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Abstract From Master's Work
“Solution of Industrial Problems Using Dynamic Programming Technique”
- Introduction. Dynamic programming concept.
- General problem definition for classical resource allocation problem.
- Research urgency
- Research objective and tasks
- Existing researches review
- Prospective scientific novelty and practical value
Introduction. Dynamic programming concept.
Dynamic programming constitutes mathematical apparatus, developed for effective solution of some problems of mathematical programming. This class is characterized by a possibility to divide operation into several interconnected stages.
Dynamic programming models can be used for example in inventory management, during scarce resources allocation among possible new ways of their usage, during planning current and capital repair of complicated equipment, etc.
For determining dynamic programming essence let’s examine a problem.
Imagine some operation O which consists of several subsequent stages, for example industry activity during several years. Let the number of stages be m. The gain (operation Z effectiveness) far total operation consists of gains from separate stages.
where zi- gain from i-th stage.
Operation O is a conducted process, which means that me can choose parameters which impact on its result, at each stage we choose decision from which depends gain from current stage and from the whole operation. Such decisions are called stage decisions.
All this decisions in aggregate make operation control. We denote it by letter x, and stage controls we denote by letters õ1, õ2, ... , õm: õ=õ(õ1, õ2, ... , õm). We need to find such a control x, which turn the gain Z to maximum:
The control x* at which maximum gain is reached is called optimal control. It consists from stage controls õ*=õ*(õ1*, õ2*, ... , õm*).
Maximum gain reached by this controlk we denote this way:
Where X - acceptable solutions set.
The most simple way of solution this problem is simple enumeration. When the problem size is not large, this method is acceptable. But in practice low-dimensional problems are rare. Dynamic programming often helps to solve a problem for which an estimation algorithm requires much time.
Bellman principle of optimality is in the basis of all dynamic programming problems. It is formulated in such way:
an optimal path has the property that whatever the initial conditions and control variables (choices) over some initial period, the control (or decision variables) chosen over the remaining period must be optimal for the remaining problem, with the state resulting from the early decisions taken to be the initial condition.
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General Problem Statement for Classical Resource Allocation Problem.
Funds in size S are appropriated for development. We consider n objects of capital investment, income f(x) from investment current sum is known for every object. We need to allocate resources among n objects (enterprises, projects) to gain maximum total income.
For mathematical model composition we proceed from the assumptions:
- Income from every object does not depend from investments in other objects
- Income from every object is evaluated in the same monetary unit
- Total income equals to the sum of incomes received from every project.
Such statement is a reductive model of real process of investment resources allocation. It does not take into consideration many facts:
- Presence of informal criteria that cannot be estimated numerically (e.g. co-ordination with general enterprise strategy, social or ecological importance of the project).
- Level of project risks
- Other factors.
In connection with necessity of risks consideration during forming of investment portfolio, stochastic dynamic programming appeared which deals with probabilistic observations. It is used in different branches. One or the most deeply researched is risky investments management.
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Research urgency
During the process of development, also as economical conditions change, all enterprises stumble with necessity of perfection their economical structures. Enterprises reconsider their executive systems, implement new technologies, reorganize business on the basis of new conceptions or re-engineering. Hands-on situation is the reason of necessity of forming new methodical basis and practical recommendations for building new systems of financial management working out. Especially it concerns investment activity which is the key condition for enterprise development.To the top
Research objective and tasks
Work’s objective is the research of methods of mathematical optimization of enterprise investment activity.This objective I reached by such tasks:
- Analysis of methods and models of dynamic programming used for financial activity optimization
- Modeling investment activity of an enterprise and its optimization using dynamic programming method.
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Existing researches review
The study of dynamic programming dates from Richard Bellman, who wrote the first book on the subject (1957) and gave it its name. A very large number of problems can be treated this way. A number of mathematicians and economists research this problem.To the top
Prospective scientific novelty and practical value
Prospective scientific novelty of the work is the result of the perfection of decision-making model by taking into consideration new factors.Practical value of this research is the result of possibility to use this model in investment planning process of an enterprise.
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