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Methodology of a substantiation of rational variants of the armament programme on the basis of utility theory
Material posted: Publication date: 03-11-2010

The monograph is devoted to one of the topical issues of the organization development of new weapons – planning of the actions envisaged by the Program of weapons. It is a logical continuation of the work. In the same way as previously, considers the activities of planning and ordering of on middle management, i.e. an organization to achieve its objectives, has the ability to carry out the distribution of the allocated financial resource on various aspects of their operations.

However, significantly changing how their overall objectives and the total value of the resource that is the parent planning body and shall be communicated periodically in the form of appropriate targets, consider ordering on control is not feasible.

But only now is considered somewhat different range of issues. With planning the research, the emphasis shifted to the planning to direct development and creation of new defence equipment. On the one hand this simplifies the consideration of the problem, as it decreases the influence of various uncertainties associated with the long-term forecasting, on the other – complicates it, since it is necessary additionally to take into consideration many essential features of real production of weapons and military equipment (AME). These features got its methodological reflection and the used mathematical apparatus, which is significantly different from that which applied in the works.

Before You start reading the book, You have the right to expect some answers. What is the main idea of the proposed approach? For what purpose this book is written? Who is the reader for whom it is intended?

The answer to the first question is this.

In the formation of the arms Programme offers not just one of the possible variants of development of the IWT system, and rational option. The term "rational" indicates that there should be a procedure of choosing the best in some sense the solution. At present, there are only two groups of mathematical methods, the subject of which is the choice in a certain sense the best solutions are mathematical programming and optimal control. But, on the other hand, the optimal control problem is very often solved the so-called direct methods, the essence of which is to note the original problem to a mathematical programming problem of high dimension.

Thus, almost clearly come to the conclusion that the formal part of the methodology justification of the armament Programme should be based on the methods of mathematical programming, i.e. on the methods of solving optimization problems.

As you know, the key point in the formulation of the problem mathematical programming is to establish the objective function being optimized and the definition of its dependence on the aggregate independent variable, control variables. Let us ask ourselves: what are the main property should have a target function in the optimization problems of weapons Programmes?

Consider the very simplified case when you need to form a rational plan for the development of only one type of VVT (for example, air defense systems medium range) taking into account the possibility of only one modernization of existing facilities. This modernization can be carried out through the introduction of the results of one research or experimental development (R & d), for which you need to spend some of the funds.

On the one hand, customers of IWT may not spend funds on R & d and limited to simple incremental number of already developed systems. This case is similar to a military threat or a relatively small increase in the efficiency of the modernized systems.

On the other hand, if the military threat is still small, but at the expense of modernization can significantly increase the combat capabilities of the ATA, then customers it is advisable to spend part of the funds for research and development. Reducing the production rate of existing weapons, in order subsequently to obtain a more effective weapon systems and in General over the plan period to raise the combat capabilities of VVT on a higher level.

From these simple considerations it is logical to conclude that the objective function of the program of development of this type of AME may not be focused on tasks only a specific point in time. It must take into account the combat capabilities of the weapon system (consisting of existing and modernized systems) to different points in time, i.e. it must map to a single system of measurements belonging to different time intervals of the ability of IWT and needs. In other words, the objective function of the program of development of IWT should be integrated at the time the form

For a given dynamics the total funding and external threats as the optimized parameters can be, for example, the start and end of R & D. Under certain research and development costs, these parameters completely determine the dynamics of AME funding for the development of different generations and, therefore, the time dependence of the quantities AME of all generations.

When studying the plurality of different funds of IWT, to improve each can be conducted not one, but several R & d the task of forming the armament Programme and becomes significantly complicated optimization problem of high dimension, especially if you consider this optimization for the financing of maintenance activities, capital construction and maintenance of IWT. But the basic idea of the proposed approach remains unchanged – it is necessary to form the integral indicator of the quality of the Program, taking into account the dynamics of military threats and the relative efficiency levels of different generations of VVT all the considered types, and then to its optimization. The formation of such integral indicator can be performed by methods of the theory of utility. These methods are based on combining the advantages of formal and heuristic approaches enable to recover the target function when making decisions under conditions of uncertainty.

Thus, the basic idea of the proposed approach is to establish the basic parameters of the armament Programme from the solution of the problem of mathematical programming for high-dimensional target function which is integrated in time figure.

Now we will try to answer the second group of questions posed above.

Deep systemic crisis of the economy and the defence industries, in particular, put on the agenda the issue of improving program planning. The more the gap becomes assigned to the Armed Forces and the tasks allocated on their resources, the more urgent the development of such methods of planning of development of IWT system that contribute to resource concentration on the really important areas and stop spraying on the secondary.

The successful solution of tasks of improvement of quality of program-target planning of development and creation of the WME may contribute to the accelerated development of information technologies, which allows to store and process large amounts of information, and a significant increase of computing possibilities. This is a positive premise. On the other hand, focus only on traditional research methods and a significant reduction in their overall level associated with a sharp reduction in the number of scientific staff of the special institutions dealing with issues of defense, is currently a limiting factor in developing the theoretical basis for addressing this urgent problem.

This monograph was written in order to try to fill this gap. A possible variant of such a methodological apparatus, which, given the characteristics of the process of modern planning, would be compatible with the existing computer technology. The authors summarize and critically comprehended experience program planning ordering offices in one of the Armed Forces of the Russian Federation.

It is especially necessary to note following circumstance. Today, when Russian Armed Forces are able to purchase only certain individual samples of modern weapons, it would seem that you can do without the proposed methods. However, the analysis of some foreign programmes of modernisation shows that in the next few years Russia is likely to have to give this issue much more attention. It was then proposed in methods and techniques will be extremely important. It is therefore appropriate now to publish them.

Your readers we see, first of all employees ordering of departments that deal directly with issues of development of IWT and from which depends largely on how effectively the state's point of view will be spent allocated funds. So one of the tasks of the book is to develop not only the appropriate theoretical apparatus, but the wording is clear and understandable practical recommendations for creating in the Contracting Department of the Ministry of defense of the current layout, and then sample the expert and design support system of making decision on the justification of proposals to be considered.

I hope that this important work a great help ordering offices will provide specialized scientific organizations of the Ministry of defence, engaged in the evaluation of prospects of development of weapons. Therefore, this book is also for them.

At the organization of AME development is important to establishing rapport between employees ordering departments and representatives of industry organizations. Success in joint practical activities will largely depend on their coordinated evaluation of current and future problems in the General context of solving national problems, not only issues of a single firm. We hope that the present book may help to establish closer understanding between vendors and customers of IWT.

Finally, on its basis you can create a study aid for students and adjuncts of military schools and academies to organize a special course on program planning of development of IWT. Question study and application of economic methods in the organization of production of weapons is long overdue and demands its speedy solution, if of course we want in the future to have qualified professionals that could, in the public interest initiative, competently and in a timely manner to respond to changes in economic situation, on top without waiting for special instructions and orders.

We have tried to present the material as more coherent and self-contained. With this purpose, the first section in sufficient detail the main provisions of the theory of utility, and in the sixth – foundations of mathematical programming. This is done for the following reason. We have seen how some experts with the literature. Many of them, when faced with a difficult questions that the book doesn't have a direct answer, and only contains a link to another source of information, often lay to read the book and begin to study another until the moment when it encounter another such link – and the process is repeated. As a result, the interest to the original source of information is largely lost. That's the kind of relationship to our book, we would like to avoid – we hope that it can be read without distraction from beginning to end.

The book consists of an introduction, ten chapters, conclusions and applications.

The introduction reviews the object and subject of research is formulated and studied scientific problem.

The object of the study is the arms Program, which is formed to increase the level of technical equipment of one of the branches of the Armed Forces. Subject of research – methods of forming a rational variants armament Programme and the main regularities peculiar to these rational options.

Consider the scientific problem is to develop the theoretical foundations of the methodologies for the justification of rational funding levels provided by the armament Programme of activities taking into account the main features of the development and production of AME.

The first section provides a list of the main normative documents regulating the development and creation of new AME, from the point of view of system analysis is considered one of the key documents – the arms Program. The following describes the position of utility theory on the basis of which in the most General form set form of indicator of the quality of the Program. The fundamental issue here is to build such a procedure, which allows to obtain an objective decision, to the maximum extent protected from manipulation by any member of the Program development process. It is shown that the methods of utility theory and, in principle, allow to meditators the procedure of formation of collective decisions.

The second section is devoted to the development of the methodology for determining the explicit form of the integral indicator that determines the quality of the Program depending on the quantitative composition of the VVT. The technique is based on the specification established in the previous section the overall shape of the figure in relation to the development of IWT system. At the end of the section is the formalization of a scientific problem and provides a mathematical programming problem for optimization of the armament Programme. As the objective function for this problem, we use an established indicator of quality.

The third section addresses the method of determining the dynamics of changes in the technical capacity of one of the types of VVT. It is based on the solution of the system differential and algebraic equations, and checking the logical conditions that determine the development of one of the types of VVT.

In the fourth section these results apply to the whole Program of armaments; it also specifies the vector (set) of governors (optimized, variable) parameters, changing quantities which can improve the quality score of the Program.

The generalization of the results obtained for the analysis of the case when, along with technical equipment of the troops takes into account measures to combat preparation and maintenance of maintenance of IWT personnel, contained in the fifth section of the book. The obtained system of nonlinear differential equations allows to relate the dynamics of change in technical level with the change in the number of different categories of servicemen.

Consideration of possible numerical methods for solving mathematical problems of nonlinear mathematical programming is contained in the sixth section. According to the analysis of the selected rational method Program optimization weapons.

The seventh section presents the algorithm for calculation of indicator of the quality of the armament Programme and the results of the numerical solution of the optimization problem of the justification of the armament Program, which takes into account only the technical aspects of IWT. The analysis of the obtained results.

The contents of the next – eighth section is almost similar to the previous except for the fact that we are considering a solution of another optimization problem of higher dimension. It is further taken into account in implementation of combat training and personnel maintenance. In addition, in the eighth section presents proposals for the creation, testing and application of expert design system for the validation and adjustment in the dynamic regime of sentences in the weapons Programme, the estimation of the basic configuration settings of the system.

In cases where the annual amount of funding falls below some critical level, most R & d can be excluded from the armament Program. In this case it degenerates into a set of short-term (usually annual) plans of orders, deliveries and repair of IWT. Such a situation is not typical for the Ministry of defense, but peculiar to some law enforcement agencies of Russia, for example the Federal border service (FPS).

To optimize short-term plans for the development of IWT can be applied simpler methods that take into account the possibility of linearization used functional dependencies by examining the changes of parameters close to the initial points.

Finally, the ninth section deals with the development of rational structures of specialized scientific organizations involved in the development of the armament Programme. From the quantitative point of view to examine the implications of various options of reducing the number of employees of these organizations.

In the conclusion the main conclusions.

The annexes contain supporting materials.











1.1. Regulatory documents governing the development and creation of new weapons and military equipment.


1.2. The arms program as an object of systematic study...


1.3. The main provisions of the theory of utility.


1.3.1. The theory of one-dimensional utility


1.3.2. The usefulness of multicriteria problems


1.3.3. The usefulness of time-dependent


1.4. The aggregation of individual preferences and group decisions






2.1. Ways to ensure Armed Forces with new systems (samples) VVT.....................................................................


2.2. Quality indicator development programme of the same type of weapons............................................................................


2.3. An indicator of the quality of the armament Programme. Formalization of the problem of substantiation of rational variants of the armament Programme


2.4. Methods of paired comparisons (analysis of hierarchies)






3.1.The differential equation of the development of weapons of the same type: one R & d project, three generations of VVT (analytical approach)


3.2. Refinement method based on the solution of differential equations


3.3. The system of equations for the analysis of the development of IWT of the same type






4.1. Development odnogruppnikami schema: single type arms, four generations, two R & d


4.2. The system of equations and initial conditions to determine the time dependence of technical potential of IWT and its components for odnogruppnikami scheme


4.3. Multi-group approximation — generalization odnogruppnikami scheme to the case of multiple types of weapons


4.4. The list of optimizable parameters in the formulation of the problem in the multi-group approximation. The source data for the complexes (samples) VVT


4.5. A diagram and description of algorithm of calculation of integral indicator of the quality of the armament Programme. The definition of the dimension of the optimization problem






5.1. The method of cost accounting for capital construction
military facilities


5.2. The method of determining the cost of maintaining personal


5.3. The method of determining the costs of measures for fighting


5.4. The system of equations and initial conditions to determine
the time dependence of technical potential of IWT and
its components for odnogruppnikami scheme


5.5. The list of optimized parameters in the formulation
tasks in multi-group approximation. Extended system
the original data set (samples) to account for the VVT
new factors






6.1. Formulation and classification of mathematical programming problems. Basic definitions


6.2. Sufficient conditions for the existence of the solution of a nonlinear programming problem. The main problem of convex programming. The Kuhn-Tucker Theorem


6.3. The method of sequential optimization without constraints (penalty method). The method of barrier functions


6.4. Methods one-dimensional optimization


6.4.1. Methods Fibonacci and "the Golden section"


6.4.2. The method of half division


6.4.3. The method of parabolas


6.5. Methods of optimization of functions of many variables with no constraints (unconstrained optimization methods


6.5.1. Methods of order zero — coordinatewise descent (ascent)


6.5.2. Gradient methods of the first order


6.5.3. Methods of the second order


6.6. Methods of optimization of functions of many variables with constraints


6.6.1. The method of conditional gradient


6.6.2. Method gradient projection


6.6.3. Method of possible directions


6.7. Heuristic methods for solving an integer mathematical programming


6.7.1. Method of successive increments


6.7.2. The method of normalized functions for solving problems of integer linear programming


6.7.3. The method of normalized functions for the solution of problems of integer convex programming


6.7.4. Concluding remarks and possible generalizations of the method of normalized functions


6.8. Pre-selection of optimization methods for solving the problem of forming a rational armament Programme






7.1. The algorithm and the structure of the computer program for
the numerical solution of optimization problems on justification
rational variants of the armament Programme


7.2. The limits of applicability and possibilities of the developed


7.3. The results of the synthesis solution of the armament Programme
(the optimization problem of the 1st kind






8.1. The results of the synthesis solution of the armament Programme
given the costs of personnel and major military construction projects (the optimization problem of the 2nd kind


8.2. Expert design system for the substantiation of rational
variants armament Programme: proposals for the establishment,
trial operation, application. The main parameters






9.1. Basic theoretical principles of the method


9.2. The technique of determination of rational values of total number of staff in the military research organization


9.3. Analysis of the properties of solutions for General cases, exponential production function










Annex 1. Preferences in the problem of decision-making


Annex 2. The formal scheme of the method of eigenvalues


Appendix 3. An error estimate for the numerical solution of differential equations by Euler's method


Annex 4. Direct methods in variational problems

Annex 5. Efficiency optimization of the objective function by different groups of parameters


Annex 6. Questions of solution of problems of mathematical programming


Annex 7. Evaluate the magnitude of the economic effect from the scientific and technical support for R & d


Annex 8. The arrays of input data




THE LIST OF REFERENCES..........................................................




Probably already irretrievably gone are the days when the country's defense spent so much funds, how much was requested. To replace costly mechanism comes another. It assumes carrying out a comprehensive analysis of the ratio of the solved tasks of the Armed Forces and spent their solutions resources with the subsequent selection of priority areas of development. And it is not only in the economic crisis, which has exacerbated the need for a systematic study of this issue has resulted in a more clearly identify its relevance. In economically developed Western countries, where the economy is much better than in Russia, a similar approach is used already in 60‑ies. For example, in the United States under the leadership of administrator General of the company "Ford motors" R. McNamara, who later became Minister of defense, with the participation of scientists of the RAND Corporation was a system of "planning –programming – budgeting" [7]. The system was intended to improve the efficiency of use of financial resources while managing the development of science and technology in the interests of the Armed Forces.

Therefore, the relevance of the application of economic methods in the planning of development and creation of the ATA is due not only to our country are experiencing economic difficulties. It is necessary, first of all, to improve the efficiency of the functioning of the economy in General and the Armed Forces in particular. Because from an economic point of view, defense spending is an investment in the security of the state, and they affect the interests of practically all members of society, then simple logic leads to the necessity of applying economic categories in planning the development of weapons. The dilemma of "objective costs" will exist in any society – rich or poor – at all system levels. Moreover, from its correct decision will depend largely on whether there will be after a time poor society rich or not.

It should be recognized that currently, economic methods have not yet received due development when planning the development and creation of modern Russian weapons. The truth is you have already created a significant backlog in assessing and forecasting expected expenditures on research and development (R & d), the cost of creating prototypes and production models of various types of IWT. On the other hand, by methods of mathematical modeling is used to assess the effectiveness of individual tools and IWT groups, which sometimes allows to estimate the expected levels achieving the set targets. All this is already or will be created in the near future. But what yet – so this is middleware element, which would allow to connect among themselves, to integrate into a single set of tasks and resources to optimize the cost distribution by key events determining the development of weapons systems. Attempts to develop theoretical methods justification of the armament Programme, there have been several ([8-16]), but to the complete, internally consistent result is not possible.

Analysis of the causes of these failures (let's call a spade a spade) has led us to the following main conclusions. Most of the methods focused on the use of formal settlement schemes. For such ultralarge systems as a weapon system of the Russian Federation, the consistent use of this purely formal approach evokes the need to build mathematical models with a very large volume of data, which is prohibitive not only for existing but also for advanced models of computing equipment. Should be modeled all possible situations not only today but also the future, and mathematical problems were formulated and solved not in a deterministic and, at least in the game setting, taking into account all possible risks. But even if on the basis of such mathematical models will be determined by some generalized indicators of the quality of the weapons, they then have to optimize, and the number of independent (optimization) variables in this case can reach a few hundred-thousand units. Naturally, to solve such problems only have numerical methods, and the magnitude of the characteristic time of the decision can only be guessed at; some very optimistic estimates [1] show that a single solution may require several years.

Possible simplify mathematical models and reduce the dimension of the problem the fundamental problem, unfortunately, is not solved, as it remains unclear how to compare with each other, "to bring to a common denominator", the levels of particular challenges related to different moments of time. This is a fundamental issue, since the rearmament Programme must take into account the date of the planned and ongoing activities.

So, the traditional approach to the formation of economically justified programs of development of weapons for a long period based only on formal methods, is untenable, the problem is not linked to resources. It is therefore necessary to find a method of mathematical description, which would have the following properties:

a) allowed obtaining solutions in situations with an uncertain (probabilistic) outcome;

b) would use the positive side developed formal schemes and mathematical models, it was possible to seamlessly incorporate modeling results into the General scheme of the method;

C) admits the existence of the generalized indicator, which correctly took into account the importance of the events separated by significant time intervals;

g) used an acceptable amount of input data, compatible with memory modern computational tools;

d) allows to obtain results in a reasonable time (not more than several days of continuous operation of the computer).

The analysis showed that such a methodological apparatus with seemingly contradictory properties really exists. Is – utility theory ([17-19]).

The main provisions of this theory will be presented in the first section. So now we not going to talk about it. Just to say that combine acceptable accuracy results with less complexity of its receipt helps the use of additional expert information from those responsible for its solution, or from a group of such persons.

However, the methods of utility theory is not simply conventional ways of processing examination results, type of voting methods. They involve the use when examinations of all available results obtained during mathematical modeling situations (to enhance the predictive capacity of experts), take into account the attitude of decision makers towards risk are protected from "dictatorial" arbitrariness in the decision, allow you to obtain not just point estimates, and to present the result in the form of special functional expression of utility functions. This makes possible their natural combination with formal schemes and, on that basis more reliable result. Thus, the overall method becomes combined, expert-estimated. In addition, methods of utility theory enable us to reach an integral indicator of the quality of the armament Programme; after that, only one step to investigate the possibilities of its further optimization and the formation of rational variants of the Program.

In order to emphasize more clearly that the relationship between the expert and formal approaches in the proposed combined expert-design method, let us briefly describe the scheme.

As the main physical parameter that determines the usefulness of the ATA deals with the number Ni of weapon systems of different generations. Thus, from the outset, we reject the rather widespread approach, when the parameters of the utility function explicitly includes tactical and technical characteristics of weapons [10-11]. In our opinion, this will allow to obtain more reliable expert judgement about the usefulness of IWT in the various groups. Why? Because this information is mainly tactical (operational-strategic) level. It takes into account possible uncertainty of opponent's behavior and are associated with these risks. The relevant experts – people familiar with tactics, strategy and operations planning at various levels; they are much easier to navigate in the quantitative composition of the various complexes (samples) and types of IWT than in long lists of tactical and technical characteristics. Naturally, any expert will produce a more accurate response to plain language to him.

Methods of the theory of utility based on properties of utility functions is also a generalized form of the integral indicator of the quality of the armament Programme. However, to calculate it we need to get the functional relationship of a different type (time dependence of the potential of IWT. It is therefore necessary to additionally determine the dynamics of the quantitative composition of VVT different generations. For this a system of differential and algebraic equations and logical conditions, which takes into account the main features of the production of modern weapons. Solving this system, we can obtain the dependence of Ni(t), which uniquely determines the dynamics of the usefulness of different types of VVT, as well as the value of the integral indicator of the quality systems of arms, i.e. in other words – an indicator of the quality of the armament Programme.

It follows that the values of Ni serve as supplementary in some sense intermediate variables to facilitate obtaining the final result with the lowest error.

Thus, the definition of quality indicator Program is conducted in three stages.

First, the expert-estimated methods based on the results of mathematical modeling is established based on the potentials of IWT from the quantitative composition of the Ni complexes and weapons of different generations.

Then only formal methods is determined by the time dependence of Ni(t) amounts of complexes VVT different generations. Substitution of these values and the subsequent integration of the results over time of the numerical (formal) method defines the value of the indicator of the quality of the Program.

Optimization of this indicator of the quality of and access to rational variant armament Programme is also produced formal methods.

Indeed, the rational option – that means the best option compared to other possible. Scientific discipline, the subject of which is to determine the best (according to the accepted indicators of quality) variants – this is the theory of optimal control and mathematical programming. But it should be noted that the numerical solution of optimal control problems of high dimension with a complex structure of restrictions is carried out usually by methods of mathematical programming. It is therefore natural that the mathematical apparatus justify the rational variants of weapons Programs widely used formal methods of mathematical programming.

Which component in this design scheme is heuristic or formal, it's up to the reader. For our part, we believe that with the combination reached a reasonable compromise, the "Golden mean" in the aggregation of the two different approaches. It is advisable not to oppose them, and so to construct a scheme of mutual mates, so they complement each other. Therefore, the main task that we set before the start of work on the monograph is to draw attention of readers to the possibility of applying methods of the theory of utility and combined expert-the design of circuits on their basis for the rational justification of the methods of distribution of financial resources in the formation of the arms Programme.

But to commit to writing this monograph, we have been prompted another circumstance.

Lately in the practice of software planning of development of IWT have been some changes for the worse. A large number of so-called "concepts" which, while claiming to be the main policy documents, however, are developed outside of all rules of system analysis.

According to the logic and the sense of the words, concepts can be considered, for example, as the first stage of development of the armament Programme. They are designed to prior, meaningful formulation of the problem and the emergence of target plants. It is clear that in conditions of severe financial constraints it was not possible to develop strength equal to all the type of VVT; often one can only be created by the termination of funding of others. So the next step should be formed of weapons Programs as a mechanism for rational balancing of goals with available resources.

It is logical also to elaborate the concept and at last, the final stage of the development of weapons Programs; in this case it will contain the main findings programme planning and to identify ways to implement them.

But what happens in reality? These "concepts" that are often developed by people who are far from understanding the essence of program planning and not owning the relevant knowledge and methodological apparatus, in fact, supersede the armament Programme. Many of them are formulated at primitive level. They are vague, on some points have slogans: "speed, to ensure to surpass" – who could argue with that? Need concrete figures and concrete suggestions for effective set of tools for the implementation of targets under the given constraints.

We know these "experts" who can write (at least they think they can) any concept within one, maximum two weeks. The result of this work is obvious in advance is formed such versions of the weapons Programs that are either not balanced or is not economically feasible, or both combined. That is, the theory and practice of program planning of development of IWT exist independently from each other. But in the end, incorrect planning destroys the image effect on the development of weapons systems – we are seeing in recent years.

Many of the concepts are not just useless; they are harmful as incorrect and unnecessarily redundantly assessing the development priorities of certain types of VVT, provide opportunities for advocacy and education of the systemic biases, imbalances, and as a consequence to the loss of efficiency of weapons.

Therefore, the second objective of this book we see the establishment of the theoretical foundations of this methodological apparatus, which would allow the formation of the armament Program on a real economic basis, to contribute to this important cause sound economic sense. We were not limited to theoretical conclusions, but also provided specific practical recommendations for the creation of the expert design support system of decision-making on justification of the proposals in armaments Program (section 8). Its use can allow to significantly improve the efficiency of software development planning systems of arms.

A few words about the form of presentation.

At the beginning of each section of the book we will briefly lay out the main solve it problems and tasks that the reader had the opportunity to get up to speed, to understand the essence of the question. The presentation of the material is deliberately chosen to avoid addressing the cumbersome computations; this will allow you to focus on the informative part of the problem and to facilitate its understanding. Therefore, in presenting the main text, we have tried to avoid the "disease" of excessive, sometimes completely unjustified matematicheskaya, which is characteristic of some works. All the necessary mathematical calculations, the reader will find in the annexes.

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