CC BY
23TECHNICAL SCIENCES
MODELING OF DIGITALIZATION PROJECT MANAGEMENT SYSTEMS BASED ON
STOCHASTIC NETWORKS
Bushuyev S.
Doctor of technical science, Professor Head of Department of Project Management, Kyiv National University of Construction and Architecture, Ukraine
Bushuieva V.
Candidate of technical science (PhD), Associated Professor Department of Project Management, Kyiv National University of Construction and Architecture, Ukraine
Zasukha I.
Post Graduate student, Department of Project Management, Kyiv National University of Construction and Architecture, Ukraine
Abstract
The formation of the process block of stochastic networks is focused on a complex human-oriented process with the result at the output in solving and implementing in time synergistic complex tasks in the systematic management of digitization projects within the state existing industries. There are computer technologies for finding solutions to approximation problems using stochastic networks, but the cost of such calculations is usually too high. Both the collection of the necessary numerical data and the construction of a stochastic network are very expensive. Therefore, the use of stochastic network methods is justified in solving large-scale and important problems. An additional condition for the proper use of the discussed technology is the availability of appropriate information systems, which continuously accumulate data on the subject area and similar design work required to calculate or estimate the probabilities. The recalculation of stochastic networks is quite complex due to the cumbersome mathematical apparatus. There are attempts to use simpler mathematical expressions for calculations or to simplify complex formulas. actual execution.
Keywords: digitization, digitization projects, stochastic networks, approximation.
Introduction. Our research is related to the development of a formalized model of the formation of the process unit of the digital project management system, where stochastic networks are widely used in practice [1, 2]. GERT technology, designed to describe and study a variety of scientific and technical measures, measures in the actual implementation of the actual tasks in their actual performance, allows you to determine the duration (or characteristics of another kind) and the probability of implementation of sequences of events. The application of GERT network models for planning and management of research and design work, and in fact their implementation allows:
- provide much greater compactness than when using other methods;
- rank decisions on the probability of success;
- quickly determine the impact of new information about the values of the parameters on the final event through the use of computer technology;
- apply simulation to evaluate research and design activities;
- to determine the exact execution of actual works with possible ways of their deviation and to receive concrete terms of the end of actual performance of works;
- dynamically optimize the modeled structures;
- modify the analyzed processes;
- to create visual graphical representations of the process of implementation of the whole event (as well as when using traditional technologies, such as CPM and PERT).
The method of graphical evaluation and revision of programs (GERT), allows you to take into account the risk of changes in the composition of work at the occurrence of certain events or the results of previous work. The GERT network model can create branch points or selection points, after which several independent chains of work are planned, not all of which are performed.
The GERT method allows to determine the expected duration (OT) of the project on the basis of three probabilistic estimates of time. The network model is a reliable network that takes into account the possibility of different composition of the project.
Thus, it is possible to take into account not only the risks (uncertainty) at the level of individual works, but also at the level of the project as a whole.
The accounting of risks affecting the duration of work is carried out in the same way as in the PERT method, i.e. based on the results of the calculation of the weighted average duration estimate on the basis of three estimates issued by experts.
The result of GERT modeling will be several graphs that take into account the probability of different durations and the uncertainty of the project.
The complexity of the problem of finding a solution is solved by using a simulation solution based on the Monte Carlo method.
The purpose of the article. The aim is to study the formation of the process unit using stochastic networks in the digital project management system.
Modelling of digitalization project management. At the decision of the set task and as a result of the researches carried out by me, in the directions of the field of researches and software development for a system of management of projects of digitalization - it is possible to use such methods of acceleration of realization of all project at the minimum expenses.
Let consider the following method, PERT-COST technology.
Considering this method, it is necessary to consider and assume that it is impossible to reduce the duration of the work included in the project. However, the use of additional tools and resources usually speeds up some work, for example, when using more powerful equipment or by attracting additional staff. However, such actions require additional costs and are not always economically justified. If the company is threatened with significant financial sanctions if the project is not completed or if the project is associated with high fixed costs, then raising additional funds will be the right decision. However, how to determine what work can be accelerated, how much will such acceleration cost, will it reduce the performance of a particular job and the time of the project as a whole? The ideal solution would be to develop a method of accelerating the implementation of the entire project at minimal cost. To do this, a modification of PERT technology, called PERT-COST. It is designed to find the optimal reduction in project duration in PERT networks at minimal cost.
Optimization of the duration of the project is based on the principle of reducing the time of work, according to which the greatest reduction is the duration of those critical works, the implementation of which is associated with the lowest costs. When using PERT-COST technology, the cost curve will be a function of several parameters of each specific work (a, m, b, te, s). In addition, PERT-COST technology operates with the following notation:
ten - the expected normal duration of work n, which corresponds to the lowest cost Kn;
tegr - the expected maximum duration of work n, i.e. the minimum time of work,
due to technical and technological constraints, which correspond to the marginal cost Kgr
According to the conditions of PERT-COST technology, the cost of work depends linearly on its duration, so they increase in proportion to the change in duration from normal to the limit value. For each work, you can calculate the average cost gradient S, which characterizes the increase in the cost of work caused by reducing its duration by one unit of time:
Sn =
Kgr—Kr
=
bn _ bgr _
t = t = 2'
^en ^egr
From these relations it is possible to deduce values of a standard deviation for reduction of duration of performance of work n:
G =
36
■*tí
ten tegr
An additional condition of the considered technology is the constancy of the relations of durations of each concrete work, which are equal to: - for optimistic duration:
egr
- for pessimistic duration:
where tes - reduced duration of work (after the emergence of additional costs).
The cost of accelerating each specific work is calculated as multiplying the gradient of the cost of this work by the number of units of time for which the duration of the work was reduced:
Kn (ten - tes) * Sn.
The following stages of application of PERT-COST technology can be distinguished:
1. Determining the completion date and critical path based on the expected normal durations of work (by analogy with PERT technology).
2. Selection of critical works and calculation of cost gradients for them.
3. Exclusion from the set of critical works for which the average cost gradient does not exist, i.e. ten -
tegr.
4. The beginning of the process of reducing the duration of work on the critical work that has the lowest cost gradient.
5. Reducing the duration of work by as many units of time as possible, taking into account two restrictions:
- the maximum duration of this work tegr,
- the emergence of a new critical path - if the time reserve in the sequence of non-critical work disappears.
6. If there are two or more critical paths in the network, the durations should be reduced by the same amount on all parallel critical paths.
7. The shortest duration of the project is achieved when the duration of all works that lie on the critical path, reach the value of tegr. Further reduction of the duration of the project is no longer possible.
8. Acceleration costs at each stage are calculated as multiplying the cost gradient of a particular job by the number of units of time for which this critical work was reduced: Kn = (ten - tes) * Sn. The total cost of accelerating the implementation of the project is the sum of the costs of accelerating individual works.
8. Acceleration costs at each stage are calculated as multiplying the cost gradient of a particular job by the number of units of time for which this critical work was reduced: Kn = (ten - tes) * Sn. The total cost of accelerating the implementation of the project is the sum of the costs of accelerating individual works.
Case study
In the already mentioned project "Constructive development of product X", the company's management set a task for the project team to create a new product as soon as possible, for which it allocated additional funds. The company was forced to increase costs, as the
2
1
a
n
expected conclusion on the market of a new product of the main competitor can greatly weaken the position.
The maximum duration of each tegr work was assessed in consultation with the specialists responsible
for the implementation of specific stages of the project so as to actually bring them closer to the technically and technologically acceptable limits. The obtained estimates are presented in the sixth column of Table 1.
Table. 1.
Activity Estimated time te tegr Kn Kegr
a m b
A 2 3 4 3,00 2 100 120
B 7 8 10 8,17 6 600 800
C 3 4 5 4,00 2 250 350
D 1 1 2 1,17 1 150 160
E 1 2 4 2,17 1 150 200
F 3 5 8 5,17 3 100 120
G 4 6 9 6,17 4 800 1000
H 10 12 15 12,17 8 1200 1800
I 2 2 2 2,00 2 100 100
J 4 5 6 5,00 3 200 300
K 12 16 21 16,17 10 1400 2000
L 7 9 10 8,83 6 900 1100
M 1 1 1 1,00 1 150 150
N 2 3 4 3,00 2 100 110
O 2 3 4 3,00 2 300 350
P 5 7 10 7,17 4 650 800
Q 1 2 2 1,83 1 100 110
R 7 8 10 8,17 6 500 700
S 2 2 3 2,17 2 200 210
T 1 2 3 2,00 1 100 120
U 4 5 7 5,17 4 400 550
PA30M - - - - - 8450 11150
In addition, the project team calculated the normal costs of each work Kn, i.e. the cost of their implementation during the expected period, as well as the marginal cost Kgr - the highest cost of work during the deadline with maximum use of forces and means. Next, for each work was calculated the average cost gradient, as well as the ratio of optimistic and pessimistic durations. The results of these calculations are presented in table 2.
The critical path of the project is a sequence of works A-B-C-F-G-H-K-L-P-R-S-U with an expected duration of Te = 86.36 weeks. When using the criterion of the minimum cost gradient S, the reduction will start with the work F and will consistently cover the work: A, P, C, S, L, B, G, R, K, U, H. The expected duration of the measure after each step is reduced by more than the difference between the expected and the maximum duration of the relevant work. In the table. 3.P. the sequence of reducing the duration of work and the estimated additional costs are presented.
Further reduction of the duration of the project is impossible, as the duration of all works that lie on the critical path A-B-C-F-G-H-K-L-P-R-S-U have reached critical values. The standard deviation of the expected duration of the project implementation 6. The decreases after each step, as the value of the standard deviation 6s for reduced works also decreases, even if for other works that are on the critical path, the standard deviation 6s will remain unchanged. Table 4 presents the calculation of the expected durations of the various paths of the project "Constructive development of product X" after each step of reducing work.
After each reduction of the next works the way A-B-C-F-G-H-K-L-P-R-S-U remained critical, therefore it was possible to continue reduction of the works lying on it. In other projects, new critical pathways may emerge in the process of reduction; in such cases, work on these or parallel critical paths should be reduced.
Tab.2.
Calculation of cost gradients and ratios of durations of project works
Activity Estimated time te tegr Kn Kgr Average gradient of cost Kgr - Kr Sn = Relation a ri= Relation b te
a m b
ten - tegr te
A 2 3 4 3,00 2 100 120 20 0,67 1,33
B 7 8 10 8,17 6 600 800 92 0,86 1,22
C 3 4 5 4,00 2 250 350 50 0,75 1,25
D 1 1 2 1,17 1 150 160 60 0,86 1,71
E 1 2 4 2,17 1 150 200 43 0,46 1,85
F 3 5 8 5,17 3 100 120 9 0,58 1,55
G 4 6 9 6,17 4 800 1000 92 0,65 1,46
H 10 12 15 12,17 8 1200 1800 144 0,82 1,23
I 2 2 2 2,00 2 100 100 - - -
J 4 5 6 5,00 3 200 300 50 0,80 1,20
K 12 16 21 16,17 10 1400 2000 97 0,74 1,30
L 7 9 10 8,83 6 900 1100 71 0,79 1,13
M 1 1 1 1,00 1 150 150 - - -
N 2 3 4 3,00 2 100 110 10 0,67 1,33
O 2 3 4 3,00 2 300 350 50 0,67 1,33
P 5 7 10 7,17 4 650 800 47 0,70 1,40
Q 1 2 2 1,83 1 100 110 12 0,55 1,09
R 7 8 10 8,17 6 500 700 92 0,86 1,22
S 2 2 3 2,17 2 200 210 60 0,92 1,38
T 1 2 3 2,00 1 100 120 20 0,50 1,50
U 4 5 7 5,17 4 400 550 129 0,77 1,35
Pa30M - - - - - 8450 11150 - - -
Table 3.
Reduction of duration of works and costs of reduction, as well as the duration of the project._
Step Action Reduction costs Project duration (Weeks)
1 CKoponeHHH tF Ha 2,17 Kf = 2,17 x 9 = 19,53 Te= 84,19
2 CKoponeHHa tA Ha 1,00 Ka = 1 x 20 = 20,00 Te= 83,19
3 CKoponeHHH tp Ha 3,17 Kp = 3,17 x 47 = 148,99 Te= 80,02
4 CKoponeHHa tc Ha 2,00 Kc = 2 x 50 = 100,00 Te= 78,02
5 CKoponeHHa ts Ha 0,17 Ks = 0,17 x 60 = 10,20 Te= 77,85
6 CKoponeHHa 4 Ha 2,83 Kl= 2,83 x 71 = 200,93 Te= 75,02
7 CKoponeHHH tB Ha 2,17 Kb = 2,17 x 92 = 199,64 Te= 72,85
8 CKoponeHHH tG Ha 2,17 Kg= 2,17 x 92 = 199,64 Te= 70,68
9 CKoponeHHH tR Ha 2,17 Kr= 2,17 x 92 = 199,64 Te= 68,51
10 CKoponeHHH tK Ha 6,17 Kk = 6,17 x 97 = 598,49 Te= 62,34
11 CKoponeHHa U Ha 1,17 Ku = 1,17 x 129 = 150,93 Te= 61,17
12 CKoponeHHa fe Ha 4,17 Kh = 4,17 x 144 = 600,48 Te= 57,00
Table 4.
Calculation of expected durations of different project paths after each step of work reduction
Path / Step 0 1 2 3 4 5 6 7 8 9 10 11 12
A-B-C-E-G-H-J-N-Q-R-S-T 57,9 57,9 56,9 56,9 54,9 54,7 54,7 52,5 50,34 48,17 48,17 48,17 44,00
A-B-C-E-G-H-J-N-Q-R-S-U 61,0 61,0 60,0 60,0 58,0 57,9 57,9 55,7 53,51 51,34 51,34 50,17 46,00
A-B-C-E-G-H-K-L-P-R-S-T 80,2 80,2 79,2 76,0 74,0 73,9 71,0 68,9 66,68 64,51 58,34 58,34 54,17
A-B-C-E-G-H-K-L-P-R-S-U 83,4 83,4 82,4 79,2 77,2 77,0 74,2 72,0 69,85 67,68 61,51 60,34 56,17
A-B-C-E-G-H-K-M-O-P-R-S-T 75,4 75,4 74,4 71,2 69,2 69,0 69,0 66,9 64,68 62,51 56,34 56,34 52,17
A-B-C-E-G-H-K-M-O-P-R-S-U 78,5 78,5 77,5 74,4 72,4 72,2 72,2 70,0 67,85 65,68 59,51 58,34 54,17
A-B-C-E-G-H-I-S-T 41,9 41,9 40,9 40,9 39,9 38,7 38,7 36,5 34,34 34,34 34,34 34,34 30,17
A-B-C-E-G-H-I-S-U 45,0 45,0 44,0 44,0 42,0 41,9 41,9 39,7 37,51 37,51 37,51 36,34 32,17
A-B-C-F-G-H-J-N-Q-R-S-T 60,9 58,7 57,7 57,7 55,7 55,5 55,5 53,3 51,17 49,00 49,00 49,00 44,83
A-B-C-F-G-H-J-N-Q-R-S-U 64,0 61,9 60,9 60,9 58,9 58,7 58,7 56,5 54,34 52,17 52,17 51,00 46,83
A-B-C-F-G-H-K-L-P-R-S-T 83,2 81,0 80,0 76,9 74,9 74,7 71,9 69,7 67,51 65,34 59,17 59,17 55,00
A-B-C-F-G-H-K-L-P-R-S-U 86,4 84,2 83,2 80,0 78,0 77,9 75,0 72,9 70,68 68,51 62,34 61,17 57,00
A-B-C-F-G-H-K-M-O-P-R-ST 78,4 76,2 75,2 72,0 70,0 69,9 69,9 67,7 65,51 63,34 57,17 57,17 53,00
A-B-C-F-G-H-K-M-O-P-R-S-U 81,5 79,4 78,4 75,2 73,2 73,0 73,0 70,9 68,68 66,51 60,34 59,17 55,00
A-B-C-F-G-H-I-S-T 44,9 42,7 41,7 41,7 39,7 39,5 39,5 37,3 35,17 35,17 35,17 35,17 31,00
A-B-C-F-G-H-I-S-U 48,0 45,9 44,9 44,9 42,9 42,7 42,7 40,5 38,34 38,34 38,34 37,17 33,00
A-D-E-G-H-J-N-Q-R-S-T 46,9 46,9 45,9 45,9 45,9 45,7 45,7 45,7 43,51 41,34 41,34 41,34 37,17
A-D-E-G-H-J-N-Q-R-S-U 50,0 50,0 49,0 49,0 49,0 48,9 48,9 48,9 46,68 44,51 44,51 43,34 39,17
A-D-E-G-H-K-L-P-R-S-T 69,2 69,2 68,2 65,0 65,0 64,9 62,0 62,0 59,85 57,68 51,51 51,51 47,34
A-D-E-G-H-K-L-P-R-S-U 72,4 72,4 71,4 68,2 68,2 68,0 65,2 65,2 63,02 60,9 54,68 53,51 49,34
A-D-E-G-H-K-M-O-P-R-S-T 64,4 64,4 63,4 60,2 60,2 60,0 60,0 60,0 57,85 55,68 49,51 49,51 45,34
A-D-E-G-H-K-M-O-P-R-S-U 67,5 67,5 66,5 63,4 63,2 63,2 63,2 63,2 61,02 58,85 52,68 51,51 47,34
A-D-E-G-H-I-S-T 30,9 30,9 29,9 29,9 29,9 29,7 29,7 29,7 27,51 27,51 27,51 27,51 23,34
A-D-E-G-H-I-S-U 34,0 34,0 33,0 33,0 33,0 32,9 32,9 32,9 30,68 30,68 30,68 29,51 25,34
A-D-F-G-H-J-N-Q-R-S-T 49,9 47,7 46,7 46,7 46,7 46,5 46,5 46,5 44,34 42,17 42,17 42,17 38,00
A-D-F-G-H-J-N-Q-R-S-U 53,0 50,9 49,9 49,9 49,9 49,7 49,7 49,7 47,51 45,34 45,34 44,17 40,00
A-D-F-G-H-K-L-P-R-S-T 72,2 70,0 69,0 65,6 65,9 62,9 62,9 62,9 60,68 58,51 52,34 52,34 48,17
A-D-F-G-H-K-L-P-R-S-U 75,4 73,2 72,2 69,0 69,0 66,0 66,0 66,0 63,85 61,68 55,51 54,34 50,17
A-D-F-G-H-K-M-O-P-R-S-T 67,4 65,2 64,2 61,0 61,0 60,9 60,9 60,9 58,68 56,51 50,34 50,34 46,17
A-D-F-G-H-K-M-O-P-R-S-U 70,5 68,4 67,4 64,2 64,2 64,0 64,0 64,0 61,85 59,68 53,51 52,34 48,17
A-D-F-G-H-l-S-T 33,9 31,7 30,7 30,7 30,7 30,5 30,5 30,5 28,34 28,3 28,34 28,34 24,17
A-D-F-G-H-I-S-U 37,0 34,9 33,9 33,9 33,9 33,7 33,7 33,7 31,51 31,5 31,51 30,34 26,17
Table 5.
Calculation of total costs to reduce the duration of the project
Step Te K A K 6Te
0 86,36 0,00 0,00 2,5004
1 84,19 19,53 19,53 2,4119
2 83,19 39,53 39,53 2,3991
3 80,02 188,52 188,52 2,2973
4 78,02 288,52 288,52 2,2791
5 77,85 298,52 298,72 2,2782
6 75,02 499,72 499,65 2,2484
7 72,85 699,29 699,29 2,2290
8 70,68 898,93 898,93 2,1303
9 68,51 1098,57 1098,57 2,1031
10 62,34 1697,06 1697,06 1,7418
11 61,17 1847,99 1847,99 1,7129
12 57,00 2448,47 2448,47 1,5936
Table 5 presents the expected duration of the project "Constructive development of product X" after each step of reducing work, along with the associated additional costs. The last column shows the calculated standard deviations of the duration of the project (the square root of the sum of the squares of the standard deviations of the duration of the work, are on the critical path).
Now, let consider the technology of GERT.
To achieve the goals of any project requires the correct interaction of contractors and optimal use of resources in each phase of its implementation. To do this, there must be a possibility of redistribution of funds in accordance with the current situation, and such redistribution must be rational. In this case, it may be useful to use stochastic networks, as an example of which we consider the GERT network. These networks are, of
course, more complex than deterministic networks (used, in particular, in CPM and PERT technologies), but they allow us to consider different dependencies between events in the same network, as well as freely choose in the project implementation paths that differ from certain in advance.
Stochastic network technologies can be used in all situations where PERT technology is used. Construction of a stable PERT network and triple evaluation of the characteristics of each of its arcs, as a rule, simplify the described reality. Stochastic network technologies provide much greater and more diverse opportunities for reality analysis. Technologies based on stochastic networks introduce probabilistic types of events in the form of logical combinations of works by the "OR" operation, which allow to consider alternative solutions [1, 3].
Used in PERT technology, three estimates of the duration of the work, often reflecting three aspects of the problem (e.g., labor, material, cost), obscure the picture of the use of specific resources. Attempts to improve PERT technology have led to several new methodological proposals. Representation of various networks became possible due to the introduction of a new type of event. This event differs from the ones used in CPM and PERT networks in that the moment of its occurrence allows to determine the beginning of only one work, and not several at a time, as previously predicted. H. Eisner's concept, which consists in the introduction into the network of decision-making units and the possibility of various exits from events, allowed to begin to create networks for sets of activities, which so far, given the alternative nature of decisions required separate planning.
However, the introduction of events of this type was also insufficient. Soon SE Elmaghraby [1, 2] proposed a further expansion of the set of logical intranet dependencies and the main types of events and mathematical relations caused by these dependencies. He introduced the GAN (Generalized Activity Networks) network. SE Elmagrabi proposed to further increase the flexibility of network technology by introducing separate event outputs. In GAN technology, an event consists of two parts: the input side and the output side. In addition to the advantages in the form of alternative outcomes of events, the discussed technology allows to streamline the probabilities of work and determine the parameters that more accurately characterize these works, such as the duration of their implementation. The advantage of GAN technology is also the admissibility of cycles and feedback in the network.
Among the technologies based on stochastic networks, the technology GERT (Graphical Evaluation and Review Technique) deserves attention. It uses as elements of the algebra of graphs SE Elmagrabi and the GAN network. Of particular importance is the accounting in the network of alternative works in the case when it describes a project of a stochastic (random) nature. In the implementation of such projects there are various perturbations, as a result of which the characteristics of specific arcs of the network (i.e. parameters describing the relevant work) take on new, different from the planned values. In addition, such perturbations force changes to be made to the network in the form of new
arcs and vertices representing alternative works. Experience with the use of network methods shows that it is more effective to enter into the network all possible alternative works before the start of the project than to adjust the network in the process of its implementation. The discussed technology can be applied to any complex projects and in unclear management situations. Examples include research or design work.
The procedure for applying GERT technology can be divided into the following stages:
1. Description of the project by a stochastic network.
2. Collection of numerical data characterizing each arc of the network.
3. Minimization of the constructed stochastic network.
4. Transformation of the replacement network (or function) into a form that allows you to determine the duration and probability of project implementation, as well as the calculation of these durations and probabilities.
5. Analysis and evaluation of results obtained through network simplifications.
Project description by stochastic network. Used in stochastic networks, the concept of "event" is expanded compared to its previous understanding in technologies based on networks of deterministic structure. There are two main types of events in the GERT network: deterministic, presented in the form of a circle, and probable, presented in the form of a "drop". It should be noted that all arcs (branches) that come from deterministic events, corresponds to the probability of realization p = 1, i.e. they must be performed in order for the project to be recognized as implemented. At the same time, from all branches coming from the probabilistic event, only one, which has a certain probability p, will be sufficient to bring the project to a logical conclusion.
In [3, 4, 5] proposed a generalized GAN network and a typology of network vertices. Logical forms of vertices from the input side in GAN networks are defined as follows:
- a vertex of type "I" for events that occur if and only when all their previous work is completed, i.e. the first work "AND", the second work "AND" and so on. In this case, the logical operation "AND" is implemented. For example, the completion of work on the design of the prototype, the manufacture of elements of the prototype and the choice of technology for its assembly allow you to properly assemble the prototype;
- top type "OR" - for events that occur when at least one of their previous work is completed; the timing of such events is due to the shorter duration of the work leading to them. The logical operation "OR" is implemented. As an example: meeting the expectations of one of the designed products can, but should not interrupt further research. Upon receiving positive results from the application of one technology in the production process, research can be continued to create an alternative technology, which is characterized, for example, less resource consumption;
- top type "exclusive OR" - for events that occur if and only when exactly one of the previous works will be performed. Events are preceded by mutually exclusive
events. The moment of occurrence of the event means the completion of one and only one work. The logical operation "exclusive OR" is implemented. For example, the acquisition of a license makes it impossible to change the documentation without the consent of the owner of the technology, and the development of its own optimal design eliminates the need to purchase a license.
Logical forms of vertices from the output side in GAN networks are practically limited to two types:
- "AND" - the so-called deterministic output associated with events, the occurrence of which means the performance of all subsequent work on such events. There must be one or more jobs after the event node. The moment of the event means the possibility of starting all these works. The logical operation "AND" is implemented. For example, the moment of creation of a prototype allows to begin research of this prototype.
- "OR" - the so-called probabilistic outcome associated with events, the occurrence of which entails the performance of at least one of the subsequent works. From several initial works one or several can be realized. As an example: the use of one or more concepts of
product development to continue design developments. In the case of alternative solutions, the sum of their probabilities at the output-solver must be equal to step 1.
For these logical forms of network vertices, depending on the proposed graphic notation. All practically applied types of vertices are shown in fig. 5.
Both inputs and outputs of events of deterministic networks have a logical form "AND". In a generalized GAN network, the logical description of the input side determines the condition of the event. The description of the output side can be crucial. After the event in accordance with this function is determined by the fate of subsequent work. The nature of the decisions made is considered deterministic, because the implementation of the network depends on the implementation of all subsequent work. If the description of an exit from an event allows to realize only some works, then in relation to the following works it is possible to speak only about their probable performance. This network is called a stochastic network, it is based on GERT technology. The sum of the probabilities corresponding to each branch coming from the probabilistic event must be equal to step 1.
Fig.5. Characteristics of vertices used in stochastic networks
Along with the presented models of events, called indirect events, in network models GERT it is possible to allocate initial (initial) and final events. The initial event can be deterministic or probabilistic [7, 8, 9]. Final events are always deterministic. A characteristic feature of GERT models - contours, called feedback or loops. They indicate that certain actions or events may be performed or occur more than once. The GERT network indicates the number of repetitions of work (counter)
[10, 11, 12]. A contour begins in a so-called statistical event, and an event in which the contour leads to a normal duration of work is called a selected event [13, 14, 15,16].
Conclusions
The solution of GERT network models by the method of sequential network reduction in practice is very time consuming. The complexity of the problem of
finding a solution for a GAN-type network has led to the need to use for this purpose simulation solutions based on the Monte Carlo method. An example of such a computational approach is the technology GERTS (Graphical Evaluation and Review Technique Simulation).
The scheme of application of this technology is as follows:
1. We use random number generators available in software packages on most computers, with the following purpose:
a) for nodes with alternative outputs, we generate random numbers according to the probability distribution at these outputs; these numbers uniquely define the subnet, which is one of the possible options for the event;
b) for each work of the subnet obtained in paragraph a), we generate a random number according to the probability distribution that characterizes the duration of this work.
2. Considering the data received in item 1 as determined, we calculate characteristics characteristic of us, for example, term of completion of the project and time reserves. For this purpose, technologies are used that correspond to deterministic models, in particular CPM. The results obtained at this stage are recorded in computer memory. The presented steps are repeated a given number of times to obtain fairly accurate estimates of the parameters of interest to us. Examples of parameters calculated by GERTS technology include the probabilities of realization of final events, as well as the average values and variances of the distribution of the timing of these events.
Stochastic networks are widely used in practice. A very important step for the formation of a process unit using stochastic networks in the public sector digital project management system is GERT technology, which is designed to describe and study various scientific and technical measures, to determine the duration (or characteristics of another kind) and probabilities of followers. Application of GERT network models for planning and management of research and design works allows:
- provide much greater compactness than when using other methods;
- rank decisions on the probability of success;
- quickly determine the impact of new information about the values of the parameters on the final event through the use of computer technology;
- apply simulation to evaluate research and design activities;
- dynamically optimize the modeled structures;
- modify the analyzed processes;
- to create visual graphical representations of the process of implementation of the whole event (as well as when using traditional technologies, such as CPM and PERT).
References
1. Elmaghraby S.E. An Algebra for the Analysis of Generalized Activity Networks / Management Science. — 1964. — № 3.
2. Elmaghraby S.E. The Theory of Networks and Management Science. Part 2 / Management Science. — 1970. — № 3.
3. Mohamed, S., McCowan, A.K. (2001) «Modelling project investment decisions under uncertainty using possibility theory». Int. J. Project Management, 19, pp. 231—241.
4. Liang, G.S. and Wang, M.J. (1991) «A fuzzy multi criterion decision making for facility site selection». International Journal of Production Research, 29, pp. 2313—2330.
5. Chan, D.Y. «Application of extent analysis method in fuzzy AHP». European Journal of Operation Research, (1996) № 95, pp. 649—655.
6. Lee, J.W. and Kim, S.H. «Using analytic network process and goal programming for interdependent information system project selection». Computers & Operations Research, (2000) № 27, pp. 367—382.
7. Kahraman, C, Cebeci, U. and Ruan, D. «Multiattribute comparison of catering service companies using fuzzy AHP: the case of Turkey». International Journal of Production Economics, (2004) № 87, pp. 171— 184.
8. Huang X. (2007) «Optimal project selection with random fuzzy parameters». Int. J. Production Economics, 106, pp.513—522.
9. Lai, Y.J., Lai, H.C. (1993) «Possibilistic linear programming for managing interest rate risk». Fuzzy Sets and Systems, 54, pp. 135—146.
10. Bondar A., Bushuyev S., Bushuieva V., Bushuyeva N., Onyshchenko Bondar A., Bushuyeva N., Bushuyev S., Onyshchenko S. Modelling of Creation Organisational Energy-Entropy, (2020): IEEE 15th International Conference on Computer Sciences and Information Technologies (CSIT), Zbarazh, Ukraine, (2020): 141-145
11. Saaty T. (1990) «How to make a decision: The Analytic Hierarchy Process». European Journal of Operational Research, 48, pp. 9—26.
12. Bondar A, Bushuyev S., Onyshchenko S., Tanaka H. Entropy Paradigm of Project-Oriented Or-ganizatio ns Management (2020) Proceedings of the 1st International Workshop IT Project Management (ITPM 2020) Volume 1. Lviv, Ukraine, February 18-20, (2020), CEUR Workshop Proceedings (CEUR-WS.org), (2020): 233-243
13. Bushuyev S., Bushuieva V., Onyshchenko S., Bondar A. Modeling the Dynamics of Information Panic in Society. COVID-19 case (2021); Proceedings of The Fourth International Workshop on Za-porizhzhia, Ukraine, April 27, 2021.; CEUR Workshop Proceedings, 2021, 2864, 400-408
14. Bondar A., Bushuyev S., Bushuieva V., Onyshchenko S. Complementary strategic model for managing entropy of the organization (2021); Proceedings of the 2nd International Workshop IT Project Management (ITPM 2021) Slavsko, Lviv region, Ukraine, February 16-18, 2021. CEUR Workshop Proceedings, 2021.
15. Bushuyev, S., Bushuiev, D., Bushuieva, V. Modelling of emotional infection to the information system management project success Advances in Intelligent Systems and Computing, (2021), 1265 AISC, pp. 341-352
16. Bushuyev, S., Babayev, J., Bushuiev, D., tion SMART Government Projects 2020 IEEE Euro-Kozyr, B. Emotional Infection of Management Innova- pean Technology and Engineering Management Summit, E-TEMS (2020)
СРАВНИТЕЛЬНЫЙ АНАЛИЗ РОТОРНЫХ СИСТЕМ СИНХРОННЫХ МАШИН НА
ПОСТОЯННЫХ МАГНИТАХ
Кириллов И.В.
АО «УАПО», инженер-конструктор Апальков Р.Г. АО «УАПО», инженер-схемотехник Борисоглебский Н.А. НИУМЭИ, студент Иванов А. С. НИУ МЭИ, ст. преп.
COMPARATIVE ANALYSIS OF ROTOR SYSTEMS OF PERMANENT MAGNET SYNCHRONOUS
MACHINES
Kirillov I.
JSC "UAPO", design engineer Apalkov R. JSC "UAPO", circuit engineer Borisoglebsky N. NRUMPEI, student Ivanov A. NRU MPEI, st. Prep
Аннотация
Данная работа посвящена рассмотрению и анализу конструкций ротора синхронных машин с постоянными магнитами. Рассматривается два вида расположения постоянных магнитов на роторе: радиальное и тангенциальное расположение постоянных магнитов. Анализируется каждый вид на предмет воздействия пульсации момента на ротор синхронной машины.
Abstract
This work is devoted to the consideration and analysis of the rotor designs of synchronous machines with permanent magnets. Two types of arrangement of permanent magnets on the rotor are considered: radial and tangential arrangement of permanent magnets. Each type is analyzed for the impact of the pulsation of the rotor of a synchronous machine.
Ключевые слова: Синхронная машина, синхронная машина с постоянными магнитами, СМПМ, радиальное поле, тангенциальное поле, пульсация момента.
Keywords: synchronous motor, permanent magnet synchronous motor, PMSM, radial field, tangential field, moment pulsation.
Использование синхронных машин с использованием в качестве возбудителя постоянные магниты (СМПМ) в последнее время все более широко применяются. Объясняется это появлением редкоземельных материалов, таких как самарий-кобальтовые сплавы и соединения NdFeB (неодим-железо-бор) [1]. Использование постоянных магнитов позволяет выполнять машины с лучшими массога-баритными показателями, а также исключает электрические потери на возбуждение, что положительно сказывается на коэффициенте полезного действия [2].
Недостатком в использовании синхронных машин с постоянными магнитами является отсутствие возможности регулирования потока возбуждения. В основном синхронные машины с постоянными магнитами используются в системах регулируемого электропривода. В таких машинах пульсации
электромагнитного момента часто оказываются весьма значительными.
Пульсации момента, обусловленные дискретностью статорной и роторной частей магнитной системы машины, принято называть зубцовыми. Зуб-цовые пульсации момента могут негативно сказываться на характеристиках электрической машины. Следует заметить, что действие постоянных магнитов не прекращается при отключении питания машины. Следовательно, этот эффект присутствует и при обесточенной обмотке якоря,
Существует множество вариантов конструкций ротора с постоянными магнитами, которые можно разделить на два основных типа: с поверхностной установкой постоянных магнитов на роторе (рисунок 1) и конструкции со встроенной установкой магнитов на роторе (рисунок 2). Как правило, в первом случае магниты имеют радиально
