**3. Methodology**

DEA, as a type of multi-criteria decision analysis (MCDA) method, has mainly been applied for evaluation of relative e fficiency. Additionally, it has been used as a benchmarking tool rather than choosing alternatives as the best solutions or directions in traditional decision making [4]. For measuring energy and environment e fficiency, in the literature, radial and non-radial models are the two most widely used in DEA [21]. Radial DEA models proportionally decrease the amount of inputs and outputs, which may have weak discriminating power [6], lead to partial ranking in which most of the DMUs have the same score of e fficiency [45], as well as occurrence of di fficulties in ranking the environmental performance of e fficient DMUs [20]. When including the environmental variable in the model, e fficiency measuring is a challenging task because the environmental pollutant need not increase or decrease proportionally with outputs or inputs [19], and, consequently, non-radial DEA has a higher discriminating power than radial in the environmental performance thanks to non-proportional adjustments of di fferent inputs/outputs in comparing DMUs [4]. Radial models also need to especially treat a negative or zero value in a data set; they do not have the property of "translation invariance" so cannot directly handle zero [46]. In addition, they do not provide information regarding the e fficiency of the specific inputs and outputs included in the process [18,20,47]. To overcome such weaknesses, non-radial models have been developed are widely used in empirical research [21,48]. According to Lui et al. [21], the non-radial DEA model also causes less bias.

In this paper, a two-step methodology for the evaluation of transport *EEE* of EU countries has been employed. In the first step, the non-radial DEA model proposed by Wu et al. [6] has been used for the evaluation of transport *EEE*. The proposed non-radial model provides the use of decision makers' specified weights, di fferent non-proportional adjustments, and proportionally decreases several energy inputs and undesirable outputs simultaneously to the degree possible. However, based on the fact that the DEA method, in the case of the same e fficiency of two DMUs, cannot rank DMUs and provide evaluation of DMUs with simultaneously minimization and maximizations of inputs and outputs, we have proposed a Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) as an MCDA method for benchmarking the alternatives—i.e., decision making units (DMUs), detecting the best practices based on alternative rank and evaluation of transport *EEE*.

Hence, the TOPSIS method has been proposed to rank DMUs, and simultaneously compare efficiency scores vs. DEA results. Based on the content of TOPSIS—i.e., consideration of DMUs from di fferent viewpoints (for example, through inputs and outputs that are presented as cost criterion and a beneficial criterion) this method was introduced for evaluation and ranking of DMUs for monitoring changes of *EEE*. Consequently, for this purpose the following research hypothesis was defined: Any similarity between the results of the evaluation and analyzing of *EEE* through the application of non-radial DEA model and TOPSIS method does not exist.

In these terms, questions that this paper endeavors to answer involve changes to *EEE* for EU transport sectors and the suitability and applicability of the TOPSIS method in the evaluation of *EEE*. Therefore, the objective of this paper is not to study factors of *EEE*, but rather to evaluate the *EEE* for EU transport sectors using the non-radial DEA model and consider the utility of the TOPSIS method regarding evaluation of *EEE*.

### *3.1. A Brief Description of DEA Method*

The DEA method was proposed by Charnes et al. [49] and presents a non-parametric frontier approach for evaluating the relative e fficiency of a set of entities, DMUs, with multiple inputs and outputs [9,10,50]. A major stated advantage of DEA is that it does not require prior assumptions regarding underlying functional relationships between inputs and outputs [4] and weights for input and output is calculated based on the input oriented Charnes, Cooper and Rhodes (CCR) DEA model [4] that can be written as: minθ;*s*.*t X*λ ≤ θ*xi*, *Y*λ ≥ *yi*, λ ≥ 0, where X and Y represent a set of vectors of inputs and outputs, respectively. θ represents a goal function of technical e fficiency where θ ∈ [0, 1]. Based on the result, θ indicates how much an evaluated entity could potentially reduce its input vector while holding the output constant. The presented CCR model exhibits the constant returns to scale (CSR), but with additional constraint λ = 1, the CCR model becomes the classical Banker, Chames and Cooper (BCC) model that allows the variant to return to scale (VRS) [4,51].

### 3.1.1. DEA Method for EEE Evaluation

DEA is strongly related to production theory, where raw materials and resources are treated as inputs, while products are treated as outputs in the production process [5,9]. Then, in the production process, in terms of evaluation of *energy and environmental <sup>e</sup>*ffi*ciency*, desirable and undesirable outputs, are jointly produced by consuming both energy and non-energy inputs, where *x*, *e*, *y* and *u* are vectors of non-energy inputs, energy inputs, desirable outputs, and undesirable outputs, respectively. The joint production process can be represented as *T* = (*<sup>x</sup>*,*e*, *y*, *<sup>u</sup>*);(*<sup>x</sup>*,*<sup>e</sup>*) can produce (*y*, *<sup>u</sup>*).

Based on that let's assume that there are K DMUs, and each DMU uses *n* non-energy inputs and *l* energy inputs in order to produce *m* desirable outputs and *j* undesirable outputs denoted respectively as *x* = (*<sup>x</sup>*1*K*, ... , *xnK*), *e* = (*<sup>e</sup>*1*l*, ... , *xLK*), *y* = (*ymK*, ... , *ymK*), *u* = *<sup>u</sup>*1*K*, ... , *uJK*. Then, environment DEA production technology T exhibiting constant returns to scale (CRS) and incorporating undesirable outputs can be written as:

$$T = \{ (\mathbf{x}, e, y, u) : \sum\_{k=1}^{K} \lambda\_k \mathbf{x}\_{nk} \le \mathbf{x}\_{n\prime}, n = 1, \dots, N \} \tag{1}$$

$$\sum\_{k=1}^{K} \lambda\_k e\_{lk} \le e\_{l\prime}, l = 1, \dots, L,\tag{2}$$

$$\sum\_{k=1}^{K} \lambda\_k y\_{mk} \ge y\_{m\prime} m = 1, \dots, M\prime \tag{3}$$

$$\sum\_{k=1}^{K} \lambda\_k u\_{j\overline{k}} = u\_{j\prime} \, j = 1, \ldots, l\_{\prime} \tag{4}$$

where λ*k* ≥ 0, *k* = 1, ... ,*K*.

Based on this, T reference technology, radial model, modified-radial, and non-radial models such as the Russell measure model, tone's slack-based model, range adjusted model and directional distance function model are used in energy efficiency and carbon emission efficiency in the literature. Additionally, there are four types of returns to scale (*RTS*) such as constant *RTS* (*CRS*) which is the most commonly used *RTS* category, non-increasing *RTS* (*NIRS*), non-decreasing *RTS* (*NDRS*) and variant *RTS* (*VRS*), where each of them reflects reference technology [5].

There are several DEA-type models, radial and non-radial, for pure energy efficiency evaluation with consideration of undesirable outputs, some of which can be used for estimating potential energy saving [9]. The radial model aims at reducing energy inputs as much as possible for the given level of non-energy inputs, plus desirable and undesirable outputs. Since the radial model has weak discriminating power in energy efficiency comparisons and does not consider energy mix effects, non-radial models for energy efficiency evaluation is also proposed in [8,9]. Therefore, the application of non-radial DEA models for energy efficiency evaluation considering undesirable outputs and maximized energy-saving potential, all under *CRS*, *NIRS* and *VRS* were presented in [8]. For example, if in the model (M) instead of limitation (5) we write *Kk*=<sup>1</sup> λ*k* ≤ 1, *Kk*=<sup>1</sup> λ*k* ≥ 1 or *Kk*=<sup>1</sup> λ*k* = 1, we receive non-radial model under *NIRS*, *NDRS*, and *VRS*, respectively. However, their non-radial models also attempt to reduce energy inputs as much as possible for the given level of non-energy input, desirable and undesirable outputs. In other words, their non-radial models do not consider reduction of undesirable outputs.

### 3.1.2. Non-Radial DEA Model for EEE Evaluation

Radial and non-radial DEA models for evaluating DMUs' total-factor *energy and environment e*ffi*ciency* have been presented in Wu et al. [6]. To overcome all disadvantages of the presented radial model, following [52,53], in [6] the radial DEA model has been extended to the following non-radial model (M) for *energy-environment e*ffi*ciency* evaluation as:

$$EEEI = \min \frac{1}{2} (\frac{1}{L} \sum\_{l=1}^{L} \theta\_l + \frac{1}{I} \sum\_{j=1}^{J} \theta\_j) \tag{5}$$

s.t.

$$\sum\_{k=1}^{K} \lambda\_k \mathbf{x}\_{nk} \le \mathbf{x}\_{n0}, n = 1, \dots, N \tag{6}$$

$$\sum\_{k=1}^{K} \lambda\_k \mathfrak{e}\_{lk} \le \theta\_l \mathfrak{e}\_{l0}, l = 1, \dots, L \tag{7}$$

$$\sum\_{k=1}^{K} \lambda\_k y\_{mk} \ge y\_{m0\prime} m = 1, \dots, M \tag{8}$$

$$\sum\_{k=1}^{K} \lambda\_k \mu\_{j\mathbf{k}} = \theta\_j u\_{j0\prime} \; j = 1, \dots, J \tag{9}$$

λ*k* ≥ 0, *k* = 1, ... ,*K*.

The model (M) will be used in this paper for *EEE evaluation* of EU transport sectors. The main advantage of the non-radial model (M) is that it proportionally decreases several energy inputs and undesirable outputs as much as possible for the given level of non-energy inputs and desirable outputs. The optimal values of *energy-environment e*ffi*ciency index* (*EEEI*) are in the interval between 0 and 1. An entity with a higher value of *EEEI* has better *EEE* in terms of other entities. However, if the entity has *EEEI* equal to 1 it means that entity is the best, located on the frontier, and could not reduce energy input and undesirable output. Another benefit of the model is that (M) can consider energy input mix effects and undesirable outputs mix effects in the evaluation of *EEE* [6]. Such non-radial model (M) is suitable for *EEE* evaluation because it has a relatively strong discriminating power and capability to expand desirable outputs, simultaneously reducing undesirable outputs. Additionally, benefit lies in the fact that unified efficiency can be calculated through DM specified weights assigned to each of these two efficiency scores and depends on the preferences between energy use and environment protection performance. However, we have retained the weights as in the paper Wu et al. [6] and both are set to 1/2. These weights point to the similarity of the model (M) with TOPSIS method. Based on the all above pointed out simultaneous benefits in comparison to other non-radial DEA models and the fact that *EEE* evaluation in this paper couldn't be considered to be a dynamic change over time, we have chosen non-radial DEA model (M) by Wu et al. [6] for evaluating *energy-environment <sup>e</sup>*ffi*ciency*.

### *3.2. Background of the TOPSIS Method*

In this paper, the TOPSIS method proposed by Hwang and Yoon [54] has been employed as a decision-making tool to aid DMs in trade-off the whole DMUs. In the literature, this method has received much interest from researchers and practitioners that confirmed a wide range of real-world applications across different fields and specific sub-areas [55]. This method is based on the assumption that the selected alternative is to be at the least possible distance from the ideal positive solution and ideal negative solution. As one of the best and most frequently used methods, MCDM implies overall assessment, comparison, and ranking of alternatives. DEA divides DMUs into efficient and inefficient [49]. However, the question is, which of these efficient DMUs can be located in the higher position [56]. Based on that, it can be concluded that total discrimination of the DEA method can be low in some cases, especially in terms of differentiating efficient DMUs.

Therefore, our paper has included the TOPSIS method for finding the best alternative—i.e., for ranking and solving the drawbacks of the DEA method. Moreover, besides the fact regarding the grea<sup>t</sup> variety of existing DEA ranking methods, ranking DMUs such as cross-efficiency, super-efficiency, benchmarking, statistical techniques and so on, all consider DMUs only from one point of view—i.e., input-oriented or output-oriented views [56].

Consequently, an additional reason for selection of TOPSIS for *EEE* evaluation and ranking of DMUs is based on the content of TOPSIS—i.e., DM intention to rank DMU with the best ranking score

closer to the positive ideal and to have the greatest distance from the negative ideal solution, and the ability of consideration of DMUs from both pessimistic and optimistic viewpoints—i.e., inputs and outputs, such as a cost and benefit criterion [56,57].

After application of DEA, the TOPSIS method was used to evaluate and rank DMUs to present the behavior of DMUs. For our purpose, the TOPSIS method has been employed for road, rail and air transport sectors following the steps in [7,43]:


3. Calculation of the weight normalized decision matrix as *V* = *vijn*×*m* = *wjrijn*×*m*, where *wj* is a weight given to criteria from DM and sum of weights *mj*=<sup>1</sup> *wj* = 1. This method is appropriate for decision making which is based on criteria of different importance.

