LERCH-GROSSMAN PIT OPTIMIZATION

Intelligent Mining Software Solutions IMS - Lerch-Grossman Pit Optimization

IMS 1.0Produced by W.S. Mart and G. MarkeyFor MineMap Pty Ltd

Intelligent Mining Software Solutions IMS - Lerch-Grossman Pit Optimization Copyright 2013 by William Seldon Mart and Geoff Markey. All rights reserved.

Table of ContentsINTRODUCTION4MODEL REQUIREMENTS6Block Models6Setting Up Dollar Value Models6Economic Data7Laminar Models7RUNNING THE LERCH-GROSSMAN MODULE8The Input Cells Page9The Ore Price Page11The Ore Recoveries Page13The Processing Costs Page15The Ore Modifiers Page18The Overburden Modifiers Page20If the default modifier is 0 and there no entries in the modifiers list then focus assay does not use ore modification.The Mining Costs Page21The Required Profit Margin Page24The Slope Modelling Page26The Assay Flagging Page29The Recovery Flagging Page31The Output Page32APPENDIX A: THE PIT OPTIMIZATION ALGORITHM34Basic Concepts34The Algorithm35APPENDIX B: FILE FORMATS36APPENDIX C: ERROR MESSAGES37REFERENCES38

INTRODUCTIONAn open-cut mining operation can be viewed as a process where the exposed surface of a mine is continuously deformed. One of the factors to be considered in the planning of such an operation is how the pit should be designed in order to maximise the financial return while satisfying requirements such as safe wall slopes. This optimization is provided by the IMS Lerch-Grossmann module.The IMS Lerch-Grossmann module uses the algorithm described by Helmut Lerchs and Ingo Grossmann (Lerchs & Grossmann, 1965) and modified by Louis Caccetta and Lou Giannini (Caccetta & Giannini, 1986).The algorithm is best illustrated by the physical analogue in which each block in a model has a downward and upward force applied to its centre. Here the downward force represents the costs involved in block mining while the upward force represents the value of the mineral in the block.The downward force of a given block is dependent on all the other blocks above it that need to be removed to get the block in question out of the ground.The upward force of all blocks is the value of the block based on its mineral content and the mineral unit value applied to the blocks of the model. The optimum contour is established where the forces equalise within the model.The algorithmic equivalent of the above is implemented by the Lerch-Grossman module using three important assumptions:1. The cost of mining each block does not depend on the sequence of mining.2. The desired wall slopes and pit outlines can be approximated by removed blocks.3. The objective of the optimisation is to maximise total undiscounted profit.With these assumptions in mind the Lerch-Grossman module assigns a cell value based on the unit of the mineral assessed. A cell is defined as ore if

This generates a cut-off grade for the bench. Processing costs are then applied to ore cells after the cut-off is defined. If the resultant cell value is less than the cut-off value, after mining costs are removed, then the waste removal cost is assigned to the cell to indicate that it is waste. If more than one ore type (mineral type) is extracted, the cumulative value is used.

Assay cut-offs/block dollar values are determined by the equations below:Grade Cut-off (unit/t) = Processing cost ($/t) / Recovery (%) x Ore Price ($/unit)Equation 1: Calculation of grade cut-offRaw Cell Value ($) = [Assay (unit/t) x Tonnes (t) x Recovery (%) x Ore Price ($/unit) x Ore Proportion (%)] Modifiers ($/T)Equation 2: Calculation of raw cell valueCell Processing ($) = [Tonnes (t) x Processing cost ($/t)]Equation 3: Calculation of cell processingCell Value ($) = Raw Cell Value ($) - Cell Processing Value ($)Equation 4: Calculation of cell valueFinal Cell ($) = Cell Value ($) [Tonnes (t) x Ore Mining Cost ($/t)]Equation 5: Calculation of final cell value

If then the cell is assigned the value of (i.e. a model cell cannot cost more to mine than the basic cost of mining).

MODEL REQUIREMENTSBlock ModelsThe Lerch-Grossman module works on block models only. These models must have an elevation for each cell and the number of assays depicting the content of the minerals being evaluated for that cell.The natural surface will inevitably vary over the region of the model so it is good practice to first mine out or partially mine out cells that would otherwise be above the natural surface (use the Grade Tonnage Reporting module or the menu item). The cells that are mined out in this process are the air blocks. The Lerch-Grossman module recognises these air blocks and discards them from the pit processingNOTE:The Lerch-Grossman module does not check if the benches of the model have the same thickness. This is a requirement. It is therefore the responsibility of the user to prepare a valid model with constant bench thickness.If the model contains dollar values as one of its assays then the Lerch-Grossman module will use the dollar values contained in each cell value instead of calculating that value. The dollar value contained in each cell of the model must be for the entire tonnage of the cell. A positive value indicates a cell that returns a profit while a negative value indicates a cell that is waste and is normally calculated as the product of mining cost and cell tonnage.

Setting Up Dollar Value ModelsA dollar value can be computed for each cell by using SQL commands from a database of the model. You would normally do this if you have a more complex dollar computation than that allowed for by the Lerch-Grossman modules costing parameters. The steps to produce a dollar value using this method are: 1. Load the model and export the cells that have ore to an ASCII text file ().

2. Create or open a Microsoft Access database.

3. Import the ASCII file into the database as a comma delimited text file (CSV).

4. Create queries in Microsoft Access to massage the data as required and export it to an ASCII file.

5. Create a new model definition (). This model must have the same base points, number of cells and cell sizes as the original model.

6. Create an empty block model from the definition ().

7. If the waste dollar value is constant with depth then it can be assigned to the model using the menu item.8. If the waste dollar value varies by depth then use the menu item to create a range of values for the background waste (according to bench depth). This option uses a value assigned to a polygon that encompasses the model area and injects that value into all cells for a bench. The value may change for each bench.

9. Import the database data ().

NOTE:Every cell in the model has a valid dollar value. Failure to assign dollar values to cells will result in miscalculation of the optimum pit.

Economic DataEconomic data should be prepared before running the Lerch-Grossman module (probably by way of an external program). This data must have for every bench/block in the model the net profit that would be obtained from mining that block once it has been exposed. It is not necessary to provide a cost for the uncovering of each block since this is one of the factors automatically taken into account by the program. Waste blocks should be assigned negative economic values.

Laminar ModelsOptimization of laminar models can be performed indirectly by exporting the model to a block model (). A resultant quality of this conversion is the RD of LG Ore Percent assay (see LAMINAR MODEL OPERATIONS) which is used by the Lerch-Grossman module.

RUNNING THE LERCH-GROSSMAN MODULE1. Load the required block model by dragging it to the 3D View Pane. This block model must have the surface mined off (e.g. via ).

2. Select the menu item (Figure 1) to display the first page of the Lerch-Grossman parameters wizard.

Figure 1: Starting the Lerch-Grossman pit optimization

The Input Cells Page

Figure 2: Input cells page1. Enter the area of the model defined by the minimum and maximum rows and columns and the number of benches.2. Enter the number of cells to be bulked together to form a larger cell size. If you do not wish to bulk cells together then enter 1 for each direction. See the notes below for more information.3. Enter the average relative density of ore. This value is only used if relative density is not modelled already or if the modelled relative density is too low (< 0.1).4. Press next to go to the next page in the wizard.

The optimised pit produced by the Lerch-Grossmann algorithm always has a crest within the region that you choose so no generated pit will break through the side of the model. Consequently if the region or model area is too small, an underestimated optimised pit contour will result.

If the model is too large, and the optimisation data cannot fit into RAM, then the Lerch-Grossman algorithm will slow down considerably. To offset this, you may wish to bulk the model cells together as shown in Figure 3.

Figure 3: Grouping the cellsThis will have the effect of reducing the number of cells (albeit large ones) that need to be processed, and hence make it possible for all the optimisation data to fit in RAM. Bulking starts from the minimum row / column and from the deepest bench. If the bulked cells break out beyond the limits of the model, as a result of the bulking parameters not being even multiples of the number of rows, columns and benches, then that portion of the breakout is considered to be air. This is acceptable because the crest of the pit will always be computed to be inside the row and column limits.

The Ore Price Page

Figure 4: Fixed ore price1. Enter the cut-off profit value. This will usually be 0.0 since higher values will high-grade the deposit.

2. Select an assay from the assay list to focus on that assay.

3. Enter the price for that assay in dollars per unit (the unit depends on the quality: e.g. gold would be $/gram). A blank value is converted to 0.0. The price can be fixed or variable as follows:

a. Fixed price: Enter the fixed price as the default price and leave the list above it empty (Figure 4).b. Variable prices: Select the quality which will provide the ranges (can be the same as the focus assay) and press the button to enter the range and price for each range (Figure 5). A default price is also required. Up to 10 model qualities can use variable prices.

Figure 5: Variable ore prices

The Ore Recoveries Page

Figure 6: Fixed recovery1. Select an assay from the assay list to focus on that assay.

2. Enter the recovery for that assay. A blank value is converted to 0.0. The recovery can be fixed or variable as follows:

a. Fixed recovery: Enter the fixed price as the default price and leave the list above it empty (Figure 6).

b. Variable recoveries: Select the quality which the ranges will be based on (can be the same as the focus assay) and press the button to enter the range and recovery for each range (Figure 7). A default recovery is also required. Up to 10 model qualities can use variable recoveries.

Figure 7: Variable recoveriesVariable recoveries are useful if qualities had been modelled (i.e. weathered, transitional, fresh rock) that affect ore recovery. Note, however, if variable recoveries are used then the cut off grades will not be reported in the report file.

The Processing Costs PageProcessing costs are covered by any activities associated with the extraction of the key target mineral out of ore material. Generally these costs will be associated with milling, extraction, rehandling, administration and haulage to the mill to name a few.Processing costs can be variable or fixed:1. Fixed processing costs:a. Select Fixed processing cost for each assay (Figure 8)b. Enter the processing cost for each assay in the list. Enter 0.0 for any assays not considered for optimisation. This method is recommended for multi-element optimisation.

Figure 8: Fixed processing costs

2. Variable processing costs:a. Select Variable processing cost by ranges of ( Figure 9).b. Select the quality that will provide the ranges.c. Enter all the required ranges and costs in the fields provided. Note that these processing costs will apply to all assays being considered for optimisation.

Figure 9: Variable processing costsIf variable processing costs are used then the cut off grades will not be reported in the report file.

The Ore Modifiers PageOre modifiers can be positive or negative cost modifiers, based solely on a block tonnage basis and it will apply to ore cells. For instance, you can set values to penalise a block based on the content of a particular model quality.1. Select an assay to set the focus on that assay.

2. Select the quality which will provide the ranges (can be the same as the focus assay).

3. Press the button to enter the range and modifier for each range (Figure 10). Up to 10 model qualities can use variable modifiers.

4. Enter a default modifier.

Figure 10: Ore modifiersIf the default modifier is 0 and there no entries in the modifiers list then focus assay does not use ore modification.

The Overburden Modifiers PageOverburden modifiers can be positive or negative cost modifiers, based solely on a block tonnage basis and it will apply to waste cells.1. Select an assay to set the focus on that assay.

2. Select the quality which will provide the ranges (can be the same as the focus assay).

3. Press the button to enter the range and modifier for each range (Figure 10). Up to 10 model qualities can use variable modifiers.

4. Enter a default modifier.

Figure 11: Overburden modifier dialog boxIf the default modifier is 0 and there no entries in the modifiers list then focus assay does not use ore modification.The Mining Costs PageThe mining cost is the cost of removing material (ore or waste) from the pit only (i.e. drill and blast, transport of material from floor to pit crest). The ore material may be more difficult to mine with respect to the ore and as a result, can introduce extra costs.

Figure 12: Mining costs There are three methods of entering mining costs (Figure 12):1. Enter an ore cost and waste cost for each bench.2. Select the button to enter global costs to a nominated bench and all benches below it (Figure 13).

Figure 13: Global mining costs3. Select the button to select a text file containing the required costs (Figure 14). The text file must be a comma separated file with the format:

ore cost,waste cost,field 3,field 4,

Only the first two fields are required. If the model benches are required for reference then use field 3. No header lines should be included in the file.

Figure 14: Mining costs from a CSV fileThese methods can be used in conjunction if required. For example, you can import a file to fill all the bench cost values and then modify a few benches.

The Required Profit Margin PageIf you have assigned dollar values to the model (the model consists of qualities of type Dollar value), you may want to produce a series of nested pits. Nested pits are created by first producing a high-grade pit, caused by a large profit requirement per cell. This pit is then mined from the model using the Grade Tonnage Reporting module. By repeating the optimisation with a smaller cut-off (a minimum required profit value), the next nested pit is created.The cut-off values are specified as follows (Figure 15):1. Enter the profit margin required on each cell. If you are not producing nested pits then the cut off value will normally be 0.0. Higher values will high-grade the deposit.

2. Enter the cost of removing an ore cell as waste if that cell does not pass the profit margin.

3. Select the quality that contains the dollar value.

Figure 15: Required profit margin dialog box

The dollar value of every cell is compared against the cut-off value. If the dollar value is less than the cut-off then the cell is assigned a waste cost.Only one dollar quality can be used for optimisation. You can, however, have a number of dollar values in the model to examine discounted cash flow.

The Slope Modelling PageTo specify the slopes and bearings of the pit walls ( Figure 17):1. Enter the number of benches for slope modelling (i.e. the number of benches that are required to approximate the angles used in the slopes of the wall) that best fit the cell size and your choice of angles.2. Press the button to add a bench group.3. Select the bench group from the Bench groups list to focus on that bench group.4. Enter all the required gradients and bearings (up to eight) for the selected bench group.

Figure 16: Slope modelling as a single bench group5. Repeat steps 1 through 3 for each additional bench group ( Figure 17).

Figure 17: Slope modelling as multiple bench groupsThe slope angles start from the floor of the bench level. The slope gradient should be the overall angle of the pit wall including catch berms and one or more passes of the haul road with a batter included for each horizontal plane.Bearing angles start from North, with East being 90o, South 180o and West 270o. These define the area where the corresponding wall slope is generated. The program interpolates the slope angles between supplied bearings.

E.g. if the supplied bearings are at 30o and 240o, with slopes at 50o and 60o respectively then the interpolated slope gradient at the 82.5o bearing mark is 52.5o. At 145o it changes to a slope gradient of 55o and so on. This achieves a smooth transformation from one slope gradient to another.The pit wall slopes can only be approximated since the Lerch-Grossmann algorithm either removes a cell or leaves it behind as waste. The accuracy of the approximation is directly affected by the cell size.NOTE:The time taken to initialise the block dependencies is exponentially proportional to the number of regions and bearings defined.

The Assay Flagging PageAssays in the input model can be "flagged" to indicate various parameters either symbolically or with real values. This can then be used for reporting in the Grade Tonnage Reporting module or for displaying in the Plan View Publisher and Section View Publisher modules.Additionally, assays of type Lerch dollar value can be used to store various properties of the optimization.It is recommended that the assays chosen for flagging should not be the assays used in the optimization. Instead create additional assays specifically for this purpose.

Figure 18: Assay flagging dialog box1. Perform assay flagging on the input modelSelect the assay to be flagged and enter the flag values. Different values are entered for ore and waste cells inside the pit since low grade ore cells can sometimes be treated as waste if they are uneconomic and may need to be treated separately in subsequent operations.Any model cell that is inside the optimum pit will have the selected assay value overwritten with one of the flag values. Cells outside the optimum pit will remain unchanged.2. Perform assay flagging on the model cellSelect the assay to be flagged and enter the flag values. The selected assay in every model cell will be overwritten with one of the flag values.3. Cell PropertiesAssays of type Lerch dollar value are used to store various properties resulting from the optimization. The available properties are: revenue, cell value, ore mining cost, waste mining cost and processing cost.

The Recovery Flagging PageAssays of type Lerch recovery can be used to flag assay recoveries (Figure 19). This can then be used for reporting in the Grade Tonnage Reporting module or for displaying in the Plan View Publisher and Section View Publisher modules.

Figure 19: Recovery flagging dialog box

The Output PageAn output model and various reports can be specified on this page (Figure 20).

Figure 20: Output dialog box1. Enter the name of a model that will contain the optimized pit or leave the name blank if the model is not required.

The model produced consists of a single stratigraphic unit. The cell elevations in this model depict the economic depth of that part of the model according to the costing parameters specified. The model can be contoured in the PLAN VIEW PUBLISHER.

WARNING: Do not provide a name that is the same as an existing model because the existing model will be overwritten.2. Enter the name of the report file or leave the name blank if this report file is not required. If the extension of the file is RPT (e.g. report.rpt) then a plain text file is produced. If the extension is HTML, however, then a rich report is produced and can be viewed in any HTML browser.

3. Enter the name of the Microsoft Excel report file or leave the name blank if this report file is not required.

4. Enter the name of the Microsoft Word report file or leave the name blank if this report file is not required.

5. Enter a description if required. This is printed after the header of the report.

6. Control the level of detail by selecting the sections to display in the reports.

7. The RPT or HTML report can be displayed when the optimization is completed.

8. Invalid or small relative densities can be displayed in the progress dialog if required.

APPENDIX A: THE PIT OPTIMIZATION ALGORITHMThis appendix provides an overview of the algorithm as presented in Lerchs and Grossmann's paper (Lerchs & Grossman, 1965). It is assumed that the reader is at least partially familiar with its contents.Basic ConceptsThe Lerchs-Grossmann algorithm makes use of the fact that the block model of an orebody can be represented as a weighted directed graph in which the vertices represent blocks and the arcs represent mining restrictions on other blocks. The graph is said to contain the arc (x,y) if the mining of block x is dependent upon the removal of block y, hence the use of the term "directed".The profit resulting from the mining of any given block is represented by an appropriate vertex weight (the economic value of the block). It is due to the presence of these economic values that the graph is said to be "weighted".The closure of this weighted directed graph is defined as the set of vertices of the graph such that, if x is a member of this set and (x,y) is an arc of the graph, then y must also be a member of this set. That is, if block x is designated as forming part of the optimum pit and block y must be removed in order to obtain access to block x, then block y must also be part of the optimum pit. This definition ensures that a closure of the graph always represents a feasible pit contour and, as a result, the problem of determining the optimum pit contour is equivalent to the graph theoretical problem of finding a closure of maximum weight for the graph.A directed graph is said to be a tree if its underlying graph is connected (there are no breaks in it) and there are no cycles (circular block dependencies).A rooted tree is a tree with one distinguished vertex called the root. The graph obtained by deleting the arc (x,y) from a rooted tree has two components: the component of the graph that contains the root vertex of the tree, and the remainder. The remainder is referred to as a branch and is said to be supported by the arc (x,y).The end of this arc (either x or y) that lies within this branch is known as the root of the branch. If y is the root of the branch then the arc (x,y) is called a plus arc (p-arc). Otherwise, the arc is called a minus arc (m-arc).The branch is said to be strong if either of the following conditions are true:1. The arc (x,y) is a p-arc and the sum of the weights of the vertices in the branch is greater than zero, or2. The arc (x,y) is an m-arc and the sum of the weights of the vertices in the branch is less than or equal to zero.The arc (x,y) is classified as strong or weak according to the status of the branch that it supports. Any given vertex in the tree is said to be strong if there is at least one strong arc on the unique path joining that vertex to the root vertex of the tree.A rooted tree is referred to as being normalised if the root vertex of that tree is common to all arcs that are strong.The Lerchs-Grossmann algorithm is based on two theorems:1. The maximum closure of a normalised tree is the set of that tree's strong vertices.2. If, in a directed graph, a normalised tree can be found such that the set of strong vertices in this tree constitutes a closure of the graph, then the set of strong vertices is the maximum closure of the graph.The AlgorithmThe Lerchs-Grossmann algorithm is summarised below:1. Construct a normalised tree. This can be done by setting up a tree, which has an arc set, consisting of the arcs from the root vertex of the tree to every other vertex in the tree, with the appropriate weight applied to each vertex.2. Identify the set of strong vertices in the tree and search the graph for an arc that has a start vertex which is strong and an end vertex which is not. This is equivalent to finding an arc which causes the set of strong vertices in the tree not to be the maximum closure of the graph. If no such arc can be found, this tree is the key to the optimum contour and the process terminates.3. Identify the unique path from the start vertex of the arc found in Step 2 to the root vertex of the tree and determine the last arc on this path. Construct a new tree by replacing this arc with the one found in Step 2.4. Normalise the new tree. Loop back to step 2.

APPENDIX B: FILE FORMATSB.1Economic ValuesSYMBOLIC:ECONCONTENTS:Economic values of every block in the modelPREP. BY:User (probably by way of an external program)FORMAT:Direct access, unformatted 8-byte recordsRecord number X contains the economic value of block X in double precision (REAL*8).B.2Working Tree FileSYMBOLIC:WORK or WRKCONTENTS:Working TreePREP. BY:Lerch-Grossman module (temporary file)FORMAT :Direct access, unformatted 12-byte records made up of one 4-byte integer (next vertex on the path from this vertex to the root vertex) and one 8-byte double precision float (the weight supported by the vertex).Record number X contains the next vertex on the path from the vertex X to the root vertex and the weight supported by vertex X.B.3ResultsSYMBOLIC:RSLT or RESCONTENTS:Coordinates of every block that forms part of the optimum pit.PREP. BY:Lerch-Grossman moduleFORMAT :Sequential access, unformatted 6-byte records. The three fields in the record contain the x, y and z coordinates respectively of a block in the optimum pit. (The coordinates are given with respect to IMS model coordinate system).

APPENDIX C: ERROR MESSAGESOPALMN 001 (Fatal):Internal Accuracy Failure, Results Are Meaningless.Occurrence:This message is issued if the block to which all the blocks that are not to be included in the optimum pit are attached is included in the optimum pit. The results file will not be created.Action:The value of the variable NEVER should be checked. It should be, compared to the economic values assigned to the blocks in the model, a very large negative double precision floating point number. If the value appears to be satisfactory or if it cannot be increased any further, try scaling the economic values assigned to the blocks.OPVECT 001 (Fatal):No Block Dependencies Were Able To Be Established.Occurrence: This message issued when no block dependency vectors have been generated.Action:Check the specification of the slope restrictions.

REFERENCESCaccetta, L., & Giannini, L. (1986). Optimisation Techniques for the Open Pit Limit Problem. Bull. Proc. Australas. Inst. Min. Metall, Volume 291, No 8.Lerchs, H., & Grossmann, I. (1965). Optimum Design of Open-Pit Mines. Transactions, C.I.M. Volume LXVII, 17-24.

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