operations research) I. о. - a relatively new area, Short story a cut goes back to the beginning of the Second World War. This exact mate. science contains a well-defined set general principles, to-rye provide researchers with a plan for the implementation of scientific research operations. It includes the following stages. 1. Formulation of the problem. 2. Construction of mate. a model representing the system under study. 3. Obtaining a solution from a given model. 4. Verification of the model and the solution obtained from it. 5. Establishing control over the decision. 6. Practice. solution implementation: implementation. Formulating the problem Serious attention must be paid to defining the general nature of the problem and, more importantly, to the objectives of the research. These goals should be formulated in behavioral terms in order to minimize or eliminate ambiguity and ambiguity. Time should also be allocated to correctly prioritize realistically achievable goals. Too long a list of goals can cause potential difficulties in achieving them, especially if these goals are not clearly linked in a logical sequence. Building a mathematical model The second phase of research with t. Sp. And about. assumes a description of the model. The purpose of the model is to represent the real world... In I. about. such models are symbolic, expressed in mat. terms. The classical equation E = mc2 is a typical example of math. models. Traditional forms for such models are algebraic equations, to-rye not only mean. are more economical than verbal formulations, but also entail the thoroughness and precision of definition necessary for a clear expression and understanding of individual elements and their interrelationships. The most important task in building such a model is a clear and precise development and definition of the target function. This function expresses the relationship between independent and dependent variables. Getting a solution from a given model The third phase is to find a solution. As a rule, it is desirable to find the optimal or the best solution, but it should be borne in mind that such a solution will have value only in the context of the model in question. Since the model is only a representation of a real-world problem, there are many situations in which the optimal solution may not be associated with the best choice ... However, when the optimal solution is combined with less optimal or more realistic alternative solutions, with the possibility of subsequent testing against a real problem, there are certain benefits to using the optimal solution. One of these benefits is associated with the definition at the end of the study. the relative distance between this ideal solution and the accepted alternative. A by-product of this methodology of using I. o. is the assumption that less optimal solutions can be considered as stepping stones on the way to achieving the goal. This method of successive approximations can lead the operations researcher to more fruitful results. There are many checkmates. procedures for obtaining solutions in the I.O. model. These procedures are based on applications of probability theory. Validation of the model and the solution derived from it Validation of the model and the solution involves the implementation of two steps. The first is a thorough analysis of all elements of the model, incl. rechecking of its algebraic factors for the presence of simplistic cosmetic errors, which can affect the validity. Dr. an even more important step involves redefining the relationships between the model and the prerequisites that were originally used to develop this model. A more systematic verification plan also includes the use of ist. data that can be easily entered into the model so that an experimental (prototype) solution can be obtained. These data should be carefully reviewed to ensure the validity of the check by the operations investigator. It should be noted that as long as this model is practically developed on the basis of the previous history. data and needs, she may behave very differently in the future. Dr. a common mistake is the introduction of factors into the model, to-rye were not presented in the ist. database. Establishing Control The fifth stage, establishing control over the decision, emerges from the reuse of the model. Control over the model is established when the operations researcher admits discrepancies in the values ​​of ist. data and recognizes that these discrepancies can affect the relationships between model elements and the resulting solutions. Dr. an important step can be the development of constraints on the selected bases. parameters of the model to establish a range of acceptable values ​​based on real data. Implementing the Model The final step is to introduce the real data model. Practice. the implementation of the model is associated with the obvious step of introducing real data and obtaining a solution to a real problem. In addition, it seems also important to assess the proximity of the real solution to the truth. solutions obtained earlier, as well as the consequences of this decision for improving the ways of operating the model. These steps provide an important connection between the mat. nature I. o. and practical. research results. Ultimately, these decisions and their managerial implications are used by an experienced I.O. to fine-tune the model for possible future use. See also Research Methodology R.S. Endrulis

Operations research- a science engaged in the development and practical application of mathematical, quantitative methods to justify decisions in all areas of purposeful human activity (effective organizational management).

General features of operations research

In each problem, we are talking about some kind of event that pursues a specific goal.

Some conditions have been set that characterize the situation (including the means that we can dispose of).

Within the framework of these conditions, it is required to make such a decision so that the planned event is in some sense the most profitable.

Features of Operations Research

A systematic approach to the analysis of the problem posed means that a particular problem should be considered from the point of view of its influence on the criterion of the functioning of the entire system.

The greatest effect can be achieved only with continuous research, ensuring continuity in the transition from one problem to another, arising in the course of solving the previous one.

Although the goal of operations research is to find an optimal solution, due to the complexity of calculating combinatorial problems, it is limited to finding a “good enough” solution.

Operational research is carried out in a comprehensive manner, in many areas. To conduct the study, operating groups are created:

Basic Operations Research Concepts

OPERATION - any controlled (that is, depending on the choice of parameters) event, united by a single concept and aimed at achieving a goal.

SOLUTION - any definite choice of parameters depending on us.

OPTIMAL SOLUTIONS - decisions on one or another basis are preferable to others.

THE PURPOSE OF THE STUDY OF THE OPERATION is a preliminary quantitative substantiation of optimal decisions.

ELEMENTS OF A SOLUTION - parameters, the combination of which forms a solution.

PERFORMANCE INDICATOR (TARGET FUNCTION) is a quantitative criterion that allows you to compare different solutions in terms of efficiency and reflects the target orientation of the operation (W => max or W => min).

The best solution is the one that maximizes the achievement of the goal.

The concept of the mathematical model of the operation

A schematic, simplified description of the operation using one or another mathematical apparatus, reflecting the most important features of the operation, all the essential factors on which the success of the operation mainly depends.

Direct and inverse problems of operations research

DIRECT PROBLEMS answer the question of what will happen if, under the given conditions, we take some kind of solution x X, i.e. what will be the performance indicator W (or a number of indicators).

To solve such a problem, a mat is built. a model that allows one or more performance indicators to be expressed in terms of given conditions and elements of a solution.

INVERSE PROBLEMS answer the question of how to choose a solution x in order for the efficiency indicator W to reach the maximum (minimum).

These tasks are more difficult. They can be solved by a simple enumeration of solutions to direct problems.

But when the number of variants of solutions is large, methods of directed enumeration are used, in which the optimal solution is found by performing successive "attempts" or approximations, each of which brings us closer to the desired optimal one.

1. The subject and objectives of the study of operations in the economy. Basic concepts of the theory of operations research.

The subject of operations research is an organizational management system or organization that consists of a large number units interacting with each other are not always consistent with each other and can be opposite.

The purpose of operations research is to provide a quantitative justification for the decisions taken on the management of organizations.

The solution that turns out to be most beneficial for the entire organization is called optimal, and the solution that is most beneficial to one or several departments will be suboptimal.

Operations Research is the science concerned with the development and practical application of methods for the most optimal management of organizational systems.

An operation is any measure (system of actions), united by a single concept and aimed at achieving a goal.

The purpose of operations research is a preliminary quantitative justification of optimal decisions.

Any definite choice of parameters depending on us is called a solution. Optimal solutions are those that are preferable for one reason or another over others.

The parameters, the combination of which forms a solution, are called solution elements.

The set of feasible solutions are given conditions that are fixed and cannot be violated.

Performance Indicator is a quantitative measure that allows you to compare different solutions for performance.

2. The concept of network planning and management. The network model of the process and its elements.

The method of working with network diagrams - network planning - is based on graph theory. Translated from Greek, graph (grafpho - I write) represents a system of points, some of them are connected by lines - arcs (or edges). This is a topological (mathematical) model of interacting systems. Graphs can be used to solve not only network planning problems, but other problems as well. The network planning method is used when planning a complex of interrelated works. It allows you to visually represent the organizational and technological sequence of work and establish the relationship between them. In addition, it allows you to ensure the coordination of operations of varying degrees of complexity and identify the operations on which the duration of the entire work (i.e. organizational measures) depends, as well as focus on the timely execution of each operation.

The basis of network planning and management is the network model (CM), which simulates a set of interrelated activities and events that reflect the process of achieving a certain goal. It can be presented in the form of a graph or a table.

Basic concepts of the network model:

Event, work, journey.

Events are the results of one or more jobs. They do not have a length of time.

A path is a chain of successive activities connecting the start and end vertices.

The duration of the journey is determined by the sum of the durations of its constituent works.

3. Building and organizing a network diagram.

A network model is used as a model reflecting the technological and organizational relationships of the construction and installation process in network planning and management systems (SPM).

A network model is a graphical representation of processes, the execution of which leads to the achievement of one or more set goals, indicating the established relationships between these processes. The network graph is a network model with calculated timing parameters.

The structure of a network diagram, which determines the interdependence of activities and events, is called its topology.

Work is a production process that requires time, labor and material resources, which, when executed, leads to the achievement of certain results.

Dependency (fictitious work) that does not require time is depicted with a dashed arrow. Dummy work is used in network graphics to represent the relationships between events and activities.

The network schedule applies the time, cost and other characteristics of the work.

Long-term work - the time of execution of this work in working days or other units of time, which are the same for all network activities. The duration of the work can be either a definite (deterministic) or a random variable specified by the law of its distribution.

The cost of work is the direct cost required to complete it, depending on the duration and conditions of this work.

Resources are characterized by the need for physical units required to carry out a given job.

Quality, reliability and other indicators of work serve as additional characteristics of work.

An event is the fact of the completion of one or several works, which is necessary and sufficient for the beginning of one or several subsequent works. Each event is assigned a number called a code. Each job is identified by two events: the start event code, denoted by i, and the end event code, denoted by the letter j.

Events that have no previous work are called initial events; events that do not have subsequent are final.

1 The direction of building a network can be of a different nature. The network schedule can be built from the initial event to the final and from the final to the initial (initial), as well as from any of the events to the initial or final.

2 When building a network, the following issues are resolved:

What work (work) needs to be done to start this work;

What work is advisable to perform in parallel with this work;

3 The initial network schedule does not take into account the duration of the activities that make up the network.

4 The shape of the graph should be simple and visually easy to understand.

5 There can be only one job between two events. In the construction of buildings and structures, work can be performed sequentially, in parallel or simultaneously, partly in series, and partly in parallel, as a result of which various dependencies are formed between individual works.

The numbering (coding) of events is performed after the end of the network construction, starting from the initial event to the final one.

4. Critical path of network graphics. Time reserves. Early and late dates of events and works in the network.

In a network, there can be several paths between start and end events. The path with the longest duration is called critical. The critical path determines the total duration of the work. All other paths have a shorter duration, and therefore the work performed in them has reserves of time.

The critical path is indicated on the network diagram by thickened or double lines (arrows).

Two concepts are of particular importance in drawing up a network diagram:

Early start of work is a period before which it is impossible to start this work without violating the accepted technological sequence. It is determined by the longest path from the initial event to the beginning of this work.

Late completion of work is the latest completion date that does not increase the total duration of work. It is determined by the shortest path from this event to the completion of all work.

Early completion - the deadline before which it is impossible to finish this work. It is equal to the early start plus the duration of the given work.

Late start is a period after which it is impossible to start this work without increasing the total duration of construction. It is equal to the late completion minus the duration of the given work.

If the event is the end of only one work (that is, only one arrow is directed at it), then the early end of this work coincides with the early start of the next one.

The general (full) reserve is the maximum time for which it is possible to delay the execution of this work without increasing the total duration of the work. It is defined by the difference between late and early start (or late and early finish - which is the same).

Private (free) reserve is the maximum time for which it is possible to delay the execution of this work without changing the early start of the next one. This reserve is possible only when the event includes two or more activities (dependencies), i.e. two or more arrows (solid or dotted) are directed towards it. Then, only for one of these works, the early completion will coincide with the early start of the subsequent work, for the rest it will have different meanings. This difference for each job will be its private reserve.

5. Dynamic programming. Bellman's optimality and control principle.

Dynamic programming is one of the most powerful optimization techniques. Specialists of different profiles deal with the tasks of making rational decisions, choosing the best options, and optimal management. Dynamic programming occupies a special position among optimization methods. This method is extremely attractive due to the simplicity and clarity of its basic principle - the principle of optimality. The sphere of application of the principle of optimality is extremely wide; the range of problems to which it can be applied has not yet been fully delineated. From the very beginning, dynamic programming acts as a means of practical solution of optimization problems.

In addition to the principle of optimality, the main method of research, an important role in the apparatus of dynamic programming is played by the idea of ​​immersing a specific optimization problem in a family of similar problems. Its third feature that distinguishes it from other optimization methods is the shape of the final result. Application of the principle of optimality and the principle of immersion in multistep, discrete processes lead to recurrent-functional equations for the optimal value of the quality criterion. The resulting equations make it possible to consistently write out the optimal controls for the original problem. The benefit here is that the problem of calculating control for the entire process is broken down into a series of simpler problems of calculating control for individual stages of the process.

The main drawback of the method is, in the words of Bellman, the "curse of dimension" - its complexity increases dramatically with the increase in the dimension of the problem.

6. The problem of the distribution of funds between enterprises.

We can say that the procedure for constructing an optimal control by the dynamic programming method falls into two stages: preliminary and final. At the preliminary stage, for each step, the CCA is determined, which depends on the state of the system (achieved as a result of the previous steps), and the conditionally optimal gain at all remaining steps, starting from the given one, also depends on the state. At the final stage, the (unconditional) optimal control is determined for each step. Preliminary (conditional) optimization is performed step by step in reverse order: from the last step to the first; final (unconditional) optimization - also step by step, but in a natural order: from the first step to the last. Of the two optimization stages, the first is incomparably more important and laborious. After the end of the first stage, the implementation of the second difficulty does not present: it remains only to "read" the recommendations already prepared at the first stage.

7. Statement of the linear programming problem.

Linear programming is a popular tool for solving economic problems that are characterized by the presence of one criterion (for example, to maximize income from the production of products due to the optimal choice of the production program, or, for example, to minimize transportation costs, etc.). For economic tasks, resource constraints (material and / or financial) are characteristic. They are written in the form of a system of inequalities, sometimes in the form of equalities.

From the point of view of predicting acceptable price intervals (or sales volumes) within the framework of the generalized nonparametric method, the use of linear programming means:

The criterion is the MAX price of the next product from the group of interest f.

The controlled variables are the prices of all products from group f.

The limitations in our forecasting problem using the generalized nonparametric method are:

a) a system of inequalities (constraints on the rationality of consumer behavior) (see 4.2. Forecasting within the framework of the generalized nonparametric method);

b) the requirement of non-negativity of controlled variables (in our forecasting problem, we require that prices for products from group f do not fall below 80% of the price values ​​at the last time point);

c) budget constraint in the form of equality - the requirement for the constancy of the amount of costs for the purchase of products from group f (taking into account 15% inflation, for example).

8. A graphical method for solving linear programming problems.

The graphical method is based on the geometric interpretation of a linear programming problem and is used mainly in solving problems in two-dimensional space and only some problems in three-dimensional space, since it is rather difficult to construct a polyhedron of solutions, which is formed as a result of the intersection of half-spaces. It is generally impossible to graphically depict the problem of a space of dimension more than three.

Let the linear programming problem be given in two-dimensional space, that is, the constraints contain two variables.

Find the minimum value of a function

(2.1) Z = C1x1 + C2x2

a11x1 + a22x2 b1

(2.2) a21x1 + a22x2 b2

aM1x1 + aM2x2 bM

(2.3) x1 0, x2 0

Assume that system (2.2) under condition (2.3) is consistent and its solution polygon is bounded. Each of inequalities (2.2) and (2.3), as noted above, defines a half-plane with boundary lines: ai1x1 + ai2x2 + ai3x3 = bi, (i = 1, 2, ..., n), х1 = 0, х2 = 0 ... Linear function (2.1) with fixed values ​​of Z is the equation of a straight line: C1x1 + C2x2 = const. Let us construct the polygon of solutions of the system of constraints (2.2) and the graph of the linear function (2.1) for Z = 0 (Fig. 2.1). Then the posed problem of linear programming can be given the following interpretation. Find the point of the solution polygon at which the straight line C1x1 + C2x2 = const is the support line and the function Z at the same time reaches a minimum.

The values ​​Z = C1x1 + C2x2 increase in the direction of the vector N = (C1, C2), therefore we move the straight line Z = 0 parallel to itself in the direction of the vector X. From Fig. 2.1 it follows that the straight line twice becomes a reference with respect to the solution polygon (at points A and C), and the minimum value takes at point A. The coordinates of the point A (x1, x2) are found by solving the system of equations of lines AB and AE.

If the solution polygon is an unbounded polygonal region, then two cases are possible.

Case 1. The straight line С1х1 + С2х2 = const, moving in the direction of the vector N or opposite to it, constantly intersects the solution polygon and at no point is a reference to it. In this case, the linear function is not bounded on the solution polygon, both above and below (Fig. 2.2).

Case 2. A straight line, moving, nevertheless becomes a reference with respect to the polygon of solutions (Fig. 2.2, a - 2.2, c). Then, depending on the type of the region, the linear function can be bounded from above and unbounded from below (Fig. 2.2, a), bounded from below and unbounded from above (Fig. 2.2, b), or bounded both from below and from above (Fig. . 2.2, c).

9. Simplex method.

The simplex method is fundamental to linear programming. The solution of the problem begins with consideration of one of the vertices of the polyhedron of conditions. If the vertex under study does not correspond to the maximum (minimum), then go to the neighboring one, increasing the value of the goal function when solving the problem to the maximum and decreasing it when solving the problem to the minimum. Thus, the transition from one vertex to another improves the value of the goal function. Since the number of vertices of the polytope is limited, then in a finite number of steps it is guaranteed to find the optimal value or to establish the fact that the problem is unsolvable.

This method is universal, applicable to any linear programming problem in canonical form... The system of restrictions here is the system linear equations, in which the number of unknowns is greater than the number of equations. If the rank of the system is r, then we can choose r unknowns, which we express in terms of the remaining unknowns. For definiteness, we assume that we have selected the first consecutive unknowns X1, X2, ..., Xr. Then our system of equations can be written as

The simplex method is based on a theorem called the fundamental theorem of the simplex method. Among the optimal plans for a linear programming problem in the canonical form, there is necessarily a support solution to its system of constraints. If the optimal plan of the problem is unique, then it coincides with some support solution. There are a finite number of different support solutions of the system of constraints. Therefore, the solution to the problem in the canonical form could be sought by enumerating the support solutions and choosing among them the one for which the value of F is greatest. But, firstly, all the support solutions are unknown and they need to be found, and, secondly, in real problems there are a lot of these solutions and direct enumeration is hardly possible. The simplex method is a certain procedure for directed enumeration of support solutions. Based on some previously found support solution according to a certain algorithm of the simplex method, we calculate a new support solution, on which the value of the objective function F is not less than on the old one. After a series of steps, we arrive at a reference solution, which is the optimal plan.

10. Statement of the transport problem. Methods for defining baselines.

There are m points of departure ("suppliers") and n points of consumption ("consumers") of some identical product. For each item are defined:

ai - production volumes of the i-th supplier, i = 1,…, m;

вj - demand of the j-th consumer, j = 1,…, n;

сij - the cost of transportation of one unit of products from point Ai of the i-th supplier, to point Bj - of the j-th consumer.

For clarity, it is convenient to present the data in the form of a table, which is called the table of transportation costs.

It is required to find a transportation plan in which the demand of all consumers would be fully satisfied, while there would be enough supplies of suppliers and the total transportation costs would be minimal.

A traffic plan is understood as the volume of traffic, i.e. the quantity of goods to be transported from the i-th supplier to the j-th consumer. To build a mathematical model of the problem, it is necessary to enter m · n pieces of variables хij, i = 1,…, n, j = 1,…, m, each variable хij denotes the volume of traffic from point Ai to point Bj. The set of variables X = (xij) will be the plan to be found based on the problem statement.

This is a condition for solving closed and open transport problems (ZTZ).

Obviously, for the solvability of Problem 1, it is necessary that the total demand does not exceed the production volume of suppliers:

If this inequality is strictly fulfilled, then the task is called "open" or "unbalanced", if, however, then the task is called "closed" transport task, and will have the form (2):

Balance condition.

This is a condition for solving closed transport problems (ZTZ).

11. Algorithm for solving the transport problem.

Application of the algorithm requires compliance with a number of prerequisites:

1. The cost of transporting a unit of product from each point of production to each point of destination must be known.

2. The stock of products at each point of production must be known.

3. The needs for products at each point of consumption must be known.

4. The total supply must be equal to the total demand.

The algorithm for solving the transport problem consists of four stages:

Stage I. Presentation of data in the form of a standard table and search for any acceptable allocation of resources. Allocation of resources is called admissible, which allows satisfying all demand at the points of destination and taking out the entire stock of products from the points of production.

Stage 2. Checking the obtained resource allocation for optimality

Stage 3. If the obtained allocation of resources is not optimal, then the resources are redistributed, reducing the cost of transportation.

Stage 4. Rechecking the optimality of the obtained resource allocation.

This iterative process is repeated until an optimal solution is obtained.

12. Models of inventory management.

Despite the fact that any inventory management model is designed to answer two main questions (when and how much), there are a significant number of models that are built using a variety of mathematical tools.

This situation is due to the difference in the initial conditions. The main basis for the classification of inventory management models is the nature of the demand for stored products (recall that from the point of view of a more general gradation, we are now considering only cases with independent demand).

So, depending on the nature of demand, inventory management models can be

deterministic;

probabilistic.

In turn, deterministic demand can be static, when the intensity of consumption does not change over time, or dynamic, when reliable demand can change over time.

Probabilistic demand can be stationary, when the probability density of demand does not change over time, and non-stationary, where the probability density function changes with time. The above classification is explained in the figure.

The simplest is the case of deterministic static demand for products. However, this kind of consumption is quite rare in practice. The most complex models are non-stationary models.

In addition to the nature of demand for products, when building inventory management models, many other factors have to be taken into account, for example:

terms of order fulfillment. The duration of the procurement period can be constant or be a random variable;

replenishment process. Can be instantaneous or distributed over time;

the presence of restrictions on working capital, storage space, etc.

13. Queuing systems (QS) and indicators of their effectiveness.

Queuing systems (QS) are systems of a special type that implement multiple execution of the same type of tasks. Such systems play an important role in many areas of the economy, finance, production and everyday life. As examples of CMO in financial and economic; the sphere can include banks of various types (commercial, investment, mortgage, innovative, savings), insurance organizations, state joint stock companies, companies, firms, associations, cooperatives, tax inspectorates, audit services, various communication systems (including telephone exchanges), loading and unloading complexes (ports, freight stations), gas stations, various enterprises and service organizations (shops, information bureaus, hairdressing salons, ticket offices, currency exchange offices, repair shops, hospitals). Systems such as computer networks, systems for collecting, storing and processing information, transport systems, automated production areas, production lines, various military systems, in particular air or missile defense systems, can also be considered as a kind of CMO

Each QS includes in its structure a certain number of service devices, which are called service channels (devices, lines). The role of channels can be played by various devices, persons performing certain operations (cashiers, operators, hairdressers, salespeople), communication lines, cars, cranes, repair crews, railway tracks, gas stations, etc.

Queuing systems can be single-channel or multi-channel.

Each QS is designed to service (fulfill) a certain flow of claims (claims) arriving at the input of the system, for the most part, not regularly, but at random times. Serving applications, in this case, also does not last permanently, in advance known time, and random time, which depends on many random, sometimes unknown to us, reasons. After servicing the request, the channel is released and is ready to receive the next request. The random nature of the flow of requests and the time of their servicing leads to uneven workload of the QS: at other times, unserved claims can accumulate at the QS input, which leads to an overload of the QS, and sometimes, with free channels at the QS input, there will be no claim, which leads to an underload of the QS, i.e. e. to the idleness of its channels. The requests accumulating at the entrance of the QS, either "become" in the queue, or due to the impossibility of further stay in the queue, leave the QS unserved.

Indicators of the effectiveness of the functioning of the pair "CMO - consumer", where the consumer is understood as the entire set of applications or some of their source (for example, the average income brought by the CMO per unit of time, etc.). This group of indicators turns out to be useful in cases where some income received from servicing claims and the cost of servicing are measured in the same units. These indicators are usually quite specific and are determined by the specifics of the QS, the serviced applications and the discipline of service.

14. Equations of dynamics for probabilistic states (Kolmogorov equations). Limiting probabilities of states.

Formally differentiating the Kolmogorov – Chapman equation with respect to s for s = 0, we obtain the direct Kolmogorov equation:

Formally differentiating the Kolmogorov - Chapman equation with respect to t at t = 0, we obtain the inverse Kolmogorov equation

It should be emphasized that for infinite-dimensional spaces the operator is no longer necessarily continuous, and may not be defined everywhere, for example, to be a differential operator in the space of distributions.

In the event that the number of states of the system S is finite and from each state it seems possible to pass (in a given number of steps) to each other state, then marginal probabilities states exist and also do not depend on the initial state of the system.

In fig. shows a graph of states and transitions that satisfy the stated condition: from any state, the system can sooner or later go to any other state. The condition will not be met when changing the direction of arrow 4-3 on the graph in Fig, but to the opposite.

Let us assume that the stated condition is satisfied, and, therefore, the limiting probabilities exist:

The limiting probabilities will be denoted by the same letters as the probabilities of states, while they mean numbers, and not variables (functions of time).

It is clear that the limiting probabilities of states should add up to unity: Consequently, a certain limiting stationary mode is established in the system at: let the system change its own states in a random way, but the probability of each of these states does not depend on time and each of them is realized with some constant probability, which is the average relative time that the system is in this state.

15. The process of death and reproduction.

By the continuous-time Markov process of death and reproduction we mean such an r.p. which can take only non-negative integer values; changes in this process can occur at any time t, while at any time it can either increase by one or remain unchanged.

The multiplication fluxes λi (t) will be called Poisson fluxes leading to an increase in the function X (t). Accordingly, μi (t) are flows of death leading to a decrease in the function X (t).

Let us compose the Kolmogorov equations by the graph:

If the thread is finite state:

The system of Kolmogorov equations for the process of death and reproduction with a limited number of states has the form:

The process of pure reproduction is such a process of death and reproduction, in which the intensities of all death streams are equal to zero.

The process of pure death is a process of death and reproduction in which the intensities of all reproduction streams are equal to zero.

16. Queuing systems with failures.

The simplest of the considered problems in the framework of the queuing theory is the model of a single-channel QS with failures or losses.

It should be noted that in this case the number of channels is 1 (). This channel receives a Poisson stream of customers, the intensity of which is equal to. Time affects intensity:

If a request arrives on a channel that is currently not free, it is rejected and is no longer listed in the system. The service of applications is carried out during a random time, the distribution of which is realized in accordance with the exponential law with the parameter:

17. Queuing systems with waiting.

A claim arriving at a time when the channel is busy enters the queue and awaits service.

System with limited queue length. First, suppose that the number of seats in the queue is limited to m, i.e., if a customer arrives at the moment when there are already m customers in the queue, it leaves the system unserved. Further, letting m tend to infinity, we obtain the characteristics of a single-channel QS without restrictions on the queue length.

We will number the states of the QS according to the number of claims in the system (both serviced and awaiting servicing):

- the channel is free;

—The channel is busy, there is no queue;

- the channel is busy, one request is in the queue;

—The channel is busy, k - 1 customers are in the queue;

- the channel is busy, t applications are in the queue.

18. Methods of decision-making in a conflict. Matrix games. Pure and blended strategy games.

A matrix game is a finite zero-sum game of two players in which the payoff of player 1 is specified in the form of a matrix (the row of the matrix corresponds to the number of the applied strategy of player 2, the column corresponds to the number of the applied strategy of player 2; at the intersection of the row and column of the matrix there is the payoff of player 1, appropriate strategies).

For matrix games, it is proved that any of them has a solution and it can be easily found by reducing the game to a linear programming problem.

A zero-sum two-player matrix game can be thought of as the following abstract two-player game.

The first player has m strategies i = 1,2, ..., m, the second has n strategies j = 1,2, ..., n. Each pair of strategies (i, j) is assigned a number aij, which expresses the payoff of player 1 at the expense of player 2 if the first player accepts his i-th strategy, and 2 - its j-th strategy.

Each of the players makes one move: player 1 chooses his i-th strategy (i =), 2 - his j-th strategy (j =), after which player 1 gets a payoff аij at the expense of player 2 (if аij

Each strategy of the player i =; j = is often called pure strategy.

Definition. A player's mixed strategy is a complete set of probabilities of applying his pure strategies.

Thus, if player 1 has m pure strategies 1,2, ..., m, then his mixed strategy x is a set of numbers x = (x1, ..., xm) satisfying the relations

xi³ 0 (i = 1, m), = 1.

Similarly, for player 2, who has n pure strategies, the mixed strategy y is a set of numbers

y = (y1, ..., yn), yj ³ 0, (j = 1, n), = 1.

Since each time a player uses one pure strategy excludes the application of another, pure strategies are inconsistent events. Moreover, they are the only possible events.

A pure strategy is a special case of a mixed strategy. Indeed, if in a mixed strategy any i-th pure strategy is applied with probability 1, then all other pure strategies are not applied. And this i-th pure strategy is a special case of a mixed strategy. To maintain secrecy, each player applies his strategies regardless of the choice of the other player.

19. Geometric method for solving a matrix game.

The solution of games of the size 2xn or nx2 allows a clear geometric interpretation. Such games can be solved graphically.

On the XY plane, along the abscissa, we plot the unit segment A1A2 (Figure 5.1). Each point of the segment is associated with some mixed strategy U = (u1, u2). Moreover, the distance from some intermediate point U to the right end of this segment is the probability u1 of choosing the strategy A1, the distance to the left end is the probability u2 of choosing the strategy A2. Point A1 corresponds to pure strategy A1, point A2 - to pure strategy A2.

At points A1 and A2, we will restore the perpendiculars and we will postpone the winnings of the players on them. On the first perpendicular (coinciding with the OY axis), we show the payoff of player A when using strategy A1, on the second - when using strategy A2. If player A applies strategy A1, then his payoff for strategy B1 of player B is 2, and for strategy B2 it is 5. Points B1 and B2 correspond to numbers 2 and 5 on the OY axis. Similarly, on the second perpendicular we find points B "1 and B" 2 (wins 6 and 4).

Connecting points B1 and B "1, B2 and B" 2, we get two straight lines, the distance from which to the OX axis determines the average payoff for any combination of the corresponding strategies.

For example, the distance from any point of the segment B1B "1 to the OX axis determines the average payoff of player A for any combination of strategies A1 and A2 (with probabilities u1 and u2) and strategy B1 of player B.

The ordinates of points belonging to the broken line B1MB "2 determine the minimum payoff of player A when he uses any mixed strategies. This minimum value is the largest at point M, therefore, this point corresponds to the optimal strategy U * = (,), and its ordinate is equal to the price of the game v ...

We find the coordinates of the point M as the coordinates of the point of intersection of the straight lines B1B "1 and B2B" 2.

To do this, you need to know the equations of the straight lines. You can compose such equations using the formula for the equation of a straight line passing through two points:

Let's compose the equations of the straight lines for our problem.

Line B1B "1: = or y = 4x + 2.

Direct B2B "2: = or y = -x + 5.

We get the system: y = 4x + 2,

Let's solve it: 4x + 2 = -x + 5,

x = 3/5, y = -3/5 + 5 = 22/5.

So U = (2/5, 3/5), v = 22/5.

20. Bimatrix games.

A bimatrix game is a finite game of two players with a nonzero sum, in which the payoffs of each player are specified by matrices separately for the corresponding player (in each matrix, the row corresponds to the strategy of player 1, the column corresponds to the strategy of player 2, at the intersection of the row and column in the first matrix is ​​the payoff of the player 1, in the second matrix is ​​the payoff of player 2.)

For bimatrix games, a theory of optimal player behavior has also been developed; however, such games are more difficult to solve than ordinary matrix games.

21. Statistical games. Principles and criteria for making decisions under conditions of complete and partial uncertainty.

In operations research, it is common to distinguish between three types of uncertainties:

ambiguity of goals;

the uncertainty of our knowledge about the environment and the factors acting in this phenomenon (the uncertainty of nature);

uncertainty about the actions of an active or passive partner or adversary.

In the above classification, the type of uncertainties is considered from the standpoint of one or another element of the mathematical model. So, for example, the ambiguity of goals is reflected in the formulation of the problem on the choice of either individual criteria or the entire vector of the beneficial effect.

On the other hand, the other two types of uncertainties mainly affect the compilation of the objective function of the constraint equations and the decision-making method. Of course, the above statement is rather arbitrary, as, indeed, any classification. We present it only with the purpose of highlighting some more features of uncertainties that must be borne in mind in the decision-making process.

The point is that, in addition to the classification of uncertainties considered above, one must take into account their type (or "genus") from the point of view of the relation to randomness.

about this feature, stochastic (probabilistic) uncertainty can be distinguished, when unknown factors are statistically stable and therefore represent the usual objects of probability theory - random variables (or random functions, events, etc.). In this case, all the necessary statistical characteristics (distribution laws and their parameters) must be known or determined in the formulation of the problem.

An example of such tasks can be, in particular, a system for the maintenance and repair of any type of equipment, a system for organizing thinning, etc.

Another extreme case may be non-stochastic uncertainty (according to E.S. Ventzel - "bad uncertainty"), in which there are no assumptions about stochastic stability. Finally, we can talk about an intermediate type of uncertainty, when a decision is made on the basis of any hypotheses about the laws of distribution of random variables. In this case, the decision maker must bear in mind the danger of the discrepancy between his results and real conditions. This misalignment hazard is formalized using risk ratios.

Risk decision making can be based on one of the following criteria:

expected value criterion;

combinations of expected value and variance;

known limit level;

the most likely event in the future.

Under operation means any event, united by a single idea and direction to achieve a certain goal.

An operation is always a managed event, i.e. the choice of parameters that characterize the way of its organization depends on us.

Any definite choice of parameters depending on us will be called decision.

Optimal solutions are those that, for one reason or another, are preferable to others.

The main task of operations research is preliminary quantitative justification of optimal solutions... Operations Research does not aim to fully automate decision making. The decision is always made by a person. The mission of operations research is to produce quantitative data and recommendations that make it easier for a person to make decisions.

Along with the main task - justification of optimal decisions - Operations research includes other tasks:

Comparative assessment of various options for organizing the operation,

Assessment of the influence of various parameters on the operation,

Research on bottlenecks, ie. elements, the malfunction of which has a particularly strong effect on the success of the operation, etc.

These auxiliary tasks become especially important when the given operation is considered not in isolation, but as a constituent element of the whole. systems operations. A “systematic” approach to the tasks of operations research requires taking into account the interdependence and conditionality of a whole range of activities, ie. make the final decision taking into account the role and place of this operation in the system.

Under efficiency operation is understood as the degree of its adaptability to the task facing it.

In order to judge the efficiency of an operation and compare differently organized operations in terms of efficiency, you need to have some numerical evaluation criterion or performance indicator.

The sequence of actions in the research of operations.

1. The research goal is formulated and the problem statement is developed.

2. To apply quantitative methods in any field, it is always required to construct a mathematical model of the phenomenon. Based on the analysis of the properties of the original, this model is built.

3. After building the model, the results are obtained on it.

4. They are interpreted in terms of the original and transferred to the original.

5. By means of comparison, the simulation results are compared with the results obtained by direct examination of the original.

If the results obtained using the model are close to the results obtained in the study of the original, then with regard to these properties, the model can be considered adequate to the original.

During the design and operation of ACS, tasks often arise related to the analysis of both quantitative and qualitative patterns of their functioning, determination of their optimal structure, etc.

Direct experimentation on objects for solving these problems has a number of significant disadvantages:

1. The set operating mode of the object is violated.

2. In a full-scale experiment, it is impossible to analyze all the alternative options for constructing a system, etc.

It is advisable to solve these problems on a model separated from the object and implemented on a computer.

When modeling information systems, mathematical models are widely used.

The method of mathematical modeling is a way to study various objects by compiling an appropriate mathematical description and calculating, on its basis, the characteristics of the object under study.

It is necessary to build a mathematical model. It formally reflects the process of functioning of the original and describes the basic laws of its behavior. In this case, all secondary, insignificant factors are excluded from consideration.

Complex systems are the object of mathematical modeling. A complex system is called a certain way organized and purposeful functioning set of a large number of information-related and interacting elements under the influence of external factors.

There are 4 main stages of modeling systems on a computer:

Building a conceptual model of the system and its formalization;

Algorithmization of the system model and development of a simulation program;

Obtaining and interpretation of preliminary simulation results;

Checking the adequacy of the model and system; model correction

The main calculation of the quality indicators of the functioning of the system based on the simulation results, the implementation of the model.

Lecture 3. Basic concepts of the method expert assessments... Formation of expert groups. Polling procedures. Ranking methods, pairwise comparisons, relative scale assessment.

Operations research

Operations research(IO) (eng. Operations Research, OR) is a discipline engaged in the development and application of methods for finding optimal solutions based on mathematical modeling, statistical modeling and various heuristic approaches in various fields of human activity. Sometimes the name is used mathematical methods of operations research.

Operations Research is the application of mathematical, quantitative methods to inform decisions in all areas of purposeful human activity. Operations research begins when one or another mathematical apparatus is used to justify decisions. Operation- any event (system of actions), united by a single concept and aimed at achieving some goal (for example, activities of tasks 1-8, listed below, will be operations). An operation is always a controlled event, that is, it depends on the person how to choose the parameters that characterize its organization (in a broad sense, including the set of technical means used in the operation). Decision(successful, unsuccessful, reasonable, unreasonable) - any specific set of parameters depending on a person. Optimal- a solution that is preferable for one reason or another. Purpose of Operations Research- preliminary quantitative substantiation of optimal solutions based on the performance indicator. Decision making itself is outside the scope of operations research and falls under the purview of the responsible person (s). Solution elements- parameters, the totality of which forms a solution: numbers, vectors, functions, physical characteristics, etc. If the elements of the solution can be disposed of within certain limits, then the given ("disciplining") conditions (restrictions) are fixed immediately and cannot be violated (carrying capacity , dimensions, weight). These conditions include funds (material, technical, human) that a person has the right to dispose of, and other restrictions imposed on the solution. Their combination forms many possible solutions.

Examples: A plan is drawn up for the transportation of goods from the points of departure A 1, A 2,…, A m to the points of destination B 1, B 2,…, B n. Elements of the solution are numbers x ij showing how much cargo will be sent from the i-th point of departure A i to the j-th point of destination B j. Solution - a set of numbers x 11, x 12,…, x m1, x m2,…, x mn

The future relationship between IO and (complex) systems theory is not entirely clear.

Typical tasks

Taken from different areas of practice

1. Enterprise supply plan
2. Construction of a section of the highway
3. Sale of seasonal goods
4. Snow protection of roads
5. Anti-submarine raid
6. Selective product inspection
7. Medical examination
8. Library service

Some examples of problem statements related to IO:

• Scheduling tasks such as Open Shop Scheduling Problem, Flow Shop Scheduling Problem, Job Shop Scheduling Problem. en: Job shop scheduling ) etc.

A characteristic feature of operations research is a systematic approach to the problem posed and analysis. The systems approach is the main methodological principle of operations research. It is as follows. Any problem that is being solved should be considered from the point of view of its influence on the criteria for the functioning of the system as a whole. Operations research is characterized by the fact that each problem can lead to new challenges. An important feature of operations research is the desire to find the optimal solution to the problem (the principle of "optimality"). However, in practice, such a solution cannot be found for the following reasons:

1. lack of methods that make it possible to find a globally optimal solution to the problem
2. limited existing resources (for example, limited computer time), which makes it impossible to implement accurate optimization methods.

In such cases, they are limited to the search for not optimal, but rather good, from the point of view of practice, solutions. We have to find a compromise between the efficiency of solutions and the cost of finding them. Operations Research provides a tool for finding such trade-offs.

IO is mainly used by large Western companies in solving production planning problems (controlling, logistics, marketing) and other complex tasks. The use of IO in economics makes it possible to lower costs or, to put it differently, to increase the productivity of an enterprise (sometimes several times!). IO is actively used by the armies and governments of many developed countries to assess the combat effectiveness of weapons, military equipment and military formations, develop new types of weapons, solve complex problems of supplying armies, advance armies, develop war strategies, develop interstate trade mechanisms, predict development (for example, climate ), etc. The solution of complex problems of increased importance is carried out by IO methods on supercomputers, but the development is carried out on simple PCs. AI methods can also be applied in small businesses using a PC.

History

At the beginning of the war, combat patrolling of Allied aircraft to detect enemy ships and submarines was disorganized. The involvement of Operations Research Specialists in the planning made it possible to establish patrol routes and flight schedules that minimized the likelihood of leaving the target undetected. The recommendations received were used to organize patrols over the South Atlantic in order to intercept German ships with military materials. Of the five enemy ships that broke through the blockade, three were intercepted on the way from Japan to Germany, one was discovered and destroyed in the Bay of Biscay, and only one managed to escape thanks to careful camouflage.

After the end of World War II, operations research teams continued their work in the United States and British Armed Forces. Publication of a number of results in the open press caused a surge of public interest in this area. There is a tendency to use methods of research of operations in commercial activities, in order to reorganize production, transfer industry to a peaceful track. Millions of dollars are allocated for the development of mathematical methods for researching operations in the economy.

In Great Britain, the nationalization of certain types of industry has created the opportunity for economic research on the basis of mathematical models on a national scale. Operations research has come to be used in the planning and execution of several government, social and economic events. For example, studies conducted for the Food Ministry predicted the impact of government pricing policies on the family budget.

In the United States, the introduction of operations research methods into the practice of economic management was somewhat slower - but even there, many concerns soon began to attract specialists of this kind to solve problems related to price regulation, increasing labor productivity, accelerating the delivery of goods to consumers, etc. control methods belonged to the aviation industry, which could not help but keep pace with the growing demands on the air force. In the 1950s-1960s, societies and operations research centers were created in the West, publishing their own scientific journals; most Western universities include this discipline in their curricula.

The greatest contribution to the formation and development of new science was made by R. Akof, R. Bellman, J. Danzig, G. Kuhn, T. Saati (English) Russian , R. Cherman (USA), A. Kofman, R. Ford (France) and others.

L. V. Kantorovich, B. V. Gnedenko, M. P. Buslenko, V. S. Mikhalevich, N. N. Moiseev, Yu. M. Ermolaev, N.Z.Shoru and others.

For his outstanding contribution to the development of the theory of optimal use of resources in economics, Academician L.V. Kantorovich, together with Professor T. Koopmans (USA), was awarded the Nobel Prize in Economics in 1975.

see also

Notes (edit)

Literature

• Hemdi A. Taha. Operations Research: An Introduction. - M .: Williams, 2007 .-- 912 p. - ISBN 0-13-032374-8
• Degtyarev Yu. I. Operations Research: A textbook for universities in the field of ACS. - M .: Higher school, 1986.
• A. A. Greshilov Mathematical methods of decision making. - M .: MSTU im. N.E. Bauman, 2006 .-- 584 p. - ISBN 5-7038-2893-7

Links

• Operations research in the Open Directory Project (dmoz) link directory.

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