**4. An Application for Local and Global Environmental Efficiency Analysis of Power Plants**

This section applies the proposed approach for environmental efficiency analysis of 46 power plants in 21 provinces of Iran. Provinces were assumed as plants at the local level, and the country was assumed as firm at the global level. Total assets were assumed as inputs, electricity production was taken as the desirable output, and pollution was assumed as an undesirable output. Table 2 reports the descriptive statistics of the data. We are willing to share our unnamed data set and codes for those who wish to replicate the results of this research.

**Table 2.** Statistical summary of data.


In the first analysis, we assessed the classical local and global performance of all power plants. The results are reported in Table 3.

We could observe some important facts confirming the proposed methodology and theorem. Most of the production units were found to be efficient at the local level, considering both classical and environmental productions. However, this was not the case at the global level. The classical global efficiency measures were found to be higher than or at least equal to the classical local efficiency measures. We had the same observation when we considered environmental technologies. This is a rational observation since the global environment is a more competitive space, and an under-evaluation unit needs to compete with more rivals at the global level. The same observation appears when we compared the environmental performance vs. the classical performance. This is also an expectable observation since if a production unit is technically efficient and not necessarily

environmentally efficient, it does not matter at the local or global level. The local and global analysis is performed at the province and country-level, respectively. This shows that when considering the efficiency status of production units classical or environmental behavior may not reveal the whole picture of the production behavior. Thus, decision-makers are highly encouraged to consider the production behavior of DMUs at both local and global levels. Considering both classical and environmental production technology at the local and global levels, we reported the proposed indices in Table 4. Using this report, we analyzed and tracked the classical and environmental performance of production units at the local and global levels.


**Table 3.** Classical and environmental efficiency measures at the local and global level.


**Table 4.** Local-Global classical and environmental efficiency indices using a single scaling factor.

The local-global index shows only a few power plants that have a unity measure. Thus, we had just these power plants that have a resistant performance at the global level. P21, P26, and P33 are technically efficient both in the local and global environment. This result provides valuable information in the process of target setting for decision-makers. These power plants that are efficient at both local and global levels may be used for target setting instead of those that are efficient only at the local or global level. Next, we analyzed the environmental local-global index and again found a few power plants with independent

environmental performance, regardless of the local or global level. We observed for P25, P26, and P33 that their environmental local-global index is equal.

In contrast with the previous analysis, we observed more production units with a unity measure of the global environmental index. This shows that the environmental performance was better managed at the local level, and policymakers need more attention towards managing the global environmental issue. Such information may be used in the process of environmental target setting. Another interesting observation is the similarity of the local-global index in the classical and environmental space. We observed that these indices are almost similar (second and third column). This shows that power plants' technical and environmental performance have the same pattern when considering the local and global levels. Therefore, the geographical location does not affect the technical and environmental performance of power plants. In the next analysis, we looked at the case from a different angle by the local and global environmental proposed indices. We are interested in measuring the environmental effects at the local and global levels. These indices are reported in the fourth column and fifth column of Table 4. We observed that at the local levels, we have a few power plants with a greater than equal value. This shows that there is a potential for environmental improvement for those power plants. However, when we looked at the global index, we observed only one power plant with such a situation. More deep investigation revealed that this power plant is owned by a border province with an old generation technology that struggles with providing gas and has used fuel for electricity production in some situations. In order to consider the desirable and undesirable output simultaneously, we used models (5) and model (6) for the local and global performance assessment in the subsequent analysis. The results are reported in Table 5.

We found less efficient power plants when we used the joint model, not only in classical production but also in environmental production. This was also an expectable observation; when we considered just desirable or undesirable output separately, we had an easier job reaching the efficient frontier rather than when considering both desirable and undesirable outputs simultaneously. For the efficiency measure of power plants using a mixed model, that efficient power plants using this model were also efficient using model 3 and model 4 at the local and global level. This could be found by comparing the second and third columns of Table 5 and peers in Table 3. Note that we considered the scaling factor for both desirable and undesirable outputs in the mixed models of (5–6) while we had no scaling factor on undesirable output in the models (3–4). Using mixed environmental efficiency measures of power plants at the local and global level, we calculated the uniform environmental local-global index for all power plants reported in the sixth column of Table 5. We had only two power plants P26 and P34, which were resistant in the local and global environmental assessment. This emphasizes a more competitive space and a high potential for environmental improvement at the global level. Decision-makers and any environmental planning should consider this at the country level. Uniform factor analysis considers the simultaneous improvement of both desirable and desirable output; thus, we found more potential improvement, including desirable output enlargement and undesirable output reduction, in this analysis. However, the scaling factor of desirable and undesirable output may not be uniformly considered in the previous analysis. Thus, in the following analysis, we considered the non-uniform but joint scaling factor for the desirable and undesirable outputs. To this end, we used mixed models (7) and model (8) at the local environmental and global environmental levels, respectively. Table 6 lists the result of new efficiency measures and updated index regarding the local-global performance of power plants associated with new measures using models with non-uniform scaling factors.

The first observation was more discrimination power using models with the nonuniform scaling factor. Regarding the fact that the later model can be considered as a generalized model compared with models with a uniform scaling factor, we could expect this observation. More potential for environmental improvement was found using models with non-uniform scaling factors. We also saw that only the three power plants P21, P26, and P33 are environmentally resistant globally. Deeper analysis shows that these power plants are classically efficiency efficient in both the local and global space. More investigation reveals that these three power plants are classically efficient and environmentally efficient at both local and global levels. On the other hand, these are the only power plants that gained the unity value for all measures and associated indices. This fact shows that those power plants performing well in a classical and environmental manner can be the most favorable targets for other power plants at both local and global levels. However, the average local-global environmental index was about six, which emphasizes considering the local environmental performance and global environmental performance in any environmental planning at the local or country level. This emphasizes the classical and environmental efficiency at the local level and performs and indicates the global analysis of the classical and environmental efficiency status of production units.


**Table 5.** Local-Global classical and environmental efficiency indices using a uniform scaling factor.


**Table 6.** Local-Global classical and environmental efficiency indices using a non-uniform scaling factor.

#### **5. Conclusions**

The current paper proposes new models for environmental assessment and localglobal analysis of pollution generating production units. In the application section, proposed models are used for Iranian power plants' local and global classical and environmental performance. However, the theoretical foundation and associated indices introduced in this paper can be used in any type of local-global analysis that is involved with environmental aspects. The proposed models put one step forward in contrast with classical

efficiency analysis. It is highly recommended to utilize all developed indices for having a broad picture of classical and environmental performance in any performance analysis. One may perform well at the local level or may have an acceptable performance using classical models, but deeper analysis on the global level or considering environmental issues may provide better insight into the production. The current paper considers the production technology assuming convexity and constant returns to scale assumptions. Extending to other production technology types may not be a straightforward task, and we are still working on this. Investigating the production's scale effects is another important aim that can be achieved in a future research line.

**Author Contributions:** Conceptualization, M.G.; Data curation, M.G.; Formal analysis, F.T.-H.; Methodology, M.G.; Project administration, M.G.; Software, M.G.; Supervision, F.T.-H.; Validation, F.T.-H.; Writing—original draft, M.G.; Writing—review & editing, F.T.-H. All authors have read and agreed to the published version of the manuscript.

**Funding:** Farhad Taghizadeh-Hesary acknowledges the financial support of the JSPS KAKENHI (2019–2020) Grant-in-Aid for Young Scientists No. 19K13742 and Grant-in-Aid for Excellent Young Researcher of the Ministry of Education of Japan (MEXT).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are gathered from province and country sources. In the country source, we used the annual report of the energy ministry, and in the local level, we used power plants data of provinces.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A**

**Proof of Theorem 1.** Consider plant "*o*" of firm j denote it by *P<sup>j</sup> <sup>o</sup>*. The classical efficiency of this plant is gauged by the optimal value of model (1), that is, *ϕ<sup>j</sup> Lo* and its environmental efficiency measure is *ϕ<sup>j</sup> ELo* that is the optimal value of model (3). Let (*λ<sup>j</sup>* , *ϕj ELo*) be the optimal solution of model (3); thus, we have

$$\sum\_{k=1}^{K\_j} \overline{\lambda}\_k^j x\_{ik}^j \le x\_{io}^j i = 1, 2, \dots, m$$

$$\sum\_{k=1}^{K\_j} \overline{\lambda}\_k^j y\_{rk}^j \ge y\_{ELo}^j y\_{ro}^j r = 1, 2, \dots, s$$

$$\sum\_{k=1}^{K\_j} \overline{\lambda}\_k^j z\_{hk}^j = y\_{ho}^j h = 1, 2, \dots, P$$

$$\overline{\lambda}\_k^j \ge 0k = 1, 2, \dots, K\_j$$

Ignoring the third set of constraint from the above constraint set, we have

$$\sum\_{k=1}^{K\_j} \overline{\lambda}\_k^j \mathbf{x}\_{ik}^j \le \mathbf{x}\_{io}^j i = 1, 2, \dots, m$$

$$\sum\_{k=1}^{K\_j} \overline{\lambda}\_k^j y\_{rk}^j \ge \varrho\_{ELo}^j y\_{ro}^j r = 1, 2, \dots, s$$

*λ j <sup>k</sup>* <sup>≥</sup> <sup>0</sup>*<sup>k</sup>* <sup>=</sup> 1, 2, ... , *Kj* and this means that (*λ<sup>j</sup>* , *ϕj ELo*) is a feasible solution for the model (1) that implies *ϕ<sup>j</sup> Lo* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> ELo*, where *<sup>ϕ</sup><sup>j</sup> Lo* is the optimal value of the classical model of (1), namely, the classical efficiency measure of the plant under evaluation.

**Proof of Theorem 2.** Consider *P<sup>j</sup> <sup>o</sup>* then model (1) finds the classical efficiency of this plant, that is, *ϕ<sup>j</sup> Lo*. Let (*λj*∗, *<sup>ϕ</sup><sup>j</sup> Lo*) <sup>∈</sup> <sup>R</sup>*Kj*+<sup>1</sup> <sup>+</sup> be the optimal solution of model (1), then if satisfies associated constraint set of

$$\sum\_{k=1}^{K\_j} \lambda\_k^{j\*} \mathbf{x}\_{ik}^j \le \mathbf{x}\_{io}^j i = 1, 2, \dots, m$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^{j\*} y\_{rk}^j \ge \varphi\_{Lo}^j y\_{ro}^j r = 1, 2, \dots, s$$

$$\lambda\_k^{j\*} \ge 0k = 1, 2, \dots, K\_j$$

Using this, we have a feasible solution for the model (2) that gauges the plant's efficiency under evaluation, considering all firm plants. Observe that (*λj*∗∗, *ϕ<sup>j</sup> Lo*) <sup>∈</sup> <sup>R</sup> *J* ∑ *j*= *Kj*+1 +

satisfies the following constraint set

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_j} \lambda\_k^{j \ast \ast} x\_{ik}^j \le x\_{io}^j i = 1, 2, \dots, m$$

$$\sum\_{j=1}^{I} \sum\_{k=1}^{K\_j} \lambda\_k^{j \ast \ast} y\_{rk}^j \ge y\_{Lo}^j y\_{ro}^j r = 1, 2, \dots, s$$

$$\lambda\_k^{j \ast \ast} \ge 0k = 1, 2, \dots, K\_{j\nu} j = 1, 2, \dots, J$$

where *λj*∗∗ *<sup>k</sup>* <sup>=</sup> *<sup>λ</sup>j*<sup>∗</sup> *<sup>k</sup>* for the firm that owned plant "*o*" and *<sup>λ</sup>j*∗∗ *<sup>k</sup>* = 0 for other firms. This implies *ϕj Go* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> Lo*, that is, the classical efficiency of *<sup>P</sup><sup>j</sup> <sup>o</sup>* within its firm is not greater than its efficiency measure when considering all firms.

**Proof of Theorem 3.** Mathematically, we can provide a similar argument to the proof of Theorem 2 to prove this theorem. However, we may also look at the problem from a production technology view. The production space for measuring the global environmental efficiency measure is larger than the production space for measuring the local environmental efficiency measure. This provides a broader production set when we consider the global production. Therefore we cannot expect lesser output efficiency measures in such production space compared with the local production space.

**Proof of Theorem 4.** Similar to the proof of Theorem 1, if we consider the model (4) that gauges the global environmental efficiency of *P<sup>j</sup> <sup>o</sup>* then its optimal solution satisfies the following set of constraints.

$$\sum\_{j=1}^{f} \sum\_{k=1}^{K\_j} \lambda\_k^{j\*} x\_{ik}^j \le x\_{io}^j i = 1, 2, \dots, m$$

$$\sum\_{j=1}^{f} \sum\_{k=1}^{K\_j} \lambda\_k^{j\*} y\_{rk}^j \ge \varphi\_{EGo}^{j\*} y\_{ro}^j r = 1, 2, \dots, s$$

$$\sum\_{j=1}^{f} \sum\_{k=1}^{K\_j} \lambda\_k^{j\*} z\_{hk}^j = y\_{ho}^j h = 1,2,\dots,P$$

$$\lambda\_k^{j\*} \ge 0k = 1,2,\dots,K\_{j\*}j = 1,2,\dots,J$$

where (*λj*<sup>∗</sup> *<sup>k</sup>* , *<sup>ϕ</sup>j*<sup>∗</sup> *EGo*), *k* = 1, 2, ... , *Kj*, *j* = 1, 2, ... , *J* is the optimal solution of model (4). If we consider the first and the second set of constraints in the above system of in-equality, then we reach the following

$$\sum\_{j=1}^{J} \sum\_{k=1}^{K\_j} \lambda\_k^{j\*} \mathbf{x}\_{ik}^j \le \mathbf{x}\_{ho}^j \mathbf{i} = 1, 2, \dots, m$$

$$\sum\_{j=1}^{J} \sum\_{k=1}^{K\_j} \lambda\_k^{j\*} y\_{rk}^j \ge y\_{EGo}^{j\*} y\_{ro}^j r = 1, 2, \dots, s$$

$$\sum\_{j=1}^{J} \sum\_{k=1}^{K\_j} \lambda\_k^{j\*} z\_{hk}^j = y\_{ho}^j h = 1, 2, \dots, P$$

$$\lambda\_k^{j\*} \ge 0k = 1, 2, \dots, K\_{j\*} j = 1, 2, \dots, J$$

This implies that (*λj*<sup>∗</sup> *<sup>k</sup>* , *<sup>ϕ</sup>j*<sup>∗</sup> *EGo*)is a feasible solution of model (2) and therefore *<sup>ϕ</sup>j*<sup>∗</sup> *Go* ≥ *θ j*∗ *EGo*, that is, the global efficiency of a plant is not greater than its global environmental efficiency.

**Proof of Theorem 5.** Assume *P<sup>j</sup> <sup>o</sup>* is efficient using the mixed model of (5), thus we have

$$\sum\_{k=1}^{K\_j} \lambda\_k^{\*j} x\_{ik}^j \le x\_{io}^j i = 1, 2, \dots, m$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^{\*j} y\_{rk}^j \ge (1 + \rho^\*) y\_{ro}^j = y\_{ro}^j r = 1, 2, \dots, s$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^{\*j} z\_{hk}^j = (1 - \rho^\*) z\_{ho}^j = z\_{ho}^j h = 1, 2, \dots, P$$

$$\lambda\_k^{\*j} \ge 0k = 1, 2, \dots, K\_j$$

where, (*λ*∗*<sup>j</sup>* , *ϕ*∗)=(*λ*∗*<sup>j</sup>* , *ϕj UELo* − 1) is the optimal solution of mixed model (5). This implies (*λ*∗*<sup>j</sup>* , *ϕ*∗)=(*λ*∗*<sup>j</sup>* , *ϕj UELo*)=(*λ*∗*<sup>j</sup>* , 1) is a feasible solution of model (3), that is,

$$\sum\_{k=1}^{K\_j} \lambda\_k^{\*j} \mathbf{x}\_{ik}^j \le \mathbf{x}\_{io}^j \mathbf{i} = 1, 2, \dots, m$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^{\*j} \mathbf{y}\_{rk}^j \ge \mathbf{q}\_{ILLo}^j \mathbf{y}\_{ro}^j = \mathbf{y}\_{ro}^j r = 1, 2, \dots, \mathbf{s}$$

$$\sum\_{k=1}^{K\_j} \lambda\_k^{\*j} z\_{lk}^j = z\_{ho}^j \mathbf{h} = 1, 2, \dots, P$$

$$\lambda\_k^{\*j} \ge 0 \mathbf{k} = 1, 2, \dots, K\_j$$

and this implies that *ϕ<sup>j</sup> ELo* <sup>≥</sup> *<sup>ϕ</sup><sup>j</sup> UELo* = 1.
