Modelos de simulación para sistemas de manufactura

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ESCUELA SUPERIOR DE INGENIERÍA MECÁNICA Y ELÉCTRICA

SECCIÓN DE ESTUDIOS DE POSGRADO E INVESTIGACIÓN UNIDAD CULHUACÁN

MODELOS DE SIMULACIÓN

PARA SISTEMAS DE MANUFACTURA

TESIS

Que para obtener el grado de:

DOCTOR EN CIENCIAS EN

COMUNICACIONES Y ELECTRÓNICA

Presenta:

M. EN C. ANDRIY SADOVNYCHYY

Director de tesis:

DR. VOLODYMYR PONOMARYOV

MÉXICO, D.F. DICIEMBRE 2009

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NATIONAL POLYTECHNIC INSTITUTE

PROFECIONAL SCHOOL OF MECHANICAL AND ELECTRICAL ENGINEERING

SECTION OF GRADUATE STUDIES AND RESEARCH UNIT CULHUACÁN

ANALYSIS AND SIMULATION OF MULTILEVEL DISTRIBUTED TECHNOLOGICAL SYSTEMS AND

MANUFACTURINGS

DISSERTATION To receive:

DOCTOR OF PHILOSOPHY

”EN COMUNICACIONES Y ELECTRÓNICA”

Presented by:

M. ANDRIY SADOVNYCHYY

Head of dissertation:

PhD. VOLODYMYR PONOMARYOV

MEXICO, D.F. DICIEMBRE 2009

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ACKNOWLEDGEMENTS

TO NATIONAL COUNCIL ON SCIENCE AND TECHNOLOGY:

For their support, that has permitted me to reach this purpose.

TO NATIONAL POLYTECHNIC INSTITUTE:

For that, it has given me a quality education, especially ESIME Culhuacan, and to all its teachers.

TO TEACHERS AND COLLEAGUES OF THE POSTGRADUATE SCHOOL ESIME CULHUACAN:

Most especially thanks to committee of Doctoral program on reviewing of this thesis.

TO DOCTOR VOLODYMYR PONOMARYOV:

For manuals and directions during researches of this work, without which would not be possible to finish this project successfully.

TO MY PARENTS

With deep love, respect and gratitude, as the proof of my infinite gratitude for those efforts and brought a sacrifices. I am thankful for they always offered the support and love when I needed.

I sincerely hope that they feel my professional success as their own.

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Abstract.

In this work, the organization structure of Automated Manufacturing Systems (AMS) is analyzed.

The methods of analysis of production systems are investigated. Two class of analyzing methods were analyzed: analytical methods and simulation methods.

Analytical methods make it possible to optimize the industrial systems.

However, the development of mathematical model needs severe formalization of an industrial system. In case of uncertainty of some parameters of industrial system, the formalization either is inconvenient, or absolutely is not possible.

Using the simulation methods it is possible to design the models of various industrial systems. Model’s modification is not complex, so in the model, it is possible to use stochastic and uncertainty parameters.

Different software, which realized simulation methods, is investigated. It was concluded, that they mainly realized the queuing theory methods, thus they can analyze only technological system of AMS.

Therefore, the new method is proposed. It can analyze the production systems taking into account both technical and technological subsystems. This makes it possible facilitate the using of a simulation model.

The proposed method is based on some of analysis methods. For present model of technical systems, it uses the queuing theory and for model of technological systems it uses Regular Schemes of Algorithms (RSA).

RSA is good for description of the control algorithms as the diagrams.

Graphic representation of algorithm is easily translated in a semantic form, which can be proceeding by elementary interpreter.

Using the queuing theory, it is possible to design the models of the various industrial systems. With increase of complexity of the manufacturing system its imitating model is increased. If a parameter of an industrial

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system’s element is changed it is necessary to change only parameter of the corresponding element of model. Disadvantages of the queuing theory are that the powerful computing means are necessary for modeling.

To simplify the design of the simulation model some basic modules were constructed and implemented. Its internal structure and working algorithms were developed. The manager module was labored for work control of basic modules.

Proposed approach is useful for simplified design of the simulation models of the manufacturing systems. It proposes a flexible way to describe production processes in AMS using high level program language and proposes user friendly interface for design model of a technical system of AMS. It permits to change independently each a part of a model (model of technical system and technological system), thereby it facilitates the development and usage of simulation model like an analysis tool.

The simulation experiments are realized. They expose some advantages of the proposed method.

The proposed method decreases number of operation (more than half) when the simulation model of manufacturing system should be changed.

The efficiency of the proposed method is more than half.

If more technological processes are in the manufacturing system, the more effectiveness can be achieved from application of the proposed method.

Little drawback of the proposed method is rather more time that is necessary to design a simulation model at initial stage of the manufacturing system analysis.

Therefore the developed method and engineered software are more effective than existent methods and software and permits to accelerate the process of manufacturing systems analysis.

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INTRODUCTION PURPOSE

PROBLEMS

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Introduction

The modern market is characterized by big dynamism and big variety of a consumer demand. Demands for quality, for cost and other characteristics of the goods can vary cardinally during a short time. The customer demands not universal goods but strictly fulfilling to his demand.

For satisfaction of a consumer demand, modern companies should release the goods of wide range. Besides, they should react to fluctuations of a consumer demand promptly. All this makes a number of demands to modern organization of manufacturing and administrative structures of companies.

Therefore, the development of mathematical methods and models, which are oriented to the analysis of new schemes of organization and management of computer-aided manufacturing in engineering industry, is an actual problem.

Existing methods of analysis of production systems do not present various arranging solutions of manufacturing in full, do not take into account the complex dynamics of an assembly processes, and do not permit to calculate a rational stocks of the materials and semi finished products.

Therefore, the development of the new class of methods and the models, which should be based on system representations of machine-building manufacturing, spatio-temporal simulation modeling and for algorithmization of manufacturing methods is necessary.

Automation of technological processes, application of the computer- aided manufacturing systems that permit to overhaul manufacturing for fabrication of new products quickly and effectively is a basis of intensification of manufacturing.

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Appearance of atomized manufacturing systems (AMS) characterizes the qualitatively new stage of integrated automation in engineering industry. The basis of this stage is wide circulation of information technologies and computers. Development and implantation AMS is an objective demand of a time and is stipulated by the further developing and perfecting of the process equipment and resources of automation and by information control technologies.

Purpose

The purposes of this work are:

- design of the research and develop models and algorithmic methods of the manufacturing processes;

- analysis of the computerized machine-building manufacturing and suggestion of the way for increase of its efficiency using methods and models, which should be proposed.

Problems

To achieve the purpose stated above it is necessary to solve the following problems:

- analyze the state of the art of new methods of organization and management of machine-building manufacturing;

- classify main technological processes of engineering industry;

- investigate the state of art of methods for analysis of the automated manufacturing systems;

- realize the formalization the representations of technological processes for problems of automated management;

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- design the structural representations and the control mechanism for simulation of technological processes in engineering industry;

- develop the simulation models for the analysis and calculation of characteristics of machine-building manufacturing;

- generate principal requirements to the architecture and subsystems of the automated control system of manufacturing;

- create the special software for check of the developed approaches and carry out the simulation of a technological assembly process;

- analyze the simulation results;

- present practical recommendation for application developed methods for analysis of other discrete systems

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CHAPTER 1 STATE OF THE ART

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1. State of the art

1.1. The analysis of the automated technological processes and systems in engineering industry

The intensification of manufacturing of engineering industry is connected with wide automation of technological processes, implantation of the computer-adding manufacturing systems that permit rapidly and effectively to overhaul manufacturing for production of new products.

In this connection, development of the methods, which make it possible to automate analysis of AMS at stage of designing, modernization and reconstruction, is actual.

Any analysis of manufacture system is made on the basis of a model.

The model is designed with the help of one of methods or their combinations.

Let present the classifying AMS by various character and on this basis it can be considered the features AMS in engineering industry from the point of view of effect of these features on the design process of the formalized model of manufacturing.

The basic components for constructing AMS have modular structure [1], [2].

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1.1.1. Organization structure of manufacturing system

From the point of view of organization structure the AMS, it is possible to classify them as Fig. 1. Organization structure of the AMS.Fig. 1 shows:

Fig. 1. Organization structure of the AMS.

The supreme level of automation is the automated manufacturing system (AMS) consisting from several manufacturing units, consolidated by means of the automated control system and by the automated transport-warehouse system. The organizational level of AMS consists of the automated line (AL) and the automated section (AS).

Flexible manufacturing units in AMS are serviced by the automated transport systems and warehouses. They (automated transport systems and

Automated Manufacturing System (AMS)

Automated Line (AL)

Automated Section (AS).

Warehouse Unit (WU)

Transport Unit (TU)

Automated Manufacturing Unit (AMU).

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warehouses) serve for accumulating and transportation of materials, details, semi manufactured articles and work tool among separate AMU.

Automated manufacturing units AMU, which are included in automated sections (AS) and in the automated lines (AL), can have different operating speed, therefore, for the coordination of their operation each a unit has the input and output storage devices, permitting to store semi manufactured articles before and after an execution of a processing. The machining process in AS and AL has sufficient differences.

In AS, the robots and the overload devices displace the workpieces to the machine units. Installation of semimanufactured article and its return are fulfilled by robot-manipulator. After a termination of machining, the detail gets in an output storage device. After that, the detail is transmitted in an in- process warehouse. Then, from warehouse, the detail transmits to next machine units for execution of the following operation, foreseen by an operation-routing sequence.

In AL, as a rule, the semi manufactured article from a warehouse is feeding to an input storage first, and then following to AMU.

Warehouse units make the automated warehouse complexes that organized by a block-sectional principle.

The transport unit includes the soft transport facilities, overloading, reception and distribution facilities and hardware components of control. It relocates production parts according to the program generated by a manufacturing control system or by commands of terminal devices at autonomous operation [3].

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1.1.2. Levels of automation of manufacturing system

Three types AMS can be selected depending on a degree of automation of manufacturing [1], [4]:

- I type (full-scale production) – the sections with the numerical control (NC) machine and multioperational rigs;

- II type (mass production) – for machining of the small group of the mechanically homogeneous details;

- III type (single-unit production) – widely universal AMS.

Automated manufacturing systems of the first type intended for serial and a small-scale production. The automated control system fulfils the functions of planning and preparation of control programs for NC machine.

The universal and NC machines, as well as the non-NC machines, form this system. The set-up of rigs on machining of new details and substitution of work pieces on the rig is fulfilled by the operator.

The details having minor differences are handled by AMS of the second type. The design similarity of details permits to work out operation-routing sequence, having minor differences for each of subgroups of details.

Operation-routing sequence permits to group the equipment by the type of machining.

Widely universal AMS, intended for machining of small series of heterogeneous details refer to the third group. Modification of equipment of this group is very difficult.

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1.1.3. Classification of the technological processes

Each technological process can be presented like a set of technological operations [2]. Technological operations can be classified like principal and auxiliary. The principal operations are processing and assembly. The auxiliary are transportation and warehousing. Fig. 2 shows the structure of technological process.

Fig. 2. Structure of technological process.

1.1.4. Interaction of the organization structure and technological processes

Let’s see how to interact the units of organization structure and technological process.

Technological processes

Processing

Discrete processing

Discrete-continuous processing

Assembly Transportation Warehousing Principal operations Auxiliary operations

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1.1.4.1. Discrete processing

In AL, workpieces from a warehouse move to the input store of the first AMU. After servicing in the first AMU workpieces follow to the next one until all operations specified by a technological route will be executed.

Machining of details in AS includes the following functional operations:

- preparation, - warehousing;

- transportation;

- placement in the input store;

- machining;

- placement in the output store;

- warehousing.

Machining duration of the detail і in AP, organized on a principle of site (AS), is defined [2]:

 









gi

j

ij ij

ij

i p a S

F

1

2

1  

 , (1.1)

where gi– quantity of operations of processing of the detail і; pij– duration of transportation of the detail і on a warehouse at operations j; σ1 – in-process queue in the input store; σ2 – in-process queue in the output store; Sij – time in- process queue in an interoperation warehouse between the i and j operations of processing; aij – duration of processing of the detail i at the operation j; γ – in- process queue, caused by breakages of the equipment.

Machining process in AL a little bit others:

- preparation,

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- warehousing;

- placement in the input store of AMU і;

- machining;

- placement in the output store of AMU і, - transportation;

- warehousing.

In this case there are no time in-process queue in the output store (σ2) and no time in-process queue in an interoperation warehouse between the i and j operations of processing (Sij). So from equation 1.1 we have machining duration of the detail і in AL:

 









gi

j

ij ij

i p a

F

1

`   , (1.2)

where p_ij^` - duration of transportation details i from AMU (j-1) to AMU j.

1.1.4.2. Discrete-continuous processing

Drawing electrodeposit is a typical example of such process.

The basic features of this technological process are:

- continuity, that practically excludes idle times between execution of two consecutive operations above a suspension bracket with details;

- duration of processing of a suspension bracket in a bath should not fall outside the limits the given range (allowable time of processing).

The time diagram of detail i can be presented so:

 

 







i i

g

j

g

j ij ij

i

a p

F

1 1

1

, (1.3)

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where: gi– quantity of operations of processing of the detail і; pij– duration of transportation of the detail і on a warehouse at operations j; aij – duration of processing of the detail i at the operation j.

1.1.4.3. Common characteristics discrete and discretely- continuous processes

AMS with discrete and discreet-continuous types of technological processes, have a number of common characteristics:

- parallelism in processing details, i.e. the simultaneous machining of details by one or different technologies (multinomenclate);

- multiposition (multiphase) processing – presence in a route of several processing with different duration;

- transportation of details between separate AMS;

- equipment failures and occurrence of spoilage.

In case of parallel processing, the details by different technologies can provoke the conflict situations. These situations appear when on the same AMU different details pretend. In discrete AMS the machining of details in AMU occurs according to the priorities of their service. In this case, the details having low priority are located in an input storage. In AMS with discretely-continuous technological processes, such situations are not allowed in general. They should be eliminated already at a stage of construction of graphs of the equipment loading.

1.1.4.4. Transportation

The choice of structure of the automatic transport system provokes a question: How many working positions can attend the transport devices, the

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industrial robots or the transport carriages [5]. If a service range is too big, therefore there are conditions for the loss of time connected to waiting of transport. The reduction of a service range is resulting by the augmentation of an amount of transport facilities, decrease of coefficient of their loading and rise in price of the system. Depending of type and delivery capacity of transport tools, and a type of technological process, the time is spent for transportation can be either insignificant, or commensurable with time for machining. The module structure of AMS provokes the necessity of the operation coordination of its separate units by a top level control system, and the transport system is the coordinator that correlates the rhythm of reception - output of details with the rhythm of the equipment operation [6].

On the basis of above-stated for AMS with discrete and discretely- continuous technological processes, it is possible to select the following major factors, which influence on a productivity:

- amount of AMU - generecity of AMU;

- duration of operations on separate AMU;

- reservoir of the entry and output storages of AMU;

- topology of the transport system;

- coordination of operation of separate units control programs for the transport system;

- defect of separate operations;

- refusals of the equipment;

- features of routes of technological processed.

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1.1.4.5. Assembly and adjustment

Duration of assembly item i is defined as:

,

 

 n

F_i (1.4)

where  – a step of movement of the conveyor; n – quantity of the conveyor positions; g – duration of some unforeseen (casual) factors.

The assembly line represents a number of positions, on each of which there are stores with the details which necessary for assembly. The line goes according to the given rhythm (step) of work r. The condition of assembly products k on position i expressed an inequality:

, , 1

, _ik

ijk

ij d j n

d   _(1.5)

where dij – quantity detail j on position i; dijk – norm of the charge detail j for product k on position i; nik – quantity of types of the details which are included in product k on position i.

Observance of a rhythm of work of an assembly line depends on reliability of work of the conveyor and duly delivery of details and the units necessary for assembly on positions of the conveyor.

Duration of workpiece i passage on assembling line is defined as:

,

1











n

j j

i n

F    (1.6)

where  j – duration of additional adjustment or repair on positions j.

The specificity of adjustment process is that a defect of the unit can be detected in any position of the technological process. Therefore, after the expiration of time allocated on operation of adjustment (a step of a line) on position i, the unit, if it is serviceable, is transferred on following position (i+1); and in the case of detection of defect in the output store of the position i

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pending additional adjustment. Upon termination of additional adjustment the unit is transferred to the input store of the position (i+1), from which transferred on a position for performance operation (i+1) of adjustment. As on a line of adjustment and installation the defect comes to light and corrected directly on positions normal functioning of a line depends on: presence of enough of workplaces for additional adjustment; duration of additional adjustment of details.

1.1.5. Example of manufacturing system

Here, it is represented an example of a manufacturing process that produces an electrical connector for high voltage lines.

Fig. 3. Manufacturing process.

There are some flows in the production system: material, information, financial, transport and others.

There are some flows (see Fig. 3) in the production system: material, information, financial, transport and others. There are flat, rod, and screw in the material flow to produce the connector. They are arrived from supplier. At

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the manufacturer site, rods of aluminum are cut into shafts with a given length, flats are bored and ground. Each finished product is then produced by assembling a shaft, a flat, and two screws. The products will be packaged and delivered to customers by a transporter.

Customer demands are formed information flow. When the quantity of the finished product (Stock 4) decreases is below a defined value, an assembly order with a given batch size will be released to the assembly line of the manufacturer. When quantity of inventory in the stocks is below of defined value, a purchase order with a given batch size will be placed to corresponding supplier. Each purchase or assembly order contains the information such as ordering time and order quantity, which are determined by an inventory policy involved. The inventory policies of all four stocks are periodic review batch ordering policy.

Customers’ pays are representing financial flow. Customers pay the manufacturer within a given time period after receiving finished products. The manufacturer pays its suppliers similarly. The suppliers and the manufacturer pay their transporters within a given time period after the completion of a delivery.

The model’s design and analysis of this manufacturing system will be explained in the next part of the thesis.

Among set of parameters of the AMS, it is possible to allocate ones what have deterministic character (for example, duration of processing, a step of movement of the conveyor), or stochastic (a nature) character (for example, expectation of clearing AMU or transport, duration of transportation, in-

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automated manufacturing systems it is necessary to use methods, which can take into account stochastic nature of their parameters.

1.2. Methods for automated manufacturing system analysis

The concept “the manufacturing system” is used for wide class of the objects - from company up to a department, a section, group of machines.

According to their models of manufacturing, the systems can represent the goals of a different level. For example, optimization of planning of company activity is a highest level of representation (company as a whole), but operative - scheduling of the department it is a lower level.

There are two principal types of analysis methods: analytical methods and methods of imitation modeling. The methods are divided by type of models, which are result of their implementation. So, the result of the analytical methods implementation is a mathematical model, and the result of imitation modeling methods is a simulation model.

1.2.1. Analytical methods

The monographs [3], [7] are devoted to the problems of design of mathematical models of manufacturing systems.

As the subjects of the given work are closely connected with the modeling of AMS activity, therefore from all diversity of mathematical models we consider only the models, which are intended for research of the manufacturing process.

In this section, the next methods are considered:

- operative-scheduling planning;

- conditional;

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- characteristics definite;

- organizational-technological conditional;

- graph models;

- method of power balance;

- method of rational resource allocation;

- Language of regular schemes of algorithms.

1.2.1.1. Operative-scheduling planning

The tasks of operative-scheduling planning and management of the production are most developed and known [8].

The diversity of calendar tasks is determined by organizational and technological conditions of production. For example, like a dividing of production for machining and assembly, continuous and not continuous organization forms. Differences in settings of the planning tasks are connected to rhythm of product lines, the volume of in-process reserves, and the type of a motion of details (sequential or parallel-sequential etc.).

The planning tasks that are solved by means of the calendar models can be classed as follows:

- by conditions of the target setting;

- by the way of characteristics definition;

- by organizational-technological conditions of production for which the calendar model develops.

By the conditions of the target setting, there are selected direct and inverse problems of calendar simulation.

The direct task of calendar modeling is used for determining such feasible solution , which fulfils of the limitations accepted for model, and

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also a number of additional conditions (for example, optimal somewhat); thus, all model units and their parameters are considered as given. An instance of such task is the task of scheduling of execution of determined operations at the given manufacturing performances.

In case of the solution of an inverse problem, the purpose of simulation is definition of some unknown characteristics of model, which in a direct problem are known. As a rule, such value of model parameters, which provides existence of the whole class of the feasible solutions{X_s}, is searched. Inverse problems in the greater extent are problems of designing than the analysis. An example of such problem can be definition of composition and characteristics of the equipment of a production section.

The method of the characteristics definition of model performances can be deterministic or stochastic. In the case of stochastic model for an input stream of details, the probability distributions principles per the assortment and the moments of arrival are set. The performance of the equipment operation (breakdown, worker absence, absence of equipment etc.) and the defective goods magnitude during modeling can be set stochastically.

It is possible to select two types of the calendar models corresponding to different organizational-technological conditions of production: linear and network. Calendar models with linear structure of operations can be used for modeling the machining department. The assembly department is the typical object, for which the network calendar models are applied.

The rigorous optimum solution of a calendar simulation problem in the general case does not exist, though for their research the various mathematical

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tooling used: linear and a dynamic programming, special methods of discrete mathematics etc. [9].

Solution methods of the calendar modeling.

Let's consider the most typical problems of calendar modeling and the methods of their solution.

Problem of distribution.

There are M of warehouses and each of them maintains resources, total number of which is presented by following equation:

 

 M i Xi

X

1 . (1.7)

There are also N items of consumption, which request these resources, and cost of delivery of resources from a warehouse i in item j is determined by the function gij⁽xij⁾, where xij – an amount of supplied resources in all items of consumption.

 



j i gij xij

X E

,

min )

( )

( . (1.8)

Nonlinear character of function gij(xij) does not permit to use the simplest methods, like the method of a dynamic programming and its modification [10].

The solution of the nonlinear distributed problem of calendar simulation permits to arrange workers to working place, to generate the employee group, to control a work-in-process. In this case, Nj - an amount of machines,xi - an amount of operations (details),gij(xij) - duration (cost) of execution.

Functional equation for N items of consumption and for arbitrary amount of warehouses M is given by:

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)],

;...;

; (

) (

...

) , ( )

( min[

) ,..., ,

(

, 1 1

, 2 2 , 1 1 1

1

1 , ,

, 1 ,

1

2 , 2 ,

1 , 1 1

2 1

N M M

N N

N

M

i N i N

N M N

M N M

N N

N M

N

x x

x x x x

x r

g x

g

N x g x

g x

x x



















(1.9)

With following limitations:

. , 1 , 0

; ...

,

, ,

2 ,

1

M i

x x

r x

x x

i N i

N N M N

N









(1.10) Practical usage of the functional equation (1.9) already for М=3 meets serious obstacles. Really, if the number of quantum values xi is Mxi, then the computer memory size, which is necessary for implementation of a method of a dynamic programming, will be:

,

1





M

i

Mx

i

W 

_(1.11)

or, forMxi=100, and for anyone i and М=2 this size is proportional10⁴, but for М=3 will be equal10⁶.

The problem of selection (assignment)

This problem can be considered as a special case of a nonlinear distributive problem [11]. There are n warehouses, and in each of which the unit of a resource is stored, and there are n items of consumption, which are requesting one unit of a resource, and numbersaij, which are defining a cost of delivery of unit of a resource from a warehouse i in the item of consumption j are known, then the problem is named as a problem of selection.

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 

 







n

i

n

j ij ij

n

i n

j ij ij ij

x x

x a x

E

1 1

1

. min )

(

(1.12)

The variable xij takes on a value 1 or 0.

Application of the exhaustive methods demands the analysis n! of alternate solutions. For acceleration of the analysis process it is possible to use the dynamic programming method.

If n=10 that by means of the exhaustive method it is necessary to analyze about 4*10⁶ alternate solutions. If the dynamic programming method is used, then the number of examined alternatives is7*10³.

Optimum search problem (traveling salesman problem)

In [12], there is presented the classical statement of optimum search problem (traveling salesman problem). Some problems of formulation of the operation schedule and development of the graph of machine utilization can be reduced to this problem.

The traveling salesman problem consists of the following: the amount n cities and distance between them are given. The traveling salesman leaving from some city i, should visit each of (n-1) other cities only once and return to the initial city, having overcome minimum distance. Mathematically, it can be presented as follows: the square matrix of weights A (rank n) is given.

It is required to define the commutation i₁,i₂,i₃,...,i_n_₁,i_n,i_j i_q j  q_, which would minimize a functional:

.

1

1 ,

, ₁

1









n

j

i i i

i _n

a

_j _j

a

_(1.13)

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Classical traveling salesman problem is a characteristic representative of extremum problems of combinatory type. Its solution by means of the simple exhaustive search results in necessity of the analysis ( 1)!

2

1 n of alternatives, that it is possible for n, not exceeding 10. Application of a method of a dynamic programming permits to solve traveling salesman problem forn20. The basic method for the traveling salesman problem solution, which determine the optimum solution, is the branch and bound algorithm. However, this method has essential deficiencies, which do not permit to solve a traveling salesman problem for n35 40.

Publications touching the calendar modeling

Obviously, all diversity of problems of calendar simulation can not be presented by the listed above settings. This explains a lot of publications devoted to these problems. We consider some of them below.

The book [13] gives exposition of scheduling on an operationally - network models by means of the branch and bound algorithm, but also specifies the restricted possibilities of application of this method owing to high-cube computation. In the same work the probability operationally - network models are presented and analytical dependences for the elementary and multiphase Markov webs are obtained.

In the book [14], the simulation of graphs of loading of the equipment with usage of methods of minimization of the non-synchronism and cyclic ranking, resolving factors and index estimations is described. These methods can be effectively enough used for the solution of calendar problems in a full- scale production.

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The presence of the big number of random factors in technological processes is reason of construction of probability calendar models. So, in [15], the results of application of statistical methods for prediction of indexes of manufacturing activity are considered. The basis model is:

), , 0 .~ (

);

, 0 .~ ( ,

);

, 0 .~ ( ,

1 1























V N

V N S

t t

t

t t

t t t















 (1.14)

Wheret - a current value of the predicted process;

t – a current value of a trend;

t – a current value of a trend inclination angle;

St – a current value of the current factor;

t – observable noise;

t – perturbation of a trend;

t – a possible trend inclination angle.

Random components t, t, t are assumed independent and normally distributed with a zero value of average of distribution and known dispersions:

V e, V g, V d.

Method of the exponential suspended regression is used for prediction on the basis of the design model. In this method, the in-season factor is accepted as fixed and is ignored. However, in this case, the system is insensitive to noise and to unsteady errors that carries on to serious errors in the prognosis.

The carried out analysis of scheduling permits to present the following conclusions:

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- its reduction to more prime models decrease the extent of adequacy to real plant;

- the solution of the simplified models by means of the linear and dynamic programming methods and by the network and probability methods in most cases does not give satisfactory results.

1.2.1.2. Graph models

For the presentation of the technological processes structure and its generalized algorithms of control, graph models are frequently used. In these models, a set of the system units (blocks, manufacturing operations, states of control) are nodes of a graph, and material and information streams between units of the system are marked out by arcs (digraph). By means of the conversions of graph models the preliminary arrangement and calculation of the required technological equipment, an estimation of delivery capacity and loading of the equipment is carried out. Thus, the presentation of the technological processes as graphs [16] permits to create the generalized graph corresponding to the maximum loading of the equipment. The equation for the calculation of the maximum equipment loading can be written as follows:





















M

j N j

i ij N

i ij

a a S M

1 1

1

1  , (1.15)

where: M is amount of a graph nodes; N is number of details in treated group;

aij is a time that expends equipment for machining "i" detail on "j"

operations; _j is the coefficient which is taking into account extent of importance of underload of the "j" equipment.

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Graph models are visual models and easy for understanding process’s essence. But, their analytic form is complex for analysis and with the model extension the model’s visualization is decreasing.

1.2.1.3. Method of power balance

On the preliminary design stage of AMS for determination of the amount of equipment, the volumetric balance model or a method of power balance is used [17]. For such models, it is necessary that the common time is needed for machining of all size of emitted details, did not exceed common fund of a time in use of the equipment.

In this case, the amount of machinery frequently is determined, outgoing from an average exhaust stroke of details:

year use

avg N

К

T  F⁰ , (1.16)

where F0 – fund of a time in use of the equipment; Kuse– an activity factor of the equipment; Nyear – the annual program of turnout.

Calculations by such simplified models can be made only if it is beforehand known what operations are executed on the determined equipment.

More exact sampling of the equipment and also fixing of rigs behind determined an operation reduces a method of power balance to a problem of linear or discrete programming.

1.2.1.4. Method of rational resource allocation

The AMS rational resource allocation problems are close to a method of power balance. Here, the allocation of total amount of operations on units of rigs is considered [13]. In this case, input information is allocation of the

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labor-intensiveness obtained technological processes analysis. Reliability of machine tools is estimated with the account of factor of its planned use (i.).

Then, the problem of definition of a machinery optimum configuration is formulated as follows:

  

  

    





n

j

n

j

M

i

j j

x j ij ij ij ij

j

jx y i M y a k F j n x x N

C i j

1 1 1

, 0

0

; , 1 ,

; , 1 , 1

min; _ , (1.17)

where aij – labor-intensiveness of operations i on the universal unit; kij – relative productivity unit j for execution of operations i; Cj – average reduced for one year costs for operation unit j; yij – a share of labor-intensiveness type of work i with executed by unit j.

For the solution of discrete programming problems, the procedures of branches and bounds are used. For solving the big dimensions problem, the application of known integer programming methods is hampered and consequently heuristic algorithms are offered.

1.2.1.5. Regular schemes of algorithms

Language of the regular schemas of algorithms (RSA) permits [18]:

- visually, uniquely and strictly logically to describe operating and events by the way of set of algorithms;

- to realize equivalent transforming of algorithms from any description languages in RSA and on the contrary;

- to realize the identical transforming of algorithms for their minimization;

- accessible and easy to realize structural synthesizing of AMS;

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- rather simple to automate the process of analysis and synthesizing of interactive control systems.

The RSA language vocabulary is







 











X X Y Y 0 1 X Y

R , , , , , , , , , where Y = {yi} – a set of main operators of controlling’s algorithms;

X = {xk} – a set of branchings' conditions in algorithms;

 – empty operator, defines termination of algorithm;

1,0 – logical “true” and “false” in algorithms conditions;









X X Y

Y, , , – the signatures of RSA basic operators, they describe branching’ rules in the control algorithms (multiplying, conjunction, disjunction, iteration).

The signatures of basic operators make it possible adequately and function completely to describe all possible branching in the AMS control algorithms. Graphical representation of each basic signature is show in Table 1.

Thus, it is possible to receive internal formalizing models for simulation.

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Table 1 RSA basic operators Operations of

transferring

graphical representation

representation in a RAS

signature of operators Sequential

execution of the operators

3 2

1 Y Y

Y

R  

multiplying



Y

Parallel execution

of the operatives R



Y₁Y₂



conjunction



Y

Conditional branching

x 2 1 x

Y Y R(  )



 0 X 1

disjunction



X

Cyclic execution of the operators



1 2



x

x Y Y

R 



 0 X 1

iteration





Let’s consider application RSA on the example, which is shown in Fig. 3.

For this manufacturing process control operators (Y _i R^Y), where are: Y1

– delivery two screw to the assembly unit; Y2 – delivery flat to the assembly unit; Y3 – delivery rod to the assembly unit; Y4 – assembly electrical connector; Y5 – delivery electrical connector to the packaging unit; Y6 – pack 4 electrical connector in one pack; Y7 – delivery pack of electrical connectors to the customer.

Y2 Y3

Y1

Y2

Y1

Y2

Y1

Y1 Y2 X 0

1

0 1

X

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Branchings’ conditions (X _i R^X) are: X1 – there are less than 4 electrical connectors in the packaging unit; X2 – there are not enough packs of electrical connectors in the customer.

So, this manufacturing process can be described in RSA language like:

 

 

2 1 1

2

X X 7 6 5 4 3 2 X 1

X Y Y Y Y Y Y Y

R       ,

where



Y1Y2 Y3



are parallel delivery workpieces to assembly unit, because each delivery no depends from others.

 

¹

1

X 7 6 5 4 3 2

X Y1Y Y Y Y Y Y conditional branch in control algorithm.

As each pack contains four electrical connectors, it is impossible to pack if there are less than four connectors in the packaging unit. If packaging unit don’t have enough connectors – assembly one more.

The iteration

     

2 1 1

2

X X 7 6 5 4 3 2 X 1

X Y Y Y Y Y Y Y shows operations for make electrical connectors and pack them until customer don’t have enough packs.

Advantage of RSA is that the description of the control algorithms is realized as diagrams. It permits to present the work of logic of the equipment more demonstrable. Graphic representation of algorithm is easily translated in a semantic form. For work with semantic record of language, it is enough to use the elementary interpreter. Disadvantage of this language is that it permits to describe only control processes without a feedback.

Analytical methods permit to optimize industrial systems. Calculation of mathematical model can be made quickly. It permits to analyze of industrial system in short terms. However, the design of the mathematical model needs

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severe formalization of industrial system. In case of uncertainty of some parameters of industrial system, the formalization either is inconvenient, or absolutely is not possible. Time expenses for formalization in some cases exceed advantages from the fast performance of the analysis by mathematical methods. In case of change of structure of industrial system, it is necessary to alter the most part of mathematical model.

1.2.2. Imitation modeling methods

There are many imitation modeling (simulation) methods, but most of them are variation of Queuing theory or Petri nets.

Some accepted Petri nets extensions are:

- colored;

- hierarchy;

- prioritized;

- timed;

- stochastic.

1.2.2.1. Queuing theory

The methods of queuing theory (QT), first of all, treat to dynamic methods [19], [20]. AMS models designed on QT basis, can were used on early development cycles of the manufacturing system. By means of the given class models are determined a series of the important performances of AMS operation dynamic: carrying capacity, loading basic and optional equipment, sizes of queues etc. Obtained estimations are averaged and are used for description of the static mode of the system (asymptotic estimations).

General queuing theory concept

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In general, queuing systems may be characterized by complex input process, service time distribution and queue disciplines. In practice, such queuing processes and disciplines are often not amenable to analysis by analytics methods. But they can be analyzed using simulation methods like queuing models. Queuing model is commonly made in the context of packet switching networks like the Internet that are based on the store and forward principle. Typically, packets on their ways to their destinations arrive at a router where they are stored and further forwarded according to addresses in their headers. One of the most fundamental elements of queuing model is the single-server queue.

Shorthand notation is commonly used in the queue theory. It called Kendall notation [21], for such single queue models describes the arrival process, service distribution, the number of servers and the buffer size as follows: arrival process / service distribution / number of servers / waiting room

Commonly used characters for the first two positions in this shorthand notation are: D (Deterministic), M (Markovian – Poisson for the arrival process or Exponential for the service time), G (General), GI (General and independent), and Geom (Geometric)..

For example, M/M/1 denotes a single-server queue with Poisson arrival process and exponential service time with infinite buffer. M/G/k/k denotes a k-server queue with no additional waiting room accept at the servers with the arrival process being Poisson.

One of the important measures for queuing systems performance is the utilization – Uˆ . It is the proportion of time that a server is busy on average. In

Modelos de simulación para sistemas de manufactura

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