The use of mathematical methods for decision making in optimal pricing strategy for enterprises Текст научной статьи по специальности «Экономика и бизнес»

UDC 658:338.5+519

N. Shevchenko,

PhD (Economics),

E. Gudkova,

Donbas State Machine-Building Academy, Kramatorsk

THE USE OF MATHEMATICAL METHODS FOR DECISION MAKING IN OPTIMAL PRICING STRATEGY FOR ENTERPRISES

Problem Statement. At the present stage of transformation of economic system of Ukraine the pricing policy of the company plays an important role in maintaining the required level of the enterprise competitiveness. Increased competition and lowering solvency of the population is forcing companies - manufacturers to make more and more concessions to consumers and middlemen in marketing with the help of the pricing policy. In addition, the objective factor of increasingly important role of pricing policy is that the advertising efficiency goes down because of the rising costs of advertising and tightness in the media. Therefore, a growing number of companies resort to the effective pricing policy, as a means which can effectively support the company's competitiveness.

Analysis of Researches and Publications. Sufficient attention is given to the problem of pricing (I.D. Paderin, G.Yu. Kramarenko, V.I. Yanchuk, G.V. Yan-chuk, L.A. Shkvarchuk et al.). Classical methods of pricing (cost and market) are often used in practice. Pricing methods and models develop, modify and adapt according to conditions the price is formed in. The variability of the external operating environments (political and economic) determines the topicality of choosing the optimal pricing strategy under uncertainty and risk.

The objective of the study is to develop a model of the optimal pricing strategy for industrial enterprise in the conditions of uncertainty and risk.

Presentation of the Main Material. The company may form a numerous. possible pricing strategies, such as:

- Strategy 1 - the formation of prices with cost method, taking into account 20% of the profits;

- Strategy 2 - the formation of prices with cost method, taking into account 0% of the profits;

- Strategy 3 - the formation of prices in order to accumulate their own funds.

The choice of the company will depend on the projected sales volume for the next period and the company's management attitude to the economic risk.

According to [1-2] Gross profit PV can be defined as the difference between revenues B and expenses Z. We assume that:

B _ IX _ • a • P1 ) ;

Z _ Xn ■ Un + Zn,

(1) (2)

where Zn - the fixed costs of production; Un - variable cost for one unit; Xn - the number ofunits produced in the n-th period;

m

PV_B-Z_ I X

r _ 0 n-r r

where P(1) - the price of one item.

a ■ P

■ P(1).

X .U + Z n n n

(3)

Profit Index p is determined by the ratio of the profit PV to the cost C:

IX ■ a ■ P(1)

_pv _ b - z _ b -1 _ r=o n-r r p _ c ~ z ~ z ~ Xn ■ u„ + Zn

-1.

(4)

n n

P(1) =

X. ■ U + Z„

(5)

I Xn-

When X_

= const = X , P0(1) = Un .

n ' 0 n X

When p ^ 0 the expression for the price C (1) takes the form:

p(i) _(1 + p)•[[ • U„ + Z„]

(6)

At the profit of zero in the period n = 0 the price U (1) is determined this way:

I X„-r ■ ar

When Xn-r _ const _ Xn, P(1) _ (1 + p)

U +-

X„

Relation (7) and (8) are used to determine p and C (1), providing the enterprise accumulation of new own funds for the period n at the expense of the profit, and take into account the rate B of income tax:

rnp

r_0

r _0

r_0

(1) x • u + z + c i(1-B )

p(l) _ n n n n V "up S _

m '

Z xn-r ■ a

r _0

fx • U + Z + C l(1-B )

I n n n n V I up S I

p _

-1.

(7)

(8)

^Xn ■ U+zn

To determine the minimum level of retail and wholesale prices, on the basis of the method used for pricing at the enterprise, the ration is used (for a cost-based method for setting the price):

Pf = ( + Z„m) • 1,2 , (9)

where P(1) - the minimum retail price of one item, UAH; Zdc - the direct costs of one item, UAH; Zoverh -the overhead costs of one item, UAH:

Zdc _

(Zsal b + 0ded — Zsal )

S + —, V '

(10)

s _ Z _ tí V-N. p

I 1 AAA 1

Zoverh

(12)

where Zv

Zoverh — dir + Zad min + Zmarket

1000 ,

. - direct overhead costs are determined

1000

where Zsal b - basic salary at the rate of 1000 units, UAH$ Oded sal - deductions from the basic salary,

UAH; S - the cost of materials for the production of a given output, UAH; V - the volume of production of one kind, items.

k k f V ■ N

1000 ' '' (11) where S - the cost of I -th kind of materials, UAH; k - the number of kinds of materials; N - the consumption rate of I- th kind of material for 1000 units, units; P - the cost of i -th kind of material, UAH. Overhead costs are determined by the formula:

as the percentage of the basic salary, UAH; Zadmin - administrative costs, determined as a percentage of the basic salary, UAH; Zmarket - marketing costs, determined as a percentage of the basic salary, UAH.

Minimum wholesale price is set at the level:

Pip, =( + 0,3 • ZQverh )• 1,2, (13)

where Pp1 - minimum wholesale price of one unit, UAH.

It was noted earlier that the choice of pricing strategy depends on the strategic objectives of the enterprise and the fact, what management's expectations on the level of demand will be. This situation can be represented as a statistical game where players on the one hand are the company's management, and on the other hand - the level of demand (the planned volume of sales) for the company's products.

The level of demand is characterized as possible states of the economic environment. The strategies are considered as the possibility of setting the price at the level of 20%, 0% and 40%, respectively. Then, it is advisable to use the gross profit from sales of products as the elements of the payment matrix, the matrix of the game, taking into account the risk of receiving less profit and the storage costs of the product, provided that it will not be marketed during the year.

The elements of the payment matrix atJ will be determined by the formula:

_ Pj - Zj _ dj ■ Z P(1) • V - ( r + Z—h ) • (1 - dj ) • Z PP(1) • V,

(14)

where P - the gross income from sales, UAH; Z - the expenditures, related with the risk of receiving less profit, and the storage costs of the unsold products, UAH; dj - demand for products with j - th state of the economic

environment, the share; P(1) - The price of unit of products of l - th kind and i - th strategy, UAH; Vl - planned annual sales volume of products of l-th kind, units.; L - the total number of kinds of products of enterprise; Ri -the risk of receiving less profit, share; Z _h - the share of the storage costs of unsold goods during the year, share.

The storage costs of the unsold products are assumed to be 27% of the total cost of unsold products.

Under the risk of receiving less profit we mean insurance of working capital in order to ensure the efficient organization of the production process.

Initial data for the determination of the payment matrix are represented in the table 1.

The criteria for selection of the optimal pricing strategy of the enterprise are:

- Bayes criterion, providing an alternative choice, taking into account the unequal probability of

economic environment states and expressing a choice of optimistic point of view;

- Laplace criterion that characterizes alternatives from the standpoint of average gain;

- Wald criterion, which gives an idea about the best alternative in extreme conditions.

Bayes criterion for a positive evaluation of the functional F = F+:

:B+ ( s,

P) _ max B+ (s ; P)

S.ES

(15)

s

Table 1

Initial data

The volume of sell- The volume of sell- The volume of The volume

Strategy ing 100% ing 75% selling 50% of selling 20%

Z _ h z _ h z _ h R z _ h

Strategy 1 (profit 20%) 0,10 0,20 0,10 0,20 0,10 0,20 0,10 0,20

Strategy 2 (profit 0%) 0,05 0,20 0,05 0,20 0,05 0,20 0,05 0,20

Strategy 3 (profit 40%) 0,25 0,25 0,25 0,25 0,25 0,25 0,25 0,25

Probability of demand (variant 1) 0,10 0,45 0,35 0,10

Probability of demand (variant 2) 0,25 0,25 0,25 0,25

where B+ (s;P) = ^pa - Bayesian estimation of so-

j=1

Pj « pj =~, j =1n. n

(16)

lutions (strategies) s (i _ 1. m) ;

p - probability of j - th state of the economic

Wald criterion defines the optimal strategy in the worst conditions:

s : a = max a = max min a..

•0 •) • s ^ S Ñ tj

environment; ^ (17)

s = (s1,•••, sm) - a set of possible strategies. where 0j <e0( j = 1~n) - a set of states of the economic

Laplace criterion differs from the Bayes criterion environment

in probability vector only: Payment matrix and results of selection of alterna-

tives are shown in table 2.

Table 2

Selection of the optimal pricing strategy

The volume of selling 100% The volume of selling 75% The volume of selling 50% The volume of selling 20% Average income (B) Average income (L) Minimum income (B)

Strategy 1 (profit 20%) 399439,26 269621,50 139803,74 -15977,57 208607,15 198221,73 -15977,57

Strategy 2 (profit 0%) 335458,39 230627,64 125796,90 0,00 181357,19 172970,73 0,00

Strategy 3 (profit 40%) 469641,75 293526,09 117410,44 -93928,35 210751,73 196662,48 -93928,35

The probabilities of the states of the economic environment (Bayes criterion) 0,1 0,45 0,35 0,10

The probabilities of the states of the economic environment (Laplace criterion) 0,25 0,25 0,25 0,25

According to the criterion of Bayes the optimal strategy of pricing is that one, that takes into account 40% of the profit. In monetary terms the profit will be:

s. :max(208607,15; 181357,19; 210751,73) _ 210751,73 UAH.

By the criterion of Laplace the optimal strategy of pricing is the strategy, that takes into account 20% of the profit. In monetary terms the profit makes up:

s. :max(198221,73; 172970,73; 196662,48)_ 198221,73. UAH

By the criterion of Wald the optimal strategy of pricing is the strategy, that takes into account 0% of the profit. In monetary terms the profit makes up:

s :maxmin(-15977,57;

h s,es ejSo

Conclusions. Selection of the optimal strategy depends on the projected volume of selling for the next period and the company's management attitude to the economic risk.

The proposed mathematical model of pricing based on different rates of profit and seasonality described by

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EKOHOMÍHHHH BÍCHHK ^OHÔacy № 4(46), 2016

0; - 93928,35) = 0 UAH.

varying the volume of sales, in different conditions of the economic environment, will make it possible to design different pricing scenarios, allowing to increase the efficiency of the price policy of the enterprise.

References

1. Падерш 1.Д. Методи цiноутворення на про-мисловому пiдприeмствi в умовах перехвдно! еко-HOMiKH / 1.Д. Падерiн // Економiка промисловостi. -2005. - № 3. - С. 127-135. 2. Крамаренко Г.Ю. Цшоутворення в умовах ринкових реформ / Г.Ю. Крамаренко // Фшанси Укра!ни. - 2005. - № 4. -С. 54-60. 3. Янчук В.1. Основи статистичного моделювання формування цiни екологiчно чисто! проду-кцп / В.1. Янчук, Г.В. Янчук, О.В. Янчук. // Збiрник наукових праць ВНАУ. Серiя: Економiчнi науки. -2011. - №1 (48). - С. 78-85. 4. Шкварчук Л.О. Щни i цiноутворення: Навч. посiбник. / Л.О. Шкварчук. -3-е вид., виправл. - К.: Кондор, 2005. - 214 с. 5. Вгг-лшський В. В. Моделювання економiки: навч. по-сiбник / В. В. Вгглшський. - К.: КНЕУ, 2003. -408 с.

Шевченко Н. Ю., Гудкова К. Ю. Викорис-тання математичних методiв прийняття р1шемь при формуванш оптимально"! щновоТ стратеги пiдприeмства

Вiдмiчено, що розвиток, модифiкацiя i адапта-цiя методiв i моделей цшоутворення ввдбуваеться з урахуванням умов, в яких формуеться цiна. Зазна-чено, що мiнливiсть зовнiшнiх умов функщону-вання пiдприемств (полiтичних i економiчних) ви-значае актуальнiсть питання вибору оптимально! щ-ново! стратегi! в умовах невизначеност i ризику. Поставлено завдання розробки моделi формування оптимально! щново! стратегi! виробничого тдпри-емства в умовах невизначеностi i ризику iз сформовано! множини стратегш. Наголошено, що вибiр пiдприемства залежатиме вiд прогнозованого об'ему реалiзацi! на майбутнш перiод i вiдношення керiв-ництва пiдприемства до економiчного ризику. Вка-зана ситуацiя представлена у виглядi статистично! гри, де гравцями виступають з одного боку керiвни-цтво пiдприемства, а з шшого - рiвень попиту (пла-нований об'ем реалiзацi!) на продукцiю тдприемс-тва. Вибiр оптимально! стратегi! пропонуеться здш-снювати за допомогою критерпв прийняття рiшень: Байеса, Вальда i Лапласа.

Ключовi слова: цiнова стратегия, моделювання, ризик, прибуток, статистична гра, критерп прийняття рiшень.

Шевченко Н. Ю., Гудкова К. Ю. Использование математических методов принятия решений при формировании оптимальной ценовой стратегии предприятия

Отмечено, что развитие, модификация и адаптация методов и моделей ценообразования происходит с учетом условий, в которых формируется цена. Указано, что изменчивость внешних условий функционирования предприятий (политических и экономических) определяет актуальность вопроса выбора оптимальной ценовой стратегии в условиях неопределенности и риска. Поставлена задача разработки модели формирования оптимальной ценовой стратегии производственного предприятия в условиях неопределенности и риска из сформированного множества стратегий. Отмечено, что выбор предприятия будет зависеть от прогнозируемого объема реализации на будущий период и от отношения руководства предприятия к экономическому риску. Указанная ситуация представлена в виде статистической игры, где игроками выступают с одной стороны руководство предприятия, а с другой - уровень спроса (планируемый объем реализации) на продукцию предприятия. Выбор оптимальной стратегии предлагается осуществлять с помощью критериев принятия решений: Байеса, Вальда и Лапласа.

Ключевые слова: ценовая стратегия, моделирование, риск, прибыль, статистическая игра, критерии принятия решений.

Shevchenko N., Gudkova E. The use of mathematical methods for decision making in optimal pricing strategy for enterprises

It is noted that the development, modification and adaptation of methods and pricing models takes place according to the conditions in which the price is formed. It is indicated that the variability of the external conditions of functioning of enterprises (political and economic) determines the relevance of the issue of choosing the optimal pricing strategy under uncertainty and risk. The task of developing a model of the optimal pricing strategy of industrial enterprise in the conditions of uncertainty and risk of the generated sets of strategies. It is indicated that the choice of the enterprise will depend on the projected sales volume for the next period and the ratio of the company's management to the economic risk. This situation is presented in the form of a statistical game where players act on the one hand the company's management, and on the other - the level of demand (the planned volume of sales) in the company's products. Choosing the optimal strategy is proposed to carry out with the help of decision criteria: Bayes, Wald and Laplace.

Keywords: pricing strategy, modeling, risk, profit, statistical game, the criteria for decision-making.

Received by the editors: 07.12.2016

and final form 28.12.2016

Економiчний вюник Донбасу № 4(46), 2016