Strategy of manufacturing programme optimization in manufacturing system.
Balczak, Stanislav ; Hrubina, Kamil ; Wessely, Emil 等
1. INTRODUCTION
In the field of technological processes, the primarily goal of the
firms is accurate and reliable fulfilment of customers' demands and
according to such approach to production planning, the firms belong to
those using Just-in-Time manufacturing system. The manufacturing system
is required to accept customers' demands and at the same time to
fulfil the firm's targets, including cost minimisations. In the
analyzed firm, a manufacturing programme is elaborated based on
customers' requirements and to ensure cost minimization is a
demanding problem. By means of a mathematical model it is possible to
verify various strategies designs of manufacturing control as well as
the implementation of the manufacturing programme. Based on the analysis
of an actual manufacturing process and a mathematical description
resulting from it, a deductive, deterministic model is designed. The
advantage of a deductive model is that its utility is not limited by an
experiment extent. A deductive mathematical model applied to a
manufacturing process is limited by the rules, laws, eventually
manufacturing conditions valid in a given period of time.
2. OPTIMIZATION OF PRODUCTS
The manufacturing programme analyzed in the paper contains plastic
parts manufacturing used in white goods. Parts pressing are provided by
the most modern technology. The firm receives orders much time in
advance and completed products are supplied in regular ( relatively
short) time intervals in a volume that corresponds to the amount of
products ordered for the following week. Not later than one working day
before the beginning of the week (on Friday) the products have to be
prepared. Since the manufacturing is automated, it is possible to
optimize the products preparation for the customers and products taking
delivery by the customers. The aim is to minimize time (time cost) spent
on the order processing and delivery of ordered goods to the customers.
Optimization of products arrangement for a taking delivery by a customer
will be modelled by the Dantzig's Northwest Corner Method. We are
searching for such amounts [x.sub.ij] which have to be supplied from the
i-th manufacturing plant to the j-th consumer so that a total time cost
[q.sub.ij] is minimum, where time [q.sub.ij] is determined based on the
relation (1). The designed problem is a combination of the problem
related to the manufacturing planning and products distribution, where
the cost function will express the solution target--time consumption
(time cost) minimization with the products distribution. We will
minimize the cost needed to process orders [q.sub.ij], a total value is
expressed by the cost function
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
3. TIME SPENT TO PROCESS THE ORDER AND PRODUCT DISTRIBUTION
In the defined relation
q = e.(v/a) + y.ET + d + z.c + (b - i).c + r + {y - c) (1)
q is time spent to process a file card( known in practice as time
of a card circulation in a loop) (Macura, 2003). The variables meaning
and some standard values of variables of the relation (1): v time spent
on manufacturing, r possible repair, z number of preparatory plants, T =
v/A time of all ordered products, manufacturing (in practice called tact
time), A customers' total demands, a customer's demands, b
number of pieces /packages, c cycle, d version change, w 10% time needed
to change the version manufacturing time, E batch dimension, e = d/(wc)
batch dimension, e/b number of file cards batch dimension (Macura,
2005).
4. A GENERALIZED PROCEDURE OF A DEDUCTIVE MATHEMATICAL MODEL DESIGN
APPLIED TO A MANUFACTURING PROGRAMME
The model presented in the paper expresses substantial elements and
properties of a part of a manufacturing programme. It is not possible to
include all the properties because they may affect parts of the
manufacturing programme as well as unknown processes and effects. We
investigate the relations between the inputs and outputs of the
manufacturing programme. In the first stage of the model designing, we
analyze the manufacturing programme, in this case we determine the
substance of products supply to the customers based on the demand. We
specify the effects that are considered principal when designing the
mathematical model. We use simplifications of some events, make
decisions about individual phases and the extent in which they will
participate in the resulting model. The simplification phase is
expressed by means of several suppositions:
* the process is divided into simpler parts that are analyzed
individually taking into .consideration their mutual connections,
* we suppose independence on the effects of external temperature,
pressure, etc.
* we suppose that the elements of the manufacturing programme are
of a homogeneous and isotropic material,
* losses are neglected,
* in a certain defined area, nonlinear dependences are
linearization,
* investigated dependences between the values are of an empiric
character.
5. PROCESSES MODELING
The aim of the paper is to present the designing of mathematical
model by formulas (1) of their application to the solution of selected
optimization problems. The solution to the defined problem represented
by a model often consists in finding an extreme of the criterion
function, i.e. the point, eventually n-components of a vector in which
the function attains its maximum or minimum. Such a vector represents
the best variant of the problem under solution of the given set of
variants. In this sense we speak about optimization and optimization
methods. Thus, optimization in informatics means searching for "the
best solution to the defined problem". In case of incomplete
information about a composite system, eventually a regulated process,
this refers to the system with incomplete information, therefore the
methods of a new scientific branch--artificial intelligence--have to be
used. Models of operational research are systems which are a simplified
image of objective reality. They are of a mathematical character, i.e.
they contain equations, inequalities, differential equations and
differential-difference equations. Such modeling method belongs to
mathematical modeling; mathematical models themselves, considering the
form of uncertain variables occurrence, can belong to some of the
following categories of models: deterministic, stochastic, strategic,
adaptive and fuzzy. A criterion function an objective function with
these models represents the extent of the evaluation of the achieved
goal which was defined for the given system (Jadlovska et al (2005). The
methods and the means of artificial intelligence include:
* Solution methods for problems with constraints
* Genetic algorithms
* Neural networks theory
* Expert systems
* Methods of chaos theory
Nowadays, based on Darwin's theory of evolution, evolutionary
algorithms are being developed very fast. They are the essential
instrument of informatics and modern numerical mathematics applied to
the solution of complex optimization problems. (Hrubina & Jadlovska,
2002). When assessing a mathematical model, a real phenomenon is
simplified, schematized and the scheme obtained is expressed in
dependence on the phenomenon complexity by means of a selected
mathematical apparatus. A model has to reflect all most important
factors affecting the process, it has to provide a sufficiently true
description of both quantitative and qualitative properties of the
process that is being modeled. A mathematical description of the model
structure according to the character of the process is a system of
linear, non linear, differential or difference equations which reflect a
mutual influence of different parameters. In the mathematical
description of the equation of one type they do not exclude the
occurrence of equations of another type. Mathematical modeling is not in
contradiction to physical modeling, it is rather its replenishment.
Physical modeling is not determined to analyze specific properties of a
mathematical description, it is used to assess the objects adequacy
based on the comparison of the values of some determining complexes in
mathematical equations. With mathematical modeling, the process is under
investigation. Accordingly, various parameters of a mathematical model
that is modeled on a computer are changed. This enables to obtain
information about various variants of the investigated process very
fast. Within a reasonable time it is possible to realize optimum
variants of a model which means to carry out a mathematical model
optimization. Mathematical modeling is much cheaper than physical
modeling regardless it expresses money costs or time costs
6. CONCLUSION
The advantages of the manufacturing programme investigation by
means of the mathematical model lead to the assumptions of failure
elimination that could occur with the experiments, e.g. products
distribution. The model expressing the principal properties of the
manufacturing programme, eventually its part, satisfies the basic
criterion of a quality and applicability which is in accordance with a
real situation in manufacturing. In the following stage of designing a
general mathematical model of the process, a system of mathematical
relations and equations is created. A simulation programme requires a
selection of the method applied to the model equations solution, a
design of a suitable algorithm of a solution and a programme. An
important stage of the solution is the model verification, check-up
tasks solution, analysis of their results as well as the suitability
evaluation of their application to a given purpose. Based on the
complete procedure, it is obvious that the design of products
distribution scheduling is proposed in order to minimize the supply time
and increase the profit of the firm.
7. REFERENCES
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Vienna
Macura, D. (2003). Ordinary Differential Equations. 1. issue.
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