*3.4. Grey Forecasting Results*

In this study, the authors used the GM (1, 1) model to forecast the business situation of two enterprises selected to implement a strategic alliance. This forecast result provides a solid basis for managers to make decisions using a complete picture of the business situation of partners in the period 2021–2024. The authors used the data of IN1 of LO7 to explain in detail the calculation steps of the forecast data of enterprises. The steps to calculate the total forecast are performed as follows:

The statistics on IN1 of LO7 in the period 2017–2020 were used to build the original value chain as follows:

$$\chi^{(0)} = (1, 421, 715, 1, 625, 789, 1, 947, 165, 2, 346, 510)$$

The authors use the cumulative addition method to build the value chain of X(1) :

⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ X(1) (1) = X(0) (1) = 1, 421, 715 X(1) (2) = X(0) (1) <sup>+</sup> <sup>X</sup>(0) (2) = 3, 047, 504 X(1) (3) = X(1) (2) <sup>+</sup> <sup>X</sup>(0) (3) = 4, 994, 669 X(1) (4) = X(1) (3) <sup>+</sup> <sup>X</sup>(0) (4) = 7, 341, 179

$$\chi^{(1)} = (1, 421, 715, 3, 047, 504, 4, 994, 669, 7, 341, 179)$$

The authors use the value chain of X(1) to calculate the mean Z(1) :

$$\begin{cases} Z\_{(2)}^{(1)} = (1, 421, 715 + 3, 047, 504) \times 0.5 = 2, 234, 609.5\\ Z\_{(3)}^{(1)} = (3, 047, 504 + 4, 994, 669) \times 0.5 = 4, 021, 086.5\\ Z\_{(4)}^{(1)} = (4, 994, 669 + 7, 341, 179) \times 0.5 = 6, 167, 924.0 \end{cases}$$

After calculating the mean Z(1) , the authors set up the following system of equations:

$$\begin{cases} 3,047,504+2,234,609.5 \times \mathbf{a} = \mathbf{b} \\ 4,994,669+4,021,086.5 \times \mathbf{a} = \mathbf{b} \\ 7,341,179+6,167,924.0 \times \mathbf{a} = \mathbf{b} \end{cases}$$

From the values in the above system of equations, the authors set up the matrices and find the coefficients a, b by the method of least squares:

$$\mathbf{B} = \begin{bmatrix} -2, 234, 609.5 & 1 \\ -4, 021, 086.5 & 1 \\ -6, 167, 924.0 & 1 \end{bmatrix}; \mathbf{Y}\_{\mathbf{N}} = \begin{bmatrix} 3, 047, 504 \\ 4, 994, 669 \\ 7, 341, 179 \end{bmatrix}; \begin{bmatrix} \mathbf{a} \\ \mathbf{b} \end{bmatrix} = (\mathbf{B}^{\mathrm{T}}\mathbf{B})^{(-1)}\mathbf{B}^{\mathrm{T}}\mathbf{Y}\_{\mathbf{N}} = \begin{bmatrix} -0.1833 \\ 1, 213, 958 \end{bmatrix}$$

With the coefficients a, b, the authors built the equation of the GM (1, 1) model:

$$\frac{\mathbf{dx}\_{(\mathbf{k})}^{(1)}}{\mathbf{d}\_{(\mathbf{k})}} - \text{ } 0.1833 \mathbf{x}\_{(\mathbf{k})}^{(1)} = 1 \text{ } 213 \text{ } 958$$

The formula for calculating the forecast values is set up as follows:

$$
\hat{\mathbf{X}}\_{(k+1)}^{(1)} = \text{ } \text{ } \text{\textquotedblleft}{43 \text{\textquotedblright}{}} \text{\textquotedblright}{} \text{\textquotedblleft}{} \text{\textquotedblright}{} \text{\textquotedblleft}{} \text{\textquotedblleft}{} \text{\textquotedblright}{}$$

Substituting the values of k in turn, the authors obtain the values of Xˆ (1) (k+1) as in Table 12.

> **Table 12.** Values of Xˆ (1) (k+1).


By the same calculation methods, the authors obtain forecast values that reflect the business situation of logistics enterprises participating in the strategic alliance in the period 2021–2024, which are shown in Tables 13 and 14, below.

To ensure the reliability of the forecast results, the authors used MAPE to recheck and the results are as follows: MAPELO7 = 3.96%; MAPELO10 = 2.06%. This result shows that the predictive values have very high accuracy (<10%). The forecast results provide managers of enterprises with an overview of the business situation of enterprises in the period 2021–2024. Therefore, managers of enterprises can use the forecast results in this study as a solid basis for making decisions about implementing alliances to bring high results for the business and sustainable development.

The cumulative method is used to calculate the forecast values below:

$$\begin{array}{l} \hat{\mathbf{X}}\_{(1)}^{(0)} = \mathbf{X}\_{(1)}^{(1)} = 1, 421, 715\\ \hat{\mathbf{X}}\_{(2)}^{(0)} = \hat{\mathbf{X}}\_{(2)}^{(1)} - \hat{\mathbf{X}}\_{(1)}^{(1)} = 1, 618, 417.07\\ \hat{\mathbf{X}}\_{(3)}^{(0)} = \hat{\mathbf{X}}\_{(3)}^{(1)} - \hat{\mathbf{X}}\_{(2)}^{(2)} = 1, 944, 055.03\\ \hat{\mathbf{X}}\_{(4)}^{(0)} = \hat{\mathbf{X}}\_{(4)}^{(1)} - \hat{\mathbf{X}}\_{(3)}^{(1)} = 2, 335, 213.87\\ \hat{\mathbf{X}}\_{(5)}^{(0)} = \hat{\mathbf{X}}\_{(5)}^{(1)} - \hat{\mathbf{X}}\_{(4)}^{(1)} = 2, 805, 076.87\\ \hat{\mathbf{X}}\_{(6)}^{(0)} = \hat{\mathbf{X}}\_{(6)}^{(1)} - \hat{\mathbf{X}}\_{(5)}^{(1)} = 3, 369, 479.92\\ \hat{\mathbf{X}}\_{(7)}^{(0)} = \hat{\mathbf{X}}\_{(7)}^{(1)} - \hat{\mathbf{X}}\_{(6)}^{(1)} = 4, 047, 445.21\\ \hat{\mathbf{X}}\_{(8)}^{(0)} = \hat{\mathbf{X}}\_{(8)}^{(1)} - \hat{\mathbf{X}}\_{(7)}^{(1)} = 4, 861, 822.33 \end{array}$$

**Table 13.** Forecast results of LO7 (calculated by researcher).



**Table 14.** Forecast results of LO10 (calculated by researcher).

#### **4. Conclusions**

The COVID-19 pandemic has greatly affected the production and business situation of enterprises around the world. Enterprises in different industries and of different sizes have different levels of influence. In this study, the authors used the Malmquist productivity index to assess the impact of the pandemic on logistics enterprises in Vietnam and the super-slack-based model to evaluate the rankings for businesses in two steps before and after alliance implementation. The rankings are used to choose the optimal alliances for logistics enterprises in Vietnam. The alliances can help businesses save on transportation costs, loading and unloading costs, storage costs, and labor costs, while helping businesses increase revenue and profit. The alliances can help businesses promote their strengths and create competitive advantages. In addition, the authors used the Grey forecasting model to provide managers of enterprises with a complete picture of their partners' business situation in the period 2021–2024 as a solid basis for decision making.
