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O.R. Applications

The new Fundamental Tree Algorithmfor production scheduling of open pit mines

Salih Ramazan

Rio Tinto, GPO Box A42, 152–158 St. Georges Tce, Perth, WA 6000, Australia

Received 10 July 2005; accepted 31 December 2005Available online 20 March 2006

Abstract

The problem of annual production scheduling in surface mining consists of determining an optimal sequence of extract-ing the mineralized material from the ground. The main objective of the optimization process is usually to maximize thetotal Net Present Value of the operation. Production scheduling is typically a mixed integer programming (MIP) typeproblem. However, the large number of integer variables required in formulating the problem makes it impossible to solve.To overcome this obstacle, a new algorithm termed ‘‘Fundamental Tree Algorithm’’ is developed based on linear program-ming to aggregate blocks of material and decrease the number of integer variables and the number of constraints requiredwithin the MIP formulation. This paper proposes the new Fundamental Tree Algorithm in optimizing production sched-uling in surface mining. A case study on a large copper deposit summarized in the paper shows substantial economic ben-

efit of the proposed algorithm compared to existing methods.Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Large scale optimization; Fundamental Tree Algorithm; Production scheduling; Open pit mine optimization

1. Introduction

Solving the annual production scheduling prob-lem is crucially important for surface mining since

it determines the rate and quality of the productioninvolving large cash flows, which can be hundreds of millions of dollars in magnitude. The schedulingproblem is determining the optimal sequence of extracting the mineralized material from theground, so that the total net present value (NPV)from the operation will be maximized, subject to aset of operational and physical constraints.

In mining operations, a mineralized zone is firstidentified through exploration which involves dril-ling, mapping and geological interpretation. Anorebody model is then developed to numerically

represent the mineral deposit by dividing the fieldinto representative three dimensional rectangularblocks. Each block is assigned attributes of thedeposits such as the grades of elements that areestimated using information obtained by samplingdrill cores. These attributes are used to estimatethe economic value of each block, which is the rev-enue expected to be generated from selling themetal contained, minus the cost of mining andprocessing.

0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2005.12.035

E-mail address: [email protected]

European Journal of Operational Research 177 (2007) 1153–1166

www.elsevier.com/locate/ejor

mailto:[email protected]

mailto:[email protected]



Subsequently, the orebody model containing thedeposit attributes and economic expected value areused to determine final pit limits, which are the lim-its of the deposit up to which it is economic to mine.Hochbaum and Chen (2000) provide detailed dis-

cussion of commonly used methods in finding ulti-mate pit limits.Annual production scheduling is a decision prob-

lem, that is; which blocks within the final pit limitsshould be mined in which year to maximize the totalNPV within a set of defined constraints. The miningblocks are aggregated into larger volumes that eachof them can be mined out in a year. The problem isfinding the best way to aggregate these blocks sothat NPV can be maximized when these volumesare mined.

The main physical constraint in open pit mining

is the slope requirement; all the blocks on top pre-venting mining of a given block must be mined.Fig. 1 shows a cross-sectional view of a hypothetical2D orebody model containing eight blocks whichare square in shape. The block identification num-bers are written on the left-top corner and expectedeconomic values are written in the center. If the safeslope angles are assumed to be 45°, blocks 1, 2 and 3would have to be excavated, or mined, before exca-vating block 6, or blocks 2, 3 and 4 have to be minedbefore block 7. In actual mining operations, the

blocks that have to be mined before mining a givenblock are identified by using a cone template whoseside angles are equal to the required slope angles forthe deposit. If a deposit has different slopes at differ-ent regions, multiple cone templates are used. Fur-ther discussions and illustrations of the slopeangles on a cone can be found in Ramazan (2001).

There are often too many blocks, over 100,000,within the final pit limits to determine the optimumannual production schedule. To reduce the size of the problem, it is a common implementation to par-tition the material within the final pit limits intosmaller volumes called ‘‘pushbacks’’ using one of the existing methods including Ramazan (1996),Seymour (1995) and Whittle (1988). The pushbacks

are used as a guide in determining annual produc-tion scheduling for the duration mine life.

Mixed integer and linear programming modelsare recognized as having significant potential foroptimizing production scheduling in open pit mines,

in which the objective is to maximize total dis-counted profit. However, MIP formulation of theproduction scheduling problem for open pit minesrequires too many binary variables making it verydifficult or impossible to solve. For example, if thereare 10,000 mining blocks in a pushback to be sched-uled over 3 years, it will require 30,000 binary vari-ables to generate the MIP formulation. This wouldmake it very difficult or even impossible to solve theMIP formulation.

Recognizing the strength of operations researchtechniques in optimization process, Johnson (1968)

developed an LP scheduling model. He appliedDantzig and Wolfe (1960) decomposition principlesto decompose the model into a master problem anda set of subproblems, which are solved using a max-imum network flow (maxflow) algorithm developedby Johnson (1968). However, this LP approach useslinear variables and leads to the mining of fractionalblocks. Dagdelen (1985) used the Lagrangiandecomposition method to decompose and solve alarge MIP problem. The drawback of the approachis that the Lagrangian method might not always

converge to an optimum solution if the Lagrangemultipliers cannot result in a feasible solution. Gers-hon (1983) presented an LP approach together withMIP models for optimizing mine scheduling, whichallowed partial block mining on the condition tothat the entire block preceding it has been mined.The author suggests that the models for optimizingproduction scheduling of open pit mines require toomany binary variables and cannot be solved.

Alternative efficient methodologies for long-termproduction scheduling that reduce the required num-ber of binary variables are presented in Ramazanand Dimitrakopoulos (2004). The maximum reduc-tion of the binary variables with the proposedmethods is down to the number of ore blocks, orpositive economic value blocks, in the model timesthe production periods minus one. However, thisreduction may not be sufficient for some large openpit mines. Tolwinski (1998) proposes a method thatcombines the blocks on the same bench, termed‘‘atoms’’, and uses the Lerchs and Grossmann(LG) method (Lerchs and Grossmann, 1965) togenerate pushbacks combining these atoms. The

approach generates a production schedule using

-1 -2 -2 -2 -2

+6 +3 +4

52 41 3Level 1

6 7 8Level 2

Fig. 1. A cross-sectional view of a small 2D block model.

1154 S. Ramazan / European Journal of Operational Research 177 (2007) 1153–1166



dynamic programming. However, combining blocksinto atoms may strongly reduce any possibility of theoptimal solution depending on the size of the atomswhich is not mentioned. The Milawa algorithm dis-cussed in Whittle (2000) considers a few benches at

each pushback as a variable and uses a search tech-nique called the ‘‘step and stride’’ algorithm dis-cussed in Warton (2000) to identify the regions of high value, rather than identifying individual miningblocks. This is a heuristic approach and does notguarantee an optimal solution. Godoy and Dimit-rakopoulos (2004) applied simulated annealing opti-mization method for scheduling a large gold mine.Although the method seems promising, it does notexplicitly include grade blending constraints in theirmodel for application to mines with blending prob-lems. Dimitrakopoulos and Ramazan (2004) devel-

oped an LP model that considers maximizing thechance of grade blending requirements and the feasi-bility of mining operations providing equipmentaccess to the blocks. This LP model needs more test-ing in terms of satisfying the sequencing constraintsfor the open pit mines that have significant depth, ormultiple blocks vertically.

In this paper, a new algorithm called ‘‘Fundamen-tal Tree Algorithm’’ is proposed to reduce the num-ber of binary variables required in MIP formulationsand the number of constraints within MIP for opti-

mizing annual production scheduling in open pitmines. The algorithm involves a newly developed lin-ear programming model formulation to combineblocks into Fundamental Trees. A set of combinedblocks is termed a ‘‘Fundamental Tree’’ if the com-bined blocks have the three properties: (i) can bemined without violating the slope constraints; (ii)the total economic value of the combined blocks asa fundamental tree must be positive; and (iii) a Fun-damental Tree (FT) cannot be partitioned into smal-ler trees without violating (i) and (ii). It should benoted that all the blocks within the pushback consid-ered for optimization must belong to a FundamentalTree. It is shown in this paper why the LP model gen-erates the Fundamental Trees (FTs) with the definedproperties. After generating the FTs for a given min-eral deposit, an MIP model modified from tradition-ally known MIP formulations is used to generateannual production schedules for mining.

2. Fundamental Tree Algorithm

The proposed Fundamental Tree Algorithm con-

sists of five steps that are schematically illustrated in

Fig. 2. The FT Algorithm is implemented to theblocks within a pushback that has to be determinedusing one of the true optimizing methods such asLerchs and Grossmann’s method as implementedby Whittle (1988), or Maxflow Algorithm by John-

son (1968). If the pushback is designed using a heu-ristic method such as implementations of thefloating cone method by Lemieux (1979), as pre-sented in Ramazan (1996) or Wang and Sevim(1993), the LP model formulations in Step 3, whichis discussed in the next section, would be infeasibleas discussed in Section 4.

The economic value of a block i (V i ) is calculatedbased on the expected net revenue (NRi ) to b egained from selling the contained metal withinblock i . If PCi is the processing cost, the value of the block is

V i ¼NRi À MCi À PCi if NRi > PCi

ÀMCi if NRi 6 PCi

; ð1Þ

where MCi is the cost of mining block i .Some definitions and explanations are provided

here to contribute clarification of the proceedingsections in the paper. If V i > 0, the block i is saidto be a ‘‘positive value block’’. Otherwise, it is saidto be a ‘‘negative value block’’. If a negative value

Step 5

Start from the ultimate pit limits or a pushback

Step 6

Generate the solution network

Step 7

Step 4

Step 3

Step 2

Step 1

Stop

Generate a starting network

No

Yes

Current Solution =Previous Solution?

Set up and solve the LP formulation

Assign coefficients to all the positive blocks

Find the cone value for all the positive blocks

Fig. 2. Steps of the Fundamental Tree Algorithm.

S. Ramazan / European Journal of Operational Research 177 (2007) 1153–1166 1155



block, say j , preventing mining two positive valueblocks, say i and k , under j and neither of the posi-tive value blocks have enough value to mine j

(V i + V j < 0, and V k + V j < 0), revenue from bothof the blocks i and k may need to support the cost

of miningj

if V

i +V

k +V

j > 0. When more thanone positive value block supports the mining costof a negative value block, it is called ‘‘joint support’’in mining. If there is a block, say j , that is located ontop of another block, say i , preventing block i to bemined, j is said to be an ‘‘overlying block’’ of i and i

is said to be an ‘‘underlying block’’ of j . The blocksthat are sent to the processing plant to extract themetal from the rock, NRi > PCi , are called ‘‘oreblocks’’ and otherwise ‘‘waste blocks’’. It shouldbe noted that there are more complex methods forclassifying a block as ore or waste, but this defini-

tion is used herein during scheduling process forsimplicity.

2.1. Steps of the FT Algorithm

The steps of the FT Algorithm illustrated sche-matically in Fig. 2 are discussed below:

Step 1. Generate a network for the blocks within apushback. In the network, the blocks arerepresented by nodes and the mining slope

requirement is represented by the arcs.The arcs in the network represent the nodeprecedence relationship within the pit. Anarc is set from each positive value node toall the overlying negative value nodes onthe upper levels that have to be removedbefore accessing the underlying positivevalue node considered for removal. Thisnode precedence relationship within thenetwork is determined by using one or morecone templates as discussed previously.

Step 2. Find the cone value CVi for each node i

having positive value within the network.The economic values of all the nodes con-nected to node i with an arc set from nodei are summed. This total economic valueof the nodes is said to be the cone valueof node i .

Step 3. Assign coefficients to the positive valuenodes according to their cone value. Thismay also be considered as ranking of thesenodes by elevation, or level. On the topmost level where one or more positive value

nodes exist, the node with the highest cone

value is assigned to 1, and the second high-est cone value node is assigned to 2, and soon. For instance, if there are 3 positivevalue nodes on that level, the node withthe smallest cone value is assigned to 3.

Then, the ranking process moves downone level. If there are some positive valuenodes on that level, the node with the high-est cone value is assigned to 4. Otherwise, alower level is searched for positive valuenodes. The process is performed for allthe positive value nodes within the network.If two or more positive value nodes on thesame level have the same cone value (tiecondition), the coefficients are assigned ran-domly; two nodes must not be assigned thesame coefficient. For example, nodes i and j

are on the same level, have the same conevalue and C max is the highest coefficientassigned previously, C i is set to C max + 1and C j is set to C i + 1. Since nodes i and j

have exactly the same cone values, theresults would be the same if coefficients of i and j are exchanged.

Step 4. Set up the linear programming formulationas discussed in Section 3. After the problemformulation is ready, it can be solved usinga commercially available solver.

Step 5. If the number of trees obtained is the same asthe number of the trees obtained from theprevious solution, go to Step 7 as the prob-lem is considered to be at optimal, and thealgorithm is stopped. If the number of treesis higher than the previous solution, keep thecurrently found connections between nodesand go to Step 6 to generate a new networkto be used for iterating the algorithm. Ini-tially, one can assume that whole networkis 1 tree. Usually, two or three iterationsare required for convergence.

Step 6. After obtaining a solution from LP modelin Step 5, a new network is generated forthis new solution. To generate the new net-work, first the arcs in the previous networkhaving no flow in the LP solution aredeleted. After arc deletion, the nodes thatare connected to each other are identified.The deleted arcs that exist in the previousnetwork between the connected nodes areadded again to the network. After generat-ing the network, go back to Step 2.

Step 7. Stop.




2.2. Application of steps of the FT Algorithm

Application of the Fundamental Tree Algorithmis discussed using an example that can be consideredas a cross-sectional view of some blocks on two con-

secutive elevations, or levels, given in Fig. 1. Thenode numbers are written on the left top corner of each block and the expected economic value frommining and processing a block is written at the cen-ter of each block.

Step 1. A network is generated as shown in Fig. 3.Since nodes 6, 7 and 8 have positive eco-nomic values, the arcs are set from thesenodes to the nodes on the upper level. Forsimplicity of the illustration, it is assumedthat the blocks are the same in size and

have to be mined with 45° slope angle inall directions. The arcs in the figure showthe node precedence relationships basedon the slope angle requirement. For exam-ple, in order to mine block 6, blocks 1, 2and 3 must be mined, or to mine block 7,blocks 2, 3 and 4 must be mined, and so on.

Step 2. If the economic value of node i is V i , thenCV6 = V 6 + V 1 + V 2 + V 3 = +6 À 1 À 2 À 2CV6 = +1. Similarly, CV7 = À3, CV8 = À2.

Step 3. Coefficients C i are assigned to positive

value nodes according to CVi value andthe levels where nodes are located. SinceCV6 is greater than CV7 and CV8, C 6 isset to 1; and since CV8 > CV7 C7 is set to3 and C 8 is 2.

Step 4. The initial problem formulation and itssolution are given in Fig. 4a and the net-work representation of the solution is inFig. 4b. The LP formulation, terms andnotations used on the figure are explained

in the next section. Fig. 4b shows that thereare now two trees, which is greater than theinitial one tree considering the initialassumption that starting network is a tree.Now the algorithm moves to Step 6.

Step 6. After obtaining a solution from the LPmodel in Step 5, a new network (seeFig. 4b) is generated. To generate the newnetwork, first the arcs in the previous net-work having no flow in the LP solutionare deleted. After arc deletion, the nodesthat are connected to each other are identi-

fied. The deleted arcs that exist in the previ-ous network between the connected nodesare added again to the network. For exam-ple, in Fig. 4b, if node 3 were connected tonode 7 by an arc, all the arcs in the initialnetwork would have to be kept since allthe nodes would belong to the same tree.That means although there is no flow on

436

-2-2-2

8

s

76

5432

-2

1

-1

t

Fig. 3. Network representation of the 2D block model in Fig. 1.

A source node s and a sink node t are added.

Minimise f 61+f 62+f 63+3f 72+3f 73+3f 74+2f 83+2f 84+2f 85

Subject Tof s6 <= 6f s7 <= 3f s8 <= 4f 1t = 1.001f 2t = 2.001f 3t = 2.001f 4t = 2.001f 5t = 2.001

f 61-f 1t = 0f 62+f 72-f 2t = 0f 63+f 73+f 83-f 3t = 0f 74+f 84-f 4t = 0f 85-f 5t = 0f s6-f 61-f 62-f 63 = 0f s7-f 72-f 73-f 74 = 0f s8-f 83-f 84-f 85 = 0

Variable Valuef S6 5.003000f S7 0.002000f S8 4.000000f 1t 1.001000f 2t 2.001000f 3t 2.001000f 4t 2.001000f 5t 2.001000

f 61 1.001000f 62 2.001000f 63 2.001000f 72 0.000000f 73 0.000000f 74 0.002000f 83 0.000000f 84 1.999000f 85 2.001000

Fig. 4a. The LP fundamental tree problem formulation on theleft-hand side in the first iteration and the solution on the right-

hand side.




the arc from node 8 to node 3 in the currentsolution, the arc would still be kept sincenodes 3 and 8 would be members of thesame tree. After generating the network,go to Step 2.

Step 2. The cone value of node 6 CV6 isunchanged, 1. Since node 2 and 3 belongto different tree than node 7, there is noarc between node 7 and nodes 2 and 3 in

the current network. So, the value of nodes2 and 3 are not considered in calculating thecone value of node 7. Since there is an arcfrom node 7 to only node 2, CV7 = +3 À2 = +1, CV8 = +4 À 2 À 2 = 0.

Step 3. Coefficients C i are assigned to positivevalue nodes according to new CVi values.C 6 is set to 1; and since CV7 > CV8 C 7 isset to 2 and C 8 is 3.

Step 4. The iterated problem formulation and itssolution are given in Fig. 5a and the net-work representation of the solution is inFig. 5b. The iterated LP formulation isexplained in the next section.

Step 5. Since the number of trees from the currentsolution 3 is greater than the previous num-ber of trees 2, the algorithm should nor-mally move to Step 6. However, sincethere is no tree containing more than onepositive value node in the results, it isimpossible for any tree to be partitionedinto sub-trees and repetition can be skippedin this case by going to Step 7.

Step 7. Stop the algorithm.

In the final solution, the trees have the threedefined properties of the fundamental trees as dis-cussed in Section 1. Each of the trees has positivetotal economic value; trees can be removed withoutviolating the slope constraints starting from the firsttree, then second tree and finally third tree; andnone of the trees can be partitioned into sub-trees

without violating the slope requirement and con-

Tree II

Tree I

36

-2-2-2

8

4

-1

s

76

5432

-2

1

t

Fig. 4b. The network representation of the initial solutioncontaining two trees surrounded by dashed lines.

Minimise f 61+f 62+f 63+2f 74

+3f 84+3f 85

Subject Tof

s6<= 6

f s7 <= 3f s8 <= 4f 1t = 1.001f 2t = 2.001f 3t = 2.001f 4t = 2.001f 5t = 2.001f 61-f 1t = 0f 62 -f 2t= 0f 63 -f 3t = 0f 74+f 84-f 4t = 0f 85-f 5t = 0f s6-f 61-f 62-f 63 = 0f s7 -f 74 = 0f s8 -f 84-f 85 = 0

Variable Value

f S6 5.003000f S7 2.001000f S8 2.001000f 1t 1.001000f 2t 2.001000f 3t 2.001000f 4t 2.001000f 5t 2.001000f 61 1.001000f 62 2.001000f 63 2.001000f 74 2.001000f 84 0.000000f 85 2.001000

Fig. 5a. The LP fundamental tree problem formulation on theleft-hand side in the first iteration and the solution on the right-hand side.

Tree II Tree III

Tree I

436

-2-2-2

8

-1

s

76

5432

-2

1

t

Fig. 5b. The network representation of the solution containingthree fundamental trees surrounded by dashed lines.




taining a total positive value. It is noted that treenumbers do not necessarily correspond to thesequence in which trees should be mined althoughit is the case in this small example. The miningsequence of the trees is determined using an MIP

production scheduling model that will be discussedin Section 5.

3. The LP formulation to generate fundamental trees

This section discusses the LP formulation of theFundamental Tree Algorithm. The objective func-tion is minimization of arc connections in thenetwork weighted by the assigned ranks. The objec-tive function is expressed as

MinXn

i

Xw

j C i f ij; ð2Þ

where C i is the coefficient for node i , which is for n

positive value nodes discussed in the previous sec-tion; f ij is the flow from node i to node j ; j is the in-dex for the negative value nodes connected to node i

with an arc coming from i . If there is a flow over anarc, the arc is kept on the network to be generatedfrom the output of this LP model.

If there is one or more positive value nodes onlevel 1, there is no need to include them in the for-

mulation because there are no arcs formed fromthese nodes. Since they already possess the definedproperties of the fundamental tree, they are onlyconsidered in calculating cone values at Step 2,but not in the LP formulation.

The objective function is constructed in a waythat the arcs will be set from high cone value nodesto support the negative nodes above them. In agiven level, it is considered that the highest conevalue node, say node i , has the highest chance of supporting all the negative value nodes above nodeI, preventing node i from being mined. Therefore, if the arcs to be constructed are started from the high-est cone value node (lowest objective coefficient),considering the model constraints, the number of

joint supports for negative value nodes will be min-imized. This ranking, or coefficients, in the objectivefunction together with the model constraints has themost effect in establishing the third property of FT,which an FT cannot be partitioned into sub FTs.The coefficients also have some role in making theFTs obey slope constraints although the effect isnot as direct as in Eq. (4). Since the arc connections

are prioritized from higher cone value nodes, the

Fundamental Trees are generated in a way thathigher value blocks become feasible for miningbefore the lower value blocks for the MIP schedul-ing model. This is a desirable condition for NPVmaximization objective of annual production

scheduling.A positive value node is limited in flow capacityto its value, which is constrained from source node,s, to positive nodes. The physical meaning of theseconstraints for mining is that the expected revenuefrom mining an ore block cannot support themining waste where the cost is higher than the rev-enue. The constraint formulation is expressed asbelow:

f si 6 V i for all i’s, ð3Þ

where f si is the flow sent from source node, s to nodei , vi is the economic value of block i , which is set foronly positive value nodes.

Constraints are imposed on the arcs going fromnegative valued nodes to sink node t to ensure thatthe cost of mining waste blocks are justified by theincome to be generated from mining the ore blocks.A small extra value n is also applied to negativevalue nodes to ensure that precedence relationshipwill not be violated by the trees. n is set to a verysmall number that will not be ignored by the solver

such as 0.001. These constraints ensure that mini-mum economic value of a tree is greater than, orequal to n, which is strictly positive. This is the firstpre-defined property of a Fundamental Tree.

Without using n value, if an overlying negativenode is fully supported by an underlying positivevalue node, the total value of negative and positivenodes could be zero without requiring a joint sup-port. That would not only generate zero value treesviolating the defined property of FundamentalTrees for being strictly positive, but also cause vio-lation in slope requirements, which is further dis-cussed in Section 4.

Note that if n is set to a number too high, the LPmodel may not produce fundamental trees with thedefined properties. The total of the added n valuesfor all the overlying connected nodes should be keptbelow 1. Otherwise, some trees may violate the lastpre-defined property of a Fundamental Tree; one ormore trees may be partitioned into sub-trees havingthe first two pre-defined properties of fundamentaltrees if n value is set too high. Since the economicvalues in mining are sufficiently large, thousands

in magnitude, they are rounded to integer values.




The functionality of this rounding is discussed inSection 4, Property II. This constraint formulationis expressed as

f jt ¼ ÀV j þ n; ð4Þ

where n is a small positive decimal number; V j is thevalue of the negative value node j ; and t is the sinknode.

The total flow coming to a negative value nodemust be equal to the flows leaving that node. If the number of positive value nodes that arcs areset from towards the negative value node j is O j ,then the mass balance constraints around each neg-ative value node is expressed as

XO j

i¼1

f ij À f jt ¼ 0. ð5Þ

The total flow coming to a positive value node i

from the source node must be equal to the total flowleaving that ore block. If the number of wasteblocks overlying positive node i is W i , the mass bal-ance constraints for the positive value nodes areexpressed as

f si ÀXwi

j¼1

f ij ¼ 0. ð6Þ

The initial LP formulation and solution for theexample network model given in Fig. 3 are illus-trated in Fig. 4a. Fig. 4b is generated by deletingthe arcs that are not used by the LP model fromthe initial network. The iterative LP formulation isgenerated using the current network of the systemas stated in Section 2.1 shown in Fig. 4b. The LPmodel and the solution are given in Fig. 5a. Thesolution network in Fig. 5b is generated by deletingthe arcs with no flow on them from the previous net-work. In the example, the number of binariesrequired is reduced from 8 to 3 by the use of FT.

After generating the fundamental trees for agiven orebody model, the annual production sched-uling can be formulated as an MIP model treatingeach tree as a block having certain attributes dis-cussed in Section 5.

4. Properties of the Fundamental Trees generated

by the FT Algorithm

The Fundamental Trees have three major prop-erties by definition as given earlier in Section 1.

These properties are discussed in this section.

Property 1. A tree obtained by the FT Algorithm

always has a total cumulative value greater than

zero. Eq. (3) ensures that the positive value nodes

cannot send more flow to the overlying negative value

nodes than their own value. Eq. (4) ensures that all

the negative value nodes are totally supported by

underlying positive value nodes. For a given tree,

constraints in Eqs. (3) and (4) can be written as

follows:

Xi

V i PXi

f si;

X j

ðÀV j þ nÞ ¼Xi

f si.

Therefore,

XiV i P

X j

ðÀV j þ nÞ;

whereP

iV i is the sum of all the positive value nodes

in a tree, and P

jðÀV j þ nÞ is the sum of all the neg-

ative value nodes in the tree plus the small epsilon as-

signed to those nodes. Therefore, the minimum value

of a tree is the sum of all the epsilons assigned to

the negative value nodes within the tree. Since n is al-

ways a positive number, the total economic value of a

tree generated by the FT Algorithm is always greater

than zero. In the given example, the first and second

trees both have a cumulative value of +1 and the third tree has a value of +2.

The property of the trees to be positive may raisethe question: what happens if a tree has negativevalue. It is stated in Section 2 that the FT Algorithmis implemented to the blocks within a pushback thathas to be determined using one of the true optimiz-ing methods. This means that the total value of allthe blocks, or nodes, considered must be positive.If all the blocks are connected to each other, form-ing one tree, the total value is positive and the entire

pushback forms a feasible solution for the LP for-mulation. Therefore, there is at least one feasiblesolution to the LP formulation although it is notthe optimal solution in most cases.

Property 2. Slope constraints are not violated when

fundamental trees are removed from the network one

at a time respecting the sequencing between them. It is

shown in Property 1 that all the overlying negative

value nodes connected to positive value nodes in the

initial network must be supported by underlying

positive value nodes having arcs towards the negative

value nodes, due to the model constraints. Since




Property 1 provides a feasible solution to the Funda-

mental Tree LP problem, all the overlying negative

value nodes will be supported.

The arcs in the initial network are set from eachpositive value node to all its overlying negative

value nodes which are forced to be fully supported.In a situation where a positive value node, say i ,cannot fully support all its connected overlyingnodes, the overlying node currently considered forsupport will require extra support from anotherpositive value node, say k . Due to the small n, theflow capacity of the positive value node i cannotbe fully consumed for supporting overlying negativenodes without requiring support from another posi-tive value node k . It is because of the fact that theflow capacities (economic values of blocks) of

the nodes are rounded to integers before settingthe LP formulations whilst n is a small decimalnumber. Therefore, either the positive value node i

will have to be able to support all the overlying neg-ative nodes, or it will have to require support fromanother positive value node k to the same overlyingnegative value node where the flow capacity of thenode i is fully consumed. This characteristic of nmakes the resultant trees always obey the slopeconstraints.

Property 3. A tree found by the FT Algorithm

cannot contain a subset of trees that also can be a fundamental tree. If a sub-tree exists that is also a

fundamental tree, it means there is a set of one or

more positive value nodes in the sub-tree that has

sufficient flow capacity to support all its overlying

waste. If this were so, the LP solution would have

identified the sub-tree as a fundamental tree due to the

optimality of the objective function. It is always better

to send the flows from the smaller coefficient node in

the objective function than sending some of the flows

from a higher coefficient node since each coefficient in

the objective function is unique.

5. Optimization of annual production scheduling

using fundamental trees

MIP scheduling formulations of the trees aredone by treating each FT as a block having certainamount of ore tons with average grade of commod-ities and possibly some waste tons. Since binaryvariables are assigned to the FT’s instead of blocks,the number of sequencing constraints, given in Eqs.

(8) and (9) in Section 5.1, are also reduced signifi-

cantly. Traditionally, sequencing constraints arewritten for all the blocks below the first level inthe model whilst they are written only for FTs thatare substantially less than the number of blocks withthe proposed method. The substantial reduction in

MIP problem size by applying FT Algorithmreduces the required solution time significantly andthat makes it possible to apply MIP model to largeopen pit mines.

5.1. MIP production scheduling formulation with

Fundamental Trees

The optimization model is maximization of theNPV, or discounted economic value, to be gener-ated from the mining operation and hence for

scheduling N Fundamental Trees over Y years, theobjective function can be expressed as

MaxXY p ¼1

X N i¼1

fðDEV piÞ Ã Oip þ ðDWC pi Ã W ip Þg; ð7Þ

where Oip is defined as a binary variable for p < Y ,valued 1 if the ore ton in FT i is scheduled for period p, and 0 otherwise; when p = Y , Oip is the percent-age of the block scheduled in year p; W ip is a linearvariable representing the tons of the waste to bemined at period p from fundamental tree i ; DEV

pi discounted economic value to be generated frommining and processing all the ore tons in year p

from FT i ; DWCip discounted average cost of min-ing one ton waste from FT i in year p. If the sche-dule of the mine until the last year is optimizedusing binary variables, the last year’s variables donot need to be defined as binary since the sloperequirement will be satisfied from the previousyears. Further discussions on this issue can be foundin Ramazan and Dimitrakopoulos (2004).

If the economic discount rate is d , discounting

factor in year p, DF p, can be calculated by

DF p ¼ 1=ð1:0 þ d Þ p

. ð7aÞ

The economic value to be generated by processingall the ore tonnage at time 0 from FT i , DEV0i ,can be determined by

DEV0i ¼X j

V j;

where j is the index for all the ore blocks that be-longs to FT i . The objective function coefficient

DEV pi can be found by




DEV pi ¼ DEV0i Ã DF p . ð7bÞ

Similarly, the cost of mining the waste materialcan be calculated using Eq. (1) and DWCip is deter-mined by

DWCip ¼ DWCi0 Ã DF p ; ð7cÞ

where DWCi0 ¼ ðP

hV hÞ=T Wi, which is the cost of mining one ton of waste material; h is the indexfor waste blocks that belongs to FT i and T Wi isthe total tons of waste material within FT i .

The number of binary variables (NB) requiredfor the MIP model using FTs is equal to the numberof FTs multiplied by one less the total time period tobe scheduled, which can be expressed as, NB =N * (Y À 1).

It is very important to maintain correct slopeangles for the walls surrounding mined out areasin open pit mines. The slope cannot be too steepto prevent the side walls collapsing towards thezones of mining activity and it cannot be too shal-low so as to avoid extracting too much waste mate-rial which can substantially increase the miningcost. Precedence of the FTs in mining operationscan be identified using cone templates as in movingcone method (Lemieux, 1979). If j is the index forthe fundamental trees that have to be removedbefore being able to remove tree i in period p, thesequencing formulations is written for each j asfollows:

Oip ÀX p t ¼1

O jt 6 0. ð8aÞ

The sequencing formulation can also be writtenas in Eq. (8b), which is 1 constraint for all overlyingFTs instead of one constraint for each overlying FT.However, this type of formulation containing fewerconstraints in this way does not always reduce solu-tion time compared to Eq. (8a). The two types of

formulations are discussed in Ramazan and Dimit-rakopoulos (2004) in terms of solution times. If N i is the number of overlying FTs for FT i , Eq. (8b)is expressed as

N iOip ÀX N i j

X p t ¼1

O jt 6 0. ð8bÞ

Another set of constraints that may also be con-sidered as sequencing is that before mining the oretonnages from a given FT, all the waste tonnages

must be mined out in that FT. If T Wi is the total

tons of waste material within tree i , these constraintscan be set as follows:

T WiOip ÀX p t ¼1

W it 6 0. ð9Þ

To achieve a possible low cost at the processingplants, the ore material sent to the processing plantis sometimes constrained to have a grade (gram perton, ounce per ton, or % metal contained) withina range of values. Assume that G i is the averagegrade of FT i , and T Oi is the total ore tonnagein FT i , G min and G max are the periodical minimumand maximum grade range requirements at theprocessing plant. Then, the constraint formulationsfor n-fundamental trees and a period p is expressedas

P N

i¼1G iT OiOip

T OiOip

P G min;

which can be expressed in linear form as

X N i¼1

ðG i À G minÞT OiOip P 0; ð10aÞ

X N i¼1

ðG i À G maxÞT OiOip 6 0. ð10bÞ

The processing plant is limited by the amount of ore material it can process during a time period. Alower bound may also be applied in some cases toensure that the processing plant will not be idle,and to avoid or reduce undesirable fluctuations inthe production. For n-FTs and a period p, the limi-tations of the processing mill capacity as lower andupper bounds can be written as follows:

X N i¼i

T OiOip P PCmin; ð11aÞ

X N

i¼i

T OiOip P PCmax; ð11bÞ

where PCmin and PCmax are the minimum and max-imum capacity of mill.

Since the production scheduling is performedwithin the final pit, or a pushback, constraints areneeded to ensure that all the material consideredin optimization will be mined out. These constraintsensure that ore tons in each FT must be mined in asingle period:

XY

p ¼1

Oip ¼ 1. ð12Þ




Stripping ratio is defined as the ratio of totalamount of waste material mined to the amount of ore produced. At some mining operation, strippingratio is also limited at a maximum value. If the strip-ping ratio is SR, for a period p considering n-FT,

the constraints not to exceed a given stripping ratiolimit is defined as

SR ¼

P N

i¼1W ip P N

i¼1T OiOip

.

Then

X N i¼1

W ip À SRmax

X N i¼1

T OiOip 6 0; ð13Þ

where SRmax is the maximum stripping ratio al-

lowed during any year.The total amount of material mined during ayear cannot be more than the total mining capacityof all the equipments, MCap. This requirement inopen pit mining operations is expressed by

X N i¼1

ðW ip þ T Oi Ã Oip Þ 6 MCap. ð14Þ

In some of the mining operations, the total ton-nage of waste to be stripped out in a period isrestricted not to exceed a waste stripping amount

WCmax. For n-FTs and a given period p, the formu-lation is

X N i¼1

W ip 6 WCmax. ð15Þ

6. Application of the MIP model with

fundamental trees

The application of MIP model in annual produc-tion scheduling of open pit mines is shown using theexample containing three FTs. Although some of the constraints in this application may seem redun-dant, or the result may also be obvious, all theconstraints assumed to exist in this case are writtenfor illustration. The FTs determined previouslyare illustrated in Fig. 6 representing them as blocks.The scheduling requirements for the mine are setout in Table 1. Assuming a 10% economic dis-count rate, discount factors for the two years arefound as shown in Eq. (7a); DF1 = 1/(1.10)1 =0.91 and DF2 = 1/(1.10)2 = 0.83. Using these fac-

tors, discounted economic values for each FT and

each time period p can be calculated by Eq. (7b)as below:

DEV11 ¼ DEV10 Ã DF1 ¼ 6 Ã 0:91 ¼ 5:46

for FT ¼ 1; p ¼ 1;

DEV12 ¼ DEV10 Ã DF2 ¼ 6 Ã 0:83 ¼ 4:98

for FT ¼ 1; p ¼ 2;

DEV21 ¼ DEV20 Ã DF1 ¼ 3 Ã 0:91 ¼ 2:73

for FT ¼ 2; p ¼ 1;

DEV22 ¼ DEV20 Ã DF2 ¼ 3 Ã 0:83 ¼ 2:49

for FT ¼ 2; p ¼ 2;

DEV31 ¼ DEV30 Ã DF1 ¼ 4 Ã 0:91 ¼ 3:64

for FT ¼ 3; p ¼ 1;

DEV32 ¼ DEV30 Ã DF2 ¼ 4 Ã 0:83 ¼ 3:32

for FT ¼ 3; p ¼ 2.

Discounted cost of mining one ton of wastematerial for the example problem is calculated asfollows:

DWC01 ¼ 5=30 ¼ 0:167; DWC02 ¼ 2=10 ¼ 0:2

and DWC03 ¼ 2=10 ¼ 0:2.

Using Eq. (7c),

DWC11 ¼ 0:167 Ã 0:91 ¼ 0:152;

FT1 FT2 FT3Total Ore Value ($) 6 3 4Total Waste Cost ($) 5 2 2Ore Tons 18 10 10

Waste Tons 30 10 10Grade (%) 4.4 3.8 4.5

FT I

FT II

FT III

Fig. 6. Fundamental trees for the example block model with thegiven attributes.

Table 1Production scheduling requirements for the example miningproblem

Description Values

Total mining capacity 50 tonsMinimum ore input for processing plant (mill) 10 tonsMaximum ore input for processing plant (mill) 20 tonsMaximum stripping ratio (waste/ore) 2.0:1.0Minimum average grade at the mill plant 4.0%Maximum average grade at the mill plant 4.5%




and similarly,

DWC12 ¼ 0:138; DWC21 ¼ 0:182;

DWC22 ¼ 0:166; DWC31 ¼ 0:182 and

DWC32 ¼ 0:166.

By inserting the calculated values to the objectivefunction in Eq. (7), the following objective functionis obtained for the example problem:

Max5:46O11 þ 4:98O12 þ 2:73O21 þ 2:49O22

þ 3:64O31 þ 3:32O32 À 0:152W 11 À 0:138W 12

À 0:182W 21 À 0:166W 22 À 0:182W 31 À 0:166W 32.

The sequencing constraints between the secondtree and the first tree given in Eq. (8a) for the exam-ple are expressed as O21 À O11 6 0, that means if the ore from tree 2 is taken in period 1, the orefrom tree 1 must also be taken in period 1. The sameconstraint for the second period is written as O22 ÀO11 À O12 6 0, that means if the ore from tree 2 isexcavated in period 2, the ore from tree 1 must beexcavated in period 1, or period 2. Similarly, theconstraints for the third tree is written as O31 ÀO21 6 0 and O32 À O21 À O22 6 0. Since the con-straints from second tree to first tree are written,the constraints between the third tree and first treeare not necessary.

The constraints in Eq. (9) to represent therequirement of mining waste material within eachtree before being able to mine ore material are writ-ten for the example problem as 30 * O11 À W 11 6 0,which means if the ore within FT1 is mined in thefirst period (O11 = 1), then, the waste variable of the first tree in the first period, W 11, must be equalto the waste tons in that tree, which is 30 tons.

The same constraint for the second period is writ-ten as 30 * O12 À W 12 À W 11 6 0, which means if the ore material in the first tree is mined in the sec-

ond period, waste material in that tree can be minedin the first period, or second period. The same con-straints for the 2nd and 3rd FTs can be written sim-ilarly as follows:

10 Ã O21 À W 21 6 0;

10 Ã O22 À W 21 À W 22 6 0;

10 Ã O31 À W 31 6 0;

10 Ã O32 À W 31 À W 32 6 0.

The grade constraints given in Eqs. (10a) and

(10b) are illustrated for the problem as below:

ð4:4 À 4:0Þ Ã 18 Ã O11 þ ð3:8 À 4:0Þ Ã 10 Ã O21

þ ð4:5 À 4:0Þ Ã 10 Ã O31 P 0;

ð4:4 À 4:0Þ Ã 18 Ã O12 þ ð3:8 À 4:0Þ Ã 10 Ã O22

þ ð4:5 À 4:0Þ Ã 10 Ã O32 P 0;

ð4:4 À 4:5Þ Ã 18 Ã O11 þ ð3:8 À 4:5Þ Ã 10 Ã O21þ ð4:5 À 4:5Þ Ã 10 Ã O31 6 0;

ð4:4 À 4:5Þ Ã 18 Ã O12 þ ð3:8 À 4:5Þ Ã 10 Ã O22

þ ð4:5 À 4:5Þ Ã 10 Ã O32 6 0.

The processing capacity is assumed to be limitedwith an upper and lower bound in this model. Therequirements for this processing plant can beexpressed using Eqs. (11a) and (11b) as

18O11 þ 10O21 þ 10O31 P 10;

18O12 þ 10O22 þ 10O32 P 10;

18O11 þ 10O21 þ 10O31 6 20;

18O12 þ 10O22 þ 10O32 6 20.

The constraints in Eq. (12) ensures that an FTmust be mined in one of the periods. We suggestthese equality constraints be written before otherconstraints due to efficiency of the problemalthough we are following a different order in thisexample to follow the equations given in Section5.1. The constraints are applied for the exampleproblem as

O11 þ O12 ¼ 1;

O21 þ O22 ¼ 1;

O31 þ O32 ¼ 1.

Eq. (13) representing the limit on the strippingratio in each period is applied to the example prob-lem as

W 11 þ W 21 þ W 31 À 2 Ã 18 Ã O11

À 2 Ã 10 Ã O21 À 2 Ã 10 Ã O31 6 0;

W 12 þ W 22 þ W 32 À 2 Ã 18 Ã O12

À 2 Ã 10 Ã O22 À 2 Ã 10 Ã O32 6 0.

The total capacity of the equipments in removingore and waste material are limited during a givenyear at a maximum of 50 tons. This requirementgiven in Eq. (14), written for the example problemas

W 11 þ W 21 þ W 31 þ 18 Ã O11

þ 10 Ã O21 þ 10 Ã O31 6 50;

W 12 þ W 22 þ W 32 þ 18 Ã O12

þ 10 Ã O22 þ 10 Ã O32 6 50.




Solving the MIP problem results in mining FT 1during the first year (O11 = 1 and W 11 = 30), andFT 2 and FT 3 are mined during the second year(O22 = 1, O32 = 1, W 22 = 10 and W 32 = 10). Theobjective value of the MIP is 3.39, which is the opti-

mal NPV that can be generated from mining theexample orebody model. Using the FT Algorithm,the number of binaries is reduced from 16 to 6.

7. Case study

One of the case studies is performed on a largecopper mine in Southern Peru that contains about1 million blocks in the orebody model and hastwo ore processors, leaching process and mill plant.The proposed Fundamental Tree Algorithm re-duced the number of blocks within the ultimate pit

limits from 38,457 to 5512 fundamental trees. TheMIP scheduling model is applied to develop a pro-duction schedule over 8 years for the mine. Thiswas impossible to perform without applying theFundamental Tree Algorithm due to large numberof binary variables and large number of sequencingconstraints required in formulating the problem. Inone of the pushbacks scheduled over 4 years, thenumber of blocks were about 12,600. This wouldnormally require 37,800 binary variables in the tra-ditional MIP formulations, but FTA combined the

blocks into 1640 trees, resulting the MIP schedulingmodel containing 4920 binary variables being solvedin about 36 minutes with 5% gap.

The number of constraints is also significantlyreduced by using FTs instead of blocks in theMIP formulation. Traditionally, sequencing con-straints would be required for about 12,350 blocksexcluding the top level. The proposed LP algorithmrequires the sequencing constraints for only 1640FTs, which is about 7.5 times fewer constraints.

In mining, a gap less than 10% is often accepted asa good solution considering the uncertainty inestimated block grades used as input to themodel. For this single project, the method produced$25M ($7%) higher NPV than the best NPV gener-ating schedule obtained among three commonlyused traditional software packages includingMINTEC’s M821V, Earthworks’ NPV scheduler,and Whittle’s Milawa schedulers in the Four-Xprogram. The NPV values are calculated afterdesigning the haul roads and smoothing the pits thatmakes the profit achievable through the actual min-ing operation. A detail description of the proposed

scheduling process, calculations and various maps

can be found in Ramazan (2001) and the three tradi-tional scheduling details are given in Bernabe (2001).

8. Conclusions

In this paper, a mathematical programmingmodel is developed using linear variables to produceFundamental Trees for any given orebody modelfor open pit mines. This model does not includeany integer variables so that it solves large problemswithout running into time constraints. The numberof integer variables required for the MIP schedulingformulation is decreased substantially using the fun-damental trees. Since the number of FTs producedis significantly less than the number of blocks inthe model, the number of constraints are also sub-stantially less than the traditional MIP scheduling

formulations. Therefore, large open pit productionscheduling can be optimized by MIP modelling forany objective such as maximizing NPV of a givenmine project, or to solve hard ore quality and gradeblending problems.

It should be noted that alternative solutions maybe available to find the fundamental trees, whichmeans the same number of fundamental trees maybe determined with different configurations of con-nected blocks. However, determining all the possi-ble configurations of the trees and measuring their

effect on the scheduling are not performed. A signif-icant difference in the scheduling results for differentblock configuration of fundamental trees is notexpected due to the structure of the FTA that makesthe higher cone value blocks feasible to mine beforethe other blocks.

Acknowledgements

The author express special thanks to Prof. Rous-sos Dimitrakopoulos for reviewing this paper. The

project was partly sponsored by Natural ResourcesResearch Institute of Minnesota.

References

Bernabe, D., 2001. Comparative analysis of open pit minescheduling techniques for strategic mine planning of a coppermine in Southern Peru. M.Sc. Thesis, Colorado School of Mines, Golden CO.

Dagdelen, K., 1985. Optimum multi-period open pit mineproduction scheduling by Lagrangian parameterization.Ph.D. Thesis, Colorado School of Mines, Golden, CO.

Dantzig, G.B., Wolfe, P., 1960. Decomposition principle for

linear programs. Operations Research 8 (1), 101–111.




Dimitrakopoulos, R., Ramazan, S., 2004. Uncertainty basedproduction scheduling in open pit mining. SME Transactions316.

Gershon, M.E., 1983. Optimal mine production schedulingevaluation of large scale mathematical programmingapproaches. International Journal of Mining Engineering 1,315–329.

Godoy, M., Dimitrakopoulos, R., 2004. Managing risk and wastemining in long-term production scheduling. SME Transac-tions 316.

Hochbaum, D.S., Chen, A., 2000. Performance analysis and bestimplementations of old and new algorithms for the open-pitmining problem. Operations Research 48, 894–913.

Johnson, T.B., 1968. Optimum open pit mine productionscheduling. Ph.D. Dissertation, University of California,Berkeley, CA.

Lemieux, M., 1979. Moving Cone Optimizing Algorithm. Com-puter methods for the 80’s in the mineral industry. In: WeissA. (Ed.), SME-AIME, pp. 329–345.

Lerchs, H., Grossmann, I.F., 1965. Optimum design of open-pit

mines. Transactions on CIM LXVII, 47–54.Ramazan, S., 1996. A new push back design algorithm in open pitmining. M.Sc. Thesis, Colorado School of Mines, Golden,CO.

Ramazan, S., 2001. Open pit mine scheduling based on funda-mental tree algorithm. Ph.D. Thesis, Colorado School of Mines, Golden, CO.

Ramazan, S., Dimitrakopoulos, R., 2004. Recent applications of operations research and efficient MIP formulations in openpit mining. SME Transactions 316.

Seymour, F., 1995. Pit limit parameterization from modified 3DLerchs–Grossmann algorithm. SME Preprint Number: 95-96.

Tolwinski, B., 1998. Scheduling production for open pit mines.In: Proc. 27th Internat. APCOM Symp., pp. 651–662.

Wang, Q., Sevim, H., 1993. An alternative to parameterization infinding a series of maximum-metal pits for productionplanning. In: Proc. 24th Internat. APCOM Symp., pp. 168– 175.

Warton, C., 2000. Add value to your mine through improvedlong term scheduling. In: Whittle North American MinePlanning Conference, CO.

Whittle, J., 1988. Beyond optimisation in open pit design. In:Proc. Canadian Conf. on Computer Applications in theMineral Industries, Rotterdam, pp. 331–337.

Whittle, D., 2000. Proteus environment: Sensitivity work madeeasy. In: Whittle North American Mine Planning Conference,CO.

Salih Ramazan holds a PhD degree in mining engineering with aminor in computer science and mathematics, and MSc degree in

mining engineering from Colorado School of Mines, USA, andME degree in Geostatistics from Ecole Nationale Superieure desMines de Paris, France. He is specialized in operations researchmethods especially development of efficient large size mixedinteger and stochastic programming models and applications toproduction scheduling of large open pit mines. He has worked asa research scientist for 4 years in developments of new stochasticoptimization methods and algorithms at the University of Queensland, Australia, and is currently a mine planning engineerat Kalgoorlie Consolidated Gold Mines, Australia.


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