1. Introduction. 2. The Probable Stope Algorithm

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1 1. Introdution Optimization in underground mine design has reeived less attention than that in open pit mines. This is mostly due to the diversity o underground mining methods and omplexity o underground mining parameters, whih do not allow the development o general optimization tools, espeially, sotware tools or omputer programs. However, ew algorithms are available or optimization o ultimate stope boundaries, whih are either tailored or a speii mining method (i.e. rigorous algorithms) or lak rigorous mathematial proo and ail to guarantee the true optimum (i.e. heuristi algorithms). Fewer sotware tools are available based on these algorithms. The heuristi approahes inlude the Floating Stope tehnique o Datamine and the Maximum Value Neighborhood (MVN) algorithm. Both algorithms provide 3D analysis o the solution and an be applied to any underground mining methods. Nevertheless, they lak mathematial proo and ail to guarantee the true optimum solutions. All these algorithms apply on an eonomi blok model o the ore body. Based on the Floating Stope tehnique a sotware tool has been developed by Datamine Sotware Company [1], whih is the most popular and proessional sotware applied to the problem. A non-ommerial program alled Stope Limits Optimizer (SLO) has been developed based on the MVN onept using Fortran 90 ode. An appliation o the SLO or stope limits optimization on a hypothetial 3D model has been reported [2]. Rigorous algorithms inlude the Dynami Programming solution, the Branh and Bound tehnique and the Probable Stope algorithm, whih are developed or ew underground mining methods and supported by mathematial reasoning. The dynami programming tehnique provides a 2D analysis and has been adjusted only or blok aving method. A Fortran IV ode was prepared or this algorithm and has been run over some hypothetial models [3]. The branh and bound tehnique provides a 1D solution to the optimization o stope geometry and determines the start and the end o a stope in one diretion. Several ommerial, mathematial programming omputer sotware pakages provide Branh and Bound tehnique ailities. MPS and LINDO are two examples o the most popular pakages [4]. An appliation o this tehnique has been reported by MPS sotware or optimization o stope layout in the Pea Ridge iron mine [5]. The Probable Stope algorithm provides a 2D analysis and enjoys omplying with all underground mining methods used or tabular deposits [6]. A Visual C++ program is developed to implement the Probable Stope algorithm, whih is alled Stope Boundaries Optimizer (SBO). This paper, desribes the SBO program in details and the orresponding algorithm briely. 2. The Probable Stope Algorithm The Probable Stope Algorithm (PSA) is based on the Dynami Programming tehnique, whih is supported by mathematial proo. The algorithm is applied on a 2D onventional eonomi model. So, it is best suited or vein type or tabular deposits. In at, harateristis o the ore body in the third dimension are projeted on the model plane, i.e. omposited into one blok. The PSA omplies with all underground mining methods used or tabular deposits [6]. 2.1 The Modeling o the Mining Area The PSA applies on a 2D onventional ixed eonomi blok model o the mining area, whih is onstruted using the geologi, tehnial and eonomi data, olleted through the oneptual design. This is alled, in this paper, the primary model. Then through two stages, the probable stope (PS) model and the integrated probable stope (IPS) model are derived rom the primary model. The algorithm is then implemented on the IPS model. Ι) The Primary Model The primary model is a 2D onventional ixed eonomi blok model, whih may be thought o as a 2D matrix, where eah entry, m y,x, represents the net eonomi value o a blok loated in the y th row and the x th olumn. The matrix has X rows representing the total number o bloks in the diretion o ore body s strike and Y olumns representing the total number o bloks in the ore body s dip diretion. Figure 1 illustrates a simple example o suh primary model ontaining 10 bloks in the strike diretion and our bloks in the dip diretion. 8 ۷۷

2 A Computer Program to Optimize Stope Boundaries Using Probable Stope Algorithm SL (,),x and m y,x is the net value o the blok loated at row y and olumn x o the primary model. Figure 1: An example o the primary model Π) The PS Model In the PS model the stope height onstraint is onsidered. A probable stope in eah olumn inludes a set o adjaent bloks in the same olumn o the primary model whih satisy the minimum height o the stope. In order to make the PS model, all probable stopes in eah olumn that satisy the minimum height onstraint, are reviewed and their total eonomi values are alulated. The number o suh probable stopes in eah olumn o the model depends on the stope height onstraint (i.e. the restrition in the dip diretion). Consider that the stope height is restrited to a minimum o h bloks, the total number o probable stopes may be obtained through Equation (1): Y NS x, h = ( y h + 1), 1 x X (1) y= h where NS x,h is the number o probable stopes with a minimum o h bloks loated at the x th olumn, h is the minimum stope height measured in the number o bloks, x is the olumn number, X is total number o olumns, y is the row number and Y is total number o rows. The loation o eah probable stope in eah olumn, x, is deined as SL (,),x, where SL is the stope loation and and whih orm a pair, represent the loation o the loor and eiling o the stope in olumn x. For example, i the stope ontains three adjaent bloks rom the irst, seond and third rows o olumn x (i.e. bloks with net eonomi values m 1,x, m 2,x and m 3,x ), then its loation is addressed by SL (0,3),x. Figure 2 shows the loation o eah o the three probable stopes in olumn x or a model with our rows. The net eonomi value o eah stope may be obtained through Equation (2): M (, ), x = y= m y+ 1, x For 1 x X, 0 Y-h, h Y Where M (,),x is the net value o the stope loated at (2) Figure 2: Loation o probable stopes in the x th olumn o primary model [6] One the net value o probable stopes are alulated, the probable stope model may be onstruted. The total number o olumns in this model is the same as that o the primary model and its total number o rows is one more than the number o probable stopes in eah olumn (i.e. NS x,h+1 ). Figure 3 shows the probable stope model o the example shown in Figure 1, assuming a three blok height restrition. The lowest row indiates the net value o stopes at the loation SL (0,0),x, whih are in at virtual stopes with zero (blok) height. In at the PS model ontains, in eah row, the eonomi value o one probable stope (M (,),x ). Figure 3: The PS model ΙΙΙ) The IPS Model The IPS model is onstruted by imposing the minimum stope length onstraint. The minimum length o the stope depends on ators suh as the mining method and the equipment used. Eah blok o the IPS model ontains the maximum eonomi value o the stope ending at that blok, while satisying the minimum length onstraint. I the minimum stope length equals l bloks, l virtual olumns are added to the let o the PS model to provide easible stopes, otherwise none o the bloks loated in the irst l bloks o the model may orm a stope with a minimum length o l bloks. The net value o all stopes, whih are not entirely 9 ۷۶

3 loated inside the primary model (those with olumns j=1 to j= l) is set to a very large negative value. Another restrition that must be onsidered is the maximum variation o the elevation o both the loor and the eiling o the stope rom one olumn to the next. This onstraint is expressed as the number o bloks. Consider that SL (,),x represents the stope loation at olumn x, then the stope loation at olumn (x+1) may assume one o the under the ollowing onditions: + r 0, + r Y, ( + r) ( + r ) h r = 0,1,2,..., n r = 0,1,2,..., n,, n < h n < h Where r is an integer number representing the loor level variation onstraint, r is an integer number representing the eiling level variation onstraint, n is maximum number o bloks o the loor level variation onstraint, n is maximum number o bloks o the eiling level variation onstraint. When dealing with variation in the stope levels, are should be taken not to violate the minimum height onstraint. The expression ( + r) ( + r ) h has been inluded in the list o onstraints above to guarantee that this ondition is satisied. Imposing the level variation onstraint, the maximum net value o stopes with the minimum allowed length may be alulated through the ollowing relations [6]: i, = 0 MSV 0 (3) = 0 ( 0,0), j = i 0, 0 -u MSV(,),j = M ( Subjet to: + r 0 r = 01,, 2,...,n,),j-l l-1 r = 1 max, r = 01,, 2,...,n Where 1 j < l { M } ( + r,+ r),j-l r+ 1 Where l j J, + r I, ( + r) ( + r) h where MSV (,),j is the maximum eonomi value o the stope with minimum allowed length, ending to SL (,),j, u is a large positive number, l is the minimum stope length allowed, M (,),j is eonomi value o the stope at SL (,),j and r is a ounter. In at, eah ell o the IPS model represents the maximum net value o a stope with the minimum length and ending to that ell. Figure 4 illustrates the inal model orresponding to the PS model o Figure 3. (4) 2.2 The PSA Logi The PSA is proposed to optimize the stope boundaries, whih is run on the IPS model desribed above. A Dynami Programming tehnique is employed using a reursive ormula, similar to that used or open pit limit optimization. However, the ormula onsists o two riteria. By applying the algorithm to eah stope at a speiied loation, the eonomi value o all probable mining areas ending to that stope are alulated and the maximum one is speiied or the IPS model. Figure 4: The IPS model developed using Figure 3 The reursive ormula is as ollows [6]: 0 Where 1 j < l P( 0,0), j = max { P( + r, + r), j 1, P( 0, 0 ),j-1 } Where l j J (5) - u P max (, ), j = + MSV Subjet to : + r r = 0,1,2,..., n { P, P } ( + r, + r), j l (,),j 0, + r I, + NSVP (0,0), j l (,),j r = 0,1,2,..., n Where 1 j l Where l < j J, ( + r) ( + r ) h where P (,),j is the maximum net value o the mining area ending to the stope loated at SL (,),j, P (0,0),j is the maximum net value o the mining area ending to the stope loated at SL (0,0),j and NSVP (,),j is new stope value parameter. The value o NSVP (,),j is alulated only at two irumstanes. The irst one is when the algorithm starts and there is no stope ormed yet. The seond one is when the limits o the stope is passed and a new stope is going to be ormed. The value o NSVP (,),j is assigned zero, otherwise [6]. Figure 5 illustrates the appliation o the algorithm on the inal model or n =0, n =1 and l=3, with the orresponding limits. As Figure 5 shows, the optimum limits provide the maximum value o 32 units [6]. (6) 10 ۷۵

4 A Computer Program to Optimize Stope Boundaries Using Probable Stope Algorithm Figure 5: An appliation o the PSA [6] 3. The SBO Program A omputer program is developed to implement the PS algorithm desribed above, using Visual C++ omputer programming language. The program is alled Stope Boundaries Optimizer (SBO); the program is implemented on a 2D onventional eonomi blok model o the ore body to loate the optimum mining area, whih provides the maximum proit. The algorithm, whih is the base or developing the SBO program, is apable o deining the onstraints imposed to all onventional underground mining methods used or mining tabular deposits. These onstraints inlude the minimum length and height o the stope. The struture o the SBO program is designed suh that the optimization proess is divided into a number o steps. Firstly, the PS model is onstruted. Then, the PS model is modiied to build the IPS model and reord all relevant inormation. Next, the Probable Stope algorithm is run over the IPS model. The implementation begins rom the let o the IPS model and ontinues to the right. During the run o the program, the inormation resulted rom proessing the model, whih shows the eonomi values o the proessed areas are reorded or urther use. The SBO may be run on usual personal omputers The Input Data The input data related to the primary model are provided with a text ile, onsisted o three olumns. These olumns represent the X-oordinate, Y-oordinate and the net eonomi value o eah blok (BEV) o the primary model. Table 1 illustrates an example o the input ile to the SBO program. These piees o inormation are provided based on the eonomi model shown in Figure 2 and are saved as a text ile alled INPUT01.DAT. (For saving the spae, some parts o this ile have been omitted). The SBO input ile may be saved as any name and extension. Table 1: The SBO input ile orresponding the primary model, shown in Figure 1 File ame: INPUT01.DAT X Y Values : : : : : : Ater saving the data related to the eonomi model in a text ile, the program may be run to omplete the input inormation. The SBO program reeives three types o inormation through a dialogue box menu ormed by visual C++ ailities. The irst group inludes the general inormation suh as the projet title, the name o the input ile, in whih blok addresses and eonomi values are stored and the name o the output ile. The seond type o input inormation is related to the struture o the primary model and the system o addressing in the input ile, inluding the distane between two adjaent bloks (entre to entre), in both the strike and dip diretions. The third type o inormation reers to the stope speiiations based on the seleted mining method, inluding the minimum length and height onstraints o the stope and the maximum variation o the elevation o both the loor and eiling o the stope rom any olumn to the next one. Figure 6 shows the sub-menu o the SBO menu-bar or entering the input data and type o output data. In the irst setion, the names o both the SBO input and output iles are presented or the example, shown in Figure 1. Blok dimensions in the strike and dip diretions have been assigned 10 and 15 units, respetively. The minimum stope length and height are both assumed to be three bloks. The maximum variations in the level o the loor and eiling are assigned zero and one blok, respetively. Ater entering the irst group o inormation, the preliminary proess o inormation is begun. The proess inludes reading the input ile and 11 ۷۴

5 ontrolling the reorded inormation. I there is any mistake in the input data, a onvenient error message is prompted to the user or any data orretion. The most important mistakes here are inding two or more reords with various values but idential addresses. Figure 6: The SBO menu or entering input data and seleting output ormat. Ater it is made sure that the input data are errorree (data veriiation), the program reeives the seond type o inormation. Then, the SBO proeeds to proess the seond type o data. The input data are heked to be error-ree and a onvenient message is prompted to the user to either orret the input data or proeed to the next step. The most important ause o halting the proess is lak o ompatibility o the size o the bloks with the size o the model in both the X and Y orthogonal diretions, whih results in non-integer number o bloks. I so, an adjustment should be made to orret the input data. Provided that input data are veriied and aeptable, the program omputes the ranges o values that the minimum stope length and height and the maximum variation o the stope level (or both the loor and the eiling) may assume. These ranges are then given to the user to selet a hoie among them. The user may then enter the third group o inormation. One all groups o inormation are entered through the main menu, the SBO program reates the output ile and prompts the name and path o the output ile to the user The Output Results Ater all required data are entered and veriied, the SBO reates the primary, PS and IPS models as some matries to ind the eonomi values o various mining areas. These matries are reated to; only, provide the means to ontrol the orretness o the input data. All results are stored in a ile, whose name and extension are already deined through the main menu, shown in Figure 6. As shows in the igure, two ormats are available to rom the results and user may selet one o them. Example o these piees o inormation reorded in the output ile (OUTPUT01.DAT) exluding the primary, PS and IPS matries, generated by SBO, are shown in Table 2 and 3. The most important expeted results o using SBO are the determination o the optimum boundaries o the mining area and its net value. In order to ind the orresponding limits, the SBO program generates the matrix, shown in Table 2. This matrix orresponds the primary model as well and its entries have the value o 1 or 0. All bloks that should be mined to satisy the onstraints and orm the optimum stope boundaries are assigned the value 1. Other bloks that should be let unmined are assigned the value 0. Thereore, the ultimate stope boundaries are deined on the primary model. The output ile o the SBO program also ontains a list o bloks, whih all within the optimized boundaries. The list inludes inormation suh as the X and Y oordinates and the net eonomi values o the bloks. Table 3 shows an example o this list, presented at the end o the SBO output ile. As shown, the total value o the mining boundaries equals 32 units. Table 2: The optimized stope boundaries The Optimum Stope Boundaries ۷۳

6 A Computer Program to Optimize Stope Boundaries Using Probable Stope Algorithm Table 3: List o bloks o the optimum stope boundaries The Mineable Bloks X Y Value Total Value= A Numerial Example In order to validate the SBO program, a model setion previously optimized using other tehniques, is studied. Figure 7 shows a 2D eonomi blok model employed by Riddle (1977) to optimize the stope limits, using a Dynami Programming algorithm, provided that the deposit is mined using blok aving method. Mining onstraints were assumed the same as what used by Riddle to ompare the results. The minimum length and height onstraints were assigned two bloks and the maximum variation in the loor and eiling level were assigned zero and one blok, respetively. The provided inormation about the primary model is reorded in a ile named INPUT02.DAT and the input data are shown in Table 4. Figure 7: A worked example o the primary model Some parts o the results or the worked example, reorded in the SBO output ile, are shown in Table 5 and 6. Table 6 shows the optimized stope boundaries as a matrix orresponding the primary model by 1 and 0 entries. All entries with value o 1 reer to bloks that should be mined and other entries that have the value o 0 indiate to bloks that should be let in modeled mining area. Table 6 shows a list o bloks that all within the optimized boundaries. As shown in the end o the ile, the maximum value o the stop boundaries equals 77 units. In order to ompare the results obtained using SBO with those obtained using Riddle s Dynami Programming tehnique, the optimized layout o the stope imposed on the primary model using both alternatives are presented in Figure 8. A omparison between the results shows that using the SBO program provides a stope with a higher net value as well as ore ontent. Table 4: The SBO input data or the worked example Projet Title A Numerial Example Input File Name INPUT02.DAT Output File Name OUTPUT02.DAT Strike Interval 1.00 Dip Interval 1.00 Minimum Stope Height 2 Minimum Stope Length 2 Maximum Floor Level Variation Constraint 0 Maximum Ceiling Level Variation Constraint 1 Table 5: Stope boundaries o implementing SBO on the worked example The Optimum Stope Boundaries ۷۲

7 Table 6: List o bloks o the optimized stope boundaries or the worked example The Mineable Bloks X Y Value Total Value=77.00 a) Optimization layout using SBO b) Optimization layout using Riddle s program [3] Figure 8: SBO vs. Riddle s program 5. Conlusions The SBO program has been written by visual C++ ode, based on the probable stope algorithm. The SBO program reates a probable stope model and integrated probable model, respetively, onsidering the tehnial onstraints o the underground mining method, inluding the minimum length and height o the stope; determines eonomially optimum stope boundaries using the probable stope algorithm; and reords the outputs in a data ile, inluding the maximum values o all probable stope boundaries, the address o all bloks, whih all within the optimum limits and their total value. The SBO omplies with all underground mining methods used or tabular deposits. The SBO is user riendly and its appliation to optimal stope design is very simple and also an be operated with very little training. 6. Reerenes [1] C. Alord, "Optimization in Underground Mine Design", in 1995 Proeedings o the 25th International Symposium on the Appliation o Computers and Operations Researh in the Mineral Industry, The Australasian Institute o Mining and Metallurgy, Melbourne, pp [2] M. Ataee-pour, "A Heuristi Algorithm to Optimize Stope Boundaries", Ph.D. dissertation, Dept. Mining. Eng., University o Wollongong, [3] J. Riddle, "A Dynami Programming Solution o a Blok-Caving Mine Layout", in 1977 Proeedings o the 14th International Symposium on the Appliation o Computers and Operations Researh in the Mineral Industry, Soiety or Mining, Metallurgy and Exploration, Colorado, pp [4] J. Ovani, and D. Young, "Eonomi Optimization o Stope Geometry Using Separable Programming with Speial Branh and Bound Tehniques", in 1995 Third Canadian Conerene on Computer Appliations in the Mineral Industry, H. Mitri (ed.), Balkema, Rotterdam, pp [5] J. Ovani and D. Young, "Eonomi Optimization o Open Stope Geometry", in 1999 Proeedings o the 28th International Symposium on Computer Appliations in the Minerals Industries, K. Dagdelen (ed.), Colorado Shool o Mines, pp [6] S. E. Jalali and M. Ataee-pour, "A 2D Dynami Programming Algorithm to Optimize Stope Boundaries", in 2004 Proeedings o the 13th symposium on Mine Planning and Equipment Seletion, (eds. M. Hardygora et al.), Rotterdam, Balkema, pp ۷۱