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GHIFARI RAZI PERENCANAAN DAN PERMODELAN TAMBANG

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Floating Clone Methods

Floating cone pit optimisation has been applied to block models. The method

is has many variations but the gist is as follows. One can think of it using two block

models (three dimensional arrays of values). The problem model, which initially

contains the values for each block, and a solution model, initally all zeros. The

problem model is searched for a block which, if mined with its conical overburden

will yield a profit.When such a block is found it is 'mined' -moved to the solution

model solution and zero values placed in the problem model.

Taking a two dimensional case with a slope of 45 degrees, this image showsa the [problem set at the start of a program. The first cone with a positive value to be

found is shaded yellow. Note that the +2 block is covered by three -1 blocks. The

yellow blocks will now be moved to the 'solution'

The program may proceed to search for more profitable cones to move to the

solution, perhaps until no more can be found. This process will, unfortunately, move

blocks to the solution that should not be there. So the next stage is to search the

solution file for blocks that, placed at the top of a cone will have a negative value. If

such cones are found they are moved back to the problem model.


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The above diagram now represents the 'solution model' these are the blocks

that have been found to be in positive cones in the first step. Searching this solution

model for an inverted cone with a negative value we find the blue zone. This must be

moved back to the problem model.

Returning to the problem model we find that there is now a new block that

can profitably be mined. When this is removed there will be neither any positve cones

that can be found in the problem set or any negative (inverted) cones that can be

found in the solution set. In this case the algoritm has found the best possible pit.


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Here we see the final 'problem model'on the left and 'solution model' on the

right. Of course, it is more efficient to use one array for the values and devise some

means for flagging which are in the problem and which are in the solution, such as an

array of boolean values.

There is no one 'floating cone'algorithm. How should you search for a

positive cone? When should you switch and look for negative cone in the solution

model? Floating cone methods are potentially very good at representing pit slope

angles. However, they have two drawbacks. Firstly they are not guaranteed to find the

best optimal pit.

Here for example, the optimum pit is shaded yellow. However no single

block generates a positive cone. Probably you can imagine an improvement to the

algorithm. Perhaps instead of looking at the cones over single blocks, you combine

positive valued blocks on the same level and consider their combined overburden.

Would this slow down the search several fold? Might it add in blocks that were not

really economic but that the negative block removal program took a long time to

remove?


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Here again the optimum pit is shaded yellow and no single block generates a

positive cone. The blue square, for example, has a cone with a value of -1.2.

Imagining a floating cone algorithm that copes with multiple ore bodies with

intersecting overburdens is difficult.

That you may not find the best pit would seem to be a killer criticism of

floating cones methods. On the other hand where it is used on massive deposits,

where values tend to change incrementally between adjacent blocks, where mineral

concentrations are assumed to vary gradually through the rock mass rather than

change abruptly at lithologic boundaries this may be less of a serious concern. On the

other hand they have a reputation for being slow. Clearly there are many choices to

make when designing a floating cone algorithm. I have no doubt that some worked

well for certain types of geology.

If the grade of the base is above the mining cutoff grade, the expansion is

projected upward to the top level of the model as in Fig. 8. The resulting cone is

formed using the appropriate pit slope angles. If the total revenues are greater than the

total costs for the blocks in the cone, the cone has a positive net value and is

economic to mine.


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A second block is then examined, as shown in Fig. 9. Each block in the

deposit is examined in turn as a base block of a cone.


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Lerchs-Grossman method

In 1965 Lerch & Grossman produced a paper describing a 3-D algorithm

based on graph theory which could be specifically applied to the optimization of

open-pit mine designs. At the time, most computers were incapable of performing the

large quantities of iterative calculations required by the method. A 2-D algorithm was

also described which, though effective on sections, lost its optimized quality when

sections were combined. The 3-D algorithm therefore remained the preferred option.

In 1986 the Whittle 3-D open-pit optimization package was launched by

Whittle Programming Ltd. This package utilized the Lerch-Grossman algorithm for

the first time in a commercial software application. Since then the package has

undergone a number of changes andimprovements.

The purpose of an optimization package is to produce the most cost effective

and most profitable open-pit design from a block model of an orebody. It must be

capable of rapidly analysing alternatives accurately; it must be able to carry out

sensitivity analyses of different components to assess financial risk; it must allow

frequent updates to take into account changes in costs etc. and it must allow the

production of interim designs. In todays mining world, a hastily produced or non-

rigorous design is financially unacceptable as the design determines the potential for

profit or loss. To develop this further it is essential not just to make a profit but to

maximize the return on investment. This paper appraises the ability of the Whittle 3-D

package to satisfy this objective.

The two-dimensional Lerchs-Grossman method will design on a vertical

section the pit outline giving the maximum net profit. The method is appealing

because it eliminates the trial-and-error process of manually designing the pit on each

section. The method is also convenient for computer processing.


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The results must still be transferred to a pit plan map and manually

smoothed and checked. The example in Fig. 5 represents a vertical section through a

block model of the deposit.

There are three steps in Lerchs-Grossman method:

Step 1: Add the values down each column of blocks and enter these numbers into the

corresponding blocks in Fig. 7.

Step 2: Start with the top block in the left column and work down each column.

Step 3. Scan the top row for the maximum total value. For example the optimal pit

would have a value of $13. This is the total net return of the optimal pit. The

Fig. 7 shows the pit outlined on the section.


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REFERENSI

http://www.agt.net/public/nstuart/pan/poalgos.htm

http://sp.lyellcollection.org/cgi/content/abstract/63/1/179

http://www seam.ustb.edu.cn/UploadFile//20090516095155640.ppt

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t3 Renmod h1c106097 Ghifari Razi