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An algorithm for the anchor points of the PPS of the BCC model

Abstract

Anchor points play an important role in data envelopment analysis theory and applications; they delineate the efficient part of the production possibility set (PPS) frontier. In this paper, we propose an approach for finding the anchor points of the PPS of the Banker, Charnes and Cooper (BCC) model. This approach is based on a variant of super-efficiency models and their duals. The necessary and sufficient conditions for the characterization of the anchor points are also provided. Finally, the applicability of the proposed model is illustrated with some numerical examples.

Introduction

Data envelopment analysis (DEA) is an approach for measuring the relative efficiency of a set of decision-making units (DMUs) with multiple inputs and outputs; it was first introduced by Charnes, Cooper, and Rhodes (CCR) (Charnes et al. 1978) and then extended by Banker et al. 1984. DEA has been used in various environments and in numerous applications [Kou et al. (2021), Zha et al. (2020), Castelli et al. (2004), Chao et al. (2021), Zhang et al. (2021), and references therein]. Anchor points form a subset of the extremely efficient DMUs. More precisely, an anchor point in DEA is an extremely efficient DMU for which some inputs can be increased and/or some outputs can be decreased without penetrating the interior of the production possibility set. An anchor point is, therefore, an extreme element of the production possibility set that lies on the transition between the strong efficient frontier and the “free-disposability”(unbounded face) part of the boundary. Anchor points play a major role in defining the shape of the DEA production possibility set. They were first identified by Allen and Thanassoulis (2004), who generated unobserved DMUs to extend the DEA efficient frontier (Thanassoulis and Allen 1998). Anchor points also allow identifying unobserved DMUs, as in the work of Thanassoulis et al. (2012). Allen and Thanassoulis (2004) proposed a special procedure for identifying anchor points. The scheme applies specifically to DEA models with one input and multiple outputs under constant returns to scale, and requires assumptions on the geometry of the efficient frontiers; such assumptions are not verified. Therefore, the procedure is not guaranteed to always identify all anchor points. Bougnol and Marie-Laure (2009) identified anchor points based on their geometrical properties, focusing on production possibility sets with variable returns. In this method, the PPS of the BCC model is projected on all coordinate hyperplanes, and an extremely efficient DMU is considered an anchor point if and only if it belongs to the boundary of at least one simple projection. Rouse (2004) used this approach to identify healthcare service prices. Mostafaee and Soleimani-damaneh (2014) used sensitivity analysis techniques to find the anchor points and presented some conditions for their characterization. They made a connection between DEA and sensitivity analysis in linear programming theory to identify anchor points, Akbarian (2021) and Soleimani-damaneh and Mostafaee (2015) studied anchor points using non-convex technologies. Bani et al. (2020) used a mixed-integer linear program to identify the anchor points of the PPS of the BCC model. Koushki and Soleimani-damaneh (2019) considered anchor points as special properly nondominated solutions of multiobjective optimization problems and provided two algorithms to check whether a given feasible solution is an anchor point. Akbarian (2017) proposed to identify the anchor points of the PPS of the CCR model in the multiple inputs and outputs case through testing all CCR-efficient DMUs by a variant of super efficiency models [refer to Krivonozhko et al. (2015), Bougnol (2001) and Shadab et al. (2020) for more details and applications]. In this paper, by analogy with the work of Mostafaee and Soleimani-damaneh (2014), we establish a connection between the anchor points and super-efficiency models and propose a novel method to identify anchor points of the PPS of the BCC model by testing all BCC-efficient DMUs with a variant of the super-efficiency models (5 and 6) and their duals (8 and 9) (after eliminating the BCC-inefficient DMUs from the PPS). In addition, we present a new characterization of anchor points and the necessary and sufficient conditions for an extremely efficient DMU to be an anchor point. Some useful facts related to the properties of models (5), (6), (8) and (9) are proved. In addition, three numerical examples are provided.

Background

Consider a set of \({{\varvec{n}}}\) DMUs associated with \({{\varvec{m}}}\) inputs and \({{\varvec{s}}}\) outputs. In particular, each \(DMU_{j}=(X_j, Y_j)\) \((j\in J=\{1,\ldots , n\})\) consumes an amount \(x_{ij}(>0)\) of input \({{\varvec{i}}}\) and produces an amount \(y_{rj}(>0)\) of output \({{\varvec{r}}}\). The production possibility set T, \(T\subset \big \{(X, Y)|X\in E^{m}, Y\in E^{s}, X\geqslant 0, Y\geqslant 0\big \}\) is based on postulate sets, which are briefly explained (Banker et al. 1984; Yu et al. 1996). One of the most representative DEA models for evaluating the relative efficiency of a set of DMUs is the BCC model, proposed by Banker et al. (1984). The production possibility set (PPS) of the BCC model can be defined as follows:

$$\begin{aligned} T_v=\left\{ (X, Y)|X\geqq \sum _{j\in J}\lambda _{j}X_{j},\ Y\leqq \sum \limits _{j\in J}\lambda _{j}Y_{j},\ \sum \limits _{j\in J}\lambda _{j}=1,\ \lambda _{j}\ge 0,\ j\in J\right\} . \end{aligned}$$

The support set of a supporting hyperplane is the face of a polyhedral set. The PPS of the BCC model has bounded and unbounded faces. Unbounded faces constitute the free disposability part of the frontier.

A facet of a k-dimensional polyhedral set is a \(k-1\) dimensional face. The input-oriented BCC model, corresponding to \(DMU_{k}\), \(k\in J\), is given by

$$\begin{aligned} \begin{array}{rlllc} \min &{} \ \theta -\epsilon \left( \sum \limits _{i=1}^{m}s_{i}^{-}+\sum \limits _{r=1}^{s}s_{r}^{+}\right) \\ s.t.&{} \sum \limits _{j\in J}\lambda _{j}y_{rj}-s_{r}^{+}=y_{rk},&{}r=1,\ldots ,s\\ &{} \sum \limits _{j\in J}\lambda _{j}x_{ij}+s_{i}^{-}=\theta x_{ik},&{}i=1,\ldots ,m\\ &{} \sum \limits _{j\in J}\lambda _{j}=1&{}\\ &{}\lambda _{j}\ge 0,&{}{j\in J}\\ &{}s_{i}^{-}\ge 0,&{}i=1,\ldots ,m\\ &{}s_{r}^{+}\ge 0,&{}r=1,\ldots ,s\\ &{}\theta &{}free \end{array} \end{aligned}$$
(1)

In addition, the output-oriented BCC model corresponding to \(DMU_{k}\), \(k\in J\), is as follows:

$$\begin{aligned} \begin{array}{rlllc} \max &{} \ \varphi +\epsilon \left( \sum \limits _{i=1}^{m}t_{i}^{-}+\sum _{r=1}^{s}t_{r}^{+}\right) \\ s.t.&{} \sum _{j\in J}\lambda _{j}y_{rj}-t_{r}^{+}=\varphi y_{rk},&{}r=1,\ldots ,s\\ &{} \sum _{j\in J}\lambda _{j}x_{ij}+t_{i}^{-}= x_{ik},&{}i=1,\ldots ,m\\ &{} \sum _{j\in J}\lambda _{j}=1&{}\\ &{}\lambda _{j}\ge 0,&{}{j\in J}\\ &{}t_{i}^{-}\ge 0,&{}i=1,\ldots ,m\\ &{}t_{r}^{+}\ge 0,&{}r=1,\ldots ,s\\ &{}\varphi &{}free \end{array} \end{aligned}$$
(2)

where \(\epsilon\) is a non-Archimedean small and positive number and \(s_{i}^{-}\), \(s_{r}^{+}\), \(t_{i}^{-}\) and \(t_{r}^{+}\), \(i=1,\ldots ,m\), \(r=1,\ldots ,s\) are the slack variables belonging to \({\mathbb {R}}^{\ge 0}\). Note that \(s_{i}^{-}\) and \(t_{i}^{-}\) represent the input excesses and \(s_{r}^{+}\) and \(t_{r}^{+}\) represent the output shortfalls. Here, \(\theta\), \(\varphi\) and \(\lambda _{j}\), \(j\in J\) are real numbers and \(\lambda _{j}\in {\mathbb {R}}^{\ge 0}\). Models (1) and (2) are called envelopment forms (with a non-Archimedes number).

Let \((\theta ^*, {{\textbf {s}}}^{+*}, {{\textbf {s}}}^{-*})\) and \((\varphi ^*, {{\textbf {t}}}^{+*}, {{\textbf {t}}}^{-*})\) be the optimal solutions for models (1) and (2), respectively. \(DMU_k\) is said to be strongly efficient (BCC-efficient) if and only if either (i) or (ii) occur.

  1. (i)

    \(\theta ^*=1\) and \({\textbf {1s}}^{+*}\)+\({\textbf {1s}}^{-*}\)=0

  2. (ii)

    \(\varphi ^*=1\) and \({\textbf {1t}}^{+*}\)+\({\textbf {1t}}^{-*}\)=0

\(DMU_k\) is said to be weakly efficient if and only if either (v) or (iv) occur.

  1. (xxii)

    \(\theta ^*=1\) and \({\textbf {1s}}^{+*}\)+\({\textbf {1s}}^{-*}\ne\) 0

  2. (xxiii)

    \(\varphi ^*=1\) and \({\textbf {1t}}^{+*}\)+\({\textbf {1t}}^{-*}\ne\) 0

Note that if \(\theta ^*<1\) and \(\varphi ^*>1\) then \(DMU_k\) is an interior point of the PPS.Footnote 1

Each interior DMU and weakly efficient DMU in the BCC model is said to be BCC-inefficient DMU.

The Efficient Frontier is the set of all points (real or virtual DMUs) with an efficiency score equal to unity. The efficient frontier can be divided into two categories:

i) Strongly efficient frontier is the set of all (real or virtual) strongly efficient (BCC-efficient) DMUs.

ii) Weakly efficient frontier, in which all its relative interior points (real or virtual DMUs) are weakly efficient DMUs.

\(DMU_k=(X_k, Y_k)\) is said to be non-dominated if and only if there is no \(DMU=(X, Y)\) (real or virtual), such that \((-X_k, Y_k)\ge (-X, Y)\) and \((-X_k, Y_k)\ne (-X, Y)\).

We denote the set of BCC-inefficient, non-extreme, and extreme DMUs as F, E and \(E^*\), respectively. These three subsets partition the set J. Set \(E^*\) is also called the frame of J. The frames are important in DEA because they build the PPS of the DEA models, and the exclusion of any of them alters the shape of the PPS. The PPS of the BCC model with one input and one output is depicted in Fig. 1. In Fig. 1, \(J=\{DMU_1, DMU_2, DMU_3, DMU_4, DMU_5\}\), \(F=\{DMU_4, DMU_5\}\), \(E=\{DMU_6\}\) and \(E^*=\{DMU_1, DMU_2, DMU_3\}\).

Fig. 1
figure 1

Extreme, non-extreme and anchor DMUs

The dual of models (1) and (2) (without \(\epsilon\), i.e., \(\epsilon=0\)), which are called multiplier forms, are similar to models (3) and (4), respectively:

$$\begin{aligned} \begin{array}{rlllc} \max &{} \sum _{r=1}^{s}u_{r}y_{rk}+u_{0}\\ s.t.&{} \sum _{r=1}^{s}u_{r}y_{rj}-\sum _{i=1}^{m}v_{i}x_{ij}+u_{0}\le 0,&{}j=1,\ldots n\\ &{} \sum _{i=1}^{m}v_{i}x_{ik}=1,&{}&{}\\ &{}u_{r}\ge 0,&{}r=1,\ldots ,s\\ &{}v_{i}\ge 0,&{}i=1,\ldots ,m\\ {} &{}u_{0}&{}free\\ \end{array}\end{aligned}$$
(3)
$$\begin{aligned} \begin{array}{rlllc} \min &{} \sum _{i=1}^{m}v_{i}x_{ik}+u_{0}\\ s.t.&{} \sum _{i=1}^{m}v_{i}x_{ij}-\sum _{r=1}^{s}u_{r}y_{rj}+u_{0}\ge 0,&{}j=1,\ldots n\\ &{} \sum _{r=1}^{s}u_{r}y_{rk}=1,&{}&{}\\ &{}u_{r}\ge 0,&{}r=1,\ldots ,s\\ &{}v_{i}\ge 0,&{}i=1,\ldots ,m\\ {} &{}u_{0}&{}free\\ \end{array}\end{aligned}$$
(4)

Note 1: \(DMU_k\) is said to be strong efficient DMU if and only if either (i) or (ii) occur:

  1. (i)

    \({\bar{u}}^{*t}y_{k}+{\bar{u}}^*_{0}=1\) and \(({\bar{u}}^{*t}, {\bar{v}}^{*t})>0\) for some optimal solutions of (3)

  2. (ii)

    \({\hat{v}}^{*t}x_{k}+{\hat{u}}^*_{0}=1\) and \(({\hat{u}}^{*t}, {\hat{v}}^{*t})>0\) for some optimal solutions of (4).

Note 2: \(DMU_k\) is said to be weak efficient DMU if and only if either (v) or (iv) occur:

  1. (xxii)

    \({\bar{u}}^{*t}y_{k}+{\bar{u}}^*_{0}=1\) and \(({\bar{u}}^{*t}, {\bar{v}}^{*t})\ngtr 0\) for all optimal solutions of (3)

  2. (xxiii)

    \({\hat{v}}^{*t}x_{k}+{\hat{u}}^*_{0}=1\) and \(({\hat{u}}^{*t}, {\hat{v}}^{*t})\ngtr 0\) for all optimal solutions of (4).

Corresponding to each BCC-efficient DMU \(DMU_j=(x_{1j}, \ldots , x_{mj}, y_{1j}, \ldots , y_{sj})\), we define virtual DMUs \(DMU^{l}_j=(x_{1j}, \ldots , x_{lj}+\alpha , \ldots , x_{mj}, y_{1j}, \ldots , y_{sj})\) and \(DMU^{q}_j=(x_{1j}, \ldots , x_{mj}, y_{1j}, \ldots , y_{qj}-\gamma , \ldots , y_{sj})\), where \(\alpha , \gamma >0\). These virtual DMUs are either interior points of the PPS of the BCC model or lie on free-disposability faces (unbounded faces). In the latter case, we refer to these virtual DMUs as ”weak efficient virtual DMUs”” or WEV DMUs. Evidently, the WEV \(DMU^{l}_j\) (\(DMU^{q}_j\)) lies on the unbounded face with \(v_l=0\) (\(u_q=0\)). If \(DMU_j\) and \(DMU^{l}_j\) (\(DMU^{q}_j\)) lie on the same hyperplane, say, \({\overline{u}}y-{\overline{v}}x+{\overline{u}}_o=0\), then \({\overline{v}}_l=0\) (\({\overline{u}}_q=0\)). In Fig. 2, the BCC-efficient DMU \(D_2=(x_{12}, x_{22}, y_{12})\) and WEV DMU \(D'_2=(x_{12}+\alpha , x_{22}, y_{12})\) lie on the same hyperplane(s); thus, \({\overline{v}}_1=0\). In addition, the extremely efficient DMU \(D_1=(x_{11}, x_{21}, y_{11})\) and WEV DMU \(D'''_1=(x_{11}, x_{21}, y_{11}-\beta )\) lie on the same hyperplane(s); thus, \({\overline{u}}_1=0\).

Definition 1

The supporting hyperplane \(H=\big \{(x,y)|\ {\bar{u}}^{t}y- {\bar{v}}^{t}x+{\bar{u}}_{0}=0, ({\bar{u}}, {\bar{v}})\ge 0, ({\bar{u}}, {\bar{v}})\ne 0\big \}\) of the PPS of the BCC model is weak defining hyperplane if and only if at least \(m+s\) extreme efficient and weak efficient (virtual or real) DMUs of the PPS lie on H. (In this case, at least one component of its gradient (normal vector) is zero).

In Fig. 1, \(H_1\) and \(H_2\) are weak defining hyperplanes.

Bougnol and Marie-Laure (2009) defined anchor DMUs as follows:

Definition 2

\(DMU_k \in E^*\) is an anchor DMU if it belongs to an unbounded face of the PPS of the BCC model.

Remark 1

By definition of anchor DMU it is obvious that \(DMU_k \in E^*\) is anchor DMU if and only if there exist some l (or q) so that \(DMU^l_k\) (or \(DMU^q_k\)) is WEV DMU (i.e. \({\overline{v}}_l=0\) (or \({\overline{u}}_q=0\))).

In Fig. 1, \(DMU_1\) and \(DMU_3\) are anchor DMUs, which lie on the unbounded faces with \({\overline{u}}=0\) and \({\overline{v}}=0\), respectively. In addition, in Fig. 2, \(D_1\), \(D_2\), and \(D_3\) are anchor DMUs. \(D_1\) lies on the three unbounded faces with \({\overline{v}}_1=0\), \({\overline{v}}_2=0\) and \({\overline{u}}=0\). \(D_2\) lies on two unbounded faces, with \({\overline{v}}_1=0\) and \({\overline{v}}_2=0\). Further, \(D_3\) lies on two unbounded faces with \({\overline{v}}_1=0\) and \({\overline{v}}_2=0\).

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 using a variant of the super-efficiency model. This is theoretically interesting because it establishes a new link between anchor points and super-efficiency models. First, we evaluate each \(DMU_{k},~(k\in J)\) using models (1) or (2). We then hold all BCC-efficient DMUs and remove the others. Suppose the set of all BCC-efficient DMUs is denoted by \(E'\)(=\(E\bigcup E^*\)). For each \(DMU_{k}=(x_{1k}, \ldots , x_{mk}, y_{1k}, \ldots , y_{sk}),~(k\in E')\), we solve the following models:

$$\begin{aligned} \begin{array}{rlllc} \min &{} \ \theta _{l}^{k}-\epsilon \left( \sum _{i=1}^{m}s_{i}^{-}+\sum _{r=1}^{s}s_{r}^{+}\right) \\ s.t.&{} \sum _{j\in E'-\{k\}}\lambda _{j}^{k}x_{lj}+s_{l}^{-}=\theta _{l}^{k} x_{lk}\\ &{} \sum _{j\in E'-\{k\}}\lambda _{j}^{k}x_{ij}+s_{i}^{-}=x_{ik},&{}i=1,\ldots ,m&{}i\ne l \\ &{} \sum _{j\in E'-\{k\}}\lambda _{j}^{k}y_{rj}-s_{r}^{+}=y_{rk},&{}r=1,\ldots ,s&{}\\ &{} \sum _{j\in E'-\{k\}}\lambda _{j}^{k}=1 \\ &{}\lambda _{j}^{k}\ge 0,&{}j\in E'-\{k\}\\ &{}s_{i}^{-}\ge 0, s_{r}^{+}\ge 0&{}i=1,\ldots ,m, &{}r=1,\ldots ,s\\ &{}\theta _{l}^{k}&{}free &{}l=1,\ldots ,m&{} \end{array} \end{aligned}$$
(5)
$$\begin{aligned} \begin{array}{rlllc} \max &{} \ \varphi _{q}^{k}+\epsilon \left( \sum _{i=1}^{m}t_{i}^{-}+\sum _{r=1}^{s}t_{r}^{+}\right) \\ s.t.&{} \sum _{j\in E'-\{k\}}\mu _{j}^{k}x_{ij}+t_{i}^{-}= x_{ik},&{}i=1,\ldots ,m&{}\\ &{} \sum _{j\in E'-\{k\}}\mu _{j}^{k}y_{qj}-t_{q}^{+}=\varphi _{q}^{k}y_{qk},&{}&{}\\ &{} \sum _{j\in E'-\{k\}}\mu _{j}^{k}y_{rj}-t_{r}^{+}=y_{rk},&{}r=1,\ldots ,s&{}r\ne q\\ &{} \sum _{j\in E'-\{k\}}\mu _{j}^{k}=1, \\ &{}\mu _{j}^{k}\ge 0,&{}j\in E'-\{k\}\\ &{}t_{i}^{-}\ge 0,&{}i=1,\ldots ,m\\ &{}t_{r}^{+}\ge 0,&{}r=1,\ldots ,s\\ &{}\varphi _{q}^{k}&{} free&{}q=1,\ldots ,s&{} \end{array} \end{aligned}$$
(6)
Fig. 2
figure 2

Weak efficient virtual and anchor DMUs

The following theorem holds for models (5) and (6):

Theorem 1

\(DMU_{k}\) is BCC-efficient if and only if for each l(or q) either model (5)(or (6)) is infeasible or \(\theta _{l}^{k*}\geqslant 1\)(\(\varphi _{q}^{k*}\leqslant 1\)).

Proof

Only if: Suppose that \(DMU_{k}\) is BCC-efficient and for some l model (5) is feasible. We demonstrate that \(\theta _{l}^{k*}\geqslant 1\). Consider the following, corresponding to \(DMU_{k}\):

$$\begin{aligned} \begin{array}{rlllc} \min &{} \ \theta _{l}^{'k}-\epsilon \left( \sum _{i=1}^{m}s_{i}^{-}+\sum _{r=1}^{s}s_{r}^{+}\right) \\ s.t.&{} \sum _{j\in E'}\lambda _{j}^{k}x_{lj}+s_{l}^{-}=\theta _{l}^{'k} x_{lk}\\ &{} \sum _{j\in E'}\lambda _{j}^{k}x_{ij}+s_{i}^{-}=x_{ik},&{}i=1,\ldots ,m&{}i\ne l \\ &{} \sum _{j\in E'}\lambda _{j}^{k}y_{rj}-s_{r}^{+}=y_{rk},&{}r=1,\ldots ,s&{}\\ &{} \sum _{j\in E'}\lambda _{j}^{k}=1 \\ &{}\lambda _{j}^{k}\ge 0,&{}j\in E'\\ &{}s_{i}^{-}\ge 0, s_{r}^{+}\ge 0&{}i=1,\ldots ,m, &{}r=1,\ldots ,s\\ &{}\theta _{l}^{'k}&{}free &{}l=1,\ldots ,m&{} \end{array} \end{aligned}$$
(7)

Now suppose that \(\theta ^{k*}(=1)\), \(\theta _{l}^{'k*}\) and \(\theta _{l}^{k*}\) are the optimal objective functions, with respect to \(DMU_{k}\), of the models (1), (7) and (5), respectively. We can easily show that \(\theta ^{k*}\le \theta _{l}^{'k*}\le \theta _{l}^{k*}\). Therefore, \(\theta _{l}^{k*}\ge 1\).

If: For proving the part “if”, we can use inequality \(\theta _{l}^{k*}\le \theta ^*\le 1\) (\(\varphi _{q}^{k*}\ge \varphi ^*\ge 1\)). The details have been removed. \(\qquad \qquad \qquad \,\qquad \qquad \qquad \square\)

We now study the dual of models (5) and (6) (without \(\epsilon\), i.e., \(\epsilon =0\)) to identify the anchor points that meet the necessary and sufficient conditions.

$$\begin{aligned} \begin{array}{rlllc} \max &{} \sum _{r=1}^{s}u_{r}y_{rk}-\sum _{i=1, i\ne l}^{m}v_{i}x_{ik}+u_{0}\\ s.t.&{} \sum _{r=1}^{s}u_{r}y_{rj}-\sum _{i=1}^{m}v_{i}x_{ij}+u_{0}\le 0,&{}j=1,\ldots n,~~j\ne k\\ &{} v_lx_{lk}=1,&{}&{}\\ &{}u_{r}\ge 0,&{}r=1,\ldots ,s\\ &{}v_{i}\ge 0,&{}i=1,\ldots ,m\\ {} &{}u_{0}&{}free\\ \end{array} \end{aligned}$$
(8)
$$\begin{aligned} \begin{array}{rlllc} \min &{} \sum _{i=1}^{m}v_{i}x_{ik}-\sum _{r=1, r\ne q}^{s}u_{r}y_{rk}+u_{0}\\ s.t.&{} \sum _{i=1}^{m}v_{i}x_{ij}-\sum _{r=1}^{s}u_{r}y_{rj}+u_{0}\ge 0,&{}j=1,\ldots n,~~j\ne k\\ &{} u_qy_{qk}=1,&{}&{}\\ &{}u_{r}\ge 0,&{}r=1,\ldots ,s\\ &{}v_{i}\ge 0,&{}i=1,\ldots ,m\\ {} &{}u_{0}&{}free\\ \end{array} \end{aligned}$$
(9)

Theorems 2, 3 and 5 provide the necessary and sufficient conditions for the existence of anchor DMUs.

Theorem 2

If for at least one l (or q), model (8) (or model (9)) is unbounded, then \(DMU_{k}\) is an anchor DMU.

Proof

Assume that model (8) is unbounded for at least for l. In this case, model (5) is infeasible by the duality theorem. First, we demonstrate, by contradiction, that \(DMU_{k}\in E^*\). Suppose \(DMU_{k}\in E\setminus E^*\). Thus, there are real or virtual units \(DMU_p=(X_p, Y_p)\) and \(DMU_q=(X_q, Y_q)\), such that

$$\begin{aligned} \begin{array}{ll} DMU_k={\overline{\lambda }}DMU_p+{\overline{\mu }}DMU_q&\end{array} \end{aligned}$$
(10)

for some \({\overline{\lambda }}>0, {\overline{\mu }}>0\), \({\overline{\lambda }}+ {\overline{\mu }}=1\).

By replacing (10) in (5) instead of \(DMU_k\), a feasible solution for model (5) can be found, which is a contradiction. Therefore, \(DMU_{k}\in E^*\). \(\square\)

We now show that \(DMU_k\) lies on the weak defining hyperplane(s) (unbounded facet), which is parallel to the lth axis of the input. Consider virtual \(DMU'_{k}\) as follows: \(DMU'_{k}=(x_{1k},\ x_{2k},\ldots , x_{lk}+\alpha ,\ldots ,\ x_{mk},\ y_{1k},\ldots \,y_{sk})\)

in which \(\alpha >0\). It is sufficient to show that \(DMU'_{k}\) is a weak efficient DMU in the output-oriented of the BCC model. Consider the following model corresponding to \(DMU'_k\).

$$\begin{aligned} \begin{array}{rlllc} \max \ \varphi \\ \ s.t.&{} \sum _{j\in E'}\lambda _{j}x_{lj}+\mu _{k}(x_{lk}+\alpha )\le x_{lk}+\alpha &{}&{}\\ &{} \sum _{j\in E'}\lambda _{j}x_{ij}+\mu _{k}x_{ik}\le x_{ik}&{}i=1,\ldots ,m, i\ne l&{}&{}\\ &{} \sum _{j\in E'}\lambda _{j}y_{rj}+\mu _{k}y_{rk}\ge \varphi y_{rk},&{}r=1,\ldots ,s&{}\\ &{} \sum _{j\in E'}\lambda _{j}+\mu _{k}=1, \\ &{}\lambda _{j}\ge 0,&{}j\in E'\\ &{}\mu _{k}\ge 0\\ &{}\varphi &{} free\\ \end{array} \end{aligned}$$
(11)

Suppose \((\lambda _{j}^{*}\ (j=1, \ldots ,n),\ \mu _{k}^{*},\ \varphi ^{*})\) is the optimal solution to the above model. We show, by contradiction, that \(\varphi ^{*}=1\).

Suppose that \(\varphi ^{*}>1\). Thus we have:

$$\begin{aligned} \begin{array}{rlllc} \sum _{j\in E'}\lambda _{j}^{*}y_{rj}+\mu _{k}^{*}y_{rk}=\sum _{j\in E'-\{k\}}\lambda _{j}^{*}y_{rj}+(\lambda _{k}^{*}+\mu _{k}^{*})y_{rk}\ge \varphi ^{*} y_{rk}>y_{rk},&{}r=1,\ldots ,s&{}\\ \end{array} \end{aligned}$$

Therefore, \((\lambda _{j}^{*}=0\ (j=1, \ldots ,n, j\ne k ),\ \mu _{k}^{*}+\lambda _{k}^{*}=1)\) cannot be the optimal solution for model (11). Thus, \(\mu _{k}^{*}+\lambda _{k}^{*}<1\) and \(\mu _{k}^{*}\ne 1\).

The constraints of model (11) can be written as follows:

$$\begin{aligned} \begin{array}{llllc} \sum _{j\in E'-\{k\}}\lambda _{j}^{*}x_{lj}\le (1-\mu _{k}^{*}-\lambda _{k}^{*})x_{lk}+(1-\mu _{k}^{*})\alpha &{}&{}&{}\\ \sum _{j\in E'-\{k\}}\lambda _{j}^{*}x_{ij}\le (1-\mu _{k}^{*}-\lambda _{k}^{*})x_{ik}&{}i=1,\ldots ,m, i\ne k&{}&{}\\ \sum _{j\in E'-\{k\}}\lambda _{j}^{*}y_{rj}>(1-\mu _{k}^{*}-\lambda _{k}^{*})y_{rk},&{}r=1,\ldots ,s&{}\\ \sum _{j\in E'-\{k\}}\lambda _{j}^{*}=1-\mu _{k}^{*}-\lambda _{k}^{*},&{}&{} \\ \end{array} \end{aligned}$$
(12)

Divide both sides of model (12) by \(1-\mu _{k}^{*}-\lambda _{k}^{*}\) and define \({\overline{\mu }}_{j}= \frac{\lambda _{j}^{*}}{1-\mu _{k}^{*}-\lambda _{k}^{*}},~j\in E'- \{k\}\). Thus, model (4) becomes

$$\begin{aligned} \begin{array}{rlllc} &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}x_{lj}\le x_{lk}+\beta &{}&{}\\ &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}x_{ij}\le x_{ik}&{}i=1,\ldots ,m, i\ne l&{}&{}\\ &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}y_{rj}> y_{rk},&{}r=1,\ldots ,s&{}\\ &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}=1, \\ \end{array} \end{aligned}$$

in which, \(\beta = (\frac{1-\mu _{k}^{*}}{1-\mu _{k}^{*}-\lambda _{k}^{*}})\alpha\). Since, \(\beta >0\) then there is \({\hat{\theta }}>1\) so that \(x_{lk}+\beta ={\hat{\theta }}x_{lk}\); therefore, we have: \(\sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}x_{lj}\le {\hat{\theta }}x_{lk}\).

In summary, the constraints of model (12) can be rewritten as follows:

$$\begin{aligned} \begin{array}{rlllc} &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}x_{lj}\le {\hat{\theta }}x_{lk}&{}&{}\\ &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}x_{ij}\le x_{ik}&{}i=1,\ldots ,m, i\ne l&{}&{}\\ &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}y_{rj}> y_{rk},&{}r=1,\ldots ,s&{}\\ &{} \sum _{j\in E'-\{k\}}{\overline{\mu }}_{j}=1, \\ &{}{\overline{\mu }}_{j}\ge 0&{}j\in E'-\{k\} \end{array} \end{aligned}$$

Therefore, \(({\overline{\mu }}_{j}\ (j\in E'-\{k\}),~{\hat{\theta }})\) is a feasible solution for model (5), which is a contradiction. This result implies that \(\varphi ^{*}=1\). However, \(DMU'_{k}\) is dominated by \(DMU_{k}\); therefore, \(DMU'_{k}\) is weak efficient. Thus, \(DMU_{k}\) lies on the weak defining hyperplane that is parallel to the lth axis of the input.

This completes the proof. There is a similar proof for the case that at least one q, model (9) is unbounded.                                     \(\qquad \qquad \qquad \square\)

Theorem 3

If for at least one l (or q), model (8) (or model (9)) has a finite optimal solution such that some of \(u_{r}^{*}\, (r=1,\ldots ,s)\) or \(v_{i}^{*}\, (i=1,\ldots ,m)\) is zero, then \(DMU_k\) is an anchor point.

Proof

Assume that model (8) has a finite optimal solution, such as \((u^*,v^*,u_{0}^*)\) for at least one l such that some \(u_{r}^{*}\) or \(v_{i}^{*}\) are zero. Then, using the first two constraints of model (8), we obtain:

$$\begin{aligned} \sum _{r=1}^{s}u^{*}_{r}y_{rj}-\sum _{i=1}^{m}v^{*}_{i}x_{ij}+u^{*}_{0}\le 0, \,\,\, \forall j\ne k, \\ v^{*}_l\ne 0, \end{aligned}$$

For \(j=k, \sum _{r=1}^{s}u^{*}_{r}y_{rk}-\sum _{i=1}^{m}v^{*}_{i}x_{ik}+u^{*}_{0}\ge 0.\) Therefore, there exists \(T\ge 0\) such that

$$\begin{aligned} \sum _{r=1}^{s}u^{*}_{r}y_{rk}-\sum _{i=1}^{m}v^{*}_{i}x_{ik}+u^{*}_{0}-T=0 \\ \sum _{r=1}^{s}u^{*}_{r}y_{rj}-\sum _{i=1}^{m}v^{*}_{i}x_{ij}+u^{*}_{0}-T\le 0, \,\,\, \forall j\ne k, \\ v^{*}_l\ne 0, \end{aligned}$$

Therefore, we reach the supporting hyperplane of the PPS that passes through \(DMU_k\) that some components of its gradient vector are zero. This completes the proof. \(\square\)

Theorem 4

Let \(DMU_{k}\) be an anchor point Assume that the weak supporting hyperplane passing through \(DMU_{k}\) is \(H^*=\{(x,y)| u^{*t}y-v^{*t}x+u^*_0=0\}\) and \(v^*_1=0\). If the dual model is not unbounded for \(l=1,\) then hyperplane \(H^*\) will still pass through \(DMU_{k}^{\theta }\).Footnote 2

Proof

Let \(H^*=\{(x,y)|u^{*t}y-v^{*t}x+u^*_0=0\}\) be the weak supporting hyperplane passing through \(DMU_k,\) such that \(v^*_1=0.\) Therefore, \(i) u^{*t}y_k-v^{*t}x_k+u_{0}^*=0,\) \(ii) u^{*t}y_j-v^{*t}x_j+u_{0}^*\le 0, \,\,\, \forall j\ne k,\) \(iii)v^*_1=0,\) and can be expressed as follows: \(\square\)

\((u^*_{1},\ldots ,u^*_{s})(y_{1k},\ldots ,y_{sk})^t-(v^*_{1},\ldots ,v^*_{m})(x_{1k},\ldots ,x_{sk})^t+u_{0}^*=0,\)

\((u^*_{1},\ldots ,u^*_{s})(y_{1j},\ldots ,y_{sj})^t-(v^*_{1},\ldots ,v^*_{m})(x_{1j},\ldots ,x_{sj})^t+u_{0}^*\le 0, \,\,\, \forall j\ne k,\) because \(v^*_{1}=0\), we have that \((u^*_{1},\ldots ,u^*_{s})(y_{1k},\ldots ,y_{sk})^t-(v^*_{1},\ldots ,v^*_{m})(\theta x_{1k},\ldots ,x_{sk})^t+u_{0}^*=0,\)

Therefore, the weakly supporting hyperplane \(H^*\) passing through \(DMU^{\theta }_k.\,\,\,\,\qquad \qquad \qquad \qquad \quad \square\)

Theorem 5

Let \(DMU_{k}\) is an anchor point; then, one of the following conditions will be fulfilled: (a) (for at least one l (or q)) Model (8) (or model (9)) is unbounded. (b) (for at least one l (or q)) Model (8) (or model (9)) has a finite optimal solution such that some of \(u_{r}^{*}\) (or \(v_{i}^{*}\)) are zero.

Proof

Assume that \(DMU_{k}\) is an anchor point. In this case, there would be a weak supporting hyperplane passing through it, such as \(H^*=\{(x,y)|u^{*t}y-v^{*t}x+u^*_0=0\}\), such that at least one \(u_{r}^{*}\) or \(v_{i}^{*}\) is zero. Without a loss of generality, we assume that \(v^*_1=0.\) Assume that a) does not hold, then we prove, by contradiction, that b) holds. Assume that models (8) and (9) have strictly positive \(u_{r}^{*}\) and \(v_{i}^{*}\) for each l and q for each optimal solution. Therefore, according to the duality theorem in the corresponding primal model, we have \(s^{-*}_{i}=0\) and \(s^{+*}_{r}=0.\) Thus, the obtained \(\theta ^*\) for each l and the obtained \(\varphi ^*\) for each q\(DMU_{k}^{\theta }\) and \(DMU_{k}^{\varphi }\)Footnote 3 are the strong efficient units such that no weak supporting hyperplane would pass through it, which is in contradiction with Theorem 4. Consequently, \(v^{*}_{i}\) and \(u^{*}_{r}\) cannot be strictly positive. Now, we assume that b) does not hold, and we prove that a) holds. Assume that models (8) and (9) are not unbounded for each l and q, then all \(v^{*}_{i}\) and \(u^{*}_{r}\) should be strictly positive; which is not possible since \(DMU_k\) is an anchor point. \(\square\)

Corollary

\(DMU_k\), \(k\in E^*\) is an anchor DMU if and only if at least one of the following conditions holds: (a) (for at least one l (or q)), Model (8) (or model (9)) is unbounded. (b) (for at least one l (or q)) Model (8) (or model (9)) has a finite optimal solution such that some of \(u_{r}^{*}\) (or \(v_{i}^{*}\)) is zero.

Proof

From Theorems 2, 3 and 5, the proof is straightforward. Now, we are in a position to combine the elements of the method.

Summary of the algorithm for finding all anchor DMUs

  • Step 1. Evaluate n DMUs with a suitable form of models (1) and (2). Hold all BCC-efficient DMUs and remove other DMUs. Put the indices of these BCC-efficient DMUs in \(E'\).

  • Step 2. Evaluate each DMU in \(E'\) with models (8) and (9).

  • Step 3. If, for some l (or q), the model (8) (or (9)) is unbounded, then, \(DMU_k\) is an anchor DMU and \(DMU_{k}\) lies on the weak defining hyperplane, which is parallel to the lth axis of input (qth axis of output).

  • Step 4. If, for each l and q, models (8) and (9) are not unbounded, then we study the zero value in \(u^*_{i} (i=1,\ldots ,m)\) and \(v^*_{r} (r=1,\ldots ,s)\). If some components of the optimal solutions are zero, we conclude that \(DMU_k\) is an anchor point, otherwise, it is not.

  • Step 5. If each DMU in \(E'\) is evaluated by models (8) and (9), then stop. Otherwise, go to step 1.

\(\square\)

Table 1 Data of numerical example 1

Numerical examples

Example 1

(Single output case) Table 1 shows the data for the four DMUs with two inputs and one output. Using Model (1), the BCC-efficient DMUs are determined to be \(D_1\), \(D_2\), \(D_3\). Therefore, \(E'=\{1, 2, 3\}\). Remove BCC-inefficient DMU \(D_4\) from the PPS and solve models (8) and (9) corresponding to BCC-efficient DMUs \(D_1\), \(D_2\) and \(D_3\).

The following results are obtained. By Theorem 2, DMUs \(D_1\), \(D_2\) and \(D_3\) are anchor DMUs. DMU \(D_3\) lies on the weak defining hyperplane parallel to the 1th axis of the input, and DMUs \(D_1\) and \(D_2\) lie on the weak defining hyperplane parallel to the 2th axis of the input.

Table 2 Example 2: Multiple input and output
Table 3 Example 2: The results of evaluation BCC-efficient DMUs by models (8) and (9)
Table 4 Example 3: DMUs’ data [extracted from (Amirteimoori and Kordrostami 2005, p. 689)]

Example 2

(Multiple outputs and inputs case)

Table 2 shows the data for the five DMUs with two inputs and two outputs. The running model (1) (or (2)) shows that \(D_1, D_2\), and \(D_4\) are BCC-efficient, and the other DMUs are BCC-inefficient. Therefore, \(E'=\{1, 2, 4\}\). Applying models (8) and (9) to each \(DMU_k\), \(k\in E'\) produces the results in Table 3. In Table 3, “‘UNB” and “B” denotes “unbounded” and “bounded,” respectively. For instance, “UNB” in the first row and second column indicates that model (8), corresponding to DMU \(D_1\) with \(l=2\), is unbounded. Therefore, according to Theorem 2, \(D_1\) is an anchor DMU and lies on the weak defining hyperplane parallel to the 2th axis of the input. Using Theorems 2 and 3, and the information in Table 3, all \(DMU_k\) \(k\in E'\) are anchor DMUs. Moreover, DMUs \(D_1\), \(D_2\) and \(D_4\) lie on the weak defining hyperplane parallel to the 1th axis of the input, and DMUs \(D_1\) and \(D_2\) lie on the weak defining hyperplane parallel to the 2th axis of the input. In addition, DMUs \(D_2\) and \(D_4\) lie on the weak defining hyperplane parallel to the 1th axis of output, and DMUs \(D_1\) and \(D_2\) lie on the weak defining hyperplane parallel to the 2th axis of the output.

Example 3

(Real word data) We evaluated the data from 20 branches of a bank in Iran using the proposed method. The data were previously analyzed by Amirteimoori and Kordrostami (2005) (see Table 4). Running the DEA model (1) (or (2)) resulted in \(E'=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 17, 19, 20\}\). Using the proposed method, all DMUs in \(E'\) are found to be anchor DMUs. Additionally, \(DIV^{1,2}_1\), \(DIV^{1,2,3}_2\), \(DIV^{1,2,3}_3\), \(DIV^{1,2,3}_{4}\), \(DIV^{1,2,3}_{5}\), \(DIV^{1,2,3}_{6}\), \(DIV^{1,2,3}_{7}\), \(DIV^{1,2}_{8}\), \(DIV^{1,2,3}_{9}\), \(DIV^{1,2,3}_{11}\), \(DIV^{1,2,3}_{12}\), \(DIV^{1,2,3}_{15}\), \(DIV^{2,3}_{16}\), \(DIV^{1,2,3}_{17}\), \(DIV^{2,3}_{19}\), \(DIV^{2,3}_{20}\) and also, \(DOV^{1,2,3}_1\), \(DOV^{2,3}_2\), \(DOV^{1,2,3}_3\), \(DOV^{1,2,3}_{4}\), \(DOV^{2,3}_{5}\), \(DOV^{2,3}_{6}\), \(DOV^{1,2,3}_{7}\), \(DOV^{1,2,3}_{8}\), \(DOV^{1,2,3}_{9}\), \(DOV^{1,2,3}_{11}\), \(DOV^{1,2,3}_{12}\), \(DOV^{2,3}_{15}\), \(DOV^{2,3}_{16}\), \(DOV^{1,2,3}_{17}\), \(DOV^{1,2,3}_{19}\), \(DOV^{1,2,3}_{20}\) DMUs are WEV DMUs.

Conclusions

Anchor DMUs form an important class of extremely efficient points in DEA and define the transition from the efficient frontier to the free-disposability portion of the convex PPS frontier. Their identification has several interesting DEA applications, such as the construction of “unobserved” DMUs to capture prior value judgments in DEA and identify DMUs that are efficient for multiple constituencies. For more properties, applications, and identification procedures, refer to Bougnol and Marie-Laure (2009), Allen and Thanassoulis (2004), Rouse (2004) and Thanassoulis et al. (2012). In this study, we establish a connection between the anchor points and super-efficiency models and provide some main theorems for the characterization of these points. Using these theoretical results, a procedure was introduced to identify anchor points. The advantage of the proposed approach is that it determines the inputs (outputs) of the anchor DMUs that can be increased (decreased) without penetrating the interior of the production possibility set. The necessary and sufficient conditions for a DMU to become an anchor DMU are stated and proved.

Availability of data and materials

Data used in this paper were extracted from Amirteimoori and Kordrostami (2005).

Notes

  1. (*)used for the optimal solution.

  2. \(DMU_{k}^{\theta }\) is the projected \(DMU_k\) in a new PPS.

  3. \(DMU_{k}^{\varphi }\) is the projected \(DMU_k\) in a new PPS.

Abbreviations

CCR:

Charnes, Cooper, and Rhodes

BCC:

Banker, Charnes and Cooper

PPS:

Production possibility set

DMU:

Decision-making unit

DEA:

Data envelopment analysis

CRS:

Constant returns to scale

VRS:

Variable returns to scale

LP:

Linear programming

WEV:

Weakly efficient virtual

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The author would like to thank the four anonymous reviewers and the editor for their insightful comments and suggestions.

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Akbarian, D., Bani, A.A., Rostamy-Malkhalifeh, M. et al. An algorithm for the anchor points of the PPS of the BCC model. Financ Innov 8, 88 (2022). https://doi.org/10.1186/s40854-022-00392-z

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Keywords

  • Data envelopment analysis (DEA)
  • Anchor point
  • Production Possibility Set (PPS)
  • Efficient and inefficient boundary