CC BY
21
2017
Perm University Herald ECONOMY
Vol. 12. No. 2
doi 10.17072/1994-9960-2017-2-234-246 UDC 330.45 LBK 65 .05 JEL Code C53
OPTIMIZATION STRATEGIES OF GUARANTEED RESULT FOR INVESTMENT PROJECTING PROCESSES
Elena V. Butsenko
ORCID ID: 0000-0003-2747-5391, Researcher ID: H-4042-2015
E-mail: evl@usue.ru
Ural State University of Economics
62, 8 March st., Ekaterinburg, 620144, Russia
Anrey F. Shorikov
ORCID ID: 0000-0003-1255-0862, Researcher ID: J-6801-2017 E-mail: afshorikov@mail.ru
Ural Federal University named after the first President of Russia B.N. Yeltsin 19, Mira st., Ekaterinburg, 620002, Russia
In current economic conditions the improvement of investment projecting process is an essential element of efficient control. Investment projecting determines a financial policy of an economic entity as it includes both the development of most profitable means of income acquisition and search for new methods and tools for spare capital investment. In this case the development of the strategy based on investment projecting optimization is an important scientific issue. In the present research matrix games and regression analysis methods are used to make an economic - mathematical control model of an enterprise investment projecting process. The scientific novelty of the research is to develop methods of planned control and to extend the application field of mathematical apparatus of game theory to solve the tasks of optimal employment of recourses under conditions of competitive economic environment. The method to optimize the control of investment projecting suggested in the article allows choosing the best investment projecting strategy. This strategy is considered to be the process of such production output of an enterprise that will be soled with the best guaranteed profitability/risk ratio. The suggested econometric method to choose an optimal control strategy of investment projecting has been used in practice. Features of the suggested method application have been investigated in the case study of a real object of investment projecting, necessary calculations have been made and the obtained results have been analysed. This analysis has revealed the method efficiency to make control decisions when realizing investment projects. The present method may become a foundation for the development of modern tools for investment projecting control in the tasks of strategic optimization. The research is of practical interest for experts in the field of investment and project planning as well as for the specialists who deal with optimal employment of recourses considering the factors of profitability and risk. The research results may be used by any economic entity that implements investment activity and there is no doubt that they will increase its activity efficiency and as a result its competitiveness.
Keyword: investment projecting, control optimization, decision-making, control strategies, optimality criteria, algorithm, pay-off matrix, compromise solution, game theory, guaranteed result.
© Butsenko E.V., Shorikov A.F., 2017
СТРАТЕГИИ ОПТИМИЗАЦИИ ГАРАНТИРОВАННОГО РЕЗУЛЬТАТА ДЛЯ ПРОЦЕССОВ ИНВЕСТИЦИОННОГО ПРОЕКТИРОВАНИЯ
Елена Владимировна Буценко
ORCID ID: 0000-0003-2747-5391, Researcher ID: Н-4042-2015 Электронный адрес: evl@usue.ru
Уральский государственный экономический университет 620144, Россия, г. Екатеринбург, ул. 8 Марта, 62
Андрей Федорович Шориков
ORCID ID: 0000-0003-1255-0862, Researcher ID: J-6801-2017 Электронный адрес: afshorikov@mail.ru
Уральский федеральный университет имени первого Президента России Б.Н. Ельцина 620002, Россия, г. Екатеринбург, ул. Мира, 19
В условиях современной экономики совершенствование процесса инвестиционного проектирования является необходимым элементом эффективного управления. Инвестиционное проектирование определяет финансовую политику хозяйствующего субъекта, поскольку предполагает не только разработку наиболее выгодных способов получения доходов, но и поиск новых методов и инструментов инвестирования свободного капитала. В этой связи разработка технологий, основанных на оптимизации инвестиционного проектирования, является весьма актуальной научной проблемой. В данной работе для построения экономико-математической модели процесса управления инвестиционным проектированием предприятия используются методы матричных игр и регрессионного анализа. Научная значимость работы заключается в развитии методов проектного управления и расширении области применения математического аппарата теории игр к решению задач оптимального использования ресурсов в условиях конкурентной экономической среды. Предложенная в ходе исследования методика оптимизации управления процессом инвестиционного проектирования предприятия позволяет реализовать выбор наилучшей стратегии инвестиционного проектирования, под которой понимается процесс формирования такого объема производства предприятия, который будет реализован при наилучшем гарантированном значении соотношения «доходность/риск». Осуществлена практическая реализация предлагаемой эконометрической методики для выбора оптимальной стратегии управления инвестиционным проектированием. На примере реального объекта инвестиционного проектирования рассмотрены особенности применения предложенной методики, выполнены необходимые расчеты и проведен анализ полученных результатов, который показал эффективность методики для выработки управленческих решений при реализации инвестиционных проектов. Данная методика может стать основой для разработки современного инструментария управления процессом инвестиционного проектирования в решении задач стратегической оптимизации. Работа имеет практический интерес для экспертов в области инвестиций и проектного управления, а также широкого круга специалистов, решающих задачи оптимального использования ресурсов с учетом параметров доходности и риска. Результаты исследования могут быть использованы любым хозяйствующим субъектом, осуществляющим инвестиционную деятельность, и, безусловно, повысят эффективность результатов его деятельности и, как следствие, уровень конкурентоспособности.
Ключевые слова: инвестиционное проектирование, оптимизация управления, принятие решений, стратегии управления, критерии оптимальности, алгоритм, платежная матрица, компромиссное решение, теория игр, гарантированный результат.
Introduction
Applied economics faces a lot of difficulties nowadays therefore it is necessary to implement its processes efficiently. Among these processes is investment projecting that elaborates and ac-
complishes not only the most favourable means to get income but also seeks new profitable options to invest available financial resources.
The development of new economic and mathematical models and methods to optimize the control of investment processes and the crea-
tion of the supporting systems of investment decision making based on these models and methods are the important condition for the modern economy operation as it improves the investment climate and increases the benefits from resource investment into promising projects.
A lot of contemporary scientists write about different issues devoted to the implementation of investment projecting. In particular, O.V. Tochilin, E.I. Markovskaya, V.P. Karev research the issues of innovative investment projecting in science-intensive industry; new original models of analysis and forecasting of planned economic systems behaviour are suggested; the issues devoted to the organization of different financial tools that may be used to finance investment projects are discussed; the peculiarities of investment decision making at enterprises and banks are revealed as well [1-6]. Some articles are devoted to control optimization based on gaming technologies [e.g., 7-10]. The further development of models and methods to optimize the control of investment projecting processes is an acute issue.
The main tasks of investment projecting, from the view point of its optimization, are results forecast and the optimization of investment projecting control based on particular qualitative criteria. The present article is devoted to the solution of these tasks. Hence, the scientific practical task and the scientific novelty of the present research are to optimize the strategy of investment projecting. The methodology and the corresponding algorithm of optimal investment projecting strategy, presented in the article, are based on simulation and research of three features of the analyzed process. The suggested strategy optimizes the guaranteed result for these features. Moreover, both statistical data, describing previous periods of enterprise operation, and market research methods are used to forecast prices and sales volume (e.g. when competitors appear, statistical data about them are added).
Methods and performance of the scientific problem of control of investment projecting process
To solve the above mentioned task we suggest using the method implemented within the frameworks of a "games with nature" model. An op-
timal strategy for investment projecting control is based on this model. Since the sales volume of production (and correspondingly the volume of products manufacture) is one of the investment projecting resources, an optimal investment projecting strategy means the formation such production volume that will be realized with the best guaranteed ratio of profitability/risk values.
To assess the quality of an enterprise development, a set of criteria that are used to choose an optimal strategy to implement investment projecting, is selected. The Wald criterion (the guaranteed result principle) that recommends to apply a maximin strategy, is pessimistic and nature is believed to behave in the worst way for an economic entity. Maximum criteria (principle of optimism) is chosen according to maximax condition, it is optimistic and nature is believed to be the most favourable for the functioning of an enterprise under consideration. The Hurwitz criterion takes an intermediate position that considers both the best and the worst nature behaviour.
It should be noted that separately each of the above mentioned criteria may not be considered to be satisfactory enough to make a final decision about the control optimization of investment projecting process, but their joint analysis allows to visualize the consequences of control decisions and to form the best strategy.
An econometric method to build a model that optimizes the control of investment projecting process has been suggested. This method is based on the theory of optimal control as well as the analysis and modification of economic methods of investment decision result forecast conducted by the authors and described in the works [11-13].
The setting of the task to make investment decision includes the aim determination, the characteristic of future condition of an investment environment and the supposed options of actions. The task solution is to select actions that guarantee the best achievement of the specified aim.
The sales revenue from a company's products is considered to be a target index. This revenue depends on sales volume, market share and gross return. These indices, in their turn, depend on marketing decisions that determine, first of all, product price.
The condition of external investment environment is determined by possible options of market condition, the expected reaction from competitors to a company action (e.g. increase/decrease of product price), suggested alteration of consumers' preference, etc.
The various options of a decisionmaker's actions (DM) are the set of investment measures. Among them are, for example, marketing decisions:
- to reduce the product price by 10%;
- to increase an ad budget by P1 million;
- to decrease the price by 5% and to increase an ad budget by P 0.5 million.
Thus, a formal task to optimize decision making during investment projecting may be formulated in the following way. The future of an external investment environment is suggested to be described with the finite number m of Xj, X2, X3, ..., Xm conditions. There is a possible alternative list from q marketing decisions of Cj, C2, C3, Cq. The aim of a decision maker is to form a control strategy that is realized by selecting of one of the possible options that will assure the best (optimal) guaranteed value of the target feature.
Control optimization method of investment projecting process
he present method may be described as an implementation of the following set of actions.
1. The possible control strategiesCi,
i e 1, q, where q is the quantity of strategies (scenarios) of an enterprise development and its operation at the market during the period under consideration are formed based on the conducted marketing research data.
A set of purposeful measures is considered to be a strategy to manage an enterprise. This set of measures is characterized by a particular pricing and market policy, expenses rate, ad budget and other factors controlled by an enterprise.
2. Possible market conditions in a k arrays form (options corresponding to different conditions of the process implementation) are formed based on the conducted marketing research data. These k arrays consist of m forecast values X(f) = (X/),X2f,X¡j),X%), j eÜ,
stment p
T1
ts
independent variables (factors) X^\ l e 1, m (e.g. these variables assess the inflation rate, products prices sold by competitors, market capacity, etc.), that correspond to possible implementation of a control strategy Ci ( i e 1, q ) of an enterprise development and its behaviour at the market during the period under consideration.
Thus, market conditions are considered to be various combinations of external factors independent from an enterprise, i.e. each Xi)
(l e 1, m ; i e 1, q ; j e 1,k ) value is a forecast value of the l-th factor of a market condition. This value corresponds to a particular control strategy Ci and a forecast option j.
3. The data necessary to make multiple regression models are organized based on the results of conducted market researches. This model is the dependence between function values (forecast values of sales volume) that correspond to forecast values of independent variables (factors) and it is used as a forecast model in the form:
Y = A(j) + a,0') x X,(j) + a{j) x X2j) + a¡j) ij i 1* 2 i 2 i ¡ i
xX¡(j) +... + a(j) x X.
( j )
(1)
where Y is a forecasted sales volume corre-
•j
sponding to the implementation of the control strategy (scenario) C •, (• e 1, q) and to the array (option) of forecasted values Xj) = (X/),Xf,X(j),XU)) (jeU) of dependent variables XlliJ), l e 1, m ; A(J) is a constant of the corresponding equation of the regression (l e 1, m ; J e 1, k ); rff) is coefficients of the corresponding equation of the regression (l e 1, m ; • e 1, q ; J e 1, k ).
Thus, various possible options of the external independent from an enterprise factors implementation are considered to be permissible market conditions. It means that in the equation (1) each X(f) (l e 1, m ; • e 1, q; J e 1, k) value is a forecast condition of the market that characterizes, for example, a particular inflation rate, a competitor's pricing policy, market capacity or other external factor independent from an enterprise. This value also corresponds to a possible implementation of a control strategy Q (• e 1, q ) and a forecast option J.
4. Forecast values of sales volume are formed to make a pay-off matrix (Table 1).
Forecast values of sales volume are calculated based on the correlation (1) by means of variation of independent variable values. These values correspond to the content of a great number of suggested control strategies C = {C,}, i e 1, q and the values of possible
market conditions described by corresponding arrays of independent variables
K j)_(Y( j) Y( j) Y( j) Y( j)ï
of control r e c , that meets the following
7 e)
= (Xj X™, xmj)), j el, k ; the pay-
off matrix of forecasted values of sales volumes Y = |y || , i e 1, q, j e 1, k is formed according to
the formula (1) (Table 1).
Table 1
Elements of a pay-off matrix of forecasted values of sa es volumes
Possible control strategies № factors array
1 2 3 • ♦ ♦ K
Ci Y11 Y12 Y13 Y 1k
C2 Y21 Y21 Y21 Y2k
C3 Y31 Y32 Y33 Y3k
Cq Yqi Yq2 Yq3 Yqk
5. On the pay-off matrix y = ||y.|| ,
i e 1, q, j e 1,k data basis that is determined by the element values from Table 1 we calculate a maximin estimation of a control strategy. This estimation determines a guaranteed upper bound (guaranteed result) of the forecasted sales value under the worst conditions of the implementation of the process under consideration: w, = maxmin Yri., also we calculate the
ie1,q je1, k j
corresponding maximin strategy of control that meets the following maximin
C(e) ^ C
conditions:
i((e) el, q : min Y ( )
je1,k i*( )
= max min Yj = W( •
(2)
ie1,q je1,k J
6. The pay-off matrix data from table 1 are used to calculate a maximin estimation of a control strategy. This estimation determines the upper bound value of forecasted sales volume when implementing the most favourable situation (a superoptimistic result): W* = maxmaxY . These data are also used to
ie1,q je1,k j
calculate the corresponding maximin strategy
(3)
maximin condition:
i(e)* e 1, q : max Y(e), = max max Yu = W*.
je1,k • ) ,j iel,q jel,k j
7. To make the compromise solution between superoptimistic estimation W* and minimax estimation W,, we calculate the Hurwitz criterion Gie value using the formula:
Gf = max[y§ x min Yl} + (1 - fi) x max Yl} ], (4)
ie1,q je1,m je1,m
where f is a fixed pessimism-optimism indicator that is determined by experts based on competitive advantages analysis of an economic entity whenfi e [0;1]. The pessimism-optimism
indicator gives the quantitative expression of a decision maker's ideas about the risk of different level, about favourable or neutral environment where the decision maker should choose an optimal strategy. f value is an optimism indicator, and (1 -f) is a pessimism one. The closer the optimism indicator f to 1 is, the closer to 0 the pessimism indicator (1 -f)is, and more optimistic and less pessimistic the decision maker is. And vice versa, when f = 1, the present indicator transforms into a maximax criterion, when f = 0, it coincides with the Wald maximin criterion. When f = 0.5, then (1 -f) = 0.5. It means that a decision maker while choosing a strategy behaves neutrally. Responsibility measure influences the choice of the optimism indicator f value. More serious consequences of wrong judgement make a decision maker insure against errors, i.e. the pessimism indicator (1 -f) is closer to 1. To determine the environment and to analyze the competitive advantages of an enterprise, a decision maker may use, for example, the pay-off matrix of Boston Consulting Group Ltd., or McKinsey Company [14; 15].
8. Based on the Hurwitz criterion (4) we make an optimal compromise control strategy Cf = C() e C that is determined by the
,-(e)
el,q index and meets the following opti-
mality condition:
i(e) e 1, q: ft x min 7( ) + (1 - ft) x max 7,, =
Je1,k '"'.j jeA,h " ).j
= max | A x min 7. + (1 - ft) x max 7] = Gf. (5 )
iel,q je l,k je \,k
9. When various options have been assessed with several criteria. we make one of the following decisions: 1) when recommendations match. it is more confident to make the best decision; 2) when recommendations contrary to each other. the final decision should be based on its advantages and disadvantages (choose the strategy that is optimal at least for two criteria); 3) if different strategies have been made for all three criteria. the values of the pessimism-optimism indicator should be modified in the Hurwitz criterion or. for example. the data about possible market condition may be changed.
Considering the above mentioned information we suggest the following formalized algorithm to make an econometric method to optimize control process of investment projecting.
Step 0. Initial data formation.
The array of possible control strategies C = {Ci}. i e 1.q is formed. where q is the quality of control strategies (scenarios) applied to plan an enterprise's investment policy and its operation during the period under consideration.
According to the data of the conducted marketing research possible market conditions are made in a form of k arrays (options corresponding to various conditions of the process implementation) that consist of m forecasted values of Xj) = (XJ. Xf. Xf. X^. j e Ü. independent variables of XII¡J). l e 1. m that correspond to possible implementation of control strategy of an enterprise investment projecting Ci (i e 1. q) and the enterprise operation at the
market during the period under consideration.
Step 1. Pay-off matrix formation.
The pay-off matrix of forecasted values of sales volumes Y = ||yjJ| . i e 1. q. j e 1. k at the
end of the period under consideration is formed. This matrix is based on (q x k) arrays of forecasted values of the chosen factors
XjJ) = (XXx2j).Xf.Xmf). j eü. that correspond to the array of admissible control strate-
gies C = {C}, i e 1, q. It means that Table 1 is completed with the values of elements, each element is calculated according to the formula (1):
Y = A(J) + ai) x X(J) + df.) x X2J) + dJ) x
ij i 1i 1i 2i 2i 3i
xX3J) +... + a3J) x X^),i e , J e U.
Step 2. Formation of a maximin control strategy.
According to (2) we calculate a maximin estimation of the control strategy W((e) = max min Y , = max W (C.), where
ie1,q je1,k J ze1,q 1
W((Ci) = min Y and corresponding maximin
1 Je1,k 11
control strategy C () e C, that meets the fol-
4 e)
lowing maximin conditions:
1((e) e 1, q : min Y(e) = max min Y = max W( (C.) =
Je1,k i» , J ie1,q Je1,k J ie1,q 1
= W.(C (e)) = W((e).
(
This strategy is calculated on the payoff matrix data Y = ||Y ||, • e 1, q, J e 1,k basis.
Step 3.Formation of a maximax control strategy.
According to (3) we calculate a maximax estimation of the control strategy W(e)* = max max Y , = max W* (C.), where
ie1,q je1,k J ie1,q 1
W *(C ,) = max Y and the corresponding maxi-
1 Je1,k J
max control strategy C (e)( e C, that meets the
following maximax conditions:
1(e)* e 1, q : max Y(e)( = max max Yy = max W* (C;) =
je1,k 1 ) ,J iel,q je1,k J ie1,q
= W *(C (,).) = W(e)(.
This estimation is based on the pay-off matrix Y = ||YJ ||, i e 1, q, J e 1, k data.
Step 4. Formation of the compromise control strategy.
For the fixed pessimism-optimism index ¡1 e [0; 1], according to (4) we calculate
the value of the Hurwitz criterion using the formula:
Gf = max[1 x min Yiy + (1 - ¡) x max Yiy ] = max Gf(Ci),
ie1,q je\,m jel,m ie1,q
where GsCC. ) = 1 x min Yy + (1 - ¡) x max Yy • We
je1,m je1,m
also calculate the optimal compromise strategy
C(e) = Cr, e C, that is determined by the
P (e) J
A e)
1, q index and meets the optimal condi-
tion:
if e iq : ß x min Y(e) + (1 - ß) x max Ye . =
jelk iß ,y V
je1,k iB ,j
= max [ß x minYV + (1 - ß) x maxYy] = max Gß(Ct)
yelk ie1,q
ie1,q je1,k
= Gf(C(e)) = Gf.
f
Step 5. Formation of the optimal control strategy.
The optimal strategy c(e) = C, , e C is
formed on the basis of organised pairs that contain the strategies formed from the solutions of optimization tasks and the forecasted estima-tions(c(e),W,(e)); (C(e),,W(e)*); (Cfe),Gf)) corresponding to them. This optimal strategy is determined by the i(e) e 1, q index from the condition system: if W*(e) = W(e> = Gf, then i(e) =
= i(e) ^ C(e) = C ■ i*
if (W(e) = Gf ) v (W(e) = W(e) *) v
v(W(e> * Gf) ),then i(e) = ,) ^ C(e) = C (e);
i*
if (W*(e) = Gf ) v (W(e) * W(e) *) v
v(W(e> = Gf) ),then i(e) = if) ^ C(e) = C (e);
f
if (W*(e) * Gf ) v (W*(e) = W(e) *) v v(W(e> = Gf ), then i(e) = i(e)* ^ C(e) = C (e),;
if W*(e) * W(e) * * Gf, then i(e) = = i(e) ^ C(e) = C
= if C = Ci(e) ■
End of the algorithm. Considering the above mentioned algorithm the flow chart of the present method to optimize the control of investment projecting process is presented in Figure.
ze nie il
Th
ta
Thus, the present algorithm is a detailed consequences of the simulation stages of the decision making process to choose the optimal control strategy of investment projecting. This algorithm may be implemented using computer simulation.
Approbation of the method to optimize the investment projecting control he practical application of the authors' method to optimize the process of investment projecting control has been approved by the results of its approbation in the business project "Tandem" Ldt. (Ekaterinburg). While choosing an optimal pricing strategy of the present enterprise we considered the marketing research results that revealed that the consumer price index is a key feature of the market condition. In this connection various inflation rates (109%, 110%, 112%, 113%, 115% from the initial rate) are considered to be the "nature" conditions.
The following possible solutions are considered:
Strategy 1: the price is 10% higher than the competitors currently have;
Strategy 2: the price is 5% higher than the competitors currently have;
Strategy 3: the price is the same as the competitors currently have;
Strategy 4: the price is 5% lower than the competitors currently have;
Strategy 5: the price is 10% lower than the competitors currently have;
We mean the average price that the competitors have.
The task is to determine an optimal pricing strategy of an enterprise in the current conditions. The earnings maximization is considered to be an optimization criterion in this case.
(W*(e) = Gp)) v (W*(e) = W (e)* y v (W(e)* * G^
-W^") ^ W(e)* ^ G^
e)
" ^ i e)
+ (W*(e) = Gp) ) v (W*(e) * W(e)*) ^V(W(e)* = gP
r ( e) = C , s
^ ( e) P
+ (W*(e) * Gp)) v (W*(e) - W^*)>-v (W(e)* - Gp)
c(e) = C f e) i
c(e) = c (
End
The flow chart of the algorithm to choose the optimal strategy of the investment projecting control
To make a pay-off matrix, we build a forecast model of sales values using the multiple regression analysis method [16-18].
This method means to determine functional dependences between the function (volume of sales) and factors (realization value,
competitors' price, prime cost, advertising expenses, etc.).
The data about volume of sales, realization value, competitors' prices, ad budget, prime cost and consumer price index collected for 24 months are the initial data and are presented in Table 2.
Table 2
Initial data to make an econometric model
Month Volume of sales, thousands of roubles Time Realization value, P Competitors' price, P Advertising expenses, P Prime cost,P Consumer price index, %
July 7992.40 1 6 8 2.049 112.42 100
August 5170.33 2 8 12 2.049 112.42 103.90
September 4212.12 3 8.3 12 2.049 112.23 107.40
October 2137.20 4 8.5 13 1.119 93.00 111.60
November 2980.69 5 9.0 13.5 1.119 93.00 115.70
December 2688.54 6 9.7 13.7 1.119 93.00 117.50
January 2298.15 7 10.0 13.7 0.84 68.85 121.50
February 1435.65 8 10.5 15 0.84 68.85 126.50
March 2190.90 9 10.9 15.5 0.84 68.85 131.40
Aipril 2930.56 10 11.4 15.5 2.53 148.75 137.40
May 4862.85 11 11.8 15.5 2.53 148.75 142.10
June 8345.56 12 12.3 15.5 2.53 148.75 147.80
July 8639.25 13 12.8 15.5 3.28 190.94 153.80
August 5458.16 14 13.3 16 3.28 190.94 156.20
September 4867.32 15 13.9 16.4 3.28 190.94 162.20
October 2540.70 16 14.1 16.4 1.79 127.93 169.20
November 2900.20 17 15.0 16.4 1.79 127.93 174.30
December 2951.43 18 15.6 16.4 1.79 127.93 181.00
January 2391.80 19 16.2 16.8 1.35 112.90 187.70
February 1413.19 20 16.9 16.9 1.35 112.90 195.60
March 2434.05 21 17.5 17.5 1.35 112.90 204.10
Aipril 3140.23 22 18.2 21.2 4.05 241.56 210.90
May 4875.29 23 19.0 21.2 4.05 241.56 213.50
June 8900.32 24 19.7 21.2 4.05 241.56 220.40
When applying the regression analysis, various possible statistically significant models have been considered. From these models in accordance to F-test, coefficient of determination and values of balance standard deviation
In accordance to formula (1) and the obtained indices values we create the following model:
we choose the most valuable factors analysing the significance of the model factors using t-index. Thus, the model presented in Table 3 describes the data more efficiently.
Table 3
Y = 6184.83 - 571.33 x X + 2420.16 x X2 --1207.06 x X3 +16524.67 x X4 - 310.87 x X
Regression statistics of forecast model
Index Index value
Multiple R 0.82
R2 0.67
Normalized R2 0.57
Standard error 1439.32
Y-intersection 6184.83
X] variable -571.33
X2 variable 2420.17
X3 variable -1207.06
X4 variable 16524.67
X5 variable -310.87
where Y is the volume of sales; X1 is time; X2 is the realization value; X3 is the competitors' price; X4 is the advertising expenses; X5 is the consumer price index.
The model we have made has the correlation coefficient r=0.82 and accurately reflects sales volume dynamics.
Table 4
Pay-off matrix and maximin criterion
Price strategies Market contitions Minimum of the line W*( e) criterion value
X X5 X, X5
Strategy 1 7005.26 7069.53 7198.07 7262.34 7390.87 7005.26 7005.26
Strategy 2 6873.36 6936.42 7062.54 7125.59 7251.71 6873.36 -
Strategy 3 6741.46 6803.31 6927.01 6988.86 7112.55 6741.46 -
Strategy 4 6609.57 6670.20 6791.48 6852.12 6973.39 6609.57 -
Strategy 5 6477.67 6537.095 6655.95 6715.38 6834.24 6477.67 -
We build a pay-off matrix (Table 4) based on the created forecast model. The matrix reflects the forecast values of sales volumes in October next season considering various strategies and "nature" conditions combination.
We calculate the maximax estimation of a control strategy using the data of the same pay-off matrix. This estimation determines the most possible upper bound of the forecasted sales volume at the most favourable condition implementation. Strategy 1 and its sales volume value equal to P 7390.87 correspond to the maximax control strategy (Table 5).
Table 5
Pay-off matrix and maximax criterion
Price strategies Mar iet conditions Maximum of the line W(e) * criterion value
Xt X, X5 X, X5
Strategy 1 7005.26 7069.53 7198.07 7262.34 7390.87 7390.873 7390.87
Strategy 2 6873.36 6936.42 7062.54 7125.59 7251.71 7251.714 -
Strategy 3 6741.46 6803.31 6927.01 6988.86 7112.55 7112.555 -
Strategy 4 6609.57 6670.20 6791.48 6852.12 6973.39 6973.395 -
Strategy 5 6477.67 6537.095 6655.95 6715.38 6834.24 6834.236 -
Thus, the maximin estimation of a control strategy determining the guaranteed upper bound of the forecasted sales volume in the worst implementation conditions of the process under consideration forms strategy 1 and its corresponding sales volume that is equal to P 7005.26.
To make a compromise solution between the pessimistic estimation by the minimax criterion W* and the optimistic estimation
by the maximax criterion W*, we calculate a value of the Hurwitz criterion GPe) for each
strategy (Table 6).
Table 6
On the present table data basis we choose a strategy which compromise solution is maximum. In our case it is strategy 1 and its corresponding Hurwitz criterion value - P 7101.66.
To compare, we make calculations using the pessimism-optimism criterion ^=0.25 (Table 7) and ^=0.5 (Table 8).
Table 7
Price strategies Compromise Hurwitz criterion G( p value
Strategy 1 7101.66 7101.66
Strategy 2 6967.95 -
Strategy 3 6834.24 -
Strategy 4 6700.52 -
Strategy 5 6566.81 -
Price strategies Compromise Hurwitz criterion Gß value
Strategy 1 7294.47 7294.47
Strategy 2 7157.12 -
Strategy 3 7019.78 -
Strategy 4 6882.43 -
Strategy 5 6745.10 -
Strategy 1 and its corresponding Hurwitz criterion value equal to P 7294.47 are the compromise solution based on the calculations with the present pessimism-optimism criterion value.
Table 8
Correspondingly, strategy 1 and its corresponding Hurwitz criterion value equal to P 7198.06 are the compromise solution when fi=0.5.
Thus, when we use the criteria under consideration (the maximin criterion forms strategy 1, the maximax criterion makes strategy 1, the Hurwitz criterion creates strategy 1) we observe the coincidence of these criteria recommendations. In this case the application of strategy 1 is more preforable as this strategy provides an enterprise with the optimal sales volume.
The application of the method developed by the authors allows an enterprise to choose an optimal price strategy considering the impact of external environment on the enterprise activity.
Conclusion
The present research poses the method to choose an optimal control strategy of investment projecting and describes its practical implementation. Features of the suggested method application are discussed in the case study of a particular investment projecting object.
The present method may become the foundation of the development of modern tools of an investment projecting control system. This system will be able to make decisions for practical implementation of particular investment projects.
Further research will be devoted to the improvement of the mathematical apparatus used in the authors' method. In particular, to make the elements of a pay-off matrix it is possible to use neural network simulation techniques as options to the mathematical apparatus of a regression analysis [19-24].
Search of a compromise solution when jff=0.5
Price strategy Compromise Hurwitz criterion Gip) value
Strategy 1 7198.06 7198.06
Strategy 2 7062.53 -
Strategy 3 6927.01 -
Strategy 4 6791.48 -
Strategy 5 6655.95 -
Acknowledgements
The work was conducted with a financial support of Russian Foundation for Basic Research (RFBR) (project № 17-01-00315 "Development of models and methods of monitoring, conditions forecast and optimization of multivariate regional social economic system control based on entropic and minimax approaches ").
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Статья поступила в редакцию 02.05.2017
Information about the Authors
Butsenko Elena Vladimirovna - Candidate of Economic Sciences, Associate Professor, Associate Professor at the Department of Business Informatics, Ural State University of Economics (62, 8 March st., Ekaterinburg, 620144, Russia; e-mail: evl@usue.ru).
Shorikov Andrey Fedorovich - Doctor of Physical and Mathematical Sciences, Professor, Professor at the Department of Applied Mathematics, Ural Federal University named after the first President of Russia B.N. Yeltsin (19, Mira st., Ekaterinburg, 620002, Russia; e-mail: afshorikov@mail.ru).
Благодарности
Работа выполнена при финансовой поддержке РФФИ (проект № 17-01-00315 «Разработка моделей и методов мониторинга, прогнозирования состояния и оптимизации управления многомерными региональными социально-экономическими системами на основе энтропийного и минимаксного подходов»).
Сведения об авторах
Буценко Елена Владимировна - кандидат экономических наук, доцент, доцент кафедры бизнес-информатики, Уральский государственный экономический университет (Россия, 620144, г. Екатеринбург, ул. 8 Марта, 62; e-mail: evl@usue.ru).
Шориков Андрей Федорович - доктор физико-математических наук, профессор, профессор кафедры прикладной информатики, Уральский федеральный университет имени первого Президента России Б.Н. Ельцина (Россия, 620002, г. Екатеринбург, ул. Мира, 19; e-mail: afshorikov@mail.ru).
Просьба ссылаться на эту статью в русскоязычных источниках следующим образом:
Butsenko E.V., Shorikov A.F. Optimization strategies of guaranteed result for investment projecting processes // Вестник Пермского университета. Сер. «Экономика» = Perm University Herald. Economy. 2017. Том 12. № 2. С. 234-246. doi: 10.17072/1994-9960-2017-2-234-246
Please cite this article in English as:
Butsenko E.V., Shorikov A.F. Optimization strategies of guaranteed result for investment projecting processes // Vestnik Permskogo universiteta. Seria Ekonomika = Perm University Herald. Economy. 2017, vol. 12, no. 2, pp. 234-246. doi: 10.17072/1994-9960-2017-2-234-246
