\documentclass[6p,12pt]{article} \renewcommand{\baselinestretch}{1} \setcounter{page}{1} \setlength{\textheight}{20cm} \setlength{\textwidth}{15cm} \setlength{\oddsidemargin}{1cm} \setlength{\evensidemargin}{1cm} \pagestyle{myheadings} \thispagestyle{empty} \date{} \usepackage{epsfig} \usepackage{amsmath, amsthm, amscd, amsfonts,graphicx} \usepackage{amssymb,fontenc} \usepackage{latexsym,wasysym,mathrsfs} \usepackage{lscape} \begin{document} \begin{center} {\Large \bf A Method for Finding the Anchor Points of BCC Model in DEA Context } \end{center} \begin{center} {\small \bf Ali Akbar Bani$^{1}$, Mohsen Rostamy-Malkhlifeh$^{1*}$, Farhad Hosseinzadeh Lotfi$^{1}$, Dariush~Akbarian$^{2}$ \footnotetext{\footnotesize $^{*}$Corresponding Author. E-mail Address: mohsen\_rostamy@yahoo.com\\ bani\_ali@yahoo.com (Ali Akbar Bani)\\ mohsen\_rostamy@yahoo.com (Mohsen Rostamy-Malkhalifeh)\\ farhad@hosseinzadeh.ir (Farhad Hosseinzadeh Lotfi)\\ d\_akbarian@yahoo.com (Dariush Akbarian)\\ }} \end{center} \begin{center} {\footnotesize $^1$Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran}\\ {\footnotesize $^2$Department of Mathematics , Arak branch, Islamic Azad University, Arak, Iran}\\ \end{center} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{proposition}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{conjecture}[thm]{Conjecture} \theoremstyle{definition} \newtheorem{dfn}[thm]{Definition} \newtheorem{example}[thm]{Example} \newtheorem{xca}[thm]{Exercise} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{Note}[thm]{Note} \begin{abstract} Anchor points are an important subset of the set of extreme efficient points of the Production Possibility Set $(PPS)$ in the Data Envelopment Analysis $(DEA).$ They delineate the efficient from the inefficient part of the $PPS$ frontier. In this paper we present a method to find anchor points of the $PPS$ of the $BCC$ model based on searching weak supporting hyperplane passing through unit under consideration. Then, one theorem is proved creating necessary and sufficient condition for identification of these points. Using numerical examples, we will demonstrate how to use the results. \end{abstract} \indent \hskip .45cm \textit{\textbf{Keywords:}} Data Envelopment Analysis; Anchor point; Extreme point; Mixed integer linear programming.\\ \bigskip \noindent DEA Subject Classification: 90C11, 90C05 \section{Introduction} Data Envelopment Analysis introduced by Farrel \cite{far} and Charnes, et al. \cite{char1}, at first, is a non-parametric technique and a useful tool to evaluate Decision Making Units $(DMUs)$ with multiple inputs and multiple outputs \cite{coop,emroz}. Today, this technique is well-known in many respects. In this method, the production function is estimated without presupposition of the production function by solving mathematical models for a group of $DMUs,$ and having information about their input and output. Therefore, we can evaluate their performance. The original model of Data Envelopment Analysis was considered by Charnes, et al. \cite{char1} with constant returns to scale $(CRS)$, and then was expanded by Banker, et al. \cite{bank3} for variable returns to scale $(VRS)$ technologies. In $DEA$ models, each model has a production boundary. In 1957, Farrel expressed the production boundary approach as the maximum of production with a certain amount of inputs \cite{far}. In fact, the performance boundary is the situation that every $DMU$ is at the best mode, and knowing it will provide useful information to the decision maker. It has been studied by several researchers. Jahanshahloo, et al. \cite{jahan1,jahan2,jahan3,jahan4}, Yu, et al. \cite{yu}, Olesen and Petersen \cite{olesen}, Wei, et al. \cite{wei}, Hosseinzadeh Lotfi, et al. \cite{hos}, Amirteimoori and Kordrostami \cite{amir2}, Davtalab Olyaie, et al. \cite{dav1,dav} are among the most important of them. Anchor points are one of the main subsets of the set of extreme efficient points of the Production Possibility Set in $DEA.$ An anchor point is an extreme efficient $DMU$ for which some inputs can be increased and/or outputs decreased without penetrating the interior of the $PPS$. Therefore, it is an extreme efficient element of the Production Possibility Set lying on the transition between the strong efficient frontier and the “free-disposability”(unbounded face) part of the boundary. As the $DEA$ context shows, the anchor points play an important role in $DEA$ theory and applications. Thanassoulis and Allen \cite{tha2} applied the concept of these points, at first, to generate, unobserved $DMUs$ and extend the $DEA$ frontier. Rouse \cite{rous} used this meaning to identify prices for health care services. These points also play an effective role in \cite{tha1} in presenting the unobserved $DMUs.$ Bougnol and Dula \cite{boug} provided another method to identify them based on the geometrical properties of anchor points. They discussed Production Possibility Set with the variable returns to scale. In their method, the $PPS$ of the $BCC$ model is projected on all coordinate hyperplanes; an extreme efficient $DMU$ is an anchor point if and only if it belongs to the boundary of at least one simple projection. Mostafaee and Soleimani-damaneh \cite{mos} suggested a manner to find the anchor points using the sensitivity analysis techniques and also presented some conditions for characterizing anchor points in \cite{mos2}. Soleimani-damaneh and Mostafaee \cite{sol} offered an algorithm different with the above-mentioned works to identify the anchor $DMUs$ in (nonconvex) FDH models. They defined the extreme unit and anchor point ideas in nonconvex technologies in their article. See also \cite{allen,boug1} for more details. Since the set of anchor points is a subset of the set of extreme efficient points, finding the extreme efficient points is a principle. There are several algorithms to do this. See \cite{char2,dula}. In this paper, we express a new method to identify $PPS$ anchor points of the $BCC$ model by searching weak supporting hyperplane passing through unit under consideration. Our proposed method needs less computational effort than Bougnol\'{}s method \cite{boug}, since we do not need project frame $DMUs$ on the boundary of simple projections one after another. Furthermore, in most of the above-mentioned articles, it is first necessary to identify the extreme efficient units of $PPS$ by models then by running the algorithm for each of the methods, the anchor points are determined from the extreme efficient elements. In other words, to achieve the goal, at least two or three or more models need to be solved, but in our approach, we will only rely on solving one model (mixed integer linear program) to reach the ultimate goal as, the advantage of our method. The rest of the article is as follows: Section 2 explains the preliminary concepts. Section 3 presents our approach to find anchor points. Sections 4 and 5 contain numerical examples and conclusion, respectively. \section{Background} Consider a set of $n\,\, DMUs$, which is associated with $m$ inputs and $s$ outputs. We apply the notation\,$(x_j,y_j) (j\in J=\{1,\dots ,n\})$\,for the observed $DMUs,$ that the first component is vectors of the inputs and the second component is vectors of the outputs. Also suppose that for each $DMU_j=(x_j,y_j), j\in J \,\text{the} \,x_j\geq0 \,\, x_j\neq0\,\,\text{are} \,(m\times 1)$\, input vectors and the \,$y_j\geq0\,\,y_j\neq0\,\,\text{are}\,(s\times1)$\, output vectors. The Production Possibility Set is the set of all technologically possible input-output combinations given by the following \cite{bank1,bank3,yu}: $$T=\{(x,y)|\text{y can be produced by x}\}.$$ One of the $DEA$ models to evaluate efficiency of set of $DMUs$ is $BCC$ model \cite{bank3}. The its $PPS$ be defined follows: \begin{align*}\label{eq.0} & T_v= \{(x,y)\,|\,x\geq \sum_{j=1}^{n}\lambda_jx_j,\quad y\leq \sum_{j=1}^{n}\lambda_jy_j,\quad \sum_{j=1}^{n}\lambda_j=1,\, \lambda_j\geq0,\quad j\in J\}, \end{aign*} The input-oriented $BCC$ model pertaining to $DMU_p,\,p\in J$ is given by: \begin{align}\label{eq.1} min \qquad &\theta-\varepsilon(\sum_{i=1}^ms_i^- + \sum_{r=1}^ss_r^+)\cr s.t.\qquad & \sum _{j=1}^{n}\lambda_jx_{ij} +s_i^-=\theta x_{ip},\quad i=1,\ldots,m,\cr &\sum _{j=1}^{n}\lambda_jy_{rj}-s_r^+=y_{rp},\quad r=1,\ldots,s,\cr &\sum _{j=1}^{n}\lambda_j=1,\cr &\lambda_j\geq 0,\quad j\in J,\cr &s_i^-\geq 0,\quad i=1,\ldots,m,\cr &s_r^+\geq 0,\quad r=1,\ldots,s,\cr & \theta\,\, is\,\, free.\cr \end{aign} Also the output-oriented \,$BCC$\,model pertaining to\,$DMU_p, p\in J$\, is given by:\\ \begin{align}\label{eq.2} max \qquad & \varphi+\varepsilon(\sum_{i=1}^mt_i^- + \sum_{r=1}^st_r^+)\cr s.t.\qquad&\sum _{j=1}^{n}\lambda_jx_{ij} +t_i^-=x_{ip},\quad i=1,\ldots,m, \cr &\sum _{j=1}^{n}\lambda_jy_{rj} -t_r^+=\varphi y_{rp},\quad r=1,\ldots,s,\cr &\sum _{j=1}^{n}\lambda_j=1,\cr &\lambda_j\geq 0,\quad j\in J,\cr &t_i^-\geq 0,\quad i=1,\ldots,m,\cr &t_r^+\geq 0,\quad r=1,\ldots,s,\cr &\varphi\,\, is\,\, free.\cr \end{align} Where $\varepsilon$\, is non-Archimedean small and positive number. The models \eqref{eq.1} and \eqref{eq.2} are called the envelopment forms of the $BCC$ model and their dual models (without $\varepsilon$\, i.e. $\varepsilon=0$) which are called the multiplier forms are as the following, respectively:\\ \begin{align}\label{eq.3} max \qquad &\sum_{r=1}^su_ry_{rp}+ u_0\cr s.t.\qquad & \sum_{i=1}^mv_ix_{ip}=1,\cr &\sum _{r=1}^su_ry_{rj}-\sum _{i=1}^mv_ix_{ij}+u_0\leq0,\quad j\in J\cr &u_r\geq 0,\quad r=1,\dots,s,\cr &v_i\geq 0,\quad i=1,\dots,m,\cr & u_0\,\, is\,\, free.\cr \end{align} \begin{align}\label{eq.4} min \qquad &\sum_{i=1}^mv_ix_{ip}- u_0\cr s.t.\qquad& \sum_{r=1}^su_ry_{rp}=1,\cr &\sum _{r=1}^su_ry_{rj}-\sum _{i=1}^mv_ix_{ij}+u_0\leq0,\quad j\in J\cr &u_r\geq 0,\quad r=1,\dots,s,\cr &v_i\geq 0,\quad i=1,\dots,m,\cr & u_0 \,\,is\,\, free.\cr \end{align} Another model that is important in $DEA,$ is additive model. The presented LP of $BCC$ performs the contraction of all inputs, in input-oriented, and the expansion of all outputs, in output-oriented; therefore, it is called radial model in $DEA$. The additive model has both the input-oriented and the output-oriented, but it is not radial and is expressed as follows: \begin{align}\label{eq.66} Max \qquad &\sum_{i=1}^ms_i^- + \sum_{r=1}^ss_r^+\cr s.t.\qquad & \sum _{j=1}^{n}\lambda_jx_{ij} +s_i^-= x_{ip},\quad i=1,\ldots,m,\cr &\sum _{j=1}^{n}\lambda_jy_{rj}-s_r^+=y_{rp},\quad r=1,\ldots,s,\cr &\sum _{j=1}^{n}\lambda_j=1,\cr &\lambda_j\geq 0,\quad j\in J,\cr &s_i^-\geq 0,\quad i=1,\ldots,m,\cr &s_r^+\geq 0,\quad r=1,\ldots,s,\cr \end{aign} The Production Possibility Set of the additive LP \eqref{eq.66} is the same as the $BCC$ model. For further explanation, consider the Figure \eqref{fig02}. Suppose that we consider the efficiency of the $DMU, \,\, D.$ A feasible movable for $D$ is the movement along $S^-$ (along the input axis, input reduction) and then moving along $S^+$ (along the output axis, increasing output). $DB$ and $BA$ must be selected to maximize their total.\\ \begin{figure}[htp] \centering \includegraphics[width=8cm]{Pic/fig2} \caption{feasible movement in additive model.} \label{fig02} \end{figure} \begin{Note}\label{not6} The additive model \eqref{eq.66} is always feasible and a feasible solution is as follows: \\ \\ $$\lambda_p=1,\,\, \lambda_j=0, \,\, j=1,\dots,n \,\, j\neq p, \,\, s_{i}^{-}=0,\,\, 1\leq i\leq m, \,\, s_{r}^{+}=0,\,\, 1\leq r\leq s,$$ \end{Note}\\ \\ Several definitions are now available. \begin{dfn}\label{d3} (Envelopment ).\, $DMU_p$ is pareto efficient or strong efficient in $BCC$ model if and only if either $(i)$ or $(ii)$ happen:\\ \\ $(i)$ For every optimal solution of model \eqref{eq.1}, \,$(\theta^*,\lambda_j^*,s^+_i^*,s^-_r^*),\, j\in J,\,i=1,\dots,m,\,r=1,\dots,s,$\, satisfies $\theta ^{*}=1$ and $s^+_i^*=0,s^-_r^*=0, i=1,\dots,m,\,r=1,\dots,s, $ \\ $(ii)$ For every optimal solution of model \eqref{eq.2} \,$(\varphi^*,\lambda_j^*,t^+_i^*,t^-_r^*),\, j\in J,\,i=1,\dots,m,\,r=1,\dots,s,$\, satisfies $\varphi ^{*}=1$ and $t^+_i^*=0,t^-_r^*=0, i=1,\dots,m,\,r=1,\dots,s, $\\ otherwise, $DMU_{p}$ is $BCC-$inefficient. \end{dfn} \begin{dfn}\label{d33} (Multiplier).\, $DMU_p$ is pareto efficient or strong efficient in $BCC$ model if and only if either $(iii)$ or $(iv)$ happen:\\ \\ $(iii)$ The optimal objective value of \eqref{eq.3} is equal to 1 and there exist some optimal solutions such that all decision variables $v_i \, , \, u_r\,\, ,\,\, 1\leq i\leq m, 1\leq r\leq s$ be strictly positive.\\ $(iv)$ The optimal objective value \eqref{eq.4} is equal to 1 and there exist some optimal solutions such that all decision variables $v_i \, , \, u_r\,\, ,\,\, 1\leq i\leq m, 1\leq r\leq s$ be strictly positive. \end{dfn} \begin{dfn}\label{d2} (Additive).\,\, $DMU_p$ is pareto efficient in additive model if and only if in model \eqref{eq.66} $$s^+^*=0 \,\,,\,\, s^-^*=0.$$ \end{dfn} \begin{Note}\label{n3} Clearly Definitions \eqref{d3}, \eqref{d33} and \eqref{d2} are equivalent. \end{Note} \begin{dfn} (Envelopment ).\,\, $DMU_p$ is weak efficient if and only if there exists an optimal solution such that either $(v)$ or $(vi)$ happen:\\ \\ $(v)$ $\theta^*=1$ and $(s^+^*,s^-^*) \neq(0,0)$ in \eqref{eq.1}\\ $(vi)$ $\varphi^*=1$ and $(t^+^*,t^-^*)\neq(0,0)) $ in \eqref{eq.2}. \end{dfn} \begin{dfn}\label{dfn2.7} (Multiplier).\,\, $DMU_p$ is weak efficient if and only if either $(vii)$ or $(viii)$ happen:\\ \\ $(vii)$ The optimal objective value of \eqref{eq.3} is equal to 1 and for all its optimal solutions, some decision variables $v_i \, , \, u_r\,\, ,\,\, 1\leq i\leq m, 1\leq r\leq s$ be zero.\\ $(viii)$ The optimal objective value of \eqref{eq.4} is equal to 1 and for all its optimal solutions some decision variables $v_i \, , \, u_r\,\, ,\,\, 1\leq i\leq m, 1\leq r\leq s$ be zero. \end{dfn} \begin{dfn}\label{d1} The pareto efficient $DMU_p$ is extreme efficient if models \eqref{eq.1} or \eqref{eq.2} have unique optimal solution with $\lambda ^*_p>0 \,\, \text{and}\,\, \lambda^*_j=0, j\in J-\{p\}.$ Otherwise, $DMU_p$ is non-extreme efficient. \end{dfn} We denote the set of extreme $BCC-$efficient $DMUs$ as $E^*$. The set $E^*$ is also called the frame of $J.$ The frames are important in $DEA$ because the $PPS$ of the $DEA$ models are constructed by them and the exclusion each of them alters the shape of the $PPS.$\\ The following notes are taken from the above definitions and their proof is straightforward. \begin{Note}\label{not2} $DMU_p$ is an inefficient unit (interior point of $PPS$ or weak efficient) if and only if objective value of additive model \eqref{eq.66} is greater than zero. \end{Note} \begin{Note}\label{not3} $DMU_p$ is an interior point of $PPS$ and inefficient if and only if in model \eqref{eq.1} and model \eqref{eq.2} $\theta^*<1$ and $\phi^*>1$ respectively. \end{Note} \begin{Note}\label{not4} $DMU_p$ is an interior point of $PPS$ if and only if objective value in model \eqref{eq.3} is smaller than 1 and in model \eqref{eq.4} is greater than 1. \end{Note} The concept of supporting hyperplane in the $DEA$ literature plays a decisive role in identification of anchor points. A hyperplane in a space of $m+s$ dimensions and passing through $DMU_p=(x_p,y_p)$ is defined as follows: \begin{align}\label{eq.8} H_p=\{(x,y)\in R^{m+s}|u^t(y-y_p)-v^t(x-x_p)=0\}, \end{align} where $u\in R^s\,\, , \,\, v\in R^m$ are normal vectors. If $u_0$ defined as: $$u_0=v^tx_p-u^ty_p,$$ Therefore, the relation \eqref{eq.8} can be expressed as: $$H_p=\{(x,y)\in R^{m+s}|u^ty-v^tx+u_0=0\}.$$ In general, a supporting hyperplane divides space into two half-spaces. If the Production Possibility Set can be located in one of the half-spaces, then $H_p$ is supporting hyperplane on $PPS$ at the point $(x_p,y_p).$ In other words, supporting hyperplane touches the $PPS$ at the $DMU_p.$ Thus, the following definition can be represented. \begin{dfn} The hyperplane $$H=\{(x,y)\in R^{m+s}|u^ty-v^tx+u_0=0,\,\,\, (u,v)\geq 0,\,\,\, (u,v)\neq 0\},$$ is supporting on $PPS$ at a point of boundary, $DMU_p=(x_p,y_p),$ if and only if both $(i)$ and $(ii)$ happen:\\ \\ $(i)\,\, u^ty_p-v^tx_p+u_0=0$\\ $(ii)\,\, \forall (x,y)\in PPS\Rightarrow u^ty-v^tx+u_0\leq0.$\\ \\ $H$ is called strong supporting if $(u,v)>0$ and called weak supporting if some of components $(u,v)$ is zero. \end{dfn} The following theorem states relation between the optimal solution of the $BCC$ multiplier form and the supporting hyperplane on $PPS.$ \begin{thm}\label{thm2.13} Let $DMU_p$ be a boundary unit of $PPS.$ In evaluation of $DMU_p$ by $BCC$ multiplier form, $(u^*,v^*,u^*_0)$ is an optimal solution if and only if $$H^*=\{(x,y)\in R^{m+s}|u^*^ty-v^*^tx+u^*_0=0\}$$ is a supporting hyperplane on the $PPS$ at $DMU_p$. \end{thm} \begin{proof} Theorem 5.1 in \cite{coop}. \end{proof} %Theorem 11: \begin{dfn} Suppose $(u^*,v^*,u_0^*)$ is an optimal solution of model \eqref{eq.3}, then according to the Theorem \eqref{thm2.13} $H^*=\{ (x,y)\in R^{m+s} | u^*^ty-v^*^tx+u_0^*=0 \}$ is a supporting hyperplane of the $T_v$. The set $F=\{ (x,y)\in R^{m+s} | u^*^ty-v^*^tx+u_0^* =0\}\cap T_v=H^*\cap T_v$ is called a face of $T_v$. \end{dfn} \begin{Note}\label{not55} A face of a polyhedral set is the support set of a supporting hyperplane. A facet of a $k$-dimensional polyhedral set is a $k-1$ dimensional face. \end{Note} \begin{Note}\label{not66} The $PPS$ of the $BCC$ model has bounded and unbounded faces. The unbounded faces make up the free-disposability part of the frontier. \end{Note} The following theorem and two corollaries play a crucial role in our approach in the next section. \begin{thm}\label{thm11} \textit{Suppose $(x_p,y_p)\in int(H^*\cap T_v)$ (where $H^*$ is the supporting hyperplane of $T_v$ passing through the $DMU_p$ and defined as follows) evaluated by the $BCC$ multiplier model. If at least one of the components $u^*$ or $v^*$ is equal to zero, then a production possibility can be found with lower input or more output dominating $(x_p,y_p).$ (The meaning of $int S$ is interior of set $S$).} $$H^*=\{(x,y)\in R^{m+s}|u^*^ty-v^*^tx+u_o=0\}.$$ \end{thm} \begin{proof} Without loss of generality, assume that $v_1^*=0.$ Since $(x_p,y_p)$ is an interior point of face $F=H^*\cap T_v$ then $$\exists \delta>0, N_{\delta}(x_p,y_p)\subseteq F$$ So, consider the following production possibility $$(x_{1p}-\delta, x_{2p}, \dots , x_{mp}, y_{1p}, \dots,y_{sp})=(\hat{x},\hat{y})$$ thereupon $$u_1^*y_{1p}+\dots+u_s^*y_{sp}-0x_{1p}-v_2^*x_{2p}-\dots-v_{m}^*x_{mp}+u_{0}^*=0$$ Since $v_1^*=0,$ so we will have for each $0\leq\acute{\delta}\leq\delta$ $$u_1^*y_{1p}+\dots+u_s^*y_{sp}-0(x_{1p}-\acute{\delta})-v_2^*x_{2p}-\dots-v_m^*x_{mp}+u_0^*=0$$ That means $(\hat{x},\hat{y})$ is on the $H^*$ and dominates $(x_p,y_p).$ Similarly, it can be expressed and proved while, one of the components $u^*$ is zero. \end{proof} \begin{cor}\label{cor1} If $DMU_p$ is a non-extreme pareto efficient unit, then in the evaluation of $DMU_p$ by multiplier form, for each optimal solution, the optimal objective value is equal to 1, and all decision variables $u_r\,\, ,\,\, v_i\,\,\,(1\leq r\leq s\,\, , \,\, 1\leq i\leq m)$ are strictly positive. \end{cor} \begin{proof} Suppose $DMU_p$ is a non-extreme pareto efficient unit. By Definition \eqref{d33}, the optimal objective value of multiplier form is obviously equal to $1$. Suppose there is an optimal solution that at least one of the decision variables $u_r\,\, ,\,\, v_i\,\,\,(1\leq r\leq s\,\, , \,\, 1\leq i\leq m)$ is equal to zero. Since $DMU_p=(x_p,y_p)$ is a non-extreme unit then, it is an interior point of face $F=H^*\cap T_v.$ Thus, according to the Theorem \eqref{thm11}, there exists a production possibility dominating $(x_p,y_p)$, which is contrary to the assumption that $DMU_p$ is pareto efficient. Therefore, all decision variables $u_r\,\, ,\,\, v_i\,\,\,(1\leq r\leq s\,\, , \,\, 1\leq i\leq m)$ are strictly positive. \end{proof} \begin{cor}\label{cor2} If $DMU_p$ is an extreme strong efficient unit then maybe some optimal solutions of $BCC$ multiplier model exist, $(u^*,v^*,u_0^*)$ such that at least one of the normal vector components $(u^*,v^*)$ is zero. \end{cor} \begin{proof} Suppose $DMU_p=(x_p,y_p)$ is an extreme strong efficient unit. Whereas $(x_p,y_p)\notin int(H^*\cap T_v)$. Therefore, $(x_p,y_p)$ does not apply at the assumptions of Theorem \eqref{thm11} and Corollary \eqref{cor1}. Therefore, all components of normal vector $(u^*,v^*)$ are not strictly positive and maybe some of these components be zero. \end{proof} The following definition introduces the concept of anchor points in $T_v.$ In fact, this property comes from Result $1$ in Bougnol and Dula \cite{boug}. \begin{dfn}\label{dfn2.9} $DMU_p=(x_p,y_p)\in E^*$ is called an anchor point if and only if it is located on a supporting hyperplane of $T_v,$ say $H_{(u,v,u_0)},$ such that at least one component of the gradient vector $(u,v)$ is zero. \end{dfn} \begin{rem} The $DMU_p\in E^*$ is an anchor DMU if it belongs to an unbounded face of the $PPS$ of the $BCC$ model. \end{rem} \section{Identifying the anchor DMUs of the PPS of the BCC model} In this section, we identify the anchor $DMUs$ of the $PPS$ of the $BCC$ model as follows. Corresponding to each $DMU_{p}=(x_{1p},\dots,x_{mp},y_{1p},\dots,y_{sp})$ $(p\in J)$, we solve the following mixed integer linear program: \begin{align}\label{eq.5} Max \qquad &\sum_{i=1}^{m}s_i^{-}+\sum_{r=1}^{s}s_r^{+}\cr s.t.\qquad \qquad &\sum_{j=1}^{n}\lambda_{j}x_{ij}+s_{i}^{-}=x_{ip},\qquad\qquad i=1,\dots,m &(1)\cr &\sum_{j=1}^{n}\lambda_{j}y_{rj}-s_{r}^{+}=y_{rp},\qquad\qquad r=1,\dots,s &(2)\cr &\sum_{j=1}^{n}\lambda_{j}=1, &(3)\cr &\sum_{r=1}^{s}u_ry_{rp}-\sum_{i=1}^{m}v_ix_{ip}+u_0=0, &(4)\cr &\sum_{r=1}^{s}u_ry_{rj}-\sum_{i=1}^{m}v_ix_{ij}+u_0\leq0, \qquad\quad j\in J, & (5)\cr &\sum_{r=1}^{s}u_r + \sum_{i=1}^{m}v_i=1, & (6)\cr &\sum_{i=1}^{m}z_i+\sum_{r=1}^{s}k_r \leq m+s-1, & (7)\cr &v_i-M_1z_i\leq 0,\qquad\qquad i=1,\dots,m, &(8)\cr &u_r-M_2k_r\leq 0,\qquad\qquad r=1,\dots,s, &(9)\cr &z_i\in \{0,1\},\qquad\qquad i=1,\dots,m, &\cr &k_r\in \{0,1\},\qquad\qquad r=1,\dots,s, &\cr &u_i, s_i^-\geq0,\qquad\qquad i=1,\dots,m, &\cr &v_r, s_r^-\geq 0, \qquad\qquad r=1,\dots,s. &\cr &\lambda_j \geq 0, \qquad\qquad j=1,\dots,n. &&\cr &u_0\,\, is\,\, free &\cr \end{align} The values of $M_1$ and $M_2$ are very large positive numbers. The model \eqref{eq.5} acts as follows:\\ \\ The constraints $(4), (5)\, \text{and}\, (6)$ guarantee that if $(\lambda^*, s^+^*, s^-^*, u^*, v^*, u_0^*)$ is a feasible solution of model \eqref{eq.5}, then in accordance with Theorem \eqref{thm2.13}, an optimal solution of the $BCC$ form will be $(u^*, v^*, u_0^*)$. Constraint $(7)$ together with constraints $(8)\, \text{and}\, (9)$ ensure that there is at least one zero component in the vector $(u^*, v^*)$. However, the three constraints $(1), (2)\, \text{and}\, (3)$ along with the objective function, help to assess the extreme $DMU.$ By Charnes, et al. \cite{char2}, if divided units of the $PPS$ into six groups, model \eqref{eq.5} for $DMUs$ of groups $NE, N\acute{E}, NF$ according to Note \eqref{not3} and Note \eqref{not4} and presence of constraints $(4), (5)\, \text{and} (6)$ will be infeasible. This mixed binary program is also infeasible for units $\acute{E}$ due to Theorem \eqref{thm11} and Corollary \eqref{cor1} and existence of constraint $(7)$. The model is feasible for members $F$ (constraints $(1), (2)\, \text{and}\, (3)$ by Note \eqref{not6} and constraints $(4), (5), (6)\, \text{and}\, (7)$ by Definition \eqref{dfn2.7} are established), and the objective function value will be greater than zero due to Note \eqref{not2}. Finally, the preceding program \eqref{eq.5} for the $DMUs$ in the set $E$ is twofold: or all supporting hyperplanes on the $PPS$ and passing through $DMU_p$ ($DMU$ under consideration) are strong efficient, which in this case will be infeasible with respect to the presence of constraint $(7)$, or there exists at least one weak supporting hyperplane including $DMU_p$ where, the model would be feasible, and the objective function value would be zero with respect to Definition \eqref{d2} (in this situation, $DMU$ is extreme unit). Therefore, in general, the following theorem stating the necessary and sufficient condition to identify anchor points, can be expressed. \begin{thm}\label{thm3.1} $DMU_p (p\in J)$ is an anchor point if and only if the model \eqref{eq.5} is feasible and its optimal objective value is equal to zero. \end{thm} \begin{proof} Let $DMU_p$ be an anchor point, in this case, by Definition \eqref{dfn2.9}, $DMU_p$ is an extreme strong efficient unit; therefore due to Definition \eqref{d33}, the $BCC$ multiplier form is feasible, and the objective function value is equal to 1, thus constraints $(4), (5),\, \text{and}\, (6)$ are established. Furthermore, $DMU_p$ is located on a supporting hyperplane of $T_v,$ say $H^*=\{(x,y)\in R^{m+s}|u^*^ty-v^*^tx+u_o=0\}$, which at least one of the components $(u^*,v^*)$ is zero. So, constraint $(7)$ also holds. Since $DMU_p$ is assumed to be an extreme strong efficient unit, then according to Definition \eqref{d3} and Note \eqref{not6}, constraints $(1), (2)\,\, \text{and}\,\, (3)$ are established, and the objective function value of model \eqref{eq.5} is zero. To prove the converse, let mixed binary program \eqref{eq.5} be feasible, and the optimal objective value be zero. Suppose $(\lambda^*, s^+^*, s^-^*, u^*, v^*, u_0^*)$ is the optimal solution for this model. From constraints $(4), (5),\, \text{and}\, (6),$ Theorem \eqref{thm2.13} is deduced $H^*=\{(x,y)\in R^{m+s}|u^*^ty-v^*^tx+u_o=0\}$ as a supporting hyperplane of the $PPS$ including $DMU_p.$ Forasmuch as presence constraint $(7),$ at least one of the components $(u^*,v^*)$ is equal to zero. Therefore, the weak hyperplane $H^*$ supports $T_v$ at $DMU_p.$ Thus, since the value of the objective function is zero according to the first three constraints and Definition \eqref{d3}, Definition \eqref{d2} and Note \eqref{n3}, unit under consideration is pareto efficient. By constraint $(7)$ and Corollaries \eqref{cor1} and \eqref{cor2}, it is inferred that, $DMU_p$ is an extreme strong efficient $DMU$ that weak supporting hyperplane $H^*$ passes through it. \end{proof} \section{Numerical Examples} In this section, we deal with explanation of the model suggested in the previous section by presentation of three numerical examples. \subsection{Example}\label{exm1} Consider a system consisting of $6\,\,DMUs$ listed in Table \eqref{table1}. According to Figure \eqref{fig01} $DMU_1, DMU_2, DMU_3\,\, \text{and}\,\, DMU_4$ are extreme efficient and $DMU_6$ is weak efficient and $DMU_5$ is inefficient. Model \eqref{eq.5} is infeasible for $\{DMU_3, DMU_4\}$ and $\{DMU_5\}$ due to the presence of constraint $\{(7)\}$ and constraints $\{(4), (5), (6), (7)\}$, respectively. Therefore, they are not anchor points. The mentioned mixed binary program is feasible for $DMU_6$ and gives weak supporting hyperplane passing through it $(y=8),$ but, the objective value is equal to 2, therefore it is not anchor. This model is feasible for $DMU_1\,\, \text{and}\,\, DMU_2,$ the objective value is equal to zero for both of them and the supporting hyperplanes passing through them are $x=2, y=8$ respectively. So, by Theorem \eqref{thm3.1} they are anchor points. \begin{table} \centering \caption{Example \eqref{exm1}, Data of input and output of DMUs.}\label{table1} \begin{tabular}{ccccccccccccccccccccc} \hline & & & DMU_1 & & &DMU_2 & & & DMU_3 & & &DMU_4 & & &DMU_5& & &DMU_6\\ \hline x& & & 2 & & & 8 & & & 3 & & & 5 & & &5 & & &10\\ y& & & 2 & & & 8 & & & 5 & & & 7 & & &2 & & &8\\ \hline \end{tabular} \end{table} \begin{figure}[htp] \centering \includegraphics[width=8cm]{Pic/fig1} \caption{Example \eqref{exm1}, $DMU_1$ and $DMU_2$ are anchor points.} \label{fig01} \end{figure} \subsection{Example}\label{exm2} We implement the presented approach for the data listed in Table \eqref{table2}. $DMUs$ have two inputs and one output. For identifying anchor points, it suffices to apply our algorithm for each of the observed $DMUs$. Here, all of them are anchor points, since the mentioned algorithm is feasible, and the objective value is equal to zero for all of them. Moreover, model \eqref{eq.5} identifies $y=10$ as weak supporting hyperplane passing through $DMU_3,DMU_4$ and $x_1=4,x_2=4$ pass through $DMU_1$ and $DMU_2$, respectively. \begin{table} \centering \caption{Example \eqref{exm2}, $DMUs$ with two inputs and one output.}\label{table2} \begin{tabular}{clllllclllclllclllcll} \hline & & & & & & DMU_1 & & & &DMU_2 & & & & DMU_3 & & & &DMU_4 & &\\ \hline x_1& & & & & & 4 & & & & 9 & & & & 8 & & & & 14 & &\\ x_2& & & & & & 9 & & & & 4 & & & & 14 & & & & 8 & &\\ y& & & & & & 7 & & & & 7 & & & & 10 & & & & 10 & &\\ \hline \end{tabular} \end{table} \subsection{Example(Empirical data)}\label{exm3} \begin{table} \caption{Example \eqref{exm3}, Data of inputs and outputs of DMUs(extracted from Amirteimoori et al. \cite{amir})}\label{table3} \begin{tabular}{clclclcllclclc} \hline Branch & & Staff & & Computers & &Space m^2& &&Deposits & &Loans& &Charge\\ &&&&terminals&&&&&&&&&\\ \hline 1& &0.9503 & &0.70& &0.1550 && &0.1900 & &0.5214& &0.2926\\ 2& &0.7962& &0.60& &1.0000 & &&0.2266 & &0.6274& &0.4624\\ 3& & 0.7982 & &0.75 & &0.5125 && &0.2283 & &0.9703& &0.2606\\ 4& &0.8651& & 0.55 & & 0.2100 & && 0.1927 & &0.6324& &1.0000\\ 5& &0.8151 & & 0.85 & & 0.2675 & && 0.2333 & & 0.7221& &0.2463\\ 6& &0.8416 & &0.65 & & 0.5000 & & &0.2069 & & 0.6025& &0.5689\\ 7& &0.7189 & & 0.60 & & 0.3500 & & &0.1824 & & 0.9000& &0.7158\\ 8& &0.7853& & 0.75& &0.1200& & &0.1250 & &0.2340& &0.2977\\ 9& &0.4756 & & 0.60 & & 0.1350 & & &0.0801 & & 0.3643& &0.2439\\ 10& &0.6782 & & 0.55 & & 0.5100 & & &0.0818 & &0.1835& &0.0486\\ 11& & 0.7112 & &1.00 & & 0.3050 & & &0.2117 & &0.3179& &0.4031\\ 12& &0.8113& & 0.65 && 0.2550 & & &0.1227 & &0.9225& &0.6279\\ 13& &0.6586 & & 0.85 & & 0.3400 & & &0.1755 & &0.6452& &0.2605\\ 14& &0.9763 & & 0.80 & & 0.5400 & & &0.1443 & &0.5143& &0.2433\\ 15& &0.6845 & & 0.95 & & 0.4500 & & &1.0000 & & 0.2617& &0.0982\\ 16& &0.6127 & &0.90 & & 0.5250 & &&0.1151 & & 0.4021& &0.4641\\ 17& &1.0000 & & 0.60 & & 0.2050 & & &0.0900& &1.0000& &0.1614\\ 18& & 0.6337 & &0.65 & &0.2350 & &&0.0591& &0.3492& &0.0678\\ 19& &0.3715& &0.70 & &0.2375 & &&0.0385 & &0.1898& &0.1112\\ 20& &0.5827& &0.55 & &0.5000 & &&0.1101& &0.6145& &0.7643\\ \hline \end{tabular} \end{table} In this subsection, we use a set of data of 20 Iranian banks to evaluate the proficiency of our algorithm (Table \eqref{table3}). The branches are assessed in terms of three inputs and three outputs defined below:\\ $I_1:$ staff, $I_2:$ computer terminals, $I_3:$ space (m^2), $O_1:$ deposits, $O_2:$ loans, $O_3:$ charge. Table 4 in Amirteimoori, et al. \cite{amir} presents the input-output data set for these 20 banks. Each of the observed $DMUs$ has been assessed by model \eqref{eq.5} and the results are summarized in Table \eqref{tab.4}. Thus, the following $DMUs$ are anchor points. $$\{DMU_1,DMU_3,DMU_4,DMU_7,DMU_8,DMU_9,DMU_{12},DMU_{15},DMU_{17},$$ $$DMU_{19},DMU_{20}\}.$$ Note that, after assessing $DMU_{10}$, we can see that model \eqref{eq.5} is feasible and weak supporting hyperplane passing through it, is $-x_2+0.55=0,$ but objective value is equal to $1.3967.$ Thus, $DMU_{10}$ is not an extreme efficient point, and therefore not an anchor point. \begin{landscape} \begin{table} \caption{Example \eqref{exm3}, Results from model \eqref{eq.5}}\label{tab.4} \begin{tabular}{clclclclc} \hline Branch&&Feasible&&Objective Value&&Weak Supporting Hyperplane && Anchor\\ \hline 1&&Yes&&0&&0.0345y_1+0.0978y_2-0.8676x_3+0.0769=0&&Yes\\ \hline 2&&No& &-& &-&&No\\ \hline 3&&Yes& &0& &0.2393y_1+0.3297y_2-0.4308x_1-0.0306=0&&Yes\\ \hline 4&&Yes&&0&&0.1135y_3-0.8864x_3+0.0725=0&&Yes\\ \hline 5&&No&&-&&-&&No\\ \hline 6&&No&&-&&-&&No\\ \hline 7&&Yes&&0&&0.1748y_2+0.1345y_3-0.3079x_1-0.3826x_3+0.1015=0&&Yes\\ \hline 8&&Yes&&0&&-x_3+0.12=0&&Yes\\ \hline 9&&Yes&&0&&0.1y_2-0.0014x_1-0.8984x_3+0.0855=0&&Yes\\ \hline 10&&Yes&&1.3967&&-x_2+0.55=0&&No\\ \hline 11&&No&&-&&-&&No\\ \hline 12&&Yes&&0&&0.1637y_2+0.0599y_3-0.0985x_1-0.6777x_3+0.0641=0&&Yes\\ \hline 13&&No&&-&&-&&No\\ \hline 14&&No&&-&&-&&No\\ \hline 15&&Yes&&0&&0.2738y_1-0.7261x_3+0.0529=0&&Yes\\ \hline 16&&No&&-&&-&&No\\ \hline 17&&Yes&&0&&0.1721y_1+0.0629y_2-0.1688x_2-0.596x_3+0.145=0&&Yes\\ \hline 18&&No&&-&&-&&No\\ \hline 19&&Yes&&0&&0.1885y_1+0.1916y_3-0.4685x_1-0.1512x_3+0.1814=0&&Yes\\ \hline 20&&Yes&&0&&0.2878y_1+0.1244y_2+0.2701y_3-0.3175x_1-0.1296=0&&Yes\\ \hline \end{tabular} \end{table} \end{landscape} \section{Conclusions} Anchor $DMUs$ are a new collection in the general classification of $DMUs$ in $DEA.$ An anchor point in $DEA$ is an extreme efficient unit for which some inputs can be increased and/or outputs decreased without penetrating the interior of the Production Possibility Set. Their identification has several interesting $DEA$ applications such as the construction of “unobserved” $DMUs$ to capture prior value judgments in $DEA$ and to identify $DMUs$ that are efficient for multiple constituencies. In this paper we express a new method to identify $PPS$ anchor points of the $BCC$ model by searching weak supporting hyperplane passing through unit under consideration. Our proposed idea needs less computational effort than Bougnol\'{}s method \cite{boug}, since we do not need project frame $DMUs$ on the boundary of simple projections one after another. Furthermore, in most of the published articles, it is first necessary to identify the extreme efficient points of the $PPS$ by models then by running the algorithm for each of the methods, the anchor points are determined from the extreme efficient points. In other words, to achieve the goal, at least two or three or more models need to be solved, but in our approach, we will only rely on solving one model to reach the ultimate goal as, the advantage of our method. Diminution in the computational commitments of the procedures given for recognition of the anchor points can be worth studying in the future. \section{Conflict of Interest} The authors declare that they have no conflict of interest. \bibliographystyle{amsplain} \begin{thebibliography}{5} \bibitem{allen}R. Allen, E. Thanassoulis, Improving envelopment in data envelopment analysis, European Journal of Operational Research. 154(2004), 363-379. \bibitem{amir}A. Amirteimoori, S. Kordrostami, Efficient surfaces and an efficiency index in DEA: A constant returns to scale, Applied Mathematics and Computation, 163(2005), 683-691. \bibitem{amir2}A. Amirteimoori, S. Kordrostami, Generating strong defining hyperplanes of the production possibility set in data envelopment analysis. Journal of Applied Mathematics Letters, 25(2012), 605-609. \bibitem{boug1}M.L. Bougnol, Nonparametric frontier analysis with multiple constituencies. Ph.D. Dissertation, The University of Mississippi, 2001. \bibitem{boug}M.L. Bougnol, J.H. Dula, Anchor points in DEA, European Journal of Operational Research. 192(2009), 668-676. \bibitem{bank1}R.D. Banker, Estimation most productive scale size using data envelopment analysis, European Journal of Operational Research. 17(1984), 35-44. \bibitem{bank3}R.D. Banker, A. Charnes, W.W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science. 30(1984), 1078-1092. \bibitem{char1}A. Charnes, W.W. Cooper, E. Rodes, Measuring the efficiency of decision making units, European Journal of Operational Research. 2(1978), 429-444. \bibitem{char2}A. Charnes, W.W. Cooper, R.M. Thrall, A structure for classifying and characterizing efficiency and inefficiency in data envelopment analysis, Journal of Productivity Analysis. 2(1991), 197-237. \bibitem{coop}W.W. Cooper, L.M. Seiford, K. Tone, Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software (2nd ed.). Springer(2007). \bibitem{dav1}M. Davtalab Olyaie, I. Roshdi, G. Jahanshahloo, M. Asgharian, Characterizing and finding full dimensional efficient facets in dea: A variable returns to scale specification. journal of the Operational Research Society. 65(2014), 1453-1464. \bibitem{dav}M. Davtalab Olyaie, I. Roshdi, V. Partovi Nia, M. Asgharian, On characterizing full dimensional weak facets in DEA with variable returns to scale technology, Optimization. 64(2015), 2455-2476. \bibitem{dula}J.H. Dula, F.J. Lopez, Algorithms for the frame of a finitely generated unbounded polyhedron. INFORMS Journal on Computing. 18(2006), 97-110. \bibitem{emroz}A. Emrouznejad, B. Parker, G. Tavares, Evaluation of research in efficiency and productivity: A thirty years survey of the scholarly literature in DEA, Socio-Economic Planning Sciences. 42(2008), 151-157. \bibitem{far}M.J. Farrell, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A. 120(1957), 253-290. \bibitem{hos}F. Hosseinzadeh Lotfi, G. R. Jahanshahloo, M. R. Moza ari, J. Gerami, Finding DEA-Efficient Hyperplanes Using MOLP Efficient Faces. Journal of Computational and Applied Mathematics, 235(2011), 1227-1231. \bibitem{jahan1}G. R. Jahanshahloo,F. Hosseinzadeh Lotfi, D. Akbarian, Finding weak defining hyperplanes of PPS of the BCC model. Applied Mathematical Modelling, 34(2010), 3321-3332. \bibitem{jahan2}G. R. Jahanshahloo, F. Hossinzadeh Lotfi, H. Zhiani Rezai, F. Rezai Balf, Finding strong defining hyperplanes of Production possibility set. European Journal of Operational Research. 177(2007), 42-54. \bibitem{jahan3}G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Zohrehbandian, Finding the piecewise linear frontier production function in Data Envelopment Analysis. Applied Mathematics and Computation, 163(2005), 483-488. \bibitem{jahan4} G. R. Jahanshahloo, A. Shirzadi, S. M. Mirdehghan, Finding strong defining hyperplanes of PPS using multiplier form. European Journal of Operational Research, 194(2009), 933-938. \bibitem{mos}A. Mostafaee, M. Soleimani-damaneh, Identifying the anchor points in DEA using sensitivity analysis in linear programming, European Journal of Operational Research. 237(2014), 383-388. \bibitem{mos2}A. Mostafaee, M. Soleimani-damaneh, Some Conditions for Characterizing Anchor Points, Asia-Pacific Journal of Operational Research. 33(2016)1650013. \bibitem{olesen}O. Olesen, N. Petersen, Identification and use of efficient faces and facets in DEA. Journal of Productivity Analysis, 20(2003), 323-360. \bibitem{rous}P. Rouse, Using DEA to set prices for health care delivery in New Zealand hospitals. Working paper, The University of Auckland, New Zealand (2004). \bibitem{sol}M. Soleimani-damaneh, A. Mostafaee, Identification of the anchor points in FDH models, European Journal of Operational Research. 246(2015), 936-943. \bibitem{tha2}E. Thanassoulis, R. Allen, Simulating weights restrictions in data envelopment analysis by means of unobserved DMUs, Management Science. 44(1998), 586-594. \bibitem{tha1}E. Thanassoulis, M. Kortelainen, R. Allen, Improving envelopment in Data Envelopment Analysis under variable returns to scale, European Journal of Operational Research. 218(2012), 175-185. \bibitem{wei}Q. Wei, H. Yan, G. Hao, (2007). Characteristics and structures of weak efficient surfaces of production possibility sets. Journal of mathematical analysis and applications, 327(2007), 1055-1074. \bibitem{yu}G. Yu, Q. Wei, P. Brockett, L. Zhou, Construction of all DEA efficient surfaces of the production possibility set under the Generalized Data Envelopment Analysis Model, European Journal of Operational Research. 95(1996), 491-510. \end{thebibliography} \end{document}