1.3 Main stages of mathematical modeling

1) Construction of the model. At this stage, given some «non-mathematical " object - a phenomenon of nature, design, economic plan, production process, etc. Thus, as a rule, a clear description of the situation is difficult. First identifies the main features of the phenomenon and the relationships between them on a qualitative level. Then found qualitative relations formulated in the language of mathematics, that is, a mathematical model. This is the hardest stage of the simulation.

2) The solution of a mathematical problem, which leads to the model. At this stage, much attention is paid to the development of algorithms and numerical methods for solving the problem on a computer with which results can be found with the required accuracy and time allowed.

3) Interpretation of the consequences of the mathematical model. Investigation, derived from the model in the language of mathematics, interpreted language accepted in the art.

4) Check the adequacy of the model. At this point it becomes clear whether the results of the experiment are consistent with the theoretical model of the consequences within certain accuracy.

5) Modification of the model. At this stage or complication of the model, make it more adequate reality or its simplification to achieve practically acceptable solution.

Requirements for models.

1. Versatility - characterizes the completeness of the display model of the studied properties of the real object.

2. Adequacy - the ability to reflect the desired properties of an object with an error no greater than specified.

3. Accuracy - estimated degree of coincidence of the characteristics of a real object and the values ​​of these parameters obtained by the models.

4. Economical - determined by the cost of computer resources of memory and time for its implementation and operation.

1.4 Mathematical Modeling and computerization or simulation models

Now, when the country come almost universal computerization, from specialists in various professions to hear sayings: "There have implemented a computer, then all problems will be solved immediately." This view is absolutely not true; by themselves without computer mathematical models of various processes cannot do anything about global computerization can only dream of.

In confirmation of the above, try to justify the need for modeling , including mathematical, reveal its advantages in human cognition and transformation of the external world , to identify existing deficiencies and go ... to the simulation , i.e. modeling using a computer . But first lets things.

First of all, to answer the question: What is a model?

Model - it is material or mentally represented object that the process of learning (study) replaces the original, retaining some of this research are important for typical properties.

Well- built models are available for research - rather than a real object. For example, not allowed to experiment with the country's economy in educational purposes, here without a model cannot do.

In summary, you can answer the question: Why do we need models? We need it in order to understand how the object (its structure, properties and laws of development, interaction with the outside world). We need it to Learning to manage the object (process) to determine the best strategy to predict effects of the object.

What is positive in any model? It allows you to gain new knowledge about the object, but, unfortunately, in varying degrees, is not complete.

Model formulated in the language of mathematics by using mathematical methods is called a mathematical model.

The starting point of its construction is usually a certain task, such as economic. Widespread as descriptive and mathematical optimization, characterizing various economic processes and phenomena, such as:

Allocation of resources;

Rational cutting;

Transportation;

Consolidations have;

Network planning.

How is the construction of a mathematical model?

First, formulate the object and purpose of the study.

Secondly, the important characteristics are allocated corresponding to the purpose.

Third, verbally describe the relationship between elements of the model. Next, the relationship is formalized. And it is calculated by a mathematical model and analysis of the resulting solution.

Using this algorithm can solve any optimization problem, including multi-criteria i.e. that which is pursued not one, but several goals, including controversial.

Here is an example. Queuing theory - the problem of formation of queues. Need to balance two factors - the cost of maintenance of servers and costs stay in line. The formal description of the model makes calculations using analytical and numerical methods. If the model is good, the answers have been found with the help of adequate modeling system, if bad, it is subject to improvements and replacements. Adequacy criterion is practice.

Optimization models, including multi-criteria, have a common property - known goal (or more goals) to achieve which often have to deal with complex systems, where it is not so much about solving optimization problems, much of the study and prediction of states depending on the elected management strategies. And here we face the difficulties of implementation of the previous plan. They are as follows:

1. Complex system contains a lot of links between elements

2. Actual system is influenced by random factors, keeping them analytically impossible

3. Opportunity to compare the original with the model exists only at the beginning and after the application of mathematical apparatus, because Intermediate results may not have analogues in the real system.

In connection with the above difficulties arising in the study of complex systems, the practice required a more flexible method, and he appeared - Simulation "Simulation modeling".

Usually, the simulation model refers to a set of computer programs, describing the operation of the individual blocks of systems and rules of interaction between them. Using random variables makes it necessary to conduct multiple experiments with simulation system (on a computer) and the subsequent statistical analysis of the results. A very common example of the use of simulation models is to solve the problem of queuing Monte Carlo method.

Thus, working with the simulation system is an experiment carried out on a computer. What are the benefits?

- Great proximity to the real system than the mathematical models ;

Block - principle allows each unit to verify its inclusion in the overall system;

- Using dependencies more complex nature, cannot be described by simple mathematical relations.

Define these advantages disadvantages

- Build a simulation model longer difficult and expensive;

- To work with the simulation system must have the appropriate class of computers;

- User interaction and simulation model (interface) should not be too complicated, comfortable and well -known;

Simulation model - building requires a deeper study of the real process , rather than mathematical modeling .

The question arises: can replace simulation optimization techniques? No, but complements them conveniently. Simulation model - a program, which implements an algorithm to optimize the management of that before optimization problem is solved.

Thus, neither a computer nor a mathematical model or algorithm to study it separately cannot solve a rather complex task. But together they are the force that allows the world to learn to manage them for the benefit of man.

Chapter 2. Practical part.

Modeling family

2.1. Prerequisites for creating a model family

In sociological research is to study the important issue of family and gender relations as modeling objects. But sociologists do not pay enough attention to the formalization and calculation of multiple variables gender adaptive behavior. And this is a serious omission in the search for answers to many questions. How families can become stable and survive in a crisis situation? As the economic transformation affects women in the social system? What adaptive actions should lead to maintaining the stability of the family? How to determine the adaptive capacity of gender consistency and personal activity of women?

It is desirable to create a multi- agent-based simulation model , with which it will be possible not only to investigate the different types of adaptive behavior and the very identity of the women and men in the economic crisis , but also by analyzing the change in gender roles to explore self-identity roles of women and men in marriage .

The first goal of our research is to demonstrate how can be studied with the help of multi-agent simulation of adaptive behavior and stability of the family.

The second goal - to test the following hypotheses:

1) Stability and adaptive behavior families during the economic crisis associated with various forms of individual and joint actions, such as personal family activity and compatibility of gender roles;

2) Balance and consistency of gender roles (the expected behavior of the individual, depending on his gender, that of gender), professional orientation and activity of women defined high adaptive capacity and stability of the family.

Comparison of the behavior of women in different situations will reveal the types of behavior that affect the choice of a particular action aimed at preserving the family.

Simulating various conditions and factors of influence on family very useful for analyzing the stability of the family. In this chapter we look at a model of adaptive behavior of the family in terms of economically unstable society. With it possible:

1) To study the behavior of women and men in terms of economic instability;

2) The results of assessments of adaptive selection;

3) Simulation and analysis of a large number of different situations;

4) The many experiments.

2.1.1 Model of adaptive behavior of the family. Formalization of the family

The concept of «family strategy" focuses on the mechanisms of formation and reproduction behavior of the family in different spheres of life and different situations. Thus, isolated family strategy in the implementation of economic , cultural and reproductive functions; strategy to overcome the crisis , as external to the family (e.g. , economic , social and cultural ) , and within - family (e.g. , associated with the transition to a new stage in the life cycle of the family or the crisis of family relations ) . Distinguish strategies specific to families with one or two working spouses, for single-parent families, to dare, belonging to different social groups. In the proposed model of family problems related material family instability and experience constant stress.

What are the factors of adaptive behavior and stability of the family? To answer this question it is necessary to determine the identity and interests of family members. On the basis of these features in the model will be split into different groups of agents.

The first group of agents consists of women who have a professional attitude. The second group consists of agents of women with a focus on family life. Modern Kazakhstan gender system demonstrates a correlation between professional orientation and successful adaptive economic behavior of the family. Many families in which women are busy with their careers, socially and economically more prosperous than those with a family -oriented women. The increasing spread of the same families, supported women's professional orientation, due to the personal activity of women. A third group of agents are women with high creative energy and adaptive abilities. In contrast, the fourth group includes women with low adaptive capacity and social insecurity.

According to the survey the majority of Kazakhstan men for major sources of identity are the work. During the economic crisis and growth of professional activity of women is an important factor of harmony and balance in gender roles. That is why the model is assumed that agents have professional male identity and the high level of tolerance to the professional activity of women.

In a situation of economic crisis is the restructuring family lifestyle with regard to adaptation to the realities of the market, an unstable society. We believe that the main strategy of the family during the economic crisis is a survival strategy. It takes place in various forms, but the main family- oriented lifestyle, involving the adoption by both men and women's traditional roles. However, this study seeks to find out what strategies adaptive behavior elects family to overcome the socio- economic crisis. If the family chooses the wrong strategy for you, but for a time it overcomes the difficulties generated by the economic crisis, but it inevitably falls on her again with renewed vigor. It is important to understand how men and women with certain gender roles, adapting to life in a crisis society. That contributes to the success of some other and prevents adaptation?

Family budget, in general, is the sum of the salaries of husband and wife. They point out the "poor" families with incomes below the subsistence minimum, and «wealthy «with revenue in excess of the subsistence minimum. And a family with a relatively high income all family «given ", mainly husbands earnings. For each family in the model set own income between men and women and family consumption.