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Abstract

One of the main important issues in Data Envelopment Analysis (DEA) is to recognize the set of anchor points which is the subset of the extreme efficient points of the production possibility set (PPS). An anchor point is an extreme efficient point which is located on the intersection of the strong efficient frontier and the weak efficient frontier. In the other word, each anchor point delineate the strong efficient frontier from the weak efficient frontier. So, if a decision making unit (DMU) is an anchor point, then there is at least one supporting hyperplane whose the gradient vector has some of zero components, and so some input\output factor does not play any role in the performance of the unit under evaluation. The concept of anchor point was used in DEA for the generation of unobserved DMUs in order to extend the DEA efficient frontier and so, this concept plays a critical role in the DEA theory and its applications. Given the importance of the anchor pints in the DEA literature, this study focuses on finding the anchor points and presents a new method to search the anchor points of the PPS under the variable returns to scale (VRS) assumption. For this purpose, we use the definition of the anchor points and present an approach to find the anchor points of the PPS. The proposed method is based on finding the weak and strong defining supporting hyperplanes passing through the unit under evaluation. The main advantage of the proposed method is that it exactly uses the definition of the anchor points to provide the approach and it is very simple to use and the anchor points can be easily identified by solving two simple models. In addition, the proposed approach is such that in addition to determining the anchor points, it also finds two important defining supporting hyperplanes on the PPS, which can be used in many problems in DEA. The potentially of the proposed method is illustrated by some numerical examples, reported in the literature to compare the proposed method with the existing methods.

Keywords

Data envelopment analysis anchor point variable returns to scale defining supporting hyperplane

Introduction

Data envelopment analysis is a non-parametric mathematical method to assess the efficiency of decision making units (DMUs) with multiple inputs and outputs. For more studies, see Refs.^1–4 DEA models assign an efficiency score between 0 and 1 to each DMU. A DMU, which has the larger the efficiency measure, shows the better performance. If the efficiency score of a unit is equal to 1, then this unit is efficient, otherwise, it is inefficient. DEA models use some axioms of the production technology to build a set, namely the production possibility set (PPS) to estimate the production function. For more details, see Refs.^5–13

One of the most important subset of the extreme efficient units in each PPS is the set of the anchor points. An anchor point is an extreme efficient unit which its inputs can be increased and its outputs can be decreased without entering the PPS. In the other word, an anchor point is an extreme efficient unit which is located on the intersection of the strong efficient frontier and the weak efficient frontier. In fact, for anchor points, there is at least one supporting hyperplane whose the gradient vector has some components equal to zero, and so some input\output factor does not play any role in the performance of the unit under evaluation. It should be noted that the anchor points play an essential role in DEA theory and its applications. For this reason, determining the anchor points in the PPS have attracted the attentions of many scholars in the DEA literature. For example, Jayekumar and Nagarajan¹⁴ introduced the anchor points and applied them to define the unobserved units and to extend the efficiency frontier.

Bougnol and Dulá¹⁵ proposed a method to find the anchor points based on the geometric properties of the anchor points. Their algorithm it is very expensive from the computational point of view. Regarding this drawback of their algorithm, Hosseinzadeh Lotfi et al.¹⁶ proposed an algorithm which identifies a main part of anchor points without solving any model and the remaining anchor points can be determined only by solving one model. Hosseinzadeh Saljooghi and Elahi Moghaddam¹⁷ proposed identifying algorithms for the anchor points and then by using their characteristics, they presented some methods to identify the anchor points.

Algorithm
Step 1. Solve model (2) for finding the BCC-efficient units and define the set $E$ as the set of the efficient units.Step 2. Solve models (5a) and (4b) for all $DM U_{o}, o \in E,$ to find two defining supporting hyperplanes with the minimum and maximum slopes among the supporting hyperplanes on $T_{v}$ at $DM U_{o} .$ Step 3. If the optimal value of model (5a) is equal to zero and the optimal value of model (4b) is positive, then this unit is an anchor point. Otherwise, it is not an anchor point.Step 4. End

Algorithm

Step 1. Solve model (2) for finding the BCC-efficient units and define the set

E

as the set of the efficient units.Step 2. Solve models (5a) and (4b) for all

DM U_{o}, o \in E,

to find two defining supporting hyperplanes with the minimum and maximum slopes among the supporting hyperplanes on

T_{v}

DM U_{o} .

Step 3. If the optimal value of model (5a) is equal to zero and the optimal value of model (4b) is positive, then this unit is an anchor point. Otherwise, it is not an anchor point.Step 4. End

Mostafaee and Soleimani-Damaneh¹⁸ presented an approach to identify the anchor points by using the sensitivity analysis techniques. Soleimani-Damaneh and Mostafaee¹⁹ introduced the extreme efficient units and the anchor points in a non-convex production technology and suggested an algorithm to find the anchor points in FDH model. Krivonozhko et al.²⁰ introduced the notion of terminal units. Moreover, some relationships were established between terminal units and other sets of units that were proposed for improving envelopment. They developed an algorithm for improving the frontier. The construction of algorithm has been based on the notion of terminal units. Jahanshahloo et al.²¹ proposed an innovative distance and energy aware optimized routing (DEA-OR) algorithm for wireless sensor nodes. Considering distance as the base factor, DEA-OR algorithm gives solution for energy efficient transmission and route failure recovery. The process is of two stages: weight based neighbor selection for routing and cost-confined greedy method for route failure recovery.

Zemtsov and Kotsemir²² applied DEA to assess the relationship between the results of patenting and resources of a regional innovation system (RIS). They used the DEA method over a long period, comparing regions to one another and over time. Mostafaee and Sohraiee²³ made use of the concept of supporting hyperplanes to provide a basic definition for exterior units and present a careful model for discovering exterior units. Also, based on the concept of supporting hyperplanes, they presented different definitions of anchor units. Also, they demonstrated the relationship between the sets of exterior, terminal, and anchor units. Koushki and Soleimani-Damaneh²⁴ introduced the concept of anchor points for the multi-objective optimization problems. They presented two approaches to recognize whether a given feasible point is an anchor point or not. Akbarian²⁵ used a variant of super-efficiency model to characterize all extreme efficient decision making units and anchor points of the PPS with the free disposability postulate. Shadab et al.²⁶ developed an algorithm to show the connection between the anchor points and the concept of congestion by using the S-shape form of the production function and the geometric features of the anchor points. Because of the set of anchor points is a subset of the set of extreme efficient DMUs, so finding the extreme efficient units is a useful method to find the anchor points. Hence, there are several algorithms to find the extreme efficient points in the literature of DEA. For more studies see Charnes et al.,²⁷ Dulá and López.²⁸ Bani et al.²⁹ presented a method to find anchor points of the $PPS$ of the BCC model based on a searching weak supporting hyperplane passing through unit under consideration.

Also, in the recent years, one of the attractive problems in optimization theory is presenting the forecasting models which many scholars in the different areas of operation research (OR), such as DEA, have been attracted to them. For example, in the DEA literature, measuring forecasting performance is a crucial issue. Karasu et al.³⁰ proposed a model based on support vector regression SVR with a wrapper-based feature selection using multi-objective particle swarm optimization (MOPSO). Their proposed model can capture the nonlinear properties and they determined the effect of MOPSO on model by comparing it with PSO. Karasu and Altan³¹ used a logistic chaotic map-based optimization algorithm to develop a method to predict the crude oil price with high accuracy. Their proposed model can capture the nonlinear dynamics in crude oil time series and the complexity of their model is less than the existing models such as PSO-LSTM model. Altan et al.³² proposed a combined model which significantly improves the forecasting accuracy. They tested the effectiveness of their model on data from the wind farm in five regions. Their proposed model can capture non-linear features of the wind speed time series. For more studies about the forecasting models, see Lim,³³ Fontalvo et al.,³⁴ An and Zhai,³⁵ and Kafi et al.³⁶

The concept of anchor point was used in DEA for the generation of unobserved DMUs in order to extend the DEA efficient frontier and so, this concept plays a critical role in the DEA theory and its applications. Hence, this study focuses on finding the anchor points in the different production possibility sets. For this purpose, we consider the production possibility set under the variable returns to scale (VRS) assumption and propose a new approach to identify the anchor points of this PPS. However, the proposed method can be easily developed to the different production possibility sets. The proposed method exactly uses the definition of the anchor points to provide the approach and so, it is based on finding the weak and strong defining supporting hyperplanes passing through the unit under evaluation. The use of the proposed approach is very simple and the anchor points can be easily identified by solving two simple models. In addition, the proposed approach is such that in addition to determining the anchor points, it also finds two important defining supporting hyperplanes on the PPS, which can be used in many problems in DEA, and this is the main contribution of the proposed method compared to the existing methods such as Bani et al.²⁹ Regarding the definition of the anchor points, if there are a weak and a strong defining supporting hyperplanes which the DMU under evaluation is located on them, then this DMU is an anchor point.

The rest of the paper is organized as follows: Section “Preliminaries and basic definitions” contains some preliminaries. Section “The proposed method for finding the anchor points” proposes an approach to determine the anchor points. A numerical example is provided in Section “Case study”. Section “Conclusion” concludes the paper.

Preliminaries and basic definitions

Consider a system of $n$ DMUs, denoted by $DM U_{j}, j = 1, \dots, n,$ where each unit consumes $m$ different inputs to generate $s$ different outputs. The $i^{th}$ input and $r^{th}$ output for $DM U_{j}$ are denoted by $x_{ij}$ and $y_{rj},$ respectively, for $i = 1, \dots, m$ and $r = 1, \dots, s .$ Also, assume that $DM U_{o}$ is the unit under evaluation.

Banker et al.² introduced the following PPS under the variable returns to scale (VRS):

\begin{array}{l} T_{v} = {(x, y) | x \geq \sum_{j = 1}^{n} λ_{j} x_{i j}, y \leq \sum_{j = 1}^{n} λ_{j} y_{r j}, \\ \sum_{j = 1}^{n} λ_{j} = 1, λ_{j} \geq 0, j = 1, \dots, n} \end{array}

(1)

They formulated the multiplier form of BCC model to assess the efficiency score of DMUs as follows:

\begin{matrix} max \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} \\ s . t . \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, j = 1, \dots, n, \\ v_{i}, u_{r} \geq 0, i = 1, \dots, m, r = 1, \dots, s, \end{matrix}

(2)

where, $v_{i}, i = 1, \dots, m,$ and $u_{r}, r = 1, \dots, s,$ are the input and output weights, respectively. Banker et al.² defined the BCC-efficient unit as follows:

Definition 2.1. The unit $DM U_{o} = (x_{o}, y_{o})$ is BCC-efficient, if and only if one of the following conditions is held:

(i) The optimal value of model (2) is equal to 1.

(ii) There is at least an optimal solution for model (2), for example, $(u_{1}^{*}, \dots, u_{s}^{*}, v_{1}^{*}, \dots, v_{m}^{*}),$ whose all components are strictly positive.

In the following, we review some basic definitions about the hyperplanes in the DEA literature.

Definition 2.2. A hyperplane $H$ is a collection of points of the form $H = {(x, y) | u^{t} y - v^{t} x + u_{0} = 0},$ where $(u, v)$ is a non-zero vector in $R^{s + m}$ and $u_{0}$ is a scalar. $(u, v)$ is called the gradient vector of the hyperplane.

Definition 2.3. A hyperplane $H$ defines two closed half-spaces as follows:

H^{+} = {(x, y) | u^{t} y - v^{t} x + u_{0} \geq 0}

and

H^{-} = {(x, y) | u^{t} y - v^{t} x + u_{0} \leq 0}

Also, $H$ defines two open half-spaces ${(x, y) | u^{t} y - v^{t} x + u_{0} > 0}$ and ${(x, y) | u^{t} y - v^{t} x + u_{0} < 0} .$ It is clear that, any point in $R^{s + m}$ lies in $H^{+}$ , in $H^{-},$ or in both.

Definition 2.4. A hyperplane $H = {(x, y) | u^{t} y - v^{t} x + u_{0} = 0}$ is called a supporting hyperplane of $T_{v}$ at $DM U_{o} = (x_{o}, y_{o}),$ if and only if $u^{t} y_{o} - v^{t} x_{o} + u_{0} = 0$ and $u^{t} y - v^{t} x + u_{0} \leq 0$ , for all $(x, y) \in T_{v} .$

A hyperplane $H$ is called the strong supporting hyperplane if $(u, v) > 0$ and it called the weak supporting hyperplane if some components of $(u, v)$ are equal to zero.

Dulá and López²⁸ proved the following relation between the optimal solution of model (2) and the supporting hyperplanes of $T_{v} .$

Theorem 2.1. Suppose that $DM U_{o}$ is located on the frontier of $T_{v} . (u^{*}, v^{*}, u_{0}^{*})$ is an optimal solution for model (2) if and only if $H = {(x, y) | u^{* t} y - v^{* t} x + u_{0}^{*} = 0}$ is a supporting hyperplane on $T_{v}$ at $DM U_{o} .$

Dulá and Bougnol²⁸ introduced the anchor points of $T_{v}$ as follows:

Definition 2.5. The BCC-efficient unit $DM U_{o} = (x_{o}, y_{o})$ is an anchor point if and only if it is located on a supporting hyperplane of $T_{v},$ namely $H,$ such that at least one component of $(u, v)$ is equal to zero.

The set of anchor points is a subset of the extreme efficient units in $T_{v}$ . in the other word, an anchor point is an extreme efficient unit which its inputs can be increased and its outputs can be decreased without entering the PPS. This means that, an anchor point is an extreme efficient unit which is located on the intersection of the strong efficient frontier and the weak efficient frontier. In fact, for anchor points, there is at least one supporting hyperplane whose the gradient vector has some components equal to zero, and so some input\output factor does not play any role in the performance of the unit under evaluation.

The next section proposes a new method to recognize the set of anchor points in $T_{v} .$ The proposed method is based of finding the units which are located on the weak and strong defining supporting hyperplanes at the same time.

The proposed method for finding the anchor points

As we said in the previous section, the concept of anchor point was used in DEA for the generation of unobserved DMUs in order to extend the DEA efficient frontier and so, this concept plays a critical role in the DEA theory and its applications. Hence, this section focuses on proposing a new method for finding the anchor points of $T_{v} .$ For this purpose, we aim to use the definition of the anchor points which shows that each anchor point is an extreme efficient unit which is located on the intersection of the strong efficient frontier and the weak efficient frontier. So, we provide an approach based on finding the weak and strong defining supporting hyperplanes passing through the unit under evaluation. As far as we know, for the first time, we focus on finding the anchor points by searching both weak and strong defining supporting hyperplanes in the DEA literature. This helps us, in addition to find the anchor points, we can determine two of the most important defining supporting hyperplanes, which are used in the different problems in the DEA literature. If a weak and strong defining supporting hyperplanes, which the unit under evaluation is located on them, are found by the proposed method, then this unit is an anchor point. In the following, we describe the proposed method in details.

Given that, each anchor point is an extreme efficient unit, so, at the first step of the proposed method, we should determine the BCC-efficient DMUs. For this purpose, we solve model (2) and define the set $E$ as the set of the BCC-efficient DMUs. After that, the anchor pints can be selected among the member of the set $E .$ In the nest step of the proposed method, regarding each anchor point lies on the weak and strong defining supporting hyperplane, so we need to determine these hyperplanes if they exist. It should be noted that the weak defining supporting hyperplane has the minimum slope among all the weak supporting hyperplanes on $T_{v}$ at $DM U_{o} .$ Also, the strong defining supporting hyperplane has the maximum slope among the strong supporting hyperplanes on $T_{v}$ at $DM U_{o} .$ With this argument, we formulate models (3a) and (3b) to determine the gradient vectors of these defining supporting hyperplanes. The proposed models are reported in Table 1.

Table 1.

The weak and strong defining supporting hyperplanes.

The weak defining supporting hyperplane	The strong defining supporting hyperplane
$\begin{matrix} min min {u_{1}, \dots, u_{s}, v_{1}, \dots, v_{m}} & (3 a) \\ s . t . \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, & r = 1, \dots, s, \\ v_{i} \geq 0, & i = 1, \dots, m . \end{matrix}$	$\begin{matrix} max min {u_{1}, \dots, u_{s}, v_{1}, \dots, v_{m}} & (3 b) \\ s . t . \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, & r = 1, \dots, s, \\ v_{i} \geq 0, & i = 1, \dots, m . \end{matrix}$

Given that, the weak defining supporting hyperplane has the minimum slope among all the weak supporting hyperplanes on $T_{v}$ at $DM U_{o},$ we formulate model (3a) which aims to find the input and output weights such that the slope of the corresponding defining supporting hyperplane on $T_{v}$ at the efficient unit be minimum. For this purpose, we minimize the minimum of all the input and output weights. According to the objective function of model (3a), it is clear that, if the optimal value of model (3a) is equal to zero, then there is a set of weights in which at least one of the input or output weights is equal to zero. Hence, there is a weak defining supporting hyperplane which the unit under evaluation lies on.

Similarly, regarding the strong defining supporting hyperplane has the maximum slope among all the strong supporting hyperplanes on $T_{v}$ at $DM U_{o},$ we formulate model (3b) which aims to find the input and output weights such that the slope of the corresponding defining supporting hyperplane on $T_{v}$ at the efficient unit be maximum. For this purpose, we maximize the minimum of the weights. According to the objective function of model (3b), it is clear that, if the optimal value of model (3b) is positive, then there is a set of weights in which all of the input and output weights are positive. Hence, there is a strong defining supporting hyperplane which the unit under evaluation lies on.

Models (3a) and (3b) are multi-objective programming problems. Given that, there is no optimal solution for multi-objective problems, hence, we try to convert these models into the single objective models. So, we define the non-negative variable $ζ = min {u_{1}, \dots, u_{s}, v_{1}, \dots, v_{m}}$ in model (3a) and formulate the single objective model (4a) which is equivalent to model (3a). According to the definition of $ζ,$ the first and the second constraints of model (4a) are clear. The third constraint of model (4a) shows that $ζ$ is exactly the minimum of $u_{1}, \dots, u_{s}, v_{1}, \dots, v_{m}$ and not just the infimum for ${u_{1}, \dots, u_{s}, v_{1}, \dots, v_{m}} .$ Similarly, we define the non-negative variable $ψ = min {u_{1}, \dots, u_{s}, v_{1}, \dots, v_{m}}$ in model (3b) and formulate the single objective model (4b) which is equivalent to model (3b). According to the definition of $ψ,$ the first and the second constraints of model (4b) are clear. Models (4a) and (4b) are reported in Table 2. All constraints of models (4a) and (4b) are clear, only the third constraint of model (4a) should be replaced by the suitable constraints. Given that in model (4a), $ζ = u_{1} \lor \dots \lor u_{s} \lor v_{1} \lor \dots v_{m},$ , so we can define the binary variables $μ_{r}, r = 1, \dots, s,$ and $w_{i}, i = 1, . . ., m,$ to guaranty that this constraint of model (4a) is held. Therefore, model (5a) which is equivalent to model (4a) can be formulated.

\begin{matrix} min ζ \\ s . t . \\ ζ \leq u_{r}, & r = 1, \dots, s, \\ ζ \leq v_{i}, & i = 1, \dots, m, \\ \sum_{r = 1}^{s} μ_{r} + \sum_{i = 1}^{m} w_{i} = 1, \\ ζ = \sum_{r = 1}^{s} μ_{r} u_{r} + \sum_{i = 1}^{m} w_{i} v_{i}, \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, μ_{r} \in {0, 1}, & r = 1, \dots, s, \\ v_{i} \geq 0, w_{i} \in {0, 1}, & i = 1, \dots, m . \end{matrix}

(5a)

Table 2.

The single objective equivalent forms of models (3a) and (3b).

The weak defining supporting hyperplane	The strong defining supporting hyperplane
$\begin{matrix} min ζ & (4 a) \\ s . t . \\ ζ \leq u_{r}, & r = 1, \dots, s, \\ ζ \leq v_{i}, & i = 1, \dots, m, \\ ζ = u_{1} \lor \dots \lor u_{s} \lor v_{1} \lor \dots v_{m}, \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, & r = 1, \dots, s, \\ v_{i} \geq 0, & i = 1, \dots, m . \end{matrix}$	$\begin{matrix} max ψ & (4 b) \\ s . t . \\ ψ \leq u_{r} & r = 1, \dots, s, \\ ψ \leq v_{i}, & i = 1, \dots, m, \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, & r = 1, \dots, s, \\ v_{i} \geq 0, & i = 1, \dots, m . \end{matrix}$

The weak defining supporting hyperplane

The strong defining supporting hyperplane

\begin{matrix} min ζ & (4 a) \\ s . t . \\ ζ \leq u_{r}, & r = 1, \dots, s, \\ ζ \leq v_{i}, & i = 1, \dots, m, \\ ζ = u_{1} \lor \dots \lor u_{s} \lor v_{1} \lor \dots v_{m}, \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, & r = 1, \dots, s, \\ v_{i} \geq 0, & i = 1, \dots, m . \end{matrix}

\begin{matrix} max ψ & (4 b) \\ s . t . \\ ψ \leq u_{r} & r = 1, \dots, s, \\ ψ \leq v_{i}, & i = 1, \dots, m, \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} + u_{0} \leq 0, & j = 1, \dots, n, \\ \sum_{r = 1}^{s} u_{r} y_{ro} + u_{0} = 1, \\ u_{r} \geq 0, & r = 1, \dots, s, \\ v_{i} \geq 0, & i = 1, \dots, m . \end{matrix}

According to the above discussion, models (5a) and (4b) determine two defining supporting hyperplanes which $DM U_{o}$ lies on both of them. Given that the anchor point is an extreme efficient unit which lies on both weak and strong defining hyperplanes, hence, if one of the defining supporting hyperplanes, determined by models (5a) and (4b), is the weak hyperplane and another is the strong hyperplane, then $DM U_{o}$ is an anchor point. This means that, if the optimal value of model (5a) is equal to zero and the optimal value of model (4b) is non-zero, then the unit under evaluation is an anchor point.

In summary, in this section, a new method for finding the anchor points of $T_{v},$ has been proposed which uses the definition of the anchor points to ensure that it finds all that units. Given that, each anchor point is an extreme efficient unit which is located on the intersection of the strong efficient frontier and the weak efficient frontier. So, for the first time, we focus on finding the anchor points by searching both weak and strong defining supporting hyperplanes in the DEA literature. Hence, in addition to find the anchor points, we can determine two of the most important defining supporting hyperplanes, which can be used in the different problems in the DEA literature. For this purpose, two multi-objective programming models, to search the weak and strong defining supporting hyperplanes, were formulated and then some techniques were applied to convert our proposed models into the single objective models. If the proposed method finds the weak and strong defining supporting hyperplanes, which the unit under evaluation is located on them, then this unit is an anchor point.

In the following, we summarize the proposed method in an algorithm for more clarity.

The next section uses two case studies, reported in Soleimani-Damaneh and Mostafaee¹⁹ and Bani et al.²⁹ to illustrate the proposed method and to compare it with the existing methods.

Case study

In this section, we apply our proposed method for two datasets, reported in Bani et al.²⁹ and Soleimani-Damaneh and Mostafaee,¹⁹ to show the potentially of the proposed method and to compare the proposed method with the existing methods.

Example 1. The dataset, reported in Bani et al.²⁹ includes 20 Iranian banks with three inputs, number of staffs $(x_{1}),$ computer terminals $(x_{2})$ , and space $(x_{3})$ to produce three outputs, deposits $(y_{1}),$ loans $(y_{2})$ , and charge $(y_{3}) .$ The data is summarized in Table 3.

Table 3.

The data of Iranian banks.

DMU	$x_{1}$	$x_{2}$	$x_{3}$	$y_{1}$	$y_{2}$	$y_{3}$	Efficiency
1	0.9503	0.7	0.155	0.19	0.5214	0.2926	1.0000
2	0.7962	0.6	1	0.2266	0.6274	0.4624	0.9691
3	0.7982	0.75	0.5125	0.2283	0.9703	0.2606	1.0000
4	0.8651	0.55	0.21	0.1927	0.6324	1	1.0000
5	0.8151	0.85	0.2675	0.2333	0.7221	0.2463	0.9240
6	0.8416	0.65	0.5	0.2069	0.6025	0.5689	0.8820
7	0.7189	0.6	0.35	0.1824	0.9	0.7158	1.0000
8	0.7853	0.75	0.12	0.125	0.234	0.2977	1.0000
9	0.4756	0.6	0.135	0.0801	0.3643	0.2439	1.0000
10	0.6782	0.55	0.51	0.0818	0.1835	0.0486	1.0000
11	0.7112	1	0.305	0.2117	0.3179	0.4031	0.7959
12	0.8113	0.65	0.255	0.1227	0.9225	0.6279	1.0000
13	0.6586	0.85	0.34	0.1755	0.6452	0.2605	0.9223
14	0.9763	0.8	0.54	0.1443	0.5143	0.2433	0.6947
15	0.6845	0.95	0.45	1	0.2917	0.0982	1.0000
16	0.6127	0.9	0.525	0.1151	0.4021	0.4641	0.8130
17	1	0.6	0.205	0.09	1	0.1614	1.0000
18	0.6337	0.65	0.235	0.0591	0.3492	0.0678	0.8956
19	0.3715	0.7	0.237	0.0385	0.1898	0.1112	1.0000
20	0.5827	0.55	0.5	0.1101	0.6145	0.7643	1.0000

The first step of the proposed method determines the efficiency score of units by solving model (2). The results are reported in the last column of Table 3. As can be seen in Table 4, the set of efficient units is $E = {1, 3, 4, 7, 8, 9, 10, 12, 15, 17, 19, 20} .$ The anchor points should be selected among the members of $E .$ Next, models (5a) and (4b) are solved to find the defining supporting hyperplanes on $T_{v}$ at $DM U_{o}, o \in E$ . The obtained results are summarized in Table 4. Columns 2 and 3 of this table report the optimal value of models (5a) and (4b), respectively. As we discussed in the previous section, if $ζ^{*} = 0$ and $ψ^{*} > 0,$ then the unit under evaluation is located on the weak and strong defining supporting hyperplanes and so this unit is an anchor point. The last column of this table reports the status of each unit.

Table 4.

The optimal value of models (5a) and (4b).

DMU	$ζ^{*}$ (optimal value of (5a))	$ψ^{*}$ (optimal value of (4b))	Anchor point
1	0.0345	0.9763	No
2	–	–	–
3	0.1257	0.5559	No
4	0.0000	0.8559	Yes
5	–	–	–
6	–	–	–
7	0.0781	0.4081	No
8	0.0000	0.3333	Yes
9	0.0000	0.5668	Yes
10	0.0000	0.8182	Yes
11	–	–	–
12	0.8113	0.9216	No
13	–	–	–
14	–	–	–
15	0.0845	0.5609	No
16	–	–	–
17	0.0000	0.8780	Yes
18	–	–	–
19	0.0000	0.6918	Yes
20	0.0000	0.3562	Yes

Therefore, the anchor points are the units 4, 8, 9, 10, 17, 19, and 20. Given that each anchor point is located on weak and strong defining supporting hyperplanes, hence, we can report the hyperplanes which at least one anchor point is located on them and this is one of the main advantages of the proposed method in this paper, compared to the existing methods in the DEA literature. Given that, we obtain the input and output weights by solving models (5a) and (4b), so, we can easily specify the corresponding hyperplanes. Therefore, the weak and strong defining hyperplanes, determined by models (5a) and (4b), are summarized in Table 5. In each row, the first hyperplane is the weak defining supporting hyperplane (some input or output weights are equal to zero) on $T_{v}$ at the anchor point and the second hyperplane is the strong defining supporting hyperplane (all input and output weights are positive) on $T_{v}$ at the anchor point, determined by models (5a) and (4b), respectively.

Example 2. This example considers the dataset, reported in Soleimani-Damaneh and Mostafaee¹⁹ This dataset has 17 DMUs which consume two inputs to produce two outputs. The data of this example, listed in Table 6.

Table 5.

The weak and the strong defining supporting hyperplane on $T_{v}$ at anchor points.

DMU	Weak hyperplane and strong hyperplane
4	$\begin{matrix} 0.0014 y_{1} - 0.0123 x_{1} - 0.9071 x_{3} + 0.2009 = 0 \\ 0.1453 y_{1} + 0.0230 y_{2} + 0.18 y_{3} - 0.1471 x_{1} - 0.0161 x_{2} - 0.8559 x_{3} + 0.0933 = 0 \end{matrix} l$
8	$\begin{matrix} 0.4633 y_{2} - 0.1541 x_{2} + 0.0072 = 0 \\ 0.0014 y_{1} + 0.3025 y_{2} + 0.234 y_{3} - 0.3333 x_{1} - 0.2403 x_{2} - 0.1411 x_{3} + 0.3183 = 0 \end{matrix}$
9	$\begin{matrix} 0.1763 y_{1} - 0.0141 = 0 \\ 0.21 y_{1} + 0.5668 y_{2} + 0.55 y_{3} - 0.1201 x_{1} - 0.3321 x_{2} - 0.0017 x_{3} - 0.1008 = 0 \end{matrix}$
10	$\begin{matrix} 0.1 y_{1} - 0.0014 x_{1} - 0.8984 x_{3} + 0.4510 = 0 \\ 0.1135 y_{1} + 0.0271 y_{2} + 0.1857 y_{3} - 0.2563 x_{1} - 0.8182 x_{2} - 0.0998 x_{3} + 0.6515 = 0 \end{matrix}$
17	$\begin{matrix} 0.3179 y_{2} - 0.1536 x_{3} - 0.2864 = 0 \\ 0.2708 y_{1} + 0.9001 y_{2} + 0.0846 y_{3} - 0.8780 x_{1} - 0.2455 x_{2} - 0.7544 x_{3} + 0.0225 = 0 \end{matrix}$
19	$\begin{matrix} 0.0725 y_{1} + 0.0805 y_{2} - 0.66 x_{1} - 0.1584 x_{3} - 0.2647 = 0 \\ 0.6918 y_{1} + 0.4179 y_{2} + 0.1857 y_{3} - 0.12 x_{1} - 0.1135 x_{2} - 0.2052 x_{3} + 0.0461 = 0 \end{matrix}$
20	$\begin{matrix} 0.6408 y_{1} + 0.2818 y_{2} - 0.4457 x_{2} - 0.0014 = 0 \\ 0.1513 y_{1} + 0.134 y_{2} + 0.3562 y_{3} - 0.2 x_{1} - 0.2502 x_{2} - 0.2653 x_{3} - 0.0893 = 0 \end{matrix}$

Table 6.

The data of units in Example 2.

DMU	$x_{1 j}$	$x_{2 j}$	$y_{1 j}$	$y_{2 j}$	Efficiency	$ζ^{*}$ (optimal value of (5a))	$ψ^{*}$ (optimal value of (4b))	Anchor point
1	26	7	12	3	1.0000	0.0000	0.0477	Yes
2	29	6	10	7	1.0000	0.0000	0.0404	Yes
3	40	8	20	6	1.0000	0.0050	0.0375	No
4	42	7	12	6	0.8869	0.1013	0.1140	No
5	45	9	18	6	0.8340	0.0145	0.0237	No
6	92	12	40	2	0.7604	0.0432	0.0833	No
7	83	11	58	3	1.0000	0.0785	0.0909	No
8	87	14	52	7	1.0000	0.1897	0.2005	No
9	149	16	61	4	0.7582	0.0423	0.0619	No
10	177	17	54	12	1.0000	0.0439	0.0588	No
11	191	19	61	11	0.9056	0.0476	0.0526	No
12	185	14	73	4	1.0000	0.0327	0.0714	No
13	186	20	85	10	1.0000	0.3504	0.3893	No
14	74	12	36	5	0.7941	0.0193	0.0295	No
15	164	22	69	8	0.8254	0.0486	0.0500	No
16	225	20	80	5	0.8750	0.0994	0.1000	No
17	108	10	27	3	0.7771	0.0325	0.0477	No

The first step of the proposed method determines the efficiency score of units by solving model (2). The results are reported in the last column of Table 6. As can be seen in Table 4, the set of efficient units is $E = 1, 2, 3, 7, 8, 10, 12, 13 .$ Next, models (5a) and (4b) are solved to find the defining supporting hyperplanes on $T_{v}$ at $DM U_{o}, o \in E$ . The obtained results are summarized in Table 6. Columns 7 and 8 of this table report the optimal value of models (5a) and (4b), respectively. If $ζ^{*} = 0$ and $ψ^{*} > 0,$ then the unit under evaluation is located on the weak and strong defining supporting hyperplanes and so this unit is an anchor point. The last column of this table reports the status of each unit.

Therefore, the anchor points are the units 1 and 20. The weak and strong defining hyperplanes, determined by models (5a) and (4b), are summarized in Table 7. In each row, the first hyperplane is the weak defining supporting hyperplane on $T_{v}$ at the anchor point and the second hyperplane is the strong defining supporting hyperplane on $T_{v}$ at the anchor point, determined by models (5a) and (4b), respectively.

Table 7.

The weak and the strong defining supporting hyperplane on $T_{v}$ at anchor points.

DMU	Weak hyperplane and strong hyperplane
1	$\begin{matrix} 0.0645 y_{2} - 0.1325 x_{1} - 0.8568 x_{2} + 0.1983 = 0 \\ 0.3461 y_{1} + 0.0087 y_{2} - 0.3021 x_{1} - 0.0853 + 0.0332 = 0 \end{matrix}$
2	$\begin{matrix} 0.3365 y_{2} - 0.2341 x_{1} + 0.0827 = 0 \\ 0.0459 y_{1} + 0.2132 y_{2} - 0.4261 x_{1} - 0.4031 x_{2} + 0.2835 = 0 \end{matrix}$

In summary, the proposed approach in this study finds the anchor points in the PPS by searching both weak and strong defining supporting hyperplanes on the PPS at the unit under evaluation. The idea of finding both weak and strong defining supporting hyperplanes on the PPS for determining the anchor points has been first introduced in this paper. As we saw in the examples, the proposed approach can easily find all anchor points by solving two models and also, it can determine two of the most important defining supporting hyperplanes which are very useful to extend the efficient frontier of the PPS.

Conclusion

The concept of anchor point was used in DEA for the generation of unobserved DMUs in order to extend the DEA efficient frontier and so they play an essential role in data envelopment analysis theory and its applications. This study focused on recognizing the anchor points in the PPS under the variable returns to scale assumption and presented a new method to find the anchor points which used the definition of the anchor points to ensure that it finds all that units. So, for the first time, we focused on finding the anchor points by searching both weak and strong defining supporting hyperplanes in the DEA literature. For this purpose, we proposed two models for determining the supporting hyperplanes with the minimum and maximum slopes among the supporting hyperplanes on the PPS at the unit under assessment. The main advantage of the proposed method is that it exactly uses the definition of the anchor points to provide the approach and it is very simple to use and the anchor points can be easily identified by solving two simple models. Also, the proposed approach is such that in addition to find the anchor points, two of the most important defining supporting hyperplanes can be found, which can be used in the different problems in the DEA literature. The potentially of the proposed method was illustrated by a numerical example, reported in the DEA literature to compare the proposed method with the existing methods.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Sevan Sohraiee

Mohsen Rostamy-Malkhalifeh

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A new method based on finding the weak and strong defining supporting hyperplanes to recognize the anchor points

Abstract

Keywords

Introduction

Preliminaries and basic definitions

The proposed method for finding the anchor points

Case study

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References