FEDERAL FISHERIES AGENCY
MINISTRY OF AGRICULTURE
KAMCHATKA STATE TECHNICAL UNIVERSITY
DEPARTMENT OF INFORMATION SYSTEMS
Topic: “SIMITATION MODELING OF ECONOMIC
ENTERPRISE ACTIVITIES"
Course work
Head: position
Bilchinskaya S.G. "__" ________2006
Developer: student gr.
Zhiteneva D.S. 04 Pi1 “__” ________2006
The work is protected by "___" __________2006. with rating______
Petropavlovsk-Kamchatsky, 2006
Introduction........................................................ ........................................................ ........................... 3
1. Theoretical foundations of simulation modeling.................................................... 4
1.1. Modeling. Simulation modeling................................................... 4
1.2. Monte Carlo method......................................................... ............................................ 9
1.3. Using the laws of distribution of random variables.................................... 12
1.3.1. Uniform distribution................................................... ................ 12
1.3.2. Discrete distribution (general case)..................................................... 13
1.3.3. Normal distribution................................................ .................. 14
1.3.4. Exponential distribution................................................................... ...... 15
1.3.5. Generalized Erlang distribution................................................................. .. 16
1.3.6. Triangular distribution................................................... ................. 17
1.4. Planning a computer simulation experiment.................................... 18
1.4.1. Cybernetic approach to organizing experimental studies of complex objects and processes.................................................... ........................................................ ............. 18
1.4.2. Regression analysis and control of model experiment. 19
1.4.3. Second-order orthogonal planning................................................... 20
2. Practical work................................................... ........................................................ ..... 22
3. Conclusions on the “Production Efficiency” business model.................................................. 26
Conclusion................................................. ........................................................ .................... 31
Bibliography............................................... ................................... 32
APPENDIX A................................................... ........................................................ .......... 33
APPENDIX B................................................... ........................................................ .......... 34
APPENDIX B................................................... ........................................................ .......... 35
APPENDIX D................................................... ........................................................ .......... 36
APPENDIX D................................................... ........................................................ .......... 37
APPENDIX E................................................... ........................................................ .......... 38
INTRODUCTION
Modeling in economics began to be used long before economics finally took shape as an independent scientific discipline. Mathematical models were used by F. Quesnay (1758 Economic table), A. Smith (classical macroeconomic model), D. Ricardo (international trade model). In the 19th century, the mathematical school made a great contribution to modeling (L. Walras, O. Cournot, V Pareto, F. Edgeworth, etc.). In the 20th century, methods of mathematical modeling of the economy were used very widely, and outstanding works of Nobel Prize laureates in economics (D. Hicks, R. Solow, V. Leontiev, P. Samuelson) are associated with their use.
Coursework on the subject “Simulation Modeling of Economic Processes” is an independent educational and research work.
The purpose of writing this course work is to consolidate theoretical and practical knowledge. Coverage of approaches and methods of using simulation modeling in project economic activities.
The main task is to study the efficiency of the enterprise’s economic activities using simulation modeling.
1. THEORETICAL FOUNDATIONS OF SIMULATION MODELING
1.1. Modeling. Simulation modeling
In the process of managing various processes, the need to predict results under certain conditions constantly arises. To speed up decision-making on choosing the optimal control option and save money on experiments, process modeling is used.
Modeling is the transfer of the properties of one system, which is called a modeling object, to another system, which is called an object model; the influence on the model is carried out in order to determine the properties of the object by the nature of its behavior.
Such a replacement (transfer) of the properties of an object has to be done in cases where its direct study is difficult or even impossible. As modeling practice shows, replacing an object with its model often gives a positive effect.
A model is a representation of an object, system or concept (idea) in some form that is different from the form of its real existence. A model of an object can be either an exact copy of that object (albeit made of a different material and on a different scale), or it can display some characteristic properties of the object in an abstract form.
At the same time, during the modeling process it is possible to obtain reliable information about the object with less time, money, money and other resources.
The main goals of modeling are:
1) analysis and determination of properties of objects according to the model;
2) designing new systems and solving optimization problems using a model (finding the best option);
3) management of complex objects and processes;
4) predicting the behavior of an object in the future.
The most common types of modeling are:
1) mathematical;
2) physical;
3) imitation.
In mathematical modeling, the object under study is replaced by the corresponding mathematical relationships, formulas, expressions, with the help of which certain analytical problems are solved (analysis is done), optimal solutions are found, and forecasts are made.
Physical models represent real systems of the same nature as the object under study or another. The most typical option for physical modeling is the use of mock-ups, installations, or the selection of fragments of an object for conducting limited experiments. And it has found its most widespread application in the field of natural sciences, sometimes in economics.
For complex systems, which include economic, social, information and other social information systems, simulation modeling has found wide application. This is a common type of analog modeling, implemented using a set of mathematical tools of special simulating computer programs and programming technologies, which, through analogue processes, allow a targeted study of the structure and functions of a real complex process in computer memory in the “simulation” mode, and optimization of some of its parameters.
To obtain the necessary information or results, it is necessary to “run” simulation models, rather than “solve” them. Simulation models are not capable of forming their own solution in the way that is the case in analytical models, but can only serve as a means for analyzing the behavior of the system under conditions that are determined by the experimenter.
Therefore, simulation is not a theory, but a methodology for solving problems. Moreover, simulation is only one of several critical problem-solving techniques available to the systems analyst. Since it is necessary to adapt a tool or method to solve a problem, and not vice versa, a natural question arises: in what cases is simulation modeling useful?
The need to solve problems through experimentation becomes obvious when there is a need to obtain specific information about the system that cannot be found in known sources. Direct experimentation on a real system eliminates many difficulties if it is necessary to ensure consistency between the model and real conditions; however, the disadvantages of such experimentation are sometimes quite significant:
1) may disrupt the established operating procedure of the company;
2) if people are an integral part of the system, then the results of experiments can be influenced by the so-called Hawthorne effect, which manifests itself in the fact that people, feeling that they are being watched, can change their behavior;
3) it may be difficult to maintain the same operating conditions each time an experiment is repeated or throughout a series of experiments;
4) obtaining the same sample size (and, therefore, statistical significance of the experimental results) may require excessive time and money;
5) when experimenting with real systems, it may not be possible to explore many alternative options.
For these reasons, the researcher should consider the appropriateness of using simulation modeling when any of the following conditions exist:
1. There is no complete mathematical formulation of this problem, or analytical methods for solving the formulated mathematical model have not yet been developed. Many queuing models that involve queuing fall into this category.
2. Analytical methods are available, but the mathematical procedures are so complex and time-consuming that simulation provides a simpler way to solve the problem.
3. Analytical solutions exist, but their implementation is impossible due to insufficient mathematical training of existing personnel. In this case, the costs of design, testing and work on the simulation model should be compared with the costs associated with inviting outside specialists.
4. In addition to assessing certain parameters, it is advisable to monitor the progress of the process over a certain period using a simulation model.
5. Simulation modeling may be the only option due to the difficulties of setting up experiments and observing phenomena in real conditions (for example, studying the behavior of spacecraft during interplanetary flights).
6. Long-term systems or processes may require timeline compression. Simulation modeling makes it possible to fully control the timing of the process being studied, since the phenomenon can be slowed down or accelerated at will (for example, studies of urban decline).
Additional benefit Simulation modeling can be considered the broadest possible application in the field of education and training. The development and use of a simulation model allows the experimenter to see and experience real processes and situations on the model. This, in turn, should greatly help to understand and feel the problem, which stimulates the process of searching for innovations.
Simulation modeling is implemented through a set of mathematical tools, special computer programs and techniques that allow using a computer to carry out targeted modeling in the “simulation” mode of the structure and functions of a complex process and optimization of some of its parameters. A set of software tools and modeling techniques determines the specifics of the modeling system - special software.
Simulation modeling of economic processes is usually used in two cases:
1. to manage a complex business process, when a simulation model of a managed economic entity is used as a tool in the contour of an adaptive management system created on the basis of information technology;
2. when conducting experiments with discrete-continuous models of complex economic objects to obtain and “observe” their dynamics in emergency situations associated with risks, the full-scale modeling of which is undesirable or impossible.
Simulation modeling as a special information technology consists of the following main stages:
1. Structural Process Analysis. At this stage, the structure of a complex real process is analyzed and decomposed into simpler interconnected subprocesses, each of which performs a specific function. The identified subprocesses can be subdivided into other simpler subprocesses. Thus, the structure of the simulated process can be represented as a graph with a hierarchical structure.
Structural analysis is especially effective in modeling economic processes, where many of the constituent subprocesses occur visually and do not have a physical essence.
2. Formalized description of the model. The resulting graphical representation of the simulation model, the functions performed by each subprocess, and the interaction conditions of all subprocesses must be described in a special language for subsequent translation.
This can be done in various ways: described manually in a specific language or using a computer graphic designer.
3. Model building. This stage includes translation and editing of connections, as well as verification of parameters.
4. Conducting an extreme experiment. At this stage, the user can obtain information about how close the created model is to a real-life phenomenon, and how suitable this model is for studying new, untested values of arguments and parameters of the system.
1.2. Monte Carlo method
Statistical tests using the Monte Carlo method represent the simplest simulation modeling in the complete absence of any rules of behavior. Obtaining samples using the Monte Carlo method is the basic principle of computer modeling of systems containing stochastic or probabilistic elements. The origin of the method is associated with the work of von Neumann and Ulan in the late 1940s, when they introduced the name “Monte Carlo” for it and applied it to solving certain problems of shielding nuclear radiation. This mathematical method was known earlier, but it found its rebirth in Los Alamos in closed work on nuclear technology, which was carried out under the code designation “Monte Carlo”. The application of the method turned out to be so successful that it became widespread in other areas, in particular in economics.
Therefore, to many specialists, the term “Monte Carlo method” is sometimes considered synonymous with the term “simulation modeling,” which is generally incorrect. Simulation modeling is a broader concept, and the Monte Carlo method is an important, but far from the only methodological component of simulation modeling.
According to the Monte Carlo method, a designer can simulate the operation of thousands of complex systems that control thousands of varieties of similar processes, and examine the behavior of the entire group by processing statistical data. Another way to apply this method is to simulate the behavior of a control system over a very long period of model time (several years), and the astronomical execution time of the modeling program on a computer can be a fraction of a second.
In Monte Carlo analysis, a computer uses a pseudorandom number generation procedure to simulate data from the population being studied. The Monte Carlo analysis procedure constructs samples from the population according to user instructions, and then performs the following actions: simulates a random sample from the population, analyzes the sample, and stores the results. After a large number of iterations, the stored results closely mimic the actual distribution of the sample statistics.
In various tasks encountered when creating complex systems, quantities can be used whose values are determined randomly. Examples of such quantities are:
1 random moments in time at which orders are received by the company;
3 external influences (requirements or changes in laws, payments of fines, etc.);
4 payment of bank loans;
5 receipt of funds from customers;
6 measurement errors.
The corresponding variables can be a number, a collection of numbers, a vector, or a function. One of the variations of the Monte Carlo method for the numerical solution of problems involving random variables is the statistical testing method, which involves modeling random events.
The Monte Carlo method is based on statistical testing and is extreme in nature, and can be used to solve completely deterministic problems such as matrix inversion, solving partial differential equations, finding extrema and numerical integration. In Monte Carlo calculations, statistical results are obtained through repeated trials. The probability that these results differ from the true results by no more than a given value is a function of the number of trials.
The basis of Monte Carlo calculations is the random selection of numbers from a given probability distribution. In practical calculations, these numbers are taken from tables or obtained through some operations, the results of which are pseudo-random numbers with the same properties as numbers obtained by random sampling. There are a large number of computational algorithms that allow you to obtain long sequences of pseudorandom numbers.
One of the simplest and most effective computational methods for obtaining a sequence of uniformly distributed random numbers r i, using, for example, a calculator or any other device operating in the decimal number system, involves only one multiplication operation.
The method is as follows: if r i = 0.0040353607, then r i+1 =(40353607ri) mod 1, where mod 1 means the operation of extracting only the fractional part after the decimal point from the result. As described in various literature sources, the numbers r i begin to repeat after a cycle of 50 million numbers, so that r 5oooooo1 = r 1 . The sequence r 1 turns out to be uniformly distributed over the interval (0, 1).
The use of the Monte Carlo method can give a significant effect when modeling the development of processes, field observation of which is undesirable or impossible, and other mathematical methods in relation to these processes are either not developed or are unacceptable due to numerous reservations and assumptions that can lead to serious errors or wrong conclusions. In this regard, it is necessary not only to observe the development of the process in undesirable directions, but also to evaluate hypotheses about the parameters of undesirable situations to which such development will lead, including the parameters of risks.
1.3. Using the laws of distribution of random variables
For a qualitative assessment of a complex system, it is convenient to use the results of the theory of random processes. Experience in observing objects shows that they operate under the influence of a large number of random factors. Therefore, predicting the behavior of a complex system can only make sense within the framework of probabilistic categories. In other words, for expected events only the probabilities of their occurrence can be indicated, and for some values it is necessary to limit ourselves to the laws of their distribution or other probabilistic characteristics (for example, average values, variances, etc.).
To study the process of functioning of each specific complex system, taking into account random factors, it is necessary to have a fairly clear understanding of the sources of random influences and very reliable data on their quantitative characteristics. Therefore, any calculation or theoretical analysis associated with the study of a complex system is preceded by the experimental accumulation of statistical material characterizing the behavior of individual elements and the system as a whole in real conditions. Processing of this material allows us to obtain initial data for calculation and analysis.
The law of distribution of a random variable is a relationship that allows one to determine the probability of the occurrence of a random variable in any interval. It can be specified tabularly, analytically (in the form of a formula) and graphically.
There are several laws of distribution of random variables.
1.3.1. Uniform distribution
This type of distribution is used to obtain more complex distributions, both discrete and continuous. Such distributions are obtained using two main techniques:
a) inverse functions;
b) combining quantities distributed according to other laws.
The uniform law is the law of distribution of random variables, which has a symmetrical form (rectangle). The uniform distribution density is given by the formula:
that is, in the interval to which all possible values of the random variable belong, the density maintains a constant value (Fig. 1).
Fig.1 Probability density function and characteristics of uniform distribution
In simulation models of economic processes, uniform distribution is sometimes used to model simple (single-stage) work, when calculating according to network work schedules, in military affairs - to model the time it takes for units to travel, the time of digging trenches and the construction of fortifications.
Uniform distribution is used when the only thing known about time intervals is that they have a maximum spread, and nothing is known about the probability distributions of these intervals.
1.3.2. Discrete distribution
The discrete distribution is represented by two laws:
1) binomial, where the probability of an event occurring in several independent trials is determined by the Bernoulli formula:
n – number of independent tests
m is the number of occurrences of the event in n trials.
2) Poisson distribution, where with a large number of trials the probability of an event occurring is very small and is determined by the formula:
k – number of occurrences of an event in several independent trials
Average number of occurrences of an event across multiple independent trials.
1.3.3. Normal distribution
The normal, or Gaussian, distribution is undoubtedly one of the most important and frequently used types of continuous distributions. It is symmetrical with respect to the mathematical expectation.
Continuous random variable t has a normal probability distribution with parameters T And > Oh, if its probability density has the form (Fig. 2, Fig. 3):
Where T- expected value M[t];
Fig.2, Fig.3 Probability density function and characteristics of normal distribution
Any complex work at economic facilities consists of many short, sequential elementary components of work. Therefore, when estimating labor costs, the assumption is always valid that their duration is a random variable distributed according to a normal law.
In simulation models of economic processes, the law of normal distribution is used to model complex multi-stage work.
1.3.4. Exponential distribution
It also occupies a very important place when conducting a systematic analysis of economic activity. Many phenomena obey this distribution law, for example:
1 time of receipt of the order at the enterprise;
2 customers visiting a supermarket;
3 telephone conversations;
4 service life of parts and assemblies in a computer installed, for example, in an accounting department.
The exponential distribution function looks like this:
F(x)= at 0 Exponential distribution parameter, >0. Exponential distributions are special cases of gamma distributions. Rice. 5 Probability density function of gamma distribution In simulation models of economic processes, the exponential distribution is used to model the intervals of orders coming into the firm from numerous customers. In reliability theory, it is used to model the time interval between two consecutive faults. In communications and computer science – for modeling information flows. 1.3.5. Generalized Erlang distribution P(t)= at t≥0; Where K-elementary sequential components distributed according to exponential law. The generalized Erlang distribution is used to create both mathematical and simulation models. This distribution is convenient to use instead of the normal distribution if the model is reduced to a purely mathematical problem. In addition, in real life, there is an objective probability of groups of requests arising as a reaction to some actions, therefore group flows arise. The use of purely mathematical methods to study the effects of such group flows in models is either impossible due to the lack of a way to obtain an analytical expression, or is difficult, since the analytical expressions contain a large systematic error due to the numerous assumptions due to which the researcher was able to obtain these expressions. To describe one of the varieties of group flow, you can use the generalized Erlang distribution. The emergence of group flows in complex economic systems leads to a sharp increase in the average duration of various delays (orders in queues, payment delays, etc.), as well as to an increase in the probabilities of risk events or insured events. 1.3.6. Triangular distribution A triangular distribution is more informative than a uniform one. For this distribution, three quantities are determined - minimum, maximum and mode. The graph of the density function consists of two straight segments, one of which increases as the X from the minimum value to the mode, and the other decreases with change X from the mode value to the maximum. The mathematical expectation value of a triangular distribution is equal to one third of the sum of the minimum, mode and maximum. The triangular distribution is used when the most probable value over a certain interval is known and the piecewise linear nature of the density function is assumed. Fig.5 Probability density function and characteristics of the triangular distribution. The triangular distribution is easy to apply and interpret, but there needs to be a good reason for choosing it. In simulation models of economic processes, such a distribution is sometimes used to model access times to databases. 1.4. Planning a computer simulation experiment The simulation model, regardless of the chosen modeling system (for example, Pilgrim or GPSS), allows you to obtain the first two moments and information about the distribution law of any quantity of interest to the experimenter (the experimenter is a subject who needs qualitative and quantitative conclusions about the characteristics of the process under study). 1.4.1. Cybernetic approach to organizing experimental studies of complex objects and processes. Experimental planning can be considered as a cybernetic approach to organizing and conducting experimental studies of complex objects and processes. The main idea of the method is the possibility of optimal control of an experiment under conditions of uncertainty, which is akin to the premises on which cybernetics is based. The goal of most research work is to determine the optimal parameters of a complex system or optimal conditions for a process: 1. determining the parameters of an investment project under conditions of uncertainty and risk; 2. selection of structural and electrical parameters of the physical installation, ensuring the most advantageous mode of its operation; 3. obtaining the maximum possible reaction yield by varying temperature, pressure and ratio of reagents - in chemistry problems; 4. selection of alloying components to obtain an alloy with the maximum value of any characteristic (viscosity, tensile strength, etc.) - in metallurgy. When solving problems of this kind, it is necessary to take into account the influence of a large number of factors, some of which cannot be regulated and controlled, which makes a complete theoretical study of the problem extremely difficult. Therefore, they follow the path of establishing basic patterns through a series of experiments. The researcher was able to use simple calculations to express the results of the experiment in a form convenient for their analysis and use. 1.4.2. Regression analysis and control of model experiment Fig.7 Example of averaging experimental results Range of values ηv in this case it is determined not only by measurement errors, but mainly by the influence of interference z j. The complexity of the optimal control problem is characterized not only by the complexity of the dependence itself η v (v = 1, 2, …, n), but also the influence z j, which introduces an element of randomness into the experiment. Dependency graph η v (x i) determines the correlation between quantities ηv And x i, which can be obtained from the results of an experiment using methods of mathematical statistics. Calculation of such dependencies with a large number of input parameters x i and significant influence of interference z j and is the main task of the experimental researcher. Moreover, the more complex the task, the more effective the use of experimental design methods becomes. There are two types of experiments: Passive; Active. At passive experiment the researcher only monitors the process (changes in its input and output parameters). Based on the observation results, a conclusion is then drawn about the influence of the input parameters on the output parameters. A passive experiment is usually performed on the basis of an ongoing economic or production process that does not allow active intervention by the experimenter. This method is inexpensive but time consuming. Active experiment carried out mainly in laboratory conditions, where the experimenter has the opportunity to change the input characteristics according to a predetermined plan. Such an experiment leads to the goal faster. The corresponding approximation methods are called regression analysis. Regression analysis is a methodological toolkit for solving problems of forecasting, planning and analysis of the economic activities of enterprises. The objectives of regression analysis are to establish the form of dependence between variables, evaluate the regression function and establish the influence of factors on the dependent variable, estimate unknown values (prediction of values) of the dependent variable. 1.4.3. Second-order orthogonal planning. Orthogonal experimental planning (compared to non-orthogonal) reduces the number of experiments and significantly simplifies calculations when obtaining a regression equation. However, such planning is only feasible if it is possible to conduct an active experiment. A practical means of finding an extremum is a factorial experiment. The main advantages of a factorial experiment are its simplicity and the ability to find an extreme point (with some error) if the unknown surface is sufficiently smooth and there are no local extrema. It is worth noting two main drawbacks of the factorial experiment. The first is the impossibility of searching for an extremum in the presence of stepwise discontinuities of the unknown surface and local extrema. The second is the lack of means of describing the nature of the surface near the extreme point due to the use of the simplest linear regression equations, which affects the inertia of the control system, since in the control process it is necessary to conduct factorial experiments to select control actions. For control purposes, second-order orthogonal planning is most suitable. Typically, an experiment consists of two stages. First, using a factorial experiment, the region where the extreme point exists is found. Then, in the region where the extreme point exists, an experiment is carried out to obtain a 2nd order regression equation. The 2nd order regression equation allows you to immediately determine control actions, without conducting additional tests or experiments. Additional experimentation will be required only in cases where the response surface changes significantly under the influence of uncontrolled external factors (for example, a significant change in tax policy in the country will seriously affect the response surface reflecting the production costs of the enterprise 2. PRACTICAL WORK. In this section we will look at how the above theoretical knowledge can be applied to specific economic situations. The main task of our course work is to determine the efficiency of an enterprise engaged in commercial activities To implement the project, we chose the Pilgrim package. The Pilgrim package has a wide range of capabilities for simulating the temporal, spatial and financial dynamics of modeled objects. It can be used to create discrete-continuous models. The models being developed have the property of collective control of the modeling process. You can insert any blocks into the model text using the standard C++ language. The Pilgrim package has the property of mobility, i.e. portable to any other platform if a C++ compiler is available. Models in the Pilgrim system are compiled and therefore have high performance, which is very important for working out management decisions and adaptive selection of options in a super-accelerated time scale. The object code obtained after compilation can be built into developed software systems or transferred (sold) to the customer, since the tools of the Pilgrim package are not used when operating the models. The fifth version of Pilgrim is a software product created in 2000 on an object-oriented basis and taking into account the main positive properties of previous versions. Advantages of this system: Focus on joint modeling of material, information and “monetary” processes; Availability of a developed CASE shell that allows you to construct multi-level models in the mode of structural system analysis; Availability of interfaces with databases; The ability for the end user of models to directly analyze the results thanks to the formalized technology for creating functional windows for monitoring the model using Visual C++, Delphi or other tools; The ability to manage models directly during their execution using special dialog windows. Thus, the Pilgrim package is a good tool for creating both discrete and continuous models, has many advantages and greatly simplifies model creation. The object of observation is an enterprise that sells manufactured goods. For statistical analysis of data on the functioning of the enterprise and comparison of the results obtained, all factors influencing the process of production and sale of goods were compared. The company produces goods in small batches (the size of these batches is known). There is a market where these products are sold. The batch size of the purchased product is generally a random variable. The business process block diagram contains three layers. On two layers there are autonomous processes “Production” (Appendix A) and “Sales” (Appendix B), the schemes of which are independent of each other because there are no ways to transfer transactions. The indirect interaction of these processes occurs only through resources: material resources (in the form of finished products) and monetary resources (mainly through a current account). Management of monetary resources occurs on a separate layer - in the “Cash Transactions” process (Appendix B). Let us introduce an objective function: the delay time for payments from the current account TRS. Main control parameters: 1 unit price; 2 volume of the batch produced; 3 the amount of the loan requested from the bank. Having fixed all other parameters: 4 batch release time; 5 number of production lines; 6 interval of order receipt from customers; 7 variation in the size of the lot being sold; 8 cost of components and materials for production of the batch; 9 starting capital in the current account; Trs can be minimized for a specific market situation. The minimum TRS is reached at one of the maximums of the average amount of money in the current account. Moreover, the probability of a risk event - non-payment of loan debts - is close to a minimum (this can be proven during a statistical experiment with the model). The first process " Production"(Appendix A) implements the basic elementary processes. Node 1 simulates the receipt of orders for the production of batches of products from the company management. Node 2 – attempt to get a loan. An auxiliary transaction appears in this node - a request to the bank. Node 3 – waiting for credit by this request. Node 4 is the bank administration: if the previous loan is returned, then a new one is granted (otherwise the request waits in the queue). Node 5 transfers the loan to the company's current account. At node 6, the auxiliary request is destroyed, but the information that the loan has been granted is a “barrier” to the next request for another loan (hold operation). The main order transaction passes through node 2 without delay. In node 7, payment for components is made if there is a sufficient amount in the current account (even if the loan is not received). Otherwise, there is a wait for either a loan or payment for the products sold. At node 8, the transaction is queued if all production lines are busy. In node 9, a batch of products is manufactured. At node 10, an additional application for loan repayment occurs if the loan was previously allocated. This application is received at node 11, where money is transferred from the company’s current account to the bank; if there is no money, then the application is pending. After the loan is repaid, this application is destroyed (at node 12); The bank received information that the loan was repaid and the company can be issued the next loan (operation rels). The order transaction passes through node 10 without delay, and at node 13 it is destroyed. Next, it is considered that the batch has been manufactured and has arrived at the finished goods warehouse. Second process " Sales"(Appendix B) simulates the main functions for selling products. Node 14 is a generator of transactions that purchase products. These transactions go to the warehouse (node 15), and if the requested quantity of goods is there, then the goods are released to the buyer; otherwise the buyer waits. Node 16 simulates goods release and queue control. After receiving the goods, the buyer transfers money to the company’s bank account (node 17). At node 18 the customer is considered served; the corresponding transaction is no longer needed and is destroyed. Third process " Cash transactions"(Appendix B) simulates accounting entries. Requests for postings come from the first layer from nodes 5, 7, 11 (Production process) and from node 17 (Sales process). The dotted lines show the movement of cash amounts on Account 51 (“Current account”, node 20), account 60 (“Suppliers, contractors”, node 22), account 62 (“Buyers, customers”, node 21) and account 90 (“ Bank", node 19). The conventional numbers roughly correspond to the chart of accounts. Node 23 simulates the work of the financial director. Serviced transactions, after accounting entries, go back to the nodes from which they came; the numbers of these nodes are in the transaction parameter t→updown. The source code of the model is presented in Appendix D. This source code builds the model itself, i.e. creates all nodes (represented in the business process block diagram) and connections between them. The code can be generated by the Pilgrim constructor (Gem), in which processes are built in object form (Appendix E). The model is created using Microsoft Developer Studio. Microsoft Developer Studio is a software package for application development based on the C++ language. After adding additional libraries (Pilgrim.lib, comctl32.lib) and resource files (Pilgrim.res) to the project, we compile this model. After compilation we get a ready-made model. A report file is automatically created that stores the simulation results obtained after one run of the model. The report file is presented in Appendix D. 3. CONCLUSIONS ON THE BUSINESS MODEL “PRODUCTION EFFICIENCY” 1) node number; 2) Name of the node; 3) Node type; 5) M(t) average waiting time; 6) Input counter; 7) Remaining transactions; 8) The state of the node at this moment. The model consists of three independent processes: the main production process (Appendix A), the product sales process (Appendix B) and the cash flow management process (Appendix B). Basic production process.
During the period of business process modeling in node 1 (“Orders”), 10 applications for the manufacture of products were generated. The average time for order generation is 74 days, as a result, one transaction was not included in the time frame of the modeling process. The remaining 9 transactions entered node 2 (“Fork1”), where a corresponding number of requests to the bank for a loan were created. The average waiting time is 19 days, this is the simulation time during which all transactions were satisfied. Next, you can see that 8 requests received a positive response in node 3 (“Issue permission”). The average time to obtain a permit is 65 days. The load on this node averaged 70.4%. The state of the node at the end of the simulation time is closed, this is due to the fact that this node provides a new loan only if the previous one is returned, therefore, the loan at the end of the simulation was not repaid (this can be seen from node 11). Node 5 transfers the loan to the company's current account. And, as can be seen from the results table, the bank transferred 135,000 rubles to the company’s account. At node 6, all 11 loan requests were destroyed. In node 7 (“Payment to suppliers”), payment for components was made in the amount of the entire loan received previously (RUB 135,000). At node 8 we see that 9 transactions are queued. This occurs when all production lines are busy. In node 9 (“Order fulfillment”), the direct production of products is carried out. It takes 74 days to produce one batch of products. During the modeling period, 9 orders were completed. The load on this node was 40%. In node 13, applications for the manufacture of products were destroyed in the amount of 8 pieces. with the expectation that the batches have been manufactured and arrived at the warehouse. Average production time is 78 days. At node 10 (“Fork 2”), 0 additional loan repayment applications were created. These applications were received at node 11 (“Return”), where the loan in the amount of 120,000 rubles was returned to the bank. After the loan was repaid, 7 refund applications were destroyed at node 12. with an average time of –37 days. Product sales process.
In node 14 (“Customers”), 26 product purchasing transactions were generated with an average time of 28 days. One transaction is waiting in the queue. Next, 25 purchasing transactions “turned” to the warehouse (node 15) to purchase the goods. Warehouse utilization during the modeling period was 4.7%. Products from the warehouse were issued immediately - without delay. As a result of the distribution of products to customers, 1077 units remained in the warehouse. products, the goods are not expected to be received in the queue, therefore, upon receipt of the order, the company can issue the required quantity of goods directly from the warehouse. Node 16 simulates the release of products to 25 customers (1 transaction in queue). After receiving the goods, customers without delay paid for the received goods in the amount of 119,160 rubles. At node 18, all processed transactions were destroyed. Cash flow management process.
In this process we are dealing with the following accounting entries (requests for execution of which come from nodes 5, 7, 11 and 17, respectively): 1 loan issued by the bank - 135,000 rubles; 2 payment to suppliers for components – 135,000 rubles; 3 repayment of a bank loan – 120,000 rubles; 4 funds from the sale of products were transferred to the current account - 119,160 rubles. As a result of these postings, we received the following data on the distribution of funds across accounts: 1) Account 90: Bank. 9 transactions have been processed, one is waiting in the queue. The balance of funds is 9,970,000 rubles. Required – 0 rub. 2) Account 51: Account. 17 transactions have been processed, one is waiting in the queue. Balance of funds – 14260 rub. Required - 15,000 rubles. Consequently, when the simulation time is extended, a transaction in the queue cannot be serviced immediately due to a lack of funds in the company account. 3) Account 61: Clients. 25 transactions processed. Balance of funds – 9880840 rub. Required - 0 rub. 4) Account 60: Suppliers. 0 transactions were serviced (the “Delivery of goods” process was not considered in this experiment). The balance of funds is 135,000 rubles. Required - 0 rub. Node 23 simulates the work of the financial director. They processed 50 transactions Analysis of the graph “Dynamics of delays”.
As a result of running the model, in addition to the file containing tabular information, we obtain a graph of the dynamics of delays in the queue (Fig. 9). Graph of the dynamics of delays in the queue in the “Calculation” node. A score of 51 indicates that the delay is increasing over time. The delay time for payments from the company's current account is ≈ 18 days. This is a fairly high figure. As a result, the company makes payments less and less often, and soon the delay may exceed the creditor's waiting time - this can lead to bankruptcy of the company. But, fortunately, these delays are not frequent, and therefore this is a plus for this model. This situation can be resolved by minimizing the payment delay time for a specific market situation. The minimum delay time will be reached at one of the maximums of the average amount of money in the current account. In this case, the probability of non-payment of loan debts will be close to a minimum. Assessing the effectiveness of the model.
Based on the description of the processes, we can conclude that the processes of production and sales of products generally work effectively. The main problem of the model is the cash flow management process. The main problem of this process is debts to repay a bank loan, thereby causing a shortage of funds in the current account, which will not allow free manipulation of the received funds, because they must be used to repay the loan. As we learned from the analysis of the “Dynamics of Delays” graph, in the future the company will be able to repay accounts payable on time, but not always within clearly specified lines Therefore, we can conclude that at the moment the model is quite effective, but requires minor improvements. Generalization of the results of statistical processing of information was carried out by analyzing the results of the experiment. The graph of delays in the “Current Account” node shows that, throughout the entire modeling period, the delay time in the node remains mainly at the same level, although delays occasionally appear. It follows that the increase in the likelihood of a situation where an enterprise may be on the verge of bankruptcy is extremely low. Consequently, the model is quite acceptable, but, as mentioned above, it requires minor modifications. CONCLUSION Systems that are complex in their internal connections and have a large number of elements are economically difficult to use with direct modeling methods and often turn to simulation methods for construction and study. The emergence of the latest information technologies increases not only the capabilities of modeling systems, but also allows the use of a greater variety of models and methods for their implementation. The improvement of computing and telecommunications technology has led to the development of machine modeling methods, without which it is impossible to study processes and phenomena, as well as build large and complex systems. Based on the work done, we can say that the importance of modeling in economics is very great. Therefore, a modern economist must have a good understanding of economic and mathematical methods and be able to practically apply them to model real economic situations. This allows you to better understand the theoretical issues of modern economics, helps to improve the level of qualifications and general professional culture of a specialist. Using various business models, it is possible to describe economic objects, patterns, connections and processes not only at the level of an individual company, but also at the state level. And this is a very important fact for any country: it is possible to predict ups and downs, crises and stagnations in the economy. BIBLIOGRAPHY 1. Emelyanov A.A., Vlasova E.A. Computer modeling - M.: Moscow State University. University of Economics, Statistics and Informatics, 2002. 2. Zamkov O.O., Tolstopyatenko A.V., Cheremnykh Yu.N. Mathematical methods in economics, M., Delo i servis, 2001. 3. Kolemaev V.A., Mathematical Economics, M., UNITI, 1998. 4. Naylor T. Machine simulation experiments with models of economic systems. – M.: Mir, 1975. – 392 p. 5. Sovetov B.Ya., Yakovlev S.A. Systems modeling. – M.: Higher. School, 2001. 6. Shannon R.E. Simulation modeling of systems: science and art. - M.: Mir, 1978. 7. www.thrusta.narod.ru APPENDIX A Business model diagram “Enterprise efficiency” APPENDIX B The process of selling products of the business model “Enterprise Efficiency” APPENDIX B Cash flow management process of the business model “Enterprise Efficiency” APPENDIX D Model source code APPENDIX D Model Report File APPENDIX E Simulation modeling method and its features. Simulation model: representation of the structure and dynamics of the simulated system The simulation method is an experimental method for studying a real system using its computer model, which combines the features of the experimental approach and the specific conditions for using computer technology. Simulation modeling is a computer modeling method; in fact, it never existed without a computer, and only the development of information technology led to the establishment of this type of computer modeling. The above definition focuses on the experimental nature of simulation and the use of a simulation research method (experimentation is carried out with the model). Indeed, in simulation modeling, an important role is played not only by conducting, but also by planning the experiment on the model. However, this definition does not clarify what the simulation model itself is. Let's try to figure out what properties a simulation model has, what is the essence of simulation modeling. In the process of simulation modeling (Fig. 1.2), the researcher deals with four main elements: computational experiment. The researcher studies a real system, develops a logical-mathematical model of a real system. The simulation nature of the study presupposes the presence logical or logical-mathematical models, described process (system) being studied. To be machine-implementable, a complex system is built on the basis of a logical-mathematical model modeling algorithm, which describes the structure and logic of interaction of elements in the system. Rice. 1.2. There is a software implementation of the modeling algorithm simulation model. It is compiled using automated modeling tools. Simulation technology and modeling tools - languages and modeling systems with the help of which simulation models are implemented - will be discussed in more detail in Chapter. 3. Next, a directed computational experiment is set up and carried out on a simulation model, as a result of which the information necessary for making decisions in order to influence the real system is collected and processed. Above we defined a system as a set of interacting elements operating over time. The composite nature of a complex system dictates the representation of its model in the form of a triple A, S, T>, where A - many elements (including the external environment); S- set of permissible connections between elements (model structure); T - multiple points in time considered. A feature of simulation modeling is that the simulation model allows you to reproduce simulated objects while preserving their logical structure and behavioral properties, i.e. dynamics of element interactions. In simulation modeling, the structure of the simulated system is directly displayed in the model, and the processes of its functioning are played out (simulated) on the constructed model. The construction of a simulation model consists of describing the structure and functioning processes of the modeled object or system. There are two components in the description of the simulation model: The idea of the method from the point of view of its software implementation was as follows. What if some software components were assigned to the elements of the system, and the states of these elements were described using state variables. Elements, by definition, interact (or exchange information), which means that an algorithm for the functioning of individual elements and their interaction according to certain operational rules can be implemented - a modeling algorithm. In addition, elements exist in time, which means that an algorithm for changing state variables must be specified. Dynamics in simulation models is implemented using mechanism for advancing model time. A distinctive feature of the simulation method is the ability to describe and reproduce the interaction between various elements of the system. Thus, to create a simulation model, you need to: The key point in simulation modeling is the identification and description of system states. The system is characterized by a set of state variables, each combination of which describes a specific state. Therefore, by changing the values of these variables, it is possible to simulate the transition of the system from one state to another. Thus, simulation is the representation of the dynamic behavior of a system by moving it from one state to another according to well-defined operating rules. These state changes can occur either continuously or at discrete points in time. Simulation modeling is a dynamic reflection of changes in the state of a system over time. So, we figured out that during simulation, the logical structure of a real system is displayed in the model, and the dynamics of interactions of subsystems in the simulated system are also simulated. This is an important, but not the only feature of the simulation model, which historically predetermined the not entirely successful, in our opinion, name of the method ( simulation modeling), which researchers more often call systems modeling. The concept of model time. Model time promotion mechanism. Discrete and continuous simulation models To describe the dynamics of the simulated processes in simulation, it is implemented mechanism for advancing model time. These mechanisms are built into the control programs of any modeling system. If the behavior of one component of the system were simulated on a computer, then the execution of actions in the simulation model could be carried out sequentially, by recalculating the time coordinate. To ensure the simulation of parallel events of a real system, some global variable is introduced (providing synchronization of all events in the system) / 0, which is called model (or system) time. There are two main ways to change t Q: When step by step method time advances with the minimum possible constant step length (A/ principle). These algorithms are not very efficient in terms of using computer time for their implementation. At event-based method(principle "special conditions") time coordinates change only when the state of the system changes. In event-based methods, the length of the time shift step is the maximum possible. Model time changes from the current moment to the nearest moment of the next event. The use of the event-by-event method is preferable if the frequency of occurrence of events is low, then a large step length will speed up the progress of model time. The event-by-event method is used when events occurring in the system are unevenly distributed on the time axis and appear at significant time intervals. In practice, the event-based method is most widespread. The fixed step method is used if: Thus, due to the sequential nature of information processing in a computer, parallel processes occurring in the model are transformed using the considered mechanism into sequential ones. This method of representation is called a quasi-parallel process. The simplest classification into the main types of simulation models is associated with the use of these two methods of advancing model time. There are continuous, discrete and continuous-discrete simulation models. IN continuous simulation models variables change continuously, the state of the modeled system changes as a continuous function of time, and, as a rule, this change is described by systems of differential equations. Accordingly, the advancement of model time depends on numerical methods for solving differential equations. IN discrete simulation models variables change discretely at certain moments of simulation time (the occurrence of events). The dynamics of discrete models is the process of transition from the moment of the onset of the next event to the moment of the onset of the next event. Since in real systems continuous and discrete processes are often impossible to separate, continuous-discrete models, which combine the mechanisms of time progression characteristic of these two processes. Problems of strategic and tactical planning of a simulation experiment. Directed computational experiment on a simulation model So we have determined that simulation methodology- This is a system analysis. It is the latter that gives the right to call the type of modeling under consideration system modeling. At the beginning of this section, we gave a general concept of the simulation method and defined it as an experimental method for studying a real system using its simulation model. Note that the concept of a method is always broader than the concept of “simulation model”. Let us consider the features of this experimental method (simulation research method). By the way, the words “ simulation", "experiment", "imitation" of one plan. The experimental nature of simulation also determined the origin of the name of the method. So, the goal of any research is to find out as much as possible about the system being studied, to collect and analyze the information necessary to make a decision. The essence of studying a real system using its simulation model is to obtain (collect) data on the functioning of the system as a result of conducting an experiment on a simulation model. Simulation models are run-type models that have an input and an output. That is, if you feed certain parameter values to the input of the simulation model, you can get a result that is valid only for these values. In practice, the researcher is faced with the following specific feature of simulation modeling. A simulation model produces results that are valid only for certain values of the parameters, variables, and structural relationships embedded in the simulation program. Changing a parameter or relationship means that the simulation program must be run again. Therefore, to obtain the necessary information or results, it is necessary to run simulation models rather than solve them. The simulation model is not capable of forming its own solution in the same way as is the case in analytical models (see computational research method), but can serve as a means for analyzing the behavior of the system under conditions determined by the experimenter. For clarification, consider the deterministic and stochastic cases. Stochastic case. A simulation model is a convenient apparatus for studying stochastic systems. Stochastic systems are systems whose dynamics depend on random factors; the input and output variables of a stochastic model are usually described as random variables, functions, processes, sequences. Let's consider the main features of modeling processes taking into account the action of random factors (the well-known ideas of the method of statistical tests and the Monte Carlo method are implemented here). The simulation results obtained by reproducing a single implementation of processes, due to the action of random factors, will be implementations of random processes and will not be able to objectively characterize the object being studied. Therefore, the required values when studying processes using the simulation method are usually determined as average values based on data from a large number of process implementations (estimation problem). Therefore, an experiment on a model contains several implementations, runs, and involves estimation based on a set of data (samples). It is clear that (according to the law of large numbers) the greater the number of implementations, the more the resulting estimates become more and more statistically stable. So, in the case of a stochastic system, it is necessary to collect and evaluate statistical data at the output of the simulation model, and to do this, carry out a series of runs and statistical processing of the simulation results. Deterministic case. IN In this case, it is enough to carry out one run with a specific set of parameters. Now let’s imagine that the goals of modeling are: studying the system under various conditions, evaluating alternatives, finding the dependence of the model’s output on a number of parameters, and, finally, finding the optimal option. In these cases, the researcher can gain insight into the functioning of the modeled system by changing the values of the parameters at the input of the model, while performing numerous machine runs of the simulation model. Thus, conducting experiments with a model on a computer involves conducting multiple machine runs in order to collect, accumulate and subsequently process data on the functioning of the system. Simulation modeling allows you to explore a model of a real system in order to study its behavior through repeated runs on a computer under various operating conditions of the real system. The following problems arise here: how to collect this data, conduct a series of runs, how to organize a targeted experimental study. The output data obtained as a result of such experimentation can be very large. How to process them? Processing and studying them can turn into an independent problem, much more difficult than the task of statistical estimation. In simulation modeling, an important issue is not only conducting, but also planning a simulation experiment in accordance with the stated purpose of the study. Thus, a researcher using simulation modeling methods always faces the problem of organizing an experiment, i.e. choosing a method for collecting information that provides the required volume (to achieve the research goal) at the lowest cost (an extra number of runs means extra computer time). The main task is to reduce the time spent on operating the model, reduce the computer time for simulation, which reflects the expenditure of computer time resources on conducting a large number of simulation runs. This problem is called strategic planning simulation research. To solve it, methods of experiment planning, regression analysis, etc. are used, which will be discussed in detail in section 3.4. Strategic planning is the development of an effective experimental design that either identifies the relationship between controlled variables or finds a combination of values of controlled variables that minimizes or maximizes the response (output) of a simulation model. Along with the concept of strategic, there is the concept tactical planning, which is associated with determining how to conduct simulation runs outlined in the experimental plan: how to conduct each run within the framework of the drawn up experimental plan. Here the problems of determining the duration of a run, assessing the accuracy of simulation results, etc. are solved. We will call such experiments with a simulation model directed computational experiments. A simulation experiment, the content of which is determined by a previously conducted analytical study (i.e., which is an integral part of a computational experiment) and the results of which are reliable and mathematically justified, is called directed computational experiment. In ch. 3 we will consider in detail the practical issues of organizing and conducting directed computational experiments using a simulation model. General technological scheme, capabilities and scope of simulation modeling Summarizing our reasoning, we can present in the most general form the technological scheme of simulation modeling (Fig. 1.3). (The technology of simulation modeling will be discussed in more detail in Chapter 3.) Rice. 1.3. Let us consider the capabilities of the simulation modeling method, which have led to its widespread use in a variety of fields. Simulation modeling traditionally finds application in a wide range of economic research: modeling of production systems and logistics, sociology and political science; modeling of transport, information and telecommunication systems, and finally, global modeling of world processes. The simulation modeling method makes it possible to solve problems of exceptional complexity, provides the simulation of any complex and diverse processes with a large number of elements; individual functional dependencies in such models can be described by very cumbersome mathematical relationships. Therefore, simulation modeling is effectively used in problems of studying systems with a complex structure in order to solve specific problems. The simulation model contains elements of continuous and discrete action, therefore it is used to study dynamic systems, when an analysis of bottlenecks is required, a study of the dynamics of functioning, when it is desirable to observe the progress of a process on a simulation model over a certain time Simulation modeling is an effective tool for studying stochastic systems, when the system under study can be influenced by numerous random factors of a complex nature (mathematical models for this class of systems have limited capabilities). It is possible to conduct research under conditions of uncertainty, with incomplete and inaccurate data. Simulation modeling is the most valuable, system-forming link in decision support systems, as it allows you to explore a large number of alternatives (decision options) and play out various scenarios for any input data. The main advantage of simulation modeling is that the researcher can always get an answer to the question “What will happen if?” to test new strategies and make decisions when studying possible situations. ...". The simulation model makes it possible to make predictions when it comes to the system being designed or when development processes are studied, i.e. in cases where no real system exists. The simulation model can provide various (including very high) levels of detail of the simulated processes. In this case, the model is created in stages, gradually, without significant changes, evolutionarily. In modern literature one can find several points of view on what simulation modeling is. Some argue that these are mathematical models in the classical sense, others believe that these are models in which random processes are simulated, and others suggest that simulation models differ from ordinary mathematical ones in a more detailed description. However, everyone agrees that simulation is applied to processes in which humans may intervene from time to time. Methods for analyzing the development of situations based on varying the values of various factors that determine these situations have become increasingly widespread. The meaning of this variation is as follows. The activities of any business entity depend on many factors, the vast majority of which are interrelated; at the same time, some factors are amenable to certain regulation, and from here, by varying the set of key parameters or their values, it is possible to simulate various situations and, thanks to this, choose the most acceptable scenario for the development of events. One of the difficulties in implementing this approach is the routineness of actions and the multiplicity of counting operations; this difficulty is eliminated by using a computer and associated software in what is known as simulation modeling. Simulation modeling - This is a formalized method (mathematics can be applied). The word "imitation" (from Lat. imatatio) means “imitation of someone or something, reproduction with possible accuracy.” The essence of simulation modeling is as follows: a specific economic situation is simulated in a computer environment. After making several calculations, you can select a set of parameters and their values, which you then try to manage (for example, accounts receivable should not go beyond a given corridor, obtaining a certain amount of profit). Simulation modeling of financial and economic activities is based on a combination of formalized (mathematical) methods and expert assessments of specialists and managers of an economic entity, with the latter prevailing. The simulation process is as follows: first, a mathematical model of the object under study (simulation model) is built, then this model is converted into a computer program. In the process of work, the indicators of interest to the researcher change: they are subject to analysis, in particular statistical processing. A simulation model is used, on the one hand, in cases where the model (and therefore the system, process, phenomenon it reflects) is too complex to allow the use of conventional analytical solution methods. For many problems of management and economics, this situation is inevitable: for example, even such well-established methods as linear programming, in some cases, provide a solution that is too far from reality and it is impossible to draw reasonable conclusions from the results obtained. The choice itself between a simulation (numerical) or an analytical solution to a particular economic problem is not always an easy problem. On the other hand, imitation is used when a real economic experiment is impossible or too complicated for one reason or another. Then it acts as a replacement for such an experiment. But even more valuable is its role as a preliminary stage, an “estimate”, which helps to make a decision about the need and possibility of conducting a real experiment. Using static simulation, it is possible to identify at what combinations of input factors the optimal result of the process being studied is achieved, and to establish the relative importance of certain factors. This is useful, for example, when studying various methods and means of economic incentives in production. Simulation modeling is also used in forecasting, since it “reduces time” and, in particular, allows, in a matter of hours, to reproduce on a computer (in aggregated terms) the development of an enterprise or a branch of the national economy for months and even years in advance. Recently it has been widely used imitation of economic processes, in which various interests such as competition in the market collide. As the business game progresses, certain decisions are made, for example: “increase prices”, “increase or decrease production output”, etc., and calculations show which of the “competing” parties is doing better and which is doing worse. Simulation modeling of economic processes is essentially an experiment, but not in real, but in artificial conditions. The criterion for the adequacy of a model is practice. When constructing a mathematical model of a complex system, difficulties may arise when the model contains many connections between elements, it has various nonlinear constraints, and a large number of parameters. Real systems are often influenced by various random factors that are difficult to take into account, so comparison of the model and the original in this case is only possible at the beginning. To overcome these difficulties, it is necessary to take into account the following rules when using simulation modeling: Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you. Posted on http://www.allbest.ru/ Course project Subject: “Modeling of production and economic processes” On the topic: “Simulation modeling of economic processes” Introduction 1.1 Concept of modeling 1.2 Concept of a model IV. Practical part 4.1 Problem statement 4.2 Solving the problem Conclusion Application Introduction Simulation modeling, linear programming and regression analysis have long occupied the top three places among all methods of operations research in economics in terms of range and frequency of use. In simulation modeling, the algorithm that implements the model reproduces the process of system functioning in time and space, and the elementary phenomena that make up the process are simulated while preserving its logical time structure. Currently, modeling has become a fairly effective means of solving complex problems of automation of research, experiments, and design. But to master modeling as a working tool, its wide capabilities and further develop the modeling methodology is possible only with full mastery of the techniques and technology for practical solution of problems of modeling the processes of functioning of systems on a computer. This is the goal of this workshop, which focuses on the methods, principles and main stages of modeling within the framework of the general modeling methodology, and also examines the issues of modeling specific variants of systems and instills skills in using modeling technology in the practical implementation of models of system functioning. The problems of queuing systems on which simulation models of economic, information, technological, technical and other systems are based are considered. Methods for probabilistic modeling of discrete and random continuous variables are outlined, which make it possible to take into account random impacts on the system when modeling economic systems. The demands that modern society places on a specialist in the field of economics are steadily growing. Currently, successful activity in almost all spheres of the economy is not possible without modeling the behavior and dynamics of development processes, studying the features of the development of economic objects, and considering their functioning in various conditions. Software and hardware should become the first assistants here. Instead of learning from your own mistakes or from the mistakes of other people, it is advisable to consolidate and test your knowledge of reality with the results obtained on computer models. Simulation modeling is the most visual and is used in practice for computer modeling of options for resolving situations in order to obtain the most effective solutions to problems. Simulation modeling allows for the study of the analyzed or designed system according to the scheme of operational research, which contains interrelated stages: · development of a conceptual model; · development and software implementation of a simulation model; · checking the correctness and reliability of the model and assessing the accuracy of the modeling results; · planning and conducting experiments; · making decisions. This allows the use of simulation modeling as a universal approach for making decisions under conditions of uncertainty, taking into account factors that are difficult to formalize in models, as well as applying the basic principles of a systems approach to solving practical problems. The widespread implementation of this method in practice is hampered by the need to create software implementations of simulation models that recreate the dynamics of the functioning of the simulated system in simulated time. Unlike traditional programming methods, developing a simulation model requires a restructuring of the principles of thinking. It is not without reason that the principles underlying simulation modeling gave impetus to the development of object programming. Therefore, the efforts of simulation software developers are aimed at simplifying software implementations of simulation models: specialized languages and systems are created for these purposes. Simulation software tools have changed in their development over several generations, from modeling languages and automation tools for model construction to program generators, interactive and intelligent systems, and distributed modeling systems. The main purpose of all these tools is to reduce the labor intensity of creating software implementations of simulation models and experimenting with models. One of the first modeling languages to facilitate the process of writing simulation programs was the GPSS language, created as a final product by Jeffrey Gordon at IBM in 1962. Currently there are translators for DOS operating systems - GPSS/PC, for OS/2 and DOS - GPSS/H and for Windows - GPSS World. Studying this language and creating models allows you to understand the principles of developing simulation programs and learn how to work with simulation models. GPSS (General Purpose Simulation System) is a modeling language that is used to build event-driven discrete simulation models and conduct experiments using a personal computer. The GPSS system is a language and a translator. Like every language, it contains a vocabulary and grammar with the help of which models of systems of a certain type can be developed. I. Basic concepts of the theory of modeling economic systems and processes 1.1 Concept of modeling Modeling refers to the process of constructing, studying and applying models. It is closely related to such categories as abstraction, analogy, hypothesis, etc. The modeling process necessarily includes the construction of abstractions, inferences by analogy, and the construction of scientific hypotheses. The main feature of modeling is that it is a method of indirect cognition using proxy objects. The model acts as a kind of cognition tool that the researcher puts between himself and the object, and with the help of which he studies the object of interest to him. Any socio-economic system is a complex system in which dozens and hundreds of economic, technical and social processes interact, constantly changing under the influence of external conditions, including scientific and technological progress. In such conditions, managing socio-economic and production systems turns into a complex task requiring special tools and methods. Modeling is one of the main methods of cognition, is a form of reflection of reality and consists in finding out or reproducing certain properties of real objects, objects and phenomena with the help of other objects, processes, phenomena, or using an abstract description in the form of an image, plan, map , a set of equations, algorithms and programs. In the most general sense, a model is a logical (verbal) or mathematical description of components and functions that reflect the essential properties of the object or process being modeled, usually considered as systems or elements of a system from a certain point of view. The model is used as a conventional image, designed to simplify the study of the object. In principle, not only mathematical (symbolic) but also material models are applicable in economics, but material models have only demonstrative value. There are two points of view on the essence of modeling: * this is a study of objects of cognition using models; * this is the construction and study of models of real-life objects and phenomena, as well as proposed (constructed) objects. The possibilities of modeling, that is, transferring the results obtained during the construction and research of the model to the original, are based on the fact that the model in a certain sense displays (reproduces, models, describes, imitates) some features of the object that are of interest to the researcher. Modeling as a form of reflection of reality is widespread, and a fairly complete classification of possible types of modeling is extremely difficult, if only because of the polysemy of the concept “model,” which is widely used not only in science and technology, but also in art and in everyday life. The word “model” comes from the Latin word “modulus”, meaning “measure”, “sample”. Its original meaning was associated with the art of building, and in almost all European languages it was used to denote an image or prototype, or a thing similar in some respect to another thing. Among socio-economic systems, it is advisable to highlight the production system (PS), which, unlike systems of other classes, contains as the most important element a consciously acting person performing management functions (decision making and control). In accordance with this, various divisions of enterprises, enterprises themselves, research and design organizations, associations, industries and, in some cases, the national economy as a whole can be considered as PS. The nature of the similarity between the modeled object and the model differs: * physical - the object and the model have the same or similar physical nature; * structural - there is a similarity between the structure of the object and the structure of the model; * functional - the object and the model perform similar functions under appropriate influence; * dynamic - there is a correspondence between the sequentially changing states of the object and the model; * probabilistic - there is a correspondence between processes of a probabilistic nature in the object and the model; * geometric - there is a correspondence between the spatial characteristics of the object and the model. Modeling is one of the most common ways to study processes and phenomena. Modeling is based on the principle of analogy and allows you to study an object under certain conditions and taking into account the inevitable one-sided point of view. An object that is difficult to study is studied not directly, but through the consideration of another, similar to it and more accessible - a model. Based on the properties of the model, it is usually possible to judge the properties of the object being studied. But not about all properties, but only about those that are similar both in the model and in the object and at the same time are important for research. Such properties are called essential. Is there a need for mathematical modeling of the economy? In order to verify this, it is enough to answer the question: is it possible to complete a technical project without having an action plan, i.e., drawings? The same situation occurs in the economy. Is it necessary to prove the need to use economic and mathematical models for making management decisions in the economic sphere? Under these conditions, the economic-mathematical model turns out to be the main means of experimental research in economics, since it has the following properties: * imitates a real economic process (or the behavior of an object); * has a relatively low cost; * can be reused; * takes into account various operating conditions of the object. The model can and should reflect the internal structure of an economic object from given (certain) points of view, and if it is unknown, then only its behavior, using the “Black box” principle. Fundamentally, any model can be formulated in three ways: * as a result of direct observation and study of the phenomena of reality (phenomenological method); * isolation from a more general model (deductive method); * generalizations of more particular models (inductive method, i.e. proof by induction). Models, endless in their diversity, can be classified according to a variety of criteria. First of all, all models can be divided into physical and descriptive. We deal with both of them all the time. In particular, descriptive models include models in which the modeled object is described using words, drawings, mathematical dependencies, etc. Such models include literature, fine arts, and music. Economic and mathematical models are widely used in managing business processes. There is no established definition of an economic-mathematical model in the literature. Let's take the following definition as a basis. An economic-mathematical model is a mathematical description of an economic process or object, carried out for the purpose of their study or management: a mathematical recording of the economic problem being solved (therefore, the terms problem and model are often used as synonyms). Models can also be classified according to other criteria: * Models that describe the momentary state of the economy are called static. Models that show the development of the modeled object are called dynamic. * Models that can be built not only in the form of formulas (analytical representation), but also in the form of numerical examples (numerical representation), in the form of tables (matrix representation), in the form of a special kind of graphs (network representation). 1.2 Concept of a model At present, it is impossible to name an area of human activity in which modeling methods would not be used to one degree or another. Meanwhile, there is no generally accepted definition of the concept of model. In our opinion, the following definition deserves preference: a model is an object of any nature that is created by a researcher in order to obtain new knowledge about the original object and reflects only the essential (from the developer’s point of view) properties of the original. Analyzing the content of this definition, we can draw the following conclusions: 1) any model is subjective, it bears the stamp of the researcher’s individuality; 2) any model is homomorphic, i.e. it does not reflect all, but only the essential properties of the original object; 3) it is possible that there are many models of the same original object, differing in the purposes of the study and the degree of adequacy. A model is considered adequate to the original object if it, with a sufficient degree of approximation at the level of understanding of the simulated process by the researcher, reflects the patterns of the functioning of a real system in the external environment. Mathematical models can be divided into analytical, algorithmic (simulation) and combined. Analytical modeling is characterized by the fact that systems of algebraic, differential, integral or finite-difference equations are used to describe the processes of system functioning. The analytical model can be studied using the following methods: a) analytical, when they strive to obtain, in a general form, explicit dependencies for the desired characteristics; b) numerical, when, not being able to solve equations in general form, they strive to obtain numerical results with specific initial data; c) qualitative, when, without having an explicit solution, one can find some properties of the solution (for example, assess the stability of the solution). In algorithmic (simulation) modeling, the process of system functioning over time is described, and the elementary phenomena that make up the process are simulated, preserving their logical structure and sequence of occurrence over time. Simulation models can also be deterministic and statistical. The general goal of modeling in the decision-making process was formulated earlier - this is the determination (calculation) of the values of the selected performance indicator for various strategies for conducting an operation (or options for implementing the designed system). When developing a specific model, the purpose of the modeling should be clarified taking into account the effectiveness criterion used. Thus, the purpose of modeling is determined both by the purpose of the operation being studied and by the planned method of using the research results. For example, a problem situation that requires a decision is formulated as follows: find an option for building a computer network that would have the minimum cost while meeting the performance and reliability requirements. In this case, the goal of modeling is to find network parameters that provide the minimum PE value, which is represented by cost. The task can be formulated differently: from several options for computer network configuration, choose the most reliable one. Here, one of the reliability indicators (mean time between failures, probability of failure-free operation, etc.) is selected as the PE, and the purpose of the modeling is a comparative assessment of network options according to this indicator. The above examples allow us to recall that the choice of performance indicator itself does not yet determine the “architecture” of the future model, since at this stage its concept has not been formulated, or, as they say, the conceptual model of the system under study has not been defined. II. Basic concepts of the theory of modeling economic systems and processes 2.1 Improvement and development of economic systems Simulation modeling is the most powerful and universal method for studying and assessing the effectiveness of systems whose behavior depends on the influence of random factors. Such systems include an aircraft, a population of animals, and an enterprise operating in conditions of poorly regulated market relations. Simulation modeling is based on a statistical experiment (Monte Carlo method), the implementation of which is practically impossible without the use of computer technology. Therefore, any simulation model is ultimately a more or less complex software product. Of course, like any other program, a simulation model can be developed in any universal programming language, even in Assembly language. However, in this case the following problems arise on the developer's path: * knowledge is required not only of the subject area to which the system under study belongs, but also of the programming language, and at a fairly high level; * developing specific procedures for ensuring a statistical experiment (generating random influences, planning an experiment, processing results) can take no less time and effort than developing the system model itself. And finally, one more, perhaps the most important problem. In many practical problems, interest is not only (and not so much) in the quantitative assessment of the effectiveness of the system, but in its behavior in a given situation. For such observation, the researcher must have appropriate “observation windows” that could, if necessary, be closed, moved to another location, changed the scale and form of presentation of the observed characteristics, etc., without waiting for the end of the current model experiment. In this case, the simulation model acts as a source of answer to the question: “what will happen if...”. Implementing such capabilities in a universal programming language is very difficult. Currently, there are quite a lot of software products that allow you to simulate processes. Such packages include: Pilgrim, GPSS, Simplex and a number of others. At the same time, there is currently a product on the Russian computer technology market that allows one to very effectively solve these problems - the MATLAB package, which contains the visual modeling tool Simulink. Simulink is a tool that allows you to quickly simulate a system and obtain indicators of the expected effect and compare them with the effort required to achieve them. There are many different types of models: physical, analog, intuitive, etc. A special place among them is occupied by mathematical models, which, according to Academician A.A. Samarsky, “are the greatest achievement of the scientific and technological revolution of the 20th century.” Mathematical models are divided into two groups: analytical and algorithmic (sometimes called simulation). Currently, it is impossible to name an area of human activity in which modeling methods would not be used to one degree or another. Economic activity is no exception. However, in the field of simulation modeling of economic processes, some difficulties are still observed. In our opinion, this circumstance is explained by the following reasons. 1. Economic processes occur largely spontaneously and uncontrollably. They do not respond well to attempts at strong-willed control on the part of political, government and economic leaders of individual industries and the country’s economy as a whole. For this reason, economic systems are difficult to study and formally describe. 2. Specialists in the field of economics, as a rule, have insufficient mathematical training in general and in mathematical modeling in particular. Most of them do not know how to formally describe (formalize) observed economic processes. This, in turn, does not allow us to establish whether this or that mathematical model is adequate for the economic system under consideration. 3. Specialists in the field of mathematical modeling, without having at their disposal a formalized description of the economic process, cannot create a mathematical model adequate to it. Existing mathematical models, which are commonly called models of economic systems, can be divided into three groups. The first group includes models that quite accurately reflect one aspect of a certain economic process occurring in a system of a relatively small scale. From a mathematical point of view, they represent very simple relationships between two or three variables. Usually these are algebraic equations of the 2nd or 3rd degree, in extreme cases a system of algebraic equations that requires the use of the iteration method (successive approximations) to solve. They find application in practice, but are not of interest from the point of view of specialists in the field of mathematical modeling. The second group includes models that describe real processes occurring in small and medium-sized economic systems, subject to the influence of random and uncertain factors. The development of such models requires making assumptions to resolve uncertainties. For example, you need to specify distributions of random variables related to input variables. This artificial operation to a certain extent raises doubts about the reliability of the modeling results. However, there is no other way to create a mathematical model. Among the models of this group, the most widely used models are those of the so-called queuing systems. There are two varieties of these models: analytical and algorithmic. Analytical models do not take into account the effect of random factors and therefore can only be used as first approximation models. Using algorithmic models, the process under study can be described with any degree of accuracy at the level of its understanding by the problem maker. The third group includes models of large and very large (macroeconomic) systems: large commercial and industrial enterprises and associations, sectors of the national economy and the country’s economy as a whole. Creating a mathematical model of an economic system of this scale is a complex scientific problem, the solution of which can only be solved by a large research institution. 2.2 Simulation model components Numerical modeling deals with three types of values: input data, calculated variable values, and parameter values. On an Excel sheet, arrays with these values occupy separate areas. Initial real data, samples or series of numbers, are obtained through direct field observation or in experiments. Within the framework of the modeling procedure, they remain unchanged (it is clear that, if necessary, the sets of values can be supplemented or reduced) and play a dual role. Some of them (independent environmental variables, X) serve as the basis for calculating model variables; most often these are characteristics of natural factors (the passage of time, photoperiod, temperature, abundance of food, dose of toxicant, volumes of pollutants discharged, etc.). The other part of the data (dependent variables of the object, Y) is a quantitative characteristic of the state, reactions or behavior of the research object, which was obtained in certain conditions, under the influence of registered environmental factors. In a biological sense, the first group of meanings does not depend on the second; on the contrary, object variables depend on environment variables. Data is entered into an Excel sheet from the keyboard or from a file in the usual spreadsheet mode. Model calculation data reproduce the theoretically conceivable state of the object, which is determined by the previous state, the level of observed environmental factors and is characterized by the key parameters of the process being studied. In the ordinary case, when calculating model values (Y M i) for each time step (i), parameters (A), characteristics of the previous state (Y M i -1) and current levels of environmental factors (X i) are used: Y M i = f(A, Y M i-1, X i, i), f() - the accepted form of the relationship between parameters and environmental variables, the type of model, i = 1, 2, … T or i =1, 2, … n. Calculations of system characteristics using model formulas for each time step (for each state) make it possible to generate an array of model explicit variables (Y M), which must exactly repeat the structure of the array of real dependent variables (Y), which is necessary for subsequent adjustment of model parameters. Formulas for calculating model variables are entered into the cells of the Excel sheet manually (see the section Useful techniques). The model parameters (A) constitute the third group of values. All parameters can be represented as a set: A = (a 1, a 2,…, a j,…, a m), where j is the parameter number, m? total number of parameters, and placed in a separate block. It is clear that the number of parameters is determined by the structure of the adopted model formulas. Occupying a separate position on the Excel sheet, they play the most significant role in modeling. The parameters are designed to characterize the very essence, the mechanism for the implementation of the observed phenomena. The parameters must have a biological (physical) meaning. For some tasks, it is necessary that parameters calculated for different data sets can be compared. This means that they must sometimes be accompanied by their own statistical errors. The relationships between the components of the simulation system form a functional unity focused on achieving a common goal - assessing the parameters of the model (Fig. 2.6, Table 2.10). Several elements are simultaneously involved in the implementation of individual functions, indicated by arrows. In order not to clutter the picture, the graphical representation and randomization blocks are not reflected in the diagram. The simulation system is designed to support any changes in model designs that, if necessary, can be made by the researcher. Basic designs of simulation systems, as well as possible ways of their decomposition and integration are presented in the section Frames of simulation systems. modeling simulation economic series III. Simulation Basics 3.1 Simulation model and its features Simulation modeling is a type of analog modeling implemented using a set of mathematical tools, special simulating computer programs and programming technologies that allow, through analogue processes, to conduct a targeted study of the structure and functions of a real complex process in computer memory in the “simulation” mode, and to optimize some its parameters. A simulation model is an economic and mathematical model, the study of which is carried out by experimental methods. The experiment consists of observing the results of calculations for various specified values of the input exogenous variables. The simulation model is a dynamic model due to the fact that it contains such a parameter as time. A simulation model is also called a special software package that allows you to simulate the activities of any complex object. The emergence of simulation modeling was associated with the “new wave” in economics and topic modeling. Problems of economic science and practice in the field of management and economic education, on the one hand, and the growth of computer productivity, on the other, have caused a desire to expand the scope of “classical” economic and mathematical methods. There was some disappointment in the capabilities of normative, balance sheet, optimization and game-theoretic models, which at first deservedly attracted the attention of the fact that they bring an atmosphere of logical clarity and objectivity to many problems of economic management, and also lead to a “reasonable” (balanced, optimal, compromise) solution . It was not always possible to fully comprehend a priori goals and, even more so, to formalize the optimality criterion and (or) restrictions on admissible solutions. Therefore, many attempts to nevertheless apply such methods began to lead to unacceptable, for example, unrealizable (albeit optimal) solutions. Overcoming the difficulties that arose took the path of abandoning complete formalization (as is done in normative models) of procedures for making socio-economic decisions. Preference began to be given to a reasonable synthesis of the intellectual capabilities of an expert and the information power of a computer, which is usually implemented in dialogue systems. One trend in this direction is the transition to “semi-normative” multi-criteria man-machine models, the second is a shift in the center of gravity from prescriptive models focused on the “conditions - solution” scheme to descriptive models that answer the question “what will happen, If...". Simulation modeling is usually resorted to in cases where the dependencies between the elements of the simulated systems are so complex and uncertain that they cannot be formally described in the language of modern mathematics, i.e., using analytical models. Thus, researchers of complex systems are forced to use simulation modeling when purely analytical methods are either inapplicable or unacceptable (due to the complexity of the corresponding models). In simulation modeling, the dynamic processes of the original system are replaced by processes simulated by a modeling algorithm in an abstract model, but maintaining the same ratios of durations, logical and time sequences as in the real system. Therefore, the simulation method could be called algorithmic or operational. By the way, such a name would be more successful, since imitation (translated from Latin as imitation) is the reproduction of something by artificial means, i.e. modeling. In this regard, the currently widely used name “simulation modeling” is tautological. In the process of simulating the functioning of the system under study, as in an experiment with the original itself, certain events and states are recorded, from which the necessary characteristics of the quality of functioning of the system under study are then calculated. For systems, for example, information and computing services, such dynamic characteristics can be defined as: * performance of data processing devices; * length of queues for service; * waiting time for service in queues; * number of applications that left the system without service. In simulation modeling, processes of any degree of complexity can be reproduced if there is a description of them, given in any form: formulas, tables, graphs, or even verbally. The main feature of simulation models is that the process under study is, as it were, “copied” on a computer, therefore simulation models, unlike analytical models, allow: * take into account a huge number of factors in models without gross simplifications and assumptions (and therefore, increase the adequacy of the model to the system under study); * it is enough to simply take into account the uncertainty factor in the model caused by the random nature of many model variables; All this allows us to draw a natural conclusion that simulation models can be created for a wider class of objects and processes. 3.2 The essence of simulation modeling The essence of simulation modeling is targeted experimentation with a simulation model by “playing” on it various options for the functioning of the system with their corresponding economic analysis. Let us immediately note that it is advisable to present the results of such experiments and the corresponding economic analysis in the form of tables, graphs, nomograms, etc., which greatly simplifies the decision-making process based on the modeling results. Having listed above a number of advantages of simulation models and simulation, we also note their disadvantages, which must be remembered when using simulation in practice. This: * lack of well-structured principles for constructing simulation models, which requires significant elaboration of each specific case of its construction; * methodological difficulties in finding optimal solutions; * increased requirements for the speed of computers on which simulation models are implemented; * difficulties associated with the collection and preparation of representative statistics; * uniqueness of simulation models, which does not allow the use of ready-made software products; * the complexity of analyzing and understanding the results obtained as a result of a computational experiment; * quite a large investment of time and money, especially when searching for optimal trajectories of behavior of the system under study. The number and essence of the listed shortcomings is very impressive. However, given the great scientific interest in these methods and their extremely intensive development in recent years, it is safe to assume that many of the above-mentioned shortcomings of simulation modeling can be eliminated, both conceptually and in application terms. Simulation modeling of a controlled process or controlled object is a high-level information technology that provides two types of actions performed using a computer: 1) work on creating or modifying a simulation model; 2) operation of the simulation model and interpretation of the results. Simulation modeling of economic processes is usually used in two cases: * for managing a complex business process, when a simulation model of a managed economic entity is used as a tool% in the contour of an adaptive management system created on the basis of information technology; * when conducting experiments with discrete-continuous models of complex economic objects to obtain and monitor their dynamics in emergency situations associated with risks, the full-scale modeling of which is undesirable or impossible. The following typical tasks can be identified that can be solved by means of simulation modeling when managing economic objects: * modeling of logistics processes to determine time and cost parameters; * managing the process of implementing an investment project at various stages of its life cycle, taking into account possible risks and tactics for allocating funds; * analysis of clearing processes in the work of a network of credit institutions (including application to mutual settlement processes in the Russian banking system); * forecasting the financial results of an enterprise for a specific period of time (with analysis of the dynamics of account balances); * business reengineering of an insolvent enterprise (changing the structure and resources of a bankrupt enterprise, after which, using a simulation model, one can make a forecast of the main financial results and give recommendations on the feasibility of one or another option for reconstruction, investment or lending to production activities); A simulation system that provides the creation of models to solve the listed problems must have the following properties: * the possibility of using simulation programs in conjunction with special economic and mathematical models and methods based on control theory; * instrumental methods of conducting structural analysis of a complex economic process; * the ability to model material, monetary and information processes and flows within a single model, in general, model time; * the possibility of introducing a mode of constant clarification when receiving output data (main financial indicators, time and space characteristics, risk parameters, etc.) and conducting an extreme experiment. Many economic systems are essentially queuing systems (QS), i.e. systems in which, on the one hand, there are requirements for the performance of any services, and on the other, these requirements are satisfied. IV. Practical part 4.1 Problem statement Investigate the dynamics of an economic indicator based on the analysis of a one-dimensional time series. For nine consecutive weeks, demand Y(t) (million rubles) for credit resources of a financial company was recorded. The time series Y(t) of this indicator is given in the table. Required: 1. Check for anomalous observations. 2. Construct a linear model Y(t) = a 0 + a 1 t, the parameters of which can be estimated by least squares (Y(t)) - calculated, simulated values of the time series). 3. Assess the adequacy of the constructed models using the properties of independence of the residual component, randomness and compliance with the normal distribution law (when using the R/S criterion, take tabulated limits of 2.7-3.7). 4. Assess the accuracy of the models based on the use of the average relative error of approximation. 5. Based on the two constructed models, forecast demand for the next two weeks (calculate the confidence interval of the forecast at a confidence probability of p = 70%) 6. Present the actual values of the indicator, modeling and forecasting results graphically. 4.2 Solving the problem 1). The presence of anomalous observations leads to distortion of the modeling results, so it is necessary to ensure the absence of anomalous data. To do this, we will use Irwin’s method and find the characteristic number () (Table 4.1). The calculated values are compared with the tabulated values of the Irvine criterion, and if they are greater than the tabulated ones, then the corresponding value of the series level is considered anomalous. Appendix 1 (Table 4.1) All obtained values were compared with the table values and did not exceed them, that is, there were no anomalous observations. 2) Construct a linear model, the parameters of which can be estimated by least squares methods (calculated, simulated values of the time series). To do this, we will use Data Analysis in Excel. Appendix 1 ((Fig. 4.2).Fig. 4.1) The result of the regression analysis is contained in the table Appendix 1 (table 4.2 and 4.3.) In the second column of the table. 4.3 contains the coefficients of the regression equation a 0, a 1, the third column contains the standard errors of the coefficients of the regression equation, and the fourth contains t - statistics used to test the significance of the coefficients of the regression equation. The regression equation of dependence (demand for credit resources) on (time) has the form. Appendix 1 (Fig. 4.5) 3) Assess the adequacy of the constructed models. 3.1. Let's check the independence (absence of autocorrelation) using the Durbin-Watson d test according to the formula: Appendix 1 (Table 4.4) Because the calculated value d falls in the range from 0 to d 1, i.e. in the interval from 0 to 1.08, then the property of independence is not satisfied, the levels of a number of residuals contain autocorrelation. Therefore, the model is inadequate according to this criterion. 3.2. We will check the randomness of the levels of a number of residues based on the criterion of turning points. P> The number of turning points is 6. Appendix 1 (Fig. 4.5) The inequality is satisfied (6 > 2). Therefore, the randomness property is satisfied. The model is adequate according to this criterion. 3.3. Let us determine whether a number of residuals correspond to the normal distribution law using the RS criterion: The maximum level of a number of residues, The minimum level of a number of residues, Standard deviation, The calculated value falls within the interval (2.7-3.7), therefore, the property of normal distribution is satisfied. The model is adequate according to this criterion. 3.4. Checking the equality of the mathematical expectation of the levels of a series of residues to zero. In our case, therefore, the hypothesis that the mathematical expectation of the values of the residual series is equal to zero is satisfied. Table 4.3 summarizes the analysis of a number of residues. Appendix 1 (Table 4.6) 4) Assess the accuracy of the model based on the use of the average relative error of approximation. To assess the accuracy of the resulting model, we will use the relative approximation error indicator, which is calculated by the formula: Calculation of relative approximation error Appendix 1 (Table 4.7) If the error calculated by the formula does not exceed 15%, the accuracy of the model is considered acceptable. 5) Based on the constructed model, forecast demand for the next two weeks (calculate the confidence interval of the forecast at a confidence level of p = 70%). Let's use the Excel function STUDISCOVER. Appendix 1 (Table 4.8) To build an interval forecast, we calculate the confidence interval. Let us take the value of the significance level, therefore, the confidence probability is equal to 70%, and the Student’s test at is equal to 1.12. We calculate the width of the confidence interval using the formula: (we find from table 4.1) We calculate the upper and lower limits of the forecast (Table 4.11). Appendix 1 (Table 4.9) 6) Present the actual values of the indicator, modeling and forecasting results graphically. Let's transform the selection schedule, supplementing it with forecast data. Appendix 1 (Table 4.10) Conclusion An economic model is defined as a system of interrelated economic phenomena, expressed in quantitative characteristics and presented in a system of equations, i.e. is a system of formalized mathematical description. For a targeted study of economic phenomena and processes and the formulation of economic conclusions - both theoretical and practical, it is advisable to use the method of mathematical modeling. Particular interest is shown in methods and means of simulation modeling, which is associated with the improvement of information technologies used in simulation modeling systems: the development of graphical shells for constructing models and interpreting the output results of modeling, the use of multimedia tools, Internet solutions, etc. In economic analysis, simulation modeling is the most universal tool in the field of financial, strategic planning, business planning, production management and design. Mathematical modeling of economic systems The most important property of mathematical modeling is its universality. This method allows, at the stages of design and development of an economic system, to form various variants of its model, to conduct repeated experiments with the resulting variants of the model in order to determine (based on specified criteria for the functioning of the system) the parameters of the created system necessary to ensure its efficiency and reliability. In this case, there is no need to purchase or produce any equipment or hardware to perform the next calculation: you just need to change the numerical values of the parameters, initial conditions and operating modes of the complex economic systems under study. Methodologically, mathematical modeling includes three main types: analytical, simulation and combined (analytical-simulation) modeling. An analytical solution, if possible, provides a more complete and clear picture, allowing one to obtain the dependence of the modeling results on the totality of the initial data. In this situation, one should move to the use of simulation models. A simulation model, in principle, allows one to reproduce the entire process of functioning of an economic system while preserving the logical structure, connections between phenomena and the sequence of their occurrence over time. Simulation modeling allows you to take into account a large number of real details of the functioning of the simulated object and is indispensable in the final stages of creating a system, when all strategic issues have already been resolved. It can be noted that simulation is intended to solve problems of calculating system characteristics. The number of options to be evaluated should be relatively small, since the implementation of simulation modeling for each option for constructing an economic system requires significant computing resources. The fact is that a fundamental feature of simulation modeling is the fact that in order to obtain meaningful results it is necessary to use statistical methods. This approach requires repeated repetition of the simulated process with changing values of random factors, followed by statistical averaging (processing) of the results of individual single calculations. The use of statistical methods, inevitable in simulation modeling, requires a lot of computer time and computing resources. Another disadvantage of the simulation modeling method is the fact that to create sufficiently meaningful models of an economic system (and at those stages of creating an economic system when simulation modeling is used, very detailed and meaningful models are needed) significant conceptual and programming efforts are required. Combined modeling allows you to combine the advantages of analytical and simulation modeling. To increase the reliability of the results, a combined approach should be used, based on a combination of analytical and simulation modeling methods. In this case, analytical methods should be used at the stages of analyzing the properties and synthesizing the optimal system. Thus, from our point of view, a system of comprehensive training of students in the means and methods of both analytical and simulation modeling is necessary. Organization of practical classes Students study ways to solve optimization problems that can be reduced to linear programming problems. The choice of this modeling method is due to the simplicity and clarity of both the substantive formulation of the relevant problems and the methods for solving them. In the process of performing laboratory work, students solve the following typical problems: transport problem; the task of allocating enterprise resources; the problem of equipment placement, etc. 2) Studying the basics of simulation modeling of production and non-production queuing systems in the GPSS World (General Purpose System Simulation World) environment. Methodological and practical issues of creating and using simulation models in the analysis and design of complex economic systems and decision-making in commercial and marketing activities are considered. Methods for describing and formalizing simulated systems, stages and technology for constructing and using simulation models, and issues of organizing targeted experimental studies using simulation models are studied. List of used literature Basic 1. Akulich I.L. Mathematical programming in examples and problems. - M.: Higher School, 1986. 2. Vlasov M.P., Shimko P.D. Modeling of economic processes. - Rostov-on-Don, Phoenix - 2005 (electronic textbook) 3. Yavorsky V.V., Amirov A.Zh. Economic informatics and information systems (laboratory workshop) - Astana, Foliant, 2008. 4. Simonovich S.V. Informatics, St. Petersburg, 2003 5. Vorobyov N.N. Game theory for economists - cyberneticists. - M.: Nauka, 1985 (electronic textbook) 6. Alesinskaya T.V. Economic and mathematical methods and models. - Tagan Rog, 2002 (electronic textbook) 7. Gershgorn A.S. Mathematical programming and its application in economic calculations. -M. Economics, 1968 Additionally 1. Darbinyan M.M. Inventories in trade and their optimization. - M. Economics, 1978 2. Johnston D.J. Economic methods. - M.: Finance and Statistics, 1960. 3. Epishin Yu.G. Economic and mathematical methods and planning of consumer cooperation. - M.: Economics, 1975 4. Zhitnikov S.A., Birzhanova Z.N., Ashirbekova B.M. Economic and mathematical methods and models: Textbook. - Karaganda, KEU publishing house, 1998 5. Zamkov O.O., Tolstopyatenko A.V., Cheremnykh Yu.N. Mathematical methods in economics. - M.: DIS, 1997. 6. Ivanilov Yu.P., Lotov A.V. Mathematical methods in economics. - M.: Science, 1979 7. Kalinina V.N., Pankin A.V. Math statistics. M.: 1998 8. Kolemaev V.A. Mathematical Economics. M., 1998 9. Kremer N.Sh., Putko B.A., Trishin I.M., Fridman M.N. Operations research in economics. Textbook - M.: Banks and exchanges, UNITY, 1997 10. Spirin A.A., Fomin G.P. Economic and mathematical methods and models in trade. - M.: Economics, 1998 Annex 1 Table 4.1 Table 4.2 Econometric modeling of the cost of apartments in the Moscow region. Study of the dynamics of an economic indicator based on the analysis of a one-dimensional time series. Linear pairwise regression parameters. Assessing the adequacy of the model, making a forecast. test, added 09/07/2011 Econometric modeling of the cost of apartments in the Moscow region. Matrix of pair correlation coefficients. Calculation of linear pair regression parameters. Study of the dynamics of an economic indicator based on the analysis of a one-dimensional time series. test, added 01/19/2011 Exploring the concept of simulation modeling. Time series simulation model. Analysis of indicators of the dynamics of development of economic processes. Abnormal series levels. Autocorrelation and time lag. Assessing the adequacy and accuracy of trend models. course work, added 12/26/2014 Studying and practicing skills in mathematical modeling of stochastic processes; study of real models and systems using two types of models: analytical and simulation. Main methods of analysis: dispersion, correlation, regression. course work, added 01/19/2016 The essence and content of the modeling method, the concept of a model. Application of mathematical methods for forecasting and analysis of economic phenomena, creation of theoretical models. Fundamental features characteristic of constructing an economic and mathematical model. test, added 02/02/2013 The division of modeling into two main classes - material and ideal. Two main levels of economic processes in all economic systems. Ideal mathematical models in economics, application of optimization and simulation methods. abstract, added 06/11/2010 Homomorphism is the methodological basis of modeling. Forms of representation of systems. Sequence of development of a mathematical model. Model as a means of economic analysis. Modeling of information systems. The concept of simulation modeling. presentation, added 12/19/2013 Theoretical foundations of mathematical forecasting of the promotion of investment instruments. The concept of a simulation system. Stages of constructing models of economic processes. Characteristics of Bryansk-Capital LLC. Assessing the adequacy of the model. course work, added 11/20/2013 Simulation modeling as a method for analyzing economic systems. Pre-project examination of a company providing printing services. Study of a given system using a Markov process model. Calculation of time to service one request. course work, added 10/23/2010 Application of optimization methods to solve specific production, economic and management problems using quantitative economic and mathematical modeling. Solving a mathematical model of the object under study using Excel.
Figure 4 shows the characteristics of the gamma distribution, as well as a graph of its density function for various values of these characteristics. 
This is a distribution that has an asymmetrical appearance. Occupies an intermediate position between exponential and normal. The probability density function of the Erlang distribution is represented by the formula:
Figure 5 shows the characteristics of the triangular distribution and the graph of its probability density function.
If we consider the dependence of one of the system characteristics η v (x i), as a function of only one variable x i(Fig.7), then at fixed values x i we will get different values η v (x i)
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Fig.9 Graph of delays in the “Current account” node.
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