word, the share of DMU o in Np is the maximum value of λo in efficient targets of DMU p. The following definition formally introduces this concept.

Definition 2. The reference frontier share of DMU o for DMU p, denoted by λpo , is defined as follows:

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λpo

)  P P P λj = 1, λj ≥ 0 (j ∈ E) . λj yj ∈ Np , λ j xj , = max λo | (

j∈E

j∈E

j∈E

¯ o = P λpo , requiring Further, the (total) reference frontier share of DMU o is defined by λ p∈IE

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that IE is not empty.

We rank efficient DMUs based on their reference frontier share. In order to achieve our goal,

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we propose the following model:

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λpo = max λo P s.t. λj xij ≤ xip j∈E P λj yrj ≥ yrp j∈E P λj = 1 j∈E m P

i=1

vi xij −

λj ≤ tj

s P

i = 1, . . . , m r = 1, . . . , s

ur yrj + u0 = dj

r=1

j∈E j∈E

dj ≤ M (1 − tj )

j∈E

vi ≥ 1, ur ≥ 1

i = 1, . . . , m, r = 1, . . . , s

u0 f ree

λj ≥ 0, dj ≥ 0

j∈E

tj ∈ {0, 1}

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(8)

ACCEPTED MANUSCRIPT Here M is a sufficiently large positive number. The constraint

P

λj = 1 assures us that for

j∈E

some j we have λj > 0. Hence, for these js we have tj = 1 and dj = 0. Now we discuss some characteristics of model (8). The following results have more importance for our purposes. Theorem 2. Model (8) is always feasible. Proof. Consider the additive model (1) for evaluating DMU p. Let λ◦j (j = 1, . . . , n) be an optimal solution of model (1). Clearly, for any inefficient DMU say DMU j we have λ◦j = 0. Hence, n o P ◦ P ◦ λj xij ≤ xip (i = 1, . . . , m) and λj yrj ≥ yrp (r = 1, . . . , s). Let L = j ∈ E|λ◦j > 0 . Fol-

j∈E

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j∈E

lowing the discussion after model (2) in an optimal solution of model (2) for evaluating DMU p m s P P we have vi◦ xij − u◦r yrj + u◦0 = 0 (j ∈ L). Therefore, a feasible solution for model (8) is as r=1

i=1

follows:

λj = λ◦j (j ∈ E),

dj = 0 (j ∈ L), m m P P dj = vi◦ xij − u◦r yrj + u◦0 (j ∈ E \ L), i=1

i=1

tj = 1 (j ∈ L) and tj = 0 (j ∈ E \ L). Hence, the proof is complete.

Theorem 3. The point (ˆ x, y ˆ) =

P

j∈E

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vi = vi◦ (i = 1, . . . , m) and ur = u◦r (r = 1, . . . , s),

λ∗j xj ,

P

j∈E

λ∗j yj



is efficient, in which λ∗j (j ∈ E) is the

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optimal solution corresponding to λj (j ∈ E) in model (8).

Proof. Again let L = {j ∈ E|λ∗j > 0}. For all j ∈ L we have d∗j =

m P

i=1

vi∗ xij −

s P

r=1

u∗r yrj + u∗0 = 0.

r=1

j∈L

Since

P

j∈L

λ∗j

= 1, we have

j∈L m P

j∈L

vi∗ x ˆi

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i=1

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Multiplying the jth constraint by λ∗j and then adding these constraints we get the following: m s P ∗ ∗ P ∗ P ∗ P P λj )u0 = 0. λj yrj ) + ( vi∗ ( λj xij ) − u∗r ( i=1

production possibility set.

−

s P

r=1

u∗r yˆr + u∗0 = 0. Hence, (ˆ x, y ˆ) is an efficient point of the

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Theorem 4. λpo = 1 if and only if DMU o dominates DMU p. Proof. Suppose that λpo = 1. Then the constraints of model (8) simplify as xio ≤ xip

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(i = 1, . . . , m) and yro ≥ yrp (r = 1, . . . , s) which obviously means DMU o dominates DMU p. Conversely suppose that DMU o dominates DMU p i.e. xio ≤ xip (i = 1, . . . , m) and yro ≥ yrp

(r = 1, . . . , s). Then we have a feasible solution with λo = 1, λj = 0 (j 6= o) and do = 0. It is

clear that λpo ≤ 1, hence λpo = 1.

One of the useful properties of an efficiency measure is the translation invariance property, i.e. an affine translation of the data does not affect the results of the model or measure. This property has most importance when dealing with zero or negative data. Another fundamental property is units invariance, i.e. the measure or model is independent of the measurement units of the inputs and outputs. Ali and Seiford [77], Lovell and Pastor [76] and Pastor [78] studied and classified some of DEA models regarding these properties. It has been noted by these researchers 10

ACCEPTED MANUSCRIPT that the additive DEA models under the variable returns to scale assumption are translation invariant, but the simple additive model is not units invariant. The main DEA models that completely have the units invariance property are: the normalized weighted additive model of Lovell and Pastor [76], the RAM model of Cooper et al. [79], the SBM model of Tone [4], the ERM model of Pastor et al. [80] and the BAM model of Cooper et al. [81]. Note that the results of a translation invariant model with and without translation are equivalent. Hence, there is no need to translate the data when working with such a model

Theorem 5. Model (8) is translation invariant.

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except for computational considerations.

Proof. Let αi (i = 1, . . . , m) and βr (r = 1, . . . , s) be constant real numbers. Translating data with respect to the corresponding amounts results in a new data as follows: xTij = xij + αi (i = 1, . . . , m), (j = 1, . . . , n) T = y + β (r = 1, . . . , s), (j = 1, . . . , n) yrj rj r

j∈E m P i=1

vi (xij + αi ) −

s P

r=1

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Now, applying model (8) to the new data, the relevant constraints turn into the following: P λj (xij + αi ) ≤ xip + αi (i = 1, . . . , m) j∈E P λj (yrj + βr ) ≥ yrp + βr (r = 1, . . . , s) ur (yrj + βr ) + u0 = dj (j ∈ E)

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The first and second sets of these constraints, with respect to the constraint the original constraints of model (8). By a change of variable as

m P

i=1

P

j∈E s P

vi αi −

λj = 1, turn into ur βr + u0 = u0 0 ,

r=1

the third set of constraints turns into its respective one in model (8). Consequently, we observe

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that model (8) is translation invariant.

Theorem 6. Model (8) is units invariant, that is the change of the measurement units of the

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inputs and outputs does not change the optimal value of the objective function. Proof. Let αi (i = 1, . . . , m) and βr (r = 1, . . . , s) be constant positive real factors for changing

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the measurement units of the inputs and outputs. The new data to be considered for evaluation are as follows:

x ˜ij = αi xij (i = 1, . . . , m), (j = 1, . . . , n)

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y˜rj = βr yrj (r = 1, . . . , s), (j = 1, . . . , n) ˜ po be the optimal objective values of model (8) with original and new data, reLet λpo and λ

spectively. For convenience, we refer to the models with original and new data as (8a) and (8b), respectively. First, we notice that some of the corresponding constraints of models (8a) and (8b) are equivalent. For example: P P λj (αi xij ) ≤ αi xip ⇔ λj xij ≤ xip (i = 1, . . . , m) j∈E j∈E P P λj (βr yrj ) ≥ βr yrp ⇔ λj yrj ≥ yrp (r = 1, . . . , s)

j∈E

j∈E

Now assume that vi∗ (i = 1, . . . , m), u∗r (r = 1, . . . , s), u∗0 and d∗j are a part of an optimal solu  tion to model (8a). Define γ = max max αi , max βr . With some simple settings, the relevant i

r

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ACCEPTED MANUSCRIPT constraints in model (8a) can be rewritten as follows: m s P P (γvi∗ /αi )(αi xij ) − (γu∗r /βr )(βr yrj ) + γu∗0 = γd∗j (j ∈ E) r=1

i=1

(γvi∗ /αi ) ≥ (γ/αi ) (i = 1, . . . , m) (γu∗r /βr ) ≥ (γ/βr ) (r = 1, . . . , s)

Considering the definition of γ we have (γ/αi ) ≥ 1 (i = 1, . . . , m) and (γ/βr ) ≥ 1 (r = 1, . . . , s).

Consequently, we can see that vi = (γvi∗ /αi ) (i = 1, . . . , m), ur = (γu∗r /βr ) (r = 1, . . . , s), u0 = γu∗0 and dj = γd∗j along with other values is a feasible solution to model (8b). Therefore ˜ po ≥ λpo . λ optimal solution to model (8b). Hence m s P P v˜i∗ (αi xij ) − u ˜∗r (βr yrj ) + u ˜∗0 = d˜∗j (j ∈ E) r=1

i=1

v˜i∗ ≥ 1 (i = 1, . . . , m)

u ˜∗r ≥ 1 (r = 1, . . . , s). Clearly

v˜i∗ αi

≥ αi (i = 1, . . . , m) and

u ˜∗r βr

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To prove the other side, let v˜i∗ (i = 1, . . . , m), u ˜∗r (r = 1, . . . , s), u ˜∗0 and d˜∗j be a part of an

≥ βr (r = 1, . . . , s). Define γ˜ = max max

r=1

≥ 1 (i = 1, . . . , m)

≥ 1 (r = 1, . . . , s).

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Multiplying the above constraints by γ˜ and imposing some settings we have: m s P P (˜ γ v˜i∗ αi )xij − (˜ γu ˜∗r βr )yrj + γ˜ u ˜∗0 = γ˜ d˜∗j (j ∈ E)

i=1 γ˜ v˜i∗ αi γ˜ u ˜∗r βr



i

1 1 αi , max r βr



Therefore, a feasible solution to model (8a) is obtained as vi = γ˜ v˜i∗ αi (i = 1, . . . , m), ur = γ˜ u ˜∗r βr ˜ po ≤ λpo . (r = 1, . . . , s), u0 = γ˜ u ˜∗ and dj = γ d˜∗ (j ∈ E). This implies that λ 0

j

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˜ po = λpo . Putting this together with the previous result, we have λ

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Therefore, model (8) is units invariant.

It is to be noted that each point lying on a facet in m+s dimensional space can be represented P tj ≤ as a convex combination of at most m + s points. Therefore, we can add the constraint j∈E

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m + s to model (8) in order to reduce the computational burden. Thus, the final model is as

AC

CE

follows:

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.

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λpo = max λo P s.t. λj xij ≤ xip j∈E P λj yrj ≥ yrp j∈E P λj = 1 i=1

vi xij −

s P

r = 1, . . . , s

ur yrj + u0 = dj

r=1

λj ≤ tj

j∈E j∈E

dj ≤ M (1 − tj ) P tj ≤ m + s

j∈E

j∈E

(9)

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j∈E m P

i = 1, . . . , m

vi ≥ 1, ur ≥ 1

i = 1, . . . , m, r = 1, . . . , s

λj ≥ 0, dj ≥ 0, tj ∈ {0, 1}

j∈E

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u0 f ree

Note that non-extreme efficient DMUs are linear combinations of some extreme efficient DMUs. This can be regarded in the reference frontier share of extreme efficient DMUs. Hence, we extend the concept of the reference frontier to account for this fact. For this purpose, let o, p ∈ E. The proper definition of the reference frontier for efficient DMU p is the singleton set containing just (xp , yp ). Now, the reference frontier share of DMU o for DMU p is defined

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according to definition 2 through replacing Np by {(xp , yp )}. p6=o

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Further, the reference frontier share of DMU o in the presence of non-extreme efficient DMUs P is defined as µ ¯o = λpo .

It is easy to see that models (8) and (9) are consistent with the new extended definition of

the reference frontier and there is no need to any modification. It is obvious that for p = o

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we have λpo = 1. Further, if DMU p is extreme efficient and p 6= o, then λpo = 0; if DMU p is non-extreme efficient and DMU o is efficient, then models (8) and (9) are equivalent to the

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following model:

AC

λpo = max λo P s.t. λj xij = xip i = 1, . . . , m j∈E P λj yrj = yrp r = 1, . . . , s j∈E P λj = 1

(10)

j∈E

λj ≥ 0

j∈E

¯ o and µ Note that in the absence of non-extreme efficient DMUs λ ¯o are equal. Therefore, in cases that no non-extreme efficient DMU exists, the ranking results of the reference frontier share by definition 2 and the new extension are identical. Remark 1. It is possible to include in the reference frontier some alternative reference domains other than dominating efficient frontier. For example, the targets domains of the preference 13

ACCEPTED MANUSCRIPT structure models of Thanassoulis and Dyson [73] and Zhu [74] or multi-objective models of Estellita Lins et al. [75] and Quariguasi Frota Neto and Angulo-Meza [82]. Note that some of these models allow decision maker to assign targets that do not dominate inefficient DMUs. Thus, for some of this domains the reference frontier share may be trivial or cannot help to discriminate among efficient DMUs. This issue needs further research. Remark 2. Since most of the ranking approaches have some problems with variable returns to scale assumption (as noted in the literature review), we focused on this special case. However, all assertions except theorem 5, i.e. the translation invariance property, are correct in the case

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of constant returns to scale. 4. Numerical examples

Example 1. In order to evaluate the applicability of the proposed approach in ranking nonextreme efficient DMUs we use artificial data that is presented in table 1. DMUs A, B, C, F and G are efficient. DMUs B and F are non-extreme efficient and DMUs D, E and H are

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inefficient. Figure 1 shows DMUs in the input and output space. The efficient frontier is the line segments AC, and CG. The reference frontier for DMU D is the thick line segments. The reference frontier share of DMU A for DMU D is determined by the maximum amount of the ˆ 0 as convex combination of efficient coefficient λA in all representations of the reference point D DMUs, i.e., ˆ 0, λA A + λB B + λC C + λF F + λG G = D

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λA + λB + λC + λF + λG = 1.

Here, λB = λF = λG = 0, λA = 3/4 and λC = 1/4. The reference frontier share of DMUs B and C for DMU D is 1, because DMUs B and C belong to the reference frontier of DMU D and

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dominate it. The reference frontier share of DMU F for DMU D is 1/2, because the maximum amount of the coefficient associated with point F in all convex combinations of the reference ˆ 00 which is the midpoint of DMUs C and F. points of DMU D is related with the point D

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The reference frontier shares of efficient DMUs are presented in table 2. The last row of P this table shows the values of µ ¯ derived by the formula µ ¯o = λpo . Note that these values p6=o

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are the total for each column excluding their own lambda value, e.g., the result for column A is

the total (2.75) minus 1 that is 1.75. The results of the new approach and ranking of efficient DMUs are given in table 3. The third column of this table shows the ranking of DMUs by new

AC

approach which implies that the new approach ranks efficient DMUs completely. The fourth and fifth columns also show the SBM super-efficiency scores [28] under the variable returns to scale assumption and their associated ranking. Finally, the sixth and seventh columns of this table present the results of the cross-dependence efficiency measure (indicated here by MCDE) [40]. As it is seen, the SBM super-efficiency model cannot help to provide a complete ranking of efficient DMUs. Note that the super-efficiency models all assign a 1 value to the non-extreme efficient DMUs. Hence, they cannot provide discrimination among these DMUs. Another point that we must note here, is that the super-efficiency models are in favor of DMUs that stand far from other DMUs, e.g. DMUs like A that consumes the least amount of the input and produces the least amount of the output. It is easy to see that, if DMU A reduces its output amount 14

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y G

F ˆ 00 D E

B

A

ˆ0 D

H

D

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C

x

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Figure 1: The reference frontier share

to a positive infinitesimal, its super-efficiency score does not change and its rank is retained. This is clearly an unreasonable phenomena. However, the objective of the new approach is the evaluation of the efficient DMUs based on their role in setting targets for other DMUs and it is likely that the central efficient DMUs provide a better target than other efficient DMUs. This

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can be considered as an advantage of the new approach.

The point that we must note about the ranking results obtained from the measure of cross-

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dependence efficiency (MCDE) is that DMUs B, F and G all gain an equal efficiency score, because the efficiency scores of inefficient DMUs do not change by removing each of these DMUs from the sample. Although the tie about DMU G may be a random event, but the case

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of tie about DMUs B and F is not an accident. In fact, the MCDE approach assigns to all non-extreme efficient DMUs an equal score. As inspections show, the cross efficiencies of DMU A with all optimal weights of DMUs F

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and G are negative. Also, the cross efficiencies of DMU B with optimal weights of DMUs F and G in the best possible case are zero. Therefore, the application of the cross efficiency in this example is meaningless. However, further considerations make clear that DMU C has the

AC

maximum cross efficiency value among DMUs. More carefully, as there are multiple optimal weights for all DMUs, by maximizing the cross efficiencies of DMUs individually, we can see that the average maximum cross efficiency of DMU C is 1 and it is the best among DMUs. A reason for this is that DMU C lies on all efficient reference facets associated to other DMUs. This is consistent with the logic behind our approach. Example 2. Consider the data of 20 bank branches in Iran as DMUs which extracted from

Jahanshahloo et al. [39] and had been used in some other applications. The data is presented in table 4. The inputs of DMUs are staff (x1 ), computer terminals (x2 ) and space (x3 ) and the outputs are deposits (y1 ), loans (y2 ) and charge (y3 ). The results of additive model (1) for determining the efficiency status of DMUs also are presented in table 4 . As it is seen, 11 DMUs 15

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Table 1: Data and results of the super-efficiency models (example 1)

x1

y1

Sup-I

Sup-O

A B C D E F G H

2 3 4 5 4 6 7 6

3 5 7 4 6 8 8.5 6

1.5000 1.0000 1.2500 0.5000 0.8750 1.0000 Infeasible 0.5833

Infeasible 1.0000 1.1667 0.5333 0.8571 1.0000 1.0625 0.7500

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DMU

Table 2: The extended reference frontier shares of efficient DMUs (example 1)

A

B

C

F

G

A B C D E F G H

1 0.5 0 0.75 0.25 0 0 0.25

0 1 0 1 0.5 0 0 0.5

0 0.5 1 1 1 1/3 0 1

0 0 0 0.5 0 1 0 1

0 0 0 1/3 0 2/3 1 2/3

2

3.83

1.5

1.67

ED 1.75

AC

CE

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µ ¯

M

DMU

Table 3: Ranking of efficient DMUs (example 1)

DMU

µ ¯

Rank

SBM Sup

Rank

MCDE

Rank

A B C F G

1.75 2 3.83 1.5 1.67

3 2 1 5 4

1.5000 1.0000 1.1667 1.0000 1.0625

1 4 2 4 3

0.7070 0.6745 0.7477 0.6745 0.6745

2 3 1 3 3

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Table 4: Data and results of the additive model (example 2)

x1

x2

x3

y1

y2

y3

ADD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.950 0.796 0.798 0.865 0.815 0.842 0.719 0.785 0.476 0.678 0.711 0.811 0.659 0.976 0.685 0.613 1.000 0.634 0.372 0.583

0.700 0.600 0.750 0.550 0.850 0.650 0.600 0.750 0.600 0.550 1.000 0.650 0.850 0.800 0.950 0.900 0.600 0.650 0.700 0.550

0.155 1.000 0.513 0.210 0.268 0.500 0.350 0.120 0.135 0.510 0.305 0.255 0.340 0.540 0.450 0.525 0.205 0.235 0.238 0.500

0.190 0.227 0.228 0.193 0.233 0.207 0.182 0.125 0.080 0.082 0.212 0.123 0.176 0.144 1.000 0.115 0.090 0.059 0.039 0.110

0.521 0.627 0.970 0.632 0.722 0.603 0.900 0.234 0.364 0.184 0.318 0.923 0.645 0.514 0.262 0.402 1.000 0.349 0.190 0.615

0.293 0.462 0.261 1.000 0.246 0.569 0.716 0.298 0.244 0.049 0.403 0.628 0.261 0.243 0.098 0.464 0.161 0.068 0.111 0.764

0.0000 1.2064 0.0000 0.0000 0.8105 0.7858 0.0000 0.0000 0.0000 1.3949 1.0994 0.0000 0.7289 1.6150 0.0000 0.9732 0.0000 0.8965 0.0000 0.0000

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DMU

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are efficient. Table 5 presents the results of the new approach, radial super-efficiency models [27], SBM super-efficiency model [28] and additive super-efficiency model [29] (indicated here by ADDSup), for the efficient DMUs. The third, seventh and ninth columns of this table show

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the ranking results of the new approach, SBM super-efficiency and additive super-efficiency, respectively. Note that the radial super-efficiency approach is not applicable for ranking due to the infeasibility in many cases. Hence, we compare the ranking results of the new approach with

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the slacks based super-efficiency models. It is seen that the ranks of some DMUs with different models are identical. However, the difference in ranks of DMUs 7 and 15 by the new approach

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and the super-efficiency models are considerable. With some cares, we find that the amount of the first output of DMU 15 is largely different from that of other DMUs. More precisely, the amount of the first output of DMU 15 is more than 7 times interquartile range above the

AC

third quartile of this output. This can be a sign of being an outlier. As it is said in section 1, the findings of Banker and Chang [62] showed that the super-efficiency approach performs satisfactorily in outlier detection rather than in ranking efficient DMUs. Example 3. In order to test the new approach with negative data we use the data taken from Sharp et al. [83] and also used by Emrouznejad et al. [84]. The data are presented in table 6. We have 13 DMUs with two inputs and three outputs. The inputs are x1 (cost) and x2 (effluent) and the outputs are y1 (saleable), y2 (methane) and y3 (CO2 ). As it is seen, the second input, the second and the third outputs are non-positive. DMUs 3, 7, 8, 11 and 13 are efficient and the other DMUs are inefficient. The reference frontier shares of efficient DMUs to inefficient ones are presented in table 7. The

17

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Table 5: Results of the new approach, super-efficiency models and ranking of efficient DMUs (example 2)

¯ λ

Rank

Sup-I

Sup-O

SBMSup

Rank

ADDSup

Rank

1 3 4 7 8 9 12 15 17 19 20

1.9034 1.3016 4.9944 5.5639 1.3697 4.2284 4.2976 2.4482 2.5488 2.3000 5.2993

9 11 3 1 10 5 4 7 6 8 2

1.1748 Infeasible Infeasible Infeasible 1.2150 1.3775 Infeasible Infeasible Infeasible 1.2796 1.2641

1.2575 1.0931 Infeasible 1.2330 Infeasible Infeasible 1.1242 5.0731 1.5132 Infeasible Infeasible

1.0584 1.0347 1.3384 1.0847 1.0717 1.3067 1.0567 1.3654 1.1275 1.0932 1.1047

9 11 2 7 8 3 10 1 4 6 5

1.0584 1.0347 1.4132 1.1843 1.0717 1.3458 1.0970 1.3654 1.1383 1.0932 1.1047

10 11 1 4 9 3 7 2 5 8 6

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DMU

Table 6: Data and results of the additive model (example 3)

x1

x2

y1

y2

y3

1 2 3 4 5 6 7 8 9 10 11 12 13

1.03 1.75 1.44 10.8 1.3 1.98 0.97 9.82 1.59 5.96 1.29 2.38 10.3

-0.05 -0.17 -0.56 -0.22 -0.07 -0.1 -0.17 -2.32 0 -0.15 -0.11 -0.25 -0.16

0.56 0.74 1.37 5.61 0.49 1.61 0.82 5.61 0.52 2.14 0.57 0.57 9.56

-0.09 -0.24 -0.35 -0.98 -1.08 -0.44 -0.08 -1.42 0 -0.52 0 -0.67 -0.58

-0.44 -0.31 -0.21 -3.79 -0.34 -0.34 -0.43 -1.94 -0.37 -0.18 -0.24 -0.43 0

M

ED

ADD

0.4756 1.1489 0.0000 9.1057 1.9649 0.9481 0.0000 0.0000 0.5900 4.1443 0.0000 2.5900 0.0000

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DMU

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¯ derived by the formula λ ¯ o = P λpo . The ranking last row of this table shows the values of λ p∈IE

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results of the efficient DMUs also are presented in table 8. As it is seen, the new approach can

AC

rank DMUs without any problem related to the existence of negative and zero data.

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Table 7: Reference frontier share of efficient DMUs to inefficient ones (example 3)

3

7

8

11

13

1 2 4 5 6 9 10 12

0.0645 0.6857 0.4823 0.7021 0.9707 0.0000 0.8571 1.0000

1.0000 0.5140 0.4403 0.6065 0.6837 0.0000 0.3693 0.8697

0.0067 0.0476 0.5981 0.0157 0.0679 0.0000 0.0400 0.1198

0.1875 0.9692 0.4229 1.0000 0.3919 1.0000 0.6958 0.9367

0.0064 0.0664 0.9722 0.0207 0.0997 0.0000 0.5309 0.1406

¯ λ

4.7625

4.4835

0.8957

5.6039

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DMU

1.8371

Table 8: Ranking of efficient DMUs (example 3)

¯ λ

Rank

3 7 8 11 13

4.7625 4.4835 0.8957 5.6039 1.8371

2 3 5 1 4

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DMU

5. Conclusion

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Discriminating among efficient DMUs is an important issue in data envelopment analysis. In most cases the models of DEA introduce more than one DMU as efficient. Therefore, a more

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comprehensive analysis is needed to discriminate among efficient DMUs. Some other information can be derived from production possibility set to carry out a more rigorous evaluation. Some examples of this information are: the influence of presence or absence of each DMU in

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evaluation of other DMUs, the role of each DMU in setting targets for other DMUs, the role of each DMU in acting as a benchmark for other DMUs, and etc. In this paper, we proposed a new approach for ranking efficient decision making units based on the share of each efficient DMU

CE

in reference frontier of inefficient DMUs. The reference frontier is the set of all possible targets that a DEA model may propose to an inefficient DMU. The efficient units that define a target are benchmarks or references that can help to improve the whole efficiency of the industry.

AC

Thus, the importance of each efficient unit can be specified by the value of its corresponding intensity variable in the assessment of inefficient units which provides the criterion for ranking and discriminating among efficient units. We propose a model that measures the maximum amount of the coefficient associated with each efficient DMU in providing targets for inefficient DMUs. As discussions and examples show, the new approach can provide a rational ranking of efficient DMUs. The new approach does not have any infeasibility or unboundedness problem. Further, it can be handled with negative data as well as positive data.

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ACCEPTED MANUSCRIPT Acknowledgments We thank three anonymous reviewers and the editor for their valuable suggestions and constructive comments on earlier version of the paper. References [1] A. Charnes, W. W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978) 429–444.

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