2.3.2. Research Methods

Malmquist Productivity Index

DEA is a linear programming method for measuring the effectiveness of multiple decision-making units (DMUs) when a production process presents a structure of multiple inputs and outputs. This method relies on the past business data of the enterprise to construct the production boundary in the non-parametric plane (the production boundary). In which, the Malmquist index (MI) evaluates the efficiency of a business at t1(x1 <sup>0</sup>, <sup>y</sup><sup>1</sup> <sup>0</sup>) and t2(x2 <sup>0</sup>, <sup>y</sup><sup>2</sup> <sup>0</sup>). This efficiency is assessed through the catch-up (CU) and frontier-shift (FS) indicators of that business [22].

$$\text{CU} = \frac{\text{Efficiency of t}\_2 (\mathbf{x}\_{0'}^2 \mathbf{y}\_0^2) \text{ with respect to the period 2 frontier}}{\text{Efficiency of t}\_1 (\mathbf{x}\_{0'}^1 \mathbf{y}\_0^1) \text{ with respect to the period 1 frontier}} \tag{1}$$

A specific example of evaluating the effectiveness of an element in the above model is shown in Figure 9:

$$\text{CU} = \frac{\text{BD}}{\text{BQ}} \times \frac{\text{AP}}{\text{AC}} \tag{2}$$

$$\varphi\_1 = \frac{\text{AC}}{\text{AP}} \times \frac{\text{AP}}{\text{AE}} = \frac{\text{AC}}{\text{AE}} = \frac{\text{Efficiency of } (\mathbf{x}\_0^1, \mathbf{y}\_0^1) \text{ with respect to the period 1 fromtier}}{\text{Efficiency of } (\mathbf{x}\_0^1, \mathbf{y}\_0^1) \text{ with respect to the period 2 fromtier}} \quad \text{(3)}$$

$$\varphi\_2 = \frac{\text{BF}}{\text{RO}} \times \frac{\text{BQ}}{\text{BD}} = \frac{\text{BF}}{\text{BD}} = \frac{\text{Efficiency of } (\mathbf{x}\_0^2, \mathbf{y}\_0^2) \text{ with respect to the period 1 fromtier}}{\text{Efficiency of } (\mathbf{x}^2, \mathbf{z}^2) \text{ with respect to the period 2 fromtier}} \quad \text{(4)}$$

$$\mathbf{x} = \frac{\text{BF}}{\text{BQ}} \times \frac{\text{BQ}}{\text{BD}} = \frac{\text{BF}}{\text{BD}} = \frac{\text{Efficiency of } (\mathbf{x}\_{0'}^2 \text{ y}\_0^2) \text{ with respect to the period 1 frontier}}{\text{Efficiency of } (\mathbf{x}\_{0'}^2 \mathbf{y}\_0^2) \text{ with respect to the period 2 frontier}} \quad (4)$$

$$\text{FS} = \varphi = \sqrt{\varphi\_1 \varphi\_2} \tag{5}$$

$$\text{MI} = \text{CU} \times \text{FS} \tag{6}$$

**Figure 9.** Catch-up (Le et al., 2020) [23].

The efficiency score of the DMU at point (x0, y0 t1 measured by frontier technology t2 as follows [22]:

$$\begin{array}{c} \delta^{\mathsf{t}\_{2}} \left( \mathbf{x}\_{0\prime} \mathbf{y}\_{0} \right)^{\mathsf{t}\_{1}}; \ (\mathsf{t}\_{1} = 1, 2; \mathsf{t}\_{2} = 1, 2) \\ \qquad \qquad \mathsf{CU} = \frac{\mathsf{s}^{2} \left( \mathbf{x}\_{0\prime} \cdot \mathbf{y}\_{0} \right)^{2}}{\mathsf{s}^{1} \left( \mathbf{x}\_{0\prime} \cdot \mathbf{y}\_{0} \right)^{1}} \end{array} \tag{7}$$

$$\text{FS} = \left[ \frac{\delta^1 \left( \mathbf{x}\_{0\prime} \text{ y}\_0 \right)^1}{\delta^2 \left( \mathbf{x}\_{0\prime} \text{ y}\_0 \right)^1} \times \frac{\delta^1 \left( \mathbf{x}\_{0\prime} \text{ y}\_0 \right)^2}{\delta^2 \left( \mathbf{x}\_{0\prime} \text{ y}\_0 \right)^2} \right]^{\frac{1}{2}} \tag{8}$$

$$\text{MI} = \text{CU} \times \text{FS} = \left[ \frac{\delta^1 \left( \mathbf{x}\_{0'} \text{ y}\_0 \right)^2}{\delta^1 \left( \mathbf{x}\_{0'} \text{ y}\_0 \right)^1} \times \frac{\delta^2 \left( \mathbf{x}\_{0'} \text{ y}\_0 \right)^2}{\delta^2 \left( \mathbf{x}\_{0'} \text{ y}\_0 \right)^1} \right]^{\frac{1}{2}} \tag{9}$$

If :

⎧ ⎪⎨ ⎪⎩ MI < 1 : The relative efficiency drops. MI <sup>=</sup> 1 : The relative efficiency t1(x<sup>1</sup> 0, y1 <sup>0</sup>) equivalent t2(x<sup>2</sup> 0, y2 0). MI <sup>&</sup>gt; 1 : The relative efficiency increases from t1(x<sup>1</sup> 0, y1 <sup>0</sup>) to t2(x<sup>2</sup> 0, y2 0).

Super-Slack-Based Model

In 2001, Tone introduced the slack-based measure model to evaluate the production and business efficiency of enterprises based on changes in production factors causing changes in corporate profits. The introduction is described as follows [24,25]:

$$\text{Minp} = \frac{1 - \frac{1}{\text{m}} \sum\_{i=1}^{\text{m}} \frac{s\_i^-}{\text{x}\_{\text{ik}}}}{1 + \frac{1}{\text{s}} \sum\_{r=1}^{\text{s}} \frac{s\_r^+}{\text{y}\_{\text{rk}}}} \tag{10}$$

Subject to :

$$\mathbf{x}\_{\mathbf{ik}} = \sum\_{\mathbf{j=1,j\neq k}}^{n} \lambda\_{\mathbf{j}} \mathbf{x}\_{\mathbf{i}\mathbf{j}} - \mathbf{s}\_{\mathbf{i}}^{-} \begin{pmatrix} \mathbf{i} = 1, \ 2, \ \dots, \ \mathbf{n} \end{pmatrix} \tag{11}$$

$$\mathbf{y}\_{\mathbf{r}\mathbf{k}} = \sum\_{\mathbf{j=1,j\neq k}}^{\mathbf{n}} \lambda\_{\mathbf{j}} \mathbf{y}\_{\mathbf{i}\mathbf{j}} - \mathbf{s}\_{\mathbf{i}}^{+} \begin{pmatrix} \mathbf{r} = 1 \ 2 \ \dots \ \mathbf{s} \end{pmatrix} \tag{12}$$

<sup>s</sup><sup>−</sup> <sup>≥</sup> 0, s<sup>+</sup> <sup>≥</sup> 0, <sup>λ</sup><sup>j</sup> <sup>≥</sup> <sup>0</sup> (<sup>j</sup> <sup>=</sup> 1, 2, . . . , n, j <sup>=</sup> <sup>k</sup> (13)

ρ shows the relative business performance of the enterprise. If 0 < ρ < 1, this reflects that the enterprise is not operating efficiently. If ρ = 1, it reflects that the enterprise is relatively efficient. However, there are many businesses in the same business field that achieve relative efficiency. Therefore, to evaluate and rank these enterprises, Tone introduced the super-slack-based model to evaluate the ranking of enterprises in the same industry. The super-slack-based model is described as follows:

$$\mathsf{Minp} = \frac{\frac{1}{\mathsf{m}} \sum\_{i=1}^{\mathsf{m}} \frac{\mathsf{y}\_i}{\mathsf{x}\_{\mathsf{ik}}}}{\frac{1}{\mathsf{s}} \sum\_{r=1}^{\mathsf{s}} \frac{\mathsf{y}\_r}{\mathsf{y}\_{\mathsf{rk}}}} \tag{14}$$

Subject to :

$$\overline{\infty} \ge \sum\_{\mathbf{j=1,j\neq k}}^{n} \lambda\_{\mathbf{j}} \mathbf{x}\_{\overline{\mathbf{j}}} \tag{15}$$

$$\mathcal{F} \le \sum\_{\mathbf{j=1,j\neq k}}^n \lambda\_{\mathbf{j}} \mathbf{y}\_{\mathbf{j}} \tag{16}$$

$$\overline{\mathbf{x}} \ge \mathbf{x}\_{\mathbf{k}}; \overline{\mathbf{y}} \le \mathbf{y}\_{\mathbf{k}} \tag{17}$$

$$\lambda\_{\mathbf{j}} \ge 0 \text{ ( $\mathbf{j} = 1$ ,  $2$ ,  $\dots$ ,  $\mathbf{n}$ ,  $\mathbf{j} \ne \mathbf{k}$ )}\tag{18}$$

In which :

⎧ ⎪⎪⎪⎪⎪⎪⎨ n : Denotes the number of DMUs.

m : Denotes the number of input indexes.

s : denotes the number of output indexes.

s+, s<sup>−</sup> : Are the output and the input relaxation variables.

λ : Denotes the weight vector.

xik, yrk : Denote the ith input and the rth output of the DMUk

## Grey Forecasting Model

⎪⎪⎪⎪⎪⎪⎩

GFM was first introduced by Professor Julong Deng in 1982. GFM focuses on the study of uncertain information systems and incomplete data sources in decision making. Grey system theory can perform research with a small sample size data set. Therefore, Grey system theory overcomes the inherent disadvantages of other forecasting methods. The authors used Grey system theory to conduct this study. After being introduced and published in the journal System & Control Letters, GFM has been successfully applied by many scientists around the world in most fields of economy and society. The process of calculating the GM (1, 1) model of GFM is shown in 6 steps in Figure 10 below:

**Figure 10.** Forecasting process (Nguyen Han Khanh 2021) [26].

Correlation Coefficient and Error

As mentioned, the authors used 4 inputs and 2 outputs to assess the impact of the pandemic on the business situation of logistics companies in Vietnam in the period 2017–2020. To test the correlation between factors, the authors used Pearson correlation coefficient (r). Testing the correlation coefficient through (r) is the best method to measure the relationship between factors used in the study because it is based on the covariance method. (r) is calculated by following formula [27]:

$$\mathbf{r} = \frac{\sum\_{\mathbf{i}=1}^{\mathrm{n}} \left(\mathbf{a}\_{\mathbf{i}} - \overline{\mathbf{a}}\right) \left(\mathbf{b}\_{\mathbf{i}} - \overline{\mathbf{b}}\right)}{\left[\sum\_{\mathbf{i}=1}^{\mathrm{n}} \left(\mathbf{a}\_{\mathbf{i}} - \overline{\mathbf{a}}\right)^{2} \sum\_{\mathbf{i}=1}^{\mathrm{n}} \left(\mathbf{b}\_{\mathbf{i}} - \overline{\mathbf{b}}\right)^{2}\right]^{\frac{1}{2}}} \tag{19}$$

The authors used the GM (1, 1) model to forecast the business situation of logistic enterprises in the period of 2021–2024. To evaluate the quality and fit of the GM (1, 1) model used in this study, the authors used MAPE to calculate error. MAPE is calculated according to the following formula [28]:

$$\begin{aligned} \boldsymbol{\delta} &= \frac{1}{n} \left[ \sum\_{i=1}^{n} \left| \frac{\mathbf{A}\_{i} - \mathbf{F}\_{i}}{\mathbf{A}\_{i}} \right| \times 100 \right]; \\ \begin{cases} \boldsymbol{\delta} &\le 10\% : \text{Excellent} \\ 10\% < \boldsymbol{\delta} &\le 20\% : \text{Good} \\ 20\% < \boldsymbol{\delta} &\le 50\% : \text{Qualified} \\ \boldsymbol{\delta} &> 50\% : \text{Unqualified} \end{cases} \end{aligned} \tag{20}$$
