A unified approach to radial hyperbolic and directional efficiency measurement in Data Envelopment Analysis_2

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A uniﬁed approach to radial, hyperbolic, and directional eﬃciency measurement

in Data Envelopment Analysis

Margar´

eta Halick´

aa, M´

aria Trnovsk´

aa,∗, Aleˇ

sˇ

Cern´

aFaculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynsk´a dolina, 842 48 Bratislava,

Slovakia

bBayes Business School, City, University of London, 106 Bunhill Row, London EC1Y 8TZ, UK

Abstract

The paper analyses properties of a large class of “path-based” Data Envelopment Analysis models through

a unifying general scheme. The scheme includes the well-known oriented radial models, the hyperbolic

distance function model, the directional distance function models, and even permits their generalisations.

The modelling is not constrained to non-negative data and is ﬂexible enough to accommodate variants of

standard models over arbitrary data.

Mathematical tools developed in the paper allow systematic analysis of the models from the point of view

of ten desirable properties. It is shown that some of the properties are satisﬁed (resp., fail) for all models in

the general scheme, while others have a more nuanced behaviour and must be assessed individually in each

model. Our results can help researchers and practitioners navigate among the diﬀerent models and apply the

models to mixed data.

Keywords: Data envelopment analysis, Radial eﬃciency measures, Hyperbolic distance function,

Directional distance function, Negative data

1. Introduction

Data Envelopment Analysis (DEA) is a non-parametric analytical method used to assess the performance

of Decision Making Units (DMU). The DEA approach deﬁnes technology sets via observed input-output

data in combination with certain axioms. Each traditional model links an eﬃciency measure and a given

technology set using a mathematical optimisation programme. By solving the programme for an assessed

DMU, one ﬁnds both the eﬃciency score and a projection1on the frontier of the technology set that domi-

nates the assessed DMU.2Russell and Schworm (2018) distinguish between two main methods of arriving

at the eﬃciency score, giving rise to two classes of DEA models appositely named slacks-based and path-

based models.

∗Corresponding author

Email addresses: halicka@fmph.uniba.sk (Margar´

eta Halick´

a), trnovska@fmph.uniba.sk (M´

aria Trnovsk´

a),

ales.cerny.1@city.ac.uk (Aleˇ

sˇ

Cern´

1Synonymously with “projection,” the literature also uses the terminology “reference unit,” “projection benchmark,” “projection

point,” or “target” in this context.

2The resulting projection need not be Pareto–Coopmans eﬃcient.

Preprint accepted in European Journal of Operational Research June 28, 2023

Published version available at 10.1016/j.ejor.2023.06.039

Attribution-NonCommercial-NoDerivatives 4.0 International License

This paper establishes a general path-based scheme built on variable returns to scale (VRS) technology

that provides a unifying mathematical framework for the well-known oriented radial models, the hyperbolic

distance function model, and the directional distance function models. The scheme oﬀers a rich menu

of projection paths for eﬃciency analysis and the special capability of handling negative data, which is

important in a variety of applications, especially in the areas of accounting and ﬁnance. In the conﬁnes of the

general scheme, we provide a systematic and comprehensive analysis of the models vis-`a-vis ten desirable

properties. Overall, our paper ampliﬁes the message of Russell and Schworm (2018) that recognising the

class to which a model belongs allows one to deduce some of the model characteristics, which is important

for the correct interpretation of models and their practical use.

The slacks-based and the path-based models are easily distinguishable from each other on the basis of

the objective function in the corresponding programmes, or, in other words, based on the way they look for

a projection. According to Russell and Schworm (2018), the models in the slacks-based class search for the

projection by “specifying the form of aggregation over the coordinate-wise slacks.” The slacks indicate the

input surplus and output shortage between the projection and the assessed DMU. The main representatives

of this class are the Slacks-Based Measure (SBM) model of Tone (2001), the Russell Graph Measure model

of F¨

are, Grosskopf and Lovell (1985), the Additive Model (AM) of Charnes et al. (1985), and the Weighted

Additive Models (WAM) including the Range Adjusted Measure (RAM) model (Cooper, Park and Pastor,

1999) and the Bounded Adjusted Measure (BAM) model (Cooper et al., 2011).

The path-based models — the main focus of this paper — search for the projection by specifying various

parametric paths which run from the assessed DMU to the boundary of the technology set. In the special

case of the radial BCC input or output-oriented models (Banker, Charnes and Cooper, 1984), the path is

deﬁned by a ray connecting the DMU to the origin in the space of inputs or outputs and thus represents

a proportional, radial contraction of inputs or expansion of outputs. In the Directional Distance Function

(DDF) model (Chambers, Chung and F¨

are, 1996a; Chambers, Chung and F¨

are, 1998), the path is determined

by a ray in a pre-assigned direction pointing from the assessed DMU towards the dominating part of the

frontier.3On this path, one then seeks the point of minimal distance to the frontier of the technology set. The

Hyperbolic Distance Function (HDF) model (F¨

are, Grosskopf and Lovell, 1985) combines together the input

and output oriented BCC models by using a hyperbolic path that allows for a simultaneous equiproportionate

contraction of inputs and expansion of outputs.

Despite the many papers on DEA, there are only a few studies analysing the properties of either class of

models in a uniﬁed framework. In the context of general economic productivity theory, a series of articles

Russell and Schworm (2008), Russell and Schworm (2011), Levkoﬀ, Russell and Schworm (2011), Roshdi,

Hasannasab, Margaritis and Rouse (2018), and Hasannasab, Margaritis, Roshdi and Rouse (2019) provided

a comprehensive analysis of eﬃciency measures over diﬀerent types of productivity sets. The main message

from these papers is that the slacks-based measures, when operating in the full input-output space, identify

the Pareto–Koopmans eﬃciency unambiguously while the path-based measures do not.

3There are also approaches, where the directional vector may not point to the dominating part of the frontier, or may even be

determined endogenously. For a discussion on these non-standard approaches we refer to Pastor et al. (2022).

In the DEA setting, Cooper, Park and Pastor (1999) and Sueyoshi and Sekitani (2009) analyse eﬃciency

measures and introduce a set of desirable properties that an ideal DEA model should satisfy. These properties

include the indication of strong eﬃciency; boundedness; strict monotonicity; unit invariance; and translation

invariance. The selected DEA models are then classiﬁed on the basis of these criteria.

In other work, Halick´

a and Trnovsk´

a (2021) analyse the properties of slacks-based models in a general

scheme that encompasses all commonly used models in this class, but also allows for the construction of

new models.4All models in the general slacks-based scheme project onto the strongly eﬃcient frontier

and therefore account for all sources of ineﬃciencies. Among them, the RAM model performs best when

measured against eight desirable properties, satisfying seven, and failing only the unique projection property

due to its linearity. Recently, Aparicio and Monge (2022) have proposed a convex generalisation of the RAM

model that falls into the general scheme of Halick´

a and Trnovsk´

a (2021) and provides a unique projection

point. Thus, the new model currently claims the top spot in terms of the number of desirable properties.

Our paper aims to redress the lack of comprehensive analysis of path-based models in the literature. The

knowledge about the path-based models is currently fragmented across many articles with varying focus.

This situation is exacerbated by the fact that the properties of, for example, DDF models depend signiﬁcantly

on the choice of direction vectors and these have not been treated systematically to date. There appears to

be a general consensus that the path-based models do not guarantee strongly eﬃcient projection points

and, therefore, their eﬃciency score is overstated; and that they are monotone but not strictly monotone.

Many authors noticed diﬃculties with super-eﬃciency measurement under variable returns to scale and

the associated measurement of productivity change over time (e.g., Briec and Kerstens, 2009). Aparicio

et al. (2016) investigated the translation invariance of DDF models. A certain type of homogeneity was

observed in the oriented radial models and the HDF model (Cuesta and Zof´

ıo, 2005). On the other hand,

DDF models have the property of homogeneity only in the case of constant returns to scale. Another type of

homogeneity (so-called g-homogeneity) was introduced to describe some properties of DDF (Hudgins and

Primont, 2007).

In this paper, we analyse path-based models in a general framework. The main building block of our

approach is a parametric path that starts at the assessed DMU and for decreasing values of the parameter

passes through dominating units in the technology set towards its frontier. The path has two main ingre-

dients: (i) a direction vector; (ii) a real-valued smooth function, whose speciﬁcations determine the path

shape and, ultimately, the model properties. The assumptions imposed on directions and path functions are

ﬂexible enough to accommodate all standard models, such as the BCC input and output-oriented models,

the HDF and DDF models, the generalised distance function model by Chavas and Cox (1999), and even

oﬀer the possibility of going beyond.

A further advantage of the proposed general framework is that it permits extension of existing models

designed for non-negative data to arbitrary data (i.e., it accommodates the input and output data, for which

some or all components are negative). As a rule, DEA models are designed for non-negative data. In

practice, however, negative data are encountered in many applications in areas such as insurance, accounting,

4The terminology “non-radial” and “radial pattern models” was used for slacks-based and path-based models, respectively.

ﬁnance, or banking. A common approach to overcoming this diﬃculty is to use translation-invariant models,

which can be applied to arbitrary data without modiﬁcations. The drawback is that only a few standard

models have the property of translation invariance. These include some of the DDF models (see Aparicio

et al., 2016) but not BCC or HDF. Therefore, many DEA studies propose procedures of varying complexity

to deal with negative data (e.g., Cheng et al., 2013; Tone et al., 2020).

The general framework for path-based models allows us to survey in one place the properties of all

standard path-based models and also oﬀer certain guidance on how to construct new models with given

properties. The desirable properties analysed in this paper include (a) unique projection point; (b) indication;

monotonicity; (g) super-eﬃciency; and (h) homogeneity. Our results divide the properties into two groups:

the properties that hold universally (resp., universally fail) for every model in the general framework; and

the properties that must be assessed individually for each model. With the help of the general framework,

we show that only the unique projection point property is satisﬁed in all models. Three other properties

(indication, strong eﬃciency of projection, and strict monotonicity) usually fail simultaneously, although

surprisingly there are very special cases where they simultaneously hold. For properties in the second group,

the article provides tools to determine whether a property holds or fails in a speciﬁc model. In particular,

the monotonicity property is satisﬁed by all standard path-based models, but the homogeneity property is

limited to only a speciﬁc type of directions and path functions. We ﬁnd that there are no trade-oﬀs between

the properties of indication, strong eﬃciency, and strict monotonicity on the one hand and the property of

homogeneity on the other.

The paper is organised as follows. Section 2 introduces basic terminology and notation concerning,

among others, the technological set and its eﬃcient frontier over general data; desirable properties of DEA

models; and standard path-based models. Section 3 proposes a general scheme for path-based models,

analyses its basic properties, and discusses its geometry. Section 4 conducts a deeper analysis of the general

scheme in light of ten desirable properties. This section develops practical criteria for each property and

illustrates them on individual standard path-based models. Section 5 extends the above analysis in two

directions: (a) properties of standard models over arbitrary data; (b) construction of new models with good

properties. Section 6 concludes.

2. Preliminaries

Let us establish some notation. Rddenotes the ddimensional Euclidean space, and Rd

+its non-negative

orthant. The lowercase bold letters denote column vectors and the uppercase bold letters matrices. The

superscript Tdenotes the transpose of a column vector or a matrix. The symbol edenotes a vector of ones.

Consider a set of nobserved decision-making units DMUj(j=1,...,n), each consuming minputs xi j

(i=1,...,m) to produce soutputs yr j (r=1,...,s). For each j=1,...,n, the data of the inputs and

outputs of DMUjcan be arranged into the column vectors xj=(x1j,...,xm j)T∈Rmof the inputs and

yj=(y1j,...,ys j)T∈Rsof the outputs. Finally, the input and output vectors of all DMUs form the m×n

input and s×noutput matrices Xand Y, i.e., X=[x1,...,xn] and Y=[y1,...,yn], respectively.

In the article, we reserve the notations i,j, and rfor the indices that go through whole index sets

{1,...,m},{1,...,n}, and {1,...,s}, respectively. We make no assumptions about the non-negativity of the

data at this point. The non-negativity requirement may follow later from other assumptions, and we shall

alert the reader whenever that is the case.

2.1. Technology set

On the basis of the given data, we consider the following technology set:

T=n(x,y)∈Rm×Rs|Xλ ≤x,Y λ ≥y,λ≥0,eTλ=1o,(1)

corresponding to variable returns to scale (VRS). Note that the common non-negativity of (x,y) is not

imposed here. Elements of Twill be called units. It follows from (1) that the closed set Thas a non-empty

interior; we shall denote its boundary by ∂T:=T \ intT.

By (xo,yo) we denote a unit from Tto be currently evaluated. For input vectors, we also use the notation

xmin,xmax,xev,xsd, where for i=1, . . . mwe set xmin

i=minjxi j,xmax

i=maxjxi j,xev

i=1

nPjxi j, and

xsd

i=q1

nPj(xi j −xev

i)2.Here xsd

iis the standard deviation of the i-th input over all DMUj,j=1,...,n.

The notation ymin,ymax,yev, and ysd is introduced analogously for the output vectors. Without loss of

generality, we assume that xmin <xmax and ymin <ymax. Otherwise, there would be components of inputs

/outputs, where all DMUs take the same value, and such components can be excluded from the analysis.

We write (x,y)≿(xo,yo) if the unit (x,y)dominates the unit (xo,yo), that is, if x≤xoand y≥yo. A

unit (x,y)strictly dominates the unit (xo,yo) if x<xoand y>yo. A unit (xo,yo)∈ T is called strongly

eﬃcient if there is no other unit in Tthat dominates (xo,yo), that is, if (x,y)∈ T dominates (xo,yo),

then (x,y)=(xo,yo).5A unit (xo,yo)∈ T is called weakly eﬃcient if there is no unit in Tthat strictly

dominates (xo,yo). Evidently, any strongly eﬃcient unit is weakly eﬃcient, and weakly eﬃcient units lie

on the boundary ∂T.

The converse is also true: every unit on the boundary ∂Tis weakly eﬃcient because the deﬁnition of T

in (1) does not impose the non-negativity assumption on the units therein. Therefore, the boundary ∂Tis

partitioned into the strongly eﬃcient frontier ∂STcontaining all strongly eﬃcient units and the remaining

part ∂WT:=∂T \ ∂ST, which consists of weakly but not strongly eﬃcient units. In this paper, we refer

to the remaining part of the boundary as weakly eﬃcient frontier. One thus has ∂T=∂ST ∪ ∂WTand

∂ST ∩ ∂WT=∅. The simple proof of the next lemma is omitted.

Lemma 1. For (xo,yo)∈ T , the following statements hold.

(a) (xmin,ymax)≿(xo,yo);

(b) (xo,yo)∈∂Tif and only if for all (dx,dy)≥0such that (xo−dx,yo+dy)∈ T , one has (dx,dy)≯0;

5This is the well known Pareto–Koopmans eﬃciency. Some authors call such units Pareto eﬃcient, or fully eﬃcient; see the

discussion in Cooper et al. (2007, p. 45).

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Aunifiedapproachtoradial,hyperbolic,anddirectionalefficiencymeasurementinDataEnvelopmentAnalysisMargar´etaHalick´aa,M´ariaTrnovsk´aa,∗,AleˇsˇCern´ybaFacultyofMathematics,PhysicsandInformatics,ComeniusUniversityinBratislava,Mlynsk´adolina,84248Bratislava,SlovakiabBayesBusinessSchool,City,Universityof...

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A unified approach to radial hyperbolic and directional efficiency measurement in Data Envelopment Analysis_2

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