Kararsız market koşullarında genetik algoritma ile sınır tenörleri optimizasyonu
Öz
Sınır tenörleri optimizasyonu, maden yataklarının değerlendirilmesinin temel bir konusudur. En geniş
anlamıyla sınır tenörü; madeni işletme, yerinde bırakma, cevher tesisine gönderme veya atık sahasına
boşaltma şeklinde karar vermede ayraç olarak kullanılan tenör oranıdır. Bir maden sahasına uygulanacak
sınır tenörü meselesine geleneksel yaklaşım, gelirin maliyeti karşıladığı tenör oranının statik bir sınır tenörü
şeklinde maden yatağının ömrü boyunca uygulanması şeklindedir. Bu yaklaşım, paranın zaman değerini göz
ardı etmekte ve maden yatağının en karlı şekilde değerlendirilmemesine neden olmaktadır. Maden yatağının
ömrü boyunca statik bir sınır tenörü tayini yerine optimum bir sınır tenörleri silsilesi uygulaması, işletmenin
indirgenmiş karını artırabilmektedir. Maden yataklarının değerlendirilmesinde, optimum sınır tenörleri
tayini çok önemli; ancak, çözümü kolay olmayan bir problemdir. Bu çalışmada, geniş çaplı optimizasyon
problemleri için uygun bir ortam sağlayan genetik algoritma kullanılarak, optimum sınır tenörleri tayini
yapılmıştır. Sınır tenörü optimizasyonu genel anlamıyla irdelenmiş, genetik algoritma optimizasyon yöntemi
hakkında detaylı bilgi verilmiş ve genetik algoritmanın sınır tenörü optimizasyonuna uygulanması amaçlı
geliştirilen bir bilgisayar programı tanıtılmıştır. Ayrıca, örnek bir maden yatağı ile ilgili veriler kullanılarak,
bu yazılım test edilmiş ve sonuçlar değerlendirilmiştir. Bu çalışma sonucunda, optimum sınır tenörlerinin
belirlenmesinde cari satış fiyatı kullanılarak yapılan geleneksel yaklaşımlar değerlendirilmiş ve bu konuda
Lane'in geliştirildiği ve sınır tenörleri optimizasyonu için sıkça kullanılan algoritma geliştirilmiştir.
Anahtar Kelimeler
Optimization of cut-off grades by means of genetic algorithms under uncertain market conditions
Abstract
Optimization of cut-off grades is a fundamental issue
for mineral deposits appraisal as it assigns the
boundaries between ore and waste over time. In its
broadest definition, a cut-off grade is any grade that
is used to separate two courses of action; to mine or
not to mine, to process or to dump. The traditional
approach to cut-off grades is to use the break-even
grade, at which revenue equals cost. This approach
completely ignores the time value of money and
usually leads to a sub-optimal valuation of the
mineral resource.
Determination of an optimum cut-off grades
schedule, instead of application of a static cut-off
grade for the life of a mine, maximizes discounted
profit. The profit from a mining operation is a direct
function of the sequences of cut-off grades and
associated ore tonnages that define the life-of-mine
production schedule. As profit varies with these
sequences there will be a sequence, or sequences,
that optimize any specified profit criterion. The most
widely used cut-off grade optimization criterion is
maximum net present value of profits. The objective
of maximizing the net present value can be achieved
by maximizing profit per unit of time. This process
necessitates applying, in the early years of
operation, the highest cut-off grade that can provide
sufficient ore to satisfy the requirements of the
processing plant. As time passes the cut-off grade
must be lowered, thereby lowering the opportunity
cost. Hence, the highest net present value is
achieved.
Genetic algorithms constitute a class of stochastic
algorithms that use a search method based on the
laws of biological genetics and natural evolution. In
this approach, individuals of a population are
represented as chromosomes and an expanded set of
genetic operations takes place. Genetic algorithms
are stochastic algorithms whose search methods are
based on the principles of biological genetics and
natural evolution. It is presumed that the potential
solution of any problem is an individual and can be
represented by a set of parameters. Genetic
algorithms are particularly suited to the solution of
large-scale optimization problems. They belong to
the class of probabilistic algorithms but are very
different from random algorithms as they combine
directed and stochastic searches. Another important
property of genetic-based search methods is that
they maintain a population of potential solutions.
Genetic algorithms can also easily escape from local
optima by using genetic operators, such as mutation.
Among the inputs for a cut-off grade optimization
procedure, the selling price of the product of a mine
is the most volatile. Therefore, any change in the
selling price of a mining product in the global
market make every economic valuations obsolete.
Basically, nobody knows how the prices will change
in the future, but can estimate. In order to make a
better mining plan, one must solve the problem of
annual dynamic cut-off grades optimally, and that
necessitates taking the volatile market conditions
into consideration. Therefore, it is accepted in this
work that uncertain selling prices of a mining
product must be added to the algorithm for finding
cut-off grades to be used. Because it is not possible
to find the most revenue without finding the best cutoff
grades scheme. Using today’s selling price of a
commodity, instead of estimated future values of
that, may sacrifice to reach the best net present
value.
In this work, by use of genetic algorithms, that give
a very robust searching medium for big scale
optimization problems, optimum cut-off grades were
determined. Optimization of cut-off grades is
mentioned, detailed knowledge is given about
application of genetic algorithms to cut-off grade
optimization, and a computer program developed for
cut-off grade optimization is given. Besides, the
software is tested by using data from a sample
mineral deposit and the results are evaluated.
This paper makes the traditional approaches to the
determination of optimum cut-off grades by using
current selling price obsolete and improves Lane’s
algorithm in this subject.
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