Construction of a Mathematical Model for Analyzing the Effec- tiveness of IT Startups Viktor Morozov1, Olga Tsesliv2, Anna Kolomiiets3, Sergey Kolomiiets4 1,3 Taras Shevchenko National University of Kyiv, 24, Bohdan Gavrilishin Str., Kyiv, Ukraine, 04116, 2 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37, Prosp. Peremohy, Kyiv, 03056 4 National transport University, 1, Mykhailа Omelianovycha-Pavlenka Str., Kyiv, Ukraine, 01010 Abstract This article is devoted to the development of a mathematical model for assessing the invest- ment risks of startup projects in the IT industry based on fuzzy set theory. The article exam- ines and analyzes information sources that show that the issue of evaluating the effectiveness of startups is not sufficiently developed. In addition, the future of startups is associated with many parameters that are very conditional and predictable at the initial stages of project con- sideration. Therefore, it is advisable to use fuzzy modeling methods to accept the project for consideration. By using the fuzzy set method, it is possible to use fuzzy variables that reflect the uncertainty of some parameters of such projects. The proposed research methodology is based on the analysis of the commercial effectiveness of projects and the use of fuzzy set methods. The main financial parameters of the project were taken into account, such as: net present value, internal rate of return. When conducting research, fuzzy parameters were used to evaluate project indicators. For this purpose, membership functions are constructed that es- tablish the degree of belonging of a fuzzy set. The trapezoid model and the specified parame- ters corresponding to the pessimistic, baseline, and optimistic scenarios are selected as the function type. The novelty of the paper is the determination of the risk indicator of a startup project, which depends on the criterion of project effectiveness. The paper proves the de- pendence of the project risk indicator on the value of the project effectiveness criterion. The proposed approach has shown its feasibility and can be used to analyze startup projects by scientists, entrepreneurs, and investors. Keywords IT projects, startups, model, project efficiency, valuation, fuzzy sets, project risks 1. Introduction Countries that have managed to establish a continuous process of generating new knowledge and innovative ideas and transforming them into innovative products are now the most efficient and have a leading role in the global economy. The experience of the USA, which brings 85% of innovative products to the market, Japan - 75%, Germany - 55%, Israel - more than 50% is indicative. Unfortu- nately, the share and innovations in the total volume of manufactured products in Ukraine do not ex- ceed 2%. Small companies such as startups create favourable conditions for innovation. Start-up is an inno- vative project for developing new products or services, formed to find a repeatable and scalable busi- ness model in conditions of extreme uncertainty. Based on an analysis of the number of startups in 137 countries, Startup Ranking has developed a ranking in which Ukraine took 42nd in 2018 (215 startups). First place went to the USA - 45 004 startups, second place in India - 5203 startups and third place in the UK with 4702 startups [1]. Trends [2] shows that 90% of new start-ups fail (the 20 top reasons for start-up failure shown in the table 1). 2. Analysis of recent research and publications Many scientists have been involved in the study of innovative development of enterprises: Stephen Blank [4], Brad Feld, Jason Mendelsohn [5]. Among Ukrainian scientists in project management, researches related to the use of a value approach for innovative projects was conducted by such scien- tists as Vilenskyi P. L. [6], Bushuev S. D., Yaroshenko F. A. [7], Tsypes G. L. [8] et.al. At the same time, the effectiveness of projects was studied in publications of Ukrainian and foreign scientists, such as Kolesnikova K.V. [9], Kononenko I.V. [10], Morozov V. V. [11], Yehorchenkova N. [12], Timin- skyi A. G. [13] and foreign scientists – Nonaka I., Takeuchi H. [14], Turner R. [15], Milosevich D. [16], Tom DeMarco [17] et al. Problems of constructing mathematical models in project management when solving problems of synthesis of management methodologies based on fuzzy input data were considered in the works of Kononenko I.V. [18], Shmatko O. [19], Hrabyna K., Shendryk V. [20]. However, the analysis of risks in such conditions was carried out in works Leonenkov A. [21] and Zhang L., Xu X., Tao L. [22]. Analysis of information sources has shown that currently, the issue of assessing the effectiveness of startups is not sufficiently developed, it is not always possible to use classical analytical methods, especially for problems with uncertainty. Also, the problems of implementing innovative cooperation in terms of maximizing the profit of all interested parties to increase their value interest are still insufficiently studied and require new developments and improvements. The purpose of the research. The purpose of the article is to develop a methodology for assessing the investment risks of IT projects in the form of a startup based on fuzzy set theory. Table 1 Analysis reasons for start-up failure [built on 3] Reasons for start-up failure % No market need 42 Ran out of cash 29 Not the right team 23 Get out competed 19 Pricing/cost issues 18 Poor product 17 Need/lack business model 17 Poor marketing 14 Ignore customers 14 Product mis-timed 13 Lose focus 13 Disharmony on team/investors 13 Pivot gone bad 10 Lack passion 9 Bad location 9 No financing/investor interest Legal challenges 8 Don’t use network/advisors 8 Burn out 8 Failure to pivot 7 3. Presentation of the main material As defined above, scientific research on the development of economic and mathematical models for analyzing the effectiveness of startups requires some improvement. Therefore, it is advisable to use fuzzy modeling. Using the fuzzy set method, fuzzy variables are constructed that reflect uncer- tainty [6, 23, 24]. The main idea of using this apparatus is that any economic indicator is interpreted as an interval indicator, defined not by a specific number, but by a certain interval, in the form of a fuzzy set. This corresponds to a situation where only the limits of the values of the indicator in which it can change are sufficiently accurately known, but there is no quantitative or qualitative information about the possibilities or probabilities of implementing its various values within a given interval. Models based on fuzzy logic are characterized by the ability to adapt to changing market conditions [25, 26]. Let's look at the implementation of the mathematical model using a specific example. Let the ini- tial investment for a particular IT-startup be about UAH 7-10 million. The project research will be carried out on the basis of present value and internal profitability [27-32]. Let's define the value of a startup project as – the difference between monetary income and initial costs: n Vk P = −I0 +  k =1 (1 + r ) k (1) where – the volume of initial investments for creating a startup; Vk – receipts and payments (profit) in the -th period; – number of periods; – discount rate in the -th period. We set project metrics as fuzzy parameters. To do this, we construct belonging functions for them, which set the degree of belonging to a fuzzy set. Based on expression (1), we define variables that we represent in fuzzy form. These are the initial investment , profit , and discount rate . Let's choose the limits of changes in the studied indicators. We set the membership functions for them in the form of trapezoidal functions. Let's create α-level sets. Constructing α-level sets, we ob- tain an approximate decomposition of the fuzzy set (Figure 1). Using operations on the α- levels, we find and obtain an approximate decomposition of the fuzzy set over the α-levels. In fact, we will build a membership function for , which we will investi- gate. Figure 1. Graph of trapezoidal membership functions A trapezoidal fuzzy number is written as . The Elements of set A are uniquely in the range , and the range – is the tolerance interval (stability interval), i.e. the elements of set a are approximately equal to any number from this segment. Arguments are called significant points of a fuzzy number . When describing a mathematical model using trapezoidal fuzzy numbers, significant points can be interpreted as pessimistic, most likely on the segment, and optimistic scenarios for the development of the situation. It is assumed that the initial investment is UAH 7-10 million, we assume the set with numerical parameters = (7; 8; 9; 10). We assume the profit set using numerical parameters V = (4; 4.5; 5.5; 6). For discount rates <21,5% the implementation of a startup project is profitable since its profitable value is >0 For the discount rate = 21,5%, the income from the implementation of a startup project is equal to invest- ment costs. This is the maximum possible discount rate at which you can invest funds without losses. Choose a discount rate r ranging from 12% to 21% with a probability value of 17%. We assume the set using numeric parameters = (0.12; 0.14; 0.18; 0.21). The trapezoidal membership function can generally be given analytically by the following expression (2). Similarly, the membership functions are constructed for , . Next, we construct an approximate decomposition of fuzzy sets , V , by α-levels. We calculate the limits of sets , V , for a given value α – confidence intervals. Choose 10 levels α on the segment [0,1]:   0; 0,1; 0,2; 0,3; 0,4; 0,5; 0,6; 0,7; 0,8; 0,9; 1 . 0, x  amin   x − amin , amin  x  a2  a2 − amin  fT ( x; amin ; a2 ; a3 ; amax ) = 1 a2  x  a3 , (2) a −x  max , a3  x  amax  amax − a3  0, amax  x where amin ; a2 ; a3 ; amax – are some numerical parameters that take arbitrary real values and ordered relationships. To calculate confidence intervals for a given value of  i i, equations of the following form are solved: I ( x )i =  i , V ( x )i =  i r( x )i =  i Confidence intervals are represented as matrices with elements Iij , , , (i=1,…,10; j=1,2). Using the confidence interval matrices I , V , r we find the function P ( I ,V ,r ) by the formula (3): n V P ( I ,V , r ) = − I +  k =1 (1 + r ) k (3) Table 2 Matrices I , V , r IL IR VL VR rL rR 1 4 10 4 6 0,12 0,21 2 4,15 9,85 4,1 5,9 0,125 0,206 3 7,3 9,7 4,2 5,8 0,13 0,202 4 7,45 9,55 4,3 5,7 0,135 0,198 5 7,6 9,4 4,4 5,6 0,14 0,194 6 7,75 9,25 4,5 5,5 0,145 0,19 7 7,9 9,1 4,6 5,4 0,15 0,186 8 8,05 8,95 4,7 5,3 0,155 0,182 9 8,2 8,8 4,8 5,2 0,16 0,178 10 8,35 8,65 4,9 5,1 0,165 0,174 The function P has a trapezoidal appearance, while Pmin = 0,571, Pmax = 0,958, P2,3 = 0,9. P3 = Pmin – a pessimistic scenario, Pmax – is an optimistic scenario. P2 , P3 – base value. Get calcula- tions of the values P , shown in (Figure 2). Fuzzy numbers are a fairly convenient way to model startup projects with ambiguous, probabilistic characteristics. When using fuzzy sets the calculation formula is P transformed as follows: Pmin , P2 , P3 , Pmax  = −I min , I 2 , I 3 , I max  +  ( Vmin ,V2 ,V3 ,Vmax  k ) n k =1 (1 + Rmin , R2 R3 , Rmax ) (4) Figure 2. Function graph P As a result of calculations, we get a trapezoidal fuzzy value of the indicator P = ( Pmin , P2 , P3 , Pmax ) . The project has a positive value if P there is more than the criterion set by investors . Where – an investment risk assessment – determination of criteria under which the resulting value of the investment process P will be lower than the established limit level. Let – be the selected limit value. In this task with fuzzy variables, we will evaluate the possibil- ity of an event P  W , that determines the risk that the project will be ineffective. Since the result of the calculation P , is a fuzzy number, the following variants of its correlation with the efficiency criterion (Figure 3) are possible. The specified areas of shapes can be found in different ways. In the most general form, the area of the shape on the interval [amin , W ] is a certain integral of the function that restricts the shape from above: W S( a2 ,amin ) =  lef dx , (5) amin where – is a function describing the left side of the trapezoidal fuzzy number membership func- tion . Since the graph of the membership function of a trapezoidal fuzzy number is a trapezoid we use the formula of a straight line passing through two points to obtain the equation of the function: lef −  2 x − a2 =  2 −  min a2 − amin Figure 3. Determining the degree of project risk: a. amin