Arch. Min. Sci., Vol. 56 (2011), No 2, p. 161–178

Arch. Min. Sci., Vol. 56 (2011), No 2, p. 161–178

Arch. Min. Sci., Vol. 56 (2011), No 2, p. 161–178
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Electronic version (in color) of this paper is available: http://mining.archives.pl
EDYTA BRZYCHCZY*
THE PLANNING OPTIMIZATION SYSTEM FOR UNDERGROUND HARD COAL MINES
SYSTEM WSPOMAGAJĄCY PLANOWANIE ROBÓT GÓRNICZYCH
W KOPALNIACH WĘGLA KAMIENNEGO
This paper presents a planning optimization system (POS) for underground hard coal mines. This POS
is a computer program called the “CPRG.SYS”, which is based on CPRG method (Metoda Centralnego
Planowania Robót Górniczych). CPRG method is developed for modeling and the optimization of mining
works in a multi-mine enterprise.
The main elements of the presented system are: a database of recent and planned longwall faces,
a knowledge base allowing automatic composition of longwall complexes, and a selection of equipment
with reference to working conditions. An integral part of the system includes calculation modules allowing the analysis of multiple variants, using network techniques and suitably prepared evolutionary
algorithms as well.
Keywords: CPRG method, planning, mining works, hard coal, knowledge base, evolutionary algorithm
W artykule przedstawiono propozycję nowoczesnego systemu wspomagającego proces planowania
robót górniczych w podziemnych kopalniach węgla kamiennego CPRG.SYS.
W skład systemu wchodzą: baza danych o wyrobiskach prowadzonych w przeszłości i wyrobiskach
planowanych, baza wiedzy umożliwiająca zautomatyzowanie procesu tworzenia kompleksów ścianowych
oraz dobór wyposażenia do warunków wyrobisk. Zasadniczą częścią systemu są moduły obliczeniowe
umożliwiające analizę wariantów z wykorzystaniem technik sieciowych oraz odpowiednio przygotowanego algorytmu ewolucyjnego.
Podstawą działania proponowanego systemu jest metoda Centralnego Projektowania Robót Górniczych (CPRG), której główne założenia opisano m.in. w (Brzychczy, 2008). Metoda ta przeznaczona
jest do modelowania i optymalizacji robót górniczych wraz z ich wyposażeniem w poszczególnych
polach eksploatacyjnych zakładów produkcyjnych (nazywanych dalej kopalniami) w wielozakładowym
przedsiębiorstwie górniczym.
Dane wejściowe do metody CPRG obejmują dane dotyczące robót górniczych prowadzonych
w przeszłości w kopalniach przedsiębiorstwa, robót planowanych do wykonania oraz dane dotyczące
wyposażenia jakim dysponuje, bądź będzie dysponować w analizowanym okresie to przedsiębiorstwo.
Do robót objętych analizą i modelowaniem przyjęto: roboty eksploatacyjne (E), oraz prace zbrojeniowe
(ZB), prace likwidacyjne (LIK).
* AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MINING AND GEOENGINEERING, DEPARTMENT
OF ECONOMICS AND MANAGEMENT IN INDUSTRY, AL. MICKIEWICZA 30, 30-059 CRACOW, e-mail: [email protected]
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Podstawowe charakterystyki techniczno-ekonomiczne planowanych robót górniczych stanowią:
• dla robót eksploatacyjnych:
o czas trwania robót t,
o postęp robót (pos)
o koszt wykonania robót (KosztE),
o wielkość wydobycia netto (Wden),
o wartość jednostkowa węgla pochodzącego z wydobycia (JWW),
• dla robót zbrojeniowych i likwidacyjnych
o czas oczekiwania na zestaw ścianowy (to),
o czas trwania robót t,
o czas ich wykonania (TZB/LIK),
o koszt wykonania robót (KosztZB/LIK).
Czas trwania robót eksploatacyjnych jest pochodną postępu tych robót i wybiegu ściany. W metodzie
przyjęto założenie, że postęp ten (wyrażony w metrach na dobę [m/d]) jest zmienną losową. Czasy trwania
robót zbrojeniowych i likwidacyjnych są wielkościami zdeterminowanymi. Koszt wykonania robót eksploatacyjnych (wyrażony w postaci funkcji) związany jest z postępem tych robót, natomiast w przypadku
robót zbrojeniowo-likwidacyjnych wiąże się z długością ściany. Wyznaczone charakterystyki (3) – (7)
pozwalają na odwzorowanie planowanych robót na sieci zdeterminowanej i wykonanie obliczeń według
opracowanego modelu matematycznego z wykorzystaniem symulacji Monte Carlo.
W wyniku przeprowadzenia kolejnych kroków metody uzyskane zostaje najlepsze rozwiązanie pod
względem zadanego kryterium (lub kilku kryteriów). Kryteria te dotyczą wyników produkcyjnych jak
i ekonomicznych przedsiębiorstwa górniczego i określono je następująco:
• minimalizacja odchylenia wartości oczekiwanej rozkładu wydobycia węgla handlowego dla
przedsiębiorstwa górniczego w badanym okresie od wielkości planowanych [t/okres],
• minimalizacja odchylenia standardowego rozkładu wydobycia węgla handlowego dla przedsiębiorstwa górniczego w badanym okresie [t/okres],
• minimalizacja wartości oczekiwanej rozkładu kosztu jednostkowego sprzedanego węgla dla
przedsiębiorstwa górniczego w badanym okresie [zł/t],
• minimalizacja odchylenia standardowego kosztu jednostkowego sprzedanego węgla dla przedsiębiorstwa górniczego w badanym okresie [zł/t],
• maksymalizacja wartości oczekiwanej rozkładu jednostkowego wyniku na sprzedaży węgla dla
przedsiębiorstwa górniczego w badanym okresie [zł/t],
• minimalizacja odchylenia standardowego rozkładu jednostkowego wyniku na sprzedaży węgla
dla przedsiębiorstwa górniczego w badanym okresie [zł/t],
Proponowany system składa się z następujących elementów:
• bazy danych o wyrobiskach prowadzonych w przeszłości,
• bazy danych o wyrobiskach planowanych i wyposażeniu,
• bazy wiedzy zawierającej reguły umożliwiające dobór wyposażenia do warunków górniczo-geologicznych przodka ścianowego oraz reguły zestawiania ze sobą maszyn i urządzeń (opracowane
z wykorzystaniem technik Data Mining),
• modułów realizujących odrębne algorytmy obliczeniowe.
Zasadniczym elementem systemu jest moduł, w którym działa odpowiednio zaprojektowany algorytm
ewolucyjny, przedstawiony na rysunku 1.
W pracy przedstawiono podstawowe elementy algorytmu ewolucyjnego: przyjętą reprezentację,
operatory genetyczne, funkcję oceny oraz sposób selekcji osobników do następnych pokoleń.
W efekcie działania systemu zostaje wybrane najlepsze, z punktu widzenia przyjętego kryterium,
rozwiązanie, na podstawie którego tworzony jest harmonogram robót górniczych dla całego wielozakładowego przedsiębiorstwa górniczego wraz z pełną charakterystyką wyników produkcyjnych oraz
ekonomicznych dla badanego okresu. Przykład wyników dla pojedynczej kopalni przedstawiono odpowiednio na rysunkach 3 i 4.
W podsumowaniu zwrócono uwagę na konieczność powstawania dedykowanych rozwiązań informatycznych wspierających modelowanie i optymalizację robót górniczych oraz wykorzystywanie w tym
obszarze doświadczeń z prowadzonej działalności górniczej.
Słowa kluczowe: metoda CPRG, planowanie, roboty górnicze, węgiel kamienny, baza wiedzy, algorytm
ewolucyjny
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1. Introduction
The planning of activities is one of the most important processes performed in each production company. Assumptions according to future production and solutions related to realized
production process often have a critical meaning for the company’s market existence. In the case
of a hard coal mining enterprise, the following elements should be taken into consideration:
• a useful minerals extraction process is a specific production form, realized in characteristic
conditions,
• production output is an effect of properly planned and executed mining works,
• an extraction process is accompanied with some risk and uncertainty, which should be
taken under consideration in production planning,
• cost-consuming equipment should be effectively used during production process.
Restructuring activities done in the last two decades in Polish hard coal mining works have
considerably improved operations, resulting in an increase in the concentration of production.
These activities are the result of changes introduced in the technical and spatial structure of the
mine. Reduction of the longwall number, simplification of the working structure and the geological-mining conditions of production, require careful actions and decisions on the part of the
designer, due to plans for future mining works.
The character of internal and external conditions of the extraction process is accompanied
with some characteristic sources of risk and uncertainty (Smith, 1999; Dunbar et al., 1998; Summers, 2000) and has significant influence on prepared production plans. The most often specified
sources of uncertainty in mining projects are (Kazakidis & Scoble, 2003):
• internal sources dictated by the deposit itself (i.e. grade distribution, ground conditions,
workforce, equipment, infrastructure, etc.)
• external sources determined by outside considerations (i.e. market prices, environmental
conditions, political risk, country risk, community relations, industrial issues legislation,
etc.).
Stochastic models for individual longwall face output were proposed in (Snopkowski, 2009).
A single trial taking into consideration the random character of mining works in hard coal mine
was undertaken in modeling and optimization method with an application of stochastic networks
(Brzychczy, 2006), in which the aspect of equipment selection in planned longwall faces was
also included. This trial forms the basis for the CPRG method, which is an integral element for
a planning optimization system for multi-mine enterprises (Brzychczy, 2008).
2. CPRG method
The developed method refers to mining works planning (especially extraction works), with
equipment selection for longwall faces in the multi-mine enterprise.
The issue of equipment selection for planned longwall faces in a mining enterprise can be
described as a quest for a longwall equipment matrix, for which the value of the objective function is optimal.
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This matrix may take the following form:
(1)
where:
Sn — the longwall face,
Zz — the longwall complex of machinery and equipment,
βzn — the indicator of involvement of longwall complex Zz in the longwall face Sn, taking
values:
β = 0 — the longwall complex will not work in planned longwall face,
β = 1 — the longwall complex will work in planned longwall face.
Derivative of shown matrix (configurations of longwall faces and equipment) are different
rates of the longwall advance. Taking into consideration the risks inherent in the mining process,
it was assumed that rate of the longwall advance is a random variable with a specified probability
density function.
It can be therefore formulated the following research problem: What equipment can/should
be used in planned longwall faces and what can/should be the rates of the longwall advance
according to optimal value of the objective function?
To solve the formulated research problem an appropriate mathematical model was constructed
that allows simulation and evaluation of various variants with an ultimate selection of the best
solution during the optimization procedure.
Input data of the CPRG method comprise data referring to mining works executed in the
past in the mining enterprise, planned longwall faces and data referring to the current or planned
equipment within the analyzed period.
The following works are included to the analysis and modeling: extraction works (E), longwall face equipment installation (ZB) and longwall face equipment removal (LIK).
Basal technical and economical characteristics of mining works comprise:
• For extraction works:
o Duration of works t, between start date and finish date, [month]
o Rate of the longwall advance [m/month]
o Cost of works [PLN/month],
o Net output [t/month],
o Unit value of extracted coal [PLN/t],
• For equipment installation and removal in longwall face:
o Time of waiting for longwall equipment [month],
o Duration of works t, between start date and finish date [month],
o Time of the works execution [month],
o Cost of works [PLN/month].
165
Duration of extraction works is a derivative of the longwall advance and length of the
longwall panel:
(2)
where:
L — the length of the longwall panel [m],
pos — the rate of the longwall advance [m/months].
Duration of the equipment installation and removal in the longwall face is a determined
value.
Cost of extraction works (expressed in form of a function) depends on the rate of the longwall advance, however, in case of equipment installation and removal works, it depends on the
longwall face length.
For extraction works in longwall face Sj in mine k, cost function is assumed as follows:
KESkj = f (pos) = B * pos + C
(3)
where:
B — the coefficient of the cost related to the longwall advance, [PLN/m],
C — the coefficient of the cost unrelated to the longwall advance, [PLN/month].
For the installation of longwall face equipment and the removal of longwall face equipment
the following cost function is assumed:
KZB(LIK)S kj = f (b) = I * ls + J
(4)
where:
I — the coefficient of the cost related to the longwall face length, [PLN/m/month],
J — the coefficient of the cost unrelated to the longwall face length, [PLN/month],
ls — the length of the longwall face Skj [m].
Gross output from extraction works is given by the following equation:
WbESkj = ls * h * γ * pos
(5)
where:
h — the height of the longwall face [m],
γ — the coal bulk density [t/m3].
Net output from extraction works is given by the following equation:
WnES kj = Wdeb * ke
where:
ke — the coefficient of extraction losses.
(6)
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The unit value of extracted coal is given as:
(7)
where:
Cw
Q
S
A
—
—
—
—
the price of base coal [PLN/t],
the coal calorific value [KJ/kg],
the coal sulphur value [%],
the coal ash grade value [%].
Calculated values allow projection of the planned works on determined network model, as
well as execution of more calculations according to mathematical model. During the Monte Carlo
simulation, the following values are determined for each variant of the mining works:
• the expected value of monthly net output distribution for the mining enterprise within
analyzed period:
[SWMPG1, SWMPG2, ,..., SWMPGlo]
(8)
where:
lo — the number of months in the analyzed period,
SWMPGi — the expected value of net output in the month i, calculated as:
(9)
and:
n — the number of iterations in Monte Carlo simulation, i = 1,2, ..., N,
WMPGi — monthly net output in the mining enterprise, given by following equation:
(10)
where:
lk — the number of mines in the mining enterprise,
WMEki — monthly net output in the mine k, calculated as:
(11)
and:
m — the number of longwall faces in the mine k.
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• the standard deviation of monthly net output distribution for the mining enterprise within
analyzed period:
[OWMPG1, OWMPG2, ,..., OWMPGlo]
(12)
where:
(13)
• the expected value of monthly cost of the mining works distribution for the mining enterprise within analyzed period:
[SKMPG1, SKMPG2, ,..., SKMPGlo]
(14)
where:
(15)
and:
KMPGi — the cost of the mining works in the mining enterprise in month i, calculated
as follows:
(16)
where:
KMki — the cost of the mining works in the mine k in month i:
(17)
where:
KPCk — the cost related to other cost centers (underground transport, ventilation,
electrical, mechanical, shafts, mechanical processing, other) in the mine k
[PLN/month]:
KPCk = Gk * WMEk + Hk
(18)
where:
Gk — the coefficient of the cost related to the net output of mine k, [PLN/t],
Hk — the coefficient of the cost unrelated to the net output of mine k [PLN/month].
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• the standard deviation of monthly cost of the mining works distribution for the mining
enterprise within the analyzed period:
[OKMPG1, OKMPG2, ,..., OKMPGlo]
(19)
where:
(20)
• the expected value of monthly unit cost of the sold coal distribution for the mining enterprise within the analyzed period:
[SKJPG1, SKJPG2, ,..., SKJPGlo]
(21)
where:
(22)
and:
(23)
• the standard deviation of monthly unit cost of the sold coal distribution for the mining
enterprise within the analyzed period:
[OKJPG1, OKJPG2, ,..., OKJPGlo]
(24)
where:
(25)
• the expected value of monthly unit profit on the sold coal distribution for the mining
enterprise within the analyzed period:
[SWJPG1, SWJPG2, ,..., SWJPGlo]
(26)
where:
(27)
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and:
WJPGi — monthly unit profit on the sold coal in the mining enterprise, [PLN/t],
calculated as:
WJPGi = CJPGi – KJPGi
(28)
where:
(29)
and:
WRMPGi — value of coal from extraction works in the mining enterprise, [PLN/month]
calculated as follows:
(30)
where:
WRMki — value of coal from extraction works in mine k:
(31)
• the standard deviation of monthly unit cost of the sold coal distribution for the mining
enterprise within the analyzed period:
[OWJPG1, OWJPG2, ,..., OWJPGlo]
(32)
where:
(33)
When defining the above characteristics it has been assumed that:
• the cost of mining works was increased by the remaining production costs,
• the volume of the output equals the volume of coal sold.
The characteristics (8), (12), (14), (19), (21), (24), (26), (32) are the base of selection of the
best solution for planned mining works in the mining enterprise.
The optimization procedure in CPRG method is based on suitably designed evolutionary
algorithm (basic terms are given in appendix). Its main elements are:
• problem representation
Single individual of the population is expressed with chromosome (marked as XO), which
is shown in Table 1.
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TABLE 1
Chromosome XO
S111
S121
…
S1j1
S211
S221
…
S2j1
…
Sk11
Sk21
…
Skj1
S112
S122
…
S1j2
S212
S222
…
S2j2
…
Sk12
Sk22
…
Skj2
…
…
…
…
…
…
…
…
…
…
…
…
…
S11m
S12m
…
S1jm
S21m
S22m
…
S2jm
…
Sk1m
Sk2m
…
Skjm
CP11
CP12
…
CP1j
CP21
CP22
…
CP2j
…
CPk1
CPk2
…
CPkj
K1
K2
PG
…
Kk
Positions of genes in the chromosome correspond to longwall faces Skj, in which extraction is planned. Longwall faces are combined in production flows CPkj, which, in turn,
belong to defined mines K, forming multi-mine enterprise PG.
Gene value corresponds to equipment (in form of longwall complex), which can be used
in conditions of longwall face Skj. According to character of the formulated problem
(scheduling of extraction works in time and space), crossing operator was omitted, and
suitably prepared mutation operator was applied.
• the mutation operator
Taking under consideration specific character of the formulated problem, characteristic
mutation matrix (XM) was introduced as input parameter of the algorithm. Distribution
functions of discrete variable describing possibility of application of equipment in longwall
face (Skj) are saved in suitable columns of the matrix. The variables are determined on the
basis of experiences from works executed in similar mining and geological/organizational
conditions. At the stage of population analysis and evaluation of individuals, equipment
in genes are represented by distribution of the longwall advance, which is determined for
each longwall face and equipment on the basis of examination of similarity of designed
extraction works and works from the past.
• the fitness evaluation
Objective function being a basis of the fitness estimation of individuals in the population,
is determined by a designer having possibility of subjective selection of one or more
criterions. Its formula could be expressed as follows:
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(34)
where:
WMPLi — planned monthly net output in the mining enterprise [t/month].
• selection of individuals
In the developed algorithm, elite selection was chosen as the method of selection of
individuals.
• end of the algorithm
Algorithm end is conditioned with lack of quality improvement of the best individual
during assumed number of generations or determined number of iterations.
• unacceptable individuals
Individual in whom time of waiting for longwall equipment is longer than determined
value (tomax) is unacceptable.
Scheme of designed evolutionary algorithm is presented on Fig. 1.
The developed method is intended to support designers in the selection of equipment for the
planned longwall faces in the multi-mine enterprise. The method can also be used in assessing
the options for opening the deposit and the order of mining works in the new seams on the new
levels, taking into account the aspect of equipment selection.
The method described above was the basis for CPRG.SYS system, which is presented in
the next section.
172
Fig. 1. Evolutionary algorithm in CPRG method
173
Fig. 1. Evolutionary algorithm in CPRG method (continued)
3. Basic elements of the CPRG.SYS system
The basic elements of the system are:
1. databases:
• database of executed works (BPST) comprising monitoring of production results (in
form of the rate of longwall advance), and monitoring of economical data (in form of
working costs) achieved during execution of mining works in determined mining and
geological conditions, with use of suitable equipment,
• database comprising characteristics of planned longwall faces (BFTR), accessible (or
planned) equipment, including other data needed for proper assessment of designed
values.
2. knowledge base comprising rules allowing selection of the equipment according to mining
and geological conditions of the longwall face, as well as rules for combining machines
and devices in longwall complexes (developed with use of Data Mining techniques).
Relations between extraction conditions and reached rate of the longwall advance are
also included in the database.
3. modules realizing separate calculation algorithms:
• MODULE I – module, in which production flows are defined on the basis of planned
works data, including start dates of production flows. Module is called MPRIME.
(Production flow comprises longwall faces between which time relations occur).
174
• MODULE II – module of production means allocation, in which, on the basis of longwall conditions and equipment data, forming of longwall complexes and equipment
allocation matrixes (MAW) are developed. Module is called MASP.
• MODULE III – equipment analysis module, in which, on the basis of developed MAW
matrix, so called modified matrix of the equipment allocation (MAF) is determined,
where possibility of application of given equipment in individual longwall faces is
estimated. This module comprises also analysis of similarity of longwalls conditions
(with use of comparative analysis), which is a base for assessment of rate of the longwall advance, which was achieved in similar mining and geological conditions in the
past. Module is called MAW.
• MODULE IV – network module, in which transformation of introduced input data
into description of nodes and arcs of determined network model, as well as illustration
of time relations between longwall faces. Module is called MS.
• MODULE V – calculation module, in which development of characteristics in chosen
aspect (working, production flows, mine, mining enterprise) is determined, including
defined time period, being coefficients of fitness function for evaluation of individuals
in designed evolutionary algorithm. Module is called MO.
• MODULE VI – evolutionary algorithm module, based on the other modules, having
fundamental meaning for the system operation. Module is called MAE.
Operation of the system and the connections between the modules are presented in the next section.
5. CPRG.SYS system operation
Suitably completed knowledge base is an element necessary for the system operation. Rules
for combination of machines and devices in longwall complexes, rules for selection of equipment
with reference to longwall face conditions, as well as relations between them and possible production results are described in this base. The rules are determined on the basis of data referring
to works executed in the past with use of Data Mining techniques (decision trees and association
rules). The functional scheme of CPRG.SYS is presented on Fig. 2.
Two next system operation stages are started at the moment of production flows defining
(module MPRIME). Data referring to designed longwall faces, production flows and mines are
relocated in place of suitable genes of the individual chromosome XO. At the same time, longwall complexes are formed from accessible equipment, and equipment allocation matrix (MAW)
is defined. The next step comprises defining of the probability of application of the equipment
in longwall faces – so called, modified allocation matrix (MAF). This matrix constitutes initial
form of mutation operator (XM) for developed MAE module. Completion of these stages is
followed by initialization of evolutionary algorithm population and next stages. In order to evaluate individuals and calculate necessary characteristics, suitable network model, which project
planned mining works in time, is created in MS module. Recalculation of designed network
model and calculation of fitness function value is made in module MO. After fitness function
value calculation the next selection stage is realized.
In the effect of the system operation, the best (or nondominated) solution from the point of
view of the assumed criterion is selected. This solution is a base for mining works scheduling
for whole multi-mine enterprise, with full characteristics of production output for the analyzed
period. An example of results for a single mine is shown in Fig. 3 and Fig. 4.
175
BPST
Knowledge
base
BFTR
MPRIME
MASP
MAW
MAE
MS
MO
Fig. 2. Functional system CPRG.SYS
Fig. 3. Monthly net output in mine X for chosen solution
RESULTS
176
Fig. 4. Equipment matrix for planned longwall faces in chosen solutions with longwall advance
6. Results and Discussion
Complexity and characteristics of elements being a planning subject in hard coal mines exclude application of a standard software (for example commercial systems of the ERP class) within
the area of technical production planning, thus suitable, dedicated tools should be implemented.
The systems developed for needs of mining works scheduling (Gembalczyk et al., 1990;
Mastej, 1994; Karbownik & Kosiński, 1997; Karbownik & Tchórzewski, 2007; Dzedzej &
Nowicki, 2008) have rather statistical character, they require detailed data, determined by the
user. These programs do not take advantage in automatic manner from knowledge about process.
Stage of future works modeling is usually limited and time-consuming (number of data should be
introduced manually), and it doesn’t take under consideration the randomness of mining works
(for example in form of random variables distribution). Because the programs in question are
rather used for scheduling, they do not posses advanced optimization procedures.
Presented planning optimization system for underground hard coal mines constitutes modern
solution within this area. CPRG.SYS system is a solution allowing modeling and optimization
of extraction works in a hard coal mines, taking into account the aspect of uncertainty and risk
and selection of equipment for the planned mining works. Actually, modules I, IV, V, VI have
been programmed (Malinowski et al., 2010). Modules II, III and the knowledge base will be
programmed during realized research project no N N524 468939.
Hard coal production is a specific process, and it plays essential role in domestic and
worldwide energetic future, thus it requires modern solutions and system presented in this paper
is one of them.
177
Acknowledgements
The paper is supported by Polish Ministry of Science and Higher Education as research
project no N N524 468939.
Appendix
The basic terms of evolutionary computation are given below:
individual
– represents or encodes a point in a search space of potential solutions to
a problem,
population
– set of individuals,
chromosome
– data structures representing the individuals of the population (i.e. strings
of binary digits or real values, matrices),
gene
– unit of a chromosome, located in a certain place called locus,
mutation operator – one of the genetic operators to be used during reproduction process of
individuals, mean replacement of a given gene by a different one,
fitness evaluation – fitness of individual is defined and captured in an objective function and
its value indicates the quality of an individual in relation to the problem
being solved,
selection
– process that allows the survival and reproduction of the fittest individuals
in detriment of the less fit ones, in elite selection µ number of the best
individuals are selected for the next generation, which ensures the best
individual, found so far, will not be lost.
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Received: 04 February 2011