50 Discrete Dynamic Model of Retail Trade Market of Computer Equipment in Ukraine Mykola Dyvak1,Vasyl Brych2, Iryna Spivak3, Lyudmyla Honchar4, Nataliya Melnyk5 1. Department of Computer Science, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chekhova str., email: mdy@tneu.edu.ua 2. Department of International Tourism and Hotel Business, Ternopil National Economic University, UKRAINE, Ternopil, 11 Lvivska str., email: v.brych@tneu.edu.ua 3. Department of Computer Science, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chekhova str., email: i.spivak@tneu.edu.ua 4. Department of Computer Science, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chekhova str., email: l.honchar@tneu.edu.ua 5. Faculty of Economics, Ivan Franko National University of Lviv, UKRAINE, Lviv, email: talya_m@ukr.net Abstract: In this article the discrete dynamic model of the CE, which are happening in this market. The modeling results retail computer market functioning in Ukraine is considered. can be taken into consideration during development and The existing distribution models are analyzed between the main implementing business policies in specific circumstances. entities in the market. A new model is proposed and an example Predicted by discrete dynamic mathematical models, the of the IT market dynamics is shown. Keywords: retail computer market; discrete dynamic model; indicators of business development make it possible to correction function; predicted indicators. adequately assess their own investment opportunities and to attract investments from the side. Mathematical models are I. INTRODUCTION used to track trends in the market. This allows you to Mathematical modeling is one of the most important tools correctly emphasize the position of advertising companies in for researching economic processes. The constant demand for order to timely implement effective marketing activities. new IT generates new types of computer equipment (CE), as Consequently, the skillful use of the market’s subject well as new IT services. These proposals are aimed not only methods of mathematical modeling in the final result gives at meeting the needs of business structures, but also at an you a number of advantages in the competition [3]. average household information consumer. The last Rapid changes in the IT-industry trends put high demands circumstance stimulates the rapid development of CE retail to the possibility of mathematical models. For example, a market, on which the main buyer is the individual consumer model that adequately reproduces sales of storage devices, of the information product. To predict the development of will not necessarily work in the case of a rapid and massive this market, it is necessary to build its dynamics model. Such transition from the use of optical disks to electronic media. models are described in the works [1, 2], where the problems This can be explained by the fact that entities in the common of their structural and parametric identification on the basis of market react differently to these changes. Some of them, data analysis are considered. Such models are called discrete which are oriented to the sale of goods in large batches, are dynamic or differential operators [1]. not able to quickly abandon the devices, which action is To construct a model of market dynamics, let’s consider based on "outdated" technologies, since there are a large the important moments of the subject modeling area. At the number of such devices in the warehouses. Obviously, the domestic CE retail market we distinguish four sellers advertising and marketing policies of such structures will be categories: consumer electronics, specialized computer aimed at reducing their stocks as quickly as possible. Other stores, mobile communication stores and В2В-sector market players, who are more mobile in the process of enterprises [3]. A narrow range of sellers in the domestic transitioning to new technologies, are pursuing a policy market of CE is conditioned by the monopoly in this area. aimed at promoting the latest devices. Therefore they are Those structures, which operate on the domestic market, play receiving competitive advantages that are not taken into the role of a distributive link, which does not define strategic account in conventional foreseen models [6]. directions of IT development and applies to advance It is also possible that new sellers will enter the market, achievements in this field. Without having their own product who bet on the latest IT technology, or some vendors will be and struggling for a part of the market share they are forced replaced by others. Individual sellers can change their to behave extremely responsibly in their own business, priorities and refuse to commerce certain types of CE. paying attention to a number of factors, on which they have Apparently, such vendor substitutions also require correction no influence [3]. of existing linear dynamic models [3], for example, by The need of planning strategies and tactics of doing introducing a nonlinear part that reflects switching processes. business, challenges the participants with complex problems The purpose of this study is to develop a new model of the of mathematical modeling of processes in the retail market of retail market of CE based on the rapid changes that take place ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 51 in IT. To achieve the goal, you need to solve these problems: is deterministic and its structure can be obtained based on a analysis of existing models of distribution of the retail market detailed analysis of the subject area. In order to take into of CE between sellers by major segments; to construct a account the changes in trends changes is offered to modify discrete dynamic model with a "switch" for describing the CE the proposed model by introducing switching functions that market, which is subjected to change of circumstances of its simulate a sharp change in market conditions. functioning; check the model for adequacy. Let’s consider the case when at a certain point in time t out on the market in a particular segment changes the number of II. TASK DEFINING subjects, namely, one of the subjects leaves the specified segment. This means that the number of non-zero vector Work [3] proposes a model that reflects the distribution of  the retail CE market dynamics between the four large components y (k ) in (1) decreases. To represent this possible suppliers (entities) by separate segments of the market. Each change we introduce into the equations of the output segment corresponds to a certain type of CE. In the example variables of model (1) an additional diagonal matrix T, which given in [3], four segments are considered, namely: personal has such a general view: computer segment (PC), laptops segment, a segment of displays and a segment of multifunctional devices (MFD).  f (t1 ) 0  0  To simulate the dynamics, the mathematical model is   selected as a linear discrete equation (differential operator) in  0 f (t 2 )  0  the form [3,6]: T =     , (2)            x ( k +1) = F ⋅ x ( k ) + G ⋅ v ( k )  0 0  f (t n )    ( k +1)  (1)  y = C ⋅ x ( k +1) , k = 0,1,2,...  1, ts < tout where x (k ) – vector of state variables, which characterize the where is the switch function f (ts ) =  , s = 1, n , n -  change in the formal state of the market; v (k ) – vector of 0, ts ≥ tout  input variables, which reflect the effect of factors on the  number of vector components y(k ) in the model (1). Then market; y(k ) – vector of output variables, which reflect the we will get: characteristics of the distribution of the market among its subjects; k – time sequence number, in which the value of the  x ( k +1) = F ⋅ x ( k ) + G ⋅ v ( k ) components of the corresponding vectors is determined; F , G, C – valid matrices of the corresponding   ( k +1)  measurements.  y = T ⋅ C ⋅ x ( k +1) , k = 0,1,2,... (3) Parameters of this model (matrix elements F , G, C ) are obtained in the form of parametric identification according to the well-known Ho-Calman algorithm [4-6]. The reason of Based on these tasks in [3], we will analyze the obtained parametric identification is the Henkel block matrix. Each model structure. At the same time, we assume that sellers block of this matrix in the case [3] is formed from the data on belonging to a certain category, namely: mobile the market share, which is occupied by the j-th element communication stores, refused to sell monitors. In fact, this ( j = 1,4 ) in the i-th ( i = 1,4 ) market segment by the results of situation was observed in 2015. In model (3) this year corresponds to the order number of the time point k = 5 . And k-th year ( k = 1,4 ). the serial number of the monitor – s = 3 . So with k = 5 we In the work [3] an assumption was made, that conditions of functioning of the retail market CE were unchanged. Model get t 3 = t out and accordingly the switching function built in this way, allowed to get a rough estimate of the f (t3 ) = 0 , and the corresponding matrix T will have the form: distribution of the market for k + 1 period among the main categories of vendors for each segment of the market. III. IMPROVED DISCRETE DYNAMIC MODEL 1 0 0 0   As shown in [3], even with the preservation of previous 0 1 0 0 T = trends in the functioning of the retail market of CE, the full 0 0 0 0 . (4) adequacy of the basic model could not be achieved. Only use   0 1  of optimization procedures allowed to get adequate values for  0 0 predicted indicators. Obviously, when market trends fluctuate, the proposed model becomes inadequate. Therefore, it is necessary to complicate the structure of the According to model (3), foreseen indicators reflecting the model, taking into account changes in the market. Such a the distribution of CE market in the segment of monitors complication can be done using the methods of structural between entities, which are left in this segment are: consumer identification [2]. However, in our case, the dynamics model ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 52 electronics– 38,5%; specialized stores – 23,7%; В2В sector – Vice versa, capturing part of the share market by specific 36,2%. subject, which previously belonged to another entity, are Obviously, that the market share, which according to described by replacing the value of the corresponding forecasts should have mobile exhibition halls (which is component in the vector v (1) on β ( β > 1 ). For a general case let’s write a vector v (1) with the 1,6%), were distributed among the remaining enterprises in the segment of monitors. Assuming that this division took place in proportion to that part, that each subject has in this correction function g (ξ1 , ξ 0 ) , where ξ 0 і ξ1 – respectively segment, then each of them received an additional share in market shares, which the subject takes before and after the amount respectively 0,62%, 0,38% і 0,6%. Consequently,  changing certain condition. Finally vector v (1) will have the the final forecast for the distribution of the retail market of following generalized form: monitors is as follows: consumer electronics – 39,12%; specialized stores – 24,08%; В2В - sector – 36,8%.  0  If we apply the optimization procedure for the initial   distribution (38,5%; 23,7%; 36,2%) and under conditions,  0  that the goal function as the sum of the shares of all sellers (1)    remaining in the segment is equal 100%, and the deviation of v =  each particle does not exceed 1,6%, then we obtain the  g (ξ1 , ξ 0 )  , (6) following final values: consumer electronics – 36,9%;   specialized stores – 25,3%; В2В - sector – 37,8%.      The difference between the two methods of estimating the  0  forecasted values of market distribution lies within the limits [2,22%; -1%]. Obviously, taking into account such clear  1, ξ1 = ξ 0 limitations in forecasting, the entity of the retail market CE  has the opportunity to more precisely define its own business g (ξ1 , ξ 0 ) = α , ξ1 < ξ 0 where . development strategy. β , ξ > ξ To reproduce changes in the retail market of CE, which are  1 0 connected with elimination or gradual abandonment of this Therefore, we’ll assume, that subject 3 has rapidly reduced market by separate subjects, into the model (1) it is its presence in the CE retail market. Let this be decreased appropriate to enter correction functions into the vector of  70%. This means that in this case, the correction function input variables v (k ) . Let's look at an example where subject 3 g (ξ1 , ξ 0 ) has value α = 0,3 . By introducing such a value into (mobile stores) dramatically reduces its presence on the the main model (1) we obtaining foreseen market shares in market in all its segments. Instead, its market share is different segments for this subject, namely: PC segment – - individually trying to be captured the subject 2 (specialized 0,4%, laptop segment – 3,2%, displays segment – 0,4%, stores). MFD segment – 1,6%. According to the Ho-Calman algorithm [4-6] model (3) If entity 2 can individually capture that market share, reproduces indicators of a particular subject, on the basis of which entity 3 has left, so this means, that the correction which it was built, on the condition, that at the starting function for it considering his total share for all segments will moment of time ( k = 1 ) the corresponding input variable is have value β = 1,15 . As a result of foreseeing, we obtain the equal to 1, and all others - equal to zero. So for the second and third subject, the vectors of the input variables are following values for different segments of the market: PC respectively: segment – 9,4%, laptop segment – 24,2%, displays segment – 27,3%, MFD segment – 24,4%.  0  0 For such values α and β , segments modeling error are     (1)  1  (1)  0  within the range limits [0,6%;4,3%]. v =  v =  0 і 1 .   (5) In terms of practice, it is unlikely, that entity 2 alone   captures all market shares, which entity 3 has left. Most  0  0     likely, this circumstance will be used by other entities. In our case, this is the subject 1 (consumer electronics) and subject 4 For all other moments of time ( k = 2,3,... ) all components (the enterprise B2B sector). If we assume, that released of the input vector must be zero. market share by subject 3 is divided between subjects 1, 2 і 4 Reduction of the share in all segments in the market of an in proportions 5:2:3, so the corresponding values of the individual entity simulating by replacing the value of the correction functions will be: β1 = 1,35 , β1 = 1,14 , β1 = 1,21 .  corresponding component in the vector v (1) , namely: the For such correction function values, after optimization value 1 is replaced by the value α ( 0 < α < 1 ). The smaller procedures use by the method [7-11] with restriction, which the value α , the faster is the process of leaving this subject are within [0,4%;4,5%], we get a forecast of market of the CE market. distribution by segments. ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 53 Fig. 1. Estimated distribution of the CE retail market. Consequently, the introduction into the model (1) the [2] N. Porplytsya, M. Dyvak, I. Spivak, I. Voytyuk, correction function allows to adequately describe the “Mathematical and algorithmic foundations for processes in the retail market of CE in case of trends changes. implementation of the method for structure identification of interval difference operator based on functioning of IV. CONCLUSION bee colony”, Proceedings of International Conference The instability of the situation on the retail market of CE CADSM’2015, Lviv, Ukraine, 2015, pp. 196-199. due to various factors, in particular the rapid development of [3] N. Melnyk, M. Dyvak, “Modeling of the dynamics IT. This complicates the simulation of those processes, which distribution model by retail computer market segments”, are happening on it, since existing models do not count Lviv national university herald. Economical series, Lviv, technological changes in IT industry, and corresponding 2016, Edition 53, pp. 150-157. changes in the structure of the market. Therefore, these [4] R.E. Kalman, P.L. Falb, M.A. Arbib, Topics in models should be modified by introducing correction mathematical system theory. McGraw Hill Book Co, functions to them. 1969. The correctional functions proposed in this research, make [5] Bowden, R.: The theory of parametric identification. it possible to predict the distribution of the retail CE market Econometrica 41, 1069–1074 (1973). for y segments between entities in the case of abrupt changes [6] Graupe, D.: Identification of systems. Technology & in the tendencies of its functioning. Namely: in the case of a Engineering (1976). certain subject's refusal to trade in a particular type of CE, [7] P.H. Stakhiv, Y.Y. Kozak, O.P. Hoholyuk, “Discrete and in the case of a rapid decrease in the presence of the macromodeling in electrical engineering and related entity in all segments of the market. fields”, monograph: Lviv Polytechnic Publishing House, In the future studies non-linearity of segmental 2014. – 260 p. redistribution of the market between entities should be taken [8] E. Rosolowski, P. Stakhiv, O. Hoholyuk, "Prospects of into account. To do this, you need to switch to a more discrete macromodels usage for calculation of electric complex form of the main model, for example, bilinear. power systems modes", Modern Problems of Radio However, such approach is suitable not for all cases. Engineering, Telecommunications and Computer Therefore, in further researches, the structure identification Science, Proceedings of the 13th International methods based on inductive approach will be used to choose Conference on TCSET’ 2016, Lviv-Slavsko, p.55-57. the model [12]. Also, it is advisable to take into account the [9] Rastrigin, L.A.: A random search. Znanie, Moscow uncertainty of given data, scilicet, their variety on different (1979). (in Russian) intervals. [10] Rastrigin L.A.: Adaptation of complex systems. Zinatne, REFERENCES Riga (1981). (in Russian) [11] I. Calishchuc, M. Dyvak, P. Stakchiv, "Identyfikacja [1] I. Voytyuk, M. Dyvak, V. Spilchuk, “Research of quality dynamicznego modelu obwodu elektrycznego na characteristics of models structure in kind of interval podstawie danych interwałowych", Przegląd Elektro- difference operator”, Proceedings of International techniczny, Nr. 2/2005, pp. 60-62. Conference CADSM’2011, Polyana-Svalyava, 2011, pp. [12] Ivakhnenko A.G., “The inductive method for self- 87. organization of models of complex systems” Naukova Dumka, 1981, 296pp. (in Russian). ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic