Elaboration of economic and mathematical model competitiveness management of a construction enterprise

In science and practice, various mathematical methods are used to solve diverse problems that cover analysis, planning and management in all sectors of the country's economy.

Management is a conscious influence of a person on objects, processes and on the people participating in them, which is carried out in order to give a certain focus to economic activity and obtain the desired results. Control methods are defined by the ways in which the subject of management affects the control object.

Management of the market economy requires the use of various economic and mathematical methods and models. The modeling method is the most important universal method of investigation. Applying it, one should remember the concept of “analogy”. The model, as a rule, differs from the object of research, but certainly has a similarity, similar to this object, primarily with respect to those characteristics that are subject to study and forecasting.

The model of a complex system is also a system (and often very complex) that has a physical embodiment, or written with mathematical notation, numbers, words, graphic images, and so on.

Thus, the model is a physical or sign system similar to the system under investigation with respect to the functional and structural characteristics that are the subject of the study.

There are various economic and mathematical models that are used in scientific research. The most common are the models of correlation-regression analysis, production functions and systems of econometric equations. As a rule, modern research in the field of economics and management is focused on the use of a single tool for data analysis. This article will consistently apply all of these models, which allows you to conduct the most complete and objective analysis of the mutual impact of production performance of a construction company.

Economic and mathematical model of competitiveness managementof a construction enterprise

With reference to the building enterprises we will use models of correlation-regression analysis, production functions and systems of econometric equations. The most promising economic-mathematical model from the point of view of further research is the system of recursive econometric equations.

In connection with the complexity and multifaceted relationships in the construction industry, the objects of analysis and management, as well as the specifics of the specific production structure or specific objectives and forms of research, we will represent the corresponding production functions in the form of a system of equations of the following kind [1]:

•, ^z1t t ) ;

•, z 2, t ^{) ;}

•’ z 3, t ^{) .}

y 1t = / 1 ( x 3tt , x 2, t - 1 , ‘ y 2, t = / 2 ⁽ x 1, t , x 3, t - 1 , , У 3, t = У i ( x 2, t , ^x 1, t - 1 ,

In this system there are three groups of variables:

• 1)    the endogenous variables x _{1, t} , x _{2, t} , x _{3, t} , for the determination of which a solution of the reduced system of equations is required;

• 2)    delayed (lagged) endogenous variables x 1_{t - 1}, x _{2 t - 1}, x _{3 t - 1}, for t -th period they are known and determined on the basis of statistical information or as a result of solving a similar system of equations for the ( t – 1)-th period;

• 3)    the exogenous variables z _{1, t} , z _{2, t} , z _{3, t} are determined outside of the given system of equations.

Variables of the second and third groups are united by the fact that their values are determined by external factors with respect to the system of equations; influencing the variables of the t -th period, they themselves are not subjected to their inverse influence. The variables of the second and third groups are predefined. The number of these variables in the equations can seriously affect the solution of the entire system of equations.

This type of system of equations is recursive [2, 3]. Equations in them are solved consistently, and not simultaneously. The first equation is solved x _{1, t} is calculated as a function of only predefined variables. Then from the second equation we define x _{2, t} as a function of predefined variables and already computed x _{1, t} . Further from the third equation, calculate x _{3, t} . In this case, the calculation of the first two equations is, in fact, the preparatory stage for the solution of the third equation, in which the variable x _{3, t} , is ultimately considered as a complex function of all other variables of the system.

As a basis for constructing an economic and mathematical model of competitiveness management of construction enterprises, it is proposed to use the following system of equations (formulas for calculating variables are given in Table 1) [4]:

У ц = ^a1 1 ^x 1, t + ^a 12 ^x 2, t - 1 + c 1 + - + a 1 m x m , t +- + S 1, t ;

“ У 2, t = b 21 У 1^ + a 21 ^x1,t - 1 + a 22 x 2, t + c 2 t + - + a 2 m x m , t +^ + £ 2, t";

_ У3, t = b31У1^ + b32 У 2t t + a31x3, t-1 + a32 x 2, t +c3 t + - + a 3 mxm, t +^ +S3, z, where y1,t is quality of production at t period;

y _{2, t} is timeliness of performance at t period;

y _{3, t} is product cost at t period;

x _{1, t} is profitability of production at t period;

x _{2, t} is a relative number of senior executives of the enterprise at t period;

x 2 t 1 is a relative number of senior executives of the enterprise at t - 1 period (lagged variable x 2 _t );

x _{3, t} is workforce productivity at t period;

x_{3 t} • is workforce productivity at t - 1 period t (lagged variable x _{3 t} );

x _{4, t} is mechanization level (a technological level of building machines and equipment) at t period;

x _{5, t} is discretization of use of resources at t period.

Thus, the work completion time depends on the required level of quality, and the cost depends both on quality and on the work completion time.

Table 1

Calculation formulas of variables

No.	Name of variable	Calculation formula	Determination of formula
1	Workforce productivity	w — = — N	W – scope of work in unit time; N – number of employees
2	Product profitability	P = Sp . 100% pr C	Sp – sales profit; C – full cost
3	Mechanization level (a technological level of building machines and equipment)	M _ Q mech N	Q mech – scope of work made by mechanical means; N – total number of employees at the enterprise
4	Relative number of senior executives of the enterprise	N R" = "N	N ex – number of senior executives of the enterprise; N – total number of employees at the enter-prise
5	Timeliness of performance	f         Tpli - Т fl TPerf pr = 1     T Т pli	Т pli , Т fi – planned and factual work completion time at an i object, days
6	Discretization of use of resources	Т dis D res    т	Тdis – time of discrete use of resources at an object; Т – total period of use of resources
7	Product cost	VS 1 Cpr V Pr r av	VS – volume of sales in monetary terms; V – total volume of work; Pr av – an average price of 1 sq.m. of apartment at local property market
8	Product quality	. UD + —R Q pr - 1 -     S	UD – cost of ultimately defective goods; WR – costs of removal faults and hidden defects on the products made earlier, i.e. costs of carrying out warranty repair

Proceeding from the constructed model and based on the Pareto [5] multicriteria optimization method, construction organizations with the best performance indicators are selected:

У 1,_t ^ max;

* y 2, t ^ max;                                                                                     ⁽³⁾

y_{3 t} ^ min.

Limitations for this system of equations are:

^x1t t >  0.1;

0 < x 2, t <  0.15;

= ⁰ <  x 3,;                                                                                                            (4)

⁰ < x 4, t ^ x-itt ;

0 < x ₅ , _t < 0.5.

In our opinion, this model allows the most objective and fully reflects the competitiveness of construction companies in current market conditions.

To solve the problem, we can use the algorithm for successive improvement of the plan (the method of successive concessions), at which the maximum is y _{1, t} ( y ₁^*) for the given constraints on у _{2, t} and у _{3, t} , then we find the maximum y _{2 t} ( y 2 ) for a given restriction to у _{3 1} and y 1_t >  y * - A 1 , and then we find the maximum y ₃ , _t ( y * ) for given constraints on y 1, _t >  y * - A 1 and y ₂ , _t >  y * - A ₂. Here A i is a concession determined by experts for each i -th indicator.

On the basis of the initial data obtained from construction enterprises, the coefficients of x _{i , t} were obtained by the methods of correlation regression analysis [6] and a system of equations was compiled:

' y 1,_t = 0.003 + 0.218 x 1, _t + 0.03 x 2,_t -0.0006 x ₃ , _t -2.919 x ₄ , _t + 1.434 x ₅ , _t ;

< y ₂ , _t = 0.009 + 0.561 x 1, _t + 0.076 x ₂ , _t - 1 -0.0015 x ₃ , _t - 1 -7.4 x ₄ , _t + 3.567 x ₅ , _t ;                            (5)

y ₃ , _t = -0.02 -1.24 x 1, _t -0.168 x ₂ , _t - 1 + 0.004 x ₃ , _t - 1 + 16.378 x ₄ , _t -7.936 x ₅ , _t .

To solve this system of equations, use the Microsoft Excel “Find Solution” tool [7]. At the first stage we optimize the first objective function - product quality:

y _{1 t} = 0.003 + 0.218 x_{1 t} + 0.03 x_{2 t} -0.0006 x_{3 t} -2.919 x_{4 t} + 1.434 x_{5 t} ^ max.                     (6)

As a result of the calculation, we get max y 1 _t = y *_t . In Fig. 1 shows the results of calculations in the first stage of optimization.

	A	В	c	D	E	F	G	H	1
1	x1	x2	x3	x4	x5
2	0,23715	0,14911]	0,29875	0,15578	0,45568
3	1	0	0	0	0	0,23715	>=	0,1
4	0	1	0	0	0	0,14911	>=	0
5	0	1	0	0	0	0,14911	<=	0,15
6	0	0	1	0	0	0,29875	>=	0
7	0	0	1	-1	0	0,14297	>=	0
8	0	0	0	0	1	0,45568	>=	0
9	0	0	0	0	1	0,45568	<=
10	Objecthre	function
11	0,218	0,03	-0,0006	-2,919	1,434	=	0,82832
12

Fig. 1. The first stage of optimization

Thus, the calculated maximum value of the quality level for given values of xi,t is maxy1t = y*t = 0.82832 ® 83 % .The closer this indicator is to the unit, the higher the level of product quality. Such a high value of the quality index speaks about the insignificant cost of the final marriage and the costs of eliminating imperfections, as well as the costs of guarantee repairs.

At the second stage, the second objective function is optimized – the timeliness of the work:

y ₂ t_t = 0.009 + 0.561 x_Vt + 0.076 x ₂ t_t - 1 -0.0015 x ₃ t_t - 1 -7.4 x ₄ , _t + 3.567 x ₅ , _t >  max.                   (7)

The result of the calculation is the definition of max y 2 _t = y 2 ₁ .

At the same time, the amount of assignment for y _{1, t} of the indicator – product quality, is: A 1 = 0.001. At this stage, the previous function can be degraded by no more than A 1 = 0.001. That is, one more restriction is introduced y 1_t < y *_t -A 1 ( y 1 ₁ <  0.82732).

As a result of the calculation, we get max y₂ t_t = y 2, _t . In Fig. 2 shows the results of calculations in the second stage of optimization.

	A	В	C	D	E	F	G		H	l
1	x1	x2		x4	x5
2	0,21897	0,13524	0,35687	0,11257	0,42114	0,79754	0,79754		<-	0,82732
							Г
3	1	0	0	0	0	0		0,21897	>—	0,1
4	0	1	0	0	0	0	F	0,13524	>-	0
5	0	1	0	0	0	0	F	0,13524	<-	0,15
6	0	0	1	0	0	0	F	0,35687	>-	0
7	0	0	1	-1	0	0	F	0,2443	>-	0
8	0	0	0	0	1	0	F	0,42114	>-	0
9	0	0	0	0	1	0	F	0,42114	<-	0,5
10	Objective	function
11	0,218	0,03	-0,0006	-2,919	1,434				0,79754
12	0,561	0,076	-0,0015	-7,4	3,567				'0,80177
13
14

Fig. 2. The second stage of optimization

That is, the maximum value of the level of timeliness of the work for given values of x_{i , t} is max y₂ t_t = y 2, _t = 0.80177 « 81 %. Ideal is the value equal to 1. It can be achieved when the planned and actual production times for the i -th facility are completely the same. In our case, the value of the timeliness of the work is at a sufficiently high level, taking into account the additional constraint ( y 1 _t <  0.82732), introduced after the first stage of optimization.

At the third stage, the third objective function is optimized: the cost of production:

y _{3 1} = -0.02 -1.24 x1₁ -0.168 x _{2 t} - 1 + 0.004 x _{3 1} - 1 + 16.378 x _{4 t} -7.936 x _{5 1} ^ min.                 (8)

As a result of the calculation, we get min y ₃ , _t = y *, _t .

At the same time, the amount of assignment for y _{2, t} of the indicator – timeliness of work performance, is: A ₂ = 0.01 At this stage, the previous function can be degraded by no more than A ₂ = 0.01. That is, another restriction is introduced y _{2 t} < y * _t -A 1 ( y _{2 1} <  0.79177).

Thus, the final calculation is as follows (Fig. 3).

	A	В	c	D	E	F	G	H	1		К
1	x1	x2	x3	x4	x5	y1
2	0,22471	0,14112	0,32214	0,13129	0,48213	1	0	0,54572	<=	0,79654
3						0	1	0,65857	<=	0,79177

4 5 6 7 8 9 10	1        0        0        0        0        0        0 0        1        0        0        0        0        0 0        1        0        0        0        0        0 0        0        1        0        0        0        0 0    0    1-1    0    0    0 0        0        0        0        1        0        0 0        0        0        0        1        0        0						0,22471 T 0,14112 T 0,14112 ' 0,32214 Г 0,19085 Г 0,48213 ' 0,48213	>= >=	0,1 0 0,15 0 0 0 0,5
11	Objective function
12	0,218      0,03	-0,0006    -2,919     1,434					=	0,54572
13	0,561    0,076	-0,0015      -7,4     3,567					=	0,65857
14	-1,24    -0,168	0,004    16,3 78    -7,9 3 6					=	0,97217
15

Fig. 3. The third stage of optimization

The final stage of the optimization process is the determination of the minimum value of the product cost index. In this study, the value of the product cost index was found to be min y₃ , _t = y * _t = 0.97217 ~ 97 % . This is the main indicator characterizing the competitiveness of the construction enterprise and its “viability”, “survival” in the market. If y * _t > 1, this means that the value of the enterprise's products is above the market average, hence, the enterprise's products are not competitive, and the enterprise needs to revise its pricing policy. On the other hand, if this indicator is too low, then there are risks of deterioration of the financial condition of the enterprise, which can lead to withdrawal from the market or even bankruptcy. In our case, the cost indicator is 97 %. This indicates the conformity of the cost of production to the current market conditions and at the same time the activity of the enterprise remains quite profitable.

Conclusion

Thus, an economic and mathematical model for managing the competitiveness of a construction enterprise has been developed. This model is based on three interrelated basic indicators of competitiveness (product quality, timeliness of work performance and product cost), combined in a system of recursive econometric equations. At the same time, the timeliness of performance depends on the required level of product quality, and the cost of production depends on the required level of product quality, and on the timeliness of the work.

Summarizing this study, it should be noted that the optimal indicators of production activity were obtained that allow construction companies to determine for themselves the most promising directions of development, to identify their strengths and weaknesses, to promote the development and adoption of sound strategically correct management decisions, which ultimately leads to increase competitiveness.