Review of Scheduling Algorithms

Review of
Scheduling
Algorithms in Open
Pit Mining
I N G . J O S E G O N Z A L E S B O R JA
FA C U LTA D D E I N G E N I E R I A G EO LO G I C A , M I N E R A Y M E TA LU R G I C A
LIMA, AUGUST 2015

Summary
- Lack of “universally accepted” methodology:
◦ Ad-hoc solutions for specific cases are useless for other cases
◦ Partially explained by commercial interests of software and consulting
companies

- Heuristic techniques are poorly understood by mine planning
practitioners:
◦ A fair comparison can’t be done without understanding the low-level details
of the algorithm
◦ Thus, the NPV criterion is incorrectly used as a proxy of superiority of a given
algorithm

- Every year a new algorithm is developed by academics and/or industry
researchers across the world. Is Peru ready to compete?

Pit scheduling problem –
Bench Phase formulation
- In a bench-phase model, the decision variables are the tonnages
mined in each available bench-phase at every period, subject to the
precedencies established between successive phases and the “bench
above” rule.
- The objective function is maximize the NPV of cash flows obtained
through the life of mine, taking into consideration mining costs,
processing costs, capital costs and revenues by period.
- Constraints may include vertical advance in benches per phase per
period, truck hours, loader hours, tonnages from specific regions of the
deposit, or period constraints such as “don’t start Phase X until Period
T”.

- This model is the most flexible and user-friendly for mining
practitioners.

Pit scheduling problem - DP
formulation
𝑉 𝑅, 𝑇 = 𝑀𝑎𝑥

𝑎𝑙𝑙 𝜔

𝑐 𝑡, 𝑟, 𝜔 +

𝑉(𝑅 − 𝑟, 𝑇 + 𝑡)
1+𝛿 𝑡

,0 ≤ 𝑟 ≤ 𝑅

Which means:
“The maximum NPV (V) of the entire reserve (R), at a time (T), can be
calculated by considering all feasible strategies () and picking the
maximum sum of the cash flow (c) of a portion of the reserve (r) and
the maximum value of the remaining reserve. A discount rate () is used
to adjust the remaining value by the time (t) taken to mine an
increment of the reserve”.

Pit scheduling problem - MILP
formulation
𝑀𝑎𝑥

𝑝𝑏𝑑𝑡 𝑦𝑏𝑑𝑡
𝑏 ∈𝐵

𝑡 ∈𝑇

𝑑 ∈𝐷

Subject to
𝑥𝑏𝜏 ≤
𝜏 ≤𝑡

𝜏 ≤𝑡

𝑥𝑏𝑡 =

𝑥𝑏 𝜏

𝑦𝑏𝑑𝑡

∀𝑏 ∈ 𝐵, 𝑏 ∈ 𝐵, 𝑡 ∈ 𝑇
∀𝑏 ∈ 𝐵, 𝑡 ∈ 𝑇

𝑑∈𝐷

𝑥𝑏𝑡 ≤ 1

∀𝑏 ∈ 𝐵

𝑡∈𝑇

𝑅𝑟𝑡 ≤

𝑞𝑏𝑟𝑑 𝑦𝑏𝑑𝑡 ≤ 𝑅𝑟𝑡
𝑏∈𝐵

𝑑∈𝐷

𝑦𝑏𝑑𝑡 ∈ 0,1 ; 𝑥𝑏𝑡 ∈ 0,1
where

𝑟 ∈ 𝑅, ∀𝑡 ∈ 𝑇

∀𝑏 ∈ 𝐵, 𝑑 ∈ 𝐷, 𝑡 ∈ 𝑇

Pit scheduling problem - MILP
formulation (cont.)
𝑡 ∈ 𝑇: set of time periods in the horizon
𝑏 ∈ 𝐵: set of blocks
𝑏′ ∈ 𝐵: set of predecessor blocks for block b
𝑟 ∈ 𝑅: set of operational resources

𝑑 ∈ 𝐷: set of destinations
𝑝

𝑏𝑑
𝑝𝑏𝑑𝑡 =
: profit obtained from processing block b when sending it
(1+ 𝛼)𝑡
to destination d at time period t; 𝛼 is discount rate

𝑞𝑏𝑟𝑑 : amount of resource r used to process block b when sent to
destination d

Pit scheduling problem - MILP
formulation (cont.)
𝑅𝑟𝑡 : minimum availability of operational resource r in time period t
𝑅𝑟𝑡 : maximum availability of operational resource r in time period t
𝑥𝑏𝑡 : binary variable equal to 1 if block b is extracted in time period t
𝑦𝑏𝑑𝑡 : the amount of block b sent to destination d in time period t

Pit scheduling problem – SIP
formulation
𝑃

𝑁

𝑀

𝑉𝑖𝑡 𝑏𝑖𝑡 −

𝑀𝑎𝑥
𝑡=1

𝑖=1

𝑡
𝑡
𝑐𝑢𝑡 𝑑𝑠𝑢
+ 𝑐𝑙𝑡 𝑑𝑠𝑙
𝑠=1

Subject to
𝑁
𝑡
𝑡
𝐺𝑠𝑖 − 𝐺𝑚𝑖𝑛 𝑂𝑠𝑖 𝑏𝑖𝑡 + 𝑑𝑠𝑙
− 𝑚𝑠𝑙
=0

◦
𝑖=1
𝑁

𝑡 − 𝑚𝑡 = 0
𝐺𝑠𝑖 − 𝐺𝑚𝑎𝑥 𝑂𝑠𝑖 𝑏𝑖𝑡 + 𝑑𝑠𝑢
𝑠𝑢

◦
𝑖=1

where

Pit scheduling problem – SIP
formulation (cont.)
𝑏𝑖𝑡 = percentage of block 𝑖 mined in period 𝑡; there are 𝑁 blocks and 𝑃
periods
𝑡 = excess of ore tonnage above the upper limit, in period 𝑡 for block
𝑑𝑠𝑢
model 𝑠, there are 𝑀 equiprobable block models
𝑡
𝑑𝑠𝑙
= deficit of ore tonnage below the lower limit, in period 𝑡 for block
model 𝑠
𝑡
𝑚𝑠𝑢
= dummy variable to balance the second equality constraint
𝑡
𝑚𝑠𝑙
= dummy variable to balance the first equality constraint

𝑉𝑖𝑡 = expected discounted value of block 𝑖 when mined in period 𝑡,
averaged among all block models

Pit scheduling problem – SIP
formulation (cont.)
𝐺𝑠𝑖 = grade of block 𝑖 in orebody model 𝑠
𝐺𝑚𝑖𝑛 , 𝐺𝑚𝑎𝑥 = minimum and maximum target grades of the ore
𝑂𝑠𝑖 = ore tonnage of block 𝑖 in orebody model 𝑠

Route map: models and
algorithms
Algorithms ↓

Models 
MSSP ®
Milawa ®
Tolwinski
COMET®
Branch and cut
Lagrangian relaxation
Fundamental trees
Ant colony optimization
Genetic algorithms
Critical Multipliers
DeepMine®

BP

MILP

DP

SIP

BP algorithms

MSSP®
- Cai and Banfield, 1979 – United States
- Used in Minesight Strategic Planner (MSSP®), now with support
discontinued
- The bottom benches are mined in fractions via linear programming
with the status of direct mill feed stockpiles considered automatically

- Also in Step 6, “ the materials mined from all pushbacks can be
allocated to available material destinations by linear programming on a
pushback to destination basis” (excerpt from Minesight for Engineers,
Mintec®).

Milawa® algorithm
- Unknown author, 1999
- Used in Whittle ®, now a product of Dassault Systèmes - France
- Variables are benches in each pushback and regions of high value are
identified with a heuristic approach
- considers two constraints per period:
◦ Minimum and maximum separation between pushbacks
◦ Maximum vertical advance

- all mining in a phase is assumed to occur at the same rate

DP algorithms

Tolwinski algorithm
- Tolwinski, 1992 - United States
- Used in NPV Scheduler®, now a product of CAE Inc. - Canada
- It combines ideas from dynamic programming with stochastic search
heuristics to produce feasible solutions to the problem.
- Dynamic programming states grows exponentially with number of
blocks, making the problem intractable for large open pits.

COMET® algorithm
- King, 2000 - United Kingdom
- Used in COMET®, a product of Comet Strategy - Australia
- Works as an add-in to Microsoft Excel®
- Simultaneous optimization of cutoff grades, dilution and comminution

- Requires a “seed” schedule from which the program iterates
- Requires pre-defined pushbacks

MILP algorithms

Lagrangian relaxation
- Dagdelen, 1985 - United States
- Used in Colorado School of Mines – Not available commercially.
- In his PhD thesis, Dagdelen solves the MILP problem with Lagrange
multipliers, but failed in guaranteeing convergence for the general case.
- Akaike (1999) and then Kawahata (2006) expands this procedure to
solve the convergence issue by using more multipliers and changing the
iteration scheme for determining the value of the multipliers.

Branch and cut algorithm
- Caccetta & Hill, 2003 – Australia
- It was used in MineMap software (Australia), but now is out of
business.
- Caccetta demonstrated rigorously that the ultimate pit obtained with
Lerchs & Grossmann is an upper bound of the MILP solution.

- Instead of the branch and bound method used in Minemax®, Caccetta
uses auxiliary heuristics to select which branch is analyzed in depth and
which one is cut, using 17000 lines of code in C++.
- A model of 200,000 blocks and 23,000 constraints produced a solution
guaranteed to be within 2.5% of the optimum in 4 hours.

Fundamental tree algorithm
- Ramazan, 2007 – Australia
- Not available commercially
- Reduces the number of binary variables required in the MILP model by
solving a LP model to find the fundamental trees, by minimizing the arc
connections in the network weighted by the assigned ranks.

- After generating the fundamental trees for a given orebody model, the
MILP model uses each tree as a block having certain attributes.
- A case study showed a reduction from 38,457 variables in the raw
MILP model to 5,512 with the use of the FT algorithm.
- Requires pre-defined pushbacks

Ant colony optimization
- Sattarvand & Niemann-Delius, 2011 - Iran/Germany
- Not available commercially
- When one ant finds a good path from the colony to a food source,
other ants are more likely to follow that path, and eventually all the
ants will follow a single path = emerging behavior.
- By repeated iterations, the pheromone values of those blocks that
define the shape of the optimum solution are increased, whereas those
of the others have been significantly evaporated.
- However, “a trial and error process might be necessary at the
beginning to set the relevant combination of parameters for each
individual case”, i.e., number of ants, amount of pheromone, and
evaporation rates.

Genetic algorithm
- Bitanshu Das, 2012- India
- Not available commercially
- Mimics natural selection where a population of candidate solutions
are mutated to increase the fitness of the solution.
- In his thesis, the author starts from a random solution performing
several crossovers, mutations and eliminations to reach the optimized
solution and shows an example for an iron ore mine.

Critical Multiplier algorithm
- Chicoisne et al., 2012 - Chile
- Not available commercially
- It solves an LP version of the MILP and applies a rounding heuristic
based on topological sorting. Then a second heuristic is applied based
on local search.

- The critical multipliers are break-point values from the ultimate pit
parameterization that define a piecewise linear profit function.
- It solves the Marvin deposit example in 12 seconds, but blocks mined
in a given period may be scattered over the pit.

DeepMine® algorithm
- Echeverría et al., 2013 - Chile
- Used in DeepMine®, a product of Boamine, Chile
- It creates multiple possible states in which the mine might be at a
particular period. Then for each of these possible states, the algorithm
develops new states, and selects the path that leads to the highest NPV.

- In order to guide the solution, the algorithm considers operational
constraints for generating extraction zones, and follows the LG ultimate
pit.
- Phases are not predefined, rather emerge from the tree of states
generated based on slope angles and minimum mining width required.

SIP algorithms

Simulated annealing
-Dimitrakopoulos and Consuegra, 2009 – Canada
- Not available commercially
- Finds a global optimum in a large discrete search space, by changing
the rate of decrease in the probability of accepting worse solutions as it
explores the solution space.
- It takes several mine production schedules corresponding to each one
of the simulated orebody models, and focus the attention to those
blocks that have less than 100% probability of being mined in a
particular period. These blocks will be accepted to the extent that they
exceed a predetermined annealing temperature.

- However, the method is computational and labor intensive, even with
the current computing power.

Final Note
We are living a change of paradigm: phase
design was considered previous to mine
scheduling. Now, it has been shown that
phases emerge from the scheduling
algorithm, leaving the phase design as a
post-process after the mine schedule is
completed.

References
- Dagdelen,K. and Johnson, T. 1986: Optimum Open Pit Mine Production Scheduling by Lagrangian
Parameterization. 19th APCOM Symposium, pp. 127-142

- Tolwinski, B. and Underwood, R. 1992: An Algorithm to Estimate the Optimal Evolution of an Open Pit
mine. 23rd APCOM Symposium, pp. 399-409
- Wharton, C. 2000: Add Value to Your Mine Through Improved Long Term Scheduling. Whittle North
American Strategic Mine Planning Conference, Colorado
- Caccetta, L. and Hill, S. 2003: An Application of Branch and Cut to Open Pit Mine Scheduling. Journal
of Global Optimization 27: 349-365
- Ramazan, S. 2007: Large-Scale Production Scheduling with the Fundamental Tree Algorithm – Model,
Case Study and Comparisons. Orebody Modelling and Strategic Mine Planning, pp. 121-127
- Wooller, R. 2007: Optimising multiple operating policies for exploiting complex resources – An
overview of the COMET Scheduler. Orebody Modelling and Strategic Mine Planning, pp. 309-316

References (cont.)
- Dimitrakopoulos, R. and Consuegra, A. 2009: Stochastic mine design optimisation based on simulated
annealing: pit limits, production schedules, multiple orebody scenarios and sensitivity analysis. Mining
Technology Vol 118 #2 p. 79-90
- Newman et al. 2010: A Review of Operations Research in Mine Planning Interfaces 40(3), pp. 222–
245, ©2010 INFORMS
- Sattarvand, J. and Niemann-Delius 2011: A New Metaheuristic Algorithm for Long-Term Open Pit
Production Planning. 35th APCOM Symposium, pp. 319-328

- Chicoisne et al. 2012: A New Algorithm for the Open-Pit Mine Production Scheduling Problem.
Operations Research 60(3), pp. 517-528
- Bitanshu 2012: Open Pit Production Scheduling Applying Meta Heuristic Approach. Thesis, National
Institute of Technology – India
- Juarez, G. et al 2014: Open Pit Strategic Mine Planning with Automatic Phase Generation. Orebody
Modelling and Strategic Mine Planning. pp. 147-154

Thank you!
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