# Information models of production systems - System theory and system analysis

## Information models of production systems

The organization of production processes is an extensive field for the study of problems of which various models based on the application of methods of mathematical programming and statistical methods have been developed. A considerable number of monographs and textbooks on the organization of production are devoted to these problems. At the same time, the methods of the theory of systems and system analysis allow in some cases to take into account more real features of production situations.

Set out in Ch. 3 information approach to system modeling allows in real conditions to refine the algorithm or simplify and speed up the procedure for finding a solution. In particular, this applies to the two classes of problems considered below.

Queuing chains. To such chains can be reduced both purely industrial (technological) processes, and processes of processing of the documentation (information) in zavodoupravleniyah and computer complexes. Usually the streams of orders (requirements) for servicing in such chains are assumed to be Poisson (described by Poisson's law), i.e. without taking into account the aftereffect, and the chains themselves are treated as Markov chains, which is only valid in a very limited number of cases.

True, sometimes real flows can be reduced to flows of Erlang, reflecting a wider class of phenomena, but this does not allow us to make broad generalizations, since it applies to more but still special cases.

Meanwhile, the simplest (Poisson) flow with intensity X is characterized by an exponential probability density distribution

which in the information terms is the material (information) current I =/(0 in the circuit (Figure 8.16, a), where the entity H = 1, the resistance m = 1/X, and capacity n = 1.

The equation of such a chain

(8.2)

under the initial conditions,/(0) = H/x = λ has a solution that coincides with the Poisson distribution.

However, the complete minimal information circuit, as has been shown, besides rigidity τ and capacity n, also has rigidity L, in which its aftereffect is expressed (Figure 8.16, b). In addition, and the capacity n, characterizing the originality of the flow, can generally have a value different from unity. Equation (8.2) for such a chain is transformed into the following:

(8.3)

Under the initial conditions it has a solution

at ; at

where - the variance of the time interval between applications.

Thus, equation (8.3) and its solutions approximate a wide class of packets of applications with different intensities λ and variances D, including, of course, the Poisson flow for which

Fig. 8.16

As a result, the description of a wide class of queuing systems with aftereffect and heterogeneity reduces to a system of equations of the type (8.3), which differ from the usual ones in Markov chains of the Kolmogorov equations by the terms Ldl/dt.

For example, if the state graph of the simplest queuing system has the form shown in Fig. 8.17, then, according to Kolmogorov, for the Markov chain we have ,

where p0 is the probability that the system is free; p r is the probability that the system is busy; λ, D } - the intensity and variance of the flow of their execution.

Under the same conditions, but in chains with aftereffect and arbitrary flows, pseudo-states also appear, since the transfer function of the circuit in Fig. 8.17, b is reduced to the product of the transfer functions of the chains in Fig. 8.17, and under the condition and

Thus, we have where

These relations, like the Kolmogorov equations, are valid for any variances within , and not only , as in the simplest flow. They make it possible to determine all the probabilities included in them for arbitrary queuing chains.

Fig. 8.17

At the same time, one can do without pseudo-states if, instead of Kolmogorov's equations, write a system

is also valid for any values ​​of the variance within the specified limits. If we remove any restrictions on the variance of the flow, then for an arbitrary case, without resorting to pseudo-states, we obtain a system of equations of the form

where - the closest to to is the larger integer;

This equation represents a generalization of the Kolmogorov equation to the case of Erlang flows of non-integer power, that is, in the case of flows with any aftereffect.

Transport and reduced tasks. The basic ways of solving such problems, which are problems of linear programming, are guided by the simplex method, which is rather cumbersome. On the basis of the information approach, the method of the total gradient was proposed [10, 74], making the procedure for finding the solution much less cumbersome.

Suppose that it is necessary to satisfy the demand for raw materials at several points of the production complex scattered over a large territory. Intermediate stockpiles of raw materials can also be scattered over this territory, so that the problem arises of optimal transportation from warehouses to consumers either by the criterion of a minimum value or by the criterion of the minimum time for transportation.

According to the theory of the information field, the stocks of raw materials in intermediate warehouses can be considered a positive material property (stock) - M, and its deficit at points of consumption can be considered negative M

Thus, the transport will be optimal in the direction of tension, i.e. gradient of the potential formed by the corresponding M field. In this case, either the cost of transporting a conventional unit of raw materials or the time of its transportation are in the role of the field potential.

The idea of ​​making a decision is illustrated in Table. 8.2 - 8.5.

For example, if you want to satisfy the specified in Table. 8.1 the need for the items B i and B 3, due to stocks in warehouses A, A2 and A3 (the prices of the respective transportations are indicated in the upper right corner of each cell in the table), then it is necessary to determine for each variant (for each cell) the total potential gradient.

This is done by adding, taking into account the sign of all the differences between the potential (price) of a given cell and the price of adjacent cells immediately adjacent to it in the row and column. The resulting total gradients are shown in the same table at the center of the cell.

Let's give a possible solution.

Example

The optimal first transport is the one that has the maximum total gradient. In this case, this gradient + 7 corresponds to the transportation of the entire stock of raw materials (20 units) from A 3 to B 2. The quantity of raw materials and the price are shown in bold in the table. The situation after this transportation is reflected in tab. 8.3, where the total gradients for the row A 2 are recounted, immediately adjacent to the discarded row A3. Now the optimal second transportation is the transportation of the necessary cargo to meet the entire demand for raw materials (18 units) B, from A, (gradient +3). The situation after the second transportation is reflected in tab. 8.4, where the total gradients for column B2, immediately adjacent to the discarded column B, are recomputed. According to Table. 8.4 the optimal third transportation is the satisfaction of the entire demand (33 units) B3 from A 2 (gradient +2).

Table 8.2

 B 1 B 2 B 3 Inventory A 1 50 A 2 40 A 3 20 Needs 18 21 33

Table 8.3

 B 1 B 2 B 3 Inventory A 1 50 A 2 40 Needs 18 1 33

Table 8.4

 B 2 B 3 Inventory A 1 32 A 2 40 Needs 1 33

Table 8.5

 B 2 Inventory A 1 32 A 2 7 Needs 1

After the third transportation, the situation is reflected, as shown in Table. 8.5, where the obvious last step is to satisfy the remaining requirement (1 unit) B2 due to A2.

Summarizing the values ​​of all transportations selected in the tables, we get 20x4 + 18x5 + 33x5 + 1x6 = 341, which is the minimum possible and coincides with the result obtained by the simplex method, but faster and easier, because it did not need to increase the size of the table due to the introduction of a fictitious need, necessary to reduce the original problem to a problem with the correct balance, which the simplex method needs.

It should be noted that since the total gradient method is oriented only to the nearest neighborhood of activity, then, on the one hand, it significantly speeds up the decision procedure, including by concurrently performing identical or comparable options, unless they are contiguous in rows or by columns, but, on the other hand, sometimes at the end of the procedure it gives errors when the nearest neighborhood turns out to be the entire activity space.

For example, if you select from the four options specified in Table. 8.4, the obvious first step is to move from A, to B v to exclude the unfavorable variant A2-B, but the total gradient method yields A, - B 2. In the first case, the objective function is 10 x 2 + 10 x 2 + 10 = 50, and in the second 20 x 1 + 10 x 4 = 60.

However, such failures can be avoided if, in addition to the maximum of the total gradient, the minimum of the sum of the excluded gradients is taken into account. So, in the first case, the first column excludes +1 - 4 = -3, and in the second + 2 + 1 = +3.

In conclusion, we note that in those special cases when the total gradient turns out to be the same for several cells in the table, one should choose a step that corresponds to the minimum price. If the prices are the same, then choose the cell that is in the row (if the stock is completely exhausted) or the column (if the demand is fully satisfied) with the highest price to exclude this unfavorable variant from further consideration.

Selecting the flexibility of the production structure. Designing and organizing the operation of flexible production systems (FMS) is a complex task related to solving technical, economic and social problems, combining into separate system of separate automated subsystems - ASNI , CAD, ACS, ASUTP, etc. When solving this problem, it is necessary to analyze the state of production, including analysis of the state of technological equipment and production areas, research of the possibilities for specialization and co-production, the state of technological preparation of production; determine the need for the introduction of GPS and to justify the effectiveness of its organization, the necessary degree of flexibility.

In conducting such studies, it is necessary to model GPS at various stages of its development - from conceptual design to technical implementation and management of technological processes.

An overview of the problems of system design of enterprises with flexible automated technology and examples of implementation of the main design stages can be found, for example, in [74].

One of the most difficult and significant stages of GPS design is the rationale for the flexibility of the production structure. To solve this problem, various models are being developed. This section considers one of the approaches to choosing the flexibility of the production structure, based on the application described in Ch. 3 information approach.

The simplest way to build a flexible production is to organize parallel technological chains (conveyor lines), each of which (Fig. 8.18, a) is able to produce its own modification of products. To switch from one article to another, it is sufficient to use the appropriate circuit (Figure 8.18, a, b or c).

Such a method takes place, for example, in the automotive industry, where parallel conveyor lines are used.

Fig. 8.18

The disadvantage of this method is that most of the equipment is simple when only one modification of the product is released at any time, which, however, is compensated by the possibility of parallel operation of all circuits when all the modifications are released simultaneously, which is usually done in planning production by distributing orders for various types of products for the planned periods, which would ensure the best loading of equipment and conveyor lines (similar distribution with a custom planning system can, for example p, implemented using models of morphological modeling - see Chapter 4).

An alternative way to build flexible production, the most common, is (see Figure 8.18, b) in the horizontal aggregation of simultaneous operations in a single complex, i.e. in the use of universal programmable machines and machining centers. In this case, for the transition from one article to another, one operation at each level must be selected so that together they form one of the possible vertical chains.

The disadvantage of this method is the simplicity of all operations except one, at each level a, b or in and in the impossibility of issuing various modifications of the product (solid and dashed arrows on Figure 8.18, b), since neither the machining centers nor the CNC machines are capable of performing more than one program at any given time. Worthiness -

tv is more in comparison with the first method of modifications with the same choice of elementary operations, since not only combinations of operations with the same alphabetic index, but also operations with different indices are possible here.

A certain share of complementarity of advantages and disadvantages of these methods leads to the idea of ​​the need for their combination in the form of a matrix (field) method of implementing flexibility (see Figure 8.18, c), when instead of complex and bulky aggregates the simplest rigid automata, to perform only one operation each, which allows, first, to combine these operations not only vertically, as in the first two methods, but also horizontally, increasing the number of product modifications (solid and dashed arrows in Figure 8. 18, c), which drastically reduces downtime of equipment.

In addition, the transition to the matrix structure and the use at each point of the technological field of only the simplest equipment, which is characterized by a relatively high reliability, on the one hand, increase the reliability of the entire system, on the other - dramatically facilitate its modernization, because the replacement of simple and cheap small-sized machines on more perfect does not require capital intervention in the production life of the enterprise and can be carried out without violating it.

Finally, the transition to field flexibility is psychologically important for the work of personnel serving this production, because, firstly, servicing simple machines is much easier than servicing CNC machines or machining centers, and secondly, the matrix structure of production unleashes a creative initiative , both workers and engineers, in terms of improvement, because it allows painless experimentation and the introduction of proposals and inventions in small things, and in general, without a radical breakdown of the process.

It is easy to see that the realization of the field method of production flexibility allows two basic schemes, to the combination of which the scheme of any real field technology is reduced.

One of them is that in the manufacture of relatively small and lightweight products, transport robots move them in the technological field from the machine to the machine along routes that depend not only on technology, but also on which of the suitable machines (machines) are free at this time.

Another scheme applicable to relatively cumbersome and heavy products is that the technological field forms fixed stationary products, and transport robots move in this field machining automata, selecting those that are free at the moment and are suitable for execution related operations.

Both these schemes place the main burden of control on computing facilities, simplifying and unloading technological equipment from complex functions, which, on the one hand, ensures high reliability and trouble-free operation of the entire system, and on the other - facilitates the setup, maintenance and repair of equipment, allowing easy and complete replacement of the failed machines or machines and their restoration in the conditions of the repair shop or site. In such conditions, the computer tools and application of the methods of the theory of mass service, optimization, and morphological modeling theoretically make it possible to almost completely load all the equipment, thereby avoiding the main scourge of flexible production (and, in general, any universal production), the downtime of most of the equipment that always accompanies its inevitable redundancy.

You can also specify the criteria that the field technology management system should be guided to ensure the optimal location of the equipment work. Ranking operations on urgency and assigning to them the appropriate PA potentials, the system must ensure at any time

where t is the total number of operations simultaneously possible in a given field; - the number of operations that have a potential .

This criterion takes into account all the factors and allows even neglecting a small number of urgent transactions for the sake of a large number of slightly less urgent ones, taking into account, of course, the corresponding restrictions on the term of fulfillment of orders.

Yet this criterion can put flexible production in difficult conditions, since it does not take into account all kinds of preventive equipment stops that must inevitably take place.

Therefore, the universal criterion must also include the time continuous equipment operation

which is more, the more attention is paid to prevention, although it in itself creates the appearance of some reduction of this time.

Given the increasing importance for the flexible field technology of optimization of all processes, the accelerated method of solving the transport problem should be applied.

In the case of a random change in product modifications and simultaneous production of several modifications, the trajectory of blanks in the technological field, which depend on the random nature of the equipment, acquires a random character, which makes it necessary to consider the organization of production as a mass service task.

Existing methods of solving such a problem, if necessary, reduce the streams of applications in such technological cycles to the simplest ones, and consider the cycles as Markov, which in fact does not correspond to the real state of affairs, since the significant aftereffect of such chains is ignored. Therefore, we should use this method, which allows us to solve this problem without doubtful assumptions and taking into account the real parameters of the process.

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