On indication, strict monotonicity, and efficiency of projections
in a general class of path-based data envelopment analysis models

Consider a set of $n$ decision-making units ${\mathrm{DMU}}_{j}$ ($j=1,\dots,n$) with observed input–output vectors $(\bm{x}_{j},\bm{y}_{j})\in\mathbb{R}^{m}\times\mathbb{R}^{s}$. The input–output data of $(\mathrm{DMU}_{j})_{j=1}^{n}$ are arranged into the $m\times n$ input and $s\times n$ output matrices $\bm{X}=[\bm{x}_{1},\dots,\bm{x}_{n}]$ and $\bm{Y}=[\bm{y}_{1},\dots,\bm{y}_{n}]$, respectively. No assumption about the non-negativity of the data is made at this point. The non-negativity requirement may follow later from other assumptions placed on the models.

Based on the given data $\bm{X},\bm{Y}$ we consider the technology set

corresponding to variable returns to scale (VRS). Note that the common non-negativity of $(\bm{x},\bm{y})$ is not imposed here. It follows from (1) that the set $\mathcal{T}$ is closed, has a non-empty interior (denoted $\operatorname{int}\mathcal{T}$) and its boundary $\partial\mathcal{T}$ satisfies $\partial\mathcal{T}=\mathcal{T}\setminus\text{int}\mathcal{T}$. Elements of ${\mathcal{T}}$ will be called units. By $(\bm{x}_{o},\bm{y}_{o})$ we denote a unit from ${\mathcal{T}}$ to be evaluated.

The point $(\bm{x}^{\mathrm{min}},\bm{y}^{\mathrm{max}})\in\mathbb{R}^{m}\times\mathbb{R}^{s}$, with elements $x^{\mathrm{min}}_{i}=\min_{j}x_{ij}$, for $i=1,\dots m$ and $y^{\mathrm{max}}_{r}=\max_{j}x_{rj}$ for $r=1,\dots s$, respectively, is called the ideal point of $\mathcal{T}$ in DEA (see, e.g., Portela et al., 2004). The ideal point typically does not belong to $\mathcal{T}$, in which case it dominates every unit in $\mathcal{T}$. The technology set, for which $(\bm{x}^{\mathrm{min}},\bm{y}^{\mathrm{max}})\in\mathcal{T}$, will be called trivial. Clearly, a trivial technology is an affine transformation of a non-negative orthant.

A unit $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$ is called strongly efficient³³3This 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). if no other unit in $\mathcal{T}$ dominates $(\bm{x}_{o},\bm{y}_{o})$, i.e., if the property that $(\bm{x},\bm{y})\in\mathcal{T}$ dominates $(\bm{x}_{o},\bm{y}_{o})$ yields $(\bm{x},\bm{y})=(\bm{x}_{o},\bm{y}_{o})$. A unit $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$ is called weakly efficient if there is no unit in $\mathcal{T}$ that strictly dominates $(\bm{x}_{o},\bm{y}_{o})$.

Evidently, any strongly efficient unit is weakly efficient, and the weakly efficient units lie on the boundary of the technology set. The converse is also true: every unit on the boundary $\partial\mathcal{T}$ is weakly efficient because the definition of $\mathcal{T}$ in (1) does not impose the non-negativity assumption on the units therein. The boundary $\partial\mathcal{T}$ is thus uniquely partitioned into the strongly efficient frontier $\partial^{S}\mathcal{T}$ containing all strongly efficient units and the remaining part $\partial^{W}\mathcal{T}:=\partial\mathcal{T}\setminus\partial^{S}\mathcal{T}$, which consists of the weakly but not strongly efficient units. In this paper we will refer to the remaining part of the boundary as the weakly efficient frontier. Thus, we have $\partial\mathcal{T}=\partial^{S}\mathcal{T}\cup\partial^{W}\mathcal{T}$ and $\partial^{S}\mathcal{T}\cap\partial^{W}\mathcal{T}=\emptyset$.

We now recall the general scheme (GS) for path-based models from Halická et al. (2024). The scheme depends on both the choice of a prescription $\bm{g}$ that defines the directional vector $\bm{g}_{o}=(\bm{g}_{o}^{x},\bm{g}_{o}^{y})\gneqq 0$ for each $(\bm{x}_{o},\bm{y}_{o})$, and the choice of real functions $\psi^{x}$ and $\psi^{y}$ that together with their domains ($\operatorname{dom}$) and images ($\operatorname{im}$) satisfy the following assumptions:

1. (A1)

   $\operatorname{dom}(\psi^{x})=(a^{x},\infty)$ with $a^{x}\in\{-\infty,0\}$ and $\operatorname{dom}(\psi^{y})=(a^{y},\infty)$ with $a^{y}\in\{-\infty,0\}$;

2. (A2)

   $\psi^{x}$ is smooth, concave, increasing and $\psi^{y}$ is smooth, convex, decreasing;

3. (A3)

   $\psi^{x}(1)=\psi^{y}(1)=1$;

4. (A4)

   $\text{im}(\psi^{x})=(b^{x},\infty)$ with $b^{x}=-\infty$ if $\bm{g}^{y}=\bm{0}$ and $b^{x}\in\{-\infty,0\}$ otherwise; $\operatorname{im}(\psi^{y})=(b^{y},\infty)$ with $b^{y}\in\{-\infty,0\}$.

The scheme is built on the technology set $\mathcal{T}$ specified in (1) and is consequently also dependent on the matrices $\bm{X},\bm{Y}$. Any choices of $(\bm{X},\bm{Y},\bm{g},\psi^{x},\psi^{y})$ that satisfy the requirements mentioned above will be referred to as admissible parameters for a model in the GS scheme.

For a fixed choice of admissible parameters $(\bm{X},\bm{Y},\bm{g},\psi^{x},\psi^{y})$, the (path-based) GS model for assessment of $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$ is defined by

The right-hand sides of (2b) and (2c), denoted by

define a smooth path $\bm{\phi}_{o}(\theta)=(\bm{\phi}_{o}^{x}(\theta),\bm{\phi}_{o}^{y}(\theta))$ in the input-output space $\mathbb{R}^{m}\times\mathbb{R}^{s}$ parametrized by $\theta\in{\cal{D}}=\operatorname{dom}(\psi^{x})\cap\operatorname{dom}(\psi^{y})$.

For any choice of admissible parameters and each $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$, the well-definedness of the programme $(GS)_{o}$ and other useful properties of the path $\bm{\phi}_{o}$ are established in Theorem 3.1 of Halická et al. (2024). According to this theorem, the path $\bm{\phi}_{o}$ passes through the point $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$ at $\theta=1$ (i.e., $\bm{\phi}_{o}(1)=(\bm{x}_{o},\bm{y}_{o})$), and for decreasing values of $\theta$ it moves towards the boundary of $T$ gradually passing through points that dominate one another. This property of the path can be formally expressed as $\bm{\phi}_{o}(\theta_{1})\succnsim\bm{\phi}_{o}(\theta_{2})$ for $\theta_{1}<\theta_{2}$, and we will refer to it as the monotonicity of the path with respect to $\theta$. Finally, the path leaves $\mathcal{T}$ at some $\bm{\phi}_{o}(\theta^{*}_{o})\in\partial\mathcal{T}$, where $\theta^{*}_{o}\leq 1$, and $\theta^{*}_{o}$ is the optimal value in $(GS)_{o}$. The optimal value $\theta^{*}_{o}$ is called the efficiency score, or alternatively, the value of the efficiency measure for $(\bm{x}_{o},\bm{y}_{o})$. The point $(\bm{\phi}_{o}^{x}(\theta_{o}^{*}),\bm{\phi}_{o}^{y}(\theta_{o}^{*}))$ on the path $\bm{\phi}_{o}$ is called the projection of $(\bm{x}_{o},\bm{y}_{o})$ in the GS model.

It is easy to see that the well-known BCC input and output models (Banker et al., 1984), the hyperbolic distance function model (HDF) (Färe et al., 1985) as well as the generic directional distance function model (DDF) (Chambers et al., 1996, 1998) can be equivalently rewritten in the form of the general scheme (2). The scheme also includes the so-called generalised distance function model (GDF) introduced by Chavas and Cox (1999). These models, taken in conjunction with the usual assumptions on the positiveness of the data, will be called the standard path-based models. The corresponding parameterizations are shown in Table 1.

The choice $g^{x}_{o}=0$ or $g^{y}_{o}=0$ for all $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$ leads to output or input oriented models, respectively. Among the standard path-based models, only DDF is formulated with general directional vectors. The BCC as well as the HDF use $g^{x}_{o}=\bm{x}_{o}$ and / or $g^{y}_{o}=\bm{y}_{o}$. A generalisation of HDF towards general directions is introduced in Halická et al. (2024).

In Halická et al. (2024), the GS models are analysed in light of ten desired properties. Three of these merit further investigation. First, we recall the precise definitions of Halická et al. (2024) abridged to suit our needs here.

• (ID)

  Identification of strong efficiency. If for a given $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$ one has $\theta_{o}^{*}=1$, then $(\bm{x}_{o},\bm{y}_{o})\in\partial^{S}\mathcal{T}$.

• (PR)

  Strong efficiency of projections. One has $\phi_{o}(\theta_{o}^{*})\in\partial^{S}\mathcal{T}$ for each $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$.

• (MO)

  Strict monotonicity. If $(\bm{x}_{o},\bm{y}_{o})\succnsim(\bm{x}_{p},\bm{y}_{p})$ for some $(\bm{x}_{o},\bm{y}_{o}),(\bm{x}_{p},\bm{y}_{p})\in\mathcal{T}$, then $\theta_{o}^{*}<\theta^{*}_{p}$.

Let us briefly revisit the reasons for studying this triplet in more detail. The findings of Halická et al. (2024) show that none of the three properties are satisfied in the entire class of GS models. Moreover, there are examples of admissible model parameters, where all three properties are met, but it also appears that in most models the three properties fail simultaneously.

The relationships among the three properties for a particular GS model with a fixed set of admissible model parameters have been partially explored in Halická et al. (2024). Since (ID) is a restriction of (PR) to the weakly efficient frontier, it is easy to see that (PR) $\Rightarrow$ (ID), or equivalently,

NOT (ID) occurs if and only if there is a unit on the weakly efficient frontier whose score is 1. Hence Halická et al. (2024, Theorem 4.20) further yields

Moreover, NOT (ID) arises whenever a unit on the weakly efficient frontier is associated with a positive direction (Halická et al., 2024, Theorem 3). The hierarchy of these properties, formulated by contra-position, is summarised in Figure 1.

It remains to establish the link between (PR) and (MO) and find examples, if any, where (ID) holds but (PR) and/or (MO) do not. This will be the subject of Section 3 with the main findings summarised in Figure 3.

A natural next question is what selection of admissible parameters leads to the success or failure of these characteristics. To this end, illustrative examples in Halická et al. (2024) indicate that the success could be connected with a special configuration of the data $\bm{X},\bm{Y}$ defining $\mathcal{T}$ together with a special choice of unit-dependent directions based on Portela et al.’s range directions

suitably modified to fit the GS scheme. This topic is addressed in Section 4.

The next example shows that (ID) may hold while (PR) fails for a certain selection of admissible model parameters.

Until now, we have considered the right-hand side of (2b) and (2c) as a (vector) function of $\theta$ for a fixed $(\bm{x}_{o},\bm{y}_{o})$, which yields a path passing through the assessed unit $(\bm{x}_{o},\bm{y}_{o})\in\mathcal{T}$; see (3). To obtain a link between (MO) and (PR), it is helpful to consider the right-hand sides of (2b) and (2c) as functions of the assessed unit $(\bm{x}_{o},\bm{y}_{o})$ for a fixed $\theta$. Observe that the right-hand sides in (3) depend explicitly on $(\bm{x}_{o},\bm{y}_{o})$ but the dependence may also be implicit via the directions $\bm{g}_{o}=(\bm{g}_{o}^{x},\bm{g}_{o}^{y})$.

Observe that Theorem 3.4 formulates a local property: if a point in $\mathcal{T}$ is not projected onto $\partial^{S}\mathcal{T}$, then one can find another point such that the pair fails to maintain strict monotonicity. Thus, the “local” formulation offers more flexibility than the following “global” corollary.

The monotonicity of GS models is related to the monotonicity of the path-flow mapping $\bm{\phi}(\bm{x}_{o},\bm{y}_{o};\theta)$, which we formalise next.

The next lemma will be useful for further analysis of strict monotonicity.

Let us note that the assumption of monotonicity of the path-flow mapping ensures the property of the so-called weak monotonicity of each model in the GS scheme as indicated by Lemma 3.9. To establish the strict monotonicity (MO) we will need the strict monotonicity of the path-flow mapping and property (PR). First, we formulate a local version of the assertion.

Theorem 3.10, too, is formulated locally, i.e., if two ordered units are both projected onto $\partial^{S}\mathcal{T}$, then their efficiency scores are strictly ordered. The local formulation once again offers more flexibility than the following global corollary.

The results of this section are summarised in Figure 3.

As shown in Figure 4, the paths in Example 4.1 connect units from $\mathcal{T}$ to the ideal point $(\bm{x}^{\mathrm{min}},\bm{y}^{\mathrm{max}})$, which ensures that the units are projected onto the strongly efficient boundary. Inspired by this example, we now characterise those directions $g_{o}$ for which the corresponding path $\bm{\phi}_{o}$ for $(\bm{x}_{o},\bm{y}_{o})$ passes through the ideal point $(\bm{x}^{\mathrm{min}},\bm{y}^{\mathrm{max}})$ at an arbitrarily chosen $\theta_{\mathrm{min}}\in[0,1)$. The choice of common $\theta_{\mathrm{min}}$ for all units in $\mathcal{T}$ then allows us to obtain comparable scores whose common lower bound is $\theta_{\mathrm{min}}$. The proof follows by a simple calculation and is therefore omitted.

Recall that $(\bm{x}^{\mathrm{min}},\bm{y}^{\mathrm{max}})$ denotes the ideal point of $\mathcal{T}$ and that for fixed $\psi^{x}$ and $\psi^{y}$, the path $\phi_{o}$ is determined by the evaluated unit $(x_{o},y_{o})$ and the direction $\bm{g}_{o}=(\bm{g}_{o}^{x},\bm{g}_{o}^{y})$ as detailed in Subsection 2.2.

We will call the direction vector defined by (13) the GS range direction.

In the case of DDF-g models, we have $\psi^{x}(\theta)=\theta$, $\psi^{y}(\theta)=2-\theta$, and $\mathcal{D}=\mathbb{R}$. On selecting $\theta_{\mathrm{min}}=0$, the GS range directions coincide with the range directions (4) of Portela et al. (2004). In the case of HDF-g models, we have $\psi^{x}(\theta)=\theta$, $\psi^{y}(\theta)=\frac{1}{\theta}$, and $\mathcal{D}=(0,\infty)$, hence $\theta_{\mathrm{min}}$ must be chosen positive. The value $\theta_{\mathrm{min}}=1/2$ yields $\bm{g}^{x}_{o}=2(\bm{x}_{o}-\bm{x}_{\mathrm{min}})$ and $\bm{g}^{y}_{o}=\bm{y}^{\mathrm{max}}-\bm{y}_{o}$.

The next lemma shows that the choice of the GS range direction ensures that both $\bm{\phi}_{o}^{x}$ and $\bm{\phi}_{o}^{y}$, the $x$ and $y$ components of the path $\bm{\phi}_{o}$, passes through the relative interior of the line segments connecting $\bm{x}_{o}$ with $\bm{x}^{\mathrm{min}}$ and $\bm{y}_{o}$ with $\bm{y}^{\mathrm{max}}$, respectively.

Figure 4 illustrates that the paths containing the ideal point of a given technology set project all units onto the strongly efficient frontier. This property, whilst true in two-dimensional technology sets, no longer holds for all higher-dimensional technologies. Indeed, the two-input, one-output example in Halická et al. (2024, Example 2 and Figure 3) shows that the DDF-g model with the GS range directions projects a unit from $\partial^{W}\mathcal{T}$ onto itself. Halická et al. (2024) conducted further numerical experiments on real data consisting of 30 units, 2 inputs, and 2 outputs, with some negative output values. Their analysis contrasts the so-called proportional directions $\bm{g}_{o}^{x}=|\bm{x}_{o}|$, $\bm{g}_{o}^{y}=|\bm{y}_{o}|$ of Kerstens and Van de Woestyne (2011) with the GS range directions for four different choices of the path function $\psi$. In their analysis, the GS range directions yield a significantly smaller proportion of units not projected onto the strongly efficient frontier but do not eliminate such units completely. Therefore, even though the GS range directions ensure the passage of all paths through the ideal point, this alone is not enough to guarantee that units are projected onto the strongly efficient frontier in the case of higher-dimensional technology sets.

It is obvious that to ensure the fulfilment of (PR) using GS range directions, it will be necessary to limit ourselves to a certain subclass of technology sets. To specify the properties of the subclass, it turned out that the facial structure of the technology as a polyhedral set will play an important role. Some characteristics of the facial structure of (10), as well as all necessary terms, are included in Appendix A. The results of the current section will finally show that the GS range directions will guarantee projecting onto strongly efficient frontier if and only if we restrict ourselves to technology sets specified in the following definition.

It is easy to see that the technology sets (1) with only one input and one output, as in Example 4.1, are ideal. A trivial example of an ideal technology in any dimension is the trivial technology, i.e., the technology containing its ideal point $(\bm{x}^{\mathrm{min}},\bm{y}^{\mathrm{max}})$. The trivial technology is a polyhedral cone with the vertex at the ideal point. Its facets — faces of full dimension — are parallel to the orthant hyperplanes. The vertex is the only strongly efficient unit of the trivial technology set.

We now provide examples of ideal and non-ideal technology sets.