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1. GOALS AND OBJECTIVES OF THE DEVELOPMENT OF THE DISCIPLINE
1.1.
Objectives: the main objective of the course "Mathematical Applications in Economics" is to
acquaint students with different approaches to the mathematical formalization of economic
entities; with different mathematical models of the economy, such as: model based on linear
algebra and mathematical analysis, various models of linear, non-linear, integer and dynamic
programming, model of economic planning on the basis of game theory, model economic
decision-making in certain conditions , partial and full of uncertainty.
1.2.
Objectives: the formation of students' skills in the analysis of the fundamental concepts of the
economy with the abstract mathematical point of view; theoretical development by students of
modern concepts and models of mathematics ; acquisition of practical skills of application of
mathematics in economics unit .
2.
PLACE OF DISCIPLINE IN THE STRUCTURE OF THE EDUCATIONAL
PROGRAM
1.1. Cycle (block) OP: FTD
1.2. Communication with other disciplines of the curriculum
A list of previous disciplines
1 .List follow disciplines, types of work
Macroeconomics
Microeconomics
Foreign language for professional
communication
Theoretical
Foundations of International
Business
Intercultural communication
Research work
The dissertation research
Microeconomics
Foreign language for professional
3. . REQUIREMENTS FOR DEVELOPMENT RESULTS OF THE DISCIPLINE
communication
Theoretical Foundations of International
competences to be formed
The expectations of the learning of the discipline:
Business
Microeconomics
Code
name
Foreign language for professional
General cultural competences (GC)
communication
Theoretical
Foundations
of International
B
GC-1 –tural
ability
to abstract
thinking , Know:
The subject of mathematical modeling, mathematical
competence
(GC)
analysis , synthesis
structure of the model and its meaningful interpretation.
To be able: To apply methods of linear algebra, mathematical
analysis, linear programming, game theory , probability theory and
mathematical statistics in modeling the economic objects
To have: The main methods of mathematical modeling in the
economy based on balance
models, optimization methods , methods of managerial
decision-making in relation to the solution of economic problems.
Professional Competence (PC)
PC -8
– ability to prepare analytical
materials for evaluation
activities in the field of
economic policy and strategic
decision-making at the micro
and macro level
Know: The main types of models: balance models, planning and
optimization based on the production of linear , non-linear , integer
and dynamic programming , the criteria for decision-making under
conditions of certainty and uncertainty partial , elements of the
theory of fuzzy sets .
To be able: Draw up mathematical models in the form of linear and
nonlinear equations , problems of linear and nonlinear
programming , optimization methods , the simplest initial value
problems, rows of distribution of random variables
To have: The methods of solving linear and nonlinear optimization ,
differential equations and initial value problems , research methods
of random variables in terms of certainty and partial uncertainty ,
fuzzy sets and conclusions based on fuzzy logic
PC -9
– ability to analyze and use Know: Methods of analysis of experimental data on the basis of the
various
sources
of theory of probability , mathematical statistics , econometric
information for economic analysis of the data elements of the theory of interpolation.
calculations
To be able: To carry out the interpolation of the experimental data
using linear and nonlinear regression , time series , Newton
interpolation polynomials , Lagrange , etc.
To have: Elements of the theory of statistical analysis of the
experiments, the experimental data on the basis of the error theory,
probability theory and mathematical statistics .
PC -10 – the ability to forecast the Know: Basic methods of forecasting of economic indicators on the
main socio - economic basis of econometric analysis: linear and non-linear and multiple
indicators of the enterprise, regression , production curves , time series .
industry , region and economy
as a whole
To be able: Assess the economic and social dimensions and spacing
of their fuzziness on the basis of mathematical statistics unit ,
econometrics , theory of fuzzy sets .
To have: Methods of risk assessment, decision-making under
conditions of partial and complete uncertainty
4.
STRUCTURE AND CONTENT OF THE DISCIPLINE
16
8
2
2
View class, module, topic and summary
competences
including
interactive hours
number of hours
4.1.Classroom classes - full-time course
lectures
Module 1 "The methods of linear and non-linear programming in the
economy"
1.1. Subject "Carrying Leontief model. linear programming problem in
the economy "
GC -1,
PC -2,
Carrying Leontief model. The main types of linear programming problems PC -4,
PC -11
(LPP) and their respective economic models: the problem of production
planning, the task of drawing up the diet, the transportation problem. The
decision LPP geometrical method. Simplex method for solving the LPP.
1.2.
Subject "Dual LPP. The transportation problem (TP) "
GC -1
Dual LPP and their economic interpretation. Parallel solving inter-dual
LPP simplex method. The transportation problem and its properties.
potential method of solving the transportation problem.
2
1.3. Subject "nonlinear programming (NP) and the corresponding economy PC -2
problem. Dynamic programming "
PC -4,
Quadratic programming in the economy. A geometric method for solving PC -11
quadratic problems. Fractional linear programming in the economy.
2
1.4. Subject "Dynamic programming in the economy. Game theory"
Dynamic programming: the problem of the distribution of investments.
Matrix Games. Antagonistic games. Nonantagonistic games.
8
2
Module 2 "Econometric models. Methods of decision making under
conditions of partial and complete uncertainty "
2.1. Subject "Econometric models. And multiple regression. Time series "
GC -1
PC -2
PC -2,
PC -4
GC -1
Linear regression. The simplest model steam regression. The concept of
multiple regression. Time series: addetivnaya and multiplicative models.
2
2.2. Subject "Methods of decision making under certainty and uncertainty
partial"
PC -2,
PC -11
Decision-making under certainty. Laplace criterion. Criteria Savage and
Wald. Pareto-optimality. Decision making under probabilistic uncertainty.
Criterion Baeza. Hurwitz criterion.
2
2.3. Subject "Decision-making in conditions of complete uncertainty. The
theory of fuzzy sets "
PC -4,
PC -11
Fuzzy sets and their application in the economy. Algebraic and logical
operations on fuzzy sets. Fuzzy matching. fuzzy inference system.
2
2.4. Subject "Fuzzy numbers. Application of fuzzy numbers to the
GC -1,
PC -11
modeling of economic processes "
The concept of fuzzy and linguistic variables. Special types of fuzzy
numbers and its application to the evaluation intervals blur exogenous
parameters of the model. Fuzzification model. Examples of applications of
fuzzy numbers to the modeling of economic processes.
16
8
2
practical lessons
Module 1 "The methods of linear and non-linear programming in the
economy"
1.1.Subject "Carrying Leontief model. linear programming problem in the
economy "
GC -1,
PC -4
Carrying Leontief model. Compilation of mathematical models in the form
of LPP for problems of production planning, drawing up a diet problem.
The decision LPP geometric method on the example of the problem of
minimizing the cost of the goods. Solution LPP via simplex tables.
2
1.2.Subject "Dual LPP. The transportation problem "
GC -1,
Dual LPP and their economic interpretation on the example of the problem PC -2,
PC -4
of production planning. Parallel solving inter-dual LPP simplex method.
The transportation problem. Construction of the initial decision by the
northwestern corner and by minimizing transportation costs. The decision
LPP potential method.
2
1.3. Subject "nonlinear programming (NP) and the corresponding
economy problem. Dynamic programming "
PC -2,
PC -4
Quadratic programming in the economy. Drawing up a mathematical
model for the problem of transportation in a quadratic dependence on the
volume of transportation costs. The solution of quadratic NP geometrical
method. The solution of linear fractional programming on the example of
the problem of minimizing the unit cost of the goods.
2
8
2
1.4. Subject "Dynamic programming in the economy. Game theory"
Dynamic programming: the problem of the distribution of investments
between enterprises of one branch. Matrix Games. The solution of the
optimization of financial investments in shares on the basis of the matrix
game theory.
Module 2 "Econometric models. Methods of decision making under
conditions of partial and complete uncertainty "
2.1. Subject "Econometric models. And multiple regression. Time series "
Linear regression. Making the simplest model steam
regressii.Proizvodstvennaya function Cobb-Douglas. Building addetivnoy
time series models.
2
Subject 2.2 "Methods of decision making under certainty and uncertainty
partial"
PC -2
PC -2,
PC -4
GC -1,
PC -4
Decision-making under certainty. Selecting otimalnogo investment project
on the basis of criteria Laplace, Savage and Wald, Pareto optimality.
Decision making under probabilistic uncertainty on the basis of criteria
Baeza and Hurwitz.
2
2.3. Subject "Decision-making in conditions of complete uncertainty. The
PC -4,
PC -11
theory of fuzzy sets "
Fuzzy sets and their application in the economy. Algebraic and logical
operations on fuzzy sets. Drawing up the membership functions of fuzzy
sets. fuzzy inference system.
2
2.4. Subject "Fuzzy numbers. Application of fuzzy numbers to the
modeling of economic processes "
PC -2
The concept of fuzzy and linguistic variables. The triangular and
trapezoidal fuzzy numbers and its application to the evaluation intervals
blur exogenous parameters of the model. Fuzzification clear model on the
example of a risk assessment to market new brand of goods. Fuzzy risk
assessment model of the company bankruptcy.
Independent work of students - full-time training
Topics, sections for an independent training: themes of essays, tests, recommendations for the
use of literature, computers, and others.
competences
number of hours
4.2.
24 Topics, sections for self-training
6
1.1. Subject "Methods for solving LPP in general. Methods for solving TP that is PC -2
not a closed "
6
6
1.2. Subject "The reduction to LPP task assignment, the optimal reduction of
production, distribution of the farmland."
1.3. Subject "Lagrange multipliers method NP solutions in general."
GC -1
GC -1
ПК-2
6
2.1. Subject Multiple Regression. The multiplicative model of construction of the PC -4,
time series. "
ПК-11
6
2.4. Subject "Models of decision-making based on fuzzy sets."
GC -1
10
Topics and issues determined by the teacher based on student interests
8
40
Preparing to set off
Total labor input of independent work (hours)
PC -2,
ПК-4,
ПК-11
GC
-1
5. ASSESSMENT TOOLS FUND
5.1.
Fund assessment tools for monitoring
Sample tasks for preparation for practical classes
Tasks 1.-2. The table contains data on the use of the balance for the period (in USD). Calculate the
required volume of gross output of each industry, if the end product is the first pro-industry is
expected to grow by 10%, the second industry - by 20%, and the third otrasli- 30%.
1.
2.
Consumption
Consumption
end
end
Production
Production
product
product
A
B
C
A
B
C
A
50
60
80
60
A
40
18
25
21
B
25
90
40
25
B
16
9
25
16
C
25
60
40
35
C
80
45
50
75
Task 3. The company produces two products A and B, a market which is unlimited. Each product
must be treated with each of the machines I, II and III. The processing time in hours for each product
A and B given below:
Product
The processing time, h
I
II
III
A
0,5
0,4
0,2
B
0,25
0,3
0,4
Time work machine I, II, III, respectively, 40, 36 and 36 hours a week. Profit from products
A and B are, respectively, 5 and $ 3.
The firm must determine the weekly rate of release of products A and B, maximizing profits.
Formulate this problem as a linear programming problem and solve it (graphically).
Task 4. Gain on products A, B, C are, respectively, 3, 4, 5 units. For each product is required when
using the machine I and II, which are available, respectively, 12 and 15 hours a day:
Machine
Time use of the machine, h
А
В
С
I
2
3
3
II
4
1
2
Find the optimal production solution.
Task 5. Two products B1 and B2 are sequentially processed on machine № 1, 2, 3, 4, 5. The machine
time for each unit of product to the machine described in the table. It also shows the profit from each
product, and the volume of production of the second kind of products should not exceed 40% of total
output. Determine the optimal release of a program that provides the best return.
Product
Machine time for the machine, min
Profit,
rubles / pcs.
1
2
3
4
5
В1
4
3
2
3
0
1
В2
2
0
6
5
4
1,5
Weekly fund of working time, min
352
240
330
420
400
Solve the problem graphically.
Task 6. The company can produce two types of products i1 and i2. On the unit output i1 spent 3
resource units and per unit of product i2 - 1 unit of the same resource. In the planning period at the
disposal of the company has 300 units of the same resource. Restriction on output of the first and
highest category of quality is as follows:  3 x1  4 x2  0 . This requires that the i1 products were
produced at least 40 units. The company wants to get the maximum profit.
Each product type i1 gives 3 dollars in profit each product type i2 gives 4 dollar profit. Solve
this problem graphically.
Task 7. Floor cleaning equipment are measured at the following three criteria: a) the cleansing
properties; b) disinfectant properties; c) irritant effects on the skin. Each of these parameters is
measured on a linear scale from 0 to 100.
The product on the market must have at least 60 units and cleaning properties of the disinfecting
properties units 60 for the respective scale. In this irritant effect on the skin should be minimized. The
final product should be a mixture of three primary cleaners, the characteristics of which are given
below:
Product features
Cleaner
Cleansing
Disinfectants
Annoying
А
90
30
70
В
65
85
50
С
45
70
10
Formulate the problem of finding the optimal mix as a linear programming problem and solve it.
Task 8. Simplex method to find the greatest value of the function f  3х1  4 х 2 under constraints:
 х1  2 х 2  4,
 х  х  3,
 1
2

 2 х1  х 2  8,
 х1  0, х 2  0.
Task 9. Trading company for the sale of goods using three types of resources: time and area
salesrooms. Resource Costs for sale one lot of each type of goods are given in the table below:
Kind of goods
Volume
Resources
resources
I
II
III
Time, man-hours
0,5
0,7
0,6
370
Area, m2
0,1
0,3
0,2
90
Profit from the sale of a consignment of I-type - 5 conv. U, II-th species -. 8 cond. U, III-th species -.
6 cond. u To determine the optimal structure of trade, the company providing the maximum profit.
Task 10. For a given LPP make the dual problem, solve it graphically.
f  6 x1  9 x2  3x3  min
 x1  2 x2  x3  2,
3 x  x  x  1,
 1
2
3

3 x1  x2  x3  1,
 x1  0, x2  0, x3  0
Tasks 11-12. Find the lowest fare in the problem, given a table that lists the number of tons of cargo
in the warehouses (left column), the number of tons of cargo that must be delivered to consumers
(upper column), as well as the corresponding tariffs.
 0 100 200 150 50 
 0 50 150 220 80 




1
3
4
1
4
3
 220 2
 160 2
11. 
12. 
90
5
2
4
1
40 5
4
1
2




 190 3

 300 3

6
2
4
2
4
5




Task 13. The plant has three groups of machines, each of which can perform processing operations
five items (operations can be performed in any order). The maximum operating time each group of
tools is respectively equal to 100, 250, 180 hours. Each operation to be performed, respectively 100,
120, 70, 130 hours. The performance of each group of machines for each operation specified matrix:
 3 5 11 10 5 


 5 10 15 3 2 6 
 4 8 6 12 10 


Determine how long and on which operation you want to use each group of machines to process the
maximum number of items.
Task 14 (The problem of irrigation). For watering the garden variety of sites on which grows
plums, apples, pears, are three wells. Wells can give respectively 180, 90, and 40 buckets of water.
Garden plots required for irrigation, respectively 100, 120 and 90 buckets of water. Distance (in
meters) from the wells to the garden plots is given in the following table:
plots
Wells
plums
apple
pears
1
10
5
12
2
23
28
33
3
43
40
39
How best to arrange watering?
Task 15. It is necessary to make product 4 by 4 sample machines (n = 4), each sample can be made
only on a single machine and each machine manufacturer be occupied only one sample. Given matrix
production efficiency of each sample on each machine (rows correspond to the samples and the
columns - machine tools):
 0

 4
 20

 10

10
15
4
30
2
5
0
0
0 

10 
0 

20 

(Production of the first sample on the first machine do not efficiently - c11 = 0, the second estimated
efficiency c12 = 10, ..., manufacturing efficiency quadruple sample estimated in the fourth machine
c44 = 20). Create the most efficient plan for product manufacturing.
Task 16. According to the plan of production company needs to produce at least 200 items. These
products can be made two technological ways. Production costs for the manufacture of products by
the first method n equal to 4n + n2, and the second method - 8n + n2. How many products have to
make in every way to the overall cost of production would be minimal.
Task 17. For the production of two types of products A and B, the company uses three types of
processing equipment, each product is processed on each of them. The processing time of the product.
Monetary costs associated with this time of equipment using resources indicated in the table.
The type of equipment
The time spent using the equipment for
processing of products (h.)
time resource
utilization
equipment (hr.)
Product А
Product В
I
2
4
 20
II
1
1
III
8
4
5
 30
Production costs of goods
2
3
(thousand rubles)
Determine how many of each type of product to be produced, so that the average cost of production of
one product was the lowest.
Tests
1. Let A = {0,5 / 3 0.8 / 4, 1/5, 1/6, 0.8 / 7 0.5 / 8, 0/9} - fuzzy set with elements of the universal set
E and supplies a plurality of M = [0,1]. What is a carrier of a fuzzy set A?
A) {3,8}
B) {5,6}
C) {4,7}
D) {3,4,5,6,7,8}
E) {9}
2. Let A={0,5/3;0,8/4;1/5;1/6;0,8/7;0,5/8;0/9} – fuzzy set with elements of the universal set E
and a variety of accessories, M = [0,1]. Determine the transition point of a fuzzy set A?
A) {3,8}
B) {3,4,5,6,7,8}
C) {5,6}
D) {4,7}
E) {9}
3.
Let A={0,5/3;0,8/4;1/5;1/6;0,8/7;0,5/8;0/9} – fuzzy set with elements of the universal set E
and a variety of accessories, M = [0,1]. Determine the height of a fuzzy set A?
A) 5
B) 6
C) 5;6
D) 1
4. Let A={0,5/3;0,8/4;1/5;1/6;0,8/7;0,5/8;0/9} – fuzzy set with elements of the universal set E
and a variety of accessories, M = [0,1]. Which of the following characteristics does not match the
fuzzy set A?
A) the set unimodal
B) a plurality of normal
C) the set has two transition points
D) the set has a height of 1
E) no right answer
5.
For two fuzzy sets A and B is designated A in operation
A) equality
B) addition
C) intersection
D) switch
E) union
6.
For two fuzzy sets A and B = Ā The designation means an operation
A) addition
B) switch
C) equality
D) crossing
E) union
7. For two fuzzy sets A and B, their difference is equal to B-A
A)
B)
C)
D)
E)
8.
9.
Ā
В∩Ā
В
В∩А
В Ā
For two fuzzy sets A and B designation A  B means an operation
A) the sum of disjunctive
B) switch
C) equality
D) crossing
E) union
Let А and В – fuzzy sets. А=0,4/х1+0,2/х2+0/х3+1/х4; В=0,7/х1+0,9/х2+0,1/х3+1/х4. Determine
the fuzzy set, the set operation А  В
A) 0,4/х1+0,2/х2+0/х3+1/х4
B) 0,7/х1+0,9/х2+0,1/х3+1/х4
C) 0,4/х1+0,2/х2+0/х3+0/х4
D) 0,6/х1+0,8/х2+0,1/х3+0/х4
E) 0,7/х1+0,9/х2+0,1/х3+0/х4
10. Let А and В – fuzzy sets. А=0,4/х1+0,2/х2+0/х3+1/х4; В=0,7/х1+0,9/х2+0,1/х3+1/х4. Determine
the fuzzy set, the set operation А∩В
A) 0,4/х1+0,2/х2+0/х3+1/х4
B) 0,6/х1+0,8/х2+0,1/х3+0/х4
C) 0,7/х1+0,9/х2+0,1/х3+1/х4
D) 0,4/х1+0,2/х2+0/х3+0/х4
E) 0,7/х1+0,9/х2+0,1/х3+0/х4
11. Let А and В – fuzzy sets. А=0,4/х1+0,2/х2+0/х3+1/х4; В=0,7/х1+0,9/х2+0,1/х3+1/х4. Determine
the fuzzy set, the set operation А  В
A) 0,4/х1+0,2/х2+0/х3+1/х4
B) 0,6/х1+0,8/х2+0,1/х3+0/х4
C) 0,4/х1+0,2/х2+0/х3+0/х4
D) 0,7/х1+0,9/х2+0,1/х3+0/х4
E) 0,7/х1+0,9/х2+0,1/х3+1/х4
12. What is the record А,, if A - fuzzy set?
A) fuzzy set closest to the fuzzy set A
B) the usual array closest to the fuzzy set A
C) fuzzy set - addition to the fuzzy set A
D) the usual array - addition to the fuzzy set A
E) fuzzy set - symmetrical to the fuzzy set A
12.
What is the characteristic function µА(xi), if µА(xi) < 0,5?
A) 0
B) 1
C) 0,5
D) 0 or 1
E) -1
13. What is the characteristic function µА(xi), if µА(xi) > 0,5?
A)
B)
C)
D)
E)
0
0,5
1
0 иor 1
-1
14. The support of a fuzzy relation R is usually called the set of ordered pairs (x, y), which supplies
function
A) is positive
B) is positive or zero
C) is zero
D) is equal to unity
E) is negative or zero
15. What operation relations between the two fuzzy sets the membership function defined by the
expression of the form max[µR1(x,y),µR2(x,y)]?
A) intersection
B) addition
C) the algebraic sum
D) algebraic product
E) union
16. What operation relations between the two fuzzy sets the membership function defined by the
expression of the form µR1(x,y)‧µR2(x,y) ?
A) intersection
B) addition
C) the algebraic sum
D) algebraic product
E) union
17. What operation relations between the two fuzzy sets the membership function defined by the
expression of the form µR1(x,y) + µR2(x,y) - µR1(x,y)‧µR2(x,y) ?
A) intersection
B) addition
C) the algebraic sum
D) algebraic product
E) union
18. The first projection of the relation R is called a fuzzy set membership function R'1 equal
max(x) [µR (x,y)]
A) min(y) [µR (x,y)]
B) min(x) [µR (x,y)]
C) max(y) [µR (x,y)]
D) sum(y) [µR (x,y)]
19. The second projection of the relation R is called a fuzzy set membership function R'2 equal
A)
B)
C)
D)
max(x) [µR (x,y)]
max(y) [µR (x,y)]
min(y) [µR (x,y)]
min(x) [µR (x,y)]
E) sum(y) [µR (x,y)]
Individual tasks
Problem 1. Given the model parameters of the actual volume of goods of local production sales Fmax,
jmsx, и t05. Using the values of the seasonality of the demand function d d(t) (see table), to find these
months sets a-level (Lu L, L2) actual sales volume of goods produced locally.
Parameter
Options
1
2
3
4
5
6
7
8
9
10
Fmax млнед. 3,2 3,3
3,5 3,6
3,7
3,8
3,9
4,0
4,1
4,2
/max
1,0 0,9
0,8 0,7
0,6
1,0
0,9
0,8
0,7
0,6
*0,5
2,0 2,5
3,0 3,5
4,0
2,0
2,5
3,0
3,5
4,0
Месяцы
1-5 2-6
3-7 9-12, 1
10-12,1, 2
И, 12, 13
12,1-4
1-5
2-6
1-3, И, 12
а
0,7 0,75
0,8 0,85
0,9
0,7
0,75
0,8
0,85
0,9
Problem 1.
1. Perform initial processing financial performance of the enterprise;
2. Find the values of membership functions of linguistic pe¬remennyh "risk of bankruptcy of the
enterprise in the I quarter" and "enterprise risk bankruptcy in the second quarter";
3. To give a verbal description of the enterprise status for the I and II quarters and compare the risk of
bankruptcy in each of these periods.
job options
Options
1
I quarter
X1
х2
Х3
х4
х5
х6
2
II
II
4
5
П
II
0,1
I
II
0,15
I
0,2
I
0,12
0,15
0,02
0,15
0,15
0,15
I
0,16
0,32
0,32
0,001
0,002
0,012
0
0,02
од
0
0
0,92
0,92
1,6
1,3
1,64
1,7
1,0
1,7
1,6
1,6
0,7
0,7
0,55
0,5
0,53
0,55
0,45
0,35
0,55
0,55
1,0
1,0
0,8
0,82
0,88
0,8
0,98
0,8
0,3
0,3
0,1
0,12
0,11
0,11
0,21
0,82
0,15
0,8
0,1
6
X1
х2
Х3
х4
х5
х6
3
7
8
0,1
9
10
I
0,16
П
0,12
I
0,2
П
0,15
I
0,15
II
0,02
I
0,15
II
0,15
I
0,15
II
0,1
0,001
0,02
0,012
0,32
0,1
0
0,32
0,002
0
0
1,6
1,0
1,64
0,92
1,7
1,6
0,92
1,3
1,6
1,7
0,55
0,45
0,53
0,7
0,35
0,55
0,7
0,5
0,55
0,55
0,8
0,98
0,88
1,0
0,82
0,8
1,0
0,82
0,8
0,8
0,1
0,21
0,11
0,3
0,15
0,1
0,3
0,12
0,1
0,11
5.2. Fund assessment tools for intermediate certification in the form of credit
Credit Questions:
1. The linear model of diversified economy (carrying Leontief model).
2. The main types of economic problems, reduced to LPP. LPP Forms and methods of information to
the main LPP.
3. The main theorem of linear programming
4. The algorithm simplex method.
5. Dual LPP and economic sense.
6. The transportation problem. Presentation of TP as a transport matrix.
7. Basic methods of finding the initial plan of operations.
8. The method of potentials to improve the initial TP plan.
9. Economic problems reducing to TP.
10. Economic problems are reduced to the NP.
11. General NP.
12. Graphical method for solving quadratic NP.
13. Graphical method for solving linear fractional NP.
14. Dynamic programming. The decision on the allocation of investment dynamic programming
methods task.
15. The concept of the game as a mathematical description of the conflict. Antagonistic and
non-antagonistic game. Game solution in the presence of a saddle point in its absence.
16. Linear regression. Correlation coefficient.
17. Time series. Addetivnaya and multiplicative model of the time series.
18. Decision-making under certainty. Laplace criterion. Regrets matrix. Criteria Savage and Wald.
The criterion of Pareto optimality.
19. Decision-making in conditions of partial uncertainty. Criterion Baeza. Hurwitz criterion.
20. Fuzzy sets and their application in the economy. Algebraic and logical operations on fuzzy sets.
21. Methods of preparation of the membership functions of fuzzy sets.
22. Fuzzy matching. fuzzy inference system.
23. The concept of fuzzy and linguistic peremennoy.Spetsialnye types of fuzzy numbers.
6. TRAINING METHODICAL AND INFORMATION MAINTENANCE OF THE
DISCIPLINE
6.1. Core books and further reading
№
1
2
3
4
References
Core books
Modeling of economic processes [Text]: studies. for students enrolled in the
special. Economy and exercise. (080100) / ed. MV Grachev N. Cheremnyh, EA
Tumanova. 2nd ed., Rev. and ext. - Moscow: UNITY-DANA, 2013. - 543 p. 10,000 copies. - ISBN 978-5-238-02329-8.
Crassus, Maksim Semenovich. Mathematics for economic specialties [Text]:
Proc. / MS Krass. 4th ed., Rev. - M.: Case, 2003. - 704 p. - ISBN
5-7749-0264-1.
Mathematical methods and models of operations research [Text]: studies. for
universities / ed. VA Kolemaeva. - Moscow: UNITY-DANA, 2008. - 592 p. 3000 copies. - ISBN 978-5-238-01325
Fomin, Gennady Petrovich. Mathematical methods and models in commercial
activities [Text]: Proc. for schools / GP Fomin. - Moscow: Finance and
Statistics, 2001. - 544 p. : Ill. - ISBN 5-279-02310-8.
Number of
copies
15
25
50
31
5
Nevezhin, VP Problems in course of collection "of economic-mathematical
modeling" [Text] / VP Nevezhin, SI Kruzhilov. - M.: Izdat. house "Gorodets",
2005. - 320 p. - ISBN 5-9584-0080-0.
additional literature
Kremer N. Sh Putko BA Higher Mathematics for economic specialties
Moskva.- Higher Education, 2008, 912 p.
Methods and models of solving economic problems: Textbook / SR
Khachatryan, MV Pinegina, V.P.Buyanov. - M .: "Exam" Publishing House,
2005. - 384 p. Alsevich VV Introduction to mathematical economics.
Constructive theory: Textbook. - M .: Editorial URSS, 2005. - 256 p.
Workshop on higher mathematics for economists: Proc. manual for schools /
ed. prof. N. S. Kremer. - Moscow: UNITY-DANA, 2005. - 423 p.
Elena Berezhnaya. Mathematical methods for modeling economic systems
[Text]: studies. manual for students enrolled in the special. "Finance and
Credit", "accountants. Accounting, analysis and audit", "World Economy" / E.
Berezhnaya, VI Berezhnaya. 2nd ed., Rev. and ext. - Moscow: Finance and
Statistics, 2006. - 432 p. : Ill. - 2000 copies. - ISBN 5-279-02940-8.
Kostevich LS Mathematical Programming: Inform.tehnologii optimal
solutions: A manual. - M: New knowledge, 2003.- 424 p.
1
2
3
4
5
30
1
2
1
4
2
6.2. Resources information and telecommunication network "Internet"
№
1
Выходные данные
Электронный мультимедийный учебник http://www.mathelp.spb.ru/
6.3..
№
1
3
List of software
output
Microsoft Office
Eviews 6.0
6.4. The list of information and information systems
№
1
7.
Name: Information Systems
Консультант +
LOGISTICS OF DISCIPLINE
Premises for all types of work, provided for by the curriculum, equipped with the necessary
specialized training furniture and technical means of education, display equipment used for lectures.
Practical exercises are conducted in computer classes, jobs that are equipped with the licensed
software and Internet access.
8.
GUIDELINES FOR THE DEVELOPMENT OF DISCIPLINE
Guidelines for the development of the discipline of "Mathematical
Applications in Economics" addressed to undergraduates of full-time training.
The curriculum in the direction of preparation "Economy" provides the following
types of activities:
- Lectures;
- practical lessons.
In the course of lectures devoted to the topical theoretical issues of
intercultural communication, makes recommendations for self-study and preparation
for practical classes.
During the practical exercises deepened and consolidated knowledge on a number
of issues discussed in the lectures.
In preparation for the practical training every undergraduate should:
- Examine the recommended textbooks;
- study lecture notes;
- prepare the answers to all the questions on the topic being studied.
In consultation with the teacher to prepare undergraduate essay, report or
report on employment.
Issues not discussed in lectures and practical classes, must be examined in the
course of self-study. Control of independent work of students on the curriculum of the
course is carried out in the course of employment by means of oral questioning or testing. In
the course of independent work each student is required to read the primary and secondary
literature on the subject under study, lecture notes complement the missing material,
excerpts from primary sources recommended. Highlight incomprehensible terms find their
value in professional or encyclopedic dictionaries.
With the implementation of various types of study used a variety (including
online) teaching methods, in particular:
- Interactive board for preparing and conducting lectures and
seminars;
To prepare for, monitoring and interim assessment of students can use
http://library.rsue.ru/ university library.
Also, students can take home the necessary books on the subscription or use the
university library reading rooms of the university.
EXPERTS' OPINION
Methodological Council direction "Economy" .
on the working program of the discipline FTD.1 “Mathematical Applications
in Economics” , provided by the curriculum Specialization 38.04.01 Economy,
Master's program 38.04.01.02 International Business. Having reviewed the
structure, content and quality of the program, the Methodical Council notes:
- presented program fully meets the requirements to the level of training
of graduates provided by the Federal Education Standard in the direction of
preparation of the economy, approved by Order of the Ministry of Education
and Science of the Russian Federation of March 30, 2015 N 321;
- interdisciplinary logic is justified, logical sequence of presentation of
educational material for the modules and topics of the planned activities in an
interactive way is observed;
- the budget of classroom time agreed with the time budget for the
various forms of self-study;
- information and methodological support, as well as the utilization of
technical aids meet the requirements of the Federal Education Standard.
Based on the above, the Council proposes to approve the work program
of the discipline " Mathematical Applications in Economics ".
Chairman of the Council: _________________
L.I. Nivorozhkina
Members of the Council: __________________
O.G. Semenyuta
________________
A.M. Ponomareva