**Model 4.** *Inventory management model with a different percentage change in the price of goods.*  **Model 4.** *Inventory management model with a di*ff*erent percentage change in the price of goods.*

is an increase in the price of goods, and if < 0, there is a decrease.

to model 2, we will receive by replacement of 1 + = (1 + ) (1 + ⁄ ):

))(

(ln(1+)−ln(1+)) − 1)

(ln(1+)−ln(1+с

(ln(1+)−ln(1+с

(ln(1+)−ln(1+)) (

(ln(1+)−ln(1+с

+

(ln(1+)−ln(1+)) − 1

(

(ln(1+)−ln(1+с

(ln(1+)−ln(1+с

(

After transformations, the equation appears as follows:

))

The minimum total costs are found as follows:

) in Formula (16) and will be multiplied by

) = 

(ln(1+)−ln(1+)) (

(ln(1 + ) − ln(1 + <sup>с</sup>

(ln(1 + ) − ln(1 + <sup>с</sup>

(ln(1 + ) − ln(1 + ))

(ln(1+)−ln(1+)) − 1 − (ln(1 + ) − ln(1 + ))

))

(

(

( )

= 

−

−

= 

In the EOQ model, the assumption of the constancy of the uncontrollable parameter of the product price ( = ) for the period [0, ] is replaced by the assumption that the price of the product changes uniformly according to the order that () = (1 + ) , ∈ [0, ]). If > 0, there In the EOQ model, the assumption of the constancy of the uncontrollable parameter of the product price (*p* = *const*) for the period [0, *T*] is replaced by the assumption that the price of the product changes

(decrease/increase) in the delivery cost by a certain number of times. A change in the price and delivery cost, according to model 1, will lead to the replacement of ln (1 + ) to ln(1 + ) − ln (1 +

ln(1+)

(ln(1+)−ln(1+с

(ln(1+)−ln(1+с

(ln(1+)−ln(1+))(

))(

(ln(1+)−ln(1+)) − 1)

))(

)) − 1)

)) − 1) 2

<sup>2</sup>(

)) − 1)

(ln(1+)−ln(1+)) − 1

)) − 1

ln(1+)

(ln(1+)−ln(1+с

2

(ln(1+)−ln(1+)) − 1)

ln(1+) −

ln(1+с )

(ln(1+)−ln(1+)) − 1)

)) − 1)

2 (

)

ln(1+)

(ln(1+)−ln(1+)) − 1)

ln(1+с )

ln(1+)

ln(1+) −

= 0

ln(1+с ) )

. The change in the cost of delivery, according

ln(1+)

(28)

(29)

(30)

To construct model 4, we will use the results from constructing model 1 (8) and model 2 (16).

uniformly according to the order that *p*(*t*) = *c<sup>p</sup>* 1 + ρ*<sup>p</sup> t* , *t* ∈ [0, *T*]). If ρ*<sup>p</sup>* > 0, there is an increase in the price of goods, and if ρ*<sup>p</sup>* < 0, there is a decrease.

To construct model 4, we will use the results from constructing model 1 (8) and model 2 (16). We represent the change (increase/decrease) in the price as a combination of two processes—the change (increase/decrease) in the price and delivery cost (model 1) and the simultaneous change (decrease/increase) in the delivery cost by a certain number of times. A change in the price and delivery cost, according to model 1, will lead to the replacement of ln(<sup>1</sup> <sup>+</sup> *<sup>r</sup>*) to ln(<sup>1</sup> <sup>+</sup> *<sup>r</sup>*) <sup>−</sup> ln 1 + ρ*<sup>p</sup>* in Formula (16) and will be multiplied by *e* ln (1+ρ*p*)*T* . The change in the cost of delivery, according to model 2, we will receive by replacement of 1 + ρ*<sup>c</sup>* = (1 + ρ*c*)/(1 + ρ*p*):

$$\begin{split} TC(t\_{\mathsf{s}}) &= c\_{\mathsf{s}} \frac{e^{(\ln(1+r)-\ln\left(1+\rho\_{\mathsf{c}}\right))t\_{\mathsf{s}}} \Big( e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathsf{c}}\right))T} - 1 \Big) e^{\ln\left(1+\rho\_{\mathsf{c}}\right)T}}{e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathsf{c}}\right))t\_{\mathsf{s}}} - 1} \\ &+ p\mu t\_{\mathsf{s}} \frac{e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathsf{c}}\right))t\_{\mathsf{s}}} \Big( e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathsf{c}}\right))T} - 1 \Big) e^{\ln\left(1+\rho\_{\mathsf{c}}\right)T}}{e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathsf{c}}\right))t\_{\mathsf{s}}} - 1} \end{split} \tag{28}$$

The minimum total costs are found as follows:

$$\begin{split} &\frac{d\nabla\mathbb{C}(t)}{dt} \\ &=p\mu\frac{e^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))t\epsilon\Big{}\Big{(}\epsilon^{\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))T}-1\Big{)}e^{\ln(1+\rho\_{\mathbb{P}})T}}{\epsilon^{\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))t\epsilon-1}} \\ &-c\_{s}\frac{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))e^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))t}\Big{(}\epsilon^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))T}-1\Big{)}e^{\ln(1+\rho\_{\mathbb{P}})T}}{\epsilon^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))t\epsilon-1}\Big{)} \\ &p\mu\mathbb{f}\_{\mathbb{S}}\frac{\left(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}})\right)\epsilon^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))t}\Big{(}\epsilon^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))T}-1\Big{)}e^{\ln(1+\rho\_{\mathbb{P}})T}}{\left(\epsilon^{(\ln(1+r)-\ln(1+\rho\_{\mathbb{P}}))t}-1\right)^{2}} = 0 \end{split} \tag{29}$$

After transformations, the equation appears as follows:

$$\begin{split} &e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathcal{P}}\right))t\_{\mathcal{S}}} - 1 - \left(\ln\left(1+r\right) - \ln\left(1+\rho\_{\mathcal{P}}\right)\right)t\_{\mathcal{S}} \\ &= \frac{C\_{\mathfrak{s}}}{p\_{\mu}} \frac{(\ln(1+r)-\ln(1+\rho\_{\mathcal{C}}))e^{(\ln\left(1+\rho\_{\mathcal{P}}\right)-\ln\left(1+\rho\_{\mathcal{C}}\right))t\_{\mathcal{S}}} \Big(e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathcal{P}}\right))t\_{\mathcal{S}}}-1\right)^{2}\Big(e^{\ln\left(1+r\right)\mathcal{T}}-e^{\ln\left(1+\rho\_{\mathcal{C}}\right)\mathcal{T}}\Big)}{\left(e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_{\mathcal{C}}\right))t\_{\mathcal{S}}}-1\right)^{2}\Big(e^{\ln\left(1+r\right)\mathcal{T}}-e^{\ln\left(1+\rho\_{\mathcal{C}}\right)\mathcal{T}}\Big)} \end{split} \tag{30}$$

The optimal time between deliveries of consignments of goods *tso* is found from the solution of the nonlinear Equation (30). In order to find an approximate solution of Equation (30), we use the first three terms of the Maclaurin series of the expansion of the function *y* = *e <sup>x</sup>* <sup>≈</sup> <sup>1</sup> + *<sup>x</sup>* + 0.5*<sup>x</sup>* <sup>2</sup> and the first term of the Maclaurin series of the expansion of the function *y* = ln(1 + *r*) ≈ *r*.

$$t\_{so} = t\_s = \sqrt{\frac{2c\_s(1+\rho\_c)^{\frac{1}{2}T}}{\left(\mathbf{r}-\rho\_p\right)p\left(1+\rho\_p\right)^{\frac{1}{2}T}}}\tag{31}$$

Consequently, to determine the optimal time between deliveries of consignments of goods *tso*, one can use Wilson's Formula (5), replacing r by the difference *r* − ρ and the constant value of the price of goods *p* with the geometric mean of the price of goods *p* for the planning period [0, *T*] (*p* = q *pp* 1 + ρ*<sup>p</sup> T* = *p* 1 + ρ*<sup>p</sup>* 1 2 *T* ) and the constant value of the product price *c<sup>s</sup>* by geometric mean value of the product price *c<sup>s</sup>* for the planning period [0, *T*] (*c<sup>s</sup>* = q *cscs*(1 + ρ*c*) *<sup>T</sup>* = *cs*(1 + ρ*c*) 1 2 *T* ).

Let <sup>α</sup>*<sup>p</sup>* <sup>=</sup> ln 1 + ρ*<sup>p</sup>* / ln(1 + *r*) and α*<sup>c</sup>* = ln(1 + ρ*c*)/ ln(1 + *r*); then Equation (31) can be represented as follows:

$$t\_{\rm so} = \frac{(1+\rho\_c)^{\frac{1}{4}T}}{\left(1+\rho\_p\right)^{\frac{1}{4}T}} \frac{t\_{\rm tw}}{\sqrt{1-\alpha\_p}}\tag{32}$$

√(1 + )

of the product price ̅

represented as follows:

= (1 + )

1 2 

or or

$$t\_{\rm so} = (1+r)^{\frac{1}{4}(a\_{\rm c} - \alpha\_p)T} \frac{t\_w}{\sqrt{1-\alpha\_p}} \tag{33}$$

(1 + ) =

The dependence of the optimal time between deliveries of consignments of goods *tso* on α*<sup>c</sup>* and α*<sup>p</sup>* is shown in Figure 7. The dependence of the optimal time between deliveries of consignments of goods on and is shown in Figure 7.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 11 of 19

the first term of the Maclaurin series of the expansion of the function = ln (1 + ) ≈ .

for the planning period [0, ] (̅ = √

(1 + ) 1 4 

(1 + )

=

= = √

The optimal time between deliveries of consignments of goods is found from the solution of the nonlinear Equation (30). In order to find an approximate solution of Equation (30), we use the first three terms of the Maclaurin series of the expansion of the function = ≈ 1 + + 0.5<sup>2</sup> and

2

(1 + <sup>с</sup> ) 1 2 

> 1 2

(31)

(32)

by geometric mean value

(1 + ) 1 2 ).

(r − )(1 + )

Consequently, to determine the optimal time between deliveries of consignments of goods , one can use Wilson's Formula (5), replacing r by the difference − and the constant value of the price of goods with the geometric mean of the price of goods ̅ for the planning period [0, ] (̅ =

) and the constant value of the product price

Let = ln (1 + )⁄ln (1 + ) and = ln (1 + )⁄ln (1 + ) ; then Equation (31) can be

1 4 

 √1 −

**Figure 7.** The dependence of the optimal time between deliveries of consignments of goods on and . **Figure 7.** The dependence of the optimal time between deliveries of consignments of goods *tso* on α*<sup>c</sup>* and α*p*.

The change in the dependence of the total costs ( , = −3, ) with a decrease in the cost of delivery ( = −3) on the time between deliveries of consignments of goods at different values as well as the dependence of the minimum total costs (, = −3, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares) are shown in Figure 8. The change in the dependence of the total costs *TC ts* , α*<sup>c</sup>* = −3, α*<sup>p</sup>* with a decrease in the cost of delivery (α*<sup>c</sup>* = −3) on the time between deliveries of consignments of goods *t<sup>s</sup>* at different α*<sup>p</sup>* values as well as the dependence of the minimum total costs *TC tso*, α*<sup>c</sup>* = −3, α*<sup>p</sup>* on the optimal time between deliveries of consignments of goods *tso* for different α*<sup>p</sup>* values (black line and black squares) are shown in Figure *Symmetry* **2020** 8. , *12*, x FOR PEER REVIEW 12 of 19

**Figure 8.** The dependence of the total costs ( , = −3, )) with a decrease in the cost of delivery ( = −3) on the time between deliveries of consignments of goods at different values and the dependence of the minimum total costs (, = −3, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares). **Figure 8.** The dependence of the total costs *TC ts*, α*<sup>c</sup>* = −3, α*<sup>p</sup>* with a decrease in the cost of delivery (α*<sup>c</sup>* = −3) on the time between deliveries of consignments of goods *t<sup>s</sup>* at different α*<sup>p</sup>* values and the dependence of the minimum total costs *TC tso*, α*<sup>c</sup>* = −3, α*<sup>p</sup>* on the optimal time between deliveries of consignments of goods *tso* for different α*p* values (black line and black squares).

dependence of the minimum total costs (, = 0, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares) are shown in

as well as the dependence of the minimum total costs (, = 3, ) on the optimal time between deliveries of consignments of goods for different values (black line and black

dependence of the minimum total costs (, = 3, ) on the optimal time between deliveries of

on the time between deliveries of consignments of goods

The change in the dependence of total costs (

**Figure 9.** The dependence of total costs (

( = 3) on the time between deliveries of consignments of goods

consignments of goods for different values (black line and black squares).

delivery ( = 3) on the time between deliveries of consignments of goods

Figure 6.

squares) are shown in Figure 9.

, = 0, ) at constant delivery cost ( = 0)

, = 3, ) with an increase in the cost of

, = 3, ) with an increase in the cost of delivery

for different values and the

for different values

for different values as well as the

**Figure 8.** The dependence of the total costs (

( = −3) on the time between deliveries of consignments of goods

The change in the dependence of total costs *TC ts* , α*<sup>c</sup>* = 0, α*<sup>p</sup>* at constant delivery cost (α*<sup>c</sup>* = 0) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different α*<sup>p</sup>* values as well as the dependence of the minimum total costs *TC tso*, α*<sup>c</sup>* = 0, α*<sup>p</sup>* on the optimal time between deliveries of consignments of goods *tso* for different α*<sup>p</sup>* values (black line and black squares) are shown in Figure 6. The change in the dependence of total costs ( , = 0, ) at constant delivery cost ( = 0) on the time between deliveries of consignments of goods for different values as well as the dependence of the minimum total costs (, = 0, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares) are shown in Figure 6.

dependence of the minimum total costs (, = −3, ) on the optimal time between deliveries

, = −3, )) with a decrease in the cost of delivery

at different values and the

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 12 of 19

The change in the dependence of total costs *TC ts* , α*<sup>c</sup>* = 3, α*<sup>p</sup>* with an increase in the cost of delivery (α*<sup>c</sup>* = 3) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different α*<sup>p</sup>* values as well as the dependence of the minimum total costs *TC tso*, α*<sup>c</sup>* = 3, α*<sup>p</sup>* on the optimal time between deliveries of consignments of goods *tso* for different α*<sup>p</sup>* values (black line and black squares) are shown in Figure 9. The change in the dependence of total costs ( , = 3, ) with an increase in the cost of delivery ( = 3) on the time between deliveries of consignments of goods for different values as well as the dependence of the minimum total costs (, = 3, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares) are shown in Figure 9.

**Figure 9.** The dependence of total costs ( , = 3, ) with an increase in the cost of delivery ( = 3) on the time between deliveries of consignments of goods for different values and the dependence of the minimum total costs (, = 3, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares). **Figure 9.** The dependence of total costs *TC ts*, α*<sup>c</sup>* = 3, α*<sup>p</sup>* with an increase in the cost of delivery (α*<sup>c</sup>* = 3) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different α*<sup>p</sup>* values and the dependence of the minimum total costs *TC tso*, α*<sup>c</sup>* = 3, α*<sup>p</sup>* on the optimal time between deliveries of consignments of goods *tso* for different α*p* values (black line and black squares).

Model 4 is a generalization of models 1–3:

for α*<sup>c</sup>* = α*<sup>p</sup>* we obtain model 1; for α*<sup>p</sup>* = 0 we obtain model 2; and for α*<sup>c</sup>* = 0 we obtain model 3.

Further, we provide some examples of the differences that arise when using models 1–4 and the modified EOQ model.
