Raising of task of optimization
For planning and prognostication of issue of products the methods of analysis of the systems, mathematical
design, optimization, expert estimations and new information technologies are used.
That parameter which determines the degree of perfection of decision of arising up problem comes to light for the
decision of task of optimization. This parameter is usually named an objective function or criterion of quality. In economic tasks
it, as a rule, maximization of income. The aggregate of sizes which determine an objective function is further set. Finally, all
limitations which must be taken into account at the decision of task are formulated. After it a mathematical model, consisting in
establishment of analytical dependence of objective function from all arguments and analytical formulation of concomitant to the
task of limitations, is built. So, let it is set as a result of formalization of the applied task, that objective function

,
(1)
where great number X is generalization of limitations, it is named in a number of feasible solutions. The creature
of problem of optimization consists in a search on a great number X – great number of feasible solutions of such decision
, at which objective function f arrives at the least or most value (2).
(2)
As criteria of optimization choose the followings: limitations will be the maximal loading of equipment
and minimum use of resources, and by an objective function – arrived maximum (3). On the basis of it get a necessity to us decision,
i.e. plan of issue of products.

,
(3)
Decision of planning task by genetic algorithms
Genetic algorithms are analytical technologies, created and adjusted by nature for millions of years of
its existence. They allow deciding the tasks of prognostication, classification, search of optimum variants, and quite
irreplaceable in those cases, when in ordinary conditions the decision of task is based on intuition or experience, but not
on its strict (in mathematical sense) description.
Let some difficult function (objective function), depending on a few the variables, is given, and it is required
to find such values of variables which the value of function is maximal at. Tasks are such named the tasks of optimization
and meet in practice very often.
One of examples is a task of planning of issue of products. In this task variables are volumes of output of products,
and a function which needs to be maximized, is total profit of enterprise. Also there is value of costs of realization of products,
all expenses on every good, norms of charges and fund of time.
We will make an effort decide this task, applying known to us natural methods of optimization. We will examine
every variant of issue of products (set of values of variables) as an individual, and profitableness of this variant – as
adjusted of this individual. Then in the process of evolution (if we will manage it to organize) the adjusted of individuals
will increase, and, will appear more and more profitable variants of plans. Stopping an evolution in some moment and
choosing the best individual, we will get the good enough decision of task.
A genetic algorithm is a simple model of evolution in nature, realized as a computer program. Both the
analogue of mechanism of genetic inheritance and analogue of natural selection is used in it. Biological terminology is thus
saved in the simplified kind.
To model an evolutional process, we will generate in the beginning casual population, a few individuals with the
casual set of chromosomes (numerical vectors). A genetic algorithm imitates the evolution of this population as cyclic
process of crossing of individuals and digenesis. A life cycle of population is a few casual crossings (by means of crossing-over)
and mutations as a result of which to population some amount of new individuals is added.
A selection in a genetic algorithm is a process of forming of new population from old, old population
perishes whereupon.
A selection in a genetic algorithm is closely related to principles of natural selection in nature.
After a selection to new population the operations of crossing-over and mutation are again used, after
again there is a selection, et cetera.
Thus, the model of selection determines how it is necessary to build population of next generation. As a rule,
probability of participation of individual in crossing undertakes to his proportional adjusted. So urgent strategy of elitism,
at which a few best individuals pass to the next generation without changes, is often used, not participating in a
crossing-over and selection. In any case, every next generation it will be on the average better previous. When the adjusted
of individuals stops to be increased, a process is stopped and as a decision of task of optimization take the best from the found
individuals.
Going back to the task of construction of optimum plan, it is necessary to explain the features of realization
of genetic algorithm in this case:
- Individual = variant of decision of task = set from m chromosomes of Xj, where m is an amount of wares, producible an enterprise;
- A chromosome of Xj = a volume of output of products j = 16, it is a bit record of this number.
Because the volumes of products are limited, not all values of chromosomes are possible. It is taken into account
during the generation of population. The mechanisms of crossing-over (crossings) and mutation will realize elective part, and a
selection of the best decisions is the gradient lowering.
That, if on some great number a difficult function is set from a few the variables, then a genetic algorithm is the
program which for possible time finds a point, where a value of function is enough close to the maximally possible value. Choosing
an acceptable checkout, get the best decisions which can be got for this time.
Stand a genetic algorithm generates initial population casual appearance. Work of genetic algorithm presents an
iteration process which proceeds until the set number of generations or any other criterion of stop will not be executed. In every
generation of genetic algorithm a selection will be realized proportionally adjusted, crossing-over and mutation.
At first, a proportional selection appoints probability to every structure Ps(i) equal to attitude of its adjusted
toward total adjusted of population:
(4)
Description of the got and planned results
On results for analysis of description of subsystem of planning and prognostication of issue of products:
- specified and complemented planning principles in the conditions of indefiniteness, and also the concept of the
adaptive planning of the production program of enterprise is entered;
- authentication of factors, which having influence on planning of issue of products on an enterprise and reflecting
of a particular branch specific is conducted;
- entrance information, necessary for the construction of subsystem, is certain.
Methodological basis for research was made by positions of analysis of the systems, methods of planning of the production
program of enterprise, methods of mathematical statistics, economics and mathematics methods and models, expert methods, methods of
acceptance of administrative decisions in the conditions of indefinite.
In works of many authors the attempts of account of indefiniteness are done by the use of different methodological
approaches. Offered approaches present both scientific and practical interest. At the same time we came to the conclusion, that it
does not exist enough universal methods of forming of plan of issue of products enterprises, in which all factors, providing
efficiency of this process, would be complex taken into account.
Unfortunately, classic methods are ineffective in many practical tasks. It is related to that it is impossible full enough
to describe reality by the small number of model parameters, or a model calculation requires too much time and calculable resources.
From the lacks of traditional methods to the last active development of the analytical systems of new type goes 10 years. In their
basis are technologies of artificial intelligence, imitating natural processes, such as activity of neurons of brain or process of
natural selection. Most popular and tested from these technologies there are neuron networks and genetic algorithms which can be used
for the decision of task of planning and forecasting of products issue on an enterprise.
As a result of implementation of work the algorithm of work of subsystem of planning and forecasting of issue of products
was developed in the conditions of sewing enterprise join-stock company of the closed type "DOTI" (Figure 1).
Creation of the program, which is realizing the developed algorithm of work of subsystem, is also planned in-process.
At writing of this abstract of thesis master's degree work is not yet completed. Final completion:
December, 2009. Complete text of work and materials on the topic can be got for an author or his leader after the indicated date.