MSC 91B38, 91B74 DOI: 10.14529/mmp170407 ... - kpi.kharkov.ua · 1National ecThnical University...

MSC 91B38, 91B74 DOI: 10.14529/mmp170407

MODEL OF CONVEYER WITH THE REGULABLE SPEED

Î.Ì. Pihnastyi1, V.D. Khodusov2

1National Technical University Kharkiv Polytechnic Institute, Kharkov, Ukraine2Karazin Kharkiv National University, Kharkov, UkraineE-mail: [email protected], [email protected]

This article is devoted to mathematical modelling of the production �ow lines of the

conveyor-type. Here is an analytical method for calculating the parameters of a production

line with a regulated speed of the movement of the subjects of the labour along the conveyor

developed. The description of the parameters of the state of the production line is made in

the one-moment approximation using partial di�erential equations. There has been derived

a solution that determines the state of the parameters of the production line for a given

technological position as a function of the time. The transitional period during which the

initial condition of the distributing of the subjects of the labour along a conveyor has the

in�uence on the state parameters of a production line is certain. The developed method

of the calculation of the �ow parameters of the production line allows designing control

systems of the production line of the conveyor-type with a regulated rate of the movement

of the subjects of the labour.

Keywords: conveyor; a subject of labour; production line; PDE-model of the production;

parameters of the state of the production line; technological position; transition period;

production management systems.

Introduction

To model industrial systems with a �ow-based method of the organising production,in the vast majority of the cases, four main types of models are using: discrete-eventmodels (DES-model) [1], models of the queuing [2], the �uid model [3] and the PDE �model [4, 5], that use the partial di�erential equations. A detailed comparative analysisof these models was done in the articles [4, 6, 7]. The PDE�models are most demand atthe moment when designing control systems for �ow lines that operate in transient modesor with variable capacity [8]. Production systems with a variable output of the productsinclude in oneself some production systems that use the conveyor type of the organisationof the production (Fig. 1) [9�11]. It should also be noted that the PDE-models are verye�ectively used to describe the industrial production of semiconductor products [4�6],rolled steel [12]. In this paper, we will consider in detail the model of a conveyor line witha regulable speed using partial di�erential equations. The control of the speed of the beltof a separate conveyor changes the statistical characteristics of the �ow at the output.It leads to a change in the amount of input �ow on subsequent conveyors, which a�ectstheir power consumption [13]. Uneven loading of rock along the belt conveyor directlya�ects the transportation cost of the rock and determines the dynamics of the system as awhole. When designing the control systems of the conveyor, it is required to the plannednormative volumes of the rock mining. In fact, in real production, the �ow of the rockvaries in time. These changes are very signi�cant even during the daily period of the rockmining (Fig. 2) [14]. Fig. 2 demonstrates the percentage of the time, during which aconveyor works in one of several speed modes, certain relative speed g = v/Vn, where v

64 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 4, pp. 64�77

ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ

Fig. 1. Characteristics of the conveyor line Neyveli Lignite Corporation, India [9]

Fig. 2. Chart of a relative speed of the belt conveyer for a day long: a)mine WESTFALEN,Germany; b)mine KWK ANNA, Poland [14]

actual speed of the conveyor, Vn is the maximally possible speed of the conveyor duringthe day. The speed of the conveyor belt should correspond to the level that ensures theoptimum work of the conveyor line. The uneven distribution of the loaded rock along thelength of the conveyor has a signi�cant e�ect on the law control the speed of the conveyor.This e�ect consists in the fact that the process of the changing the speed of the conveyorbelt is performed with a delay, the value of which is determined the linear density of thedistribution the rock along the conveyor line. It can be assumed that the value of delay isproportional to the length of the conveyor and is inversely proportional to the speed of thebelt. Although a large number of research on conveyor lines, the problem of regulating thespeed of belt conveyors, which reduce energy consumption at transporting, is currentlyvery actual [11]. Exclusive attention deserves the problem of the transport dependenceand operating costs on changing the distributing of the rock load along the conveyor andthe speed of the conveyor belt. In this regard, this article is devoted to the study of thedistribution of the linear load of the rock along the conveyor on the speed of the conveyorbelt.

1. PDE-Model of the Production Line

For investigating the parameters of conveyor lines the basic models are discrete-eventmodels and queuing models [10, 11, 13]. These models are not e�ective for modellingproduction lines because they do not allow describing the distribution subjects of thelabour along the production line [4, 7, 8]. For designing control systems for modern

Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèåè ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2017. Ò. 10, � 4. Ñ. 64�77

65

Î.Ì. Pihnastyi, V.D. Khodusov

production lines, at the moment a new class of models uses � PDE-model [4�6]. PDE-models take into account the stochastic nature of the interact of the technologicalequipment and the subject of labour during technological processing [5, 6, 16] as wellas their distribution by technological positions. We will consider the ideal productionsystems, which are characterised by the absence of loss of the subjects of the labour as aresult of the production process (there are no defective products). The system of equationsthat determines the behaviour of the parameters of the production line in a one-momentdescription has the form [4, p. 70; 5, p. 4585]:

∂ [χ]0 (t, S)

∂t+∂ [χ]1 (t, S)

∂S= 0, (1)

[χ]1 (t, S) = [χ]1ψ (t, S), [χ]0 (t0, S) = Ψ (S), (2)

where [χ]0 (t, S) is the density of the distribution of the subjects of the labourby technological positions (WIP: work-in-progress); [χ]1ψ (t, S) is the capacity of theprocessing of the subjects of the labour on a technological operation; [χ]1 (t, S) is therate of processing of the subjects of the labour by technological positions at a time t. Theposition of the subject of the labour in the technological route is characterized by thecoordinate S ∈ (0;Sd). When using the coordinates of the cost space, Sd corresponds thefull self-cost of the product. The initial condition determines the number of the subjectsof the labour at a time t on each technological operation. The solution of the system ofthe equations (1),(2) allows us to calculate the parameters of the production line.

2. PDE-Model of the Conveyor Line

A widespread method of organizing a �ow production is the use of conveyor lines withthe variable speed of the belt movement [10,11,13,14]. A conveyor with a variable speed ofthe belt is used to reduce the costs of the transporting the rock (in the mines, [10,11,13]), aswell as to synchronize the output of the product with the existing demand (in industrialenterprises, [4, 5]). A characteristic feature of the modelling of the conveyor line for anindustrial enterprise is that the subjects of the labour move along the conveyor with thesame speed. Similarly, for mine conveyors, the rock transport speed at an arbitrary locationof the conveyor is equal to the speed of the belt. In this connection, the system of equationsdescribing the motion of the rock along the conveyor line has the form:

∂ [χ]0 (t, S)

∂t+∂ [χ]1 (t, S)

∂S= δ (S)λ (t),

∞∫−∞

δ(S)dS = 1, (3)

[χ]1 (t, S) = [χ]0 (t, S) a(t), [χ]0 (t0, S) = H (S)Ψ (S), H(S) =

{0, if S < 0;

1, if S > 0.(4)

The technological position with the coordinate S = Sd corresponds to the degreeof readiness of the subject of the labour, that is, the state to which the subject of thelabour must correspond when leaving the conveyor line according to the production andtechnological documentation. Parameters [χ]0 (t, S) and [χ]1 (t, S) (4) are related each otherby a coe�cient a = a(t) that determines the speed of the conveyor belt. For de�niteness,



we assume that the speed of the belt during the day has the form:

a(t) = a0 − a1cos

(2π

t

TS

), a0, a1 > 0, a0 − a1,> 0, (5)

where a0 is an average conveyor speed during the duration TS ( e.g. during the day); a1 isthe amplitude of the change of the speed during the duration TS. Here we consider the casewhere the speed of the conveyor at the start of the day is a(t)|t=nTS = a0−a1, increases withthe time and reaches the maximum value in the middle of the day a(t)|t=(n+ 1

2)TS= a0+a1,

then decreases to its start value a(t)|t=(n+1)TS= a0 − a1. The right side of the equation

speci�es the source of the supply of the rock on the �rst technological operation at thepoint with the coordinate S = 0.

3. PDE-Model of Conveyor Line in Dimensionless Form

The state of �ow parameters of the conveyor line will be described using dimensionlessvariables and parametrs

τ = t/Td, ξ = S/Sd, (6)

H(ξ)ψ (ξ) =H (S)Ψ (S)

Θ, γ (τ) =

λ (t)

Θ

TdSd, g(τ) =

a(t)TdSd

, (7)

θ0 (τ, ξ) =[χ]0 (t, S)

Θ, Θ = max

Ψ(S),

t∫t0

λ(η)dη

. (8)

Taking into account the notation introduced, we write down the balance equation (3)� (4) in the dimensionless form:

∂ θ0 (τ, ξ)

∂τ+ g(τ)

∂ θ0 (τ, ξ)

∂ξ= δ (ξ) γ (τ), (9)

θ0 (τ0, ξ) = H (ξ)ψ (ξ) . (10)

We write for the system (16) the system of characteristic equations (9):

dξ

dτ= g(τ), ξ|τ=τ0 = æ, (11)

d θ0 (τ, ξ)

dξ= δ(ξ)

γ (τ)

g(τ), θ0 (τ0, ξ)|τ=τ0 = θ0 (τ0,æ) = H(æ)ψ(æ). (12)

The solution of the equation (11) can be written

ξ −∫g(τ)dτ − C1 = 0, ξ −G(τ)− C1 = 0,

∫g(τ)dτ = G(τ), C1 = const,

ξ|τ=τ0 = æ ⇒ C1 = æ−G(τ0), ξ = G(τ)−G(τ0)+æ or ξ =

t∫t0

g(τ)dτ+æ. (13)


67


We are going to express from the equation (13) the time τ

τ = G−1(ξ +G(τ0)− æ

)and substitute the resulting expression in the equation (12). This will allow us to write

θ0 (τ, ξ) =

∫δ (ξ)

γ (τ)

g(τ)dξ + C2 =

∫δ (ξ)

γ(G−1(ξ +G(τ0)− æ)

)g(G−1(ξ +G(τ0)− æ)

)dξ + C2 =

= H(ξ)γ(G−1(G(τ0)− æ)

)g(G−1(G(τ0)− æ)

) + C2 = H(ξ)γ(G−1(G(τ)− ξ)

)g(G−1(G(τ)− ξ)

) + C2, C2 = const. (14)

The integration constant C2 is determined from the initial condition (12)

θ0 (τ0,æ) = H(æ)γ(G−1(G(τ0)− æ)

)g(G−1(G(τ) − æ)

) + C2 = H(æ)ψ(æ) (15)

where from

C2 = H(æ)ψ(æ)−H(æ)γ(G−1(G(τ0)− æ)

)g(G−1(G(τ0)− æ)

) . (16)

We substitute the obtained expression for the integration constant (16) in (12), obtain asolution for (9) � (10)

θ0 (τ, ξ) = H (ξ)γ(G−1(G(τ)− ξ)

)g(G−1(G(τ)− ξ)

) −H(æ)γ(G−1(G(τ0)− æ)

)g(G−1(G(τ0)− æ)

) +H(æ)ψ(æ). (17)

We can use (13) to represent the solution in the following form

θ0 (τ, ξ) =

(H(ξ)−H

(ξ −

τ∫τ0

g(τ)dτ

))γ(G−1(G(τ)− ξ)

)g(G−1(G(τ)− ξ)

)+

+H

(ξ −

τ∫τ0

g(τ)dτ

)ψ

(ξ −

τ∫τ0

g(τ)dτ

). (18)

The solution (18) satis�es the initial condition (10)

θ0 (τ0, ξ) =

(H(ξ)−H

(ξ −G(τ0)

))γ(G−1(G(τ0)− ξ)

)g(G−1(G(τ0)− ξ)

) +H(ξ)ψ(ξ) = H (ξ)ψ(ξ), (19)

θ(τ, 0) =

{γ(τ)/g(τ), if τ > τ0;

H(0)ψ(0), if τ = τ0.(20)



4. Analysis of the Solution

We assume that the initial distribution of the rock along the conveyor line is given atthe time τ0 = 0

θ(0, ξ) = ψ(ξ) = H(ξ)(1− ξ). (21)

The intensity of the arrival of the rock on the conveyor line is started at time τ0 = 0 andhas a constant value

γ(τ) = H(τ − τ0) = H(τ). (22)

In accordance with the results of the research (Fig. 2, [14]), the conveyor speed isrepresented by a periodic function with a period TS= 24 hours:

g(τ) = g0 − g1 cos (ωSτ) , g0 =a0TdSd

, g1 =a1TdSd

, τS =TSTd, ωS =

2π

τS. (23)

The �rst integral (13) has the form

ξ = G(τ)−G(τ0) + æ, G(τ) = g0τ–g1ωS

sin (ωSτ), G(0) = 0. (24)

Taking (21) � (24) into account, the solution q can be written as follows

θ0 (τ, ξ) =

(H(ξ)−H

(ξ – g0τ +

g1ωS

sin (ωSτ)

))γ(G−1(g0τ–

g1ωSsin (ωSτ)−ξ)

)

g

(G−1(g0τ–

g1ωSsin (ωSτ)−ξ)

)+

+H

(ξ – g0τ +

g1ωS

sin (ωSτ)

)ψ

(ξ – g0τ +

g1ωS

sin (ωSτ)

). (25)

Let us determine the time of the transition period Ttr. If the working time of the conveyorline is longer than the duration of the transition period ∆τ = (τ − τ0) ≥ Ttr, then thesolution (25) will not depend on the initial conditions (21). For a period of time equal tothe duration of the transition period, all the rock that was on the conveyor belt will beunloaded from the conveyor

ψ

(1 – g0τ +

g1ωS

sin (ωSτ)

)= 0,when ∆τ = (τ − τ0) = τ ≥ Ttr. (26)

The duration of the transition period is determined by solving equation

1− g0Tpr −g1ωSsin (ωSTpr) = 0. (27)

If Td >> TS, the transition period can estimate the next value 1−g0Tpr ≈ 0, Tpr ≈ 1/g0.The distribution of the rock along the conveyor belt for the steady state is described byformula

θ0 (τ, ξ) = g

(G−1

(g0τ–

g1ωS

sin (ωSτ)−ξ))−1

, τ ≥ Ttr. (28)


69


Fig. 3. Distribution of the rock along the conveyor for times τ = (0, 0; 0, 1; 0, 2; ...; 0, 9; 1, 0)

Fig. 3 shows the solution for the parameters g0 = 2, g1 = 1, τS = 0, 5 that determinesthe speed of the conveyor belt. A detailed representation, which clearly demonstrates thedistribution of the rock along the conveyor belt for di�erent times, is shown in Fig. 4.The distribution of the rock at the initial time (Fig. 4.1) is determined by the initialcondition (10). The distribution of the rock along the conveyor, shown in Figs. 4.1 � 4.5,corresponds to the transient mode of the working of the conveyor line. For the conveyoroperation times exceeding the duration of the transition period Tpr (Figs. 4.6 � 4.10), theinitial distribution does not participate in the formation of the distribution of the rockalong the conveyor line. This corresponds to the steady state of the conveyors' work (28).For a steady state, a local accumulation of the rock with a period of τS is observed, whichnegatively a�ects the performance characteristics of the conveyor [10,11,13]. The dispersionof the distribution during a period is determined by the value of the variable componentof the speed g1, or more precisely by the relationship g1/g0. The distribution of the rock attimes τ = (0, 0; 0, 1; 0, 2; ...; 0, 5) is determined by the source of the rock input (22) and thedisplacement of the initial distribution of the rock (21) along the conveyor line with thespeed g(τ). Fig. 5 shows the values of the amount of rock for a particular technologicalposition, which is determined by the coordinate S. A detailed representation determiningthe change in the unit density of the rock as a function of time for a certain technologicalposition on the conveyor is shown in Fig. 6. Fig. 6.1 demonstrates the amount of rock inrelation to the time that enters the conveyor line. This quantity is formed by the functionof the intensity of the arrival of the rock λ(τ) (22), on the one hand, and the speed of themovement of the conveyor belt g(τ) (23), on the other hand. Fig. 6.10 shows the amount ofthe rock depending on the time that is shipped from the conveyor line. This is an importantcharacteristic that determines the competitive ability of the production system [4�8]. This�gure clearly demonstrates that the output �ow of rock from the conveyor line is greatextenting determined by the law of the speed of the movement of the conveyor belt. Thisis an important circumstance that should be taken into account when designing enterprisemanagement systems with a �ow-based method of the production organization. And thiscan be done using the PDE-model, as shown here. The speed control of the conveyor beltallows not only to save the resources used in the work of the conveyor, as correctly pointedout in [10, 11, 13], but also makes it possible to obtain revenue by ensuring the ful�lmentof the optimal portfolio of orders [4�8]. Also attention should be paid to the fact that the



Fig. 4. Detailed representation of the distribution of the rock along the conveyor for timesτ = (0, 0; 0, 1; 0, 2; ...; 0, 9; 1, 0)


71


Fig. 5. Distribution of the rock along the conveyor for the technological positionS = (0, 0; 0, 1; 0, 2; ...; 0, 9; 1.0)

amount of the rock at the output is determined by its amount of the input with somedelay, which is a function of the speed of the conveyor belt.

5. Calculation of the Duration of the Production Cycle

The duration of the production cycle is an important characteristic of the productionsystem. Let us demonstrate the calculation of this characteristic below. The characteristicequation (24) determines the trajectories of the movement of individual objects along thetechnological route of the conveyor line (Fig. 7). As one would expect, the movement of asingle object along the technological trajectory is carried out at a constant speed equal tothe speed of the conveyor line. Equation (24) allows you to calculate the duration of theproduction cycle. The duration of the production cycle is equal to the time interval forwhich the object of labour passes the path from the �rst technological position to the last.Calculation of the duration of the production cycle for enterprises with a �ow-method oforganizing production is given in [16]. The value the duration of the production cycle fora conveyor-type production line can be de�ned as follows

τd =

1∫0

dξ

g(τ). (29)

Products that arrived on the conveyor line at the time τ0, leave the conveyor line at timeτ with a delay τd = τ − τ0. This delay is equal to the duration of the production cycle.The delay depends on the time at which the rock enters the conveyor, it is not constantvalue.

6. Conclusions and Further Prospects of Development and

Improvement of PDE-Models of Production Systems

The obtained results of the research are basic for the development of control systemsfor the production of the conveyor type. It is shown that the distribution of the rock alongthe conveyor belt is determined by the speed of the conveyor belt. The in�uence of the



Fig. 6. Detailed representation of the distribution of the rock along the conveyor for thetechnological position S = (0, 0; 0, 1; 0, 2; ...; 0, 9; 1, 0)


73


Fig. 7. Family of characteristics

initial and boundary conditions on the parameters of the state of the conveyor line isconsidered. Along with the advantages in describing complex production systems of �owtype, the use of PDE-models is associated with a number of di�culties, one of which isthe construction of a closed system of equations of the production process. In this paper,the construction of a closed system of equations is solved by using an additional equationthat determines the speed of the conveyor (23). An important result of this work is themethod of calculating the duration of the production cycle (29), based on the use of thecharacteristic equation (24). The duration of the production cycle is not a constant but isdetermined by the speed of the conveyor belt. A further perspective of the developmentof the issue discussed in this paper is the construction of a control system for a conveyorline with a variable speed of movement of the belt.

References

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3. Milling P.M. System Dynamics as a Structural Theory in Operations Management.Production and Operations Management, 2008, vol. 17, no. 3, pp. 373�384.

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5. Lefeber E. Modeling, Validation and Control of Manufacturing Systems. Proceeding of the

2004 American Control Conference. Massachusetts, 2004, pp. 4583�4588.

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7. Pihnastyi O.M. [Review of Control Models of the Production Lines of the ManufacturingSystems]. Belgorod State University Scienti�c Bulletin, 2015, no. 34/1, pp. 137�152. (inRussian)

8. Pihnastyi O.M. [Analysis of the Models of Transition Process Controlled Manufacturing].Belgorod State University Scienti�c Bulletin, 2015, no. 35/1, pp. 133�144. (in Russian)

9. SIMINE for Conveyors. Siemens. Available at: https://goo.gl/Ku90xp (accessed April 12,2017).



10. Semenchenko A., Stadnik M., Belitsky P., Semenchenko D., Stepanenko O. [The Impactof an Uneven Loading of a Belt Conveyor on the Loading of Drive Motors and EnergyConsumption in Transportation]. Eastern-European Journal of Enterprise Technologies, 2016,vol. 4, no. 1 (82), pp. 42�51. (in Russian)

11. Kondrahin V., Stadnik N., Belitskii P. [Operating Parameters Statistical Analysis for theBelt Conveyor in Mine]. Donetsk National Technical University Scienti�c Works, 2013, vol. 2,pp. 140�150. (in Russian)

12. Bambach M., H�ack A., Herty M. Modeling Steel Rolling Processes by Fluid-Like Di�erentialEquations. Applied Mathematical Modelling, 2017, vol. 43, pp. 155�169.

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of Ukraine, 2005, vol. 7, pp. 66�71. (in Russian)

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Received April 10, 2017

ÓÄÊ 658.51.012 DOI: 10.14529/mmp170407

ÌÎÄÅËÜ ÊÎÍÂÅÉÅÐÀ Ñ ÐÅÃÓËÈÐÓÅÌÎÉ ÑÊÎÐÎÑÒÜÞ

Î.Ì. Ïèãíàñòûé1, Â.Ä. Õîäóñîâ2

1Íàöèîíàëüíûé òåõíè÷åñêèé óíèâåðñèòåò ≪Õàðüêîâñêèé ïîëèòåõíè÷åñêèéèíñòèòóò≫, ã. Õàðüêîâ, Óêðàèíà2Õàðüêîâñêèé íàöèîíàëüíûé óíèâåðñèòåò èì. Â.Í. Êàðàçèíà, ã. Õàðüêîâ, Óêðàèíà

Ñòàòüÿ ïîñâÿùåíà ìàòåìàòè÷åñêîìó ìîäåëèðîâàíèþ ïðîèçâîäñòâåííûõ ïîòî÷íûõ

ëèíèé êîíâåéåðíîãî òèïà. Ðàçðàáîòàí àíàëèòè÷åñêèé ìåòîä ðàñ÷åòà ïàðàìåòðîâ ïî-

òî÷íîé ëèíèè ñ ðåãóëèðóåìîé ñêîðîñòüþ äâèæåíèÿ ïðåäìåòîâ òðóäà âäîëü êîíâåéåðà.

Îïèñàíèå ñîñòîÿíèÿ ïàðàìåòðîâ ïîòî÷íîé ëèíèè âûïîëíåíî â îäíîìîìåíòíîì ïðè-

áëèæåíèè ñ èñïîëüçîâàíèåì óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ. Ïîëó÷åíî ðåøåíèå,

îïðåäåëÿþùåå ñîñòîÿíèå ïàðàìåòðîâ ïîòî÷íîé ëèíèè äëÿ çàäàííîé òåõíîëîãè÷åñêîé

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ðèîäà, â òå÷åíèå êîòîðîãî íà÷àëüíîå óñëîâèå ðàñïðåäåëåíèÿ ïðåäìåòîâ òðóäà âäîëü

êîíâåéåðà îêàçûâàåò âëèÿíèå íà ïàðàìåòðû ñîñòîÿíèÿ ïðîèçâîäñòâåííîé ëèíèè. Ðàç-

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ïðîåêòèðîâàòü ñèñòåìû óïðàâëåíèÿ ïðîèçâîäñòâåííûìè ëèíèè êîíâåéåðíîãî òèïà ñ

ðåãóëèðóåìîé ñêîðîñòüþ äâèæåíèÿ ïðåäìåòîâ òðóäà. Îðèãèíàëüíîñòü ïîëó÷åííûõ ðå-

çóëüòàòîâ çàêëþ÷àåòñÿ â óëó÷øåíèè PDE-ìîäåëåé ïðîèçâîäñòâåííûõ ñèñòåì êîíâåéåð-

íîãî òèïà, èñïîëüçóåìûõ äëÿ ïðîåêòèðîâàíèÿ âûñîêîýôôåêòèâíûõ ñèñòåì óïðàâëåíèÿ

ïðîèçâîäñòâîì.

Êëþ÷åâûå ñëîâà: êîíâåéåð; ïðîèçâîäñòâåííàÿ ëèíèÿ; ïðåäìåò òðóäà; ïîòî÷íàÿ

ëèíèÿ; PDE-ìîäåëü ïðîèçâîäñòâà; ïàðàìåòðû ñîñòîÿíèÿ ïîòî÷íîé ëèíèè; òåõíîëî-

ãè÷åñêàÿ ïîçèöèÿ; ïåðåõîäíîé ïåðèîä; ñèñòåìû óïðàâëåíèÿ ïðîèçâîäñòâîì.


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Ëèòåðàòóðà

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2. Gross, D. Fundamentals of Queueing Theory / D. Gross, C.M. Harris. � New York: Wiley,1985.

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7. Ïèãíàñòûé, Î.Ì. Îáçîð ìîäåëåé óïðàâëÿåìûõ ïðîèçâîäñòâåííûõ ïðîöåññîâ ïîòî÷íîéëèíèè ïðîèçâîäñòâåííûõ ñèñòåì / Î.Ì. Ïèãíàñòûé // Íàó÷íûå âåäîìîñòè Áåëãîðîä-ñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà. � 2015. � � 34/1. � Ñ. 137�152.

8. Ïèãíàñòûé Î.Ì. Àíàëèç ìîäåëåé ïåðåõîäíûõ óïðàâëÿåìûõ ïðîèçâîäñòâåííûõ ïðîöåññîâ/ Î.Ì. Ïèãíàñòûé // Íàó÷íûå âåäîìîñòè Áåëãîðîäñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòå-òà. � 2015. � � 35/1. � Ñ. 133�144.

9. SIMINE for Conveyors. Siemens. � URL: https://goo.gl/Ku90xp (äàòà îáðàùåíèÿ: 12 àïðå-ëÿ 2017 ã.)

10. Ñåìåí÷åíêî, À.Ê. Âëèÿíèå íåðàâíîìåðíîñòè çàãðóæåííîñòè ëåíòî÷íîãî êîíâåéåðà íàíàãðóæåííîñòü ïðèâîäíûõ äâèãàòåëåé è ýíåðãîçàòðàòû íà òðàíñïîðòèðîâàíèå / À.Ê. Ñå-ìåí÷åíêî, Í.È. Ñòàäíèê, Ï.Â. Áåëèöêèé, Ä.À. Ñåìåí÷åíêî, Å.Þ. Ñòåïàíåíêî //Âîñòî÷íî-åâðîïåéñêèé æóðíàë ïåðåäîâûõ òåõíîëîãèé. � 2016. � V. 4, � 1 (82). � Ñ. 42�51.

11. Êîíäðàõèí, Â.Ï. Ñòàòèñòè÷åñêèé àíàëèç ýêñïëóàòàöèîííûõ ïàðàìåòðîâ øàõòíîãî ëåí-òî÷íîãî êîíâåéåðà / Â.Ï. Êîíäðàõèí, Í.È. Ñòàäíèê, Ï.Â. Áåëèöêèé // Íàó÷íûå òðóäûÄîíåöêîãî íàöèîíàëüíîãî òåõíè÷åñêîãî óíèâåðñèòåòà. � 2013. � � 2/26. � Ñ. 140�150.

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13. Ïðîêóäà, Â.Í. Èññëåäîâàíèå è îöåíêà ãðóçîïîòîêîâ íà ìàãèñòðàëüíîì êîíâåéåð-íîì òðàíñïîðòå ÏÑÏ ≪Øàõòà "Ïàâëîãðàäñêàÿ"≫ ÏÀÎ ÄÒÝÊ ≪Ïàâëîãðàäóãîëü≫ /Â.Í. Ïðîêóäà, Þ.À. Ìèøàíñêèé, Ñ.Í. Ïðîöåíêî // Ãîðíàÿ ýëåêòðîìåõàíèêà è àâòî-ìàòèêà. � 2012. � � 88. � C. 107�111.

14. www.bartec.kz (2017). � URL: http://www.bartec.kz/�les/mining/for-conveyance.pdf (äàòàîáðàùåíèÿ: 12 àïðåëÿ 2017 ã.)

15. Ïèãíàñòûé Î.Ì. Ñòîõàñòè÷åñêîå îïèñàíèå ýêîíîìèêî-ïðîèçâîäñòâåííûõ ñèñòåì ñ ìàñ-ñîâûì âûïóñêîì ïðîäóêöèè / Â.Ï. Äåìóöêèé, Â.Ñ. Ïèãíàñòàÿ, Î.Ì. Ïèãíàñòûé // Äî-êëàäû Íàöèîíàëüíîé àêàäåìèè íàóê Óêðàèíû. � 2005. � Ò. 7. � Ñ. 66�71.

16. Ïèãíàñòûé Î.Ì. Ðàñ÷åò ïðîèçâîäñòâåííîãî öèêëà ñ ïðèìåíåíèåì ñòàòèñòè÷åñêîé òåî-ðèè ïðîèçâîäñòâåííî-òåõíè÷åñêèõ ñèñòåì / Î.Ì. Ïèãíàñòûé, Â.Ä. Õîäóñîâ // ÄîêëàäûÍàöèîíàëüíîé àêàäåìèè íàóê Óêðàèíû. � 2009. � Ò. 12. � Ñ. 38�44.



Îëåã Ìèõàéëîâè÷ Ïèãíàñòûé, äîêòîð òåõíè÷åñêèõ íàóê, ïðîôåññîð, êàôåäðà≪Êîìïüþòåðíûé ìîíèòîðèíã è ëîãèñòèêà≫, Íàöèîíàëüíûé òåõíè÷åñêèé óíèâåðñèòåò≪Õàðüêîâñêèé ïîëèòåõíè÷åñêèé èíñòèòóò≫ (ã. Õàðüêîâ, Óêðàèíà), [email protected].

Âàëåðèé Äìèòðèåâè÷ Õîäóñîâ, äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,êàôåäðà ≪Òåîðåòè÷åñêîé ÿäåðíîé ôèçèêè è âûñøåé ìàòåìàòèêè èì. À.È. Àõèåçåðà≫,Õàðüêîâñêèé íàöèîíàëüíûé óíèâåðñèòåò èì. Â.Í. Êàðàçèíà (ã. Õàðüêîâ, Óêðàèíà),[email protected].

Ïîñòóïèëà â ðåäàêöèþ 10 àïðåëÿ 2017 ã.


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