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Виробничий процес розглядаеться в системi 4Б, в якш фактори виробництва, що використовуються в чаы, е обектом взаемопов'язаного i взаемозалежного процесу. Встановлено, що кожному збшьшенню обсягу робт AYi вiдповiдае прир^т часу АТ i несктченному числу збшь-шень обсягу робт AYi вiдповiдае несктченне число збшь-шень часу АТ'. Встановлено, що суттсть часу в конкретному виробничому процеы проявляеться в тому, що система векторiв часу колтеарна сонаправлена вiдповiд-ног системi векторiв обсягiв робт
Ключовi слова: виробничий процес, час у просторi 4Б,
виробничi фактори, коллтеартсть векторiв, прир^т часу □-□
Производственный процесс рассматривается в системе 4Б, в которой факторы производства, используемые во времени, являются объектом взаимосвязанного и взаимозависимого процесса. Установлено, что каждому приращению объема работ AYi соответствует приращение времени АТ и бесконечному числу приращений объема работ AYi соответствует бесконечное число приращений времени АТ'. Установлено, что сущность времени в конкретном производственном процессе проявляется в том, что система векторов времени коллинеарна и сонаправлена соответствующей системе векторов объемов работ
Ключевые слова: производственный процесс, время в пространстве 4Б, производственные факторы, коллинеарность векторов, приращение времени -□ □-
UDC658.5.012.1
|DOI: 10.15587/1729-4061.2016.86535
ANALYSIS OF ROLE OF TIME IN THE PRODUCTION PROCESS IN A 4D SPACE
V. S. Borovik
Doctor of Technical Sciences, Professor, Аssistant to the Head of Volgograd, ^airman of the scientific-technical expert council Volgograd City Duma E-mail: borovikv@mail.ru V. V. Borovik PhD, Associate Professor Department of construction of transport facilities Volgograd State Technical University Academicheskaya str., 1, Volgograd, Russia, 400074 E-mail: borovikvv70@mail.ru
1. Introduction
An intention to explain a special role of time in the development of processes, taking place in production systems (PS) under conditions of introducing advanced technologies, leads to a number of assumptions, which deserve serious attention [1, 2].
In contemporary knowledge, the concept of "time" as the initial and the undefined, in practice rests on the intuition of a researcher, on his non-reflected professional experience, on the elements of frequently subconscious ideas. It is necessary that time in the production process (PP) should become the object of a comprehensive study. Time, which has the properties of vector, such as direction and magnitude, can be accepted as the initial premise of this study [3].
The problems, connected with an increase in the effectiveness of using production resources over time, are of undoubted interest. The need for the solution of this problem was noted in work [4]. In particular, relative to the introduction of time parameter into the production function (PF): "If we want to base our own theory of production on the theory of "roundabout" process of Jevons-Bohm-Bawerk-Taussig, we may introduce time directly into production function, after writing down: x=y(v1, v2...vn; t)..." [4]. Subsequent studies are directed to giving dynamism to PF, first of all, by introduction of time factor into it [5].
Nonlinear dynamic PF, which considers the fluctuations of production factors (resources volume) over time, more adequately describes the actual production process. In the nonlinear dynamic model y=f(t , Xj (t)...xn(t)), where Xj (t) reflects the dynamics of a change in the specific production
factor depending on time. Parameter t is a time independent variable, which implicitly reflects the influence of all disregarded factors on the result of indicator y.
However, in light of contemporary concept of space and time, its consideration only as an independent variable implicitly would mean substantial narrowing the possibilities of prognostic calculations when introducing advanced technologies.
There are sufficiently clear tendencies to examine production process in dependence on the joint influence of different factors, which creates prerequisites for more accurate predicting calculations of the production development. According to the specialists of the USA: " A market of 3D and 4D technologies is expected to be worth USD 127.84 billion in 2016 and to grow on average by 16.17 % in the period between 2016 and 2022. The year 2015 was taken as basic for analysis and forecasted period was between 2016 and 2022. The market is segmented based on technology, end users, industry and geography" [6].
2. Literature review and problem statement
The most widespread research is conducted in the area of optimization of the time role in the cost of an object (creation of product). For example, paper [7] gives technical and economic substantiation of the technology-designing stages of a complex of works, connected with building a motor road for the purpose of determining the influence of time on the costs of works. The project calculation included correction of risk costs, expected costs, current risk costs and final
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actual costs of construction completion. It was established that depending on the combination of different factors and conditions, an increase in the cost of the project implementation may amount up to 18.5 %, and the term of the project completion delays by 56 % [7]. The calculation of influence of totality of factors, which depend on the time of works completion, with the technical and economic substantiation is extremely promising.
Control of construction projects, as a rule, taking into account uncertainty and risk, may affect the implementation of time compromise - costs. Paper [8] proposes the model of multi-criterion optimization, which makes it possible to reach a compromise in costs and time with sufficient reliability. The results show that the person, who makes a decision, with the aid of regulating time and costs on the basis of the Pareto principle, establishes a preferable version of construction with a different acceptable risk level.
When implementing construction projects, as a rule, there appear some situations, connected with the need for performing unplanned types of work, which are necessary to complete within a rather tight time range. Solution of these problems inevitably involves an increase in the costs of work. Algorithm LLY, using linear and net graphs of solving the problem of time-costs at the assigned time range of putting an object into operation, is proposed [9]. However, in the work under consideration, an attempt to connect the production process and time in one model is not traced.
There is an experience [10] of solving the problem of optimization of selecting the material of a building structure - steel or reinforced concrete - on the basis of the Microsoft Project software. The results show that application of steel constructions makes it possible to save 55.3 % of construction time, with an increase in costs by 23.1 % [10]. In spite of the established connection between costs and time, the problem of the process of optimization is not set in the work.
In work [11], the optimization of using time and means is proposed to perform applying a mathematical model, based on fuzzy sets. The graph of construction and performing of unforeseen works, as well as delays in material supply, is developed. It is important that the model on the basis of verbal information includes the optimization of parameters of time, costs and supply with building materials. Under these conditions, the priorities of activity, which have minimum values with floating point, are accurately formulated. The function of belonging, including the costs as the objective of mathematical model, is developed, which makes it possible to operationally find the critical path of decreasing long delays of time and costs [11].
Paper [12] examines solution of the problem of effective combination of construction objects by means of achieving a compromise between time and costs as a promising strategy. Under conditions of limited resources the problem is formulated as linear integral program. Effectiveness and usefulness of the proposal is proved by the application of thirty objects. However, the authors emphasize that combination is an exclusively risky measure and consider it necessary to expand the model and to introduce the random estimation relative to several parameters [12]. However, the analyzed work did not establish any dependences, which allow the implementation of prognostic calculations when combining the construction objects.
Some attempts have been made to solve the production problems as a process in multidimensional space, in particu-
lar, 4D space with involvement of time. For example, according to a number of authors [13, 14], the application of a 4D model is a useful alternative to the project of tools planning, such as the CPM of networks and histograms in technical and economic substantiation. There appears an opportunity for a larger number of specialists to understand the process and rapidly identify potential problems, as well as to foresee possible time-and-space conflicts and problems. The need for improving 4D tools, which should include histograms, lists of components and annotation in their graphic user interface, is underlined. However, the authors do not give the examples of problem visualization in 4D [13].
The visualizations of transport projects as an effective method of exchanging the information between the project stakeholders are examined in [14]. In the examined applications, the visualization is used primarily for transmission of information about a geometric construction, or a photo-realistic image of transport projects. The use of 4D visualization is also effective in the implementation of road projects processes for facilitating joint decision making on planning the construction and the traffic of works. However, 4D visualization is limited to highlighting some kinds of works or operations in various colors. This approach was used for the construction of a section of a large-scale highway design in Dallas, Texas.
With an increase in the complexity of contemporary construction projects, there is an imperative need for a higher degree of computer utilization for the purpose of reaching effective planning and management [15]. The authors demarcate the previous developments and the implementation of the prototypes of the four-dimensional site managing model (4DSMM). It is proposed to perform planning on the three-dimensional computer building model (axonometric model), and specialists are offered an opportunity to plan viewing the graphs of the construction process simulation in color for the assumed period of time. Color comes out as the fourth parameter in 4D, where a specific period of time, connected with a specific stage or kind of works, corresponds to a definite color. However, time is not connected into the united model, which makes it possible to optimize resources consumption.
The 4D simulation method was developed for helping designers and better understanding the consequences of implementation of motor roads construction projects. In particular, the most important signs of influencing the environment, namely the spatial areas with continuous information and progression in the course of time, are visualized. The method was used for support of the Dutch project of highway extension [16]. In comparison with the 2D method, the proposed simulation method provides for the integral prospect of evaluating three-dimensional changes and the influence of the project on the environment in the course of time.
It should be noted that in the analyzed works, the 4 D use is accomplished by providing the visual image of an object (in perspective and axonometry - 3D) with highlighting in color the sections characterizing stages and kinds of works, performed within a definite period of time. The authors of research in this field pay considerable attention to developing software of the 4D designing process. As an example we may take Fig. 1, which reflects the sequence of changes, caused by the project influence on the environment [17]. It is noted that the shortcoming of such approach is that the detailing of separate elements is impossible (slopes, trenches, canals, etc., Fig. 1).
Fig. 1. Missing slopes of surfaces
4. Essence of time in the production process
For explaining the interrelation between time and a production process, we will turn to PF, for example, of the form [18]:
Y = C0 nx,(1)
i=i
where Y is the calculated indicator (for example, the volume of work, etc., in real material or value terms); xj , i = i,n are the factors (resources, for example, fixed funds, materials, labour), influencing Y (in real material or value terms); aP i = i,n are the «weights», characterizing contribution of xj into Y; C0 is the coefficient, characterizing the joint influence of the factors, not taken into account by the model.
The given analysis shows that calculations "costs - time" and attempts to consider them in space and time (4D) do not have systems character. 4D visualization by highlighting in color of stages or kinds of works, carried out at specific time, serves as an addition to an increase in clarity of a project's graphic information. The problem is that PP is not examined in the system, in which the result of resource using over time is not an object of an interconnected and interdependent process. The examination of the problem in 4D space, in which time is reflected only in color, does not include entering the metric space - geometric interpretation of space-and-time.
In connection with this, the sense of hypothesis is in the fact that, probably, obtaining the geometric interpretation of space-and-time with examining the problems of using resources and time as an interconnected and interdependent process will make it possible to solve the problems of forecasting results of the production activity at a qualitatively higher level.
Fig. 2. 3D space model on the basis of PF, reflecting the surface with control when introducing advanced technology, characterized by different combinations of resources consumption at moving from level 1U to level 2U
3. The purpose and objectives of the study
The purpose of the study is an increase in accuracy of the prognostic calculations of production activity results when introducing advanced technologies. The goal is supposed to be achieved due to including into 4D model of the basic production resources, consumed over time and incorporated into a united model of interconnected and interdependent factors.
To achieve the set goal, the following tasks were to be solved:
- to determine the initial model for including the parameter of time in the production process, developing in a 4D space;
- to establish the essence of time in production process in interrelation and interdependence with the results of using resources;
- to prove, using specific example, the effectiveness of a 4D model during the calculations, connected with estimating the results of resource using over time in the course of creating a product.
Let us examine a three-dimensional graphic model (Fig. 2) of the PF form (1). The function is most accessible for understanding in view of the possibility of its visual image in three-dimensional space [19, 20]. Curves 1Y and 2Y connect the points with the identical numerical values of volume of work. Their projections 1Y1, 2Y1, 1Y2, 2Y2, 1Y3, 2Y3 are isoquants.
As it is evident in Fig. 1, from point A, characterized, for example, by volume of works 1Y, volume 2Y may be achieved in the specific PS via the implementation, for example, of new technology by the infinite set of combinations of labor and other resources. For example, point B, C and D, show three variants of reaching the volume of works, characterized as 2Y. Depending on possibilities of the PS, qualification of personnel, quality control, social conditions, resources quality, purpose of production system, tasks, etc., one or another variant of resources combination is selected.
Let us examine three versions of transition form 1Y to a higher level 2Y, provided by the implementation of the new technology. Let us examine vectors AB, AC and AD. Each of them has its own correspondent combination of resources
use: ax4, Ax2, Bxp Bx2, Cx4, Cx2, Dxp Dx2. We will consider the variant of AB vector optimum from mathematical positions, as ABA2y and it is the shortest distance between two curves y1 and y2. The curve - isoquantum 2U - connects points with numerical marks, which characterize the identical volume of works (y) at different combinations of production resources (X1 h X2). From Fig. 2 it is seen that AB<AC and AB<AD, and the variants of entering level y2 are not equivalent by volumes of consumed resources. For example, entering point C is accompanied by a decrease in resource consumption X4, (Bx1>Cx1), and by an increase in the volume of resources X2, as Cx2>Bx2. Analogous situation is connected with entering point D. However, in this situation there is an increase in consumption of resource X1 and a decrease in X2.
When simulating a process in 4D space, it is necessary to use a symbolic record of certain transformations, which stand, as a rule, for an actual technological process. If transformations of coordinates from the fixed system to the moving one, and vice versa, are performed, this means that the transmission of saved information about an actual motion and an actual method of transformation of parameters takes place. In other words, each form of mathematical record in this case has its own correspondent method of motion and its own method of information transmission [21].
Let's assume that the start of a certain PP was fixed. Taking into account that time is connected with resources and their motion in space, the "past" of a process coincides with its beginning and time "flows" together with the process of converting resources in space. Hence it is possible to draw a conclusion that time conditionally flows in the same direction with the changes in an observed object. It is accepted to characterize motion as a generalized concept
using vectors magnitudes, and it may be assumed that the "own" time of a process will be directed collinearly with the vector of "displacement". Let us show it graphically, using the geometric interpretation of space-and-time, proposed by G. Minkowski (Fig. 3) [22].
Under the actual conditions of production, control in space and time begins from point C (from achieved result) in the direction of point D (planned result - for example, creation of a product). The actual geometric development of the process takes place in the course of time from C to D'. Let us note that specific time, the planned time of creating a product, corresponds to this process. Depending on the processes, connected with control and developing over time, point D of vector CD describes a fairly complicated trajectory (Fig. 3).
This is explained by the fact that the system of mobile equilibrium tends to change so as to reduce the effect of external influence to minimum [23]. Under the influence of a totality of factors, developing over time and not taken into account by the project, (changing conditions, delay in reaction of the managing system to changes in the production process, including overregulation [24], and other reasons) the displacement trajectory of the vector of control over time in general form can be presented, for example, in the form of the undulating surface ( Fig. 3). The surface is formed on one side by straight line CC', which coincides with the planned direction of the displacement vector in space, and is characterized by Y1, previously achieved volume of product manufacturing. On the other side it is formed by the curve, characterizing the conditionally actual trajectory, described by point D, the end of vector CD, directed toward reaching the production volume of product Y2 at point D'.
Fig. 3. Spatial model, illustrating temporary cross-sections of Minkowski space for vector of control CD (transformation of CD into CD) taking into account the reaction of vector of control to influence of external and internal factors over time in PS
Then for further consideration of the process, there appears the need for introduction time for completion of works, not taken into account by the project. Each executor needs his own time to perform each operation or a totality of operations.
Let us unite the axes, used for quantitative characteristic of resources, into one axis OX. We will obtain Fig. 4, convenient for examination.
Fig. 4. Illustration of vectors of using resources over time
AR-R2-R4.
AY=AR- VYL-R2-R! - VY |
In this case,
V = ^ e,
3x2
3x,
which proves the vector character of motion, connected with PP.
Specific scalar Q-f (x, y, z) corresponds to each point of space x, y, z, and specific vector X=^(x, y, z), Y-y(x, y, z) and Z-X(x, y, z) is applied to each point of space. Scalar energy is converted into vector force by the course of time [26].
Let us turn to dependence:
in work in indissoluble connection with production factors, the question of examination it as a resource in models of PP planning is obvious.
It must be noted that reaching AY is always accompanied by reaching the infinite number of other AYP i=1, 2,...a>. For example, the completion of volume of works AY is accompanied by the wear of machines, a change in natural conditions, volumes of building materials, changing soil temperature, etc. ATj corresponds to each AYr Consequently, the infinite number of ATj corresponds to the infinite number of AYr
Hence, we can draw a conclusion. In expression (3) time, connected to specific PP, is a system of interdependent and interconnected vectors A^, and AYr Then, for example, it is necessary that the vector of assigned directive time should be collinear and co-directed to the vector of performing this volume of works. Or, correspondingly, the vector of performing this volume of works corresponds to the vector of time of performing this volume. If this condition is not satisfied, it will lead to disagreement of PP and an increase in the costs for creating this product. Undulating surface, formed by the displacement of vector CD over time (Fig. 3) can serve as an illustration of an increase in costs for creating this product. This increase is characterized by the difference between the area of the plane, illustrating the designed PP, and the area of undulating surface, which illustrates its implementation over time.
Let us examine formula (3) in more detail.
Matrix Q(t) in general case with taking into account non-uniformity of its elements over time AT = |At1, At2...Atn| may be written down in the form:
Q =
x't (t); x' (^....x' (tn)
Then formula (3) in scalar writing will take the form:
AY = £ KqP; AY; = KqPl J dd^dtj +
i=l
t'
dt
o
AY=Q(t) ATj,
(2)
+ KqP2 J dfdt2 +... + KqPn J ^dtn.
o dt2
o dtn
(4)
where AY is the increment in volume of works; Q(t) is the productivity; AT is the period of time in question. The increment in volume of works is equal to the numerical value, created within the specific time interval. However, it is necessary to draw a fundamental conclusion. Taking into account that AY is a vector, Q is a scalar, formula (2) should be represented in the form:
AY = Q(t) ATj,
(3)
since for retaining the equality in (2), it is necessary that ATj should be a vector (tensor of the 1st rank). Hence follows A^^ AYi, i. e. the vectors of the A^ and of an increase in gains of works volume AYi, ideally are collinear and are co-directed. Consequently, the time, connected with PP, will be as uneven as the volume of production and productivity.
It also follows from formula (3) that the higher the labor productivity, the lower time consumption and the higher the gains of works volume. This is a manifestation of properties of time to be compressed and extended in PP (together with AY and Q(t)). Taking into account that time is examined
Kq is the coefficient of system (enterprise) functioning, determined by the formula:
XX;
l
where Yf is the actually completed volume of works; Xj are the resources, which participate in the product creation; j is the size of sample j=1...n
X, • n
X, • n
Fli n ' H2i n ' H3i n
X, • n
X Xli X X2
X X3
Pj is the coefficient, characterizing the use of resources; n is the size of sample.
According to actual production conditions in formula (4) t1, t2...tn may coincide or not coincide with the planned values. In they do not coincide, all upper limits of integration are selected, obviously, from the relationship: th= -min {t1, t2, ...,tn}, as the system works only while its elements
function. Then the values of integrals, as of function th, will change depending on specific production conditions and, correspondingly, |AY| will change, which is not considered in traditional calculations with the help of PF.
5. Consideration of results: an example of calculation
Let us examine, for example, the calculation of parameters of PF of the form (1) of an enterprise. Where X4, X2, X3 are resources: fixed capital of an enterprise, materials and labor costs, respectively. Y is the volume of works, carried out by accepted resources (all parameters are assigned in terms of value). First, calculations of parameters PF according to traditional diagram were performed, the results of which are given in Table 1, columns 2-6. The values of yr PF, the actual volume of work, were obtained. Then they were compared with actual Yf.
All subsequent calculations of Yr4D were performed according to the diagram, proposed below, which considers the
development of production process over time. The differences A1 and A2, characterizing the deviation of calculated volumes of work by the 4D model (Yr4D) and calculated volumes of works by the PF model (YrPF) are the initial information for determining root- mean-square errors of the calculation of unknown parameters y, volumes of works.
In order to take into consideration the influence of time, let us introduce the derivatives with respect to time, interpreted as the rate of change in the value of function X4 = f(t4); X2 = f(t2); X3 = f(t3). Let us select, for example, the function of the form: Xi = atj\ The function reflects the rate of the consumption of the ith resource. Function X, = atk is selected on conditions of simplification of integrating process. For specific calculations, the form of the function is determined on the basis of possibilities of obtaining initial information, specific character of PP, necessary for accuracy and other conditions. In our case, these will be the functions:
X4 = aiyJt!'; X2 = a2^t2; X3 = a3yJti.
Table 1
Calculations of perspective volume of works considering the factor of time
№ of entry Fixed assets X1 mln, Ru Materials X2 mln RUB Labour X3 mln, RUB Volume of work of actualYf mlnRUB Volume of work of calculated YrPF mlnRUB SXj mln, RUB K, ß1 ß2 ß3 Yt 4D mln, RUB Yf -Yt4D=A Yf -Yt4D=A
thousand RUB A2 thousand RUB A2
1 8767 21042 2014 35071 35009 31823 1,1 1,01 0,98 0,96 35078 -7,4 54,8 62 3844
2 8792 21318 2019 35218 35111 32129 1,1 1,01 0,99 0,97 35362 -145,6 21199,4 105 11025
3 8657 20047 1995 34113 34608 30729 1,1 0,99 0,93 0,95 34128 -15 225 -495 245025
4 8798 22109 2102 35307 35397 33009 1,07 1,01 1,02 1,00 35150 150,7 22710,5 -90 8100
5 8771 21126 2017 35111 35940 31914 1,1 1,01 0,98 0,96 35101 9,6 92,2 71 5041
6 9026 23874 2115 36208 36248 35015 1,04 1,04 1,1 0,92 36062 146,5 21462,2 -40 1600
7 9134 23525 2127 36451 37030 34786 1,04 1,05 1,09 1,02 36145 306 93606 -571 326041
8 8992 22457 2118 35824 36486 33567 1,07 1,02 1,04 1,01 35733 91 8281 -662 438244
9 8974 22311 2115 35626 35410 33499 1,07 1,03 1,03 1,01 35595 31 961 216 46656
10 8949 22186 2103 35317 35234 33238 1,06 1,03 1,03 1,00 35240 -6 36 83 6889
11 7537 17526 1823 28614 28328 26936 1,06 0,87 0,81 0,87 28451 162,8 26503,8 291 84681
12 7849 19004 1902 30417 29969 28755 1,06 0,90 0,88 0,91 30416 1,2 1,4 448 00704
13 9124 23577 2219 38668 38503 34920 1,1 1,05 1,09 1,06 38322 345,7 119508,5 165 7225
14 8013 19976 1958 31118 30878 29947 1,04 0,92 0,93 0,94 31209 -91,1 8299,2 240 7600
15 7916 18878 1897 29875 30341 28691 1,04 0,81 0,88 0,91 29932 -57 3249 -466 17156
16 8527 20007 1999 34187 33922 30533 1,1 0,98 0,93 0,96 33629 55,8 3113,6 265 70225
17 8391 19875 1876 33211 33485 30141 1,1 0,97 0,92 0,90 33158 52 2704 -274 75076
18 9318 24119 2508 40506 40131 35945 1,1 1,07 1,12 1,2 39547 95,9 9196 375 40625
19 9105 23965 2417 37782 38403 35487 1,06 1,05 1,11 1,16 37607 175 30625 -621 85641
20 8579 20196 1993 34525 34117 30768 1,1 0,99 0,94 0,95 33939 586 343396 408 66464
21 7958 19994 1915 29976 30342 29867 1,01 0,91 0,93 0,92 30179 -203 41209 -366 33956
22 9213 23341 2203 38715 37998 34757 1,1 1,06 1,08 1,05 38158 557 310249 417 173889
23 8979 22513 2107 35826 36334 33599 1,07 1,03 1,04 1,01 35826 0 0 -508 258064
24 9452 24768 2617 41532 41344 36837 1,13 1,00 1,15 1,25 40701 831 553536 188 35344
25 7923 18986 1926 29868 29506 28835 1,04 0,91 0,88 0,92 29954 -86 7396 362 131044
26 8122 19834 1983 31415 31648 29939 1.05 0,93 0,92 0,95 31996 19 361 -233 54289
27 9247 23452 2218 38719 38250 34917 1,1 1,06 1,09 1,07 38441 278 77284 469 219961
28 8255 20189 2004 32814 32389 30448 1,08 0,95 0,94 0,96 32969 -155 24025 425 180625
29 9008 22989 2128 37699 37482 34125 1,1 1,03 1,07 1,02 37565 134 17956 217 47089
30 9296 23577 2300 38589 38982 35173 1,1 1,07 1,09 1,10 38605 -155 24025 -393 154449
Let the planned time tpl of completing the works makes up 12 decades.
1. Let us determine the kind of function:
pi
xi = J a1 ^Mt = 8689,
X1 is the average value of volume of fixed assets within the analyzed period (determined from Table 1, column 2). Then
a^^l1/ = 8689.
2_ 3'
Then
a1 = 3 ■ 8689—= 313,57.
1 2 12л/12
So: X 4= 313,5^^/tl. 2. Similarly:
p1
Х2 = J a2^/t2dt2 = 21558,7,
X2 is the average value of costs of materials. fa^l'o2 = 21558,7;
a, = 321558,7—= 778;
2 2 12V12
X2 = 778^.
tpi
3. X3 = J a^>/t7dt3 = 2090,6;
3aaVt331
= 2090,6
are characterizes the average value of labor costs: a3 = -2090,6—= 75,4;
3 2 12V12
X - = 75, 4/-.
Let us determine the coefficients P1i, P 2i, P3i. included in the formula (4). For example:
Pii = 8767 = 1,14; Fli 8689
21042 ß2i= 21042 = 0,98; 21 21558,7
ß3l=-^ = 0,96. 31 2090,6
Yr4D = Kq £ Pi J Xidt = K1P1J Xidti +
i=1 ^ 0 j 0
tj ti tj +K2P2 J X 2dt2 + K3P3 J X 3dt3 + KnPn J X ndtn.
000
The results of calculations Yr4D are given in column 14 of Table 1. Column 16 contains data of comparison of the results, obtained by the developed method, and those, obtained by the traditional PF - Ypf.
The evaluation of accuracy of the results showed that root-mean-square error of determining volumes of works by the 4D model: o4D =±252 thousand RUB. Accuracy of calculation according to the model, which does not consider mutual and interdependent influence of time and resources, comprised oPF = ±361 thousand RUB. The difference comprises 109 thousand RUB, or 43 %.
The advantage of the proposed development is an increase in accuracy of prognostic calculations, which consider the systems development of the production process overtime, approximated to actual conditions of resources consumption.
An increase in accuracy of calculations contributes to an improvement in the quality of planning the implementation of advanced technologies, to an increase in effectiveness of using resources and, therefore, to an increase in the effectiveness of the production process.
The conducted studies, based on the previously completed works [19, 22], dedicated to a qualitatively new approach to engineering development of 4D space concept of the special theory of relativity, result into the need for the solution of the problems of actual production process, connected with appearance of non-collinearity of vectors of time and gains in volume of works.
6. Conclusions
1. As a result of the conducted studies, we developed the 4D model, including basic production resources and the outcome of production activity, which makes it possible to systemically analyze the development of production process over time from the positions of special theory of relativity.
2. It was established that the essence of time in a specific production process is manifested in the fact that the system of vectors of an increment in time ДТ!, is collinear to the corresponding vectors of an increment in volume of works AY (i=1, 2, ...<»). In this case, the vector of an increment in time АТ! corresponds to each vector of an increment in volume of works AYj. The infinite number of vectors of increments in time АТ! corresponds to the infinite number of vectors of increments in volume of works AYr It was shown that the non-uniformity of production process causes "compressibility" and "stretching" of time parameter along with changes in increments in volume of work AY and productivity Q(t).
3. The performed example of calculation, connected with determining the prospective volume of works in 4D space, including basic production resources and time, showed the possibility of a considerable (up to 40 %) increase in accuracy of determining the required parameter.
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