CC BY
30Tengiz Magradze
Doctor of Philosophy in energy and electrical engineeering, Georgian technical University, Tbilisi, Georgia DOI: 10.24411/2520-6990-2020-11511 MATHEMATICAL MODELING IN THE ENTERPRISE MANAGEMENT
Abstract
In the conditions of market relations, company success functioning and competitiveness can be provided to the enterprise only by effective, scientifically based management system for its production activities. Today, the company by own should to determine and predict the external environment preferences, range of products and services, prices, suppliers, sales markets and much more, also to be able quickly, and most importantly, correctly respond to any changes in the external and internal environment, and in accordance with them to adjust its activity. And it means that the company management should always look for new original moves in the management.
In the article it is described modern mathematical models in enterprise management with some examples and relevant conclusions were done.
Introduction
In the conditions of market relations, company success functioning and competitiveness can be provided to the enterprise only by effective, scientifically based management system for its production activities. Today, the company by own should to determine and predict the external environment preferences, range of products and services, prices, suppliers, sales markets and much more, also to be able quickly, and most importantly, correctly respond to any changes in the external and internal environment, and in accordance with them to adjust its activity. And it means that the company management should always look for new original moves in the management.
The management as a modern enterprise management system involves creation of conditions which are necessary for its effective functioning and development. It is about such management organization that is generated by the objective necessity and market relations laws of the economy. The aspect of modern management is focus on rational organization of the enterprise management.
The requirement for modeling in management is due to a number of reasons: complexity of many
organization structures, the inability of experiments conduct in real life and future-oriented.
I. Basic concepts of modeling theory in management
Model in general sense (the generalized model) is a specific object, created for purpose of obtaining or storing information (in form of mental image, description by symbolic means or material system), that reflects properties, characteristics and relations of the original object for arbitrary nature which are essential for problem being solved by the subject.
For decision theory, models, expressed in words or formulas, algorithms, and other math tools, are most useful.
Modeling, as you know, is able to replace experiment in the economy system.
It is the reason of widespread modeling use in the economy, turning it into one of main directions in the increasing management efficiency. It should be noted that more modern the enterprise management system (ACS TP, ICS) - than less discreteness, more reliable the model can be considered continuous.
Model (from lat. modulus) is a copy or analog of the studied process or phenomenon that reflects the essential properties of given process or phenomenon from perspective of study objectives.
Figure1. Model building algorithm
At formation of mathematical model, knowledge about an object is transferred to model. Then mathematical model of the object is under formation, solving by mathematical means. After that, new information about model, transferred to the object, can
be obtained. And, ultimately, new information about the object is verified. The algorithm for mathematical model formation is presented in Figure 1.
Modeling in relation to management involves the study of processes and phenomena, regarding which
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managerial decisions are made by formation and studying its models.
The requirement for modeling is due to a number of reasons: complexity of many organizational structures, the inability to conduct experiments in real life, and future-oriented.
The inability of conducting experiments in real life also requires the modeling use, since before investing into the production of new products, it is necessary to prove the possibility of releasing given products, possible demand etc. For instance, before choosing a place for the construction of new automobile plant, it should be taking into account possible availability of labor, linkages with related businesses, finished products transportation etc. It would be absurd to solve these problems empirically by plant construction at every possible place.
The orientation on future is complicated by fact that the future cannot be observed, but it can be modeled in various ways and considered alternative solutions. In the management it is customary to distinguish physical ("portrait"), analog and mathematical models.
At the organization level, modeling has certain characteristics. In particular, the rational models can be distinguished such as choice of alternatives with maximum benefit for the organization; models of the organizational and limited rationality, when the head manager is limited in decision-making by certain capabilities of the organization: resources, time, capabilities of performers, etc.; models of personality-limited rationality, when the manager's personal concerns or doubts become restrictions in decision making.
The mathematical modeling of the economic phenomena and processes in order to optimize management processes is an area of scientific and practical activities that has received powerful incentive to develop during and immediately after the Second World War.
The important issue is the accounting for uncertainty. It occupies the main place in probabilistic and statistical models of the economic and socioeconomic phenomena and processes. The problems of stability (tolerances of the source data and model preconditions) for the socio-economic models are considered in the literature.
The economic and mathematical management methods can be divided into several groups:
- Optimization methods,
- Methods that take into account uncertainty, primarily probabilistic and statistical,
- Methods for design and analysis of simulation models,
- Conflict analysis methods (game theory).
In all given groups, static and dynamic performances can be identified. In presence of time factor, different equations and methods are used. The mathematical modeling is applied to managerial decisions adoption and it allows us to describe the object under study by the mathematical means, i.e. to form a mathematical model of this object, to calculate
given model on computer and choose the optimal solution.
We discuss necessity to take into account the effect of loyalty in organization managing for modern conditions. The loyalty is understood as honest, conscientious attitude to something or to someone. The management database, formed on loyalty, was laid down by Harvard professor Joshua Royce in 1908. He is the author of "Philosophy of Loyalty" book, where the concept of "loyalty" was first scientifically defined.
Under proposed verbal model, the business loyalty is considered from point of view for three independent basic aspects: customer loyalty, employee loyalty and investor loyalty. Every time, the word "loyalty" means something different:
- commitment (from the customers view point),
- diligence (from the employees view point),
- mutual trust, respect and support (from the investors view point).
Henry Ford said that "an organization cannot work without profit ... otherwise it will die. However, the organization formation just for sake of profit...
means leading it to certain death, since it will not have incentive to exist."
The basis of loyalty model under consideration is not profit, but the attraction of additional amount of customers, a process that consciously or unconsciously underlies most successful organizations. The target number of customer's formation permeates all sectors of company's business. The forces that drive relations between customers, employees and investors are called loyalty forces. The success criterion is whether buyers come back to buy more or do they go somewhere else, i.e. if they show loyalty.
As specific model of control process, we consider model of time distribution between knowledge acquisition and development of skills. All knowledge consists partly of "information" ("pure knowledge") and partly of "skill" ("know how"). Skill is mastery as the ability to use given information you have for your goals achievement; the skill can also be characterized as combination of certain proficiency; ultimately, the skill is the ability to work methodically.
Let x(t) is the information amount, accumulated by the student by time t ("pure knowledge"), y(t) is cumulative skills amount: as ability to reason, solve problems and understand the presented material by teacher; u(t) is time proportion, allotted for knowledge accumulation in time interval (t; t + dt). It is natural to assume that increasing x(t + dt) - x(t) in student's knowledge is proportional to spent time u(t)dt and the accumulated skills y(t). Therefore,
dx(t)
dt
= k1u(t)y(t)
(1)
where the coefficient h > 0 depends on the individual characteristics of student. The increase in knowledge over the same time is proportional to spent time (1 - u(t))dt, the available skills y(t) and knowledge x(t). Therefore,
dy(t) dt
= k2(l-u(t))x(t)y(t) (2) II. Design-practical part with the economic prediction examples
Example 1. Let's suppose that according to the management of company (enterprise), the sales volume of its products is closely related to the national income of the country. Then for sales prediction it is very useful to have a forecast of continuous path of change in national income, although this variable is usually measured only once a year. The continuous model allows the obtaining such a prediction from discrete observations of the economic variables over the past period of time.
Example 2. Let it be required to determine what amount should be put into the bank in given interest rate (20% per annum) in order to receive $ 12,000 in a year?
Introducing formal notation for the quantities appearing in the problem:
the initial money amount is - Mo, the final amount of money is - m¡, the interest rate - R and writing down ratio between them
Mi=M°[l+iy (3)
Find the required value from the solution of main equation for the model
Mi _ $ 12 000
Mn =■
1,2
= $ 10 000 (4)
influence for each factor on y. In this case, the price of a new car is 28,000 credit units for x1=0, x2=0. Despite the fact that price for this car after 3 years of exploitation and mileage in 50 thousand km already will be 10,000 credit units.
Example 4. Let determine what was previous output of the plant if, as result of technical re-equipment, the average labor productivity increased by 20% and the plant began to produce 12,000 credit units.
Introducing formal notation for the quantities appearing in the problem: the initial release - Qo the final release - Qi
the percentage of performance improvement - R Writing down the ratio between them (following from definition of average labor productivity Q/L)
^ot^+^H^+iy (6)
' L0 L - L0 J y - 100y
let's find the desired value from the solution of main equation of the model
<2i
Ço =
1+T
12000 1,2
= 10000
(7)
The econometric model is partition of the explained (dependent) variable into the explained (regression) and random, and the estimation of parameters distribution of random component.
In econometric model formation, it is necessary to select factors that significantly affect the dependent variable; and to choose mathematical function that describes relation between factor and the resulting variable. As such functions, one can choose linear, square, logarithmic function of one variable. Statistically significant model is widely used in result predicting.
Example 3. Let the price Y for car is function of the variables x1 h x2:
y=28000-1000x1-0,3x2, where y - is the expected price for car (veiling monetary unit) (5)
x1 - vehicle's exploitation (in years); x2 -mileage (in thousand km).
The formed mathematical model and its meaningful interpretation make it possible to identify the process of pricing a car and determine degree of
Comparing the obtained models and results, we can notice that mathematical form of the model
*i=*o[l+iy (8)
Even the numerical values of the quantities, included in it, are the same in both cases, however, the economic situation, described by the model, depending on management decisions; the economic results are completely different. Thus, the same mathematical models and methods can be used by the enterprise management to solve completely different economic problems.
Example 5. The profit realization is one of main objectives of the entrepreneurial and commercial activities. The profit is starting point for price determining of the company. The company's price is a comprehensive indicator that characterizes profitability, prospects and company position in the market. The company's price is calculated by the formula:
PC = P / R x 100 - W, (9)
where PC - company's price; P - annual profit; R -is average rate of loan interest; W - balance sheet valuation of the firm (company).
The initial data:
Table 1.
R
R
100
Company (firm) Balance sheet valuation Annual profit Average rate of interest Company's price
1 8 000 5 000 10 42 000
2 10 000 2 000 10 10 000
3 4 000 1 000 10 6 000
Example 6. The economic analysis is also often interpreted as modeling method. This method includes all methods for assessing costs and the economic benefits, as well as relative profitability of the enterprises. The most common economic model is based on breakeven analysis, i.e. the point at which total revenue is equated with total costs. From this point, company becomes profitable.
The break-even analysis availably is presented in M. X. Mescon, M. Alberta, F. Hedouri "Fundamentals of Management" book. The American scientists in particular write that the break-even point (BEP) refers
a situation, in which total income (torevenue - TR) becomes equal to total costs (TC). For the break-even point determination, three main factors should be noticed: selling price of production unit (unit-price - P), variable costs per unit of production (VC) and total fixed costs per unit of production (TEC). Moreover, total fixed costs per unit of production can be represented as product of the breakeven point by difference between price and variable costs per unit of production in form of the formula:
TFC = BEP x (p - VC), (10)
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where TFC - total fixed costs per unit of output; BEP - breakeven point; P is selling price of a unit in production; VC - variable costs per unit of output.
By transforming this formula, you can get breakeven model (BEP), it should be presented in form of the following equation:
BEP = TFC / (P - VC). (11)
We illustrate this model with approximate calculation.
The initial conditions:
The unit price of conditional product (P) - 200 $.
Variable costs (VC) - 80 $.
Total fixed costs (TEC) - 3 500 000 $.
Determination of breakeven point (BEP)
Decision:
BEP= TFC / ( P - VC);
BEP = 3 500 000 / (200-80) = 42 000 equivalent unit product. (12)
Response: after the sale of 42 thousand of equivalent unit product, this product will begin to bring profit to manufacturers.
The break-even point determination is relatively simple model, but it provides significant amount of useful information. By comparing the break-even point and the sales estimate, you can immediately determine whether the project will be profitable and what is risk proportion. Thus, in our example, if 70 thousand units of equivalent unit product are sold, then profit will be made, and if it is supposed to sell only 40 thousand units, then it is better to abandon this project, since it will bring losses.
Conclusions
1. The complexity of many organization structures necessitates its mathematical modeling and requires simplification of reality using models, thereby increasing the ability of person to make the right decisions.
2. Under mathematical model formation, knowledge about object is transferred to the model. Then mathematical model of the object is built, which is solved by mathematical means. After that, new information about the model that can be transferred to the object can be obtained. And, ultimately, new information about the object is verified.
3. The mathematical modeling in relation to the adoption of managerial decisions allows us to describe the object under study by mathematical means, i.e. form mathematical model of this object, calculate given model on computer and choose the optimal solution.
Reference
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