The DEA is a non-parametric mathematical programming technique that calculates the efficiency frontier and indicates which Decision-Making Units (DMUs) are on the frontier, and which are not. In this analysis, a radial model with windows of two or more periods is used.

The DEA technique identifies a frontier that envelopes the data and that is used to evaluate the performance of all of the entities under analysis. The term Decision-Making Unit (DMU) represents any entity that is to be evaluated as part of a homogeneous collection, that is, all DMUs utilises and produce similar inputs and outputs. The result of this process is a set of performance scores that ranges between zero and unity, with the unity value representing the maximum efficiency. Moreover, the DEA identifies the reasons for inefficiency in each input and output for all DMUs. The set of efficient DMUs serves as a benchmark to improve future performance on inefficient DMUs.

This section begins by presenting two concepts associated with the environmental protection assessment that derive from the application of the DEA methodology to the environment, and which have been proposed by Sueyoshi and Goto (2012a, 2012b, 2012c, 2012d, 2012e, 2014, 2015) to measure Natural and Managerial efficiency of various DMUs.

The first concept refers to "Natural disposability", which indicates that DMUs consider that the input vector must be reduced in order to reduce undesirable outputs, whilst increasing the desirable output vector, if possible (2012a, 2012b, 2012c). The second concept, ‘Managerial disposability’, describes the opposite situation, in which the DMU increases the input vector in order to reduce the undesirable output vector, for which it needs to employ innovative technology to produce such an effect, while increasing the desirable output vector, if possible.

In the DEA literature, the concept of “managerial disposability” or “managerial efficiency” exposes the capacity of a DMU to optimize several outputs on a simultaneous basis (eg, increasing or at least maintaining desirable outputs while minimizing undesirable outputs), while inputs are at least maintained or increased (Exposito and Velasco, 2018). The multiple optimization problem applied in DEA (with desirable and undesirableoutputs) is usually achieved through the introduction of “innovation” into the way that inputs are used to obtain outputs, thereby allowing undesirable outputs to be reduced while still augmenting (or at least maintaining) desirable outputs (Sueyoshi and Goto 2011).

To describe the concepts of Natural and Managerial disposability using an axiomatic expression, X\(\)∈ \({R}_{m}^{+}\)must be considered as the input vector, G ∈ \({R}_{s}^{+}\)as the desirable output vector, and B ∈ \({R}_{h}^{+}\)as the undesirable output vector. All three are column vectors, whose components are all positive.

The concepts of Natural and Managerial disposability are specified by the following production factor vectors, under constant returns to scale (RTS) and constant damages to scale (DTS), respectively, (where DTS is the parallel economic concept to RTS for the case of undesirable outputs):

It is important to note that in the radial approach it is possible to incorporate unified inefficiency scores in the computational framework.

In production economics, there are several formulations of indices (such as the Laspeyres and Paasche price indices) as Törmquist and Fisher indices that analyse which is more appropriate when calculating indices of input and output quantities. Total Factor Productivity (TFP) (Coelli 2005) and the Malmquist Caves, Christensen and Diewert 1982) index link efficiency and productivity using a distance function based on productivity measures. The Malmquist Index examines an occurrence of a frontier shift across multiple periods.The Malmquist-Luenberger (M-L) index, introduced by Chung, Färe,, andGrosskopf (Chung, Färe and Grosskopf 1997), is particularly useful when measuring productivity and environmental performance associated with undesirable outputs. The distance function of the M-L index is characterised by the directional distance function (DDF) (Chambers, Chung and Färe 1996).

In this paper, the desirable and undesirable outputs are incorporated in order to address the environment assessment. It is therefore important to unify desirable and undesirable outputs in measurement indices, such as ecotechnology innovation for pollution reduction.

The concept of Natural and Managerial disposability is analysed on a time horizon through the MI values (Sueyoshi, Goto, Wang 2017).

## 2.1.1. Malmquist Index: Natural Disposability

The Malmquist indexes (MI) allow obtain interperiod changes in relative efficiency to be evaluated. (Sueyoshi et al. 2017). These indexes are estimated for alternative time periods or temporal ‘windows’, which help us to evaluate changes and trends in the medium and long term. This methodology allows for the dynamic assessment of the capacity of each country to achieve specified objectives, compared to the remaining countries

Positive change in the index (MI) under natural disposability signifies potential economic growth.

Firstly, no occurrence of frontier crossover is considered across different periods; that is to say, an efficiency frontier shifts without a frontier crossing occurring between the two periods. The MI between two periods (*z-*th: base and \(t\)–th: specific) can be specified using the following expression [24]:

$${IN}_{z}^{t}=\sqrt{\frac{{\text{U}\text{E}\text{N}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}} \frac{{\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{N}}_{t}^{R}}}$$

1

where \({\text{U}\text{E}\text{N}}_{z}^{R}\) is the Unified Efficiency in the \(z\)-th base period, and \({\text{U}\text{E}\text{N}}_{t}^{R}\) is the Unified Efficiency in the 𝑡-th base period. \({\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}\) is the Inter-temporal Unified Index from the *t*-th period to the \(z\)-th period and \({\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}\) is the Inter-temporal Unified Index from the z-th period to the \(t\)-th period. \({\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}\) and \({\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}\) may become more or less than unity, since they depend on a frontier structure (including a crossover) between the two periods. They are therefore intertemporal unified indicators. They are measured under Natural (N) disposability under constant RTS. Here, no occurrence of the frontier crossover between the two (*z*-th and *t*-th) periods is assumed.

Secondly, the possible occurrence of frontier crossover is considered and the MI may be specified, using the following expression (Sueyoshi, Goto, Wang 2017):

$${INC}_{z}^{t-1\&t}=\sqrt{\frac{{\text{U}\text{E}\text{N}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to t-1\&t}^{R}} \frac{{\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}}}$$

2

where \({\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to t-1\&t}^{R}\) is the Inter-temporal Unified Index from the \(z\)-th period to the \(t-1\&t\)-th period, and \({\text{U}\text{E}\text{N}}_{\text{t}\to t-1\&t}^{R}\) is the Unified Efficiency from the \(t\)-th period to the \(t-1\&t\)-th period.

Frontiers encountered in consecutive periods may intersect for various reasons. These intersections may be due either to low production in subsequent periods or to the implementation of new technologies that elevate environmental efficiency due to time lags in obtaining results.

In order to avoid infeasible solutions on index measurement, it is necessary to assume constant RTS. All efficiency and index measures are obtained by radial measurement under Natural disposability.

Each of the values applied in (1) and (2) are reached by linear programming problems, which are explained below.

*No occurrence of frontier crossover*

These measures are formulated by the following radial models:

The degree of unified efficiency \({\text{U}\text{E}\text{N}}_{t}^{R}\) of the *k-th* DMU in the *t* period \((t=z+1,\dots ,T)\) is measured by the following model under Natural disposability:

(P1)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

where \({d}_{i}^{x-}, 1\le i\le m\), \({d}_{r}^{g}, 1\le r\le s\) and \({d}_{f}^{b}, 1\le f\le h\) are the corresponding slack variables related to inputs, desirable outputs, and undesirable outputs, and these constitute decision variables in (P1). The column vector of unknown variables \({\lambda =(\lambda }_{1},\dots ,{\lambda }_{n})\), called the structural or intensity variables, are considered as decision variables in (P1). The scalar value, \(\xi ,\) unrestricted, represents a unified inefficiency measure. The degree of inefficiency is measured as the distance between an efficiency frontier and an observed vector of desirable and undesirable outputs. The scalar value \({ϵ}_{s}\) is an Archimedean number and indicates the relative importance between the inefficiency measure and the sum of all slack variables, and it is taken as a sufficiently small number. The superscript \(\vartheta\) has the value 0 for the Natural disposability, and \({J}_{t}\) stands for all DMUs in the \(t\)-th period.

Adjustments to the data range, R, in model (P1) are determined by the upper and lower bounds of the production factors as follows:

$${R}_{i}^{x}={(m+s+h)}^{-1}{\left(max\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{r}^{g}={(m+s+h)}^{-1}{\left(max\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{f}^{b}={(m+s+h)}^{-1}{\left(max\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

The degree of \({\text{U}\text{E}\text{N}}_{t}^{R}\) of the *k-th* DMU in the *t* period is measured by:

$${\text{U}\text{E}\text{N}}_{t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

3

The level of unified inefficiency under natural disposability is calculated in Eq. (3) between brackets in \({UEN}_{t}^{R}\) and the unified efficiency is attained by subtracting the level of inefficiency from unity. This can either be less than unity, which shows inefficiency, or be unity, thereby indicating full efficiency.

The degree of \({\text{U}\text{E}\text{N}}_{z}^{R}\), with respect to the k-th DMU in the period \(z\), is measured by replacing \(t\) with \(z\) in the Model (P1), where the superscript (\(*\)) indicates that it is the optimum of (P1).

The degree of \({\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}\) with respect to the \(k\)-th DMU in the \(t\)-th period, projecting from the \(t\)-th period to the \(z\) period, is determined by the following model:

(P2)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{jϵ{J}_{z}}{x}_{ijz} {\lambda }_{jz}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{z}}{g}_{rjz} {\lambda }_{jz}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{z}}{b}_{fjz} {\lambda }_{jz}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jz}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of the \({\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}\) index is measured by:

$${\text{I}\text{U}\text{I}\text{N}}_{\text{t}\to \text{z}}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

4

The degree of \({\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}\), with respect to the \(k\)-th DMU in the \(z\)-th period, projected from the \(z\)-th period to the \(t\)-th period, is determined by the following model:

(P3)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{(-1)}^{\vartheta }{d}_{i}^{x-} ={x}_{ikz }; \forall kϵ{J}_{z} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt}-{d}_{r}^{g} - \xi {g}_{rkz} ={g}_{rkz}; \forall kϵ{J}_{z} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkz}; \forall kϵ{J}_{z} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of the \({\text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}\) index is measured by:

$${ \text{I}\text{U}\text{I}\text{N}}_{\text{z}\to \text{t}}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

5

*Possible occurrence of frontier crossover*

The possible occurrence of frontier crossover can now be considered, and for the MI to obtain it, it is necessary to solve the following two models (P4) and (P5):

The degree of \({\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}\) on the *k*-th DMU in the *t*-th period is measured by the following model:

(P4)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{jϵ{J}_{t-1\&t}}{x}_{ijt-1\&t} {\lambda }_{jt-1\&t}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t-1\&t}}{g}_{rjt-1\&t} {\lambda }_{jt-1\&t}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t-1\&t}}{b}_{fjt-1\&t} {\lambda }_{jt-1\&t}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt-1\&t}\ge 0; j=1,\dots ,n;specific t; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of \({\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}\), with respect to the \(k\)-th DMU in the \(t\)-th period, is determined as follows:

$${\text{U}\text{E}\text{N}}_{t\to t-1\&t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

6

The degree of \({\text{I}\text{U}\text{I}\text{N}}_{z\to t-1\&t}^{R}\) on the k-th DMU in the t-th period is measured by the following model:

(P5)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{jϵ{J}_{t-1\&t}}{x}_{ijt-1\&t} {\lambda }_{jt-1\&t}+{{(-1)}^{\vartheta }d}_{i}^{x-} ={x}_{ikz }; \forall kϵ{J}_{z} ; i=1,\dots ,m$$

$$\sum _{jϵ{J}_{t-1\&t}}{g}_{rjt-1\&t} {\lambda }_{jt-1\&t}-{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkz}; \forall kϵ{J}_{z} ; r=1,\dots ,s$$

$$\sum _{jϵ{J}_{t-1\&t}}{b}_{fjt-1\&t} {\lambda }_{jt-1\&t}+{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkz}; \forall kϵ{J}_{z} ; f=1,\dots ,h$$

$${\lambda }_{jt-1\&t}\ge 0; j=1,\dots ,n;specific t; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x-}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

The degree of \({\text{I}\text{U}\text{I}\text{N}}_{z\to t-1\&t}^{R}\), with respect to the \(k\)-th DMU in the \(z\)-th period, is determined as follows:

$${\text{I}\text{U}\text{I}\text{N}}_{z\to t-1\&t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x-*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

7

## 2.1.2. Malmquist Index: Managerial disposability

If there is a shift in the frontier, then potential progress occurs due to technological development and improved management in the periods used. The alteration of the index (MI) under managerial disposability also implies the potential prevention of industrial pollution.

The MI with a frontier shift between two periods may be represented as follows:

$${IM}_{z}^{t}=\sqrt{\frac{{\text{U}\text{E}\text{M}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{M}}_{\text{z}\to \text{t}}^{R}} \frac{{\text{I}\text{U}\text{I}\text{M}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{M}}_{t}^{R}}}$$

$${IMC}_{z}^{t-1\&t}=\sqrt{\frac{{\text{U}\text{E}\text{M}}_{z}^{R}}{{\text{I}\text{U}\text{I}\text{M}}_{\text{z}\to t-1\&t}^{R}} \frac{{\text{I}\text{U}\text{I}\text{M}}_{\text{t}\to \text{z}}^{R}}{{\text{U}\text{E}\text{M}}_{t\to t-1\&t}^{R}}}$$

It should be noted that, in order to obtain this index (Managerial), it is only necessary to replace the superscript \(\vartheta\), giving the value 1 instead of zero (Natural), and to replace all \({d}_{i}^{x-}\) with \({d}_{i}^{x+}\) in the (P1), (P2), (P3), (P4), and (P5).

These models can easily be generalised for windows of three or more periods (Sueyoshi, Goto, Wang 2017). This paper presents data collected with windows of several periods.

For instance, the following models indicate the levels of potential performance, due to the frontier shift. All data is also pooled into a single balanced panel data, for the purpose of measuring the level of unified inefficiency under Natural disposability, and each *k* DMU in the period \(t\).

(P6)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{{(-1)}^{\vartheta }d}_{i}^{x+} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt} -{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt} +{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x+}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

where\({R}_{i}^{x}={(m+s+h)}^{-1}{\left(max\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}\)

$${R}_{r}^{g}={(m+s+h)}^{-1}{\left(max\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{f}^{b}={(m+s+h)}^{-1}{\left(max\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

where the superscript \(\vartheta\) has the value 1 for the Managerial disposability and \({J}_{t}\) stands for all DMUs in the \(t\)-th period.

The degree of \({\text{U}\text{E}\text{N}\text{T}}_{t}^{R}\) of the *k-th* DMU in the *t* period is measured by the following:

$${\text{U}\text{E}\text{N}\text{T}}_{t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

8

And similarly, the level of unified inefficiency under Managerial disposability and each *k* DMU in the \(t\)-th period.

(P7)\(Max \xi +\epsilon \left[\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b}\right]\)

$$s.t. \sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{x}_{ijt} {\lambda }_{jt}+{{(-1)}^{\vartheta }d}_{i}^{x+} ={x}_{ikt }; \forall kϵ{J}_{t} ; i=1,\dots ,m$$

$$\sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{g}_{rjt} {\lambda }_{jt} -{d}_{r}^{g} - \xi {g}_{rkt} ={g}_{rkt}; \forall kϵ{J}_{t} ; r=1,\dots ,s$$

$$\sum _{t=z}^{T}\sum _{jϵ{J}_{t}}{b}_{fjt} {\lambda }_{jt} +{d}_{f}^{b} + \xi {b}_{fkt} ={b}_{fkt}; \forall kϵ{J}_{t} ; f=1,\dots ,h$$

$${\lambda }_{jt}\ge 0; j=1,\dots ,n;t=2,\dots ,T; \xi \text{U}\text{n}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{c}\text{t}\text{e}\text{d};{d}_{i}^{x+}\ge 0;i=1,\dots ,m$$

$${d}_{r}^{g}\ge 0;r=1,\dots ,s; {d}_{f}^{b}\ge 0; f=1,\dots ,h$$

where\({R}_{i}^{x}={(m+s+h)}^{-1}{\left(max\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{x}_{ij};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}\)

$${R}_{r}^{g}={(m+s+h)}^{-1}{\left(max\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{g}_{rj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

$${R}_{f}^{b}={(m+s+h)}^{-1}{\left(max\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}-min\left\{{b}_{fj};jϵ{J}_{z}\cup {J}_{z+1}\cup \dots {J}_{T}\right\}\right)}^{-1}$$

where the superscript \(\vartheta\) has the value 1 for the Managerial disposability, and \({J}_{t}\) stands for all DMUs in the \(t\)-th period.

The degree of \({\text{U}\text{E}\text{M}\text{T}}_{t}^{R}\) of the *k-th* DMU in the *t-th* period is measured by:

$${\text{U}\text{E}\text{M}\text{T}}_{t}^{R}=1-\left[{\xi }^{*}+\epsilon \left(\sum _{i=1}^{m}{R}_{i}^{x}{d}_{i}^{x+*}+\sum _{r=1}^{s}{R}_{r}^{g}{d}_{r}^{g*}+\sum _{f=1}^{h}{R}_{f}^{b}{d}_{f}^{b*}\right)\right]$$

9

It is interesting to note that the combination of Malmquist indices under natural and managerial disposability provides a total potential on how to achieve the optimal level of sustainability under these two different concepts of disposability.