**2. The Proposed Model of Lot Management with Time-Variant Cost Parameters**

Inventory management is understood as the definition of optimal controllable parameters (time between deliveries *t<sup>s</sup>* (time set up) and *q* (quantity, or optimal order size)) of logistics processes, at which the minimum total costs (TC) for the purchase, delivery, and storage of goods is achieved for a certain planned time interval [0, *T*]. If the uncontrollable parameters of the logistics process (purchase price *p*, delivery cost *c<sup>s</sup>* , (cost set up) daily demand µ, and daily interest rate *i* (*r* = *i*/100%) are known and constant throughout the entire planning interval, this problem can be solved by using Wilson economic-mathematical model EOQ (Economic Order Quantity):

$$TC(t\_s) = pD + \frac{c\_s T}{t\_s} + \frac{1}{2}c\_h Dt\_s \tag{1}$$

where *D* is the demand for the period (time interval) [0, *T*] (*D* = µ*T*) and *c<sup>h</sup>* is the cost of storing a unit of goods per day (holding cost).

The optimal time between deliveries (*tso*) and optimal order quantity (*qo*) are found according to the Wilson formula:

$$t\_{\rm so} = t\_w = \sqrt{\frac{2c\_sT}{c\_hD}}\tag{2}$$

$$q\_{\ o} = \mu t\_{\text{so}} \tag{3}$$

Slesarenko and Nestorenko [49], and Nestorenko et al. [50] proposed the modified EOQ model:

$$TC(t\_s) = (c\_s + p\mu t\_s) \frac{(1+r)^{t\_s} \left((1+r)^T - 1\right)}{(1+r)^{t\_s} - 1} \tag{4}$$

The optimal time between deliveries is found by the following formula:

$$t\_{\rm so} = \sqrt{\frac{2c\_{\rm s}}{rp\mu}}\tag{5}$$

The formula coincides with Wilson's Formula (2) if the storage cost is expressed as a percentage of the unit price (*c<sup>h</sup>* = *p*µ).

If the parameters of the logistic process change, the optimal solution is recalculated using Wilson's Formula (5), taking into account the changes. Based on the available information, it is possible to build forecasts for further economic processes of behavior. The use of this information in economic and mathematical models leads to an increase in their adequacy and accuracy.

Zeng et al. [10] proposed models of inventory management that allow for determining the optimal values of parameters in the case when it is known that daily demand has a linear trend (µ(*t*) = µ + ω*t*, *t* ∈ [0, *T*]). To find those parameters, it is necessary to use Wilson's Formula (5), replacing the constant value of daily demand µ with the arithmetic mean of daily demand µ for the planning period [0, *T*] (µ = µ + 0.5ω*T*).

We further construct economic and mathematical models of inventory management that allow for determining the values of optimal controlled parameters in the case when it is known that uncontrolled cost parameters (delivery cost and/or price) have uniform relative trends (*cs*(*t*) = *cs*(1 + ρ*c*) *t* , *p*(*t*) = *p* 1 + ρ*<sup>p</sup> t* , *t* ∈ [0, *T*]).

**Model 1.** *The inventory management model with a simultaneous equal percentage change in the costs of delivery and prices (inflationary model).*

In the EOQ model, the uncontrollable cost parameters as the cost of delivery (*c<sup>s</sup>* = *const*) and price (*p* = *const*) for the period [0, *T*] will be replaced by the assumption that the cost of delivery and the price simultaneously change uniformly with equal percentage change (*cs*(*t*) = *cs*(1 + ρ) *t* , *p*(*t*) = *p*(1 + ρ) *t* , *t* ∈ [0, *T*]). It is an inflationary process when ρ > 0 and a deflationary one when ρ < 0.

The logistics process of purchasing, delivering, and storing goods with constant time between deliveries can be described by the following formula:

$$\begin{split} TC(t\_{\mathsf{s}}) &= (c\_{\mathsf{s}} + p\mu t\_{\mathsf{s}})(1+r)^{\mathsf{nt}\_{\mathsf{s}}} + \left(c\_{\mathsf{s}}(1+\rho)^{t\_{\mathsf{s}}} + p(1+\rho)^{t\_{\mathsf{s}}}\mu t\_{\mathsf{s}}\right)(1+r)^{(\mathsf{n}-1)t\_{\mathsf{s}}} + \cdots \\ &+ \left(c\_{\mathsf{s}}(1+\rho)^{(\mathsf{n}-1)t\_{\mathsf{s}}} + p(1+\rho)^{(\mathsf{n}-1)t\_{\mathsf{s}}}\mu t\_{\mathsf{s}}\right)(1+r)^{t\_{\mathsf{s}}} \end{split} \tag{6}$$

where *n* is the number of deliveries of consignments of goods for the period [0, *T*] (*n* = *T*/*ts*). Replacing it, we get the following:

$$(1+r)^{j t\_s} = e^{\ln\left(1+r\right)j t\_s}, \ (1+\rho)^{j t\_s} = e^{\ln\left(1+\rho\right)j t\_s}, \ j = \overline{1, n}$$

After performing arithmetic transformations, we get the following:

$$TC(t\_s) = (c\_s + p\mu t\_s)e^{\ln\left(1+r\right)T} \left(1 + e^{(\ln\left(1+\rho\right) - \ln\left(1+r\right))t\_s} + \dots + e^{(n-1)\left(\ln\left(1+\rho\right) - \ln\left(1+r\right)\right)t\_s}\right) \tag{7}$$

Using the formula for the sum of the first members of a geometric progression, we get the formula for total costs:

$$TC(t\_s) = (c\_s + p\mu t\_s) \frac{(e^{\ln(1+r) - \ln(1+\rho))T} - 1)e^{\ln\left(1+\rho\right)T}}{e^{(\ln\left(1+r\right) - \ln\left(1+\rho\right))t\_s} - 1} \tag{8}$$

to :

The minimum total cost is obtained as follows:

$$\begin{split} \frac{d\mathrm{TC}(t\_{\mathrm{s}})}{dt\_{\mathrm{s}}} &= \,^{\mathsf{p}}\mu \frac{(e^{(\ln(1+r)-\ln(1+\rho))T}-1)e^{\ln(1+\rho)T}}{e^{(\ln(1+r)-\ln(1+\rho))t\_{\mathrm{s}}}-1} - (\ln(1+r) \\ &- \ln(1+\rho))(c\_{\mathrm{s}}+\mathrm{p}\mu t\_{\mathrm{s}}) \frac{e^{(\ln(1+r)-\ln(1+\rho))t\_{\mathrm{s}}}(e^{(\ln(1+r)-\ln(1+\rho))T}-1)e^{\ln(1+\rho)T}}{\left(e^{(\ln(1+r)-\ln(1+\rho))t\_{\mathrm{s}}}-1\right)^{2}} \\ &= 0 \end{split} \tag{9}$$

After transformations, the equation is as follows:

$$e^{(\ln(1+r)-\ln(1+\rho))t\_s} - 1 = (\ln(1+r) - \ln(1+\rho))\left(\frac{c\_s}{p\mu} + t\_s\right) \tag{10}$$

The optimal time between deliveries of consignments of goods *tso* is found from solving the nonlinear Equation (10). In order to find an approximate solution to Equation (10), we use the first three terms of the Maclaurin series [51] 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*.

$$t\_{\rm so} = t\_{\rm s} = \sqrt{\frac{2c\_{\rm s}}{(\rm r} - \rho)p\mu} \tag{11}$$

Therefore, to determine the optimal time between deliveries of consignments of goods *tso*, one can use Wilson's Formula (5), replacing *r* with the difference *r* − ρ.

Let α = ln(1 + ρ)/ ln(1 + *r*) ≈ ρ/*r*. Then, Equation (11) can be written as follows:

2

$$t\_{s0} = \frac{t\_w}{\sqrt{1 - a}}\tag{12}$$

= , = 1,2,3 … (13)

The dependence of the optimal time between deliveries of consignments of goods *tso* on α is shown in Figure 1. When α ≥ α*max* = 1 − *t* 2 *w T*2 , it is necessary to purchase in the volume *q<sup>o</sup>* = µ*T* and to deliver the goods once for the entire planning period of the logistic process. When α ≤ α*min* = 1 − *t* 2 *<sup>w</sup>*, it is necessary to purchase and deliver goods every day in the amount of *q<sup>o</sup>* = µ. When α*min* < α < α*max*, it is necessary to purchase and deliver goods in *tso* = <sup>√</sup> *tw* 1−α days and in volume *q<sup>o</sup>* = µ*tw* √ 1−α = *qw* √ 1−α . shown in Figure 1. When ≥ = 1 − 2 , it is necessary to purchase in the volume = and to deliver the goods once for the entire planning period of the logistic process. When ≤ = 1 − 2 , it is necessary to purchase and deliver goods every day in the amount of = . When < < , it is necessary to purchase and deliver goods in = √1− days and in volume = √1− = √1− .

**Figure 1.** Dependence of the optimal time between deliveries of consignments of goods on *α*. **Figure 1.** Dependence of the optimal time between deliveries of consignments of goods *tso* on α.

Change in the dependence of total costs ( , ) on the time between deliveries of consignments of goods for different values of *α* as well as the dependence of the minimum total Change in the dependence of total costs *TC*(*t<sup>s</sup>* , α) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different values of α as well as the dependence of the minimum total costs *TC*(*tso*, α) on

= √ 2 

This option on making decision is not optimal (differs from (11)). The result of using this option

= √

for different values of *α* (white squares) is also shown in Figure 2

2(1 + )

(1 + )

Note: If we determine the values of the controlled parameters at *i* moment ( = 1,2,3 … ) of decision-making according to Wilson's Formula (5), we get the same time between deliveries equal the optimal time between deliveries of consignments of goods *tso* for different values of α (black line *Symmetry*  and black squares) are shown in Figure **2020** 2. , *12*, x FOR PEER REVIEW 6 of 19

**Figure 2.** Dependence of total costs ( , ) on the time between deliveries of consignments of goods for different *α* values; dependence of the minimum total costs (, ) on the optimal time between deliveries of consignments of goods for different values *α*; and the value of the total costs (, ) for the time between deliveries of the consignment for different *α* values. **Figure 2.** Dependence of total costs *TC*(*ts*, α) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different α values; dependence of the minimum total costs *TC*(*tso*, α) on the optimal time between deliveries of consignments of goods *tso* for different values α; and the value of the total costs *TC*(*tw*, α) for the time between deliveries of the consignment *tw* for different α values.

**Model 2.** *The inventory management model with the percentage change in the cost of delivery.* Note: If we determine the values of the controlled parameters at *i* moment (*i* = 1, 2, 3 . . .) of decision-making according to Wilson's Formula (5), we get the same time between deliveries equal to *tw*:

$$t\_{si} = \sqrt{\frac{2c\_s(1+\rho)^{t\_{si}}}{rp(1+\rho)^{t\_{si}}\mu}} = \sqrt{\frac{2c\_s}{rp\mu}} = t\_{w\_i} \text{ } i = 1, 2, 3\dots \tag{13}$$

(14)

(16)

(17)

the cost of delivery; if <sup>с</sup> < 0, there is a decrease. Then, the logistics process of purchasing, delivering, and storing goods with constant time between deliveries can be described by the following formula: This option on making decision is not optimal (differs from (11)). The result of using this option for different values of α (white squares) is also shown in Figure 2

#### ( ) = ( + )(1 + ) + ( (1 + <sup>с</sup> ) + )(1 + ) (−1) + ⋯ + ((1 + с) (−1) + )(1 + ) **Model 2.** *The inventory management model with the percentage change in the cost of delivery.*

where *n* is the number of deliveries of consignments of goods for the period [0, ] ( = ⁄ ). Replacing it, we get the following: (1 + ) = ln(1+), (1 + с) = ln(1+с), = 1̅̅̅,̅̅ In the EOQ model, the assumption about constancy of the uncontrolled parameter delivery cost (*c<sup>s</sup>* = *const*) for the period [0, *T*] is replaced by the assumption in which the delivery cost changes uniformly according to the regularity *cs*(*t*) = *cs*(1 + ρ*c*) *t* , *t* ∈ [0, *T*]. If ρ*<sup>c</sup>* > 0, there is an increase in the cost of delivery; if ρ*<sup>c</sup>* < 0, there is a decrease.

After performing arithmetic transformations, we get the following: ( ) = ln(1+) ((1 + (ln(1+с )−ln(1+)) + ⋯ + (−1)(ln(1+с )−ln(1+))) Then, the logistics process of purchasing, delivering, and storing goods with constant time between deliveries can be described by the following formula:

$$\begin{split} TC(t\_s) &= (c\_s + p\mu t\_s)(1+r)^{\text{int}\_s} + \left(c\_s(1+\rho\_c)^{t\_s} + p\mu t\_s\right)(1+r)^{(n-1)t\_s} + \cdots \\ &\quad + \left(c\_s(1+\rho\_c)^{(n-1)t\_s} + p\mu t\_s\right)(1+r)^{t\_s} \end{split} \tag{14}$$

)) (

)) − 1) 2

ln(1+))(

ln(1+) −

(

ln(1+) − 1)

ln(1+) − 1)

ln(1+с ) )

2

= 0

ln(1+) − 1

 (ln(1+)−ln(1+с )) ( ln(1+) − ln(1+с ) ) + ln(1+) ( ln(1+) − 1) where *n* is the number of deliveries of consignments of goods for the period [0, *T*] (*n* = *T*/*ts*).

( ) = (ln(1+)−ln(1+с Replacing it, we get the following:

ln(1+)(

ln(1+) − 1)

))

(

ln(1+) − 1

(ln(1 + ) − ln(1 + <sup>с</sup>

( )

= 

−

$$(1+r)^{\dot{t}t\_s} = e^{\ln\left(1+r\right)\dot{t}\_s}, \\ (1+\rho\_c)^{\dot{t}t\_s} = e^{\ln\left(1+\rho\_c\right)\dot{t}\_s}, \\ \dot{j} = \overline{1,n}$$

)) − 1

(ln(1+)−ln(1+с

After performing arithmetic transformations, we get the following:

$$\begin{split} T\mathbb{C}(t\_{\mathsf{s}}) &= e^{\ln(1+r)T} (c\_{\mathsf{s}} (1 + e^{(\ln\left(1+\rho\_{\mathsf{c}}\right) - \ln\left(1+r\right))t\_{\mathsf{s}}} + \dots + e^{(n-1)(\ln\left(1+\rho\_{\mathsf{c}}\right) - \ln\left(1+r\right))t\_{\mathsf{s}}}) \\ &+ p\mu t\_{\mathsf{s}} (1 + e^{-\ln\left(1+r\right)t\_{\mathsf{s}}} + \dots + e^{-\ln\left(1+r\right)(n-1)t\_{\mathsf{s}}})) \end{split} \tag{15}$$

Using the formula for the sum of the first members of a geometric progression, we get the formula for total costs:

$$TC(t\_s) = c\_s \frac{e^{(\ln(1+r) - \ln(1+\rho\_c))t\_t} \left( e^{\ln\left(1+r\right)T} - e^{\ln\left(1+\rho\_c\right)T} \right)}{e^{(\ln\left(1+r\right) - \ln\left(1+\rho\_c\right))t\_s} - 1} + p\mu t\_s \frac{e^{\ln\left(1+r\right)t\_t} \left( e^{\ln\left(1+r\right)T} - 1 \right)}{e^{\ln\left(1+r\right)t\_s} - 1} \tag{16}$$

The minimum total costs is as follows:

$$\begin{split} \frac{d\mathbf{T}\mathbf{C}(t\_{\mathrm{s}})}{dt\_{\mathrm{s}}} &= p\mu \frac{\epsilon^{\ln(1+r)t\_{\mathrm{s}}} \left(\epsilon^{\ln(1+r)T} - 1\right)}{\epsilon^{\ln(1+r)t\_{\mathrm{s}}} - 1} - \ln(1+r)p\mu t\_{\mathrm{s}} \frac{\epsilon^{\ln(1+r))t\_{\mathrm{s}}} \left(\epsilon^{\ln\left(1+r\right)T} - 1\right)}{\left(\epsilon^{\ln\left(1+r\right) - \ln\left(1+\rho\_{\mathrm{c}}\right)} \right) \epsilon^{\ln\left(1+r\right)t\_{\mathrm{s}}} \left(\epsilon^{\ln\left(1+r\right)t\_{\mathrm{s}} - 1} \right)^{2}} \\ &- \mathbf{c}\_{\mathrm{s}} \frac{\left(\ln(1+r) - \ln(1+\rho\_{\mathrm{c}})\right) \epsilon^{\ln\left(1+r\right) - \ln\left(1+\rho\_{\mathrm{c}}\right) \left(\epsilon^{\ln\left(1+r\right)T} - 1\right)^{2}} \frac{\left(\epsilon^{\ln\left(1+r\right)T} - 1\right)^{2}}{\left(\epsilon^{\ln\left(1+r\right) - \ln\left(1+\rho\_{\mathrm{c}}\right)\right) t\_{\mathrm{s}}}} = 0 \end{split} \tag{17}$$

After transformations, the equation is as follows:

$$\begin{split} & \varepsilon^{\ln(1+r)t\_{\mathcal{S}}} - 1 - \ln(1+r)t\_{\mathcal{S}} \\ &= \frac{\varepsilon\_{\mathcal{S}}}{p\mu} \frac{(\ln(1+r) - \ln(1+\rho\_{\mathcal{E}})) \left(\varepsilon^{\ln(1+r)t\_{\mathcal{S}}} - 1\right)^{2}}{\varepsilon^{\ln(1+\rho\_{\mathcal{E}})} \left(\varepsilon^{(\ln(1+r)-\ln(1+\rho\_{\mathcal{E}}))t\_{\mathcal{S}}} - 1\right)^{2}} \frac{\left(\varepsilon^{\ln(1+r)T} - \varepsilon^{\ln\left(1+\rho\_{\mathcal{E}}\right)T}\right)}{\left(\varepsilon^{\ln\left(1+r\right)T} - 1\right)} \end{split} \tag{18}$$

In model 2, the optimal time between deliveries of consignments of goods *tso* is also found from the solution of the nonlinear Equation (18). In order to find an approximate solution of Equation (18), 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>* 2 and the first term of the Maclaurin series of the expansion of the function *y* = ln(1 + *r*) ≈ *r*.

$$t\_{\rm so} = t\_{\rm s} = \sqrt{\frac{2c\_s(1+\rho\_c)^{\frac{1}{2}T}}{\rm r\mu}}\tag{19}$$

Consequently, to determine the optimal time between deliveries of consignments of goods *tso*, one can use Wilson's Formula (5), replacing the constant value of the delivery cost *c<sup>s</sup>* with the geometric mean of the delivery cost *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* ).

$$t\_{so} = (1 + \rho\_c)^{\frac{1}{4}T} t\_w \tag{20}$$

Let α*<sup>c</sup>* = ln(1 + ρ*c*)/ ln(1 + *r*); then Equation (20) can be represented in this form:

$$t\_{\rm so} = (1+r)^{\frac{1}{4}\alpha\_{\rm c}T}t\_w\tag{21}$$

The dependence of the optimal time between deliveries of consignments of goods *tso* on α*<sup>c</sup>* is shown in Figure 3.

(1 + <sup>с</sup> ) 1 2 ).

After transformations, the equation is as follows:

 ln(1+с )(

geometric mean of the delivery cost ̅

ln(1+) − 1 − ln(1 + )

(ln(1 + ) − ln(1 + <sup>с</sup>

)) (

(ln(1+)−ln(1+с

<sup>=</sup> <sup>=</sup> <sup>√</sup>

one can use Wilson's Formula (5), replacing the constant value of the delivery cost

= (1 + <sup>с</sup>

= (1 + )

Let = ln (1 + )⁄ln (1 + ); then Equation (20) can be represented in this form:

ln(1+) − 1)

)) − 1) 2

In model 2, the optimal time between deliveries of consignments of goods is also found from the solution of the nonlinear Equation (18). In order to find an approximate solution of Equation (18), we use the first three terms of the Maclaurin series of the expansion of the function = ≈ 1 + + 0.5<sup>2</sup> and the first term of the Maclaurin series of the expansion of the function = ln (1 + ) ≈ .

2

Consequently, to determine the optimal time between deliveries of consignments of goods ,

) 1 4 

1 4 α 

(1 + <sup>с</sup> ) 1 2 

r

2

(

ln(1+) −

(

for the planning period [0, ] ( ̅ = √

ln(1+с ) ) (18)

(19)

with the

(20)

(21)

is

(1 + <sup>с</sup> ) =

ln(1+) − 1)

= 

**Figure 3.** The dependence of optimal time between deliveries of consignments of goods *tso* on α*c*.

**Figure 3.** The dependence of optimal time between deliveries of consignments of goods on . The change in the dependence of total costs ( , ) on the time between deliveries of consignments of goods for different values as well as the dependence of the minimum total The change in the dependence of total costs *TC*(*t<sup>s</sup>* , α*c*) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different α*<sup>c</sup>* values as well as the dependence of the minimum total costs *TC*(*tso*, α*c*) on the optimal time between deliveries of consignments of goods *tso* for different α*<sup>c</sup>* values (black line and black squares) are shown in Figure *Symmetry* **2020** 4. , *12*, x FOR PEER REVIEW 8 of 19

costs (, ) on the optimal time between deliveries of consignments of goods for different

**Figure 4.** The dependence of total costs ( , ) on the time between deliveries of consignments of goods for different values and the dependence of the minimum total costs (, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares). **Figure 4.** The dependence of total costs *TC*(*ts*, α*c*) on the time between deliveries of consignments of goods *t<sup>s</sup>* for different α*<sup>c</sup>* values and the dependence of the minimum total costs *TC*(*tso*, α*c*) on the optimal time between deliveries of consignments of goods *tso* for different α*<sup>c</sup>* values (black line and black squares).

#### **Model 3.** *Inventory management model with a percentage change in the price of goods.*  **Model 3.** *Inventory management model with a percentage change in the price of goods.*

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 is an increase in the price of goods, and if < 0, there is a decrease. To construct model 3, we will use the results from constructing model 1 (8) and model 2 (16). 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 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.

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 (model 2) by the same number of times. A change in the price To construct model 3, 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

(ln(1+)−ln(1+)) (

(ln(1+)−ln(1+))(

(ln(1+)−ln(1+)) − 1)

cost of delivery, according to model 2, we make a replacement 1 + = 1 (1 + ⁄ ):

ln(1+) − 1)

ln(1+) − 1)

(ln(1+)−ln(1+)) − 1)

2

(ln(1+)−ln(1+)) − 1

(

ln(1+) − 1

(

( )

= −

+ 

−

= 0

) = 

ln(1 + )

ln(1+)(

ln(1+)(

(ln(1 + ) − ln(1 + ))

After transformations, the equation is as follows:

ln(1+) − 1)

The minimum total cost is found as follows:

(

(ln(1+)−ln(1+))(

+

ln(1+)

(ln(1+)−ln(1+)) − 1)

(ln(1+)−ln(1+)) − 1)

(ln(1+)−ln(1+)) − 1

ln(1+)

2

. To compensate for the change in the

ln(1+)

ln(1+)

(22)

(23)

(decrease/increase) in the delivery cost (model 2) by the same 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* . To compensate for the change in the cost of delivery, according to model 2, we make a replacement 1 + ρ*<sup>c</sup>* = 1/(1 + ρ*p*):

$$\begin{split} TC(t\_s) &= c\_s \frac{e^{\ln\left(1+r\right)t\_s} \left(e^{\ln\left(1+r\right)T} - 1\right)}{e^{\ln\left(1+r\right)t\_s} - 1} \\ &+ p\mu t\_s \frac{e^{\left(\ln\left(1+r\right) - \ln\left(1+\rho\_p\right)\right)t\_s} \left(e^{\left(\ln\left(1+r\right) - \ln\left(1+\rho\_p\right)\right)T} - 1\right) e^{\ln\left(1+\rho\_p\right)T}}{e^{\left(\ln\left(1+r\right) - \ln\left(1+\rho\_p\right)\right)t\_s} - 1} \end{split} \tag{22}$$

The minimum total cost is found as follows:

$$\begin{split} & \frac{d\overline{\mathcal{T}\mathcal{C}(t\_{\varepsilon})}}{dt\_{\varepsilon}} \\ &= -c\_{\mathrm{S}} \frac{\ln(1+r)\epsilon^{\ln\left(1+r\right)t\_{\mathrm{S}}} \left(\epsilon^{\ln\left(1+r\right)T} - 1\right)}{\left(\epsilon^{\ln\left(1+r\right)t\_{\mathrm{S}}} - 1\right)^{2}} \\ &+ p\mu \frac{\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right)t\_{\mathrm{S}}} \left(\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right))T} - 1\right) \epsilon^{\ln\left(1+\rho\right)T}}{\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right)t\_{\mathrm{S}}} \left(\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right))t\_{\mathrm{S}}}} \\ & -p\mu t\_{\mathrm{S}} \frac{(\ln(1+r)-\ln\left(1+\rho\right))\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right))t\_{\mathrm{S}}} \left(\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right))T} - 1\right) \epsilon^{\ln\left(1+\rho\right)T}}{\left(\epsilon^{\ln\left(1+r\right)-\ln\left(1+\rho\right))t\_{\mathrm{S}}} - 1\right)^{2}} \\ &= 0 \end{split} \tag{23}$$

After transformations, the equation is as follows:

$$\begin{split} &e^{(\ln(1+r)-\ln(1+\rho\_p))t\_s} - 1 - (\ln(1+r) - \ln(1+\rho\_p))t\_s \\ & \qquad = \frac{c\_s}{p\mu} \frac{\ln(1+r)e^{\ln\left(1+\rho\_p\right)t\_s} \Big(e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_p\right))t\_s} - 1\Big)^2 \Big(e^{\ln\left(1+r\right)T} - 1\Big)}}{\left(e^{\ln\left(1+r\right)t\_s} - 1\right)^2 \Big(e^{(\ln\left(1+r\right)-\ln\left(1+\rho\_p\right))T} - 1\Big) e^{\ln\left(1+\rho\_p\right)T}} \end{split} \tag{24}$$

We repeat the previously mentioned procedure: the optimal time between deliveries of consignments of goods *tso* is found from the solution of the nonlinear Equation (24). In order to find an approximate solution of Equation (24), 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*.

$$t\_s = \sqrt{\frac{2c\_s}{(r - \rho\_p)p e^{\frac{1}{2}\ln\left(1 + \rho\_p\right)T}\mu}}\tag{25}$$

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* ).

Let <sup>α</sup>*<sup>p</sup>* <sup>=</sup> ln 1 + ρ*<sup>p</sup>* / ln(1 + *r*); then Equation (25) can be represented as follows:

$$t\_{so} = \frac{t\_{w}}{\left(1 + \rho\_p\right)^{\frac{1}{4}T} \sqrt{1 - \alpha\_p}}\tag{26}$$

or

$$t\_{\rm so} = \frac{t\_w}{(1+r)^{\frac{1}{4}\alpha\_p T} \sqrt{1-\alpha\_p}}\tag{27}$$

 .

√(1 + )

or

The dependence of the optimal time between deliveries of consignments of goods *tso* on α*<sup>c</sup>* is shown in Figure 5. The dependence of the optimal time between deliveries of consignments of goods on is shown in Figure 5.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 9 of 19

ln(1+)(

( − )

Let = ln (1 + )⁄ln (1 + ); then Equation (25) can be represented as follows:

(1 + )

(1 + ) 1 4 

=

=

<sup>2</sup>(

We repeat the previously mentioned procedure: the optimal time between deliveries of consignments of goods is found from the solution of the nonlinear Equation (24). In order to find an approximate solution of Equation (24), we use the first three terms of the Maclaurin series of the expansion of the function = ≈ 1 + + 0.5<sup>2</sup> and the first term of the Maclaurin series of the

2

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, ] (̅ =

1 4 

√1 −

√1 −

1 2

ln(1+)

ln(1+) − 1)

(ln(1+)−ln(1+)) − 1)

(ln(1+)−ln(1+)) − 1)

2 ( ln(1+) − 1)

(24)

(25)

(26)

(27)

ln(1+)

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

ln(1 + )

(

= √

= 

expansion of the function = ln (1 + ) ≈ .

= (1 + )

1 2 ).

**Figure 5.** The dependence of the optimal time between deliveries of consignments of goods on **Figure 5.** The dependence of the optimal time between deliveries of consignments of goods *tso* on α*c*.

The change in the dependence of total costs ( , ) on the time between deliveries of consignments of goods for different values as well as the dependence of the minimum total costs (, ) on the optimal time between deliveries of consignments of goods for different The change in the dependence of total costs *TC ts* , α*<sup>p</sup>* 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>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** 6. , *12*, x FOR PEER REVIEW 10 of 19

**Figure 6.** The dependence of total costs ( , ) on the time between deliveries of consignments of goods for different values and the dependence of the minimum total costs (, ) on the optimal time between deliveries of consignments of goods for different values (black line and black squares). **Figure 6.** The dependence of total costs *TC ts*, α*p* on the time between deliveries of consignments of goods *<sup>t</sup><sup>s</sup>* for different <sup>α</sup>*<sup>p</sup>* values and the dependence of the minimum total costs *TC tso*, α*p* on the optimal time between deliveries of consignments of goods *tso* for different α*<sup>p</sup>* values (black line and black squares).
