Theory of optimal resource allocation. Summary: Application of linear programming methods in military affairs

As you know, in the practice of economic activity, the choice between different options (plans, decisions) involves the search for the best. When the hostess goes to the market to buy meat, and the designer seeks to find the best way to place the machines, they are looking for options that require a minimum of costs or a maximum of result, taking into account certain restrictions (money, resources, time).
Solving such a problem can be difficult, especially when there are a large number of options. The time and costs of choosing the optimum are not always justified: the costs of searching and sorting through options may exceed the gain achieved.
As practice shows, experience and intuition are not enough to justify the optimal solution.
138
More reliable and effective method- use of mathematical (quantitative) approaches and calculations. However, mathematical approaches and justifications were ignored for a long time by theorists who made the "weather" in economics. Many important works were frozen, the publications of mathematical economists were hampered and limited. And yet, during that period, mathematical research continued, even in the face of persecution of mathematicians, brilliant results were achieved.
One of the most significant and striking achievements in the field of economic and mathematical research was the discovery by Leonid Vitalievich Kantorovich (1912-1986) of the Linear Programming Method. Linear programming is the solution of linear equations (first-degree equations) by compiling programs and applying various methods for their sequential solution, which greatly facilitate calculations and achieve the desired results.
For the development of the linear programming method, or, as stated in the diploma of the Swedish Academy of Sciences, for “contribution to the theory of optimal resource allocation”, L. V. Kantorovich, the only Soviet economist, was awarded the Nobel Prize in Economics (1975). The prize was awarded to him jointly with the American economist Tjalling Charles Koopmans; who somewhat later, independently of Kantorovich, proposed a similar methodology.
The development of linear programming began with the search for a solution to a practical problem. Kantorovich was approached by the engineers of the plywood trust with a request to find an efficient way to allocate resources that would ensure the highest productivity of the equipment. The employees of the enterprise puzzled over how to ensure the best option for the production of plywood with five machines, eight types of raw materials. In other words, it was necessary to find a solution to a specific technical and economic problem with an objective function (“functional”) - to maximize the output of finished products.
The merit of Kantorovich is that he proposed a mathematical method for choosing the optimal variant. Solving the particular problem of the most rational loading of equipment, the scientist developed a method called the linear programming method. In fact, he opened a new section in mathematics, which has become widespread in economics.
139
chesky practice; contributed to the development and use of electronic computers.
Kantorovich was not a “pure” economist, but he understood perfectly well the importance of the maximization method with limited resources, and hence the creation of a mathematical basis for solving typical economic problems.
The conditions of the problem for the optimum and the goal to be achieved can be expressed using a system of linear equations. Since there are fewer equations than unknowns, the problem usually has not one, but many solutions. But you need to find one, according to the terminology of mathematicians, an extremal solution.
In the plywood production optimization problem, Kantorovich presented a variable that should be maximized as the sum of the cost of products produced by all machines. Limiters were presented in the form of equations that establish the relationship between all the factors spent in production (wood, glue, electricity, working time) and the amount of output (plywood) on each of the machines. For indicators of production factors, coefficients were introduced, called resolving factors, or multipliers. With their help, the task is solved. If the values of the resolving factors are known, then the desired values, in particular, the optimal volume of output, can be found relatively easily.
Kantorovich substantiated the economic meaning of the coefficients he proposed (resolving factors). They represent nothing more than the marginal costs of the limiting factors. In other words, these are the objectively significant prices of each of the factors of production in relation to the conditions of a fully competitive market.
To solve the problem for the optimum, Kantorovich used the method of successive approximations, sequential comparison of options with the choice of the best in accordance with the conditions of the problem.
Suppose we need to solve a transport problem, justify the most rational distribution of cargo flows. For example, in total, you need to transport 180 tons of cargo from three sources to three consumers, the total demand of which is also 180 tons. The difficulty is that the cargo is unevenly distributed: one supplier has 50 tons, another has 60, and the third has 70 tons .
140
The demand of consumers is also unequal, it is 40, 85 and 55 tons, respectively. The distances (shoulders) of cargo transportation are also not the same - from 1 to 6 km. The task is to draw up such a transportation plan that would meet the requirement of minimizing freight turnover (the minimum number of ton-kilometers).
How to solve this problem? In everyday practice, managers can do the monotonous work of a lengthy enumeration of possible options. Gradually they will be able to move from a transportation plan of, say, 750 t/km to a plan of 655 t/km. The search will require a lot of effort, a significant amount of calculations. Most importantly, it is difficult to establish which of the proposed options” is optimal. Suppose a variant of the plan with a cargo turnover of 575 t/km is found. But it remains unknown whether there is one or more more profitable options for the plan that require less cost.
The task becomes completely unsolvable if we move from a relatively simple scheme to setting a task for compiling a variant of transportation of one or more products (coal, cement, building materials) on a regional or country scale. Even in the case of consolidation, aggregation of initial indicators (the number of senders and recipients of mass labor), it turns out that only the network scheme will cover tens of thousands of aggregated points, and calculations and comparison of options will require such a number of operations, for which you will have to attract a little whether not the entire population of Russia.
For the first time, the work, which outlined the essence of the method proposed by Kantorovich, was published in 1939 under the title "Mathematical methods for organizing production planning." Continuing research, the scientist develops general theory rational use resources.
During the Great Patriotic War Kantorovich, being a professor at the Naval Engineering Academy in besieged Leningrad, substantiates, based on the method of linear programming, the optimal distribution of production and consumer factors. The book “Economic Calculation of the Most Expedient Use of Resources”, prepared by him in 1942, unfortunately, was not published at that time.
Later, one of his largest works, Economic Calculation of the Best Use of Resources (1959), was published.
141
In this book, as noted by the members of the Scientific Council on the Application of Mathematics in Scientific Research and Planning, an in-depth analysis of the ideas of linear programming developed by the author earlier is presented, and at the same time, for the first time, the problem of developing an optimal plan for the entire national economy as a mathematical model is posed39.
The undoubted merit of Kantorovich is the identification of dual estimates in linear programming problems. It is impossible to minimize costs and maximize results at the same time. One contradicts the other. At the same time, both of these approaches are interrelated, if, say, an optimal transportation scheme is found, then a certain price system corresponds to it. If the optimal values of prices are found, then it is relatively easy to obtain a transportation scheme that meets the requirement of optimality.
For any linear programming problem, there is a conjugate or dual problem. If the direct task is to minimize the objective function, then the dual task is to maximize it.
Dual valuations provide a fundamental opportunity to measure not only price and cost indicators, but also utility. At the same time, dual, interrelated assessments correspond to specific conditions. If the conditions change, then the estimates also change. To a certain extent, the search for the optimum is the determination of socially necessary costs, taking into account, on the one hand, labor, cost costs, and on the other hand, social needs, the usefulness of the product for consumers.
When getting acquainted with works on linear programming, one may come across some terminological subtleties. The term “resolving factors”, originally used by Kantorovich, in subsequent works receives a slightly different interpretation and a different formulation, namely, objectively determined estimates. These estimates are not arbitrary, their values are objectively determined, they are set by the specific conditions of the problem. The values of objectively determined estimates are suitable only for this task,
Kantorovich suggested calculating them when developing plans; enterprises are called upon to rely on these indicators when calculating the costs and volumes of output of certain types of products. Objectively determined estimates are adjusted to
142
depending on the ratio of demand and production volumes. Such calculations, introduced into the practice of planning and management, are designed to optimize the use of resources.
The ideas and proposals put forward by Kantorovich provided for the use of market categories in the practice of managing. In fact, at that time there was a search, the prerequisites for the conceptual basis for reforming the existing economic system were being formed.
With the active participation of Kantorovich and his closest colleagues and friends - Viktor Valentinovich Novozhilov (1892-1970), Vasily Sergeevich Nemchinov (1894-1964) in the second half of the 50s - early 60s. a national economic and mathematical school is being formed. All three continued to develop linear programming methods, built economic models, then proceeding to the development of a system of models called SOFE (systems for the optimal functioning of the economy).

Ministry of Education and Science, Youth and Sports of Ukraine

Sevastopol National Technical University

Faculty of Economics and Management

On the topic: L.V. Kantorovich: development of the theory of linear programming

in the discipline "History of Economics and Economic Thought"

Completed: Art. gr. MO-21

Kovaleva S.N.

Checked by: teacher

Kerez E.S.

Sevastopol 2009

1.2 Contribution to science

1.3 Scientific works

Conclusion

Introduction

In this essay, I will write about the activities of Leonid Vitalievich Kantorovich, an outstanding scientist of the twentieth century, about his struggle for the recognition of his economic and mathematical theories, about the initial stage of the history of linear programming, about the emergence of a new area of \u200b\u200bmathematical activity related to economic applications, which we call operations research, sometimes mathematical economics, sometimes economic cybernetics, etc., about its place and connections with the modern mathematical landscape.

1. Leonid Vitalievich Kantorovich

1.1 Biography of L.V. Kantorovich

Leonid Vitalievich Kantorovich (1912-1986) was born in St. Petersburg in the family of a doctor. His outstanding abilities manifested themselves early - at the age of 14 he entered the Leningrad State University. After graduating from Leningrad State University in 4 years, he entered graduate school. In 1932 he became an associate professor, and in 1935 a professor at Leningrad State University. In 1935 he was awarded the title of Doctor of Physical and Mathematical Sciences without defending a dissertation. In 1958 he was elected a corresponding member of the Academy of Sciences of the USSR in economics, and in 1964 - an academician. For the development of the method of linear programming and economic models, he was awarded the Lenin Prize in 1965 together with Academician V. S. Nemchinov and Professor V. V. Novozhilov. Since 1971, he worked in Moscow, at the Institute of National Economy Management of the State Committee of the Council of Ministers of the USSR for Science and Technology. 1975 - Nobel Prize in Economics (together with T. Koopmans "for his contribution to the theory of the optimal allocation of resources"). From 1976 he worked at VNIISI GKNT and the USSR Academy of Sciences, now the Institute for System Analysis of the Russian Academy of Sciences.

He was awarded 2 Orders of Lenin (1967, 1982), 3 Orders of the Red Banner of Labor (1949, 1953, 1975), Order of the Patriotic War 1st degree (1985), Order of the Badge of Honor (1944). Honorary doctorate from many universities around the world.

1.2 Contribution to science

The scientific legacy of L. V. Kantorovich is enormous. His research in the field of functional analysis, computational mathematics, the theory of extremal problems, descriptive theory of functions had a fundamental impact on the formation and development of these disciplines. L. V. Kantorovich is rightfully one of the founders of the modern economic and mathematical direction.

L. V. Kantorovich - the author of more than three hundred scientific works, which, when preparing an annotated bibliography of his writings, he himself proposed to distribute into the following nine sections: descriptive theory of functions and set theory, constructive theory of functions, approximate methods of analysis, functional analysis, functional analysis and applied mathematics, linear programming, computer technology and programming, optimal planning and optimal prices, economic problems of a planned economy.

Such an impressive variety of areas of research is united not only by the personality of L. V. Kantorovich, but also by his methodological guidelines. He always emphasized the internal unity of science, the interpenetration of ideas and methods necessary for solving a wide variety of theoretical and applied problems in mathematics and economics. One more feature his work is a close relationship with the most difficult problems and the most promising ideas of mathematics and economics of the time.

It is impossible to briefly cover the work of Leonid Vitalievich. He himself singled out two things from what was done in science: linear programming and K-spaces.

1.3 Scientific works of L.V. Kantorovich

Scientific works:

The first scientific results were obtained in the descriptive theory of functions and sets and, in particular, on projective sets.

In functional analysis, he introduced and studied the class of semi-ordered spaces (K-spaces). He put forward a heuristic principle, consisting in the fact that the elements of K-spaces are generalized numbers. This principle was substantiated in the 1970s within the framework of mathematical logic. Boolean valued analysis established that Kantorovich spaces represent new non-standard models of the real line.

He was the first to apply functional analysis to computational mathematics.

Developed a general theory of approximate methods, built effective methods solving operator equations (including the steepest descent method and Newton's method for such equations).

In 1939-40 he laid the foundation for linear programming and its generalizations.

Developed the idea of optimality in economics. Established the interdependence of optimal prices and optimal production and management decisions. Each optimal solution is interconnected with the optimal pricing system.

Kantorovich is a representative of the St. Petersburg mathematical school of P. L. Chebyshev, a student of G. M. Fikhtengolts and V. I. Smirnov. Kantorovich shared and developed the views of P. L. Chebyshev on mathematics as a single discipline, all sections of which are interconnected, interdependent and play a special role in the development of science, technology, technology and production. Kantorovich put forward the thesis of the interpenetration of mathematics and economics and sought to synthesize humanitarian and exact technologies of knowledge. Kantorovich's work has become an example of scientific service based on the universalization of mathematical thinking.

kantorovich mathematics computational descriptive

2. The origin of linear programming

Linear programming is studied by tens of thousands of people around the world. This term hides a colossal branch of science devoted to linear optimization models. In other words, linear programming is the science of theoretical and numerical analysis and solving problems in which it is required to find the optimal value, that is, the maximum or minimum, of a certain system of indicators in a process whose behavior and state is described by one or another system of linear inequalities .

One of the most significant and striking achievements in the field of economic and mathematical research was the discovery by Leonid Vitalievich Kantorovich (1912-1986) of the method of linear programming. Linear programming is the solution of linear equations (equations of the first degree) by compiling programs and applying various methods for their sequential solution, which greatly facilitate the calculations and achieve the desired results. Linear programming is studied by tens of thousands of people around the world. This term hides a colossal branch of science devoted to linear optimization models. In other words, linear programming is the science of theoretical and numerical analysis and solving problems in which it is required to find the optimal value, that is, the maximum or minimum, of a certain system of indicators in a process whose behavior and state is described by one or another system of linear inequalities .

The term "linear programming" itself was proposed in 1951 by the American economist T. Koopmans. For the development of the linear programming method, or, as stated in the diploma of the Swedish Academy of Sciences, for “contribution to the theory of optimal distribution of resources, L.V. Kantorovich was awarded the Nobel Prize in Economics (1975). The prize was awarded to him jointly with the American economist Tjalling Charles Koopmans, who somewhat later, independently of Kantorovich, proposed a similar methodology.

The development of linear programming began with the search for a solution to a practical problem. Kantorovich was approached by the engineers of the plywood trust with a request to find an efficient way to allocate resources that would ensure the highest productivity of the equipment. The employees of the enterprise puzzled over how to ensure the optimal variant of plywood production with five machines and eight types of raw materials. In other words, it was necessary to find a solution to a specific technical and economic problem with an objective function (“functional”) to maximize the output of finished products.

The merit of Kantorovich is that he proposed a mathematical method for choosing the optimal variant. Solving the particular problem of the most rational loading of equipment, the scientist developed a method called the linear programming method. In fact, he opened a new branch of mathematics, which became widespread in economic practice, and contributed to the development and use of electronic computers.

The optimal design of any linear program is automatically associated with optimal prices or "objectively determined valuations". The last cumbersome phrase Leonid Vitalyevich chose for tactical reasons to increase the "critical stability" of the term. The interdependence of optimal solutions and optimal prices is the brief essence of the economic discovery of L. V. Kantorovich.

In the plywood production optimization problem, Kantorovich presented a variable that should be maximized as the sum of the cost of products produced by all machines. Limiters were presented in the form of equations that establish the relationship between all the factors spent in production (wood, glue, electricity, working time) and the amount of output (plywood) on each of the machines.

For indicators of factors of production, coefficients were introduced, called resolving factors, or multipliers. With their help, the task is solved. If the values of the resolving factors are known, then the desired values, in particular, the optimal volume of output, can be found relatively easily.

Kantorovich substantiated the economic meaning of the coefficients he proposed (resolving factors). They represent nothing more than the marginal costs of limiting factors. In other words, these are objectively significant prices for each of the factors of production in relation to the conditions of a competitive market.

To solve the problem for the optimum, Kantorovich used the method of successive approximations, the method of successive comparison of options with the choice of the best one in accordance with the conditions of the problem.

Suppose we need to solve a transport problem, justify the most rational distribution of cargo flows. For example, in total, you need to transfer 180 tons of cargo from three sources to three consumers, whose total demand is also 180 tons. The difficulty is that the cargo is unevenly distributed: one supplier has 50 tons, another has 60 tons, and the third has 80 tons

The demand of consumers is also unequal: it amounts to 40, 85 and 55 tons, respectively. The distances - the shoulders of the transportation of goods - from 1 to 6 km, are also not the same. The task is to draw up such a transportation plan that would meet the requirement of minimizing freight turnover (the minimum number of ton-kilometers).

In everyday practice, managers can do the monotonous work of a lengthy enumeration of possible options. Gradually, they will be able to “pass” from the transportation plan of, say, 750 t/km to the plan of 655 t/km. The search will require a lot of effort, a significant amount of calculations. The main thing is that it is difficult to determine which of the proposed options is optimal. Suppose a variant of the plan with a cargo turnover of 575 t/km is found.

But it remains unknown whether there is one or more more profitable options for the plan that require less cost.

The task becomes completely unsolvable if we move from a relatively simple scheme to drawing up a variant of the transportation of one or more products (coal, cement, building materials) on a regional or country scale. Even in the case of consolidation, aggregation of initial indicators, calculations and comparison of options will require such a number of operations, for the implementation of which almost the entire population of Ukraine will have to be involved.

The linear programming method allows you to find the optimal solution. It is called linear because it is based on solving linear equations. The unknowns in them are only of the first degree; no unknown is multiplied by another unknown. Such equations reflect dependencies that can be depicted on a graph with straight lines.

A slightly different target criterion in the problem of diet (feed ration). The task is reduced to finding the optimal diet for feeding livestock or poultry. With the constant change in market prices for feed, farmers select the optimal diet at a minimum cost, making appropriate calculations on the computer.

For the first time, the work, which outlined the essence of the method proposed by Kantorovich, was published in 1939 under the title "Mathematical methods for organizing production planning." Continuing research, the scientist develops a general theory of rational use of resources.

During the Great Patriotic War, being a professor at the Naval Engineering Academy in besieged Leningrad, Kantorovich, based on the method of linear programming, substantiates the optimal placement of production and consumer factors. In 1942, he prepared the book "The Economic Calculation of the Most Expedient Use of Resources", which, unfortunately, was not published at that time.

Later, one of his largest works, Economic Calculation of the Best Use of Resources (1959), was published. In this book, as noted by the members of the Scientific Council on the Application of Mathematics in Scientific Research and Planning, an in-depth analysis of the ideas of linear programming developed by the author earlier is presented, and at the same time, for the first time, the problem of developing an optimal plan for the entire national economy as a mathematical model is posed. The undoubted merit of Kantorovich is the identification of dual estimates in linear programming problems. You cannot minimize costs and maximize results at the same time. One contradicts the other. However, both of these approaches are interrelated. If, say, an optimal transportation scheme is found, then a certain price system corresponds to it. If the optimal values of prices are found, then it is relatively easy to obtain a transportation scheme that meets the requirement of optimality.

For any linear programming problem, there is an adjoint or dual problem. If the direct task is to minimize the objective function, then the dual task is to maximize it.

Dual valuations provide a fundamental opportunity to measure not only price and cost indicators, but also utility. At the same time, dual, interrelated assessments correspond to specific conditions. If the conditions change, the estimates change. To a certain extent, the search for the optimum is the definition of socially necessary costs, taking into account, on the one hand, labor, cost costs, and on the other, social needs, the usefulness of the product for consumers.

With the direct participation of Kantorovich and his closest colleagues - V.V. Novozhilov (the author of the idea of a product-labor balance) and V.S. Nemchinov (who substantiated the global criterion for the functioning of the economy), a domestic economic and mathematical school was formed.

Conclusion

At first glance, the theories of L. V. Kantorovich were, as he himself said, adapted to a planned economy, and so on. But this is only outer side affairs. The main thing is to take into account hidden parameters (rent), a unified approach to restrictions (labor is just one of them) and everything that follows from this - make its economic applications universal and necessary now. In general, the main result of Kantorovich's great experiment is that he approached economic problems armed with the most modern mathematical tools for those years, and applied them creatively. This does not mean that his conclusions will fully work today, but it certainly means that in this respect L.V. Kantorovich was perhaps the first that the talent of a mathematician can fundamentally restructure and transform economic thought.

List of sources used

1. History economic doctrines: Tutorial/ Ed. A.G. Khudokormov. - M.: Publishing House of Moscow State University, 1994. - Part II, ch. thirty.

2. Kantorovich L.V. Economic calculation of the best use of resources. - M.: Publishing House of the Academy of Sciences of the USSR, 1959.

3. Kapustin V.F., Shabalin G.V. L.V. Kantorovich and economic and mathematical research: results, problems, prospects // Bulletin of St. Petersburg University. Ser. 5. Economy. 1996. Issue. 2.

4. Pezenti A. Essays on the political economy of capitalism. In 2 volumes - M .: Progress, 1976. T. II, ch. fourteen.

5. Shatalin S.S. The functioning of the economy of developed socialism. - M.: Publishing House of Moscow State University, 1982.

6. Shukhov N.S. Value and cost. - M.: Publishing house of standards, 1994. - Part 2, issue. 1, ch. eight.

Features of life, activity, contribution to science, economic and mathematical theories of L.V. Kantorovich. Analysis of the initial stage of the history of linear programming, the emergence of a new area of mathematical activity related to economic applications.

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Ministry of Education and Science of the Russian Federation

federal state budgetary educational institution higher professional education

Ryazan State University named after S.A. Yesenin

Test

subject: History of economic doctrines

on the topic: L.V. Kantorovich is the founder of the theory of linear programming (the theory of optimal use of resources).

Performed:

Chernova N.V.

Ryazan 2014

Introduction

1.2 Contribution to science

1.3 Scientific works

Conclusion

List of sources used

Introduction

In this essay, I will write about the activities of Leonid Vitalyevich Kantorovich, an outstanding scientist of the twentieth century. About his struggle for the recognition of his economic and mathematical theories, about the initial stage of the history of linear programming, about the emergence of a new field of mathematical activity related to economic applications, which we call operations research, sometimes mathematical economics, sometimes economic cybernetics, etc. , about its place and connections with the modern mathematical landscape.

1. Leonid Vitalievich Kantorovich

1.1 Biography of L.V. Kantorovich

1.2 Contribution to science

Scientific legacy of L.V. Kantorovich is huge. His research in the field of functional analysis, computational mathematics, the theory of extremal problems, descriptive theory of functions had a fundamental impact on the formation and development of these disciplines. L.V. Kantorovich is rightfully one of the founders of the modern economic and mathematical direction.

L.V. Kantorovich is the author of more than three hundred scientific papers, which, in preparing an annotated bibliography of his writings, he himself proposed to distribute into the following nine sections: descriptive theory of functions and set theory; constructive theory of functions; approximate methods of analysis; functional analysis; functional analysis and applied mathematics; linear programming; computer technology and programming; optimal planning and optimal prices; economic problems of the planned economy.

Such an impressive variety of research areas is united not only by the personality of L.V. Kantorovich, but also his methodological guidelines. He always emphasized the internal unity of science, the interpenetration of ideas and methods necessary for solving a wide variety of theoretical and applied problems in mathematics and economics. Another characteristic feature of his work is the close relationship with the most difficult problems and the most promising ideas of mathematics and economics of that time.

It is impossible to cover the work of Leonid Vitalievich briefly. He himself singled out two things from what was done in science: linear programming and K-spaces.

1.3 Scientific works of L.V. Kantorovich

Scientific works:

The first scientific results were obtained in the descriptive theory of functions and sets and, in particular, on projective sets.

He was the first to apply functional analysis to computational mathematics.

He developed a general theory of approximate methods, built effective methods for solving operator equations (including the steepest descent method and Newton's method for such equations).

In 1939-40 he laid the foundation for linear programming and its generalizations. kantorovich linear programming

Kantorovich - a representative of the St. Petersburg mathematical school P.L. Chebysheva, a student of G.M. Fikhtengolts and V.I. Smirnova. Kantorovich shared and developed the views of P.L. Chebyshev on mathematics as a single discipline, all sections of which are interconnected, interdependent and play a special role in the development of science, technology, technology and production. Kantorovich put forward the thesis of the interpenetration of mathematics and economics and sought to synthesize humanitarian and exact technologies of knowledge. Kantorovich's work has become an example of scientific service based on the universalization of mathematical thinking.

2. The origin of linear programming

One of the most significant and striking achievements in the field of economic and mathematical research was the discovery by Leonid Vitalievich Kantorovich of the method of linear programming. Linear programming is the solution of linear equations (equations of the first degree) by compiling programs and applying various methods for their sequential solution, which greatly facilitate the calculations and achieve the desired results.

The term "linear programming" itself was proposed in 1951 by the American economist T. Koopmans. For the development of the linear programming method or, as stated in the diploma of the Swedish Academy of Sciences, for "contribution to the theory of optimal resource allocation" L.V. Kantorovich was awarded the Nobel Prize in Economics (1975). The prize was awarded to him jointly with the American economist Tjalling Charles Koopmans, who somewhat later, independently of Kantorovich, proposed a similar methodology.

But it remains unknown whether there is one or more more profitable options for the plan that require less cost.

17 years passed before Leonid Vitalievich could see his fundamental work "Economic calculation of the best use of resources" published. This happened 6 years after Stalin's death. By that time, linear programming, as a fashionable novelty during the thaw period, began to penetrate us from the West. And then it suddenly became clear that the same duality theorem, which the Americans had just independently proved, had been proved by Professor Kantorovich back in the 1930s. Leonid Vitalievich and his students enthusiastically set about solving extreme economic problems again, but very soon they felt that nothing had really changed in Soviet life. Either at the Moskvich plant they do not implement an economical scheme for cutting expensive French body metal - due to a campaign to reduce auxiliary workers, then someone, having introduced a new method and received a fair increase in finished products, eventually lost the bonus, because he thwarted the plan for sale of scrap metal.

When it seemed that the quagmire was sucking in, and there were no hopes for the use of objectively determined assessments, Leonid Vitalievich took his soul off, writing fables.

Now we understand that under those conditions, under that decision-making system, all Kantorovich's attempts to implement the new economy were doomed. "Objective assessments" demanded the abandonment of harsh directives, and this gave rise to the danger of destroying the very building of the socialist economy.

In this book, as noted by the members of the Scientific Council on the Application of Mathematics in Scientific Research and Planning, an in-depth analysis of the ideas of linear programming developed by the author earlier is presented, and at the same time, for the first time, the problem of developing an optimal plan for the entire national economy as a mathematical model is posed. The undoubted merit of Kantorovich is the identification of dual estimates in linear programming problems. You cannot minimize costs and maximize results at the same time. One contradicts the other. However, both of these approaches are interrelated. If, say, an optimal transportation scheme is found, then a certain price system corresponds to it. If the optimal values of prices are found, then it is relatively easy to obtain a transportation scheme that meets the requirement of optimality.

For any linear programming problem, there is an adjoint or dual problem. If the direct task is to minimize the objective function, then the dual task is to maximize it.

Dual valuations provide a fundamental opportunity to measure not only price and cost indicators, but also utility. At the same time, dual, interrelated assessments correspond to specific conditions. If the conditions change, the estimates change. To a certain extent, the search for the optimum is the determination of socially necessary costs, taking into account, on the one hand, labor, cost costs, and on the other hand, social needs, the usefulness of the product for consumers.

In Moscow and Leningrad, Kantorovich became more and more uncomfortable. And most importantly, the ability to work productively has narrowed to the limit. Of course, this depressed him. And therefore, not an adventurer and not an adventurer by nature, he gladly accepted the offer of a university classmate, Academician Sobolev, to leave the capital's swamps and set off to create a new scientific center where people like him were exiled before. Novosibirsk Academgorodok in those years really became an oasis. Science flourished in it freely and incredibly vigorously, partly because youth reigned there, not only in age, but also in spirit.

Conclusion

At first glance, the theories of L.V. Kantorovich were, as he himself said, adapted to a planned economy. But this is only the outer side of the matter.

The main thing is to take into account hidden parameters (rent), a unified approach to restrictions (labor is just one of them) and everything that follows from this - make its economic applications universal and necessary now. In general, the main result of Kantorovich's great experiment is that he approached economic problems armed with the most modern mathematical tools for those years, and applied them creatively. This does not mean that his conclusions will fully work today, but it certainly means that in this respect L.V. Kantorovich was perhaps the first that the talent of a mathematician can fundamentally restructure and transform economic thought.

The scientific contribution of L. Kantorovich is the famous scientific schools in the field of functional analysis, computational mathematics, mathematical economics and optimal planning of the national economy. The mathematical programming he discovered is widely used to solve equal problems in the economy.

The linear programming method for the first time made it possible to precisely formulate the important modern economic and mathematical concept of "optimality". L. Kantorovich and his colleagues developed a system of optimal functioning of the economy (SOFE), formed models for the efficient distribution and evaluation of resources.

He gave it an economic explanation and showed its importance in economic management. It was a scientifically substantiated approach to the calculation of the numerical value of a single national economic economic indicator of the effectiveness of the use of capital investments, which was far ahead of its time.

List of references and used sources

1. History of economic doctrines: Textbook / Ed. A.G. Khudokormov. - M.: Publishing House of Moscow State University, 1994. - Part II, ch. thirty.

2. Kantorovich L.V. Economic calculation of the best use of resources. - M.: Publishing House of the Academy of Sciences of the USSR, 1959.

3. Kapustin V.F., Shabalin G.V. L.V. Kantorovich and economic and mathematical research: results, problems, prospects // Bulletin of St. Petersburg University. Ser. 5. Economy. 1996. Issue. 2.

4. Pezenti A. Essays on the political economy of capitalism. In 2 volumes - M .: Progress, 1976. T. II, ch. fourteen.

5. Shatalin S.S. The functioning of the economy of developed socialism. - M.: Publishing House of Moscow State University, 1982.

6. Shukhov N.S. Value and cost. - M.: Publishing house of standards, 1994. - Part 2, issue. 1, ch. eight.

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Ministry of Education and Science, Youth and Sports of Ukraine

Sevastopol National Technical University

Faculty of Economics and Management

On the topic: L.V. Kantorovich: development of the theory of linear programming

in the discipline "History of Economics and Economic Thought"

Completed: Art. gr. MO-21

Kovaleva S.N.

Checked by: teacher

Kerez E.S.

Sevastopol 2009

1. Leonid Vitalievich Kantorovich

1 Biography of L.V. Kantorovich

2 Contribution to science

3 Scientific works

The Birth of Linear Programming

Conclusion

Introduction

1. Leonid Vitalievich Kantorovich

1Biography of L.V. Kantorovich

L. V. Kantorovich is the author of more than three hundred scientific papers, which, when preparing an annotated bibliography of his works, he himself proposed to distribute into the following nine sections: descriptive theory of functions and set theory, constructive theory of functions, approximate methods of analysis, functional analysis, functional analysis and applied mathematics, linear programming, computer technology and programming, optimal planning and optimal prices, economic problems of a planned economy.

Such an impressive variety of areas of research is united not only by the personality of L. V. Kantorovich, but also by his methodological guidelines. He always emphasized the internal unity of science, the interpenetration of ideas and methods necessary for solving a wide variety of theoretical and applied problems in mathematics and economics. Another characteristic feature of his work is the close relationship with the most difficult problems and the most promising ideas of mathematics and economics of that time.

It is impossible to briefly cover the work of Leonid Vitalievich. He himself singled out two things from what was done in science: linear programming and K-spaces.

3Scientific works of L.V. Kantorovich

Scientific works:

The first scientific results were obtained in the descriptive theory of functions and sets and, in particular, on projective sets<#"justify">kantorovich mathematics computational descriptive

2. The origin of linear programming

Linear programming is studied by tens of thousands of people around the world. This term hides a colossal branch of science devoted to linear optimization models. In other words, linear programming is the science of theoretical and numerical analysis and solving problems in which it is required to find the optimal value, i.e., the maximum or minimum, of a certain system of indicators in a process whose behavior and state is described by one or another system of linear inequalities.

One of the most significant and striking achievements in the field of economic and mathematical research was the discovery by Leonid Vitalievich Kantorovich (1912-1986) of the method of linear programming. Linear programming is the solution of linear equations (first-degree equations) by compiling programs and applying various methods for their sequential solution, which greatly facilitate calculations and achieve the desired results. Linear programming is studied by tens of thousands of people around the world. This term hides a colossal branch of science devoted to linear optimization models. In other words, linear programming is the science of theoretical and numerical analysis and solving problems in which it is required to find the optimal value, i.e., the maximum or minimum, of a certain system of indicators in a process whose behavior and state is described by one or another system of linear inequalities.

Suppose we need to solve a transport problem, justify the most rational distribution of cargo flows. For example, in total, you need to transfer 180 tons of cargo from three sources to three consumers, the total demand of which is also 180 tons. The difficulty is that the cargo is distributed unevenly: one supplier has 50 tons, another has 60 tons, the third has 80 tons .

The demand of consumers is also unequal: it amounts to 40, 85 and 55 tons, respectively. Distances are also not the same - the shoulders of cargo transportation - from 1 to 6 km. The task is to draw up such a transportation plan that would meet the requirement of minimizing freight turnover (the minimum number of ton-kilometers).

In everyday practice, managers can do the monotonous work of a lengthy enumeration of possible options. Gradually, they will be able to “pass” from the transportation plan of, say, 750 t/km to the plan of 655 t/km. The search will require a lot of effort, a significant amount of calculations. The main thing is that it is difficult to establish which of the proposed options is optimal. Suppose a variant of the plan with a cargo turnover of 575 t/km is found.

But it remains unknown whether there is one or more more profitable options for the plan that require less cost.

For any linear programming problem, there is an adjoint or dual problem. If the direct task is to minimize the objective function, then the dual task is to maximize it.

Dual valuations provide a fundamental opportunity to measure not only price and cost indicators, but also utility. At the same time, dual, interrelated assessments correspond to specific conditions. If the conditions change, the estimates change. To a certain extent, the search for the optimum is the determination of socially necessary costs, taking into account, on the one hand, labor, cost costs, and on the other, social needs, the usefulness of the product for consumers.

Conclusion

At first glance, the theories of L. V. Kantorovich were, as he himself said, adapted to a planned economy, and so on. But this is only the outer side of the matter. The main thing is to take into account hidden parameters (rent), a unified approach to restrictions (labor is just one of them) and everything that follows from this - make its economic applications universal and necessary now. In general, the main result of Kantorovich's great experiment is that he approached economic problems armed with the most modern mathematical tools for those years, and applied them creatively. This does not mean that his conclusions will fully work today, but it certainly means that in this respect L.V. Kantorovich was perhaps the first that the talent of a mathematician can fundamentally restructure and transform economic thought.

List of sources used

1. History of economic doctrines: Textbook / Ed. A.G. Khudokormov. - M.: Publishing House of Moscow State University, 1994. - Part II, ch. thirty.

Kantorovich L.V. Economic calculation of the best use of resources. - M.: Publishing House of the Academy of Sciences of the USSR, 1959.

Kapustin V.F., Shabalin G.V. L.V. Kantorovich and economic and mathematical research: results, problems, prospects // Bulletin of St. Petersburg University. Ser. 5. Economy. 1996. Issue. 2.

Pezenti A. Essays on the political economy of capitalism. In 2 volumes - M .: Progress, 1976. T. II, ch. fourteen.

Shukhov N.S. Value and cost. - M.: Publishing house of standards, 1994. - Part 2, issue. 1, ch. eight.

L.V. Kantorovich - an economist - made an outstanding contribution to economics. His name is associated with a natural-science approach to the study of a wide range of planning problems. L.V. Kantorovich laid the foundation for the modern theory of optimal planning. A detailed presentation of the main ideas of this theory is devoted to his major monograph “Economic calculation of the best use of resources” . The core of this book is the formulation of the basic problem of production planning and the dynamic problem of optimal planning. These tasks are quite simple, but at the same time they take into account the most important features of economic planning. One of attractive qualities is that they are based on a linear programming scheme and, consequently, on a developed analytical apparatus and an extensive set of efficient computing tools, some of which were proposed by Leonid Vitalievich himself.

His contribution to the problem of pricing is significant - one of the fundamental ones, affecting, in essence, all spheres of the functioning of society. L.V. Kantorovich established a connection between prices and socially necessary labor costs. He gave a definition of the concept of optimum, optimal development, specifying, in particular, what should be understood as the maximum satisfaction of the needs of members of society. From his position on the inseparability of the plan and prices follows the dependence of socially necessary labor costs on the goals of society.

Thus, the goals of society, the optimal plan and prices are one inseparable whole. He indicated specific conditions under which the objectively determined estimates of the optimal plan coincide with the total (direct and associated) labor costs. Determining the prospects for the economy, the presence of gigantic "natural monopolies" makes it necessary to keep for them the calculation of at least reference prices, agreed both mutually and with the interests of other sectors of the economy.

Mathematical models are reflected in some courses of political economy. In the works of L.V. Kantorovich studied a number of basic problems of economic theory and management practice. Pointing out the shortcomings of the existing economic system, L.V. Kantorovich emphasized that the system of economic indicators should be unified, built on a single principle. In this regard, Leonid Vitalievich devoted a significant part of his work in this area to the development and analysis of specific economic indicators.

In the works of L.V. Kantorovich, special attention was paid to the assessment of land resources and water, the inclusion of these indicators in (procurement) prices for agricultural products. Original approaches to their calculation are proposed (combination of the method least squares and linear programming). On this basis, recommendations were made to improve the system of economic indicators and calculations in agriculture. The importance of the principles of calculation proposed by him in the emerging economic system is only increasing.

In the works of L.V. Kantorovich, the essence of the concept of the indicator of the effectiveness of capital investments is revealed, its role in the economic calculations of decision-making is shown, a method for determining the value of this standard indicator is proposed. Thus, L.V. Kantorovich gave a convincing scientific justification for the need to apply the efficiency standard and, based on the optimization approach, gave an objective way to calculate it.

In the work “Depreciation payments for optimal use of equipment” (1965) L.V. Kantorovich revealed the essence of the concept of depreciation. He showed how to improve the efficiency of equipment use by dividing depreciation payments into two types, and using an ingenious mathematical model, he showed how to determine the numerical value of the depreciation rate. This change made it possible to draw a number of fundamental conclusions about the need to adjust the accepted methodology for calculating depreciation.

Leonid Vitalievich showed special interest in the problems of transport. Even in his first economic works, a general analysis of the transport problem and the method of potentials for its solution were given. This method has been widely used in transport (railway, road, sea, air) and in centralized supply agencies for the rational attachment and rational organization of transportation. It certainly retains its importance today, along with the widely used methods of dispatch control and route calculations.

In works “On the use of mathematical models in pricing for new equipment”(1968) and " Mathematical and economic analysis of planned decisions and economic conditions for their implementation” (1971) L.V. Kantorovich studied the problem of the efficient operation of transport from an economic point of view, showed what transport tariffs should be depending on the type of transport, cargo, distances, etc. In a number of works, he also considered issues of an integrated transport system - the relationship of transport with other sectors of the national economy and distribution of traffic between modes of transport, taking into account economy and in particular energy costs. These works retain their significance to this day.

In addition to the problems of national economic planning, L.V. Kantorovich considered issues related to sectoral planning. The simplest and most frequently used is the model proposed by him, based on the transport problem. He indicated a number of more complex models, in particular, production and transport, dynamic, decomposition, in works devoted to current and future sectoral planning (“Possibilities of using mathematical methods in matters of production planning”, 1958) and others. These issues are reflected in the research by industry automated control systems.

Leonid Vitalievich paid much attention to the issues of rational use of labor. They proposed the introduction of payments by enterprises for the use of labor differentiated by profession, gender, age and territory. He also pointed to the possibilities of a scientific, quantitative approach to social problems, issues of improving the service sector, etc. The issues of economic incentives for the rational use of labor resources remain relevant even now.

Over the years, and especially in last years L.V. Kantorovich was interested in the problems of the effectiveness of technical progress, in particular, the introduction of new technology into production.

Of particular interest is the substantiation of the proposal to establish two price levels for fundamentally new products in the first years of their release. Also of great importance was the conclusion that it was necessary to value the contribution of technical progress and science to the national income more highly than was obtained by the then accepted methods of calculation (“Pricing and Technical Progress”, 1979).

L.V. Kantorovich paid great attention to the introduction of the methods he developed into economic practice. First of all, in this regard, it should be noted the cycle of works devoted to the methods of rational cutting of materials, begun by Leonid Vitalievich back in 1939 - 1942. In 1948 - 1950. these methods were introduced at the Leningrad Carriage Works named after Egorov, at the Kirov Plant and subsequently distributed at some other enterprises. A wider spread of rational cutting methods was facilitated by a number of studies carried out on the initiative of L.V. Kantorovich meetings.

Since 1964, at the suggestion of Leonid Vitalievich, a lot of work has been carried out to introduce systemic methods for calculating the optimal load of rolling mills throughout the country.

As a member of the State Committee for Science and Technology, L.V. Kantorovich carried out extensive organizational work aimed at improving the methods of planning and managing the national economy. He headed the Scientific Council of the SCST on the use of optimization calculations, was a member of many departmental councils and commissions (on pricing, transport, etc.). Leonid Vitalievich's contribution to the study of the problem of production efficiency and, in particular, the problem of the efficiency of capital investments is exceptionally great.