Mathematical Modelling as a First Stage for Logistics Systems Simulation
Mathematical modelling is presented as a principle for computer support usage in logistic processe modelling and simulation. Математические модели рассмотрены как основа для использования средств при моделировании логистических процессов. Математичні моделі розглянуто як основу для використання комп...
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2007
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| Cite this: | Mathematical Modelling as a First Stage for Logistics Systems Simulation / J. Grabara // Электронное моделирование. — 2007. — Т. 29, № 4. — С. 31-38. — Бібліогр.: 7 назв. — англ. |
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| citation_txt | Mathematical Modelling as a First Stage for Logistics Systems Simulation / J. Grabara // Электронное моделирование. — 2007. — Т. 29, № 4. — С. 31-38. — Бібліогр.: 7 назв. — англ. |
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| description | Mathematical modelling is presented as a principle for computer support usage in logistic processe modelling and simulation.
Математические модели рассмотрены как основа для использования средств при моделировании логистических процессов.
Математичні моделі розглянуто як основу для використання комп’ютерних засобів при моделюванні логістичних процесів.
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| first_indexed | 2025-11-30T15:20:41Z |
| format | Article |
| fulltext |
J. Grabara, PhD
Faculty of Management
Czestochowa University of Technology
(E-mail grabara@zim.pcz.pl)
Mathematical Modelling
as a First Stage for Logistics Systems Simulation
Mathematical modelling is presented as a principle for computer support usage in logistic
processe modelling and simulation.
Ìàòåìàòè÷åñêèå ìîäåëè ðàññìîòðåíû êàê îñíîâà äëÿ èñïîëüçîâàíèÿ êîìïüþòåðíûõ
ñðåäñòâ ïðè ìîäåëèðîâàíèè ëîãèñòè÷åñêèõ ïðîöåññîâ.
K e y w o r d s: mathematical modeling, simulation, logistics system.
Simulation is a technique for the all systems’ functioning imitation or for partic-
ular situation (economical, military or mechanical) imitation. The technique use
suitable models or tools in purpose of information achievement or didactic goals
obtaining. Simulation is defined as an art and science as well. The main aim of
simulation is whole system or particular process model creation, application of
changes and in effect researched system functioning estimation in various cir-
cumstances. Simulation is a perfect analytical technique that can really facilitate
problem-solving process (Fig. 1). Not every time simulation gives the best of
possible solutions but its effects are precious because they are results of many re-
peated research cycles. Achieved results become more real because of statistic
methods usage and computer graphics implication. It also helps non-profession-
als to understand the mechanisms of researched system.
Simulation is a technique that help determining which of possible solutions
is optimal in context of particular requirements. Financial aspect of researches
should not be forgotten, using simulation tools. It means that simulation usage
lets for optimal solution choice almost with no investments — very often there is
no necessity to build a physical system, previously. It is very important in case of
design and optimizing extraordinary expensive Flexible Manufacturing Sys-
tems. Now a day simulation usage as a tool for problem solving is one of the
most often used technique and all scientific spheres are practically the spheres of
its application. One can make analysis using simulation methods in almost all
spheres beginning from production systems, chemical and physical processes,
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 31
distribution and transportation systems, computer nets, on military projects
finishing.
The user imagination can be the only one barrier of simulation new applica-
tion. However it should be remembered that various scientific domains require
different research methods.
Mathematical modelling for simulation experiment elaboration. Distin-
guishing different ways in which a system and its’ processes might be studied, it
is rarely feasible to experiment with the actual system, because such an experi-
ment would often be too costly or too disruptive to the system, or because the re-
quired system might not even exist. For these reasons, it is usually necessary to
build a model as a representation of the real system and to study it as a surrogate
for the real system. M. Pidd [1] defines a model as follows.
A model is an external and explicit representation of part of reality as seen
by the people who wish to use that model to understand, to change, to manage,
and to control that part of reality in some way or another.
Mathematical models are ones of the models used in logistics processes
modelling and simulation. They are especially helpful because they make di-
rectly basis for usage of the computer tools in process of solution discovering.
They have the advantage of providing the possibility to evaluate numerous logis-
tics processes’ alternatives. One of the main disadvantages of mathematical
models is that they cannot precisely predict the feasibility of the modelled logis-
tics processes in real-life (in addition to the fact that such a model is only reliable
if the right assumptions underlying the model are made).
J. Grabara
32 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
System
Experiment on the
modelled system
Experiment o
real system
n the
Physical model
Mathematical
model
Analytical
solutions
Simulation
Fig. 1. System analysis techniques
Organization of modeling framework. 1. Define a modelling frame-
work. Starting with the real system, we first form a conceptual model of the sys-
tem that contains the elements of the real system that we believe should be in-
cluded in the model [2]. That is, one should identify all facilities, equipment,
events, operating rules and descriptions of behaviour, state variables, decision
variables, measures of performance, and so on, that will be part of the model.
Other authors use other terms. Van Hee [3] refer to ontology, defined as a system
of clearly defined concepts describing a certain knowledge domain. D.W.Wil-
son [4] refers to the world view or Weltanschauung, i.e. that view of the world
which enables each observer to attribute meaning to what is observed.
Consecutively, we must identify the relationships between the elements identi-
fied. From the conceptualisation of the system a logical model (or flow chart model)
is formed that contains the classification of and logical relationships among the ele-
ments of the system, as well as the exogenous variables that affect the system. A
modelling framework is defined as a set of basic modelling constructs and their pos-
sible relationships required to model the behaviour of the SC completely.
2. Devise a set of symbolic objects. The next step is to devise a set of corre-
sponding symbolic (i.e. formal) objects that can be used to represent the forego-
ing modelling constructs [5]. This includes identifying the integrity rules that go
along with those formal objects (i.e. how to use the objects).
3. Building the simulation model. Using the modeling framework and the
symbolic objects defined we develop a computer model, in a specified simula-
tion language, which will execute the logic described. Which aspects of the SC
are modelled depends on the demarcation of the SC. The decision on how much
of the real system should be included in the conceptual model to bring about a
valid representation of the real system must be jointly agreed upon by the simu-
lation analyst and the decision-makers.
Developing a simulation model is an iterative process with successive re-
finements at each stage. The basis for iterating between the different models is
the success or failure we have when verifying and validating each of the models.
Mathematical model [7]. Let us take a system of rolling mill into consider-
ation. It is assumed it consists of I assemblies and Ji passes in the i-th assembly.
The structure of the rolling mill can be presented in the matrix form:
� �
E ei j�
,
,
i = 1, ..., I, j = 1, ..., Ji, where J J
i I
i�
� �
max
1
. Matrix E elements are defined in the
following way:
e
j j J
j J j J
i j
i
i
,
,
.
�
� �
� � �
�
�
for
for
1
Consequently, matrix E elements are numbers of the passes. Non-existing
passes are marked by negative numbers. A pass parameter is its life which equals
the number of tons of material that can be passed through a new pass.
Mathematical Modelling as a First Stage for Logistics Systems Simulation
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 33
Let us assume the passes life matrix is given:
� �
G g i j�
,
, i = 1, ..., I, j = 1, ..., JI.
Elements g i j,
for j J i� have practical meaning in the above shown matrix.
Since for e ji j,
� � there are non-existent adequate passes, it can be assumed that
for such passes: e ji j,
� � . Rolling process consists in passing material through
the passes of the successive assemblies.
There are routes given for manufacturing certain products written in the ma-
trix form:
� �
D di n�
,
, n = 1, ..., N, i = 1, ..., I, where di n,
— the i-th assembly
pass number; N — the number of products. A route can omit some assemblies in
a general case. Should the n-th route not include an i-th assembly pass, then it is
assumed that di j,
= 0. A route consists of number of passes of the following as-
semblies from i = 1 to i = I. Some routes are shorter, which means they do not in-
clude passes from certain assemblies. The last pass is the decisive one for the
product type. Therefore, the maximal number of products can be the sum of all
the passes in all assemblies.
Let us assume there are M types of materials (charges) from which N types
of products can be manufactured. To allocate a charge to a product the allocation
matrix is given:
� �
A am n�
,
, m = 1, ..., M, n = 1, ..., N, where
a
n
m n,
�
1 if the -th product can be manufactured from the -th arg ,
otherwise
m ch e
0 .
�
�
Let us also introduce the charge vectorW wm� [ ], m = 1, ..., M, where wm —
the number of tons of the m-th charge type.
Let us introduce the rolling rate vectorV vn� [ ], n = 1, ..., N, where vn — the
number of the n-th product tons manufactured in a time unit.
Let us introduce the order vector Z zn� [ ], n = 1, ..., N, where zn — the num-
ber of tons of the n-th type product in state k – 1.
The order vector changes after each decision about production:
z
z x n a
z n a
n
k n
k
n
k
n
k
�
� �
�
�
�
�
1
1
, ,
, ,
if
if
where xn
k
— the number of tons of product a.
The billet mill state is defined as the matrix
� �
S si j�
,
, i = 1, ..., I, j = 1, ..., Ji,
where si j,
— the number of tons of material passed through the j-th pass of
the i-th assembly and for J j Ji � � we can write s ji j,
� � . State matrix ele-
ments must satisfy the condition: 0 � �s gi j i j, ,
, 1� �j J i . Initial state S 0
is
given. The equation of state of the production line takes a general form: S k
�
�
�f S x bk
n
k
( , , )
1
, where b — the number of an assembly assigned to replacement.
J. Grabara
34 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
The equation of state in case of production can be presented as follows:
s
s
i j
k
i j
k
,
,
,
�
�1
if material is not passed through the -th pass
of the -th assembly of rolls,
j
i
si j
k
,
�
�
1
min ( , ) .p za
k
n
k� �
�
�
1 1
otherwise
In case of the b-th assembly replacement the equation of state takes the form:
s
s x i b
i b
i j
k i j
k
n
k
,
,
, ,
, .
�
�
�
�
�
�1
0
if
if
Replacement brings about the opportunity for restarting production. To be
able to carry out further calculations the pass matrix of the rolling mill is in-
troduced: P pi j� [ ]
,
, i = 1, ..., I, j = 1, ..., Ji, where p g si j i j i j, , ,
� � , i = 1, ..., I,
j =1, ..., Ji; p ji j,
� � , j J Ji� �1,..., .
Having finished the rolling stage the billet mill flow capacity is low. Most
often, the route roll flow capacity of each product, where n = 1, …, N, equals
zero. To start the rolling stage the most worn out rolls are to be replaced by new
ones. On the basis of state S k–1
the roll pass matrix can be calculated: P k �
�
1
�
�
[ ]
,
pi j
k 1
. Residual pass of assemblies will be calculated as shown:
R pi i j
j
J i
�
�
� ,
1
.
According to the heuristic algorithm the l-th assembly is to be replaced on condi-
tion that
� � � �
�
j
i j
k
l ip R R( ) ( min )
,
1
0 .
During a rolling process, rolling time is determined: t x vn n n� / . Each assembly
replacement time cr is given as an element of the vector C.
Let us introduce the tolerance matrix
� �
H hi j�
,
, i = 1, ..., I, j = 1, ..., J,
where hi j,
— the tolerance of the j-th pass of the i-th assembly of rolls. In case of
allowing for the tolerance matrix, the equation of state takes the form:
s
s i b p h
i b
i j
k i j
k
i j
k
i j
,
, , ,
, ( ) ( ),
, ( ) (
�
� �
� �
� �1 1
0
if
if p hi j
k
i j, ,
).
�
�
�
�
1
The assembly regeneration time coefficient � is introduced. It means how
many units of the assembly flow capacity can be regenerated within the time
Mathematical Modelling as a First Stage for Logistics Systems Simulation
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 35
unit. This can be written �
�
�
maxPi
i
, i = 1, ..., I, where � i — the i-th assembly
regeneration time.
Simulators. The above assumptions are taken into account. They are the ba-
sis for creating simulators. Fig. 2 presents a general simulator of production and
replacement of rolls. The production process may be simulated in detail by
means of a production simulator shown in Fig. 3. The replacement process itself
will be checked by a simulation device.
To built simulators the following data must be input: I, J, M, N, �, A, E, H,
G, D, S0
, W, Z, V, C as well as the vector of optimization indexes Q and the al-
lowable time of the order realization T.
Summary. Modelling and simulation can make the research time and solu-
tion achieving shorter and more effective. Because of their possibilities and abil-
ity for verification of many alternative solutions for logistics processes design
and managing it is possible to choose the best solution without building the
J. Grabara
36 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
u 0
k 0
Is production
possible?
Replacement
S
k
=f (S
k �1
, i)
Qk
=Qk�1
+�Q
k
Production process
Sk
= f (S k�1
, n)
Qk
=Q k�1
+�Q
k
T
k
< T
Report
k k+1
u u +1
No No YesYes
No Yes
Input data
Scalars: I J M N, , , ,
Vectors: Q ,W V Z C, , ,
Matrixes:A B G D S, , , ,
0
,H
T
k
< T
�
�
�
�
Fig. 2. The block diagram of production and replacement
structures that is very expensive and time consuming. The computer support enable
verification of processes and plant structures in circumstances very similar to reality
that allows for avoiding many expected and unexpected situations in future opera-
tions. It can be stated that modelling and simulation are useful and effective using
computer support for processes improvement but It should be remembered that
there is some disadvantages. They are as follows: user large experience, reality sim-
plification or problems with results’ credibility estimation.
Mathematical Modelling as a First Stage for Logistics Systems Simulation
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 4 37
Data: I J M N T, , , ,
Q W Z V, , ,
A B D G S, , , ,
0
P
0
=G�S
0
R;
0
=f P(
0
, );D T
0
=0 Q;
0
=0
k 1
0
1
�
�
�
k
n
n
r
0
1
�
�
�
k
n
n
z
�
�
�
�
M
m
k
mnm
n
wa
1
1
,
0
Choice n =?
k
n
kk
k
n
k
n
k
m
M
m
nm
k
n
xQQ
rzwax
��
�
�
���
�
1
111
1
,
,,min
� �
� �
n
k
nkk
kk
kk
kk
kkk
k
n
kk
v
x
TT
DPfR
SGP
WWW
ZZZ
xSfS
��
�
��
���
���
�
�
�
�
�
1
1
1
1
,
,
TTk
�
Report
Replacement of rolls
k k +1
Yes
Yes
Yes
No
No
No
Yes
No
�
�
(
(
Fig. 3. The block diagram of the production process
Ìàòåìàòè÷í³ ìîäåë³ ðîçãëÿíóòî ÿê îñíîâó äëÿ âèêîðèñòàííÿ êîìï’þòåðíèõ çàñîá³â ïðè
ìîäåëþâàíí³ ëîã³ñòè÷íèõ ïðîöåñ³â.
1. Pidd M. Just modelling through: a rough guide to modelling // Interfaces. — 1999.—29,
¹2.— P. 123—138.
2. Hoover, S. V., Perry, R. F. Simulation; a problem-solving approach.— Reading, MA: Addi-
son-Westley, 1989.—776 p.
3. Van Hee K. M. Information systems engineering — a formal approach. — Cambridg: Cam-
bridge University Press, 1994.— 442 p.
4. Wilson B. W. Systems: concepts, methodologies and applications. Second ed. — Chichester:
John Wiley & Sons, 1993. — 337 p.
5. Date C. J. An introduction to Database Systems. 7-th Ed. Reading, MA: Addison-Westley,
2000. — 552 p.
6. Bucki R,. Marecki F. Computer-Based Simulators of Logistics Systems. — In: Selected
problems of IT Application. — Warsaw: WNT, 2004. — P. 103—111.
7. Nowakowska-Grunt J. Inventory and Capital in Dairy Industry Companies — In: Processes of
Capital Supply in Production Enterprises. H. Ko�cielniak ed.— Cz�stochowa: Wyd. WZP
Cz�stochowska, 2006. — P. 110—115.
Ïîñòóïèëà 30.03.07
J. Grabara
38 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 4
|
| id | nasplib_isofts_kiev_ua-123456789-101782 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-11-30T15:20:41Z |
| publishDate | 2007 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Grabara, J. 2016-06-07T09:01:35Z 2016-06-07T09:01:35Z 2007 Mathematical Modelling as a First Stage for Logistics Systems Simulation / J. Grabara // Электронное моделирование. — 2007. — Т. 29, № 4. — С. 31-38. — Бібліогр.: 7 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101782 Mathematical modelling is presented as a principle for computer support usage in logistic processe modelling and simulation. Математические модели рассмотрены как основа для использования средств при моделировании логистических процессов. Математичні моделі розглянуто як основу для використання комп’ютерних засобів при моделюванні логістичних процесів. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Mathematical Modelling as a First Stage for Logistics Systems Simulation Математическое моделирование как первый этап компьютерного моделирования логистических систем Article published earlier |
| spellingShingle | Mathematical Modelling as a First Stage for Logistics Systems Simulation Grabara, J. |
| title | Mathematical Modelling as a First Stage for Logistics Systems Simulation |
| title_alt | Математическое моделирование как первый этап компьютерного моделирования логистических систем |
| title_full | Mathematical Modelling as a First Stage for Logistics Systems Simulation |
| title_fullStr | Mathematical Modelling as a First Stage for Logistics Systems Simulation |
| title_full_unstemmed | Mathematical Modelling as a First Stage for Logistics Systems Simulation |
| title_short | Mathematical Modelling as a First Stage for Logistics Systems Simulation |
| title_sort | mathematical modelling as a first stage for logistics systems simulation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101782 |
| work_keys_str_mv | AT grabaraj mathematicalmodellingasafirststageforlogisticssystemssimulation AT grabaraj matematičeskoemodelirovaniekakpervyiétapkompʹûternogomodelirovaniâlogističeskihsistem |