UDC 681.51.011

MATRIX ANALYSIS IN THE ENTERPRISE MANAGEMENT SYSTEM

© 2006 A.V. Volgin1, G.E. Belashevsky2

LLC "Samara - AviaGaz"

Samara State Aerospace University

The paper analyzes various ways of using matrices in enterprise management. The relationship (connection) between the elements of two or more sets can be represented in matrix form. The composition of relations allows you to simplify the analysis of relationships between elements of sets. An example of the use of priority matrices in the enterprise management system is given.

Matrices, as an analysis tool, have long been used in the enterprise management system. Suffice it to name such quality tools as matrix charts, priority matrices, matrix analysis in Quality Function Deployment.

1. The use of matrices in management is due to the fact that almost any enterprise is characterized by a large set of objects (various equipment, divisions, suppliers, consumers), and it is difficult to describe the relationships between them with dependencies like y \u003d f (x) . Real connections are multidimensional and implicit. Matrices, on the other hand, make it possible to identify such relationships in a fairly visual form and analyze them. In the task of forming the production structure of an enterprise, a matrix of relationships between groups of parts B = ], where ^ is the number of units, can be used.

general equipment used in the processing of the 1st and] -th parts, in marketing research the technical level matrix is ​​used u = \u^], where

and y - technical level of the 1st enterprise in the ] -th market and the price matrix.

From the standpoint of mathematics, the assignment of a matrix can be interpreted as a specification of a relationship (connection) between the objects of two sets. The matrix element in this case can mean both the connection of objects (such as "yes" or "no"), and the strength of the connection, expressed as a number. In the case of three or more sets, one can build multidimensional relations and, accordingly, multidimensional matrices. However, this approach loses clarity and ease of interpretation. The complexity of the analysis of multidimensional relations

ions can be overcome with the help of relationship composition.

2. Let's assume that the company has suppliers P1 P2, ... P5, which supply materials (parts, assemblies, components) Mі, M2, M3. From these materials, the enterprise manufactures products Ib I2, ... I, for customers (consumers) Zi, Z2, ... Z5. For these sets, you can compose matrices of connections. Let, for example, relationships be established between suppliers and the materials they supply (Table 1), products and necessary materials(table 2), customers and products (table 3). The sign "x" denotes the connection of objects of two sets.

Table 1. Supplier Relationship Matrix

and supplied materials (PM)

PM Pі P2 Pz P4 P5

Table 2. Matrix of relations between products and materials (IM)

IM Mі M2 Mz

Table 3. Matrix of relationships between customers and products (PI)

ZI II I2 From From

Using the composition of the ratios given by the matrices PM, MI, and ZI, it is not difficult to compose a matrix of the ratio of PP. The PZ matrix (Table 4) shows the links established by the enterprise between suppliers P and customers Z^ So, for example, the interaction of the customer Z3 with the enterprise takes place on the product I3, which requires materials M! and M3 supplied by Pn P3 and P5.

Table 4. Relationship matrix between supplier-

Detailed scheduling of technological processes (product lines) with the help of relationship matrices simplifies the determination of added value for the customer, the profit of the enterprise and its losses.

3. The construction of an enterprise quality management system is associated with the allocation of a network of processes. The distribution of processes among the divisions of the enterprise, the implementation of the requirements of the standard, for example, ISO 9001-2000, can be carried out using matrices. Let's say the processes are highlighted: contracting, QMS documentation management, internal audit, procurement, manufacturing, monitoring customer satisfaction, and the company has divisions: marketing department, purchasing department, chief designer department, chief technologist department, production, warranty support department. Based on the results of discussion with representatives of departments, a PP matrix can be compiled (Table 5). On the other hand, dedicated processes should cover the requirements of a standard, such as ISO 9001-2000. Linking processes to ISO 9001-2000 results in a TP matrix (Table 6).

Using the composition of relations, we obtain the ISO matrix (Table 7).

us and customers (PP)

ПЗ Зі 32 Зз 34 35

Table 5. Matrix of links between processes and departments (SP)

PP matrix Marketing department Procurement department Chief designer department Chief technologist department Production Warranty support department

Contracting X X

Internal audit X

Procurement X

Manufacturing X

Table 6. Relationship of processes to ISO 9001-2000

TP matrix Quality management systems Management responsibility Resource management Product life cycle processes Measurement, analysis and improvement

Contracting X

QMS documentation management X X

Internal audit X X

Procurement X

Production X X X

Customer Satisfaction Monitoring X

ISO Matrix Marketing Department Purchasing Department Chap. designer department chap. technologist Production Warranty Support Department

Quality management systems X X

Management responsibility X X X

Resource management X

Product Life Cycle Processes X X X

Measurement, analysis and improvement X X

Obviously, with such a distribution of ISO requirements, one can expect inconsistencies in clause 5 "Management responsibility", since the quality policy is the responsibility of top management.

4. Expanding each element of the relationship matrix, for example, "Management Responsibility - Marketing Department" can be using the priority matrix underlying the hierarchy analysis method. The requirements of the ISO 9000-2000 series establish the scope and depth of the regulatory and technical documentation necessary for the functioning of the enterprise's QMS. One of the obligatory documents of the QMS of the enterprise is the policy and goals in the field of quality. The goals of the enterprise are formulated in various areas: finance, market, competition

(benchmarking), customer satisfaction, improvement of product and process performance. The goals of the entire organization should be projected (deployed, decomposed) into its divisions, so that the staff is aware of their involvement and responsibility for achieving a particular goal of the entire organization.

Planning, choosing goals, optimizing behavior in a competitive environment always require a decision at a certain stage. It became practically obvious that social processes, in particular, management processes, are poorly formalized within the classical framework.

topics. In this case, the method of analyzing hierarchies can be quite effective.

The method of analysis of hierarchies is based on the so-called priority matrix. Assume that the task is to compare the factors influencing the selected object. As a rule, the number of influencing factors is quite large, the exact dependencies are unknown, and it is practically impossible to perform the mathematical formalization of the problem. The expert also experiences difficulties in assessing the influence of factors on the object. Surprisingly, the problem is solved more easily if a pairwise comparison of the influence of factors on the object is carried out. (The bottom line is that it is difficult to answer the question how much A weighs, it is much easier to decide which is heavier: A or B)

For analytical planning of the development of an enterprise, it is necessary to describe the initial state (the “as is” position), the target state (goals) and the means to link these states. Below is an example of applying the method of analysis of hierarchies, as an object, the goal from the quality policy "Sustainable growth in enterprise profits" is selected and some factors influencing the goal are highlighted (Table 8).

Specialists - experts of the enterprise compiled priority matrices according to the selected criteria (an example is given in table 9).

Management Logistics

Planning, Procurement,

Investments, supplier relationships,

Advertising, entrance control,

Selling prices, control of resources.

Marketing strategy. Personnel and Development

production qualification,

Compliance with deadlines, staff training,

Technology, staff motivation,

Quality, creativity,

Organization of production, cost control. planning new developments

Table 9. Example of the matrix "Production"

Production Compliance with the terms of delivery of products Technology Quality Organization of production Cost control

Compliance with product delivery dates 1 5 1 3 3

Technology 1/5 1 3 1 3

Quality 1 1/3 1 3 1

Organization of production 1/3 1 1/3 1 1

Cost control 1/3 1/3 1 1 1

Scale of relationships and filling in tables 1 - equivalence of factors, 3 - dominance of one factor over another factor,

5 - strong dominance of one factor over another factor, 2.4 - possible intermediate values.

Mathematical processing of matrices consisted in finding the priority vector as an eigenvector corresponding to the maximum eigenvalue. As an example, below are the results of processing the estimates of expert N (table 10). The columns indicate the components of the priority vector by various factors, for example, according to the criterion "Management"

Priority is given to investment.

On fig. 1. The results of calculating the priorities of experts according to the above criteria are given. Goal achievement is associated with investment, quality,

planning new developments and controlling resources.

Table 10. Results of processing the estimates of expert N

Goal - Sustainable growth of the company's profit

Management Production Mat - technical supply Personnel and development

0,1084 0,3268 0,3072 0,1625

0,4198 0,1280 0,2059 0,0773

0,1084 0,2829 0,1552 0,1007

0,2356 0,1002 0,3316 0,2080

0,1279 0,1621 0,4516

Management

Production

S&I^TO o i_CO

Personnel and Development

Rice. 1. Results of calculating the priorities of experts

Knowing the distribution of priorities according to the selected criteria allows the top management of the enterprise to pursue a sound policy to achieve the goal.

Bibliography

1. Gludkin O.P., Gorbunov NM., Gurov A.I., Zorin Yu.V. Total Quality Management. - M.: Radio and communication, 1999.

2. Kuzin B., Yuriev V., Shakhdinarov G. Methods and models of firm management. - St. Petersburg: Peter, 2001.

3. Faure R., Kofman A., Denis-Papin M. Modern mathematics. - M.: Mir, 1966.

4. Saati T. Decision making. Hierarchy analysis method. / per. from English. - M.: Radio and communication, 1993.

MATRIX ANALYSIS IN ENTERPRISE EXECUTIVE SYSTEM

© 2006 A.V. Volgin1, G.E. Belachewskij2

\cSamara - Aviagas»

Samara State Aerospace University

In work various ways of matrixes application in business operation are analyzed. The relation (connection) between elements of two and more sets can be submitted in the matrix form. The composition of relations allows to simplify the analysis of connections between elements of sets. The example of use of priorities matrixes in a control system of the enterprise is the result.

Matrix analysis or the matrix method has become widespread in the comparative evaluation of various economic systems (enterprises, individual divisions of enterprises, etc.). The matrix method allows you to determine the integral assessment of each enterprise for several indicators. This assessment is called the rating of the enterprise. Consider the application of the matrix method in stages using a specific example.

1. Selection of evaluation indicators and formation of a matrix of initial data a ij, that is, tables, where the numbers of systems (enterprises) are reflected in rows, and the numbers of indicators (i = 1,2 ... .n) - systems are reflected in columns; (j=1,2…..n) - indicators. The selected indicators should have the same focus (the more, the better).

2. Compilation of a matrix of standardized coefficients. In each column, the maximum element is determined, and then all elements of this column are divided by the maximum element. Based on the results of the calculation, a matrix of standardized coefficients is created.

We select the maximum element in each column.

method scientific research properties of objects based on the use of the rules of the theory of matrices, which determine the value of the elements of the model, reflecting the relationship of economic objects. It is used in cases where the main object of study is the balance ratio of costs and results of production and economic activities and the standards of costs and outputs.

• - pseudobridge, matrix bridge

  Molecular biology and genetics. Dictionary

• - English. matrix analysis; German Matrixanalysis. In sociology - a method of studying the properties of social. objects based on the use of the rules of matrix theory...

  Encyclopedia of Sociology

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• - See: dot matrix printer...

  Glossary of business terms

• - a method of scientific study of the properties of objects based on the use of the rules of the theory of matrices, which determine the value of the elements of the model, reflecting the relationship of economic objects ...

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• - in economics, a method of scientific study of the properties of objects based on the use of the rules of the theory of matrices, which determine the value of the elements of the model, reflecting the relationship of economic objects ...

  Great Soviet Encyclopedia

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"ANALYSIS MATRIX" in books

T.N. Panchenko. Strawson and Wittgenstein. Analysis as revealing the formal structure of informal language and analysis as therapy

From the book Philosophical Ideas by Ludwig Wittgenstein author Gryaznov Alexander Feodosievich

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From the book Cartesian Reflections author Husserl Edmund

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2.6. Biosynthesis of proteins and nucleic acids. Matrix nature of biosynthetic reactions. Genetic information in a cell. Genes, genetic code and its properties

From the book Biology [ Complete reference to prepare for the exam] author Lerner Georgy Isaakovich

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From the book Great Soviet Encyclopedia (MA) of the author TSB

2.4. ANALYSIS OF REQUIREMENTS FOR THE SYSTEM (SYSTEM ANALYSIS) AND FORMULATION OF GOALS

From the book Programming Technologies the author Kamaev V A

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Matrix metering

From the book Digital Photography from A to Z author Gazarov Artur Yurievich

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Question 47 Factual and legal basis. Evidence analysis.

From the book The Author's Lawyer Exam

Question 47 Factual and legal basis. Evidence analysis. Honest, reasonable and conscientious provision of legal assistance in any form, whether it be consulting, drafting various documents, representing interests or defending

9. Science at the service of toxicology. Spectral analysis. Crystals and melting points. Structural analysis by X-ray. Chromatography

From the book One Hundred Years of Forensics author Thorvald Jürgen

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12.9. Matrix solution development method

From the book Systematic Problem Solving author Lapygin Yuri Nikolaevich

12.9. Matrix method of developing decisions Decision-making based on the matrix method is reduced to making a choice, taking into account the interests of all interested parties. Schematically, the decision process in this case looks like it is shown in Fig. 12.7. As we see, there is

4. Market research and analysis (analysis of the business environment of the organization)

From the book Business Planning: Lecture Notes the author Beketova Olga

4. Market research and analysis (analysis of the organization's business environment) Market research and analysis is one of the most important stages in the preparation of business plans, which should answer questions about who, why and in what quantities buys or will buy products

5.1. Analysis of the external and internal environment of the organization, SWOT analysis

author Lapygin Yuri Nikolaevich

5.1. Analysis of external and internal environment organizations, SWOT analysis External environment and system adaptation Organizations, like any systems, are isolated from the external environment and at the same time are connected with the external environment in such a way that they receive the resources they need from the external environment and

8.11. Matrix method RUR

From book Management decisions author Lapygin Yuri Nikolaevich

8.11. RSD matrix method Decision-making based on the matrix method is reduced to making a choice taking into account the interests of all stakeholders. Schematically, the RUR process in this case looks like it is shown in Fig. 8.13. Rice. 8.13. The RUR model by the matrix method

4. Analysis of the strengths and weaknesses of the project, its prospects and threats (SWOT analysis)

author Filonenko Igor

4. Analysis of the strengths and weaknesses of the project, its prospects and threats (SWOT-analysis) When assessing the feasibility of launching a new project, a combination of factors plays a role, and not always the financial result is of paramount importance. For example, for an exhibition company

5. Political, economic, social and technological analysis (PEST-analysis)

From the book Exhibition Management: Management Strategies and Marketing Communications author Filonenko Igor

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11.3. Matrix strategy development method

From the book Strategic Management: tutorial author Lapygin Yuri Nikolaevich

11.3. The matrix method for developing strategies Development of an organization's vision The various states of the external and internal environment of organizations explain the diversity of the organizations themselves and their actual state. The multifactorial nature of the parameters that determine the position of each

Course of lectures on discipline

"Matrix Analysis"

for 2nd year students

Faculty of Mathematics specialty

"Economic cybernetics"

(lecturer Dmitruk Maria Alexandrovna)

Chapter 3. Matrix Functions.

1. Function definition.

Df. Let the function be a scalar argument. It is required to define what is meant by f(A), i.e. we need to extend the function f(x) to the matrix value of the argument.

The solution to this problem is known when f(x) is a polynomial: , then.

Definition of f(A) in the general case.

Let m(x) be a minimal polynomial A and have a canonical decomposition such that eigenvalues A. Let the polynomials g(x) and h(x) take the same values.

Let g(A)=h(A) (1), then the polynomial d(x)=g(x)-h(x) is the annihilating polynomial for A, since d(A)=0, hence d(x) is divisible by a linear polynomial, i.e. d(x)=m(x)*q(x) (2).

Then, i.e. (3), .

Let us agree to call m numbers for f(x) such values ​​of the function f(x) on the spectrum of the matrix A, and the set of these values ​​will be denoted.

If the set f(Sp A) is defined for f(x), then the function is defined on the spectrum of the matrix A.

It follows from (3) that the polynomials h(x) and g(x) have the same values ​​on the spectrum of the matrix A.

Our reasoning is reversible, i.e. from (3) (3) (1). Thus, if the matrix A is given, then the value of the polynomial f(x) is completely determined by the values ​​of this polynomial on the spectrum of the matrix A, i.e. all polynomials gi(x) that take the same values ​​on the spectrum of the matrix have the same matrix values ​​gi(A). We require that the definition of the value of f(A) in the general case obey the same principle.

The values ​​of the function f(x) on the spectrum of the matrix A must fully determine f(A), i.e. functions having the same values ​​on the spectrum must have the same matrix value f(A). Obviously, to determine f(A) in the general case, it is sufficient to find a polynomial g(x) that would take the same values ​​on the spectrum A as the function f(A)=g(A).

Df. If f(x) is defined on the spectrum of matrix A, then f(A)=g(A), where g(A) is a polynomial that takes the same values ​​on the spectrum as f(A),

Df. The value of the function from the matrix A we call the value of the polynomial in this matrix at.

Among the polynomials from С[x], taking the same values ​​on the spectrum of the matrix A, as f(x), of degree not higher than (m-1), taking the same values ​​on the spectrum A, as f(x) is the remainder of the division of any polynomial g(x) having the same values ​​on the spectrum of the matrix A as f(x) by the minimal polynomial m(x)=g(x)=m(x)*g(x)+r(x).

This polynomial r(x) is called the Lagrange-Sylvester interpolation polynomial for the function f(x) on the spectrum of the matrix A.

Comment. If the minimal polynomial m(x) of matrix A has no multiple roots, i.e. , then the value of the function on the spectrum.

Example:

Find r(x) for arbitrary f(x) if the matrix

. Let us construct f(H1 ). Find the minimal polynomial H1 last invariant factor :

, dn-1=x2 ; dn-1=1;

mx=fn(x)=dn(x)/dn-1(x)=xn 0 nmultiple root m(x), i.e. n-fold eigenvalues ​​H1 .

, r(0)=f(0), r(0)=f(0),…,r(n-1)(0)=f(n-1)(0) .

1. Properties of functions from matrices.

Property #1. If the matrix has eigenvalues ​​(there may be multiples among them), and, then the eigenvalues ​​of the matrix f(A) are the eigenvalues ​​of the polynomial f(x): .

Proof:

Let the characteristic polynomial of matrix A have the form:

Let's count. Let's move from equality to determinants:

Let's make a change in equality:

Equality (*) is valid for any set f(x), so we replace the polynomial f(x) with, we get:

On the left, we have obtained the characteristic polynomial for the matrix f(A), expanded on the right into linear factors, whence it follows that the eigenvalues ​​of the matrix f(A).

CHTD.

Property #2. Let the matrix and the eigenvalues ​​of the matrix A, f(x) be an arbitrary function defined on the spectrum of the matrix A, then the eigenvalues ​​of the matrix f(A) are equal.

Proof:

Because the function f(x) is defined on the spectrum of the matrix A, then there is an interpolation polynomial of the matrix r(x) such that, and then f(A)=r(A), and the matrix r(A) has eigenvalues ​​according to property No. 1 which are respectively equal.

CHTD.

Property #3 If A and B are similar matrices, i.e. , and f(x) is an arbitrary function defined on the spectrum of the matrix A, then

Proof:

Because A and B are similar, then their characteristic polynomials are the same and their eigenvalues, so the value of f(x) on the spectrum of matrix A coincides with the value of the function f(x) on the spectrum of matrix B, and there is an interpolation polynomial r(x) such that that f(A)=r(A), .

CHTD.

Property number 4. If A is a block diagonal matrix, then

Consequence: If, then, where f(x) is a function defined on the spectrum of matrix A.

1. Lagrange-Sylvester interpolation polynomial.

Case number 1.

Let it be given. Consider the first case: the characteristic polynomial has exactly n roots, among which there are no multiples, i.e. all eigenvalues ​​of the matrix A are different, i.e. , Sp A is simple. In this case, we construct the basic polynomials lk(x):

Let f(x) be a function defined on the spectrum of the matrix A and let the values ​​of this function on the spectrum be. We must build.

Let's build:

Let's note that.

Example: Construct a Lagrange-Sylvester Interpolation Polynomial for a Matrix.

Let's construct basic polynomials:

Then for the function f(x) defined on the spectrum of the matrix A, we get:

Let's take, then the interpolation polynomial

Case number 2.

The characteristic polynomial of matrix A has multiple roots, but the minimal polynomial of this matrix is ​​a divisor of the characteristic polynomial and has only simple roots, i.e. . In this case, the interpolation polynomial is constructed in the same way as in the previous case.

Case number 3.

Let's consider the general case. Let the minimal polynomial have the form:

where m1+m2+…+ms=m, deg r(x)

Let's compose a fractional-rational function:

and decompose it into simple fractions.

Let's designate: . Multiply (*) by and get

where is some function that does not go to infinity at.

If we put in (**), we get:

In order to find ak3 one has to (**) differentiate twice, and so on. Thus, the coefficient aki is uniquely determined.

After finding all the coefficients, we return to (*), multiply by m(x) and get the interpolation polynomial r(x), i.e.

Example: Find f(A) if, where tsome parameter,

Let's check whether the function is defined on the spectrum of the matrix A

Multiply (*) by (x-3)

at x=3

Multiply (*) by (x-5)

In this way,is an interpolation polynomial.

Example 2

If a, then prove that

Let's find the minimal polynomial of the matrix A:

is the characteristic polynomial.

d2 (x)=1, then the minimal polynomial

Consider f(x)=sin x on the matrix spectrum:

the function is defined on the spectrum.

Multiply (*) by

.

Multiply (*) by:

Calculate by taking the derivative (**):

. Assuming,

, i.e..

So,,

Example 3

Let f(x) be defined on the spectrum of a matrix whose minimal polynomial has the form. Find the interpolation polynomial r(x) for the function f(x).

Solution: By condition f(x) is defined on the spectrum of the matrix A f(1), f(1), f(2), f(2), f(2) defined.

We use the method of indefinite coefficients:

If f(x)=log x

f(1)=0f(1)=1

f(2)=log 2f(2)=0.5 f(2)=-0.25

4. Simple matrices.

Let the matrix, since C is an algebraically closed field, then x