*Model 1*

If at the beginning of the period [0, *T*], uncontrollable parameters of the logistic process such as T = 400 days, r = 0.001, and µ = 25 units/day are known and the cost of delivery and the price increases equally during the period [0, *T*] with ρ = 0.00075, then the growth pattern will be *<sup>c</sup>s*(*t*) = <sup>400</sup> <sup>∗</sup> 1.00075*<sup>t</sup>* , *<sup>p</sup>*(*t*) = <sup>20</sup> <sup>∗</sup> 1.00075*<sup>t</sup>* , *t* ∈ [0, *T*](α = 0.75).

When using the EOQ model, excluding the increase in delivery costs and prices, the time between deliveries will be the following:

$$t\_w = \sqrt{\frac{2 \ast 400}{0.001 \ast 20 \ast 25}} = 40 \text{ days}$$

The total purchase, delivery, and storage costs for 400 days are as follows:

$$TC\_w = (400 + 20 \ast 25 \ast 40) \ast 1.001^{400} + (400 \ast 1.00075^{40} + 20 \ast 1.00075^{40} \ast 25 \ast 40) \ast 1.001^{400} + 0.001^{400} \ast 25 \ast 40$$

$$(40) \ast 1.001^{360} + \dots + \left(400 \ast 1.00075^{360} + 20 \ast 1.00075^{360} \ast 25 \ast 40\right) \ast 1.001^{40} = 0$$

$$290911 \text{ EUR}$$

When applying model 1, the time between deliveries is found by Formula (12):

$$t\_{\text{so}} = \frac{40}{\sqrt{1 - 0.75}} = 80 \text{ days}$$

The total purchase, delivery, and storage costs for 400 days are as follows:

$$TC\_0 = (400 + 20 \ast 25 \ast 80) \ast 1.001^{400} + (400 \ast 1.00075^{80} + 20 \ast 1.00075^{80} \ast 25 \ast 1.0001^{320})$$

$$(800) \ast 1.001^{320} + \dots + (400 \ast 1.00075^{320} + 20 \ast 1.00075^{320} \ast 25 \ast 80) \ast 1.001^{80} = 0$$

$$289600 \text{ EUR}$$

The savings will be as follows:

$$
\Delta TC = 290911 - 289600 = 1311 \text{ EUR}
$$

If the cost of delivery and the price decrease equally during the period [0, *T*] with ρ = −0.003, the growth pattern has the following form: *<sup>c</sup>s*(*t*) = <sup>400</sup> <sup>∗</sup> 0.997*<sup>t</sup>* , *<sup>p</sup>*(*t*) = <sup>20</sup> <sup>∗</sup> 0.997*<sup>t</sup>* , *t* ∈ [0, *T*] (α = −3).

When using the EOQ model, excluding the reduction in delivery costs and prices, the time between deliveries will be *t<sup>w</sup>* = 40 days.

The total purchase, delivery, and storage costs for 400 days are as follows:

$$\text{TC}\_w = (400 + 20 \ast 25 \ast 40) \ast 1.001^{400} + (400 \ast 0.997^{40} + 20 \ast 0.997^{40} \ast 25 \ast 40) \ast$$

$$1.001^{360} + \dots + (400 \ast 0.997^{360} + 20 \ast 0.997^{360} \ast 25 \ast 40) \ast 1.001^{40} = 164156 \text{ EUR}$$

When applying model 1, the time between deliveries is found by Formula (12):

$$t\_{\text{so}} = \frac{40}{\sqrt{1+3}} = 20 \text{ days}$$

The total purchase, delivery, and storage costs for 400 days are as follows:

$$\begin{aligned} T\mathbb{C}\_0 &= (400 + 20 \ast 25 \ast 20) \ast 1.001^{400} + (400 \ast 0.997^{20} + 20 \ast 0.997^{20} \ast 25 \ast 20) \ast \\ 1.001^{380} + \cdots &+ (400 \ast 0.997^{380} + 20 \ast 0.997^{380} \ast 25 \ast 20) \ast 1.001^{20} = 160934 \text{EUR} \end{aligned}$$

The savings will be as follows:

$$
\Delta TC = 164156 - 160934 = 3222 \text{ EUR}
$$
