Output-Oriented SBM (SBM-O-C)

The output-oriented SBM efficiency *ρ*<sup>∗</sup>*O*of DMUc = (*xc*, *yc*) is defined by [SBM-O-C] [76]:

> 1/*ρ*<sup>∗</sup>*O* = max *<sup>λ</sup>*,*s*<sup>−</sup>,*s*<sup>+</sup><sup>1</sup> + 1*s* ∑*sr*=<sup>1</sup> *s*+*r yrh*

S.t.

$$\begin{aligned} \mathbf{x}\_{ic} &= \sum\_{j=1}^{n} \mathbf{x}\_{ij}\boldsymbol{\beta}\_{j} + \ d\_{j}^{-}(i=1,\ldots m) \\ \mathbf{y}\_{ic} &= \sum\_{j=1}^{n} \mathbf{y}\_{ij}\boldsymbol{\beta}\_{j} + d\_{i}^{+}(i=1,\ldots m) \\ \boldsymbol{\beta}\_{j} &\geq \mathbf{0}(\forall j), \ d\_{i}^{-} \geq \mathbf{0}(\forall i), \ d\_{i}^{+} \geq \mathbf{0}\ (\forall r) \\ \text{of } [\mathbf{SBM} - \mathbf{O} - \mathbf{C}]\boldsymbol{\beta}^{\*}, d^{-\*}, d^{+\*}). \end{aligned} \tag{28}$$

The optimal solution

#### 3.3.4. Super-Slacks-Based Measure Model (Super SBM Model)

Tone's super SBM model [78] has proposed a slacks-based measure of efficiency (SBM model) that measures the efficiency of the units under evaluation using slack variables only. The super efficiency SBM model removes the evaluated unit DMUq from the set of units and looks for a DMU\* with inputs *xi*\*, *i* = 1, ..., *m*, and outputs *yk*\*, *k* = 1, ..., *r*, being SBM (and CCR) efficient after this removal. The super SBM model is formulated as follow:

$$\begin{aligned} \text{minimize } & \theta\_{\vec{q}}^{SRM} = \frac{\frac{1}{T} \sum\_{i=1}^{n} x\_i^\* / x\_{i0}}{\frac{1}{T} \sum\_{k=1}^{n} y\_k^\* / y\_{k0}} \\ \text{s.t.} & \\ & \sum\_{j=1}^{n} x\_{i\vec{j}} \beta\_j + d\_i^- = \gamma x\_{i0}, \ i = 1, 2, \dots, p \\ & \sum\_{j=1}^{n} y\_{r\vec{j}} \beta\_j - d\_r^+ = y\_{r0}, \ r = 1, 2, \dots, q \\ & x\_i^\* \ge x\_{i0}, \ i = 1, 2, \dots, n \\ & y\_k^\* \le y\_{k0}, \ k = 1, 2, \dots, n \\ & \beta\_k \ge 0, \ k = 1, 2, \dots, n \\ & d\_i^- \ge 0, i = 1, 2, \dots, p \\ & d\_r^+ \ge 0, r = 1, 2, \dots, q \end{aligned} \tag{30}$$

The numerator in the ratio in Equation (29) can be explained as the distance of units DMUq and DMU\* in input space and the average reduction rate of inputs of DMU\* to inputs of DMUq.

## **4. Case Study**

In this research, the authors collected 25 suppliers (DMU) in Vietnam. Information about the suppliers is shown in Table 4.


**Table 4.** Number of suppliers (DMU).

The data collection of the FANP and hierarchical structure are introduced in Figure 4.

**Figure 4.** Hierarchical structure to select best suppliers.

A fuzzy comparison matrix for all criteria is shown in Table 5.

**Table 5.** Fuzzy comparison matrix for criteria.


During the defuzzification, we obtain the coefficients *α* = 0.5 and *β* = 0.5 (Tang and Beynon) [80]. In it, *α* represents the uncertain environment conditions, and *β* represents the attitude of the evaluator is fair.

$$\mathbf{g}\_{0.5, 0.5}(\overline{\mathbf{a}\_{EMS,FS}}) = [(0.5 \times 6.5) + (1 - 0.5) \times 7.5] = 7$$

$$\mathbf{f}\_{0.5}(L\_{EMS,FS}) = (7 - 6) \times 0.5 + 6 = 6.5$$

$$\mathbf{f}\_{0.5}(\mathcal{U}\_{EMS,FS}) = 8 - (8 - 7) \times 0.5 = 7.5$$

$$\mathbf{g}\_{0.5, 0.5}(\overline{\mathbf{a}\_{EMS,FS}}) = 1/7$$

The remaining calculations are similar to the above, as well as the fuzzy number priority points. The real number priorities when comparing the main criteria pairs are presented in Table 6.



We calculate the maximum individual values as follows:

*GM*1 = (1 × 1/7 × 1/8 × 1/2)1/4 = 0.03073 *GM*2 = (7 × 1 × 1/6 × 2)1/4 = 1.2359 *GM*3 = (8 × 6 × 1 × 5)1/4 = 3.9359 *GM*4 = (2 × 1/2 × 1/5 × 1)1/4 = 0.6687 ∑*GM* = *GM*1 + *GM*2 + *GM*3 + *GM*4 = 6.1478 *ω*1 = 0.3073 6.1478 = 0.0499 *ω*2 = 1.2359 6.1478 = 0.2010 *ω*3 = 3.9359 6.1478 = 0.6402 *ω*4 = 0.6687 6.1478 = 0.1087 ⎡ ⎢ ⎢ ⎢ ⎣ 1 1/7 1/8 1/2 7 1 1/6 2 8 615 2 1/2 1/5 1 ⎤ ⎥ ⎥ ⎥ ⎦ × ⎡ ⎢ ⎢ ⎢ ⎣ 0.0499 0.2010 0.6402 0.1087 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 0.2129 0.8744 2.7889 0.4370 ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ 0.2129 0.8744 2.7889 0.4370 ⎤ ⎥ ⎥ ⎥ ⎦/ ⎡ ⎢ ⎢ ⎢ ⎣ 0.0499 0.2010 0.6402 0.1087 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 4.2665 4.3502 4.3562 4.0202 ⎤ ⎥ ⎥ ⎥ ⎦

with the number of criteria is 4, we obtain *n* = 4, and *λmax* and *CI* are calculated as follows:

$$
\lambda\_{\text{max}} = \frac{4.2665 + 4.3502 + 4.3562 + 4.0202}{4} = 4.2482$$

$$CI = \frac{4.2482 - 4}{4 - 1} = 0.0827$$

For *CR*, with *n* = 4 we obtain *RI* = 0.9:

$$CR = \frac{0.0827}{1.12} = 0.0919$$

We have *CR* = 0.0919 ≤ 0.1, so the pairwise comparison data is consistent and does not need to be re-evaluated. The results of the pair comparison between the main criteria are presented in Tables 7–11.


**Table 7.** Fuzzy comparison matrices for the criteria.



**Table 9.** Comparison matrix for the delivery and service criteria.


**Table 10.** Comparison matrix for the qualitative criteria.


**Table 11.** Comparison matrix for the environmental managemen<sup>t</sup> systems criteria.


Based on how the hierarchical structure was built, the pairwise comparison matrix was built through completing a questionnaire. Then, the received data to calculate the weight of supplier's indices and to ensure the accuracy of judged inconsistency rate and other constraints are presented.

In summary, a graphic of the DEA model for analysis of DMUs (suppliers) along with three inputs and three outputs is shown in Figure 4. The results of the FANP model for the ranking of various suppliers on qualitative attributes are utilized in the output qualitative benefits of the DEA model [71,81]. In our situation, inputs are those factors that organizations would consider as an improvement if they were decreased in value (i.e., smaller values are better), whereas outputs are those factors that organizations would consider as improvements if they were increased in value (i.e., larger is better). This is a standard approach when seeking to use DEA as a discrete alternative multiple criteria decision-making tool [71]. There are three inputs and three outputs, as shown in Figure 5.

**Figure 5.** Data envelopment analysis model.

To aid in reducing scaling errors associated with the mathematical programming software packages, the dataset is mean normalized for each factor, i.e., each value in each column is divided by that column's mean score. This normalization procedure does not change the efficiency scores of the ratio-based DEA models. As previously mentioned, to help model the analysis as inputs and outputs, instead of the standard productivity efficiency measurement approach, assume that the inputs are those factors that improve as their values decrease and the outputs are those values that improve as their values increase [71]. Raw data are provided by the case organization, as shown in Table 12.




**Table 12.** *Cont.*

## *4.1. Isotonicity Test*

The variables of input and output for the correlation coefficient matrix should comply with the isotonicity premise. In other words, the increase of an input will not cause the decreasing output of another item. The results of the Pearson correlation coefficient test are shown in Table 13.


**Table 13.** The results of the Pearson correlation coefficient.

Based on the results of Pearson correlation test, the results of all correlation coefficients are positive, thus meeting a basic assumption of the DEA model. Hence, we do not to change the input and output.

#### *4.2. Results and Discussion*

Supplier evaluation and selection have been identified as important issues that could affect the efficiency of a supply chain. It can be seen that selecting a supplier is complicated in that decision-makers must understand qualitative and quantitative features for assessing the symmetrical impact of the criteria to reach the most accurate result.

For the performance in an empirical study, the authors collected data from 25 suppliers in Vietnam. A hierarchical structure to select the best suppliers is built with four main criteria (including 15 sub-criteria). Completion of a questionnaire for analyzing the FANP model is done by interviewing experts, and surveying the managers and company's databases. The ANP model is combined with a fuzzy set, to evaluate the supplier selection criteria and define the priorities of each supplier, which are able to be utilized to clarify important criteria that directly affect the profitability of the business. Then, several DEA models are proposed for ranking suppliers. As a result, DMU 1, DMU 5, DMU 10, DMU 16, DMU 19, DMU 22, and DMU 23 are identified as efficient in all nine models, as shown in Table 7 [78], which have a conditioned response to the enterprises' supply requirements. Whereas for other DMUs, there were differences in the results, so the next research should include an improvement or review of data inputs to produce appropriate outputs, so that suppliers remain efficient. This integration model supports a grea<sup>t</sup> deal of corporate decision-making because of the

effectiveness and the complication of this model, for exactly choosing the most suitable supplier. The ranking list of 25 DMUs as shown in Table 14.


**Table 14.** Ranking list of suppliers by using nine DEA models (CCR, BCC, and SBM, Super SBM).

The optimal weights and the slacks for each DMU using nine DEA models (CCR, BCC, and SBM, Super SBM) are shown from Tables A1–A18 in appendix section.
