Classification of the models and the mathematical production programme.

Hrubina, Kamil ; Wessely, Emil ; Macurova, Anna 等

1. INTRODUCTION

The development of science, technology and manufacture requires new control methods and means that in the current stage of knowledge are expressed as "Integrated system of manufacture and enterprise control".

From the viewpoint of the system science, it is essential to investigate systems properties through the behavior of their mathematical models by means of simulations and various modeling techniques.

Mathematical models are created especially for the purpose of their utilization in the control and information systems design whose present development is enabled by informatics methods of a high quality.

It is known that cybernetics deals with the problems of description of real processes behavior and control that show the evidence of considerable complexity. Special attention is given to some difficulties that appear with their formal mathematical description. Classical formal procedures that are based on the knowledge of natural laws and numerical mathematics as well as statistics utilization have a rather limited utilization when applied to the mathematical models in the control.

From the above it follows that the application of classical procedures often leads to the simplifications that are the reason of insufficient adequacy of mathematical description (Sarnovsky et al., 1992).

Mathematical model usually failures if there is a lack in information about the controlled process. In such a case the procedures of the new scientific branch could be applied, i.e. those of the artificial intelligence whose methods are: solution of the problems with constraints, artificial neural networks and fuzzy systems (Zadeh-Polak, 1969). By means of these new procedures, the problems of processes control can be solved, in particular non-linear processes. The utilization of these methods to design and implement algorithms is a part of intelligent control (Neuschl et al., 1992).

Machining centres for the selected parts machining with adjacent times optimisation will be taken into consideration.

2. MODELING AND CLASSIFICATION OF MODELS

The models present generally a wide and miscellaneous area of means and ideas that differ from the point of view of mission and according to method of realization as well. The model is a formal or material formation that pictures some points of view of original (example system, modeled system) and that is created for it that is used at study and at intentional influence of the modeled system.

Then the modeling is a process where other system called model is attached to the searched system i.e. the object, the original according to the certain criterions.

The model is an isomorphic or homomorphism picturing objects into the choice aim quantity that transfers a competent picturing and properties of the object (Obona, 1990).

Classification of models

The problem of modeling is very wide and therefore we limit on some points of view of classification:

a) Distribution of models according to degree of abstraction

a1) Nature models (degree of abstraction zero)

a2) Physical models (the first degree of abstraction)

a3) Physical analog (the second degree of abstraction)

a4) Structure model (the third degree of abstraction)

a5) Functional models (higher degree of abstraction)

b) Distribution of models by their use in process of control

b1) Functional models

b2) Directive models

b3) Described models

b4) Structural models

c) Distribution of models by time relations

c1) Prognostic models (extra polar models)

c2) Topical models

c3) Retrospective models

Mathematical models of complicated systems

We can divide mathematical models of complicated systems, complexes with regard to the introduced classification on two classes:

a) deterministic models

b) stochastic models.

Physical models of complicated systems

The most known physical models of complexes dynamics are:

--modes passed

--diffusion models, (Hrubina, 1999; Jadlovska, 2003).

3. MATHEMATICAL MODEL OF MANUFACTURING PROGRAM

3.1 Manufacturing Program selection

Automated manufacturing systems enable to solve optimal conditions of machining and adjacent time minimisation. The task can be formulated as follows: a part with a group of slots is being manufactured. Distances between any neighbouring slots are given. The shortest closed path which will pass through each slot only once is to be selected. Mathematical model of the task resembles different cases occurring in technological processes. Intensification of cutting conditions without corresponding adjacent movements of the tool does not lead to more significant increase of machining productivity.

Process of searching for the sequence how to "detour" the slots system is characterized by its variability. It is necessary to take into consideration design peculiarities of the machine tool (range of table movements, way of changing the tool, etc) as well as design peculiarities of the part (position and dimensions of the slots). Group of slots can have the same diameter and meet the same requirements, thus they can be considered as a group. The sequence can be understood as follows: the solution is being searched gradually and the selection depends on the momentary state of the system, i.e. the previous systems. The characteristic feature of the solved task is that the slot is being determined at each step that follows after the previous one has been machined. Thus, the task is to determine the length of the optimal path from the point i to the point I, passing through, at the moment, each point [j.sub.1], [j.sub.2], ... [j.sub.k] once in random order and is not passing through any other slots axise (Etner, 1991; Douglas et al, 2005).

3. 2 Mathematical Model Design

In the process of mathematical model creation, the Euclid norm of the distance of the two points has been used. In order to find optimal variant using dynamic programming, algorithms of discrete programming have been used. With this method, optimisation process is considered to be a multistep process (Hrubina et al., 2002). Characteristic feature of this method is that the next solution always follows from the previous state (passed path to the given point):

T(i, j) = [c.sub.i,j] + [c.sub.j,1] (1)

From the above it follows, that for the each step (n steps) we are going to select the path with which the sum of step distances plus function of optimal previous values will be minimal.

Let us use denotation [f.sub.n] ([v.sub.i]) for the distance between points that corresponds to the minimisation strategy for the path from i if n-steps remain to the final slot.

Then

[f.sub.n]([v.sub.i]) = min ([c.sub.ij] + [f.sub.n-1]([p.sub.i])) (2)

Cyclic productivity of machining in machining centres will be determined from the relationship:

Q = 1/[[tau].sub.c] = 1/[summation][[tau].sub.z] + [summation][[tau].sub.x,x] (3)

Where [[tau].sub.z,z] is the basic time needed to machine all the surfaces, [[tau].sub.x,x] is the time of passive running needed to position and swing out the table, change and supply the tool.

Time of the machining cycle for all the surfaces is expressed by the relationship

[[tau].sub.c] = [summation][[tau].sub.z] + [summation][[tau].sub.x,x] (4)

where [[tau].sub.z] is the basic time needed to machine one surface, [[tau].sub.x,x] is the time of the passive running needed to machine the elementary element.

Body parts are machined from the four sides. One group of the machined slots can have the same diameter and accuracy requirements. Several groups of the same slots can be placed on each surface of the work piece. These slots can be machined using three methods.

Analysis of the three mentioned variants enables us to conclude that selection of the sequence of machining can be realised by the two basic methods: parallel and sequential. With the sequential method, each slot is being machined by the tools. Then, after position is being changed, the next slot is going to be machined. With the parallel method, each tool passes through all the slots diameter of which corresponds to the tool. After that, the tool is being changed and cycle repeats (Macurova, 2007; Macura, 2003; Tirpakova, 2000).

4. CONCLUSION

The main contribution of the paper is to generally investigate the problem of modelling and models classification. Creation of mathematical models of the manufacturing program is considered to be the introductive stage of manufacturing program optimisation.

To realise parametric and structural optimisation of manufacturing program, the database of cutting conditions, cutting tools, technical characteristics of manufacturing device is needed. Database also contains typical technological procedures of elementary surfaces machining as well as their files. Part of the manufacturing program optimisation that is being evaluated is realised with the aim to minimise the selected machining time of the part surface (4). Adequate mathematical model can be realised on PC using simplex algorithm.

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