=Paper=
{{Paper
|id=Vol-2093/paper1
|storemode=property
|title=Mathematical models for decision-making on strategic management of industrial enterprise in conditions of instability
|pdfUrl=https://ceur-ws.org/Vol-2093/paper1.pdf
|volume=Vol-2093
|authors=Loginovskiy Oleg,Dranko Oleg,Hollay Alexander
}}
==Mathematical models for decision-making on strategic management of industrial enterprise in conditions of instability==
https://ceur-ws.org/Vol-2093/paper1.pdf
MATHEMATICAL MODELS FOR DECISION-
MAKING ON STRATEGIC MANAGEMENT OF
INDUSTRIAL ENTERPRISE IN CONDITIONS OF
INSTABILITY
Loginovskiy O.V.1[0000-0003-3582-2795], Dranko O.I.2[0000-0002-4664-1335],
Hollay A.V.1[0000-0002-5070-6779]
1
South Ural State University, Chelyabinsk, Russia
2
Moscow Institute of Physics and Technology, Moscow, Russia
loginovskiyo@mail.ru
Abstract. The management of an industrial enterprise is complicated by the
high degree of instability in the world economic system at present. Therefore, it
is necessary to develop new methods and approaches to making strategic deci-
sions that allow for effective management in industry.
In this article, we propose two mathematical models of decision-making for the
strategic management of an industrial enterprise, which take into account the
conditions of instability. The first model is based on the ranking of decision cri-
teria, taking into account resource constraints. The second model is a mathemat-
ical model of an integrated assessment of the activity of an industrial enterprise,
which is considered as a procedure for conducting expert modelling of hard-to-
formalize fragments of a description of a problem. The procedure is based on
standard reporting and the use of formal methods to streamline expert assess-
ments for constructing a mathematical model of multi-criteria choice by com-
puterizing a well-known convolution principle, adapted to the number and qual-
ifications of experts, the degree of homogeneity and non-statistical uncertainty
of expert estimates. Обе модели могут быть использованы при принятии
стратегических решений по управлению промышленным предприятием.
Keywords: math modeling, industry, making decisions, strategic management,
expert assessment; Integral estimation; huperfuzzy estimation; Fishburne scale.
1 MANAGEMENT OF INDUSTRIAL ENTERPRISE IN
CONDITIONS OF INSTABILITY
Problems related to the management of industry and the economy have changed sig-
nificantly at present [38,39]. This requires their rethinking, in the new realities of the
modern world, and, most importantly, the search for new ways, methods, models and
management technologies [17,23,28] that will ensure the effective operation of large
industrial enterprises and corporations [31,32].
2
In recent years, the world economy has become instable [26]. The reason for this
became interethnic and interreligious conflicts, local wars, various kinds of sanctions
[37,9], etc. As a consequence, the conditions for doing business have deteriorated
significantly [11,21,24].
Studying scientific works devoted to the improvement of the management process-
es of industrial enterprises [47,20,8,34,36,16], it can be stated that the vast majority of
the conclusions, recommendations and proposals contained in these scientific papers
are applicable in the conditions of stable development of the world economy.
However, in conditions of global instability [22], these approaches and methods
become insufficiently effective [40] and do not allow to formulate strategic decisions
for managing the development of industrial enterprises that are adequate to the cir-
cumstances [35]. As known [4,5,46,44], among the most important problems of in-
dustrial enterprises management, the following are traditionally distinguished:
1. Formation of a rational organizational structure of industrial enterprises and
corporations [6].
2. Creation of an effective system of labor resources management, recruitment and
training of managerial and production staff [7].
3. The organization of material flows at the enterprise on the basis of the logistic
approach (optimization of cargo transportation, raw materials stock and commodity
stocks, sales of finished products on the basis of marketing analysis, etc.) [43].
4. Support the management system of an industrial enterprise on the basis of vari-
ous automated control systems for technological processes, transportation, accounting
and control of material resources, equipment maintenance and repair, etc. [25].
However, it is important to realize that in today's conditions of global military and
political, financial and economic, social instability in the world, as well as sharp drop
in incomes of manufacturing companies in international markets and low customer
purchasing power, industrial enterprises and corporations [27] can no longer receive
the same income as in previous years from the sale of their products [18]. Survival
considerations come to the forefront [1]. In this regard, the main shareholders of large
industrial enterprises and corporations are forced to look for ways to ensure profitabil-
ity and maintain the status of their companies in difficult business conditions [2,10].
As a result of all this, the emphasis in the management of industrial enterprises
shifts towards financial and economic analysis [3, 12, 13], operational and strategic
forecasting of companies' position [41]. Enterprises are looking for ways to reduce
spending by simplifying the organizational structure, reducing management staff, both
in production and other divisions [45,29].
Among the main problems can be identified:
─ Lack of investment.
─ Lack of working capital.
─ Depreciation of fixed capital, a large proportion of old equipment.
─ Insufficient introduction of new technologies.
─ High expenses.
─ Low turnover.
─ High tariffs, low solvency of customers.
3
─ Low qualification of staff, aging of staff.
─ Ineffective management system.
There are many indicators characterizing the industrial activity of the enterprise. A
set of indicators of the financial condition of the enterprise (see Table 1).
Table 1. Set of indicators of a financial condition of the enterprise.
Property Liquidity The Financial Business Evalua- Profitability
Valuation Assessment Unit Stability Assess- tion Unit Assessment
Unit ment Unit Unit
a) the share of a) maneuverability of a) coefficient of a) the turnover of a) product
fixed assets in own circulating assets concentration of funds in the profitability
assets b) total liquidity ratio equity capital calculations (in b)
b) the share c) current ratio b) coefficient of terms of turnover) profitability
of the active d) critical liquidity ratio maneuverability b) inventory of core
part of fixed e) coefficient of absolute of equity capital turnover (in activities
assets liquidity (solvency) c) structure turnover) c) return on
c) coefficient f) the share of current coefficient of c)turnover of total capital
of assets in assets long-term accounts payable d) return on
depreciation g) the share of own investments (in days) equity
of fixed assets circulating assets in d) coefficient of d) duration of the e)
d) coefficient current assets long-term operating cycle profitability
of h) the share of own borrowing (ratio e) duration of the of current
depreciation circulating assets in their "long-term financial cycle assets
of the active total amount borrowed capital - f) the rate of
part of fixed i) the share of permanent repayment of
assets inventories in current capital") receivables
e) coefficient assets e) debt capital g) turnover of
of renewal of j) share of cash and cash structure ratio own capital
fixed assets equivalents in current f) loan to equity h) turnover of
f) the asset assets ratio total capital
retirement k) share of own g) the ratio of own i) the coefficient
ratio circulating assets in funds of stability of
coverage of inventories h) I degree cover economic growth
l) stock coverage ratio i) II degree cover
m) a parity «a debt
receivable - accounts
payable»
In this article, two mathematical models for decision-making on the strategic man-
agement of an industrial enterprise in conditions of instability are proposed:
1. Model based on the ranking of decision criteria;
2. Model based on the construction of an integral indicator.
Both models can be used for making strategic decisions on the management of an
industrial enterprise.
2 MODEL BASED ON THE RANKING CRITERIAS FOR
DECISION-MAKING WITH RESTRICTIONS
The task of maximizing the target criterion can be written in the form [42]:
4
G mi xi max, (1)
i
where G - the target criterion, m - the contribution to the achievement of the target
criterion, x - scalability of the project, i – project index.
The main question: how to take into account the limitations on various resources.
The problem of linear programming is widely known:
aij xi B j .
j
(2)
where x - scalability of the project in continuous form, j - resource index, aij - the
rate of consumption of the j-th resource on the i-th product, B j - availability of j-th
resource.
In practice, the application of the linear programming is difficult due to the need to
accurately calculate the specific consumption rates of all resources The solution can
be greatly simplified if the most important restriction can be determined.
The method of one-source, one-resource optimization "Cost-effectiveness" [43] en-
sures the selection of priority directions according to the criterion:
i mi / aik max, (3)
where k - the number of the scarce resource.
If the limitation is financial resources, the criterion for selecting priority measure
will be:
i ф mi / Ii max, (4)
ф
where I - the amount of investment in the measure. Note that i is similar to the
profitability index that is used to evaluate investment projects [15].
If the limitation is human resources, the criterion for choosing priority measure
will be::
i L mi / Li max, (5)
L
where L - the amount of labor (staff) resources in the measure. Note that i is a
characteristic of labor productivity in achieving the target criterion.
3 THE MATHEMATICAL MODEL OF INTEGRATED
EVALUATION OF ACTIVITY OF INDUSTRIAL
ENTERPRISE
The comprehensive automated information system of an industrial enterprise that
ensures the unification of all information systems within the company as a whole,
without fail, should include an information system for an integrated assessment of the
activities of this enterprise [28].
The mathematical model of an integrated assessment of the activity of an industrial
enterprise can be based on such an information system.
5
An integral evaluation is the calculation as a single indicator, which unambiguous-
ly reflects the generalized, total financial and economic state of the organization at a
given moment in time. Comparing its value for any period (five years, a year or a
quarter), you can see how the state of the enterprise changes. And having analysed the
appropriate dynamics, it is possible to assess the work of the enterprise for the rele-
vant period and on this basis to formulate proposals for improving the management of
the financial and economic activities of an industrial enterprise.
To a large extent, such assessments can be carried out by ranking enterprises ac-
cording to known international methods (for example, by investment attractiveness,
solvency, creditworthiness) adapted to the peculiarities of the national economy and
the goals of stakeholders (investors, shareholders, creditors).
These methods suggest the calculation of some aggregated indicator (for example,
the Altman Z-indicator), comparing it with similar indicators of other enterprises and
then comparing them with the table values for their joint interpretation (for example,
referring to a group of financially stable enterprises for Z> 2.99).
The process of assessing the financial and economic state of an enterprise is con-
sidered as a procedure for conducting expert modelling of hard-to- formalize frag-
ments of the description of a problem situation based on standard reporting data and
applying formal methods for ordering expert assessments for constructing a mathe-
matical model of multi-criteria selection by computerizing a known convolution prin-
ciple adapted to the number and the quality of experts, degree of homogeneity and
non-statistical uncertainty of expert assessments.
3.1 Selecting metrics
The choice of indicators (Table 1) by which the integral assessment of the financial
condition of an enterprise will be calculated depending on the goals of the rating. By
their semantic purpose, the indicators are divided into several groups that will deter-
mine the structure of the aggregated estimate. When selecting indicators, the neces-
sary condition is not to use interdependent indicators. If this condition is not satisfied,
the construction of the rating by the rule of additive convolution will give an incorrect
result.
3.2 Calculation and normalisation of indicators
The procedure for expert evaluation of the financial and economic state of the enter-
prise is multi-level and involves several stages: the formulation of the decision-maker
(DM), the objectives of the expert survey; DM’s selection of the expert working
group (EWG); EWG’s the development of a detailed scenario for collecting and ana-
lyzing expert opinions (assessments), a specific type of expert information (words,
conditional grades, numbers, rankings, breakdowns or other types of objects of non-
numerical nature), the way of its formalization and methods of its analysis; selection
of experts in accordance with their competence and the formation of an expert com-
mission (EC); collection of expert information; processing of the results of the exami-
nation, including determining the consistency of expert opinions and determining the
6
maximum allowable, minimum acceptable and optimal values of financial indicators
of the enterprise; summarizing and interpreting the results obtained and preparing an
official conclusion for the decision maker.
Solving the problems of modelling and optimization is always connected with the
presence of uncertainties. Expert assessments of different specialists can vary signifi-
cantly depending on their experience, qualifications and intuition. A certain objectifi-
cation of the process of forming the desirability function can be achieved in various
ways. One of the most common is the method of aggregating the views of a group of
experts. In this technique, consideration of non-statistical uncertainty is proposed to
be performed on the basis of the apparatus of the theory of fuzzy sets. When assessing
indicators, experts set the lower ones - "pessimistic assessments", the upper ones -
"optimistic estimates" and the intervals of the most expected (possible) values of the
investigated parameters. Then, to perform operations related to the determination of
the generalized opinion of experts, procedures are used to construct particular quality
criteria based on hyperfuzzy functions.
Hyper-fuzzy sets are called fuzzy sets characterized by functions of trapezoidal
form (fuzzy intervals), the support points of which in turn are themselves indistinct
intervals of the trapezoidal shape.
Consider the situation when experts are asked to quantify the values of the refer-
ence points of the trapezoidal desirability function. It is clear that in the general case,
for each of the reference points, the experts will give differing estimates. The simplest
way to construct on their basis the function of desirability is to average the opinions
of experts. However, a significant part of the information is lost. For its preservation
and use on the basis of a set of expert estimates, we construct the membership func-
tions for each of the reference points.
Further, on the basis of the membership functions of the obtained fuzzy intervals
describing [48,19] the reference points, we construct the desired desirability function
for the quality criterion. Most often, by an indistinct interval we mean a trapezoidal
form of fuzzy value, and by an indistinct number - a triangular shape.
Fig. 2 graphically illustrates the structure of a hyper-fuzzy number on a plane. The
darker areas correspond to the greatest unanimity among experts regarding the value
of the reference points, the lighter ones to the scatter in their representations. The
most desirable value of the quality index corresponds to the maximum value of the
desirability function equal to 1, the least desirable value is the minimum value equal
to 0.
To operate with hyper-fuzzy numbers (intervals), a technique has been developed.
Practice shows that trapezoidal forms are an ample level of abstraction for formal-
izing uncertainties in most real situations.
7
Fig. 1. Representation of a hyperfuzzy number on the plane
Let us further assume that there is some particular criterion described by the desir-
ability function represented by the hyperfuzzy number GX (Fig. 2). Let further
x* X GX - some non-fuzzy number, corresponding to certain specific value of the
analyzed indicator. Then, within the framework of the formulated definitions, the
value of the hyperfuzzy function (describing the hyperfuzzy number GX ) for the
fixed argument x* is the usual trapezoidal fuzzy number G x* :
g x , g x , g x , g x , x X
G x* 1
*
2
*
3
*
4
* *
GX (6)
The last statement for the left front of the hyperfuzzy interval is illustrated graph-
ically in Fig. 2., which clearly shows that the result is also a fuzzy interval
G g1 , g 2 , g3 , g 4 . The interpretation of the result should be as follows: the most
possible values f the evaluation of the quality criterion lie in the interval g2 , g3 , and
the entire range of possible values of the criterion evaluation is g1 , g 4 .
The result of the calculation of the values of the hyperfuzzy function G x* is de-
termined in the most general situation as follows:
8
g 4 x G11 G 21 G11
g3 x G12 G 22 G12
. (7)
g 2 x G13 G 23 G13
g1 x G14 G 24 G14
Fig. 2. Display a clear number x * on the left side of the trapezoidal hyper-fuzzy number
1. Determination of the weight coefficients of the indicators
Each indicator xi is compared to an estimate of its significance. The balance sys-
tem is designed in such a way that:
n
a 1.
i 1
i (8)
Where ai is the weight of the i-th indicator; n - number of indicators; i - the num-
ber of the current indicator.
To compose a system of weights, each expert ranks the indicators in a descending
order of importance:
x1 x2 ...xi ... xn , . (9)
where xi - the indicators of the state of the enterprise.
In this case, to determine the weights of the indicators, it is suggested to use the
Fishburn scale [14], which corresponds to the maximum entropy of the available in-
formation uncertainty about the values of ai :
9
2( n i 1)
ai , (10)
n(n 1)
where ai is the coefficient of significance of the i-th indicator; i - number of the cur-
rent indicator; n - number of indicators. If the system of preferences is absent, then the
indicators are equivalent:
1
ai . (11)
n
On the basis of individual rankings of experts it is necessary to construct a general-
ized ranking. This can be done by different methods. The most correct (but also the
most time consuming) method is the "Kemeni median" method. To find the median,
first of all, you need to specify how to determine the distance between the rankings,
i.e. "Define the metric in the ranking space". After that, you need to find (build) such
a ranking, the total distance from which to all the specified expert rankings would be
minimal:
d A , X min,
m
j j (12)
j 1
where Aj is the ranking of the j-th expert; X - Kemeni median; d j Aj , X - the
distance between the ranking of the j-th expert Aj and X ; m - number of experts; j -
number of the current expert.
The desired ranking will be the Kemeni median. We note that this way we obtain a
generalized opinion of experts without rejecting any opinion, since all individual
rankings are taken into account in the construction of the median.
2. Calculation of the integral estimate
Calculation of a multi-level integrated estimate of the financial state of enterprises
is proposed to be performed according to the following formula:
m n
m n
J x* k j G x* ai G x* ai k j , x X . (13)
j 1 i 1 j 1 i 1
Where J is the integral estimate; G x* - projection of a non-fuzzy number x X
(quality score) on a hyper-fuzzy number; ai is the specific weight of the i-th index in
the j-th group; k j - specific weight of the j-th group of indicators; i - number of the
current indicator in the j-th group; j - the number of the current group of indicators;
m - number of groups of indicators; n is the number of indicators in group j.
The result of calculating the integral estimate is also a fuzzy interval. It should be
noted that the maximum width of the resulting interval (the width of the base of the
trapezoid) is much larger than the width of any of the intervals characterizing the
initial data, i.e. the solution of the problem leads to an increase in uncertainty, howev-
er, the use of fuzzy intervals allows us to determine the possible limits of the sought
10
value and determine the most probable range of the value, which gives more real re-
sults than with traditional approaches. The theory of fuzzy sets provides for this task a
convenient mathematical apparatus that allows the most complete use of information
obtained from experts.
The application of the methodology of integrated assessment is not limited to the
area of study of the financial and economic state of the enterprise. It can be used both
to evaluate specific areas of the company's activities, and to assess the entire econom-
ic activity of the company as a whole.
4 CONCLUSION
The mathematical decision-making models proposed in the article can be used to
manage industrial enterprises in conditions of instability. The advantage of the model
based on the ranking of decision criteria is the identification of priority areas for im-
proving the company's operations. While the advantage of the model based on the
construction of an integral indicator is the ability to draw conclusions about the state
of the enterprise as a whole and to consider changing this state in time to detect
trends.
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