CC BY
70
3. Grishnova, O., Polyv'iana, N., 2009. Social capital: essence, significance, interconnection with other forms of capital. Ukraina: aspekty pratsi, № 3, pp. 19-24. [in Ukrainian].
4. Yelahin, V. P., 2011. On the essence of the concept of "social capital" and its role in the process of building a social state. № 1, p. 46-55. [in Ukrainian].
5. Grishnova O., Zaichko O. 2016. Modern Technologies in Labor Economics: New Opportunities and New Challenges. Herald of the Economic Sciences of Ukraine, № 2, pp. 52-56. [in Ukrainian].
6. Kamenska, O., 2016. Conception of management of industrial enterprise human capital is in the conditions of innovative development. Ekonomichnyi visnyk Donbasu, № 3(45), p. 225-229 [in Ukrainian].
7. Zaitsev, V., 2017. The choice of the model of corporate requirements to the competence of a specialist of industrial enterprise. Ekonomichnyi visnyk Donbasu, № 2(48), p. 207-213. [in Ukrainian].
8. Campbell, J. P., Dunnette, M. D., Hough, L. M., 1990. Modeling the performance prediction problem in industrial and organizational psychology. Handbook of Industrial and Organizational Psychology. Palo Alto: Consulting Psychologists Press, Second edition, vol. 1, p. 687-732.
9. Bartram, D., Robertson, I. T., Callinen, M., 2002. Competency and individual performance. Modeling the world of work. Organizational Effectiveness: The Role of Psycholog. - Chichester: Wiley, p. 596-618.
10. Glinkowska, B., Kaczmarek, B., 2016. Zarz^dzanie mi^dzynarodowe i internacjonalizacja przedsi^biorstw. Teoria i praktyka. Wydawnictwo Uniwersytetu tódzkiego. tódz, 290 s. [in Poland].
11. Kaczmarek, B., 2016. Zarz^dzanie mi^dzykulturowe - wybrane aspekty; B. Glinkowska (red.), Internacjonalizacja przedsi^biorstw. Uwarunkowania, procesy, wyniki badaií, tódz, s. 87-95. [in Poland].
12. Glinkovska, B., Chebotaryov, V., 2016. Small and medium business in Poland and Ukraine: classification problems and initial institutional framework for transformation / B. Glinkowska (red.), Internacjonalizacja przedsi^biorstw. Uwarunkowania, procesy, wyniki badaí, tódz, s. 153-164. [in Poland].
13. Nasulea, C., Spínu, D. F., Moroianu, R. M., 2017. The Polish recipe. Visnyk Kyivskoho natsionalnoho universytetu imeni Tarasa Shevchenka. Ekonomika, 3(192), p. 34-38 [in Ukrainian]. DOI: https://doi.org/10.17721/1728-2667.2017/192-3/7.
14. Watanabe I. (1999). World-Glass management practices Enduring Methods for Competitive Success. Crisp Publications, Inc., 85 p.
15. Trompenaars, F., Woolliams, p., 2003. Business across cultures. West Sussex, England: Capstone Publishing.
16. Hofstede G., 1992. Cultures and organizations: software of the mind. London: McGraw-Hill.
17. Lewis R., 2013. Fish Can't See Water: How National Culture Can Make or Break Your Corporate Strategy, John Wiley & Sons, 312 p.
Bulletin of Taras Shevchenko National University of Kyiv. Economics, 2018; 3(198): 48-53 УДК 519.85
JEL classification: C 610
DOI: https://doi.org/10.17721/1728-2667.2018/198-3/6
E. Ivokhin, Doctor of Sciences (Physics and Mathematics), Professor,
ORCID iD 0000-0002-5826-7408 D. Apanasenko, PhD Student ORCID iD 0000-0001-8387-152X Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, V. Navrodskiy, PhD in Physics and Mathematics, Associate Professor
ORCID iD 0000-0002-0394-6593 Kyiv National University of Culture and Arts, Kyiv, Ukraine
ABOUT PRODUCTION-TRANSPORT PROBLEM REDUCTION TO THE TWO-LEVEL PROBLEM OF DISCRETE OPTIMIZATION AND ITS APPLICATION
In this study, the application of the production and transport task is considered to solve the problem of the distribution of the limited capacities of data transmission channels between different nodes of the computer network. A scheme is proposed for reducing the problem to a two-level continuous-discrete optimization problem. The model is formulated and numerical results are obtained to solve the problem of power distribution in the network of the information and computing center.
Key words: power distribution, production and transport task, discrete-continuous programming, two-level model, optimization.
Introduction. Many applied problems relate to the distribution of limited resources in hierarchical systems [17]. This, for example, can be the task of balancing the load in a homogeneous network [2], the distribution of work for parallel computers [3], the allocation of resources in the negotiation process [4], etc. The main problem here is the formalization of the problem in the form of multicriteria multi-index transport type problems with constraints in the form of linear inequalities [5-7]. In this case it is assumed that the system contains three types of elements: source, intermediate (transshipment) points and consumer (subscriber) nodes. These elements and their relations are subjects to the conditions of limited resources that affect the amount of resources circulating in the system. There are "managed" elements that determine the conditions for the "effective" functioning of the system. Each managed item defines binary relationships on a suitable allowable range of resource allocation values. Relationships are determined by the preference (goal) functions formulated for the controlled elements. Thus, in the most general case, the problem of allocating resources in a hierarchical system consists in determining the variant of the permissible
resource allocation, in which the target functions of the managed elements take extreme values. Such problems can be formally represented by multicriteria problems under linear constraints and criteria whose types depend on the type of goal functions. In [6], for example, functions of piecewise linear and quadratic form are investigated. Examples of such tasks are also the transport task with intermediate stations (warehouses) [8], distribution of the load on the data transmission channels of Internet providers [5], and production planning [7].
Block optimization principles [9] simplify the analysis, solution and meaningful conclusions of many planning and management tasks. Thanks to such methods it is possible to break up the complex production-transport task (PTT) into autonomous tasks of production planning and organization of delivery of products. At the same time, naturally, there is a need for iterative harmonization of interests of production and transport systems. On the other hand, the interpretation of the results obtained in meaningful terms often allows us to determine rational approaches to the functioning of different economic and technical systems [10].
© Ivokhin E., Apanasenko D., Navrodskiy V., 2018
The traditional production and transport task is to determine the production and transportation plan, which minimizes the total costs associated with the organization of production and transportation of the finished product to the points of consumption.
Production-transport problem and it solution
The model of planning, production and transportation is formalized in the form of a general scheme, which contains two groups of variables [9]:
f (z) + ^ (x) ^ min , (1)
Az + Bx = b , (2)
where x e X , z e Z ; A, B - given matrices; f (z), g(x) - continuous convex functions; X, Z - some convex bounded sets.
An iterative process is proposed in [2] for solving the problem (1), (2), and at each step there are solved problems containing only one of two groups of variables:
(3)
and
f (z) - vAz ^ min , z e Z
g (x) ^ min
Bx = b - Az , x e X .
(4)
(5)
The vector v in the problem (3) has an auxiliary meaning and is refined as a result of solving the problem (4), (5).
Let us assume that zs and vs is the approximation of the vectors z and v , obtained on iterations with the number s = 0,1,2,.... Then iterative processes
zs+1 = (1 -X s) zs + X szs, (6)
vs+1 = (1 - Xs)vs + XsVs, (7)
where 0 < Xs < 1, Xs ^ 0, s = 0,1,2,..., £ Xs =w,
zs - solution of the problem (3) with v = Vs, Vs - vector of
optimal conditions estimates (5) with z = zs, converge, respectively, to the solution of the problem (1), (2) and to the vector of optimal conditions estimates (2) [9].
Approach to reduction of production-transport problem
Let's take a closer look at this problem in more detail. Let's assume that producers of products (the number of which is N ) can use several production methods ( S ), each of which is characterized by different quantities of goods, as well as different cost of production and the maximum possible volume of goods of a certain type. We will assume that producers (suppliers) provide consumers (in quantity M ) with one kind of goods, and the specific cost of transportation from suppliers to consumers is known. Each consumer can meet their product needs through an arbitrary set of manufacturers.
Thus, the production-transport task is to determine the production plan and the plan of transportation with minimal transport costs in order to fully satisfy the demand of consumers.
We denote:
cpk - unit cost of production of the product by the i-th manufacturer with the help of k -th method;
cj - unit cost of transportation of products from the i-th manufacturer to the y-th consumer;
b: - the value of the demand of the y-th consumer;
aik - quantity of products manufactured by the i-th manufacturer in the k-th way;
zik - the intensity of the use of the k-th method by the manufacturer for the part of the period, which is assumed to be 1;
x:j - quantity of products transported by i-th manufacturer to the y-th consumer; i = 1, N, j = 1,M , k = 1, S.
Then the production transport problem (1)-(2) can be written in the form of a two-level optimization problem [11]:
N S
k (8)
f (z)=YLc*z* ^min
provided by
and constraints
=1 k=1
M N
g ( x) = H
C jXj ^ min (9)
j=1 i=1 N _
I x j = bj , j = 1 ^
¿=1
M S _
Ixj aikz,k , ¿= 1,N ,
j=1 k=1
^z* < 1, i = 1N,
k=1
(10) (11) (12)
zk > 0 , xj > 0 , i = 1, N, j = 1,M , k = 1, S. (13)
This means that the production and transportation of products should be organized in such a way as to ensure minimum production costs under conditions (10)-(13), in which the problem arises of obtaining a minimum-cost transportation plan (9), taking into account the restrictions (10)-(13).
Let each consumer can satisfy his need for products only at the expense of one manufacturer. Introduce the
variables =jo , i = 1,N, j = 1,M , assuming
that yi}- = 1, if the needs of the y-th consumer are satisfied by the i-th producer, and yi}- = 0 , in all other cases. Then
obtain a two-level continuous-discrete linear programming problem of this type [12]:
N S
f (z) = IICpkzik ^ min
(14)
i=1 k=1
provided by
and constraints
g(y) = IICjb;yj ^min (15)
j=1 ¿=1
I yl} = 1, j = 1, M , i=1
M S _
I jj <I ^k, 1=1N,
j=1
k=1
I zlk < 1, i = 1, N,
(16)
(17)
(18)
k=1
s=0
Zk > 0 , y1} = {0,1} , i = 1, N, j = 1,M , k = 1, S. (19)
It is proposed in [12] to solve continuously discrete linear programming problems according to the following scheme:
— the upper level problem (14), (16)—(19) is solved;
— the decision k = 1, S is fixed;
— the lower level problem (15) — (19) is solved.
Since the value of the upper-level objective function does not depend on the decision of the lower level and vice versa, this algorithm allows finding the optimal solution and the value of the corresponding goal functions for linear programming problems of both levels. At the first step of
the algorithm, an acceptable solution z0 is obtained, which will be the optimal solution of the upper-level problem. Taking this decision into account, the third step is the optimal solution for the lower-level problem.
Solving the problem of distributing the limited capacities of data transmission channels
The proposed above models of production-transport problems allow solving the problem of distributing the limited capacities of data transmission channels between various nodes of the Internet service provider's network. Suppose that there is a local computer network of the enterprise (higher education institution) that provides access to the Internet network for users. Access of users to the global network and obtaining the necessary information is made by means of several communication servers located on the territory of the information and computing center of the enterprise and connected by high-speed external communication channels with Internet providers. Server bandwidth levels lie within the bandwidth of the local network (for example, 1GB per second). Let's assume that the needs of network subscribers are known in increasing the speed of obtaining a certain amount of information. The wishes (preferences) of subscribers regarding possible volumes of increase in capacity for transferring information from the provider to the user node are specified. To implement the wishes, it is necessary to update the capacity of the switching servers of the network by deploying new, more powerful computers or by increasing the number of existing servers. In other words, it is necessary to update the server park of the Information and Computing Center (ICC), which allows increasing the total bandwidth of a group of switching servers. In this case, the value of the total server capacity, both in case of increasing the capacity of the existing computer fleet, and in the case of increasing the number of servers is assumed to be the same.
We assume that the network realizes the conditions for efficient commutation of channels (relative to their bandwidth), which are provided by programmable network devices (communication servers, routers). The structure of the network and the information distributed in it in the general case can be the most diverse. In this case, the problem of distribution of limited capacities is considered with the following limitations:
— information is distributed from the provider to subscribers (nodes) through switching servers through communication channels with bandwidth that takes into account the given bandwidth;
— each subscriber of the network is served by one switching server;
— the bandwidth for obtaining information for switching nodes and subscribers is limited both from the top (principal limitations of the provider's capabilities) and from the bottom (the minimum need for subscribers in the information received).
Obviously, the amount of payment for the use of communication channels of a certain bandwidth depends on the cost established by the providers and the used bandwidth of the external connection. Based on the available reserve bandwidth of the external connection, it is necessary to maximally increase the total bandwidth of users' communication channels by changing the total capacity of the communication servers, taking into account both the needs and wishes of subscribers (users) and the capabilities of the ICC.
Let N — a set of providers of the global network,
N2 — a set of communication servers, N3 — a set of sub-
scribers. Through Ai , i = 1,Nl , denote the maximum bandwidth of the data channel that the provider i can provide, i = 1,N ; Bj , j = 1,N2 , — the value of the maximum bandwidth of the data channel that the communica-
tion node j is capable of providing, j = 1,N2 ; Ck ,Ck ,
k = 1,N3 , — the values of the minimum and maximum bandwidth of the data channel to be provided to the subscriber k, k = 1,N3 ; tk - throughput of the k-th subscriber
station, k = 1,N3 . Then, assuming that the power distribution of the communication channels satisfies the conditions of additivity and proportionality, we can consider the distribution of a limited homogeneous resource (communication channel bandwidth) with constraints of the transport type in order to find the optimal data transfer plan. This ensures the effective functioning of the system of providing Internet access to users, which consists in finding the optimal values of data transmission capacities Ti by the i-th infor-
mation provider (provider), i = 1,N , and the optimal values of the throughput tk of using local communication
channels by the k-th user, k = 1,N3 .
Formally, the statement of this problem can be written in the form:
max t1; max t2; ... max tN , under the following conditions:
N3
Ni
I h = I4+ ;
k=1
Ti < 4 , i = 1, Nl
t , < b ;, j = l, n2
Ck < tk < c;, k=1, N3 and with constraints
N3
N1
N3
I c;<I 4+<I c; ;
(20)
(21)
(22)
(23)
(24)
(25)
k=1
i=1
k=1
where T j — the transmission capacities of the communication channels provided by the j-th communication node, j = 1,N2 .
i=1
Using of the production-transport task to find the optimal solution
Introduce the notations:
xi]k - the capacity of the communication channels
connecting the provider with the number i through the intermediate communication server j with the consumption
node k, i = 1,2, j = 1,2 (in the case of 2 servers) or j = 1,3 (in the case of 3 servers), k = 1,17 ;
Ai, i = 1,2, - maximum bandwidth provided by the
provider for connection of communication servers (both values equal 10 Gb/s);
C1J, i = 1,2, j = 1,2 ( j = 1,3 ), - the bandwidth of the
external communication channels, which provide the connection of the server y to the provider with the number i (all values are considered equal to the bandwidth of 10 Gb/s);
D]k, j = 1,2 ( j = 1,3 ), k = 1,17 , - the maximum
bandwidth of individual user connections k to the communication servers j, which initially is 260, 165, 150, 190, 275, 115, 175, 275, 155, 195, 125, 145, 90, 370, 180, 90, 150 MB Mb/s;
ai, i = 1,2, - the cost of connecting a provider with the number i within the specified bandwidth of the external communication channel.
Then the mathematical model of the problem of optimal distribution of power of communication channels with the condition of optimization of consumption volumes can be considered as a transport problem with an optimality criterion taking into account the value indicators of the use of external channels,
2 17 _
^^aixijk ^min(in the case of 2 servers), i = 1,2, (26)
j=1 k=1
or
3 17 _
^^aixijk ^min(in the case of 3 servers), i = 1,2, (27)
j=1 k=1
and the constraints, which in this case are written in the form of the following system of inequalities
2 17 _
^i^xijk ^ Ai, i = 1,2, (in the case of 2 servers) (28)
j=1 k=1
or
3 17
IIxijk < Ai, i = 1,2, (in the case of 3 servers); (29)
j=1 k=1
I xi]k < Cj , i =1,2, j =1,2 ( j =1,3 ); (30)
k=1 2
I xl]k < Djk , j = 1,2 ( j = 1,3 ), k = 1,17 ; (31)
i=1
xi]k > 0 , i = 12, j = 12 ( j = 13 ), k = U7 ; (32)
When solving the problem of optimal distribution of power of communication channels with the criterion taking into account the value indicators of the use of external channels of the type (26)-(32), it should be noted that requests for information of all users of the Internet (consumers) are provided at the expense of only one provider (supplier) In this case, the mathematical model of the problem
can be written in the form of a two-level production-transport problem (14)-(19) of the following form:
f (z) = I cl zl ^ min
(33)
provided
g(y) = IIbkyjk ^min( in the case of 2 servers) (34)
k=1 j=1
or
17 3
g(y) = ^l^lbkyjk ^ min ( in the case of 3 servers) k=1 j=1 and constraints
I yjk = 1 (fl^ 2 cepBepoB),
j=1
I y jk = 1 (in the case of 3 servers), k = 1,17 , (35)
j=1 17
Ibkyjk < Iaizi , j = 1,2 (in the case of 2 servers),
k=1
j = 1,3, (in the case of 3 servers), (36)
I zi = 1,
(37)
i =1
where ct, i = 1,2, - is the cost of connecting the provider with the number i within the specified bandwidth of the external communication channel; ai, i = 1,2, - bandwidth of the communication channel with the i-th provider (values are considered equal to 10 Gb/s); bk , k = 1,17 , - the maximum throughput of individual user connections k to the communication servers, which initially is 260, 165, 150, 190, 275, 115, 175, 275, 155, 195, 125, 145, 90, 370, 180,
Î1 —
90, 150 Mb/s; variables z, =< , i = 1,2, with z, = 1, if
! j0 !
the needs of consumers are provided by the i-th provider,
[1 —
and z{ = 0, otherwise; variables yjk =<! , k = 1,17 ,
j = 1, J, (for 2 servers), J = 3 (for 3 servers), and y k = 1, if the needs of the k-th user are provided by the j-th communication server, and y]k = 0, in all other cases;
i = 1J , k = ÏÏ7 .
The solution of continuous-discrete linear programming problems (33) - (37) occurs according to the following scheme:
- the user-server task is solved (34)-(37);
- the received solution is fixed yjk , j = 1, J, J = 2
(for 2 servers), J = 3 (for 3 servers) k = 1,17 ;
- the server-provider level problem (33)-(37) is solved for this solution.
The solution of problem was carried out on the basis of a complete search of possible connections of users with communication servers. Taking into account the condition of equal capacities of network communicators in the process of obtaining the solution, it was checked
i=1
i=1
12 £ CS = 217-1 = 216
s=0
(for
servers)
and
>2
' 17 f 17-s
4s=0 V r=0
//
(for 3 servers) connection methods.
When solving this production and transportation problem, the following results were obtained: when using 2 identical communication servers with a total capacity of 3 Gb / s, the capacity of local connections is 259, 159, 149, 166, 273, 115, 163, 274, 152, 148, 125, 144, 90, 365, 180, 89, 149 Mb / s, which coincided with the previous decisions. With the use of 3 communicators with a total capacity of 3 Gb/s, the capacity of local connections is 260, 148, 146, 190, 258, 114, 175, 266, 146, 195, 124, 143, 90, 335, 180, 89, 141 Mb/s (also coincided with previous decisions).
With the increase in the maximum values of the throughput of local connections to 280, 180, 170, 200, 290, 125, 190, 290, 170, 210, 135, 160, 100, 390, 195, 95, 165 Mb / s in case of using two communication. Optimum speeds of local connections equal to 128, 161, 160, 124, 284, 124, 152, 287, 165, 208, 134, 158, 100, 378, 194, 94, 149 Mb/s were obtained with the total throughput of 3 Gb/s, respectively, and in the case of using 3 communicators, the values of local connections 271, 146, 153, 65, 273, 123, 123, 284, 162, 206, 131, 158, 97, 378, 192, 92, 146 Mb/s, respectively.
A complete list of the results obtained for a different number of switches and their capacities is given in Table 1 (optimal solutions are highlighted).
Conclusion and discussion
In this study, the application of the production and transport task is considered to solve the problem of the distribution of the limited capacities of data transmission channels between different nodes of the computer network. A scheme is proposed for reducing the problem to a two-level continuous-discrete optimization problem. A mathematical model of the problem of power distribution of communication channels is obtained. The problem of optimal distribution of capacities with a criterion that takes into account the cost parameters of using external channels was solved, provided that requests for information from all users of the Internet (consumers) are provided at the expense of only one provider (supplier) This approach can be considered when solving various optimization problems with a hierarchical structure of the process.
Further investigation of the problem of power distribution of data transmission channels is planned to be based on the use of the traditional three-index transport problem, the use of streaming algorithms and the fuzzy approach to solving problems optimization of the distribution of limited resources. Conducting a comparative analysis of the results obtained by different methods will allow us to draw definitive conclusions about the effectiveness of the proposed approach.
References
1. Воронин A. A. Оптимальные иерархические структуры / A. A. Воронин, С. П. Мишин. - M. : ИПУ, 2003. - 214 с.
2. Elsasser, R., Monien B. and Preis R., 2002. Diffusion Schemes for Load Balancing on Heterogeneous Networks. Theory Comput. System, 35(3), pp. 305-320. DOI: http://dx.doi.org/10.1007/s00224-002-1056-4.
3. Gairing, M., Lucking, T., Mavronicolas M. and Monien B., 2004. Computing Nash Equilibria for Scheduling on Restricted Parallel Links // Proc. 36th Annual ACM Sympos. Theory Comput., 2004, pp. 613-622.
4. Dunne, P. E., 2005. Extremal Behavior in Multiagent Contract Negotiation. Jour. Artificial Intelligence Res., 23(1), pp. 41-78.
5. Prilutskii, M. Kh. Distribution of a Homogeneous Resource in Tree-Structured Hierarchical Systems // Proc. III Int. Conf. Identification of Systems and Control Problems (SICPR0'2000), Moscow, IPU, 2000, pp. 2038-2049.
6. Прилуцкий М. Х. Потоковые алгоритмы распределения ресурсов в иерархических системах [Электронный ресурс] / М. Х. Прилуцкий, А. Г. Картомин // Исследовано в России. - 2003. - 39. - С. 444-452. -Режим доступа: http://zhurnal.ape.relarn.ru/ articles/2003/039.pdf
7. Афраймович Л. Г. Многоиндексные задачи распределения ресурсов в иерархических системах / Л. Г. Афраймович, М. Х. Прилуцкий // Автоматика и телемеханика. - 2006. - 6. - С. 194-205. (Autom. Remote Control, 67:6 (2006), 1007-1016). DOI: http://dx.doi.org/10.1134/ S0005117906060130
8. Серая О. В. Многоиндексные нелинейные транспортные задачи / О. В^рая, О. И. Дунаевская // Информационные и управляющие системы на железнодорожном транспорте. - Харьков : ХГИЖ^ 2009. -№ 5. - С. 25-30.
9. Юдин Д. Б. Экстремальные модели в экономике / Д. Б. Юдин, А. Д. Юдин. - М. : Экономика, 1979. - 288 с.
10. Бурков В. Н. Введение в теорию управления организационными системами / В. Н. Бурков, Н. А. Коргин, Д. А. Новиков. - М. : Либроком, 2014. - 264 с.
11. Lukac, Z., Hunjet, D. and Neralic, L., 2008. Solving the production-transportation problem in the Petroleum Industry. Revista Investigacion Operacional, 29(1), рp.63-70.
12. 1вохш 6. В. Про розв'язок одшеТ дворiвневоТ моделi виробничо-транспортноТ задачi / 6. В. 1вохн Л. Т. Аджубей // Вюник Кшвського национального ушверситету iменi Тараса Шевченка. Серiя: Фiзико-математичн науки. - 2014. - Вип. 3. - С. 122-125.
Received: 20/01/2018 1st Revision: 25/02/2018 Accepted: 14/04/2018
Author's declaration on the sources of funding of research presented in the scientific article or of the preparation of the scientific article: budget of university's scientific project
Б. 1вохш, д-р фю.-мат. наук, проф., Д. Апанасенко, асп.
Кшвський нацюнальний ушверситет ¡меж Тараса Шевченка, Кшв, Укра'ша, В. Навродський, канд. фю.-мат. наук, доц.
Кшвський нацюнальний унтерситет культури \ мистецтв, Кшв, Укра'ша
ПРО ЗВЕДЕННЯ ВИРОБНИЧО-ТРАНСПОРТНО1 ЗАДАЧ1 ДО ДВОР1ВНЕВО1 ЗАДАЧ1 ОПТИМ1ЗАЦП ТА И ЗАСТОСУВАННЯ
Розглянуто застосування виробничо-транспортноУ задачi для розв'язання проблеми розподлу обмежених потужностей каналiв передачi даних мiж р/'зними вузлами комп'ютерноУ мереж/'. Запропоновано схему для зведення задачi до дворiвневоУ неперервно-дискретноУзадачi оптимiзацiУ. Сформульовано модель i отримано чисельн результати для розв'язання проблеми розподлу потужностей у мереж/' iнформацiйно-обчислювального центру.
Ключовi слова: розподл потужностей, виробничо-транспортна задача, дискретно-неперервне програмування, дворiвнева модель, оптимiзацiя.
Е. Ивохин, д-р физ.-мат. наук, проф., Д. Апанасенко, асп.
Киевский национальный университет имени Тараса Шевченко, Киев, Украина, В. Навродский, канд. физ.-мат. наук, доц.
Киевский национальный университет культуры и искусств, Киев, Украина
О СВЕДЕНИИ ПРОИЗВОДСТВЕННО-ТРАНСПОРТНОИ ЗАДАЧИ К ДВУХУРОВНЕВОЙ задаче оптимизации и ее применение
Рассмотрено применение производственно-транспортной задачи для решения проблемы распределения ограниченных мощностей каналов передачи данных между различными узлами компьютерной сети. Предложена схема для сведения задачи к двухуровне-
2
вой непрерывно-дискретной задаче оптимизации. Сформулирована модель и получены численные результаты для решения проблемы распределения мощностей в сети информационно-вычислительного центра.
Ключевые слова: распределение мощностей, производственно-транспортная задача, дискретно-непрерывное программирование, двухуровневая модель, оптимизация.
References (in Latin): Transliteration
1. Voronin, A. A., Mishin, S. P., 2003. Optimalnye ierarkhicheskie struktury Moscow, IPU, 214 p.
2. Elsasser, R., Monien, B. and Preis, R., 2002. Diffusion Schemes for Load Balancing on Heterogeneous Networks. Theory Comput. System, 35(3), pp. 305-320. DOI: http://dx.doi.org/10.1007/s00224-002-1056-4
3. Gairing, M., Lucking, T., Mavronicolas, M. and Monien, B., 2004. Computing Nash Equilibria for Scheduling on Restricted Parallel Links. Proc. 36th Annual ACM Sympos. Theory Comput., pp. 613-622.
4. Dunne, P. E., 2005. Extremal Behavior in Multiagent Contract Negotiation. Jour. Artificial Intelligence Res., 23(1), pp. 41-78.
5. Prilutskii, M. Kh., 2000. Distribution of a Homogeneous Resource in Tree-Structured Hierarchical Systems. Proc. III Int. Conf. Identification of Systems and Control Problems (SICPR0'2000), Moscow, IPU, pp. 2038-2049.
6. Prilutskii, M. Kh., Kartomin, A. G., 2003. Potokovye algoritmy raspredeleniya resursov v ierarhicheskih sistemah. Issledovano v Rossii, 39, pp. 444-452. - http://zhurnal.ape.relarn.ru/ articles/2003/039.pdf
7. Afraimovich, L. G., Prilutskii, M. Kh., 2006. Multiindex resource distributions for hierarchical systems. Avtomatika i telemekhanika, 6, pp. 194-205; Autom. Remote Control, 67:6 (2006), 1007-1016. DOI: http://dx.doi.org/10.1134/S0005117906060130
8. Seraya, O., Dunaevskaya, O., 2009. Mnogoindeksnyye nelineynyye transportnyye zadachi. Informatsionnyye i upravlyayushchiye sistemy na zheleznodorozhnom transporte. Khar'kov: KHGIZHT, № 5. p. 25-30.
9. Yudin D. B., Yudin, A. D., 1979. Ekstremalnye modeli v ekonomike. Moscow, Ekonomika, 288 p.
10. Burkov, V. N., Korgin, N. A., Novikov, D. A., 2014. Vvedeniye v teoriyu upravleniya organizatsionnymi sistemami: uchebnik. Izd.2. M.: Librokom, 264p.
11. Lukac, Z., Hunjet, D. and Neralic, L., 2008. Solving the production-transportation problem in the Petroleum Industry. Revista Investigacion Operacional, 29(1), pp. 63-70.
12. Ivokhin, Ye., Adzhubey, L., 2014. Pro rozv'yazok odniyeyi dvorivnevoyi modeli vyrobnycho-transportnoyi zadachi in Visnyk Ky-yivs'koho natsional'noho universytetu imeni Tarasa Shevchenka. Seriya: Fizyko-matematychni nauky, № 3, S. 122-125.
Таble 1.
User's Total Request
2 servers
260 165 150 190 275 115 175 275 155 195 125 145 90 370 180 90 150 3105 Current demands
256 146 145 35 260 114 136 268 145 0 124 143 90 340 103 89 146 2540 -195 Cap B1,B2=1270
259 140 144 90 273 115 125 274 143 0 124 143 90 365 179 89 147 2700 -195 Cap B1,B2=1350
259 156 148 119 270 115 157 273 151 53 124 144 89 360 144 89 149 2800 -142 Cap B1,B2=1400
256 156 148 154 260 115 157 268 150 123 124 144 90 340 178 89 148 2900 -72 Cap B1,B2=1450
259 159 149 166 273 115 163 274 152 148 125 144 90 365 180 89 149 3000 -47 Cap B1,B2=1500
280 180 170 200 290 125 190 290 170 210 135 160 100 390 195 95 165 3345 MAX demands
128 161 160 124 284 124 152 287 165 208 134 158 100 378 194 94 149 3000 -152 Cap B1,B2=1500
279 170 165 160 285 125 170 288 168 130 134 159 100 380 194 94 149 3150 -80 Cap B1,B2=1575
280 173 166 143 289 125 176 289 168 210 135 159 100 388 166 94 149 3210 -57 Cap B1,B2=1605
278 173 167 173 281 125 176 286 168 209 135 159 100 373 193 94 150 3240 -27 Cap B1,B2=1620
279 175 168 180 285 125 180 288 169 209 135 159 100 380 194 94 150 3270 -20 Cap B1,B2=1635
279 176 168 185 287 125 182 289 169 210 135 160 100 385 194 94 150 3288 -15 Cap B1,B2=1644
280 177 168 188 289 125 184 289 169 210 135 160 100 388 194 94 150 3300 -15 Cap B1,B2=1650
280 178 169 199 289 125 187 290 170 210 135 160 100 388 188 94 150 3312 -15 Cap B1,B2=1656
3 servers
260 165 150 190 275 115 175 275 155 195 125 145 90 370 180 90 150 3105 Current demands
260 148 146 190 258 114 175 266 146 195 124 143 90 335 180 89 141 3000 -35 Cap B1,B2,B3=1000
256 163 149 190 275 114 170 266 154 195 125 145 90 353 180 90 145 3060 -17 Cap B1,B2,B3=1020
260 165 150 190 271 115 175 273 155 195 123 145 90 363 180 90 150 3090 -7 Cap B1,B2,B3=1030
259 165 150 190 275 114 175 275 155 194 125 145 89 369 180 89 150 3099 -1 Cap B1,B2,B3=1033
280 180 170 200 290 125 190 290 170 210 135 160 100 390 195 95 165 3345 MAX demands
280 146 153 65 273 123 123 284 162 206 131 158 97 378 192 92 146 3000 -135 Cap B1,B2,B3=1000
278 154 157 199 273 123 138 281 169 209 132 153 99 355 195 94 141 3150 -52 Cap B1,B2,B3=1050
278 148 154 199 278 124 189 284 162 209 133 156 99 365 195 94 143 3210 -32 Cap B1,B2,B3=1070
280 168 164 200 280 124 165 285 167 210 134 158 100 370 195 95 145 3240 -25 Cap B1,B2,B3=1080
270 179 169 195 283 125 188 286 169 205 135 159 99 375 193 94 146 3270 -19 Cap B1,B2,B3=1090
