Implications for DEA
The world energy trend discussed in the preceding sections indicates that primary energy resources serve as inputs to produce desirable outputs (e.g., electricity) as secondary energy. In the perspective, the primary energy sources are classified into fossil and nonfossil energy categories. DEA formulations used in performance assessment for energy and environment need to be classified into the two groups. Such a group classification is based upon the fact that fossil energy sources produce Green House Gas (GHG) emissions, while nonfossil energy sources do not produce the emissions when they are used for power generation.
There are four implications for DEA development for energy and environmental assessment.
Output classification for fossil fuels
First, it is necessary to separate outputs into desirable and undesirable categories. The desirable outputs include an amount of sale and an amount of electricity, while the undesirable outputs include an amount of various GHG emissions. For example, power generations by fossil fuels produce not only a desirable output (e.g., electricity) but also an undesirable output (e.g., CO_{2}). Thus, the outputs should be classified into the two categories: desirable and undesirable outputs.
Input classification for nonfossil fuels
Second, the energy classification indicates that it is necessary for DEA to classify inputs into two categories, which could not found in a conventional use of DEA. For example, in examining the performance assessment on renewable energy sources (e.g., solar photovoltaic, wind and water power generations), inputs need to be classed into controllable variable (e.g., operational cost) and uncontrollably variables (e.g., temperature) related to weather. It is clear that solar photovoltaic power can generate any power during night and a limited amount of power during a raining season. It depends upon a weather condition, so being uncontrollable.
Direction of an input vector
Third, the world population has increased, and it is expected to reach to 11.2 billion in the year 2100.^{Footnote 1} Along with the population increase, DEA applied to energy and environmental assessment needs to increase the direction of an input vector, or energy resources, until the increase can reach to an efficiency frontier shaped by undesirable outputs. The frontier may serve as an upper limit on the increase in an input vector. The methodological implication is inconsistent with a conventional use of DEA where an input vector should decrease or maintain a current level for efficiency enhancement.
Technology development
Finally, science development and technology innovation make it possible to increase the world population. Thus, an economic growth, supported by science and technology, is essential for sustainability development in the world. Therefore, it is necessary for DEA environmental assessment to consider such technology innovation in the proposed performance assessment.
Formulations
Formulations for fossil energy
To describe the formulations for performance assessment related to fossil and nonfossil fuels, let us consider \(X \in R_{ + }^{m}\) as a controllable input vector with m components, \(Y \in R_{ + }^{z}\) as an uncontrollable input vector with z components, \(G \in R_{ + }^{s}\) as a desirable output vector with s components and \(B \in R_{ + }^{h}\) as an undesirable output vector with h components. In these column vectors, the subscript (j) is used to stand for the jth DMU, whose vector components are strictly positive.
To discuss formulations for this type of performance assessment, we need to separate outputs into desirable and undesirable categories, as mentioned previously, because this type of energy produces CO_{2} and other types of GHG emissions. The importance of the fossil energy is that it can serve as a base load.
Production and pollution possibility sets are axiomatically specified as follows:
$$\begin{aligned} P_{v}^{N} (X) & = \left\{ {(G,B) :\;G \le \sum\limits_{j = 1}^{n} {G_{j} \lambda_{j} } , \, B \ge \sum\limits_{j = 1}^{n} {B_{j} \lambda_{j} } , \, X \ge \sum\limits_{j = 1}^{n} {X_{j} \lambda_{j} } ,\sum\limits_{j = 1}^{n} {\lambda_{j} = 1} \;\& \;\lambda_{j} \ge 0\, \, (j \, = \, 1, \ldots ,n)} \right\}\;{\text{and}} \\ P_{v}^{M} (X) & = \left\{ {(G,B) :\;G \le \sum\limits_{j = 1}^{n} {G_{j} \lambda_{j} } , \, B \ge \sum\limits_{j = 1}^{n} {B_{j} \lambda_{j} } , \, X \le \sum\limits_{j = 1}^{n} {X_{j} \lambda_{j} } ,\sum\limits_{j = 1}^{n} {\lambda_{j} = 1} \;\& \;\lambda_{j} \ge 0\, \, (j \, = \, 1,..,n)} \right\}. \\ \end{aligned}$$
(1)
\(P_{v}^{N} (X)\) stands for a production and pollution possibility set under natural (N) disposability. Meanwhile, \(P_{v}^{M} (X)\) is that of managerial disposability. The subscript (v) stands for “variable” RTS, where it stands for returns to scale, because the constraint \((\sum\nolimits_{j = 1}^{n} {\lambda_{j} = 1} )\) is incorporated into the two axiomatic expressions. See Sueyoshi and Goto (2013a) for a detailed description on RTS.
The difference between the two disposability concepts is that the production technology under natural disposability, or \(P_{v}^{N} (X),\) has \(X \ge \sum\nolimits_{j = 1}^{n} {X_{j} \lambda_{j} }\) in Eq. (1), implying that a DMU can attain an efficiency frontier by reducing a directional vector of inputs. Meanwhile, that of the managerial disposability, or \(P_{v}^{M} (X),\) has \(X \le \sum\nolimits_{j = 1}^{n} {X_{j} \lambda_{j} }\) in Eq. (1), implying that a DMU, where it stands for decisionmaking unit, can attain a status of an efficiency frontier by increasing a directional vector of inputs. Meanwhile, a common feature of the two disposability concepts is that both have \(G \le \sum\nolimits_{j = 1}^{n} {G_{j} \lambda_{j} }\) and \(B \ge \sum\nolimits_{j = 1}^{n} {B_{j} \lambda_{j} }\) in their axiomatic expressions. These conditions intuitively appeal to us because an efficiency frontier for desirable outputs should locate above or on all observations on DMUs, while that of undesirable outputs should locate below or on these observations. See Sueyoshi and Goto (2012a, d, 2014a, c, 2015a, b, 2017) on a detailed description on the two disposability concepts.
Here, it is necessary to discuss that an input vector is usually assumed to project toward a decreasing direction in the previous research efforts on DEA as discussed in Sect. 5.1. The assumption, widely believed by many authors in the previous studies, is often inconsistent with the reality related to environmental protection. For example, many governments and firms consider an increase in input resources to yield an annual “growth” of a desirable output(s). Thus, the conventional framework of DEA is not consistent with the economic concept, or “economic growth,” because the previous DEA studies have implicitly assumed the minimization of total production cost. The cost concept may be acceptable for performance analysis under “economic recession” or “stagnation,” but not in many cases where industrial planning and corporate strategy are based upon their economic growths. Thus, it is easily imagined that DEA applied to energy and environment, as discussed here, is conceptually and practically different from a conventional use of DEA. The cost concept for guiding public and private entities in their strategy developments is average cost (under constant RTS) or marginal cost (under variable RTS), not the total cost, anymore. Furthermore, an opportunity cost, originated from business risk due to industrial pollutions and the other types of various environmental problems (e.g., the nuclear power plant accident at Fukushima Daiichi in Japan), has a major role in modern corporate governance issues. Such cost concepts for current policy making and modern business are implicitly incorporated in formulating the two disposability concepts, in particular in managerial disposability.
The following radial model to measure the level of unified efficiency on the kth DMU under natural disposability (e.g., Sueyoshi and Goto 2012e) is as follows:
$$\begin{array}{*{20}l} {\text{Maximize}} \hfill & {\xi + \varepsilon_{s} \left( {\sum\limits_{i = 1}^{m} {R_{i}^{x} d_{i}^{x  } } + \sum\limits_{r = 1}^{s} {R_{r}^{g} d_{r}^{g} } + \sum\limits_{f = 1}^{h} {R_{f}^{b} d_{f}^{b} } } \right)} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\sum\limits_{j = 1}^{n} {x_{ij} \lambda_{j} } + d_{i}^{x  } = x_{ik} \qquad \qquad\, (i \, = \, 1, \, \ldots \, , \, m),} \hfill \\ \hfill & {\sum\limits_{j = 1}^{n} {g_{rj} \lambda_{j} }  d_{r}^{g}  \xi g_{rk} = g_{rk} \quad \quad\, (r \, = \, 1 \, , \, \ldots \, , \, s)} \hfill \\ \hfill & {\sum\limits_{j = 1}^{n} {b_{fj} \lambda_{j} } \, + d_{f}^{b} + \, \xi b_{fk} = b_{fk}\quad \ \ \, (f \, = \, 1 \, , \, \ldots \, , \, h),} \hfill \\ \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } = 1,} \hfill \\ \hfill & {\lambda_{j} \ge 0 \, \,(j = 1, \, \ldots \, ,n), \, \xi :\;{\text{URS}}, \, d_{i}^{x  } \ge 0 \, \,(i = 1, \ldots ,m),} \hfill \\ \hfill & {d_{r}^{g} \ge 0 \, \,(r = 1, \ldots ,s) \, \& \, d_{f}^{b} \ge 0 \, \,(f = 1, \ldots ,h).} \hfill \\ \end{array}$$
(2)
Here, \(\xi\) stands for an inefficiency score of the specific kth DMU. The scalar value, listed by \(\lambda_{j}\) (j = 1, …, n), stands for the jth intensive (structural) variable. As a result of the incorporation, the surface of a production possibility set is shaped by a convex polyhedral cone under variable Returns to Scale (RTS). All slack variables are expressed by \(d_{i}^{x} \, (i = 1, \ldots ,m),\,d_{f}^{z} \, (f = 1, \ldots ,h)\) and \(d_{r}^{g} \, (r = 1, \ldots ,s)\), and \(\varepsilon_{s}\) is a prescribed very small number.
An important feature of Model (2) is that production factors are adjusted by these data ranges in the objective function. The data range adjustments are determined by the upper and lower bounds on inputs and those of desirable and undesirable outputs in the following manner:

(a)
\(R_{i}^{x} = (m + s + h)^{  1} \left( {\hbox{max} \, \left\{ {\left. {x_{ij} } \rightj = 1, \ldots ,n \, } \right\}  \hbox{min} \, \left\{ {\left. {x_{ij} } \rightj = 1, \ldots ,n \, } \right\}} \right)^{  1}\): a data range adjustment related to the ith input (i = 1,.., m),

(b)
\(R_{r}^{g} = (m + s + h)^{  1} \left( {\hbox{max} \, \left\{ {\left. {g_{rj} } \rightj = 1, \ldots ,n \, } \right\}  \hbox{min} \, \left\{ {\left. {g_{rj} } \rightj = 1, \ldots ,n \, } \right\}} \right)^{  1}\): a data range adjustment related to the rth desirable output (r = 1, …, s) and

(c)
\(R_{f}^{b} = (m + s + h)^{  1} \left( {\hbox{max} \, \left\{ {\left. {b_{fj} } \rightj = 1, \ldots ,n \, } \right\}  \hbox{min} \, \left\{ {\left. {b_{fj} } \rightj = 1, \ldots ,n \, } \right\}} \right)^{  1}\): a data range adjustment related to the fth undesirable output (f = 1, …, h).
A unified efficiency score \(\it ({\text{UEN}}_{v}^{R} )\) of the kth DMU under natural disposability is measured by
$$\it {\text{UEN}}_{v}^{R} = 1  \left[ {\xi^{*} + \varepsilon_{s} \left( {\sum\limits_{i = 1}^{m} {R_{i}^{x} d_{i}^{x  *} } + \sum\limits_{r = 1}^{s} {R_{r}^{g} d_{r}^{g*} } + \sum\limits_{f = 1}^{h} {R_{f}^{b} d_{f}^{b*} } } \right)} \right],$$
(3)
where the inefficiency score and all slack variables are determined on the optimality of Model (3). Thus, the equation within the parenthesis is obtained from the optimality of Model (3).
Shifting our research interest from natural disposability to managerial disposability (M), where the first priority is environmental performance and the second priority is operational performance, this chapter utilizes the following radial model that measures the unified efficiency of the kth DMU under managerial disposability (e.g., Sueyoshi and Goto 2012e):
$$\begin{array}{*{20}l} {\text{Maximize}} \hfill & {\xi + \varepsilon_{s} \left( {\sum\limits_{i = 1}^{m} {R_{i}^{x} d_{i}^{x + } } + \sum\limits_{r = 1}^{s} {R_{r}^{g} d_{r}^{g} } + \sum\limits_{f = 1}^{h} {R_{f}^{b} d_{f}^{b} } } \right)} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\sum\limits_{j = 1}^{n} {x_{ij} \lambda_{j} }  d_{i}^{x + } \, = x_{ik} \qquad \qquad \ \, (i \, = \, 1, \, \ldots \, , \, m), \, } \hfill \\ \hfill & {\sum\limits_{j = 1}^{n} {g_{rj} \lambda_{j} } \,  d_{r}^{g}  \xi g_{rk} = g_{rk} \qquad \, (r \, = \, 1 \, , \, \ldots \, , \, s)} \hfill \\ \hfill & {\sum\limits_{j = 1}^{n} {b_{fj} \lambda_{j} } + d_{f}^{b} + \, \xi b_{fk} = b_{fk} \, \qquad (f \, = \, 1 \, , \ldots \, , \, h),} \hfill \\ \hfill & {\sum\limits_{j = 1}^{n} {\lambda_{j} } = 1,} \hfill \\ \hfill & {\lambda_{j} \ge 0 \, (j = 1, \, \ldots \, , \, n), \, \xi :\;{\text{URS}}, \, d_{i \, }^{x + } \ge 0 \, (i = 1, \ldots ,m),} \hfill \\ \hfill & {d_{r}^{g} \ge 0 \, (r = 1, \ldots ,s) \, \& \, d_{f}^{b} \ge 0 \, (f = 1, \ldots ,h).} \hfill \\ \end{array}$$
(4)
An important feature of Model (4) is that it changes \(+ d_{i}^{x  }\) of Model (4) to \( d_{i}^{x + }\) in order to attain the status of managerial disposability. A unified efficiency score \(\it ({\text{UEM}}_{v}^{R} )\) on the kth DMU under managerial disposability is measured by
$$\it {\text{UEM}}_{v}^{R} = 1  \left[ {\xi^{*} + \varepsilon \left( {\sum\limits_{i = 1}^{m} {R_{i}^{x} d_{i}^{x + *} } + \sum\limits_{r = 1}^{s} {R_{r}^{g} d_{r}^{g*} } + \sum\limits_{f = 1}^{h} {R_{f}^{b} d_{f}^{b*} } } \right)} \right],$$
(5)
where the inefficiency score and all slacks are determined on the optimality of Model (4). Thus, the equation within the parenthesis, obtained from the optimality of Model (4), indicates the level of unified inefficiency under managerial disposability. The unified efficiency is obtained by subtracting the level of inefficiency from unity.
For nonfossil energy
Nonfossil energy sources do not produce GHGs for power generation. Thus, policy makers and individuals, who are interested in green energy, pay serious attention to the development of nonfossil energy sources such as solar photovoltaic and wind power stations. Except nuclear generation, the other nonfossil energy sources depend upon a time and a weather condition. The nuclear generation does not depend upon such an uncontrollable condition (e.g., weather), so being able to serve as a base load. However, it is widely known that the generation produces a nuclear waste.
The following mathematical structure under radial measurement can identify an efficiency score (\(\theta\)) of the specific kth DMU (Wang and Sueyoshi 2017):
$$\begin{aligned} & {\text{Minimize}}\quad \theta  \varepsilon _{s} \left( {\sum\limits_{{i = 1}}^{m} {R_{i}^{x} d_{i}^{{x  }} } + \sum\limits_{{p = 1}}^{z} {R_{p}^{y} d_{p}^{y} } + \sum\limits_{{f = 1}}^{h} {R_{f}^{b} d_{f}^{b} } } \right) \\ & \quad {\text{s}}.{\text{t}}.\quad  \sum\limits_{{j = 1}}^{n} {x_{{ij}} \lambda _{j} }  d_{i}^{{x  }} + \theta x_{{ik}} = 0\qquad (i = 1, \ldots ,m), \\ & \quad \quad \quad \quad \sum\limits_{{j = 1}}^{n} {y_{{pj}} \lambda _{j} }  d_{p}^{y} = {\text{ y}}_{{pk}} \quad \qquad \qquad(p = 1, \ldots ,z), \\ & \quad \quad \quad \quad \sum\limits_{{j = 1}}^{n} {g_{{rj}} \lambda _{j} }  d_{r}^{g} = g_{{rk}} \qquad \qquad \quad \ \ (r = 1, \ldots ,s), \\ & \quad \quad \quad \quad \sum\limits_{{j = 1}}^{n} {\lambda _{j} } = 1, \\ & \quad \quad \quad \quad \theta :{\text{URS, }}\lambda _{j} \ge 0(j = 1, \ldots ,n),\quad d_{i}^{x} \ge 0 (i = 1, \ldots ,m), \\ & \quad \quad \quad \quad d_{p}^{y} \ge 0 (p = 1, \ldots ,z)\quad \& \quad {\text{d}}_{{\text{r}}}^{{\text{g}}} \ge {\text{0}}({\text{r = 1}}, \ldots ,{\text{s}}). \\ \end{aligned}$$
(6)
where Model (6) incorporates the side constraint (\(\sum\nolimits_{j = 1}^{n} {\lambda_{j} } = 1\)) in the formulation. In Model (6), all slack variables are expressed by \(d_{i}^{x  } \, (i = 1, \ldots ,m),\,d_{p}^{y} \, (p = 1, \ldots ,z)\) and \(d_{r}^{g} \, (r = 1, \ldots ,s)\) in Model (6). The slacks are associated with theserelated data adjustment ranges. It is important to note that undesirable outputs are excluded from Model (6). The data range for \(R_{p}^{y} ,\) newly incorporated in Model (6), can be specified by its upper and lower bounds on data as discussed previously.
It is important to note that Model (6) is formulated under natural disposability, so being part of a conventional DEA framework. However, when we consider renewable energies such as solar photovoltaic generation, for example, the larger input (e.g., the degree of temperature and the number of sunshine days) produces the better performance. As a result, we should formulate it under managerial disposability. In the case, Model (6) needs to change the slacks \((  d_{i}^{x  } )\) to \(+ d_{i}^{x  } .\) and such is a necessary requirement to attain the status of managerial disposability. However, as discussed by Wang and Sueyoshi (2017), if we change such an input by the reciprocal of an original data, then Model (6) can be expressed by the natural disposability as it is. See Wang and Sueyoshi (2017) for a detailed description on the data treatment.