TY - JOUR
AU - Rostamy-Malkhalifeh,, Mohsen
AB - Abstract Inverse (DEA) is an approach to estimate the expected input/output variation levels when the efficiency score reminds unchanged. Essentially, finding most efficient decision-making units (DMUs) or ranking units is an important problem in DEA. A new ranking system for ordering extreme efficient units based on inverse DEA is introduced in this article. In the adopted method, here the amount of required increment of inputs by increasing the outputs of the unit under evaluation is obtained through the proposed models. By obtaining these variations, this proposed methodology enables the researcher to rank the efficient DMUs in an appropriate manner. Through the analytical theorem, it is proved that suggested models here are feasible. These newly introduced models are validated through a data set of commercial banks and a numerical example. 1. Introduction Data envelopment analysis (DEA) is a method for calculating the performance evaluation of the decision-making units (DMUs) of specific inputs and outputs. Charnes et al. (1978) introduced DEA for the first time to measure the efficiency of the DMUs. DEA models are based on mathematical programming for estimating the efficiency units. The relative efficiency of each one of the units will be obtained by solving a linear programming (LP) problem. The inverse DEA is introduced by Wei et al. (2000) to estimate the output variation subject to an increase in inputs and preservation of the efficiency score. Wei et al. (2000) applied the multiple-objective programming (MOP) to estimate output levels. Jahanshahloo et al. (2004a,c) and Yan et al. (2002) use the decision makers’ preferences to convert multiple-objective LP (MOLP) into a single-objective LP. In addition to the above models, there exist some other proposal like Yan et al. (2002) and Li & Cui (2008) where the input for resource allocation is estimated. Hadi-Vencheh & Foroughi (2006) introduced a model for increasing some input levels and decreasing others. Lertworasirikul et al. (2011) presented a model regarding the concurrent increases of some outputs and decreases of other outputs. Jahanshahloo et al. (2014) studied the inverse DEA using the non-radial enhanced Russell model. Jahanshahloo et al. (2015) introduced a periodic weak Pareto solution for MOLP to solving inverse DEA problems and proposed inverse DEA subject to inter-temporal dependence assumption. Hadi-vencheh et al. (2015) extended the inverse DEA models for interval data in evaluating the efficiency score. Ghobadi & Jahangiri (2015) developed fuzzy inverse DEA models. Gattoufi et al. (2014) applied inverse DEA for a case of merger and acquisition in banking in order to obtain the required level of inputs and outputs of the merged bank to reach a given efficiency score. Afterwards, Amin et al. (2017) proposed a new method to foretasted whether a merger in a market is generating a major or a minor consolidation, through inverse DEA model. Some of the latest research on inverse DEA are shown in Table 1. Table 1. Some of the latest inverse DEA research Reference Short description Amin et al. (2019) Using goal programming and inverse DEA for target setting merger Emrouznejad et al. (2018) Allocating CO2 emission quota under several assumptions Kalantary & Saen (2018) Inverse network DEA model in dynamic context for evaluation Sustainability of supply chains Amin & Al-Muharrami (2018) Using inverse DEA in merging units with negative data by inverse DEA Ghobadi (2018) Using enhanced Russell model for fuzzy data Amin et al. (2017) Using inverse DEA in merge units Ghiyasi (2017) Using inverse DEA when data prices is available for assessing cost efficiency Reference Short description Amin et al. (2019) Using goal programming and inverse DEA for target setting merger Emrouznejad et al. (2018) Allocating CO2 emission quota under several assumptions Kalantary & Saen (2018) Inverse network DEA model in dynamic context for evaluation Sustainability of supply chains Amin & Al-Muharrami (2018) Using inverse DEA in merging units with negative data by inverse DEA Ghobadi (2018) Using enhanced Russell model for fuzzy data Amin et al. (2017) Using inverse DEA in merge units Ghiyasi (2017) Using inverse DEA when data prices is available for assessing cost efficiency Open in new tab Table 1. Some of the latest inverse DEA research Reference Short description Amin et al. (2019) Using goal programming and inverse DEA for target setting merger Emrouznejad et al. (2018) Allocating CO2 emission quota under several assumptions Kalantary & Saen (2018) Inverse network DEA model in dynamic context for evaluation Sustainability of supply chains Amin & Al-Muharrami (2018) Using inverse DEA in merging units with negative data by inverse DEA Ghobadi (2018) Using enhanced Russell model for fuzzy data Amin et al. (2017) Using inverse DEA in merge units Ghiyasi (2017) Using inverse DEA when data prices is available for assessing cost efficiency Reference Short description Amin et al. (2019) Using goal programming and inverse DEA for target setting merger Emrouznejad et al. (2018) Allocating CO2 emission quota under several assumptions Kalantary & Saen (2018) Inverse network DEA model in dynamic context for evaluation Sustainability of supply chains Amin & Al-Muharrami (2018) Using inverse DEA in merging units with negative data by inverse DEA Ghobadi (2018) Using enhanced Russell model for fuzzy data Amin et al. (2017) Using inverse DEA in merge units Ghiyasi (2017) Using inverse DEA when data prices is available for assessing cost efficiency Open in new tab One of the most controversial issues in DEA is concerned with ranking DMUs. There exist many articles in the literature regarding ranking efficient units. Some methods regarding ranking DMUs are categorized as cross efficiency, super efficiency, techniques based on finding optimal weight, multi-criteria decision making (MCDM) technique, reference-based approaches, etc. The first category involves evaluation of the cross efficiency matrix where a unit is a self- and peer-evaluated. In this method, the efficiency of each DMU is evaluated |$n$| times compared with the other DMUs in the set of data. Results are provided on the cross-efficiency matrix, which its main diagonal elements are the self-efficiency for each DMU. Sexton et al. (1986) introduced the cross-efficiency method and is provided by Rodder & Reucher (2011), Jahanshahloo et al. (2011), Ramon et al. (2011), Contreras (2012), Wu et al. (2016a) and Wu et al. (2016b). An et al. (2018) incorporated the DEA and analytic hierarchy process (AHP) methods to fully rank the DMUs that consider all possible cross efficiencies of a DMU according to all the other DMUs. Furthermore, Liu (2018) consider cross efficiency intervals and their variances for ranking DMUS. Supper efficiency models are based on the idea that one unit is excluded from the production possibility set (PPS) and is assessed through the remaining other units. This idea has been a subject of study by Andersen & Petersen (1993), Mehrabian et al. (1999), Tone (2002), Jahanshahloo et al. (2004b), Amirteimoori et al. (2005), Jahanshahloo et al. (2006), Li et al. (2007), Rezai Balf et al. (2012) and Chen et al. (2013). The ranking units are based on the optimal weights obtained from the multiplier model of DEA. Washio & Yamada (2013), Wang et al. (2009), Alirezaee & Afsharian (2007), Liu & Peng (2008) and Lotfi et al. (2011) have developed this method. Oukil (2018) introduced a new approach by combining cross efficiency, weighting schemes and ordered weighted averaging (OWA) operator to provide a full ranking of DMUs. He used the property of multiple weighting schemes generated over the cross-evaluation process for developing a methodology that gives not only robust ranking patterns but also more realistic sets of weights for the DMUs. The MCDM technique has been studied by Cook & Kress (1991), Strassert & Prato (2002), Chen (2007) and Jablonsky (2012). Reference-based approaches assume the importance of efficient DMUs as references for other DMUs to assign a criterion for discriminating among efficient DMUs. Charnes et al. (1984), Jahanshahloo et al. (2007) and Chen & Deng (2011) have extended this approach. Recently, Rezaeiani & Foroughi (2018) proposed a model to measure the reference frontier share for ranking efficient units. Some of the latest research on ranking are shown in Table 2. Table 2. Some of the latest ranking research Reference Short description Liu et al. (2019) Investigate the cross-efficiency evaluation in DEA based on prospect theory de Blas et al. (2018) Applying a novel ranking method to measure dominance derived from social network analysis by detection method of self-evaluations Mufazzal & Muzakkir (2018) A new method of ranking based on proximity indexed value Hatami-Marbini et al. (2018) Introducing the SRDM measures regardless of feasibility or infeasibility of the model Liu & Wang (2018) Constructing the interval efficiency evaluation by the normalized efficiency and achieve the best normalized efficiency and the worst normalized efficiency for ranking. Lin & Chen (2018) Introducing a new modified input-oriented variable returns to scale supper-efficiency (VRS SE) model and its corresponding algorithm. Wang & Sun (2018) Dividing the problem of DMUs’ nonhomogeneity into external nonhomogeneity and internal homogeneity Salehian et al. (2018) A novel hybrid algorithm based on fuzzy analytical hierarchy process and DEA Reference Short description Liu et al. (2019) Investigate the cross-efficiency evaluation in DEA based on prospect theory de Blas et al. (2018) Applying a novel ranking method to measure dominance derived from social network analysis by detection method of self-evaluations Mufazzal & Muzakkir (2018) A new method of ranking based on proximity indexed value Hatami-Marbini et al. (2018) Introducing the SRDM measures regardless of feasibility or infeasibility of the model Liu & Wang (2018) Constructing the interval efficiency evaluation by the normalized efficiency and achieve the best normalized efficiency and the worst normalized efficiency for ranking. Lin & Chen (2018) Introducing a new modified input-oriented variable returns to scale supper-efficiency (VRS SE) model and its corresponding algorithm. Wang & Sun (2018) Dividing the problem of DMUs’ nonhomogeneity into external nonhomogeneity and internal homogeneity Salehian et al. (2018) A novel hybrid algorithm based on fuzzy analytical hierarchy process and DEA Open in new tab Table 2. Some of the latest ranking research Reference Short description Liu et al. (2019) Investigate the cross-efficiency evaluation in DEA based on prospect theory de Blas et al. (2018) Applying a novel ranking method to measure dominance derived from social network analysis by detection method of self-evaluations Mufazzal & Muzakkir (2018) A new method of ranking based on proximity indexed value Hatami-Marbini et al. (2018) Introducing the SRDM measures regardless of feasibility or infeasibility of the model Liu & Wang (2018) Constructing the interval efficiency evaluation by the normalized efficiency and achieve the best normalized efficiency and the worst normalized efficiency for ranking. Lin & Chen (2018) Introducing a new modified input-oriented variable returns to scale supper-efficiency (VRS SE) model and its corresponding algorithm. Wang & Sun (2018) Dividing the problem of DMUs’ nonhomogeneity into external nonhomogeneity and internal homogeneity Salehian et al. (2018) A novel hybrid algorithm based on fuzzy analytical hierarchy process and DEA Reference Short description Liu et al. (2019) Investigate the cross-efficiency evaluation in DEA based on prospect theory de Blas et al. (2018) Applying a novel ranking method to measure dominance derived from social network analysis by detection method of self-evaluations Mufazzal & Muzakkir (2018) A new method of ranking based on proximity indexed value Hatami-Marbini et al. (2018) Introducing the SRDM measures regardless of feasibility or infeasibility of the model Liu & Wang (2018) Constructing the interval efficiency evaluation by the normalized efficiency and achieve the best normalized efficiency and the worst normalized efficiency for ranking. Lin & Chen (2018) Introducing a new modified input-oriented variable returns to scale supper-efficiency (VRS SE) model and its corresponding algorithm. Wang & Sun (2018) Dividing the problem of DMUs’ nonhomogeneity into external nonhomogeneity and internal homogeneity Salehian et al. (2018) A novel hybrid algorithm based on fuzzy analytical hierarchy process and DEA Open in new tab Based on our literature review on ranking methods there is a lack of study for DMU’s extension and new features. To compensate this lack, we have utilized the inverse DEA approach for the aim. In the classical inverse DEA models, the DMU with existing inputs and outputs is applied for building PPS. In this article, a model is based on removing the current unit and replacing it with a revised DMU consisting of renewed inputs and outputs is introduced. The new unit is applied for building the possibility set. In this method, through MOLP model the amount of required increment of the inputs are obtained when the outputs of the unit under evaluation increase and the unit remains efficient. By obtaining this variation, this proposed method (PM) enables the efficient DMUs to be ranked in an appropriate manner. Finding a unique optimal solution for MOLP models is a challenging task; thus, a single-objective LP model is presented here to face this challenge. The inverse DEA method is the futuristic approach for ranking DMUs due to its ability to estimate the inputs changes based on any increments of the outputs so is named the growth potential. This article is structured as follows: some prerequisite from inverse DEA are given in Section 2. The correlation between inverse DEA and changing PPS is discussed in Section 3. An inverse DEA approach for ranking efficient extreme DMUs is proposed in Section 4. Numerical examples to illustrate the applicability of this PMs are presented in Section 5 and the conclusions are presented in Section 6. 2. Prerequisite A MOP is defined as follows: $$\begin{equation} \begin{array}{l} {~~~~~ Min f(x)} \\{s.t. ~~~g(x) \leqslant 0,} \\ \end{array} \end{equation}$$ (2.1) where |$ f:R^m \, \rightarrow R^q \,\,\ $| and |$ \,\ g:R^m \, \rightarrow R^p $| are the two vector functions, (i.e. |$ f(x)=(\,f_1(x),f_2(x),\dots ,f_q(x)) $|⁠, and |$ g(x)=(g_1(x),g_2(x),\dots ,g_p(x)) $|⁠). The set |$ X=\{x\in R^m: g_k(x)\leqslant 0,~~ k=1,2,\dots ,p\} $| is named the set of feasible solutions of MOP introduced by model (2.1). Provided that |$ f(.) $| and |$ g(.) $| are linear functions, model (2.1) is renamed MOLP. Usually, there is no solution |$ x \in X $| to minimize all objective functions; hence, the Pareto solutions should be considered for this purpose instead of optimal solutions. The definitions of the Pareto solution are as follows (Ehrgott, 2005): Definition 2.1 Assume |$ \bar{x} \in X $| is a feasible solution of MOP (2.1), |$ \bar{x} $| is named a Pareto solution provided that there does not exist |$ x \in X $| in a sense that |$ f_i(x) \leqslant f_i(\bar{x}) \,\ $| for all |$i=1,2,\dots ,q,$||$ f_i(x) \ < f_i(\bar{x})$| for some |$i=1,2,\dots ,q$|⁠. Definition 2.2 Assume |$ \bar{x} \in X $| is a feasible solution of MOP (2.1), |$ \bar{x} $| is named a weak Pareto solution provided that there does not exist |$ x \in X $| such that |$ f_i(x) \ < f_i(\bar{x}) \,\ $| for all |$i=1,2,\dots ,q$|⁠. Assume a set of DMUs, {|$DMU_j$|⁠: |$j = 1, 2,\dots , n $|}, which produce multiple-output |$y_{rj} (r =1, 2,..., s)$|⁠, through multiple-input |$x_{ij} (i =1, 2,..., m)$|⁠. The inputs and outputs of |$DMU_j$| are represented by vectors: |$X_{j} =(x_{1j}, x_{2j},..., x_{mj})$| and |$Y_{j} = ( y_{1j}, y_{2j},..., y_{sj})$|⁠, respectively. The PPS |$ T_{c} $| is defined as $$\begin{equation*} T_{c}=\left \{ (X,Y) \,: \, X \geqslant \sum _{j=1}^{n}X_{j} \lambda _{j} ~~,~~ Y \leqslant \sum _{j=1}^{n}Y_{j} \lambda _{j}~,~~ \lambda_j \geqslant 0,~~ j=1, \dots,n \right \}. \end{equation*}$$ For evaluating the efficiency of |$DMU_o$|⁠, |$ o \in{\{1,2,..., n\}}$| the following DEA model should be applied: $$\begin{equation} \begin{array}{l} {~~~~~ Min \theta} \\{s.t. ~~~\sum\limits _{j=1}^{n}x_{ij} \lambda _{j} \leqslant \theta x_{io}, \,\,\,\,\,\,\,\,\ i=1,2,\dots,m,} \\[12pt] {~~~~~~~\sum\limits _{j=1}^{n}y_{rj} \lambda _{j} \geqslant y_{ro}, \,\,\,\,\,\,\,\,\,\ r=1,2,\dots,s, } \\[12pt] ~~~~~~~~\lambda _{j} \geqslant 0,\ ~~~~~~~~~~~~~~j=1,2,\dots,n. \end{array} \end{equation}$$ (2.2) This model is named after Charnes, Cooper and Rhodes are abbreviated as CCR (Charnes et al., 1978). The efficiency under a constant returns to scale (CRS) assumption is measured by this model. Assume that |$\theta ^{*}$| is the optimal solution (efficiency index) of CCR model, it is obvious that |$\theta ^{*} \leqslant 1$|⁠. If |$\theta ^{*} = 1$|⁠, then |$DMU_o$| is efficient. Now consider the following question: ‘How much would the inputs increase when the outputs increase at unchanged efficiency score of |$\theta $|?’ To answer this question, assume outputs of |$DMU_o$| increase from |$ Y_o $| to |$ Y_{o} +K_o $|⁠, where the vectors |$ Y_{o}\geqslant 0 $|⁠, |$Y_{o} \ne 0 $| and |$ K_o \in R_+^s$|⁠; the inputs’ vectors |$X_{o}+\varDelta X_o$| must be estimated such that the efficiency index of |$DMU_o$| remains |$\theta ^{*}$| unchanged. The vectors |$ Y_{o} +K_o $| and |$X_{o}+\varDelta X_o$| are represented by |$ \beta _o $| and |$ \alpha _o $|⁠, respectively. The following MOLP model is applied for measuring |$ \alpha _o $| (Wei et al., 2000): $$\begin{equation} \begin{array}{l} { Min(\alpha_{1o},\alpha_{2o},\dots,\alpha_{mo})} \\{s.t. ~~\sum\limits _{j=1}^{n}x_{ij} \lambda _{j} \leqslant \theta^* \alpha_{io}, \,\,\,\,\,\,\,\,\,\,\,\ i=1,2,\dots,m,} \\[12pt] {~~~~~~\sum\limits _{j=1}^{n}y_{rj} \lambda _{j} \geqslant \beta_{ro}, ~~~~~~~~~~~~ r=1,2,\dots,s,} \\[12pt] {~~~~~~ \alpha_{io} \geqslant x_{io}, ~~~~~~~~~~~~~~~~~~~~ i=1,2,\dots,m, }\\ ~~~~~~\lambda \in \varLambda. \end{array} \end{equation}$$ (2.3) where $$\begin{equation*} \varLambda=\left\{\lambda|\lambda=(\lambda_1,\ldots,\lambda_n,\lambda_{n+1}),~\delta_1\left(\sum_{j=1}^{n+1}\lambda_j+\delta_2(-1)^{\delta_3}\nu\right)=\delta_1, ~\nu\geqslant0,~\lambda_j\geqslant 0,j=1,\ldots,n+1\right\}\!. \end{equation*}$$ It can be easily observed that |$\bullet $| if |$ \delta _1=0 $|⁠, model (2.3) is the CCR model. |$\bullet $| if |$ \delta _1=1 $| and |$ \delta _2=0 $|⁠, model (2.3) is the Banker, Charnes, and Cooper (BCC) model. |$\bullet $| if |$ \delta _1= \delta _2=1 $| and |$ \delta _3=0 $|⁠, model (2.3) is the CCR-BCC model. |$\bullet $| if |$ \delta _1= \delta _2= \delta _3=1 $|⁠, model (2.3) is the BCC-CCR model. After changing the inputs and outputs into |$\alpha ^{*}_o=X_o +\varDelta X_{o}^* $| and |$\beta _o= Y_{o}+K_o $|⁠, respectively, where |$ \alpha _{o}^{*} $| is a Pareto solution of model (2.3), the |$DMU_{o}$| is named |$DMU_{n+1}$|⁠, the efficiency of which is measured by the following model: $$\begin{equation} \begin{array}{l} {Min\ \theta} \\[4pt] {s.t. ~~~~ \sum\limits_{j=1}^{n}x_{ij} \lambda _{j}+ \lambda_{n+1} \alpha^{*}_{io} \leqslant \theta \alpha^{*}_{io},\,\,\,\,\,\,\,\ ~~ i=1,2,\dots,m, } \\[10pt] {~~~~~~~~\sum\limits _{j=1}^{n}y_{rj} \lambda _{j} +\lambda_{n+1} \beta_{ro} \geqslant \beta_{ro}, \,\,\,\,\,\,\,\,\,\,\,\,\,\ r=1,2,\dots,s, } \\ ~~~~~~~~\lambda \in \varLambda. \end{array} \end{equation}$$ (2.4) The optimal value of problem (2.4) is |$\theta ^{*}_1$|⁠, and the optimal value of (2.2) is |$\theta ^{*}$| and in case |$\theta ^{*}_1 =\theta ^{*}$|⁠, the efficiency will remain unchanged. Hadi-Vencheh et al. (2008), proved that problem (2.3) preserves the efficiency index through the following theorem. Definition 2.3 Assume |$(\alpha _{o}^{*}, \lambda ^{*}) $| is a feasible solution of problem (2.3); it is deducted that |$(\alpha _{o}^{*}, \lambda ^{*}) $| is a Pareto solution (strongly efficient solution) of problem (2.3), if and only if there is no feasible solution |$(\alpha _{o}, \lambda ) $| of (2.3) where |$ \alpha _{io}^{*} \geqslant \alpha _{io} $| for all |$i=1,2,\dots ,m$| and |$ \alpha _{io}^{*} \;> \alpha _{io} $| for at least one i. (Hadi-Vencheh et al., 2008) Definition 2.4 Assume |$(\alpha _{o}^{*}, \lambda ^{*}) $| is a feasible solution of problem (2.3), it is deducted that |$(\alpha _{o}^{*}, \lambda ^{*}) $| is a weakly efficient solution of (2.3), if and only if there is no feasible solution |$(\alpha _{o}, \lambda ) $| of (2.3) where |$ \alpha _{io}^{*} \;> \alpha _{io} $| for all |$i=1,2,\dots ,m$| (Hadi-Vencheh et al., 2008). Theorem 2.1 Assume the optimal value of problem (2.2) is |$\theta ^{*}$| and |$(\alpha ^{*}_{o}, \lambda ^{*}) $| be a strongly efficient solution of problem (2.3), then the optimal value of problem (2.4) is |$\theta ^{*}$|⁠, when the inputs of DMUs are increased up to |$X_{o}+\varDelta X_o$| (Hadi-Vencheh et al., 2008). 3. Inverse DEA and PPS change In this section, we propose a new model with consideration the changing PPS and apply it to ranking DMUs. The traditional inverse DEA models yield a new DMU. In these models, the PPS is constructed by previous DMUs, that is, it is built by |$(DMU_{1}, DMU_{2},$||$\dots ,DMU_{o},\dots DMU_{n})$|⁠, while the |$DMU_o$| is changed to a fresh unit named |$DMU^{^{\prime}}_{o}$|⁠. Hadi-vencheh et al. (2015) presented a new approach by eliminating |$DMU_o$| to obtain the inputs gain and they constructed the PPS by |$(DMU_{1}, \dots ,DMU_{o-1},$||$DMU_{o+1}\dots ,DMU_{n})$|⁠. This method is applied in interval data. In general, PPS can be constructed by substituting the ‘perturbed DMU’ with a modified unit containing the refreshed inputs and outputs. In this study, the |$DMU_o(X_{o},\,Y_{o})$| is substituted with |$DMU^{^{\prime}}_{o}(\alpha _o,\,\beta _o)$|⁠, therefore, the PPS is constructed by |$(DMU_{1}, \dots ,DMU_{o-1}, DMU^{^{\prime}}_{o},DMU_{o+1}\dots ,DMU_{n} )$|⁠. For this purpose, the following model is applied to obtain |$ \alpha _o $|⁠: $$\begin{equation} \begin{array}{l} {Min(\alpha_{1o},\alpha_{2o},\dots,\alpha_{mo})} \\[4pt] {s.t.~~\sum\limits _{j\ne o}^{n}x_{ij} \lambda _{j} \leqslant \theta \alpha_{io}, ~~~~~ i=1,2,\dots,m,} \\[10pt] { ~~~~~~\sum\limits _{j\ne o}^{n}y_{rj} \lambda _{j} \geqslant \beta_{ro}, ~~~~~~ r=1,2,\dots,s, } \\[10pt] {~~~~~~\alpha_{io} \geqslant x_{io}, ~~~~~ i=1,2,\dots,m, } \\[4pt] ~~~~~~~ \lambda \in \varLambda. \end{array} \end{equation}$$ (3.1) Note: among the existing DMUs, the unchanged one (⁠|$ DMU_o $|⁠) cannot create a new PPS. Model (3.1) under CRS is change to the following model: $$\begin{equation} \begin{array}{l} {Min(\alpha_{1o},\alpha_{2o},\dots,\alpha_{mo})} \\[4pt] {s.t.~~\sum\limits _{j\ne o}^{n}x_{ij} \lambda _{j} \leqslant \theta \alpha_{io}, ~~~~~ i=1,2,\dots,m,} \\[10pt] { ~~~~~~\sum\limits _{j\ne o}^{n}y_{rj} \lambda _{j} \geqslant \beta_{ro}, ~~~~~~ r=1,2,\dots,s, } \\[10pt] {~~~~~~\alpha_{io} \geqslant x_{io}, ~~~~~ i=1,2,\dots,m, } \\[4pt] ~~~~~~~ \lambda_j \geqslant 0 ~~~~~~~~~~~~~~~ j=1,2,\dots,n. \end{array} \end{equation}$$ (3.2) Theorem 3.1 Model (3.2) is always feasible. Proof. See Appendix A. The efficiency score of |$DMU^{^{\prime}}_{o}$| is evaluated by the following model: $$\begin{equation} \begin{array}{l} {Min \theta} \\[4pt] {s.t.~~~\sum\limits_{j\ne o}^{n} x_{ij} \lambda _{j} +\lambda_{o} \alpha^{*}_{io} \leqslant \theta \alpha^{*}_{io}, \,\,\,\,\,\,\,\,\,\ i=1,2,\dots,m, } \\[10pt] {~~~~~~~\sum\limits_{j\ne o}^{n} y_{rj} \lambda _{j} +\lambda_{o} \beta_{ro} \geqslant \beta_{ro}, \,\,\,\,\,\,\,\,\,\,\ r=1,2,\dots,s, } \\[10pt] ~~~~~~~\lambda _{j} \geqslant 0,\, \, \, \, j=1,2,\dots,n. \end{array} \end{equation}$$ (3.3) Example 3.2 Consider five DMUs |$ \{A,B,C,D,E\} $| under variable returns to scale technology according to Table 3. Assume the output of unit B increases by |$k=0.5$| amount and the increment of input |$ (\varDelta x=0.625) $| is obtained through traditional inverse DEA (model (2.3)). The revised DMU with |$ (x+\varDelta x, y+k) $| component is named |$ B^{\prime} $| as shown in Fig. 1. As can be seen, PPS does not change. Yet, if the proposed model (model (3.1)) is used for calculating input increment, then |$ (\varDelta x=1.188) $|⁠. Hence, unit B is changing into |$ B^\prime $|⁠, as shown in Fig. 2(a); therefore, PPS changes too. New PPS has been built with the participation of |$ B^{\prime} $| as shown in Fig. 2(b). In other words, unit B is removed and instead of that, unit |$ B^{\prime} $| is used to build PPS. Table 3. Inputs and outputs DMUs A B C D E |$x $| 1.5 2.5 5 4 7 |$ y $| 2 4 6 3 6 DMUs A B C D E |$x $| 1.5 2.5 5 4 7 |$ y $| 2 4 6 3 6 Open in new tab Table 3. Inputs and outputs DMUs A B C D E |$x $| 1.5 2.5 5 4 7 |$ y $| 2 4 6 3 6 DMUs A B C D E |$x $| 1.5 2.5 5 4 7 |$ y $| 2 4 6 3 6 Open in new tab Fig. 1. Open in new tabDownload slide Finding new DMU by traditional inverse DEA. Fig. 1. Open in new tabDownload slide Finding new DMU by traditional inverse DEA. Fig. 2. Open in new tabDownload slide Changing PPS. Fig. 2. Open in new tabDownload slide Changing PPS. If technology production is CRS assumption, then two cases may happen for PPS that is shown in Figs 3 and 4. First, suppose the only efficient unit is B as shown in Fig. 3. Assume the output of unit B increases by |$k=0.5$| amount, if the proposed model (model (3.2)) is used for calculating input increment, then |$ (\varDelta x=0.875) $|⁠. Hence, unit B is changing into |$ B^\prime $| as shown in Fig. 3; therefore, PPS changes too. New PPS has been built with the participation of |$ B^{\prime} $||$( OAB^{\prime}) $|⁠. Second, if B and C are efficient units, then they are on a line as a frontier. By inverse DEA model the unit B changes into |$ B^{\prime} $| that is on the same efficient frontier again and PPS is not changing (Fig. 4). Fig. 3. Open in new tabDownload slide Changing CCR-PPS. Fig. 3. Open in new tabDownload slide Changing CCR-PPS. Fig. 4. Open in new tabDownload slide CCR-PPS. Fig. 4. Open in new tabDownload slide CCR-PPS. To solve model (3.2) there exist different technique like interactive methods, goal programming and the sum weighted method. In this article, the sum weighted method is applied for this purpose. The Pareto solutions of model (3.2) preserves efficiency, that is, |$DMU^{^{\prime}}_o$| with |$(\alpha ^{*}_{o},\,\beta _{o}) $| is of the efficiency score one |$(\theta ^{*}=1)$|⁠. This model is applied only for efficient units; accordingly, the following theorem is evolved. Theorem 3.3 Assume the optimal value of problem (2.2) for |$ DMU_o $| is |$\theta ^{*}=1$|⁠, and |$(\alpha ^{*}_{o}, \lambda ^{*}) $| is a strongly efficient solution of problem (3.2), then the efficiency score of |$ DMU^{^{\prime}}_o (\alpha ^{*}_{o},\,\beta _o) $| is one. Proof. Let |$(\alpha _{o}^{*}, \lambda ^{*}) $| be a Pareto solution (strongly efficient solution) of problem (3.2) and |$(\theta ^{*}_{1}, \lambda ^{\prime}) $| be an optimal solution of (3.3), then |$\theta ^{*}_{1}=1$| must be proved. By assuming $$\begin{equation*}\bar \lambda_{j}= \begin{cases} \lambda^{*}_{j}, & \mbox{if} ~~j \ne o,\\ 0, & \mbox{if } ~~ j=o, \end{cases} \end{equation*}$$ it becomes clear that |$(\bar \lambda ,\theta ^{*}_{1} )$| is a feasible solution of problem (3.3). Since |$(\alpha _{o}^{*}, \lambda ^{*}) $| is the solution of (3.2), the following is yielded through input restrictions of problem (3.3): $$\begin{equation*}\theta^{*}_{1} \alpha^{*}_{io} \geqslant \sum_{j\ne o}^{n} x_{ij} \lambda^{\prime} _{j} + \lambda^{\prime}_{o} \alpha_{io} \geqslant \sum_{j\ne o}^{n} x_{ij} \lambda^{\prime} _{j}+ \lambda^{\prime} _{o} \,\sum _{j\ne o}^{n}x_{ij} \lambda^{*} _{j} = \sum _{j\ne o}^{n}x_{j} ( \lambda^{*} _{j} \lambda^{\prime}_{o} + \lambda^{\prime}_{j} ),\end{equation*}$$ the same procedure hold true for output restriction: $$\begin{equation*}y_{ro} \leqslant y_{ro}+k_{r} \leqslant \sum _{j\ne o}^{n}y_{rj} ( \lambda^{*} _{j} \lambda^{\prime}_{o} + \lambda^{\prime}_{j}), \end{equation*}$$ the above mentioned variables are briefed as |$ \tilde \lambda _{j} = ( \lambda ^{*} _{j} \lambda ^{\prime}_{o} + \lambda ^{\prime}_{j}) $| for |$ j \ne o $|⁠, and for |$j=o$| set |$ \tilde \lambda _{o}=0 $|⁠. Assume that |$ \theta ^{*}_{1} \;< 1$|⁠. In this case, the following two scenarios can happen: |$\bullet $| (a) if |$ \alpha ^{*}_{o} = X_{o} $|⁠, with respect to |$ \theta ^{*}_{1} \;< 1$|⁠, and |$(\tilde \lambda ,\theta ^{*}_{1} )$| as a feasible solution of problem (2.2), then the solution |$ \theta ^{*}_{1} \;< 1$| is impossible because |$ \theta ^{*}=1 $| is the optimal solution of (2.2). |$\bullet $| (b) If |$ \alpha ^{*}_{o} \gneq X_{o} $| then |$\sum \limits _{j\ne o}^{n} x_{ij} \tilde \lambda _{j} \leqslant \theta ^{*}_{1} \alpha ^{*}_{io} \leqslant \alpha ^{*}_{io} $|⁠. There exists at least one i, such that |$ \alpha ^{*}_{io} \;> x_{io} $|⁠; the following definition: $$\begin{equation*}h=min \Big\{min \Big\{ \frac{ \theta^{*}_{1} \alpha^{*}_{io} - \sum_{j\ne o}^{n} x_{ij} \tilde\lambda _{j}} { \theta^{*}_{1}} \Big\}, min\{\alpha^{*}_{io} -x_{io}: \alpha^{*}_{io} -x_{io} \;> 0 \}\Big\}, \end{equation*}$$ let $$\begin{equation*}\bar \alpha_{io} = \begin{cases} \alpha^{*}_{io}, & \mbox{if} ~~~ \alpha^{*}_{io} =x_{io}, \\ \alpha^{*}_{io} -h, & \mbox{if } ~~~ \alpha^{*}_{io} \;> x_{io}, \\ \end{cases}\end{equation*}$$ thus $$\begin{equation*}h \leqslant \frac{ \theta^{*}_{1} \alpha^{*}_{io} - \sum_{j\ne o}^{n} x_{ij} \tilde\lambda _{j}} {\theta^{*}_{1}} \, \Rightarrow \, \sum_{j\ne o}^{n} x_{ij} \tilde\lambda _{j} \leqslant \theta^{*}_{1} (\alpha^{*}_{io} - h) \leqslant \theta^{*}_{1} \bar \alpha_{io}, \end{equation*}$$ and $$\begin{equation*}\alpha^{*}_{io} -x_{io} \geqslant h \, \, \Rightarrow \alpha^{*}_{io} -h \geqslant x_{io} \, \Rightarrow \, \alpha^{*}_{io} -h \;> 0.\end{equation*}$$ Consequently, |$ (\alpha ^{*}_{o},\tilde \lambda ) $| is a feasible solution of problem (2.3) subject to |$ \bar \alpha _{io} \leqslant \alpha _{io}^{*} $| for all |$i=1,2,\dots ,m$|⁠, and |$ \bar \alpha _{io} \;< \alpha _{io}^{*} $| for some |$i=1,2,\dots ,n$|⁠; while there exists a contradiction to the assumption that |$(\alpha _{o}^{*}, \lambda ^{*}) $| is a strongly efficient solution of problem (3.2). Thus, |$ \theta ^*_1 =1$|⁠. 4. Ranking system with inverse DEA In this section, we use the inverse DEA models under CRS technology due to the following: (i) in one of the models, all outputs of a unit are increased by a similar amount, so needs a model that is stable to dimension changes. (ii) The model under VRS results in infeasible for some units while CCR model is feasible. Consider that |$ DMU_o $| with the inputs of |$ x_{io} $| (⁠|$i=1,2,\dots ,m$|⁠) and outputs of |$ y_{ro} $| (⁠|$r=1,2,\dots ,s$|⁠) is an efficient unit. Assume that |$ y_{ro} $| changes into |$ y_{ro}+k_r $|⁠, where $$\begin{equation*} \begin{cases} k_r \leqslant l_{r}, & \mbox{if }~~y_{rj} \geqslant 1 ~~ \mbox{for all j,}\\ k_r \leqslant \min\limits_j(y_{rj}), & \mbox{if } ~~ 0<y_{rj} \leqslant 1 ~~ \mbox{for all j}, \end{cases} \end{equation*}$$ where |$ l_r $| is an upper-bounded value, which the decision maker has preferred for |$ r^{th} $| output. It may note that a large value for |$ k_r $| is irrational because the potential increment of a DMU is limited. Selecting |$ k_r $| as a large number may lead the Banker, Charnes, and Cooper (BCC) model to be infeasible. The objective is to calculate minimum |$ \alpha _{o}=(\alpha _{1o}, \alpha _{2o},\dots ,\alpha _{mo}) $| such that the |$ DMU^{^{\prime}}_o $| with (⁠|$ \alpha _o,\beta _o $|⁠) remains an efficient unit, where, |$ \alpha _o=X_o+\varDelta X_o $| and |$ \beta _o=Y_o+K $|⁠. Model (3.1) is applied to determine the amount of |$\alpha _{o}$| for |$ DMU^{^{\prime}}_o$|⁠. The sum weighted method is applied to solve model (3.1) as follows: $$\begin{equation} \begin{array}{l} {z_o=Min {\sum\limits_{i=1}^{m} w_{i} \alpha_{io}}}\\{s.t.~~ \sum\limits_{j \ne o}^{n} x_{ij} \lambda _{j} \leqslant \alpha_{io}, \,\,\,\,\,\,\,\,\,\ i=1,2,\dots,m, } \\{~~~~~~\sum\limits_{j\ne o}^{n} y_{rj} \lambda _{j} \geqslant \beta_{ro}, \,\,\,\,\,\,\,\,\,\ r=1,2,\dots,s, } \\{~~~~~~\alpha_{io} \geqslant x_{io}, ~~~~~~ i=1,2,\dots,m,} \\ ~~~~~~\lambda _{j} \geqslant 0, ~~~~~~~~ j=1,2,\dots,n, \end{array} \end{equation}$$ (4.1) where |$w_{i}$| is the positive weight of the |$ i^{th} $| objective, with the assumption that |$w_{i}=1$| for |$i=1,2,...,n$|⁠. Definition 4.1 The |$DMU_j$| rank is better than |$DMU_k$|⁠, if |$ z^{*}_{j} \geqslant z^{*}_{k} $|⁠. Hence, for ranking efficient units, the following algorithm is applied: Algorithm 1 |$\bullet $| Step 1: Through model (2.2) all DMUs are evaluated and CCR efficient units are determined with the assumption that E is a set of all CCR efficient units |$\bullet $| Step 2: Assume, the vector |$ K $| is added to the outputs vectors for |$ DMU_o,~~ (o \in E) $|⁠. The increment of inputs are evaluated through model (4.1) and the objective function values |$ z^{*}_{o} $| are determined |$\bullet $| Step 3: The rank of |$ DMU_o $| (⁠|$ o \in E $|⁠) is evaluated through definition (4.1) In Step 2 of Algorithm 1, the inverse DEA approach is used for ranking DMUs due to its ability to estimate the inputs changes based on any increments of the outputs, which is named the growth potential. The outputs of |$DMU_o$| increase from |$ Y_o $| to |$\beta _{o}= Y_{o} +K_o $|⁠; the inputs’ vectors |$\alpha _{o}=X_{o}+\varDelta X_o$| is estimated through model (4.1). The newly created |$DMU^{\prime}_o=(\alpha _{o},\beta _{o})$| is called extend DMU. This method based upon the idea of measuring the distance of the extended unit from the PPS, constructed by the remaining DMUs. The distance between the extended unit and this efficiency frontier is growth potential. The important problem here is solving the MOLP through different methods, and in most cases, the MOLP will not provide an optimal solution while having Pareto solution that is not exclusive. To overcome this drawback, we can use single-objective programming based on inverse DEA (model (4.3)). Here, the objective function is built by the inputs increment. Therefore, for achieving a single-objective function, the increment of inputs should be the similar value for all inputs. Hence, all outputs of a DMU are increased by a similar amount. That means all outputs of efficient DMU are increased by k percent |$ \, (Y+k\%) \, $|⁠, then the increasing percentage of the inputs are measured. However, inputs and outputs have different measurement unit. For instance, in the bank assessment, one of the input’s is Staff (number of employees). Its measurement unit is the persons (number of people). Whereas, another input is Fixed asset and measured by the currency unit (dollar). The increasing amount a unit of them is not identical. Thus, their increment should be expressed in the ratio form or percentage. For this purpose, all data should be dimensionless or normalized. Thus, all inputs and outputs must be normalized through the following equations: $$\begin{equation} \tilde{x}_{ij}=\frac{x_{ij}}{\max\limits_ix_{ij}}, ~~~ \tilde{y}_{rj}=\frac{y_{rj}}{\max\limits_ry_{rj}}. \end{equation}$$ (4.2) The PM here is based on the inverse DEA approach. The difference between the previous method and this one is that here all outputs of a unit are increased by a similar amount. That is, all outputs of efficient units are increased by k percent |$ \, (Y+k\%) \, $|⁠, later the increasing percentage of the inputs are measured. For this purpose, the following LP is applied for efficient DMUs. $$\begin{equation} \begin{array}{l} {\alpha_o= min~ \alpha}\\[4pt] {s.t.~~\sum\limits_{j\ne o}^{n} \tilde{x}_{ij} \lambda _{j} \leqslant \tilde{x}_{io}+\alpha, \,\,\,\,\,\,\,\,\,\ i=1,2,\dots,m,} \\[10pt] {~~~~~~\sum\limits_{j\ne o}^{n} \tilde{y}_{rj} \lambda _{j} \geqslant \tilde{y}_{ro}+k, \,\,\,\,\,\,\,\,\,\ r=1,2,\dots,s, } \\[10pt] {~~~~~~\tilde{x}_{io}+\alpha \geqslant \tilde{x}_{io}, \,\,\,\,\,\,\,\,\,\ i=1,2,\dots,m,} \\[6pt] ~~~~~~ \lambda _{j} \geqslant 0, ~~~~~~~~~~~~~ j=1,2,\dots,n. \end{array} \end{equation}$$ (4.3) In model (4.3) all outputs of |$ DMU_o $| are increased by a similar amount (k), thus all outputs and inputs should be dimensionless. Hence, all data are normalized through (4.2). Definition 4.2 Assume |$DMU_j$| and |$DMU_k$| are evaluated by the model (4.3). The rank of |$DMU_j$| is better than |$DMU_k$|⁠, when |$ \alpha ^{*}_{j} \geqslant \alpha ^{*}_{k} $|⁠. Theorem 4.1 Model (4.3) is feasible and bounded. Proof. The feasibility proof is similar to that of Theorem 2 proof. If it is proved that the dual of model (4.3) is feasible, according to the duality theorem, model (4.3) is bounded. The dual of model (4.3) is given as follows: $$\begin{equation} \begin{array}{l} {\max ~ \sum\limits_{r=1}^{s} u_r(\tilde{y}_{ro}+k) - \sum\limits_{i=1}^{m} v_i\tilde{x}_{io}}\\[10pt] {s.t.~~ \sum\limits_{r=1}^{s} u_r\tilde{y}_{rj} -\sum\limits_{i=1}^{m} v_i\tilde{x}_{ij} \leqslant 0, ~~~ j=1,2,\dots,n ~~ \& ~~ j\ne o, } \\[10pt] {~~~~~~\sum\limits_{i=1}^m v_i \leqslant 1, } \\[10pt] {~~~~~~ u_r \geqslant 0, ~~ r=1,2,\dots,m,} \\[6pt] ~~~~~~ v_i \geqslant 0, ~~ i=1,2,\dots,m, \end{array} \end{equation}$$ (4.4) assume |$ u_r=0 $| for each |$ r=1,2,\dots ,s $|⁠, |$ v_1=1 $| and |$v_i=0$| for |$i \ne 1 $|⁠. It is clear that (⁠|$U,V$|⁠) is a feasible solution for model (4.4); therefore, the model (4.3) is bounded. For ranking efficient unit, the following algorithm is applied: Algorithm 2 |$\bullet $| Step 1: All inputs and outputs are normalized through (4.2) |$\bullet $| Step 2: The CCR efficient units are obtained by (2.2), with the assumption that E is a set of all CCR efficient units |$\bullet $| Step 3: Assume that the outputs of |$ DMU_O,~(o \in E) $| increase by |$ \%k $|⁠. The increment of inputs are computed through model (4.3) and the objective functions values |$ \alpha ^{*}_{o} $| are determined. |$\bullet $| Step 4: The rank of |$ DMU_o $| (⁠|$ o \in E $|⁠) is evaluated through definition (4.2) 5. Numerical examples Given the form of the inverse DEA discussed in the previous section, there are two examples of interest that must be presented these newly proposed models in order to reveal their actual applications. Example 5.1 Ranking Iranian banks Commercial bank performance evaluation is one of the research interests in DEA and several practical implications in the US, Japan, Iran, etc. (Berger & Humphrey, 1997, Holod & Lewis, 2011, Barros et al., 2012, Chitnis & Vaidya, 2016). One of the important issues arises when different DMUs reach their highest level of efficiency and their ranking is an essential objective of such studies. The application of inverse DEA, in this case, is to rank these DMUs based on different methodologies. For example Hosseinzadeh Lotfi et al. (2013) reviewed the recent ranking of DMU based methods proposed in the literature. They applied a data set of Iranian banks as a real example for verification and comparison of different models. Hosseinzadeh Lotfi et al. (2013) considered 20 commercial banks in Iran. They applied three inputs and three outputs for performance evaluation. The inputs consist of staff, computer terminal and space; the outputs consist of deposits, loan grants and banking service charges, Table 4. As observed in this table, seven units are CCR efficient, DMUs 1,4,7,12,15,17 and 20. For verification of this introduced model, assume that 0.01 is added to the efficient units’ outputs, through Algorithm 1 the efficient DMUs are ranked. Let the outputs of the efficient |$ DMU1 $| change into 0.2, 0.531 and 0.303, respectively, then through model (4.1) |$ z^*_1=0.02154 $| is obtained. If the outputs of the efficient |$ DMU4 $| are changing to 0.203, 0.642 and 1.01, respectively, then |$ z^*_4=0.6349 $|⁠. Thus, the rank of |$ DMU4 $| is better than that of the |$ DMU1 $|⁠. The rank of efficient units regarding this proposed model (model (4.1)), Mehrabian, Alirezaee, and Jahanshahloo (MAJ) model and l1-norm model are tabulated in Table 6. The results obtained through this proposed method (PM) and MAJ method are similar in the most units except for DMUs 17 and 20. Note that, the results here are more reliable due to the fact that (i) this method ranks DMUs based on the growth potential, (ii) the wasted inputs (values of slack variables) are low and (iii) the variation of outputs are controllable. This means that the decision maker has more flexibility to select different increment to gain better insight regarding input variables. Table 4. Inputs, outputs and efficiency score for Example 5.1 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ I_{3} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| |$ O_{3} \,\,\,\,\ $| CCR efficiency DMU1 0.950 0.700 0.155 0.190 0.521 0.293 1.0000 DMU2 0.796 0.600 1.000 0.277 0.627 0.462 0.8333 DMU3 0.798 0.750 0.513 0.288 0.970 0.261 0.9911 DMU4 0.865 0.550 0.210 0.193 0.632 1.000 1.0000 DMU5 0.815 0.850 0.268 0.233 0.722 0.246 0.8974 DMU6 0.842 0.650 0.500 0.207 0.603 0.569 0.7483 DMU7 0.719 0.600 0.350 0.182 0.900 0.716 1.0000 DMU8 0.785 0.750 0.120 0.125 0.234 0.298 0.7978 DMU9 0.476 0.600 0.135 0.080 0.364 0.244 0.7877 DMU10 0.678 0.550 0.510 0.082 0.184 0.049 0.2900 DMU11 0.711 1.000 0.305 0.212 0.318 0.403 0.6045 DMU12 0.811 0.650 0.255 0.123 0.923 0.628 1.0000 DMU13 0.659 0.850 0.340 0.176 0.645 0.261 0.8166 DMU14 0.976 0.800 0.540 0.144 0.513 0.243 0.4693 DMU15 0.685 0.950 0.450 1.000 0.262 0.098 1.0000 DMU16 0.613 0.900 0.525 0.115 0.402 0.464 0.6390 DMU17 1.000 0.600 0.205 0.090 1.000 0.161 1.000 0 DMU18 0.634 0.650 0.235 0.059 0.340 0.068 0.4727 DMU19 0.372 0.700 0.238 0.039 0.190 0.111 0.4088 DMU20 0.538 0.550 0.500 0.110 0.615 0.746 1.0000 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ I_{3} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| |$ O_{3} \,\,\,\,\ $| CCR efficiency DMU1 0.950 0.700 0.155 0.190 0.521 0.293 1.0000 DMU2 0.796 0.600 1.000 0.277 0.627 0.462 0.8333 DMU3 0.798 0.750 0.513 0.288 0.970 0.261 0.9911 DMU4 0.865 0.550 0.210 0.193 0.632 1.000 1.0000 DMU5 0.815 0.850 0.268 0.233 0.722 0.246 0.8974 DMU6 0.842 0.650 0.500 0.207 0.603 0.569 0.7483 DMU7 0.719 0.600 0.350 0.182 0.900 0.716 1.0000 DMU8 0.785 0.750 0.120 0.125 0.234 0.298 0.7978 DMU9 0.476 0.600 0.135 0.080 0.364 0.244 0.7877 DMU10 0.678 0.550 0.510 0.082 0.184 0.049 0.2900 DMU11 0.711 1.000 0.305 0.212 0.318 0.403 0.6045 DMU12 0.811 0.650 0.255 0.123 0.923 0.628 1.0000 DMU13 0.659 0.850 0.340 0.176 0.645 0.261 0.8166 DMU14 0.976 0.800 0.540 0.144 0.513 0.243 0.4693 DMU15 0.685 0.950 0.450 1.000 0.262 0.098 1.0000 DMU16 0.613 0.900 0.525 0.115 0.402 0.464 0.6390 DMU17 1.000 0.600 0.205 0.090 1.000 0.161 1.000 0 DMU18 0.634 0.650 0.235 0.059 0.340 0.068 0.4727 DMU19 0.372 0.700 0.238 0.039 0.190 0.111 0.4088 DMU20 0.538 0.550 0.500 0.110 0.615 0.746 1.0000 Open in new tab Table 4. Inputs, outputs and efficiency score for Example 5.1 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ I_{3} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| |$ O_{3} \,\,\,\,\ $| CCR efficiency DMU1 0.950 0.700 0.155 0.190 0.521 0.293 1.0000 DMU2 0.796 0.600 1.000 0.277 0.627 0.462 0.8333 DMU3 0.798 0.750 0.513 0.288 0.970 0.261 0.9911 DMU4 0.865 0.550 0.210 0.193 0.632 1.000 1.0000 DMU5 0.815 0.850 0.268 0.233 0.722 0.246 0.8974 DMU6 0.842 0.650 0.500 0.207 0.603 0.569 0.7483 DMU7 0.719 0.600 0.350 0.182 0.900 0.716 1.0000 DMU8 0.785 0.750 0.120 0.125 0.234 0.298 0.7978 DMU9 0.476 0.600 0.135 0.080 0.364 0.244 0.7877 DMU10 0.678 0.550 0.510 0.082 0.184 0.049 0.2900 DMU11 0.711 1.000 0.305 0.212 0.318 0.403 0.6045 DMU12 0.811 0.650 0.255 0.123 0.923 0.628 1.0000 DMU13 0.659 0.850 0.340 0.176 0.645 0.261 0.8166 DMU14 0.976 0.800 0.540 0.144 0.513 0.243 0.4693 DMU15 0.685 0.950 0.450 1.000 0.262 0.098 1.0000 DMU16 0.613 0.900 0.525 0.115 0.402 0.464 0.6390 DMU17 1.000 0.600 0.205 0.090 1.000 0.161 1.000 0 DMU18 0.634 0.650 0.235 0.059 0.340 0.068 0.4727 DMU19 0.372 0.700 0.238 0.039 0.190 0.111 0.4088 DMU20 0.538 0.550 0.500 0.110 0.615 0.746 1.0000 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ I_{3} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| |$ O_{3} \,\,\,\,\ $| CCR efficiency DMU1 0.950 0.700 0.155 0.190 0.521 0.293 1.0000 DMU2 0.796 0.600 1.000 0.277 0.627 0.462 0.8333 DMU3 0.798 0.750 0.513 0.288 0.970 0.261 0.9911 DMU4 0.865 0.550 0.210 0.193 0.632 1.000 1.0000 DMU5 0.815 0.850 0.268 0.233 0.722 0.246 0.8974 DMU6 0.842 0.650 0.500 0.207 0.603 0.569 0.7483 DMU7 0.719 0.600 0.350 0.182 0.900 0.716 1.0000 DMU8 0.785 0.750 0.120 0.125 0.234 0.298 0.7978 DMU9 0.476 0.600 0.135 0.080 0.364 0.244 0.7877 DMU10 0.678 0.550 0.510 0.082 0.184 0.049 0.2900 DMU11 0.711 1.000 0.305 0.212 0.318 0.403 0.6045 DMU12 0.811 0.650 0.255 0.123 0.923 0.628 1.0000 DMU13 0.659 0.850 0.340 0.176 0.645 0.261 0.8166 DMU14 0.976 0.800 0.540 0.144 0.513 0.243 0.4693 DMU15 0.685 0.950 0.450 1.000 0.262 0.098 1.0000 DMU16 0.613 0.900 0.525 0.115 0.402 0.464 0.6390 DMU17 1.000 0.600 0.205 0.090 1.000 0.161 1.000 0 DMU18 0.634 0.650 0.235 0.059 0.340 0.068 0.4727 DMU19 0.372 0.700 0.238 0.039 0.190 0.111 0.4088 DMU20 0.538 0.550 0.500 0.110 0.615 0.746 1.0000 Open in new tab Table 5. Results of model (4.1) for Example 5.1 DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 |$ z^*_j $| 0.02154 0.6349 0.2553 0.06670 6.2941 0.18570 0.11676 Rank 7 2 3 6 1 4 5 DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 |$ z^*_j $| 0.02154 0.6349 0.2553 0.06670 6.2941 0.18570 0.11676 Rank 7 2 3 6 1 4 5 Open in new tab Table 5. Results of model (4.1) for Example 5.1 DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 |$ z^*_j $| 0.02154 0.6349 0.2553 0.06670 6.2941 0.18570 0.11676 Rank 7 2 3 6 1 4 5 DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 |$ z^*_j $| 0.02154 0.6349 0.2553 0.06670 6.2941 0.18570 0.11676 Rank 7 2 3 6 1 4 5 Open in new tab Table 6. Ranking orders for Example 5.1 RM DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 The proposed method 7 2 3 6 1 4 5 MAJ method 7 2 3 6 1 5 4 l1-norm 7 2 3 5 1 4 6 RM DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 The proposed method 7 2 3 6 1 4 5 MAJ method 7 2 3 6 1 5 4 l1-norm 7 2 3 5 1 4 6 Open in new tab Table 6. Ranking orders for Example 5.1 RM DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 The proposed method 7 2 3 6 1 4 5 MAJ method 7 2 3 6 1 5 4 l1-norm 7 2 3 5 1 4 6 RM DMU1 DMU4 DMU7 DMU12 DMU15 DMU17 DMU20 The proposed method 7 2 3 6 1 4 5 MAJ method 7 2 3 6 1 5 4 l1-norm 7 2 3 5 1 4 6 Open in new tab Table 7. Inputs, outputs for Example 5.2 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| CCR efficiency DMU1 81 87.6 5191 205 1.000 DMU2 85 12.8 3629 0 1.000 DMU3 56.7 55.2 3302 0 0.945 DMU4 91 78.8 3379 8 0.628 DMU5 216 72 5368 639 1.000 DMU6 58 25.6 1674 0 0.585 DMU7 112.2 8.8 2350 0 0.708 DMU8 293.2 52 6315 414 0.965 DMU9 186.6 0 2865 0 1.000 DMU10 143.4 105.2 7689 66 0.956 DMU11 108.7 127 2165 266 0.448 DMU12 105.7 134.4 3963 315 0.662 DMU13 235 236.8 6643 236 0.453 DMU14 146.3 124 4611 128 0.537 DMU15 57 203 4869 540 1.000 DMU16 118.7 48.2 3313 16 0.575 DMU17 58 47.4 1853 230 0.891 DMU18 146 50.8 4578 217 0.853 DMU19 0 91.3 0 508 1.000 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| CCR efficiency DMU1 81 87.6 5191 205 1.000 DMU2 85 12.8 3629 0 1.000 DMU3 56.7 55.2 3302 0 0.945 DMU4 91 78.8 3379 8 0.628 DMU5 216 72 5368 639 1.000 DMU6 58 25.6 1674 0 0.585 DMU7 112.2 8.8 2350 0 0.708 DMU8 293.2 52 6315 414 0.965 DMU9 186.6 0 2865 0 1.000 DMU10 143.4 105.2 7689 66 0.956 DMU11 108.7 127 2165 266 0.448 DMU12 105.7 134.4 3963 315 0.662 DMU13 235 236.8 6643 236 0.453 DMU14 146.3 124 4611 128 0.537 DMU15 57 203 4869 540 1.000 DMU16 118.7 48.2 3313 16 0.575 DMU17 58 47.4 1853 230 0.891 DMU18 146 50.8 4578 217 0.853 DMU19 0 91.3 0 508 1.000 Open in new tab Table 7. Inputs, outputs for Example 5.2 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| CCR efficiency DMU1 81 87.6 5191 205 1.000 DMU2 85 12.8 3629 0 1.000 DMU3 56.7 55.2 3302 0 0.945 DMU4 91 78.8 3379 8 0.628 DMU5 216 72 5368 639 1.000 DMU6 58 25.6 1674 0 0.585 DMU7 112.2 8.8 2350 0 0.708 DMU8 293.2 52 6315 414 0.965 DMU9 186.6 0 2865 0 1.000 DMU10 143.4 105.2 7689 66 0.956 DMU11 108.7 127 2165 266 0.448 DMU12 105.7 134.4 3963 315 0.662 DMU13 235 236.8 6643 236 0.453 DMU14 146.3 124 4611 128 0.537 DMU15 57 203 4869 540 1.000 DMU16 118.7 48.2 3313 16 0.575 DMU17 58 47.4 1853 230 0.891 DMU18 146 50.8 4578 217 0.853 DMU19 0 91.3 0 508 1.000 |$DMU_{p}\,\,\, \ $| |$ I_{1} \,\,\,\ $| |$ I_{2} \,\,\,\,\,\ $| |$ O_{1} \,\,\,\,\,\ $| |$ O_{2} \,\,\,\,\,\ $| CCR efficiency DMU1 81 87.6 5191 205 1.000 DMU2 85 12.8 3629 0 1.000 DMU3 56.7 55.2 3302 0 0.945 DMU4 91 78.8 3379 8 0.628 DMU5 216 72 5368 639 1.000 DMU6 58 25.6 1674 0 0.585 DMU7 112.2 8.8 2350 0 0.708 DMU8 293.2 52 6315 414 0.965 DMU9 186.6 0 2865 0 1.000 DMU10 143.4 105.2 7689 66 0.956 DMU11 108.7 127 2165 266 0.448 DMU12 105.7 134.4 3963 315 0.662 DMU13 235 236.8 6643 236 0.453 DMU14 146.3 124 4611 128 0.537 DMU15 57 203 4869 540 1.000 DMU16 118.7 48.2 3313 16 0.575 DMU17 58 47.4 1853 230 0.891 DMU18 146 50.8 4578 217 0.853 DMU19 0 91.3 0 508 1.000 Open in new tab Example 5.2 Numerical example A classical numerical example taken from Rezai Balf et al. (2012) is applied here for evaluating the practical application of this newly introduced model (4.3). Consider 19 DMUs with two inputs and two outputs according to Table 7. First, it is required to normalize the data to feed model (4.3) for ranking efficient DMUs; this normalization is made through (4.2). Second, it is required to identify efficient units. For this purpose, model (2.2) is applied, which makes six units are CCR efficient: DMUs 1, 2, 5, 9, 15 and 19. Assume that the outputs of efficient units increase by |$ 0.1 $|⁠. Model (4.3) is used to compute the inputs’ growth rate, Table 8. The unit with larger objective function has better rank, Table 8. The rank of efficient units of the proposed (model (4.3)), MAJ, Tch. norm and Andersen and Petersen (AP) methods are tabulated in Table 9. As observed, AP method is not feasible for DMUs 9 and 19, while this PM is feasible for these two DMUs. For validation of the PM, the Spearman’s correlation has been computed and the results has been significant at the level of 0.05 as shown in Fig. 5. The correlation coefficient between the PM with MAJ and Tch. methods are 0.943 and 0.886, respectively. It confirms the existence of a good matching between the PM with MAJ and Tch. norm methods. It may note that the correlation with AP is low, because AP is not feasible for DMU9 and DMU19. In addition, the proposed inverse DEA method can be considered as futuristic approach for ranking DMUs due to its ability to estimate the inputs changes based on any increments of the output. Other methods mentioned in Table 9 lacking this feature. Table 8. Results of model (4.3) for Example 5.2 DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 |$ \alpha ^*_j $| 0.1629 0.2010 0.2445 0.0735 0.1057 0.5424 Rank 4 3 2 6 5 1 DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 |$ \alpha ^*_j $| 0.1629 0.2010 0.2445 0.0735 0.1057 0.5424 Rank 4 3 2 6 5 1 Open in new tab Table 8. Results of model (4.3) for Example 5.2 DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 |$ \alpha ^*_j $| 0.1629 0.2010 0.2445 0.0735 0.1057 0.5424 Rank 4 3 2 6 5 1 DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 |$ \alpha ^*_j $| 0.1629 0.2010 0.2445 0.0735 0.1057 0.5424 Rank 4 3 2 6 5 1 Open in new tab Table 9. Ranking orders for Example 5.2 RM DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 The proposed method 4 3 2 6 5 1 MAJ method 5 3 2 6 4 1 Tch. norm 5 2 3 6 4 1 AP 4 1 3 - 2 - RM DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 The proposed method 4 3 2 6 5 1 MAJ method 5 3 2 6 4 1 Tch. norm 5 2 3 6 4 1 AP 4 1 3 - 2 - Open in new tab Table 9. Ranking orders for Example 5.2 RM DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 The proposed method 4 3 2 6 5 1 MAJ method 5 3 2 6 4 1 Tch. norm 5 2 3 6 4 1 AP 4 1 3 - 2 - RM DMU1 DMU2 DMU5 DMU9 DMU15 DMU19 The proposed method 4 3 2 6 5 1 MAJ method 5 3 2 6 4 1 Tch. norm 5 2 3 6 4 1 AP 4 1 3 - 2 - Open in new tab Fig. 5. Open in new tabDownload slide Spearman’s rank-order correlation. Fig. 5. Open in new tabDownload slide Spearman’s rank-order correlation. 6. Conclusion In this paper, we introduce a new MOLP model for inverse DEA where the disturbed DMU is replaced with updated inputs and output units. This introduced MOLP model is applied for ranking the extreme efficient DMUs. Usually, the MOLP models lead to Pareto solution and finding a unique optimal solution is a challenging task. A single-objective LP model is proposed to overcome this drawback. Furthermore, two analytic theorems are developed to prove the feasibility of these models. This method is desirable through a practical and a numerical example extracted from the literature. The computational experiments are performed through specified data sets indicating that |$\bullet $| These introduced models are feasible for all units as proved in Theorem 3.1. |$\bullet $| These methods rank DMUs based on the growth potential. |$\bullet $| The variation of outputs are controllable. This means that the manager can be assigned the amount of increasing outputs based on available resource. Through these models, the increments of outputs are controllable, and the decision makers can manage the variation of the outputs revealing that this method can be considered as a prospective approach. Appendix. A Proof. (Theorem 2) Model (3.2) is rewritten as follows: $$\begin{equation} \begin{array}{l} { Min { (\varDelta x_{1o},\varDelta x_{2o}, \dots, \varDelta x_{mo} )}} \\[4pt] {s.t.~~ \sum\limits_{j\ne o}^{n} x_{ij} \lambda _{j} \leqslant x_{io}+\varDelta x_{io}, \,\,\,\,\,\,\,\,\,\ i=1,2,\dots,m, } \\[10pt] {~~~~~~\sum\limits_{j\ne o}^{n} y_{rj} \lambda _{j} \geqslant y_{ro}+k_r, ~~~~~~~ r=1,2,\dots,s, } \\[10pt] {~~~~~~~x_{io}+\varDelta x_i \geqslant x_{io}, ~~~~~~ i=1,2,\dots,m,} \\[4pt] ~~~~~~\lambda _{j} \ge 0, ~~~~~~~~~~~~~~~~~~ j=1,2,\dots,n. \end{array} \end{equation}$$ (A.1) Let $$\begin{equation*} \lambda_{j}= \begin{cases} 1, & \mbox{if} ~~j \ne o, \\ 0, & \mbox{if } ~~ j=o, \end{cases} \end{equation*}$$ and |$ \varDelta x_i=\sum _{j\ne o}^{n} x_{ij} $| (⁠|$i=1,2,\dots ,m$|⁠), Three cases can be derived as If |$ y_{ro}=0 $|⁠, then we can let |$ k_r=0 $|⁠. If |$ \sum \limits _{j\ne o}^{n} y_{rj}-y_{ro} \geqslant 0 $|⁠, then |$ k_r=-y_{ro}+\sum \limits _{j\ne o}^{n} y_{rj} $|⁠. 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TI - A ranking system based on inverse data envelopment analysis
JF - IMA Journal of Management Mathematics
DO - 10.1093/imaman/dpz014
DA - 2020-06-12
UR - https://www.deepdyve.com/lp/oxford-university-press/a-ranking-system-based-on-inverse-data-envelopment-analysis-0NhQzCO0aj
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -