Chapter 42

OPTIMISING OPEN PIT DESIGN AND SEQUENCING

P.A. Dowd Reader in Mining Geostatistics and Operational Research

and A.H. Onur

Research Student Department of Mining and Mineral Engineering

University of Leeds Leeds, U.K.

ABSTRACT defined by estimating the grade of each block. Having specified the technical

The authors give a corrected form of the parameters, the next stage is to convert Korobov algorithm of optimal open pit the estimated grade values into economic determination and describe a method of values by using operating, capital, optimising the mining sequence within a production and fued costs and a profit preliminary pit design. A method of value. automatically including roadways is also given. The objective of optimal open pit design is

to determine the final pit limits of an INTRODUCTION orebody and its associated grade and

tonnage, which will maximize some Most computer algorithms for open pit specified economic criterion whilst design use a block model obtained by satisfying practical operational dividing the deposit into a three- requirements. After the pit has been dimensional grid of estimated grade designed the mining operation must be values. The block dimensions are scheduled. Scheduling is essentially the determined by : development of a sequence of depletion

schedules leading from the initial 8 the size of the mining conditions of the deposit to the ultimate

equipment pit limits. Scheduling can be either short 8 the scale required to term or long term depending on the time

approximate the topography scale involved. Short term schedules cover and the irregular shape of mining rates on a daily, weekly or the orebody monthly basis; long term schedules cover the data available for yearly plans and include stripping ratios, estimating block grades ore reserves and capital investment.

There are many .feasible plans of The mineral distribution, topography, extraction which will lead to the same important geological features and other final pit shape. Each of these feasible spatial characteristics of the deposit are plans generates a production schedule described by values assigned to the blocks. over the entire mine life, which may or The distribution of the mineralisation is may not be optimal. When the NPV is

. .

23rd APCOM PROCEEDINGS

applied to all these possible pit 'schedules, the one with highest NPV is called the optimum schedule.

The objectives of this study were to ;

(1) develop t h e Korobov algorithm into . a ' , valid alternative to the Lerchs- Grossman algorithm

(2) devise a new approach to incorporating roadways in an open pit . , . ,

(3) use a dynamic programming algorithm to create possible solutions to the mine scheduling problem.

CORRECTED FORM OF THE KOROBOV ALGORITHM

The biggest disadvantages of using the Lerchs-Grossmann algorithm are computing time and the difficulty in applying variable slope angles. Many attempts have been made to overcome these difficulties, the most well-known of which is the moving cone algorithm.

The method originally devised by ~or6bov (David, Dowd and Korobov, 1974) looked very promising but it was soon shown that it does not provide the optimum solution in all cases. The main idea of the algorithm is to allocate a cone to every positive block throughout the pit and to allocate positive values within the cone against negative values within the cone (the positive blocks are said to "pay for" .the negative blocks) until no negative block remains. When this is achieved and the positive block on which a cone is based still remains positive, then this cone is accepted as a member of the optimum solution set. At the stage of establishing cones all necessary technical restrictions are used to obtain the best representation of the pit. The example in figure 1 shows how the algorithm works.

All positive blocks on the first level are removed immediately. Then on the second level block (2,2) is considered. In order to remove this block, blocks (1,1), (1,2), (1,3) from level 1 must be removed. Block (2,2) starts paying for these blocks. Blocks (1,1), 11,2) and (2,2) become 0 and so this block, thus far, is not a member of the optimum solution set. The algorithm continues with block (2,3). Blocks (1,2), (1,3), (1,4) constitute a cone for that block. Block (1,2) has been paid for previously by the positive block (2,2), block (1,4) h'as been removed and the only remaining negative block is (1,3). When

OPTlMlSlNG OPEN PIT DESIGN AND SEQUENCING 41 3

block (2,3) pays 1 unit to block (1,3) it remains positive, hence it .is included as a member of the optimum solution: Block (2,3) and its cone (1,2), (1,3) are removed. When blocks (2,3), (1,2), (1,3) are removed there is no change in the new pit value which remains +2. The next step is to start the algorithm from the beginning and determine whether the removed blocks affect any other cone at the beginning stages. In this example, block (2,2) is reconsidered after block (2,3) has been removed; block (2,2) pays for the remaining block (1,l) and remains positive 1 When blocks (2,2), (1,l) are removed the new pit value increases to +3. The same rule is applied to all remaining positive blocks on each level.

This algorithm does not alwa@give the optimum solution. ' as .shown . by the example in Figure 2.

.. -

Figure '2

In this example, blocks (1,1), (1,2) and (1,3) have been paid for by block (3,3).

. When block (3,4) pays for all the negative + blocks within its cone, it becomes +1 and it is removed. When block (3,4) is removed with the value of -1, (3,3) car? be

reconsidered and removed. Overall profit becomes zero. In this case it is better not to mine these blocks.

The error is caused by blocks which are common to both cones. It can be corrected by the following logic : "If two or more cones have blocks in common, then blocks not in common must be paid for first; common blocks are only paid for after all blocks not in common have been paid for".

This method leads to an optimum solution in shorter time than the Lerchs- Grossman algorithm and, by identifying the blocks which are members of the cones, physical and technical constraints can be applied more easily.

Method

Let A = {qjk) be the set of all blocks defining the orebody; the index k denotes the level (vertical from k = 1 at the surface) and (ij) denotes position on a level. The subset of blocks on level k is denoted by 4. Let 8, be the set of all blocks on level k which have positive values :

Let C,,, be the removal cohe on level'e corresponding to block b,, on level k (the complete removal cone is then {C,,,, e = k-1,l)). Any slope constraints can now be expressed in a completely general manner in terms of the removal cone on a particular level. For example, for cubic blocks and a slope of 45" the-removal cone on level k-1 .corresponding to b,, is :

Cmnk:l = {ai,k-l, i = m - l , m + l , j = n - l , n + l )

For any given level all possible combinations of removal cones can be formed into groups as shown in figure 3 where, for the sake of simplicity, the multiple indices have been dropped in favour of a single index referring to each of the n positive valued blocks on the level. Note that cones wholly contained

$1 4 23rd APCOM PROCEEDINGS

within another cone are removed from A - re-allocation process has been subsequent combinations. incorporated in the , algorithm to, correct

the original version and to avoid the 1 - - 2 - 3 number of possible combinations b.ecoming

unworkably large. The idea of re-allocation Cl c2 ,c3 is that every negative value block within

C, & C, C3 & C, a removal cone must first be paid for by c3 & c1 the positive block on which this cone is

-+ C, & C, & C, defined. If the positive block is still I. positive after all negative blocks have

been paid for then this cone is said to be 4 - - n a member of the optimal solution. . .

c4 ci Re-allocation can best be explainedd-by c4 & (33 C,&c, - , . means of the example shown in figure 4. C4 & (32 Cn & Cn-2 c4 & CI Q . c, & c3 & c, 1 2 3 4 5 C4 & C, & C, c4 & C, & C, cn '& c, 1 c4 & C3 & C, & C, C, & Cn-, & C,-, 2

Cn & C,-, Cn-, 3 l e v e l 1 4 5

C, & C,-, & ..... C, 2 3 l e v e l 2

. . 4

Figure 3 Figure 4

The optimal solution must consist of one of the many possible combinations of Consider the first positive block on level removal cones. The algorithm for finding 2, i.e. block (2,2,2). The removal cone this optimal combination is as follows. (cone 1) for this block is shown in figure The groups of combinations of removal 5. cones are considered in order. In each 1 2 3 group the first combination which gives a positive total revenue value is added to 1 the optimal solution; once a combination 2 has been chosen from a group all 3 combinations in subsequent groups which contain members of the chosen combination are removed. For example, if 2 I + C, is chosen from group 1 then C, & C, is removed from group 2, C, & C, and C, Figure 5 ,

& C, & C, are removed from group 3, etc. At each stage only positive combinations The .positive value (2) of block (2,2,2) is of removal cones are added to the solution allocated against the negative valued and hence the final solution always yields blocks in its removal cone until it a maximum value. becomes zero as shown in figure 6. In

OPTlMlSlNG OPEN PIT DESIGN AND SEQUENCING 41 5

figure 6 blocks (1,1,1) and (2,1,1) are assigned two new numbers, the upper number designates the cone which has paid for the block and the lower number is the new block value after allocation.

1 2 3 4 5

2 4 Figure 6

When the same rule is applied to all positive blocks the results shown in figure 7 are obtained.

1 2 3 4 5

l e v e l 1

, o

* o

l eve l 2

In figure 7, block (4,4,2), cone 9, pays only for block (5,5,1) and the value of this block remains +3. The algorithm differs from the original Korobov algorithm at this stage. In the original version this block was added directly to the optimum solution whereas in the corrected version of the algorithm a re-allocation process must be performed.

1 2 3 0

0

The idea of re-allocation is that every negative value block within a removal cone must first be paid for by the positive block of this cone. If the positive block is still positive after all negative blocks have been paid for then this cone is said to be a member of the optimal solution. Consider block (4,4,2) and its removal cone (cone 9) in figure 7. Removal cone 9 is : {(3,3,1), (3,4,1), (3,5,1), (4,3,1), (4,4,1), (4,5,1), (5,3,1), (5,4,1), (5,5,1)}.

It is evident that some of the blocks in this removal cone have already been paid for by cones 5, 6 and 8. The next step is to use these cones'to find a path which leads to any negative block (if there is one). Beginning with cone 5 a search is made for either a negative block within this cone or any other block within cone 5 which has already been paid for by another cone apart from cones 9 and 5. Figure 8 shows the removal cone for cone 5. 1 2 3 5 6

0 0 0 0 0 4 4 5 5 6 O O O O O 4 4 5 6 6

0 7 7 8 8 9 0 0 0 0 0

Within the removal cone of cone 5 there are some blocks paid for by cones 2, 3 and 4. In cone 3 there is a negative (-1) block. Block (2,3,1), paid for by cone 3, is removed from its original position and it repays to block (1,4,1). Block (2,3,1) can be paid for by cone 5 and so block (3,3,1), which is within the removal cone of cone 9, has been emptied. As shown in figure 9, block (4,4,2) can pay a value of -1 to this block.

Applying the same rules, figure 10 is obtained.

0 - 1 - 1

0

Figure 7 The next step is to find another negative

0

23rd APCOM PROCEEDINGS

block in the ,model. If there are. no more negative blocks then all contributing removal cones can be added to the optimal solution. In this example all cones (combination of 9, 8, 7, 6, 5, 4, 3, 2, 1) give the maximum vlaue of + 1 and so all blocks are mined.

C1

Computing times .

Figure 11 shows a comparison of computing times for the Korobov algorithm and our coding of the jlerchs- Grossman algorithm in ,which the clear advantage of the former over the latter is demonstrated. . - .

Figure 8 A

Figure 10

Number of blocks x103

Figure ' 11 Figure 9

OPTlMlSlNG OPEN PIT DESIGN AND SEQUENCING 41 7

INCORPORATION OF. ROADWAYS

One of the . aspects of open pit optimization which is often neglected is the incorporation of roadways in the design. A pit without roadways cannot be called optimum as roads can change the overall shape of the pit dramatically. ,In spite of the importance of the subject there have been very few reported studies of producing a suitable roadway design which maximises the profit, minimizes the additional waste which must be extracted and minimises the overall transportation costs throughout the life of the mine.

Roadways are designed to transport all the ore and waste from the bottom of the pit up to the dumping point (for waste) or to the crusher (for ore). For all levels there must be a connection to the surface. Road sizes depend totally on mining and equipment capacities. In order to maintain the specified slope'angles, bench widths are defined before the mining operation starts. Bench heights are a function of the geology and the existing mining equipment and so are rarely changed during the life of the pit unless there is a significant change in economic parameters which consequentially alters recoverability criteria. . Slope angles can be altered by changing bench widths.

created by starting them from different points at the bottom and changing the direction from clockwise to anti-clockwise. The algorithm ,starts with a . block on the pit bottom and, moving in a clockwise direction, generates .a ramp,. with the specified slope and width, to the next level; a ramp is then generated in the same direction to the next level and so on until the .surface is reached. This procedure is repeated starting from the same block but moving in an anti- clockwise direction. When the two roadways have been generated for this block the procedure is repeated for every other block on the pit bottom. Computing time can be reduced by specifying a minimum increment from one starting point to the next (eg, use only every second or third block on the pit bottom). Using a block model prevents the algorithm from producing precise road dimensions because the algorithm is always restricted by the block dimensions. To obtain a proper width a whole block must be removed, not a proportion of it.

level n-1 bench height

level n

(a) Original bench widths are not always large enough for the construction of a road. When it is decided to place a road in some part of the pit some additional excavations will be necessary. The roads are specified by their slopes as well as their widths as shown in figure 12a.. The slopes depend on the materials on which roads are built, traction factor, type of trucks and weather conditions. Slopes define the distance of travel from level n

@ n-3 to level n-1.

When an algorithm is used to obtain an optimum pit, the result may be in the form of figure 12b. By using this figure,

@-3 (b) @ starting from the lowest level of the pit several different alternative roads can be Figure 12

41 8 23rd APCOM PROCEEDINGS a .

At the end of the algorithm each alternative road can be analysed. with respect t o the number of additional blocks to be removed, reduction of the: overall profit by removing these blocks and the distance from the bottom of the pit to the dumping point.

Although the roadway design.is not a true optimizing algorithm i t . provides some helpful information to planners in the . decision making stage. The algorithm will exclude switchbacks if required. ' . Figure 13 and table 1 show a small example of the results of the algorithm applied to an optimised open pit.

alter. new total new no. pit distance blocks

value to be mined

Table 1

' s it is the last task in the mine planning ierarchy, a proper production schedule

must not only conform to both the long- range and short-range mine plans but also satisfy many practical requirements. There are many feasible mining plans which lead to the same optimal pit shape. Each of these feasible plans generates a production schedule over the entire mine life, which may or may not be optimal.

Whittle explains the complexity of the problem in his study and says that '... you can't find the pit outline with the highest total value until you knows the block values; you don't know the block values until you have worked out ' a mining sequence; and you can't work out a mining sequence until you have a pit outline ...' (Whittle, 1989). .

Mining sequences are important because they deal with the "future". Reinvestment opportunities are , employed in an equivalent but less obvious form in Discounted Cash Flow analysis. Discounted Cash Flow methods usually take one of two forms, either to compute a Net Present Value (NPV) for an investment proposal or to compute an Internal--Rate of Return. Both methods employ a discount rate. When a project has a positive NPV the project is said to be profitable. A mining sequence which has the highest NPV value is called optimum.

Mining. sequence algorithm . , . . .

As fgr as NPV is concerned, the.. best mining sequence is that which produces all. the ore in a very short time period thus making the NPV larger. However, in reality this doesn't always comply with marketing conditions, the capacity of' the processing plant' and the .capacity and amount of equipment which will be used in the pit. Thus some constraints must be applied.. There are still many different

OPTlMlSlNG OPEN PIT DESIGN AND SEQUENCING 41 9

Open pit layout without roadway .

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 1 1 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 1 2 4 0 0 0 1 2 4

1 0 0 0 0 2 0 0 0 0

0 0 0 0

0 0 0 1 2 3 3 4 4 5 6 4 4 4 4 5 5 4 3

0 0 0 0 0 0 0 1 1 1 1 1-1-11-1-1-1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1-1-1-1-1-1-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Open pit layout with roadway (dternative' 1)

Figure 13

420 23rd APCOM PROCEEDINGS.

ways of extracting the ore even with A block cannot be used twice in one stage. these constraints and the task is to But there is a technical constraint to this choose the best one which has the pattern. Suppose that we choose the highest NPV. Besides this restriction sequence 13,8,3 which means block 13 will there are several other constraints which be mined first, then block 8 and finally reduce the number of alternatives to be block 3. When the block dimensions are searched. These will be explained later. taken into account these blocks cannot

always be mined because there may not The algorithm uses dynamic programming be enough space for the equipment in the to find an optimal mining sequence under pit and it may be necessary to add some specified conditions. This technique has blocks in order to remove block 13. This been applied successfully elsewhere to depends entirely on the size of the solve similar problems, see for example equipment ~d the block dimensions. For Dowd (1976) and Dowd and Elvan (1987). example if 60 metres is the space The central idea of the algorithm is to required for loading, transporting and search all possibilities and choose the blasting access and if the block dmensions optimum one. The algorithm is: are 20m x 20m x 10m then block 13 can

only be mined with blocks 12,8,14,18 a determine the block which is together and so on. After mining block

to be mined first * 13 and its successors a search is made of • determine the-next block to the group of 8,7,3,9 or 14,9,15,19 or

be mined' in the sequence, 18,19,23,17 or 12,17,11,7. a identify -' all alternative

sequences which represent This kind of pattern limits excessive different permutations of the shovel movements as well. For example same combination of blocks. there is no chance of mining block 5 after

mining block 21. The movement of the The algorithm starts from the first level sequence is shown in figure 15. and picks one block and assumes that this is the beginning of the sequence. Then it As lower levels are encountered a cone is produces all possible permutations after defined for each block. This cone extends mining that block . Fig. 15 shows how to the surface and within this cone all the algorithm works. Suppose that we blocks are considered to be members of start mining from block 13 in this the sequence for the block. For example, example. The next block to be mined is block 19 can only be mined together with either block 12, 8, 14 or 18. blocks 5 and 7; when the same rule is

applied to block 39 the blocks which must be removed are: 2,3,4,5,6,7,8,9,17,18,19,20, 21,29,30 and 31. Blocks 6,7,19,30, are in the same sequence so these blocks are removed from the removal cone of block 39. This pattern can be changed to give any desired working slope angles.

As can be seen very easily from both figures 14 and. 15, in any. sequence when

13 t

I the number ,of stages increases the I I 1 1 8 2 14 18 number of permutation's increases at least

lf-7 : ' I ' +, four times that of the previous stage and 3 7 9 7 1 1 1 7 , 9 1 5 1 9 1 7 1 9 2 3 huge numbers of sequences could result.

This problem can be overcome by the Figure 14 following logic. At the beginning of the

OPTlMlSlNG OPEN PIT DESIGN AND SEQUENCING 421

- 4 18 8 2 0 18 n

. -do *' . + n

3 17 29 17 19 9 21 * +* +

19 31 21 17 29 39 29 31 ' 21 31

Figure 15

algorithm mining can be divided into time periods. These periods could be weeks, months or years. The program tries to find a sequence for the given time period then for the next period applies the discount rate. If any new, added stage does not satisfy the constraints then this sequence is eliminated from further searches thus reducing computing time and memory requirements. When one time period defined at the beginning succeeds, the next period starts. In every time period the following conditions apply :

The sequence in the time period must provide sufficient ore for the mill The sequence must satisfy quality constraints. This can be done by controlling the upper and lower grade limits of the ore mined during each stage. Each stage is limited by the need for adequate equipment operating room. The stripping ratio must be satisfactory for each period, stripping ratios which are too high or low are eliminated. At the beginning of the sequencing, if there is no possible way of starting up the mining operation without preproduction the user is

asked whether preproduction is requested and, if so, for how long. If preproduction is requested the operation is r e a r r a n g e d w i t h preproduction. . The program allows for the mining of extra ore for each period. This ore can be stockpiled for use when there is a shortage of ore in any stage of the sequence.

CONCLUSIONS

Computer aided open pit design is a kit to help engineers by doing all of the work previously done by hand. There have been many attempts to introduce more realistic and more reliable models during the past decade. The corrected form of the Korobov algorithm, described'in this paper, provides optimal open pit designs

. in significantly less time than the Lerchs- Grossman algorithm. The scheduling method described above is still under development but has produced some promising results. As. usual for all scheduling programs, this method requires a long time to produce reliable results. There are still sever+ points to consider such as transportation, which affects all the scheduling, and physical characteristics of the rocks and ore. However, the program can already provide important results for the decision making process.

422 23rd APCOM PROCEEDINGS '

REFERENCES

Dowd, P.A., 1976, "Application of D y n h i c and Stochastic Programming to ,Optimize Cutoff. Grades and Production Rates". Transactions of the Institution of Mining and Metallurgy, Section A, volume 85, October 1976, pp A22-A31.

Dowd, P.A. and Elvan, L., 1987, "Dynamic Programming applied to Grade Control in Sub-level Open Stoping". Transactions of the Institution of Mining and Metallurgy, Section A, volume 96, October 1987, pp A171-A178.

David, M., Dowd, P.A. and Korobov, S., 1974, "Forecasting Departure from Planning in Open Pit Design.and Grade Control". Proceedings of the twelfth APCOM symposium, Colorado School of Mines, April 1974, volume 2, pp F131- F142.

Franks, J.R., Broyles, J.E., 1981, "Modern Managerial Finaqce". Interscience Publications, Chichester, NY, USA.

Fytas, K., Pelley, C., Calder, C., 1987, "Optimization of Open Pit Short and Long Range Production Scheduling". CIM Bulletin, volume 80,. no. 904, August,

' 1987, pp 55-61.

.Kim, Y.C., 1980, "Production Scheduling Technical Overview". Computer Methods for the go's, Port City Press Inc., Baltimore,, USA, pp 610-614. - ' . ..

, . Lerchs, 'i H., ; Grossman, I.F.,. 1965? ,. "Optimum Design of Open Pit , Mines,".

:-CIM Bulletin:volume 58, J&uqy 1965,,pp . , .

47754. : , . , . ' .:. > : .,, . t i ' . ,

. ,

~ $ k h i ; h , M.P. and Borgman, i:, 1969, "Twb., and ~hree-dimensional Pit ',Design

...Qp timization . Techniques". A i)e&te of . '

il?igztal Compziting in the '-Mineral. . ' . . Fdustry. AIME, New York.' . .. . , . .,

Application of Computers in the Mining .Industries Symposium (APCOM), South African Institute of Mining and Metallurgy, Johannesburg, South Africa, pp 155-163.

Whittle, J., 1989, "The Facts and Fallacies . of Open Pit Optimization". Whittle Programming Pty. Ltd. Report, Jan. 1989

.Robjnson, R.H., And Prenn, N.B., 1972, "An Open' Pit Design Model", 10th