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A l a ! C | D E G H I I J % L M N o P Q I R s T u v W
Mine Sequences Report
1 1 st Year
2 McLaughlin04 M M M V V V V M M M M M M Q Q Q Q Y
3 Phases TP1 TP2 TP3 TP4 TPS TPS TP? TPS TP9 TP10 TP11 TP12 TP13 TP14 TP15 TP16 TP17 TP18 TP19 TP20
4 PM 1 Mined out to 1940 1920 1900 1900 1900 1900 1900 1880 1880 1860 1860 1860 1840 1840 1820 1800 1760 1720 1660 1500
6 Pt2 1 Mined out to 2 I ■ ■ 2000
8 Blue: A given phase is completed
Figure 5.10: Very short term mine plan sequences
Using the solution of the very short term mine planning starting in the fourth month,
the optimization of truck allocation using the LP model, as previously discussed in
Chapter 3, is carried out. The time periods of analysis for this LP model are twelve time
periods. These time periods are entered in General template, as illustrated in Figure 5.11.
The first four time periods are weekly, and the rest are monthly. The LP solution provides
the weekly optimal truck allocation for the fourth month, and the monthly optimal truck
allocation for eight months.
A I B c D E F G | H ' ** * L M N | O
1
2 Period? Periods Period4 Periods Period? Periods Periods Pertodt 0 Pertodt 1 Pertodt 2
3 Number of months/period 0 0 0 0 0 0 0 0 0 0 0 0
4 Number of daysAnonth 7 7 7 7 28 28 28 28 28 28 28 28
5 Number of Shifts/day 1 1 1 1 1 1 1 1 1 1 1 1
6 Number of Hours/shift 12 12 12 12 12 12 12 12 12 12 12 12l
7 Number of Hoursfpertod 84 84 84 84 336 336 336 336 336 336 336 336
8 Length of Periods (his) 84 84 84 84 336 336 336 336 336 336 336 336
9 weekly weekly weekly weekly monthly monthly monthly monthly monthly monthly monthly monthly
10
Figure 5.11: Time periods of LP model analysis
The total mine reserves that have to be moved to Autoclave, Crushed Leach, stockpile
and dump are shown in Figure 5.12. These mine reserves are summarized from the very
short term mine plan. There are four material types, where material type 1,2,3 and 4
represents materials that will be sent to Autoclave, Crushed Leach, dump and stockpile,
respectively. Besides the total mine reserves in the Pits, the materials that can be sent to
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Based on the very short term mine plan, the production targets at destinations are
known. The production targets are in term of the amount of materials that should be
delivered to each destination for each time period. The production targets of Autoclave
are discussed here. An example of the production targets at Autoclave is shown in Figure
5.14.
A I B C I D f.... E....T F G H I I J K L M N d o P
1 Atitodav Requirements
2
3
4 Period 1 Period 2 Period 3 Period 4 Period 4 Period 5 PenodS Period 6 PmodG
5 Phase Bench "“ton kton kton
6 Phasel 1880 11.9 High 11.9 High 11.9 High 11 9 High 15 High
7 1860 22 High 322 High
8 1840
9 1820
10 1800
11
12
13 Stockpile Period 1 Period 2 Period 2 Period 3 Period 4 Period 4 Period 5 Period 5 Period 6 Period 6
14 Priority Priority kton Prlorty Priority kton Priority kton
15 ioF Medium 10 Medium 10 Medium 10 Medium 50.4 Medium 55 3 Medium
16 • ■ ■as ïfcî s
Figure 5.14: An example of the production targets at Autoclave.
Figure 5.14 shows the production targets at Autoclave for six time periods. Based on
the very short term mine plan, during the first four time periods, 11.9 ktons of materials
from Pitl or Phasel should be mined from bench 1880 and delivered to Autoclave. The
priorities of satisfying the production targets are set to be “High” for the materials from
Pitl, whereas the priorities of satisfying the production targets are set to be “Medium” for
the materials from the stockpile.
There are three shovels and fifteen trucks available for this case study. The shovels
and trucks data are summarized in Table 5.3.
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Figure 5.16 shows the results of material flows and truck allocation, respectively. The
results show that the material flow at Shovel 1 is 1,251.1 tons per hour in period 1. This
amount of material flow is split to three different destinations which are Autoclave,
Crushed leach and stockpile. The material flows from Shovel 1 to Autoclave, Crushed
leach and stockpile are 119, 1007.1 and 125 tons per hour, respectively. This means that
in period one, trucks that receive material from Shovel 1 have to travel and deliver
material to Autoclave, Crushed leach and stockpile based on the amount of material flow
to the destinations. The number of trucks traveling from Shovel 1 to Autoclave, Crushed
leach and stockpile is proportionally divided according to the material flows from
Shovel 1 to these three destinations. Based on the material flows, as shown in Figure
5.16, the total number of trucks required for each time period can be determined.
Figure 5.17 shows the results of truck allocation. In period 1, the total number of
trucks required is fourteen trucks. To achieve the material flows, as shown in Figure 5.16,
five percents of fourteen trucks have to be allocated to Shovel 1 during period 1.
ARTHUR LAKES LIBRARY
COLORADO SCHOOL OF MINES
GOLDEN, CO 80401
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CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 CONCLUSIONS
This thesis successfully demonstrates how the application of mathematical
optimization tools can be used to solve complex operational problem in open pit mining.
This thesis defines a general methodology for allocating trucks to different loading
units in open pit mining operations such that daily, monthly, and yearly production goals
are met to achieve financial objectives of a mining company. The optimal assignment of
truck allocations to loading points in open pit mines is a complex problem and depends
on production requirements within the planning time horizons, type and capacity of the
equipment at hand and conditions of the haul roads between the loading points and
various destinations.
In this dissertation, the proposed solution methodology in determining truck
allocation to different shovels in a given time period is based on MIP and LP concepts of
OR. The open pit truck allocation problem is modeled within the context of MIP and LP
problems. The MIP model determines the life of mine, period by period, production
requirements in an open pit mine by considering all economic factors by maximizing
discounted cash flow subjected to all complex operational constraints of a given mine.
The LP model that is developed in this dissertation is used to determine the appropriate
allocation of trucks to loading units in all time periods in order to achieve the required
production.
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This dissertation provides the discussion in detail of the proposed MIP and LP
formulations and further describes how the uncertainties associated with coefficient of
the decision variables can be taken into account to determine their impacts on truck
allocation.
This thesis also discusses the development of a program to formulate and solve the
truck allocation LP model in order to demonstrate the application of proposed concepts to
a real world problem.
The original contribution of this thesis can be summarized as:
1) The proposed truck allocation LP model takes into account the attributes of
different types of trucks and shovels, open pit mine haul road profiles within the
frame work of multi time period problem. None of the existing truck allocation
systems has included these considerations.
2) The proposed truck allocation LP model takes into account uncertainties
associated with the input parameters, therefore it is stochastic in nature. The
existing techniques are deterministic and do not take into account the impact of
uncertainties in determining truck allocation requirements.
3) The truck assignments are based on the overall mine plan requirements
determined according the economic and operational objectives. The existing
systems perform the truck assignment based on the loading unit requirement.
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ABSTRACT
Mine planning and production scheduling is the foundation from which optimal solutions
are generated for the extraction of material while satisfying various constraints. The main
purpose of this paper is to maximize the net present value (NPV) of an open-pit coal mine
project.
The entire modeling and simulation process of this study uses commercial mine planning
software MineSight 3D from Hexagon Mining. In an open-pit coal mine, instead of modeling the
deposit using a gridded seam file which provides irregular vertical block sizes, this paper
generates a 3D block model with the required regularly sized block in all directions, allowing
implementation of the ultimate pit optimization concept.
Compared to the mineral deposit model, which classifies whole blocks as waste or ore
based on cut-off grades, coal deposit can have both waste and coal within blocks at the coal-rock
interfaces; therefore, ORE% item representing portion of coal is assigned into each block as the
critical item to calculate block economy. The ultimate pits developed represent which block to
extract, followed by selecting pushbacks produced by varying coal prices to get sufficient
increment volumes between phases.
After that, this paper presents a generated production schedule that follows a linear
program algorithm to maximize NPV. It also will comply with operational constraints such as
coal production capacity, level and phase precedence, reserve constraints and waste volume
constraints to balance production volumes between years.
A case study will also be presented involving a currently operating open-pit coal mine in
Indonesia; it concluded the mine plan optimization with a better NPV than current operational
scenarios, producing a 64% higher coal reserve than current operation for all mining areas. It
also results in an increase of US$571 million net present value (NPV) for the whole project, and
an 17.5% incremental rate of return (IRR).
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ACKNOWLEDGEMENTS
First and foremost, I would like to thank Allah the Almighty for giving me the blessing,
ability and perseverance to complete my master’s degree. May Allah's blessing go to His final
Prophet Muhammad (peace be up on him), his family and his companions. I am continuously
amazed at how Allah’s hand is involved in every aspect of my life.
I would like to present my genuine gratitude and appreciation to my thesis advisor, Dr. Kadri
Dagdelen, for his wisdom, guidance and endless support during the development of this work. His
invaluable ideas and kindness, not only during this project, but also in my master’s program
completion, greatly helped me develop my skills, especially those in mine planning. I would also
like to thank my thesis committee, comprised of Dr. Hugh Miller and Professor Andrew Pederson,
for their assistance in making this endeavor a success.
I additionally thank PT Bukit Asam for munificently funding my studies at the Colorado
School of Mines. Special thanks as well to my fellow Indonesian students for helping me in every
way during my stay in the USA.
Last but not least, I wish to thank the four most important people in my life: my husband
Riana, my daughter Hannah, my mother and my father for all their love and steadfast patience,
support and belief in me throughout the past two years. I could not have done this without them.
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CHAPTER 1
INTRODUCTION
Mine planning and production scheduling is a critical element of mining. It becomes the
foundation to generate a path to financial success for a mine; this is possible by establishing and
developing an optimal solution to extract material while satisfying various operational
constraints. Planning is based on a block model, which develops an ultimate pit limit that
represents which blocks to extract. This is followed by a production schedule describing when to
extract, and how or whether to process the blocks in compliance with operational constraints
such as mining extraction sequences, grade blending, production and infrastructure capacity.
This also includes social and environmental requirements. The end goal is to maximize net
present value (NPV) of cash flows generated during the life of the project.
The mine planning process is divided into three categories: long-term, medium-term and
short-term. These distinctions are made to consider variability in assumption parameters
including commodity pricing, inflation of a mine’s capital expenditure (capex) and operational
expense (opex) costs, declining ore grades and any other unexpected setback to ensure optimum
operational efficiency and the sustainability of mining operations.
These three terms for mine planning are not only generated based on time horizons, but
also by perspective – whether the plan is strategic or tactical. Long-term and medium-term
planning can be grouped as strategic mine planning. These deal with the factors of the operation
and the decisions that largely determine the value of the mining business. The main concern of
mine planning is developing a plan to mine out the entire mineral resource. Among the typical
elements to specify within the scope: method of mining, processing route, scale of operations,
the mining sequence and the definition of various operational cut-offs that progressively separate
the valuable fractions of the resource. Even though these variables are set at the beginning of the
business cycle, they must be reviewed periodically as internal and external conditions of the
business change.
Short-term planning, on the other hand, provides tactical guidance to operations once the
mine is developed. It encompasses the routine planning activities required for commissioning the
operation and ramping it up. In operating mines, the scope includes the continuous reworking of
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short-term production plans to incorporate new information. Tactical mine planning also deals
with budget preparation; equipment deployment and production scheduling on a yearly, monthly,
weekly and daily basis; grade and quality control; and various other routine activities.
An integrated planning process, working in a hierarchy from long-term to short-term, is
essential. The life of mine plan (LOMP) is the formally approved long-term plan for the mine
should be reviewed and updated annually as part of the planning cycle. It considers constraints
identified in shorter-term plans and evaluations from actual events. Medium-term planning,
usually produced as a five-yearl plan, is a critical link between the high-level strategies of the
LOMP and the detailed shorter-term implementation plans. While the time horizon is usually
five years, medium-term planning is generated annually, provide sufficient “look-ahead” time to
identify ways to re-align the plan. The yearly short-term plan is acutely tied to the annual budget
plan, supported by more detailed engineering work and updated quarterly with activities reported
against it on a monthly basis.
It should be ensured a plan is realistic and achievable and is aligned with corporate goal(s).
Practically said, mine planning is not only carried out at the beginning of the project, but also
done continuously throughout the life of the mine.
1.1 Coal Open-Pit Mine Planning Workflow
Figure 1.1 Coal Mine Planning Workflow (Modified from Clark and Dagdelen, 2022)
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Clark and Dagdelen (2022) said the open-pit mine planning workflow, which has been
modified to serve a stratified coal deposit as shown in Figure 1.1 above, begins with a 3D
geological block model and ends with a production schedule that serves as the input to its
financial analysis. The workflow is circular, representing that the product of a given sub-problem
becomes part of the input to solve the next sub-problem. Some steps in Figure 1.1 implement
large-scale optimization formulations, which are shown inside the dashed blue box.
Block models as digital representations of mineral resource in 3D are built from widely
spaced exploration data. In a stratified deposit such as coal, the data is composited by seam.
Hence, the nature of a stratified deposit model is having a regular size in plan view but different
thickness for each block. However, almost all open-pit optimization formulations and solution
algorithms take 3D block models of deposits as their geological inputs (Xiaowei, et al., 2021), in
which all blocks have regular sizes in X, Y and Z directions. Therefore, seam models with
different thicknesses are required to be converted into all-direction regular blocks by creating
new composite data (by bench in most cases).
After that, instead of defining a block as waste or ore, ORE% items representing portions
of coal should be assigned into each block. It is the most important component to calculate a
block’s value.
1 2 3 4 5 6 7 8 9 10 11 12 13
Topography
1
2
3
4 Coal Seam 1
5
6 Coal Seam 2
7
8
9 Coal Seam 3
Figure 1.2 Regular Size Block Model Illustration Containing Stratified Deposit Material
1 2 3 4 5 6 7 8 9 10 11 12 13
Topography
1 1% 0% 55% 70% 30% 3% 0% 0% 0% 0% 0% 0% 0%
2 25% 25% 10% 25% 60% 93% 55% 20% 1% 0% 0% 0% 0%
3 0% 1% 15% 35% 25% 10% 40% 80% 85% 50% 10% 0% 0%
4 75% 20% 3% 0% 3% 20% 30% 10% 10% 50% 90% 75% 30% Coal Seam 1
5 80% 97% 97% 55% 15% 1% 0% 5% 25% 30% 15% 15% 65%
6 0% 3% 50% 70% 100% 90% 50% 5% 0% 1% 15% 30% 30% Coal Seam 2
7 0% 0% 0% 0% 5% 30% 90% 99% 70% 20% 5% 0% 1%
8 0% 0% 0% 0% 0% 0% 1% 20% 70% 97% 98% 65% 15%
9 0% 0% 0% 0% 0% 0% 0% 0% 0% 3% 20% 65% 97% Coal Seam 3
Figure 1.3 ORE% Assigned into Each Block Representing Portion of Coal Inside Block
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of the 2D Lerch-Grossman algorithm and satisfying slope design constraint, the ultimate pit limit
can be generated by determining blocks which will be mined to achieve maximum value.
As suggested by the Lerch-Grossman 2D algorithm (Lerch & Grossman, 1965), the block
value as shown in Figure 1.6 is the sum vertically to represent the profit realized in extracting a
single column with block (i,j) at its base. The maximum possible contribution of each block to
any feasible pit is then calculated by combining the vertical cumulative value and the maximum
value of the block in the left column. If the maximum value of the first row is positive, then the
optimum pit is obtained by following from and to the left of this block. The figure below
describes the final ultimate pit with the block values representing the maximum possible
contributions of each block to any feasible pit as the dynamic programming result.
Figure 1.6 The Ultimate Pit Limit Following the Lerch-Grossman Algorithm
The mine’s ultimate pit limit is a static optimum value without considering a discounted
factor due to a reduction in the time value of money. In reality, the undiscounted value
potentially results in a suboptimal project value when the discount factor is applied. Several
studies have been done applying the discounting factor to block values prior to optimization.
However, this study use a traditional approach of generating the optimum pit based on
undiscounted block values.
The mine’s ultimate pit needs to be broken down into a production schedule over the
mine’s life. A production scheduling problem, as claimed by Gershon (1987), concerns the
sequence of removing mining blocks within the limits of the mine plan. The mining sequence
that yields the largest NPV return is the one that should be chosen. As it deals with discounted
value over time, generating higher cash flow in the early years will maximize the net present
value. Hence the idea to extract the most profitable areas in the early years, leading to the need to
produce a series of nested pits, usually called pushbacks. The nested pit shells can result from
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varying economic factors, including but not limited to commodity price, mining costs or revenue
factors (ratio of mining costs to price).
Clark and Dagdelen (2022) discussed the best-case mining sequence is shell-by-shell,
while the worst one is the bench-by-bench mining sequence. In the best-case scenario, blocks are
mined starting from the smallest pit, gradually up to the biggest pit shells, deferring the mining
of the area with the highest strip ratio as long as possible. In that way, the cost will be discounted
as much as possible – to the point that it does not have a significant impact on the net present
value. On the other hand, the worst-case scenario provides a mining sequence on a bench-by-
bench basis, where waste stripping is done well in advance but will negatively impact NPV.
Figure 1.7 below illustrates the difference between the best-case and worst-case scenarios
in an open-pit coal mine.
Figure 1.7 Best-Case and Worst-Case Mining Sequences
However, it is difficult to achieve the best-case scenario due to operational constraints. In
most cases, optimum NPV, which meets all production constraints, will fall in between the best-
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and the worst-case scheduling scenarios. It is best to select pushbacks which include sufficient
increments in volume.
The optimum production schedule for a coal mine can be generated following linear
programming or the mixed-integer linear program (MILP) method. The result is a detailed mine
plan, one which should be able to be executed in the field. This detailed mine plan becomes the
input for generating a financial analysis to measure the whole project’s value as a consideration
for assessing a mining project within a business framework.
1.2 Scope of Work
This thesis study will discuss generating an open-pit mine plan and production schedule
optimization for a multiple seam coal deposit. By following the mine planning workflow stated
above, a case study is conducted on a currently operating open-pit coal mine in Indonesia. The
study will consider existing operational conditions and constraints including throughput
capacities, change in slope design parameters and area restrictions due to land acquisitions,
existing disposal areas and river restoration. Coal price variations are analyzed to generate nested
pits for optimizing the production schedule and to evaluate the impact of changes in commodity
prices on total reserves.
To avoid the complexity arising from the production scheduling of the existing mine, it is
assumed there are plenty of areas for waste disposal, so this study will focus only on the
extraction of material from the pits. There will be three mining pits, each with specific waste and
coal destinations. Instead of determining the area’s restriction management costs and including
then as a consideration for generating pit shells, any constraint related to area’s restrictions will
be defined as a hard restriction. Simulations using higher throughput capacities than existing will
be also generated; however, the capital cost will not be assumed in this study. Coal reserve
differences, between whether to mine the slope and area restrictions and increasing the
infrastructure’s capacity, will become input for management’s considerations for further decision
making.
1.3 Methodology
The deposit selected for this study is coal deposit on the island of Sumatra, Indonesia. The
mine has been in operation for more than 20 years. Hexagon’s mine planning software package
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and its many different modules is used to perform the mine planning workflow steps. Once the
deposit’s geological database is created from exploration drillhole information, composites need
to be created using MSTorque.
The composite data then becomes input for assigning values into a 3D block model using
the inverse-distance interpolation method in MineSight 3D. A critical element to building a 3D
block model from a stratified deposit is having an ORE% item, which represents the portion of
coal within each block.
The commodity in this study is thermal coal, the major calorific value of which ranges
between 4,500 and 6,000 kcal/kg, and the coal price is retrieved from the Indonesian Coal Index
(ICI) 3 (5000 kcal/lg), obtained by the Argus Media Group and the Indonesian government’s
historical data along with Wood Mackenzie Coal Market Service’s forecasts (until 2050).
Simulations to get initial ultimate pit limits can be run using the Project Evaluator menu in
MineSight3D. This menu determines the mine’s ultimate pit using the Lerch Grossman
algorithm (Lerch & Grossman, 1965). Pit shells are generated with a depth limitation to elevation
-250 m (approximately 300-400 m below the surface) and considers hard area restrictions due to
social and technical issues. Several scenarios are resulted by varying the slope angle design to
analyze how feasible geotechnical engineering is to obtain more coal reserves – that is, whether
to apply the area restrictions mentioned above to measure the effort to manage them and produce
a series of smaller nested pits using the varied prices of coal.
Given the series of nested pits, several attempts were made to generate a mine plan using
the Mine Plan Schedule Optimizer (MPSO), also produced by Hexagon. All three mine areas
were put into one project for viewing using Viewer input, then multi-mine schedule
optimizations were run to see the animation result on the Viewer. The objective of the
optimization is to maximize NPV by fulfilling the material movement constraints, either coal
tonnage and waste volumes, considering optimum infrastructure and the equipment’s utilization
during the mine’s life.
1.4 Objectives
In dynamic mining operations, technical and non-technical parameters change alongside
the dynamics of mining itself. Change happens across all aspects, including environmental,
infrastructure, legal, economic and social conditions. The previously published life of mine plan
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is considered to be less informative due to parameter changes occurring during the mining
process; therefore, assessing and updating long-term mine plans of the existing operation is
required.
This study proposes an optimum mine plan production schedule given existing conditions
in currently operating coal mines. The methodology is compliant with linear programming multi-
mine optimization techniques, maximizing NPV while satisfying production and infrastructure
capacity constraints. The result can be used generally as a reference to generate mine plan
production schedule optimization in a coal stratified deposit, and specifically as a technical
reference in the studied coal mine operation as well as decision making for management in
regard to coal conservation and maximizing the overall project value.
1.5 Thesis Outline
For ease of reference, the rest of the thesis is divided into the following chapters:
Chapter 1 introduces the role of mine planning in the mining industry. It addresses this
thesis’ scope of work, methodology and objectives, as well as the outline.
Chapter 2 gives a brief background along with the literature review of optimization
algorithms and previous work.
Chapter 3 explain how the project was developed by following the open-pit mine planning
workflow. Starting with how to create a 3D geological model, this chapter will present,
step-by-step, the required work to generate a mine’s ultimate pit, phases and pit-by-pit
mine plans. It also describes the linear program algorithm setup used for scheduling
production through the life of the mine, and that will be analyzed financially to determine
the project’s value.
Chapter 4 presents a case study at a currently operating open pit coal mine in Indonesia.
This chapter discusses the results of an optimized production schedule in the studied mine
and its economic evaluation compared to the existing plan.
Chapter 5 addresses the conclusions and future study recommendations.
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CHAPTER 2
LITERATURE REVIEW
2.1 Ultimate Pit Limit and Scheduling Optimization Algorithms
The study of optimization in the mining industry deals with the development of algorithms
and tools to be used by mining engineers for continuous improvements of the mine planning
process for not only financial optimization of projects, but also to handle more complex and
challenging issues in mining industry.
Regarding final pit optimization, many algorithms have been developed. Lerchs and
Grossman (1965) use simple dynamic programming methods to solve two-dimensional versions
of the ultimate pit limits problem, as well as the three-dimensional graph theory for more
complex pit limit problems. This study said the dynamic programming approach becomes
impractical in three dimensions; hence, a graph algorithm was used to solve the three-
dimensional problem. Sequencing constraints were modeled as a weighted, directed graph where
vertices represent blocks and arcs represent mining restrictions. The ultimate pit limits are
determined by solving for the maximum closure of this graph.
Johnson (1968) developed the max flow-based method to solve the ultimate pit limit
problem. Johnson (1968) was also the first to apply linear programing to optimize block-by-
block production scheduling for mining operations based on decomposition technique. Gershon
(1982) also describes a linear programming application that optimizes the scheduling of mining
operations. The author used a mathematical model to determine the most profitable scheme
applicable to the entire supply chain. The program’s concept takes a life-of-mine return on
investment into account to optimize mine scheduling and equipment acquisitions, as well as
quality considerations. By accounting for pit-to-market operations, the program is able to
calculate long-, intermediate-, and short-term mine plans.
Thomas (1996) reviewed a process called the nested Lerchs-Grossman algorithm. It uses
the Lerch-Grossman algorithm to create a series of nested pits as pushbacks. A mine planner then
schedules the extraction sequence for each individual pushback and combines them to create an
overall mine schedule. The development of the pit is characterized by gradual modification of
one or more key parameters, which leads to the reduction of the economic value of each block in
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the model. As the economic value of each block is progressively increased past certain critical
values, the ultimate pit limits contour changes to enclose a smaller volume. The end result is a
series of nested pits. However, this algorithm does not overcome the gap problem between
nested pits, which can result in schedules that are not feasible as they fail to satisfy some
constraints.
Lagrangian relaxation is a tactic used to remove complicating side constraints from a
mixed integer program and transforms the problem into a more tractable formulation. Dagdelen
(1985) used Lagrangian relaxation to solve the block sequencing problem, and Akaike and
Dagdelen (1999) used Lagrangian relaxation to convert their integer-programming formulation
into one based on networks.
The underlying problem formulation has a network flow structure with a complicating
production capacity constraint. The production capacity constraint is then integrated into the
objective function, creating a long-term production scheduling problem with the same
characteristics as the final pit design problem. This relaxed problem is then solved using the
maximum closure algorithm that Lerch and Grossman applied for the three-dimensional ultimate
pit problem. The authors then use an iterative process that alters the values of the Lagrangian
multipliers until the solution to the relaxed problem meets the original capacity constraints. The
original model does contain processing and average grade constraints as well as production
capacity constraints, which are dealt with through Lagrange multipliers, resulting in a more
tractable network and structured formulation than the original problem formulation.
Ramazan and Dimitrakopoulos (2004a) also present a general description of an efficient
mixed-integer program for the open-pit mine scheduling problem. They aim to maximize the
overall discounted net present value of the mine’s ore subject to the limitations of wall slope
requirements, grade blending requirements, ore production and mine capacity.
2.2 Previous Work on Applied Mine Planning and Production Scheduling Optimization
Efforts have been widely made to achieve the most optimum NPV during the mine life.
Sevim and Lei (1998) described the simultaneous nature of open-pit mine scheduling. They
showed that long-term production planning in open-pit mines involves the simultaneous
resolution of four issues: 1) the production rate (number of blocks to be mined each year); 2) the
specific group of blocks that should be mined in a given year; 3) the cutoff grade to be used to
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determine ore and waste blocks; and 4) the ultimate pit limits. They depict these four issues
interacting in a circular fashion and propose a process that simultaneously handles all four
aspects of the problem. Ultimately, their solution methodology generates a series of nested pits,
from which the sequence with the highest NPV is chosen.
Xiaowei, et al. (2011) examined generating simultaneous optimization and production
schedules in open-pit coal mines using a dynamic sequencing method. The first step is to
generate geologically optimum final pits, or the one that contains the maximum coal quantity of
all the pits of the same volume. It should range from the smallest to the largest possible with
sufficiently small increments. Those geologically optimum final pits are only the candidates for
the final pit. The real optimum final pit should be evaluated by running production scheduling.
The dynamic programming scheme was implemented against a series of pushback designs. After
finding a state which results in the highest NPV, it traced the best transitions backward to the
first stage to indicate the optimum production schedule for the final pit.
Meagher, et al. (2014) also conducted a study to eliminate the gap problems in a pushback
design. A gap problem would occur if large section of ore was beneath a large amount of waste.
The authors formulated the pushback optimization problem as an integer programming (IP)
model. To facilitate the solution process and to overcome the gap problem, they solved the linear
programming relaxation version of the IP model first to obtain a fractional solution, to prevent
inefficient computations due to the large size of deposit data. The authors then converted those
fractional solutions into an integral one by applying a method known as “pipage rounding”; the
authors claimed their approach completely overcomes the gap problem.
Another shortcoming from the traditional pushback design is that it does not consider
discounting during the optimization. Dimitrakopoulos (2011) made a stochastic approach to
consider the risk and uncertainty during the life of the mine and take into account discounting
when optimizing the mine. The author generated a stochastic integer programming (SIP) model
which contained a minimization of the deviations together with the NPV maximization to
generate practical, feasible schedules and achievable cashflows.
Blom et al. (2018) discussed modern mixed-integer programming (MIP) based methods,
especially to create short-term mine planning. These approaches capture equipment behavior,
stockpiling, multiple processing pathways, multiple objectives, practical mining rules and mining
precedence. The study generated multiple, diverse short-term schedules while optimizing with
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respect to a customizable, prioritized sequence of objectives. For each time period t, the
remainder of the horizon is aggregated into increasingly coarse time periods. A MIP is solved
over this aggregation of time, and the activities of the focus period, t, are fixed to those in the
resulting solution. The algorithm then rolls forward to the next time period, t + 1, and repeats the
process of aggregating the remainder of the horizon and solve a MIP representation of the
scheduling problem.
The MIP of Blom et al. (2018) models take into account constraints, multiple processing
pathways for mined material, stockpiles and waste dumps, multiple truck and dig unit types,
truck cycle times and available trucking hours, constraints on the characteristics of ore fed into
each plant and stockpile, and constraints on the use of specific types of equipment across the
mine site. At predefined time periods t across the scheduling horizon, the optimizer makes
several different choices on activities performed in period t. For each of these sets of choices, a
new schedule is generated, and the algorithm proceeds to progress multiple schedules. Many of
the approaches described in this review generate short-term schedules while optimizing with
respect to multiple objectives. These objectives most commonly include the minimization of
operating costs, the minimization of deviations present in quantity and quality of produced ore
from desired targets, and maximizing the utilization of available equipment (often to achieve
maximal production or throughput).
The study done by Nwafor & Nwafor (2018) explored the use of Minex Optimizer
Programme to generate strategies for improving mining project economics at the Okobo coal
mine, Nigeria. The optimizer generated a series of pit expansions from the most valuable to the
least valuable material per tonne mined by defining a lower profit margin target. This procedure
can also develop an understanding of the economic behavior of the deposit. This series of nested
pits is then used as a guidance for creating production schedules.
Xiaowei, et al. (2021) presents a phased planning method specially designed for coal
deposits with nearly horizontal seams. Phase optimization, despite not being widely used in
open-pit planning in recent years, was demonstrated in a large coal deposit in northern China.
The study highlighted gap problems, as the size increment between two consecutive pits can be
very large and the most common shortcomings when generating schedules using nested pits and
proposed cone eliminating algorithms to overcome it.
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To generate the algorithm, this study modeled the deposit into a column model, where the
entire deposit is divided into vertical columns with square horizontal cross-sections instead of
blocks. The columns were then attributed by coal floor elevation and thickness to represent its
physical properties. This property is used to calculate the coal volumes within each column.
Other coal quality data was also added as the column’s attributes.
Given a certain number of coal production target and slope parameters which were
determined beforehand, the cone algorithm looked for a portion that contained such amounts of
coal with the highest strip ratio, save it and then eliminate it from the initial model. The
remaining parts then followed the same procedure and continued until the coal quantity in the
remaining parts were equal to or below the determined coal production target.
The mine in the study above has coal reserve estimated to be 900 Mt, with an annual
production rate of 20 Mt. The nested pits were developed to provide five years of production in
each phase to avoid complications during frequent transitions within phases. The algorithms
successfully divided the ultimate pit limit to an evenly spaced series of nested pits with small
variations. The shortcoming of this method is that the costs during the mine life are averaged,
while in actuality it fluctuates within the phases because of the quantities of rock and
unconsolidated material mined each year. Another shortcoming is that the transition between
phases happens sometimes before the previous phase is completely mined.
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CHAPTER 3
PROJECT DEVELOPMENT
This chapter provides a detailed explanation regarding project development in an open-pit
coal mine’s planning and production scheduling. The general structure to develop the project
follows the mine plan workflow as described in Chapter 1. There are many commercial software
packages that can be used to perform the steps of open-pit mine planning to determine the
production schedule and financial model for the project. This study uses MineSight 3D from
Hexagon Mining to run the entire process required for mine planning and focusing only on those
modules that work with the 3D block model file.
3.1 3D Geological Modeling
For modeling coal and layered deposits that have lateral distribution, a gridded-seam model
is usually used. Laterally, the coal deposit and surrounding area are divided into regular cells
with certain widths and lengths. At certain levels, the vertical dimension is not related to the
height, but to the stratigraphic unit of the deposit in question. Modeling is carried out in terms of
peak, bottom and thickness of the stratigraphic unit. The grades of various minerals or variables
are modeled for each layer.
However, almost all open-pit optimization formulations and solution algorithms take 3D
block models of deposits as geological inputs (Xiaowei, et al., 2021). Coal and other stratified
deposits that have characteristics of an irregular vertical-size model representing seam thickness
can also be treated as a regularly sized block by specifying the portion of coal material in a
block.
3.1.1 Block size and bench height selection
Block size and bench height selection are fundamental to building a block model. It relates
to the selective mining unit (SMU), which is defined as the smallest unit that can be mined
selectively. An ideal block size, in terms of bench height selection, is desirable for representing
the actual degrees of selectivity possible in practice. Equipment size is selected to match the
operation’s scale, and this approach is based on the premise that large equipment cannot
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generally mine small SMU sizes and also the requirement to minimize mining fleet numbers by
selection of the largest possible equipment.
Leuangthong, et al. (2004) discussed that size must somehow not only be related to the
ability of the equipment to select material but must also be based on the data available for
classification (blastholes and/or dedicated grade control drilling), the procedures used to translate
that data to mineable dig limits and the efficiency with which mining equipment excavates those
dig limits.
3.1.2 Assigning block attributes from exploration dataset
The complete dataset from the exploration phase needs to be imported as a main input to
assign values for the 3D block model. It includes seam numbers to differentiate waste layers and
every single coal seam, specific gravity, thickness and coal quality data such as calorific value,
ash, moisture, sulphur, fix carbon and volatile matter. The dataset can be imported through the
MSTorque module; MSTorque manages drillhole and blasthole data in an SQL database,
providing a platform to run procedures and calculations as well as creating composites.
Instead of creating a composite by seam, like in a gridded seam model, in a 3D block
model the composite will be generated by bench, since we need to have a regular-sized block
model in terms of bench height. The composited data is then assigned into the block model file
using the interpolation menu: Model Model Interpolation Tool. The interpolation method can
vary depending on the dataset characteristics.
When working with currently operating mines, the dataset generally has already considered
geostatistical distribution and is regularized in a lateral direction. In that case, the inverse
distance interpolation method can be used.
Stratified coal deposits, in which vertical continuity can be very different between the coal
seam(s) and interburden material, will best represent the actual seam data via inverse distance
with trend interpolation. It needs to define strike azimuth, dip and maximum distance from plane
and along trend, so the interburden (waste) layer will not be assigned any value interpolated from
the coal seam quality data.
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3.1.3 Surface topography and coal seam
As Clark and Dagdelen (2022) described, the information contained in the surface
topography file is integrated into the 3D geologic block model to designate the blocks, or
proportion of blocks, below that surface. To determine the block’s value, TOPO% item need to
be calculated to represent portion of material within a block which is located below topography.
In modeling stratified coal deposits, other than topography surface, it is required to have all
seams’ top (roof) and bottom (floor) surface to determine the proportion of coal within those
surfaces. When working with the GSM model, surfaces can be stored as GSF File 13. Because
this study works with the 3D block model, all surfaces will be saved as separate geometry files.
To import drillhole data for creating surface geometry, the point cloud menu can be used
by following these steps (one by one for all surfaces):
1) Prepare CSV files containing XYZ coordinates for each surface.
In MineSight 3D:
2) File Create Point Cloud to convert file into MS3D format (.hpc).
3) Surface Create from Point Cloud Mesher load 3D points from Point Cloud file
Calculate
Figure 3.1 3D Block Model Within Seam’s Top and Bottom Surfaces
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After creating the geometry for all surfaces, it is necessary to check whether the coal seam
resulted by the interpolation method is correctly placed within the seam roof and floor. This is
illustrated in Figure 3.1 above. Corrections can be made through the Model Calculation Tools
menu, by setting geometry filters to be between roof and floor surfaces, for each coal seam and a
seam number assigned to correlate with the filtered geometry surfaces. The comparison between
the gridded seam model and the 3D block model creation in MineSight 3D can be seen in Figure
3.2.
Figure 3.2 Comparison of Gridded Seam Model (GSM) and 3D Block Model (3DBM)
of the Coal Deposit Geological Model
3.1.4 Proportion of coal in a block
Compared to the mineral deposit model, which classifies a whole block as waste or ore
based on a cut-off grade, coal deposits can have both waste and coal within blocks at the coal-
rock interfaces, and such blocks constitute a significant portion of all blocks in the model.
Assigning ORE% item in the 3D block model File 15 will be the critical step to build the
stratified deposit model. It can be generated using the Code Model routine in the model
properties menu, layer by layer, sequentially.
The procedure captured below shows the need to set coal seam top and bottom surfaces in
the geometry set and then code total percentage between those surfaces as: 1) Ore, representing
the coal seam; and 2) Waste as inserted partial, representing the interburden (waste) layer. This
procedure should be done sequentially (either from top to bottom or otherwise, depending on
what is defined in the Surfaces tab). The option to reset items before coding should be left blank
to keep the portion code from the previous layer when generating code for material between new
surfaces.
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Figure 3.3 Model View to Code ORE% Item Representing Proportion of Coal
3.2 Production Capacities and Cost Structure
Production rate and mine life both play a big role in determining project economics.
Shorter mine life due to a higher production rate maximizes the net present value, but usually
requires a greater capital cost as larger equipment and infrastructure is required. Lower
production rates can drive a longer mine life, but also can result in higher operating costs and a
smaller footprint. However, Clark and Dagdelen (2022) stated the mine planner should also be
aware of an unrealistically short mine life generated by excessive mining rates to improve the
NPV.
Determining production capacity should also consider limitations related to infrastructure,
environmental issues, government policies and any other operational constraints. The rates will
then impact the estimations of operating, sustaining capital costs and initial capital costs. While
an economic model calculation only considers operating cost and sustaining capital cost, initial
capital cost should be included to realize the generated profit of a project. However, instead of
estimating the initial capital costs of additional required infrastructure in some cases, this study
will provide the advantages of selecting certain scenarios to consider whether to make up those
initial capital costs for management decision-making.
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3.3 Economic Block Modeling
The block economic value (EV) is one of the most important parameters in mine
evaluation. It is described as the net profit that will be generated if the specific blocks are
extracted. To determine the economic value of a mining block, one needs to determine both the
revenues generated by selling a portion of coal in the block, and the mining costs related to the
extraction and processing of coal present in the mining block.
(3.1)
The revenues gene𝐵𝐵ra𝐵𝐵t𝐵𝐵e𝐵𝐵d𝐵𝐵 b y𝐸𝐸 𝐵𝐵s𝐵𝐵el𝐸𝐸li𝐵𝐵n𝐸𝐸g 𝐸𝐸c𝐵𝐵o 𝑉𝑉al𝑉𝑉 p𝐵𝐵r𝑉𝑉e𝑉𝑉se=nt 𝑅𝑅in𝑉𝑉 𝑅𝑅a𝑉𝑉 m𝐸𝐸i𝑉𝑉n𝑉𝑉in−g b𝐶𝐶l𝐵𝐵o𝐶𝐶c𝐶𝐶k𝐶𝐶 c an be estimated by
multiplying tonnage of the portion of coal present in the mining block, a recovery factor
representing the loss of material during the mining and processing (if any) activities, and the
commodity price. It can be expressed as:
(3.2)
The 𝑅𝑅co𝑉𝑉a𝑅𝑅l𝑉𝑉 p𝐸𝐸r𝑉𝑉ic𝑉𝑉e =in 𝐶𝐶U𝐶𝐶S𝐶𝐶D𝐶𝐶/%ton 𝑥𝑥n e𝑏𝑏 𝐵𝐵c𝐵𝐵a𝐵𝐵n𝐵𝐵 v 𝑅𝑅ar𝐵𝐵y𝐵𝐵 𝑉𝑉d𝐸𝐸ep𝑉𝑉e n 𝑆𝑆d𝑆𝑆in𝑐𝑐𝑐𝑐 g𝑐𝑐 o𝑐𝑐 n𝑥𝑥 t%he 𝑅𝑅m𝑉𝑉a𝐵𝐵r𝐵𝐵k𝑅𝑅e𝑉𝑉ti𝑅𝑅n𝑅𝑅g 𝑥𝑥sa 𝐶𝐶le𝐵𝐵s𝑉𝑉 c𝐵𝐵o 𝑃𝑃n𝑅𝑅tr𝐸𝐸a𝐵𝐵c𝑉𝑉t, which is
mainly generated by coal calorific value (CV, in kcal/kg). There are cases when the coal price is
set flat within a certain range of calorific value, either as-received or on an air-dried basis.
Conversely, the price can be determined proportionally by its real calorific value, relative to a
fixed price for certain calorific values. It is also common to get gain and penalty costs by setting
a range of other coal quality parameters, such as ash and total sulphur contents of the product.
Due to the nature of the sales product’s characteristics in the case studied, the revenue
calculation setup for the next discussion will focus only on proportionalized coal price by its
calorific value on an as-received basis.
In open-pit mining, the block mining cost results only from operational cost and sustaining
capital costs, as stated earlier. The operational cost contains those costs, which relates to mining
a block from the deposit and delivering it either to the processing plant or stockpile for coal
material, or to the dump for waste material. The processing activities almost always required are
crushing, which aims to reduce the overall size of the run-of-mine (ROM) coal so it can be more
easily handled and processed, and screening to group the process particles into ranges by size.
The more complex activity is the coal washing process, which is usually used to remove
contaminants per environmental and technical considerations. It not only reduces the ash content
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Using the block SG, the block tonnage will be:
Coal = 1.26 x 94.76% x 5,000 = 5,969.75 tonnes
Waste = 2.20 x 5.24% x 5,000 = 576.63 tonnes
Block tonnage adjusted to TOPO remain the same because TOPO% is 100%
Value/Tonne = CVAR x Coal Price x Recovery x Factor
= 5,326.94 kcal/kg x 0.012 USD/(ton.kcal/kg) x 90% x 1
= 57.53 USD/tonne
Processing Cost = 28.0205 USD/ton
Mining Cost = 2.04 USD/tonne for coal
= 1.31 USD/tonne for waste
Profit/tonne = Value/tonne – Processing Cost
= 57.53 USD/tonne – 28.0205 USD/tonne
= 29.51 USD/tonne
Gross Profit = Profit/tonne x Block Tonnage
= 29.51 x 5,969.75
Processing Cost = 28.0205 USD/ton x 5,969.75 tonnes
= 167,275.32 USD
Mining Cost = (5969.75tonnes x 2.04USD/tonne)+(576.63tonnes x 1.31USD/tonne)
= 12,193.21 USD + 755.56 USD
= 12,948.77 USD
Block Value = (Gross – Mining Cost) x Bench Factor
= 163,221.38 USD (there is no bench factor assumed)
The values resulting from the above routines should be verified by auditing the blocks or
assigning those values to the block model item. To assign value into the block model item, use
Import/Export Export Values select the model item from PCF File 15.
3.4 Ultimate Pit Limit Analysis
Pit shells are generated to reflect economic design. Pit Shell Generation in Project
Evaluator uses the Pseudoflow algorithm to generate the optimal pit limits based on the Block
Model Source, Economics and Pit Shell Generation inputs. The program claims that this method
achieves the same optimal pit limits as the traditional Lerchs-Grossmann Algorithm, but in a
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Under the Restriction tab menu, there is option to limit the block in the XYZ/row-column-
level format. If there is certain area which should not be considered in generating the pit shell,
we can import the boundary to the Surface Restriction tab menu with a column to input
additional costs to mine the area. However, if those areas will not be mined, the boundary should
be categorized as a hard restriction.
The following figure presents an example of the ultimate pit boundary in a particular coal
deposit, indicated by the black line, after considering the surface restrictions that prevent it from
optimally mining to the lowest seam (shown in yellow).
Figure 3.8 Ultimate Pit Boundary at a Given Coal Deposit
3.5 Generating Pushbacks for Production Scheduling
As explained earlier chapter, the idea to create pushback is to get smaller nested pits over
the mining phases. They are generated either by varying the coal price, stripping costs or
stripping costs over coal price, as long as it can result in increment pits to differentiate the most
profitable areas to the least profitable one. The optimization engine uses the MILP (Mixed-
Integer Linear Programming) concept and can optimize NPV over the life of mine through multi-
period scheduling.
However, the biggest problem with pushback mining is the gap problem, especially in the
case of huge amounts of a deposit with the optimum pit that has a stripping rate close to the
breakeven point. The mine plan engineer needs to decide on the price variation which would
result in the best stable increments, sometimes it requires the manual cut generation due to the
gap problem. This procedure is critical to providing mining phases, which have decent volume
increments, to be scheduled periodically. For the best result, the schedule optimization will result
well if the increment has average volume in each level less than the periodic production target
volume.
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Figure 3.9 sets an example of mining increment phases generated by varying economic
parameters, such as coal price. In Project Evaluator, those increments can result from setting
cases in the destination economic setup, as explained by Figure 3.5, followed by creating cases
based on that setup in pit shell generation.
Several price assumptions can be set in the initial simulation. The selected price
assumptions are then selected to provide relatively stable volume increments between phases,
preferably lower than the targeted annual production, to be the inputs for scheduling production.
Figure 3.9 Mining Phases Generated by Varying Coal Price
The lowest price gives the smallest (yet most profitable) area to be mined, as illustrated by
Phase 1 in Figure 3.9. The highest price otherwise, also known as the ultimate pit, is the biggest
yet least profitable result with the highest strip ratio. The number of phases, which are then set as
mining precedence in the production scheduling routine, represent a prioritization for mining to
achieve the maximum net present value.
3.6 Production Scheduling
Schedule optimization based on MILP is performed using the Mine Plan Schedule
Optimizer (MPSO) module in Hexagon’s mining software. For input geometry, we need to
perform cut generation for each pushback design created prior. The software reads a cut as an
object that should be mined together at a time and it will not try to mine a part of this cut. This is
also why feasible pushbacks are needed that can provide a decent amount of material for each
level to be mined at one time.
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For long-term planning, we will only set horizontal cuts by plane (bench) so that, at one
time, the software will mine the entire level of each pushback. It will determine the cut mining
sequence that can be scheduled by the software. To implement the mining phases set up prior, we
need to run phase precedence after setting it up sequentially to mine from phase one to the last
phase. The main goal of this scheduling is to provide the required amount of coal with a
relatively stable strip ratio, as it will tie to the equipment needed each year. Volume fluctuations
can result in an unstable investment and unutilized equipment.
With the goal of maximizing the net present value (NPV) of the coal project, the mine
sequencing problem endeavors to determine the most efficient extraction schedule while meeting
sequencing and production constraints. This NPV is the sum of all extracted blocks’ discounted
net profits, which can be calculated by subtracting mining and processing costs from coal sale
revenue. For building the mathematical model, the definition of “block” here is a bit different
than is referenced with relation to the 3D block model. In this case, the block refers to the entire
material within a bench in each mining phase. The idea is to mine that deposit by using phase
and level precedence constraints.
3.6.1 Assumptions
To make the problem more manageable, the following assumptions are made:
The pushback design has been created using Pit Expansion Tools, considering the slope
design parameters as recommended by the geotechnic engineer (including face slope angle,
berm width, bench height and overall slope). As long as one level is completely mined
before going to the next, the slope constraint can be ignored.
Fractional mining of the bench is allowed. It means that a bench can be mined during
several years. However, a bench should be completely mined before going to the next
bench below it. In this case, completion of mining a bench is represented by the following
variable:
(3.4)
1,𝐸𝐸𝑖𝑖 𝑏𝑏𝑉𝑉𝐸𝐸𝐵𝐵ℎ 𝐸𝐸 𝐵𝐵𝑖𝑖 𝑝𝑝ℎ𝑉𝑉𝐶𝐶𝑉𝑉 𝑗𝑗 𝐸𝐸𝐸𝐸 𝑉𝑉 𝑚𝑚𝐸𝐸𝑅𝑅𝑉𝑉𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝑉𝑉 𝐶𝐶𝐵𝐵𝑉𝑉𝑅𝑅𝐵𝐵𝑉𝑉 𝐵𝐵
𝑅𝑅𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑤𝑤 =� 𝐸𝐸𝐶𝐶 𝐵𝐵𝐵𝐵𝐸𝐸𝑝𝑝𝐵𝐵𝑉𝑉𝐶𝐶𝑉𝑉𝐵𝐵𝑅𝑅 𝐸𝐸𝐸𝐸𝐸𝐸𝑉𝑉𝑚𝑚 𝑉𝑉𝐶𝐶 𝑝𝑝𝑉𝑉𝑅𝑅𝐸𝐸𝐵𝐵𝑚𝑚 𝐶𝐶
0,𝐵𝐵𝐶𝐶ℎ𝑉𝑉𝑅𝑅𝑒𝑒𝐸𝐸𝐶𝐶𝑉𝑉
The model only concerns with the coal throughput capacity of the mine. Since the mine
uses a mining contractor to remove waste material, it is assumed the waste volume capacity
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can be assigned following this sequence result as long as the sequence is relatively stable
over years.
The model is completely deterministic and does not consider uncertainty. We can assume
the location, mineral content and total material content of each deposit’s block is certainly
known.
3.6.2 Objective function
The MPSO module’s objective in this study is set up to maximize the net present value of
the entire mine project. The calculated NPV is the sum of all extracted blocks’ discounted net
profits. The time value of money is considered by discounting the profit based on the time period
in which the blocks are removed from the mine.
(3.5)
𝑡𝑡 𝑡𝑡 𝑡𝑡
�𝑋𝑋𝑋𝑋𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑙𝑙𝑋𝑋𝑖𝑖,𝑗𝑗,𝑘𝑘��𝑃𝑃 𝑥𝑥 𝑋𝑋𝐶𝐶𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑙𝑙�−𝑂𝑂𝑋𝑋 −𝑀𝑀𝑋𝑋𝑖𝑖,𝑙𝑙�−𝑋𝑋𝑋𝑋𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑙𝑙 𝑋𝑋𝑖𝑖,𝑗𝑗,𝑘𝑘𝑋𝑋𝑋𝑋𝑖𝑖�
𝐼𝐼 𝐽𝐽 𝐾𝐾 𝐿𝐿 𝑇𝑇
𝑁𝑁𝑃𝑃𝑉𝑉 = ∑𝑖𝑖=1∑𝑗𝑗=1∑𝑘𝑘=1∑𝑐𝑐=1∑𝑤𝑤=1 (1+𝑑𝑑)𝑡𝑡
Where
i: Benches (i = 1, …, I)
j: Mining phases (j = 1, …, J)
k: Pit sources (I = k, …, K)
l: Coal destination (l = 1, …, L)
t: Time period an activity is taken (t = 1, …, T)
d: Discount factor
XC : Portion of coal mined from bench i, phase j, mine source k and sent to
i,j,k,l,t
destination l in time period t (%)
C : Total available coal at given bench i, phase j, mine source k (tonnes)
i,j,k
XW : Portion of waste mined from bench i, phase j, mine source k and sent to
i,j,k,l,t
destination l in time period t
W : Total available waste at given bench i, phase j, mine source k (tonnes)
i,j,k
CV : Average CVAR mined from bench i, phase j, mine source k and sent to
i,j,k,l,t
destination l in time period t (kcal/kg)
P: Unit coal price $/(tonne.kcal/kg)
OC: Other cost including indirect mining cost, G&A cost, transportation to port, and
marketing cost which is applied to coal mined ($/tonne)
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MC : Coal mining cost from mine source i and sent to destination l ($/tonne)
i,l
WC: Waste stripping cost from mine source i ($/tonne)
i
3.6.3 Constraints
The objective function is restricted by both geological and operational constraints.
a) Level Precedence Constraint
The sloping requirements and sequencing constraints should be considered to keep the pit
walls from collapsing. As stated in the assumptions above, the slope degree rule can be
replaced by the requirement to take one bench completely before stepping to the bench
below it. We number benches from top to bottom, even if at a given bench there is no
material to be mined.
For ∀ i,j,k (3.6a)
𝐿𝐿 𝑇𝑇 𝑤𝑤 𝑇𝑇 𝑤𝑤
∑𝑐𝑐=1∑𝑤𝑤=1𝑋𝑋𝐶𝐶𝑖𝑖,𝑗𝑗,𝑘𝑘+1,𝑐𝑐 −∑𝑤𝑤=1𝑅𝑅𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 0 For ∀ i,j,k 3.6b)
𝐿𝐿 𝑇𝑇 𝑤𝑤 𝑇𝑇 𝑤𝑤
∑𝑐𝑐=1∑𝑤𝑤=1𝑋𝑋𝑊𝑊𝑖𝑖,𝑗𝑗,𝑘𝑘+1,𝑐𝑐 −∑𝑤𝑤=1𝑅𝑅𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 0
b) Phase Precedence Constraint
A phase can only be mined at a same or higher level than the precedence phase. However,
in practice, mining companies tend to have four to five bench differences between one
phase and another. The phase numbers in this case show the priority to be mined.
For ∀ i,j,k (3.7a)
𝐿𝐿 𝑇𝑇 𝑤𝑤 𝑇𝑇 𝑤𝑤
∑𝑐𝑐=1∑𝑤𝑤=1𝑋𝑋𝐶𝐶𝑖𝑖,𝑗𝑗+1,𝑘𝑘,𝑐𝑐 −∑𝑤𝑤=1𝑅𝑅𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 0 For ∀ i,j,k (3.7b)
𝐿𝐿 𝑇𝑇 𝑤𝑤 𝑇𝑇 𝑤𝑤
∑𝑐𝑐=1∑𝑤𝑤=1𝑋𝑋𝑊𝑊𝑖𝑖,𝑗𝑗+1,𝑘𝑘,𝑐𝑐 −∑𝑤𝑤=1𝑅𝑅𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 0
c) Reserve Constraint
This constraint ensures the total amount of material extracted is less than or equal to the
total available material, and makes sure the same bench in a given phase from each mine
pit will not be mined twice.
For ∀ i,j,k (3.8a)
𝐿𝐿 𝑇𝑇 𝑤𝑤
∑𝑐𝑐=1∑𝑤𝑤=1𝑋𝑋𝐶𝐶𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑐𝑐 ≤ 1
For ∀ i,j,k (3.8b)
𝑇𝑇 𝑤𝑤
∑𝑤𝑤=1𝑅𝑅𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 1
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d) Coal Production Constraint
The main constraint for scheduling purposes in this case study is coal throughput capacity.
To keep the business running, but with limited throughput capacity, minimum and
maximum production constraints for each period are set.
For ∀ i,t (3.9)
𝑤𝑤 𝐽𝐽 𝐾𝐾 𝐿𝐿 𝑤𝑤 𝑤𝑤
𝐶𝐶𝑃𝑃𝑖𝑖 ≤ ∑𝑗𝑗=1∑𝑘𝑘=1∑𝑐𝑐=1𝑋𝑋𝐶𝐶𝑖𝑖,𝑗𝑗,𝑘𝑘,𝑐𝑐𝐶𝐶𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 𝐶𝐶𝑃𝑃𝑖𝑖
Where
lower limit of coal production from mine i at period t
𝑤𝑤
𝐶𝐶𝑃𝑃𝑖𝑖 upper limit of coal production from mine i at period t
𝑤𝑤
𝐶𝐶 𝑃𝑃𝑖𝑖
e) Waste Production Constraint
If the mine uses a mining contractor to remove waste material, there is no specific waste
volume capacity to be assigned. However, in regards to creating a more balanced
production schedule, we need to set that up.
For ∀ i,t (3.10)
𝑤𝑤 𝐽𝐽 𝐾𝐾 𝑤𝑤 𝑤𝑤
𝑊𝑊𝑃𝑃𝑖𝑖 ≤ ∑𝑗𝑗=1∑𝑘𝑘=1𝑋𝑋𝑊𝑊𝑖𝑖,𝑗𝑗,𝑘𝑘𝑊𝑊𝑖𝑖,𝑗𝑗,𝑘𝑘 ≤ 𝑊𝑊𝑃𝑃𝑖𝑖
Where
lower limit of waste production from mine i at period t
𝑤𝑤
𝑊𝑊𝑃𝑃𝑖𝑖 upper limit of waste production from mine i at period t
𝑤𝑤
𝑊𝑊 𝑃𝑃𝑖𝑖
3.6.4 Balancing production profile
All previous constraints should be run first to evaluate the most appropriate waste capacity
to be assigned for balancing production. We only need to set the maximum capacity of waste
volume for each period. Changing volume constraints during the horizon is allowed, considering
maintenance of the same capacity for at least four to five years to maximize equipment
utilization.
There might be a case when waste volumes are too high for a certain period. We should
consider having a pre-production period by setting a minimum amount of waste to be removed
and maximum coal tonnage at a much lower capacity than required (it may be impossible to set
the maximum tonnage of coal to zero) at the same time.
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Setting up constraints can be a bit tricky; if we set too many and strict constraints, the
software may not be able to give the desire result. It is related to the input geometry cuts
explained above.
The basic idea of phase optimization is generating pits with a lower stripping ratio, either
by varying the coal price, stripping cost or stripping cost over coal price, so long as it can result
in incremental pits to differentiate the most profitable areas from the least profitable one. The
optimization engine uses the MILP (Mixed-Integer Linear Programming) concept and can
optimize NPV over the life of mine through multi-period scheduling.
As explained in the previous study, there are problems often raised in phase optimization:
the gap problem. To overcome this issue, the engineer needs to create manual increments and
evaluate if there is a significant difference between each phase. It will prevent the software from
finding the targeted production volume due to the high material within a level in a phase.
The major aspects to be considered are coal volume, strip ratio and coal quality. This study
will not take quality into account since the three areas have a close range of calorific value, and
because the price unit which is determined is USD/tonne/(kcal/kg) so it has already considered
the blending quality.
The main goal of this scheduling is to provide the required amount of coal with a relatively
stable strip ratio since it will have an impact on the equipment needed each year. We cannot have
over-fluctuating volumes because the investment cannot go up and down (leaving some
equipment unused, or vice versa).
There might be a case when optimization stops at a certain period before all available coal
is mined. This may be impacted by the gap problem, which relates to the deposit’s characteristics
and the pushback design increments created. Not maximizing the coal mined might lead to a
suboptimal NPV. Despite the shortcomings of MPSO, we keep using the software with
additional adjustments for an optimum result.
a) First, set the constraints as simply as possible to allow for the software’s completing
schedule. This study uses coal production as the basic constraint.
b) Evaluate the results by analyzing the MSReserve levelling format and assign the proper
waste volume constraints possible for implementation. The format shown by Table 3.1 is
also useful to validate satisfaction of the levels and phase precedence constraints.
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The percent or ore and waste mined under the production schedule may not always mine
100% material due to slope construction. The engineer can evaluate whether or not the
difference is acceptable in regard to the impacted volume/tonnage compared to the total
deposit.
c) Run the algorithm period by period, do the evaluation and make adjustments as needed.
The required adjustments may be to change the coal’s lower or upper limit. We can
evaluate this using the MSReserve format above to see what we have and how much the
software needs to take. That way, a decision can be made whether to decrease the lower
limit or increase the upper limit considering the mine and infrastructure’s capacity.
d) If the program stops near or at the end of the horizon, such as if material has run out, the
amount of material left is less than the minimum constraint. At that case, the minimum
constraint should be removed.
e) At certain periods, we may need to set a minimum waste volume to be mined to maintain
continuous production.
3.7 Financial Analysis
Investment decisions are important to financial management decisions because, in addition
to investing in businesses that require large capital, these decisions also contain certain risks and
directly affect the value of the company.
There are conventional and non-conventional methods to assessing the profitability of an
investment plan. In the conventional method, two benchmarks are used, payback period (PBP)
and internal rate of return (IRR). Meanwhile, the in non-conventional method, there are three
kinds of profitability benchmarks: Net Present Value (NPV), Profitability Index (PI) and Internal
Rate of Return (IRR). This study will only focus on the most common benchmarks, which are
NPV and IRR, to assess the optimization project compared to the existing one.
Net Present Value (NPV) is the present value of all future cash flows of the project. A
project is economically feasible if NPV > 0. The choice for the best project from several
mutually exclusive alternatives can be decided to the highest NPV, especially for scenarios
which have different mine lives. The formula to calculate NPV can be expressed as:
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(3.11)
𝑅𝑅𝑡𝑡
𝑁𝑁𝑃𝑃𝑉𝑉 =
(1+𝑖𝑖)𝑡𝑡
Where: NPV = Net Present Value
R = net cash flow at time t
t
i = discount rate
t = time of the cashflow
Another method to assess project alternatives is the internal rate of return (IRR). IRR is the
value of discount rate i that makes the NPV of the project equal to zero. The discount rate used to
find the present value of a benefit/cost must be equal to the opportunity cost of capital. The basic
concept of opportunity cost is essentially a sacrifice that is given as the best alternative to get a
result and benefit, or it can also state the price that must be paid to get it. IRR can also indicate
the efficiency of an investment. This method is used to create ranks for several investment
scenarios. An executed project must have an IRR higher than the minimum acceptable rate of
return (MARR), which is the minimum rate of return that investors are willing to accept. In the
case of dealing with two competing investment opportunities that involve different cost
structures, it will be best to use the incremental IRR analysis. It can also be used when one
already incurred an expense and wants to determine if it is a good decision to spend additional
funds.
Incremental IRR is calculated in exactly the same way as general IRR, but using the
differences between the cashflows of the two projects. If it is higher than the minimum
acceptable rate of return, the more expensive investment is considered the better one. However,
it still should be noted that this more expensive investment also has a higher risk; hence, an
investor must consider various factors that affect IRR before making an investment decision.
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CHAPTER 4
CASE STUDY OF AN OPEN PIT COAL MINE IN INDONESIA
4.1 Introduction of the Mine
The mine is located in Sumatra Island, Indonesia. It has been operating for more than 20
years. Due to its location in the middle of the island (with a distance of more than 200 km from
the port), the main bottleneck of production capacity at this mine is coal transportation from the
mine to the port. There are three major mine areas discussed in this study. Information regarding
coal reserves and economic evaluation presented in this chapter and discussion are factored to
maintain the confidentiality of the data.
Figure 4.1 The Studied Mine Area
The studied mine is divided into two areas, eastern and western, which are separated by a
river. Infrastructure development, such as train loading station (TLS), heavy equipment
management, electrical sources and so on are built independently for each area. As can be seen at
Figure 4.1, Mine 1 is located in the eastern area, having its own independent coal throughput
facility. Meanwhile, in the western area, there are Mine 2 and Mine 3, which share the coal
throughput facility with the flexibility of using heavy equipment adjusted to the production
required in each mine.
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4.1.1 Stratigraphy and Geological Condition of Deposit
The three mine areas (Mine 1, Mine 2 and Mine 3) lie in the same coal formation, the
Muara Enim formation. This formation was formed during Middle to Upper Miocene period,
give the deposit calorific value ranges between lignite and subbituminous.
Figure 4.2 Mine 1 Coal Stratigraphic, without scale
Mine 1 is located in the eastern part of the studied project area. The deposit has an anticline
in the southeast-northwest direction with a dip to the northwest. The coal dip in the northwest
direction varies between 10˚ and 20˚, while the east-west is steeper at about 70˚. The mine has
seven coal seams, as described in Figure 4.2, which belong to Muara Enim formation. The
average thickness of the coal seam in Mine 1 is 10 m, with the majority of calorific value ranging
between 4500-5300 kcal/kg (5000 kcal/kg on average).
The western part of the studied mine consists of Mine 2 and Mine 3. In Mine 2, the coal
deposit has dips of between 10˚ and 20˚ and there are no large faults. The average thickness of
each seam is given in Figure 4.3. The mine has shorter ranges of calorific value, between 4500
and 4900 kcal/kg, with average of 4800 kcal/kg. The lower part of Mine 2’s coal stratigraphic,
Seams D and E, are not included in the study because both seams generally do not offer coal
potential within the depth range for economical surface mining.
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Figure 4.3 Mine 2 Coal Stratigraphic, without scale
Meanwhile, the geological structure in Mine 3 is an ellipsoidal dome with the long axis
trending northeast-southwest and characterized by the distribution of layers of circular coal. The
dome was formed from the intrusion effect of andesite igneous rock. Geological interpretation
shows there are several normal faults, both major north-south trending on the eastern flank of the
dome, as well as other minor faults. Displacement due to normal faults is estimated to be around
10 m to 100 m, with the magnitude of the fault angle difficult to determine. The magnitude of the
coal seam dip varies between 5˚ and 20˚.
The presence of this intrusion makes the deposit in Mine 3 a higher calorific value, ranging
between 5,000 and 7,000 kcal/kg. However, the high-calorific value (CV) coal has already been
mined, and this study will focus only on 5,000–5,500 kcal/kg coal with an average of 5,200
kcal/kg. Seam J and EN, as shown in Figure 4.4, are the uppermost and stratigraphically
youngest sequence of the Muara Enim formation; hence, they have a lower calorific value and
are not considered in reserve calculation. In addition, similar to Mine 2, seam D and E are not
included in the study due to position to the surface, which is too deep.
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4.1.2 Mine and current operations
The biggest bottleneck of production is coal transportation, as the mines are located more
than 200 km from the port. The current capacity of the train loading stations (TLS) for all mine
areas is 26 million tonnes. With addition of 2 million tonnes of local sales, the current mine and
sales capacity of those mines is 28 million tonnes annually, with average strip ratio of 5.
Mine 1, Mine 2 and Mine 3 use a combination of the truck-and-shovel method as well as a
conveyor system. About 40% of the total production capacity is managed by the company itself
(including heavy equipment). The company keeps 60% of the total production capacity to be
mined by mining contractors with consideration of risk sharing and flexibility of production
volume change within years. These mining contracts are generally five years in length.
Figure 4.5 illustrates the general coal flow in this mine. From the mine, coal is transported
to a temporary stockpile by truck, an average distance of 2-3 km. At the temporary stockpile, a
bucket wheel excavator (BWE) reclaims the coal to be sent to the stockpile via conveyor. The
majority of the coal in the stockpile will be transferred to Port A and Port B, located 180 km and
400 km, respectively, by train. The current annual train capacity going to Port A is 5 million
tonnes, while Port B is 21 million tonnes, while the rest of the coal is locally sold. An increase to
train capacity is planned in the next five years, which was already considered when determining
the production target in this study.
The coal stockpiles are categorized based on mine product’s calorific value. The coal is
then blended during transfer to the train in such a way that each coal wagon transports a specific
quality of coal. The blending process is also carried out at the port and adjusted for market
specifications, either long-term or by spot contract.
4.1.3 The previous feasibility study
An earlier feasibility study was conducted in 2018 as part of regular data update. Since
then, there have been changes in some design parameters and additional area restrictions. Using
previous slope design parameters, it was found that several landslides occurred in the past three
years. In addition, difficult conditions caused mining activities to be carried out outside of the
long-term plan, which resulted in the emergence of disposal areas within the initial ultimate pit
boundary. This caused the need for planning adjustments and a cost analysis to find out which
optimization efforts were feasible.
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The previous study assumption used an overall slope of 20˚ on average for the three mine
areas, with a bench face angle of 50˚, berm width of 20 m and bench height of 10 m. Considering
the real events that occurred in the field, this parameter was then adjusted to be gentler with
detailed specifications discussed later.
The detailed parameters assumed by the previous study are as follows:
Table 4.1 The Previous Study’s Assumption Parameters
Description Mine Area 1 Mine Area 2 Mine Area 3
Unit Coal Price 0.011 $/(tonne.kcal.kg)
Stripping Cost 1.60 1.37 1.70
Coal Mining Cost 2.00 1.79 1.76
Other Costs, include: 28.54 28.54 28.54
- Indirect production cost
- Coal transport to port
- G&A cost
- Marketing cost
Overall Slope 20 18 15
Bench Face Angle 50 50 50
Berm Width 20 m 20 m 20 m
Bench Height 10 m 10 m 10 m
4.1.4 Previous commercial optimizer software
Almost all open-pit optimization formulations and solution algorithms take 3D block
models of deposits as their geological inputs (Xiaowei, et al., 2021). The mine previously used
Gemcom’s Minex as its optimization software. Minex is a seam or layered modeling system, the
basis of which is a 2D grid or surface model. The combination of top, bottom and thickness for
each seam make up a 3D geological model.
There is no significant difference between optimizing a pit in mineral or a stratified
deposit. Minex Pit Optimiser also uses the 3D block model and optimizes pit limits using the
Lerchs-Grossmann Algorithm. The first stage of optimization is to determine the individual
block values. The thickness of the coal seam is the critical factor to calculate the portion of coal
within a block, and thus determine the block’s value. The base seam of the model should be
introduced, since the optimizer will read it as the logical pit floor. Therefore, material below the
base seam will not be assigned a cost. Referring to the figure below, W7 will not be used for the
calculation.
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Figure 4.6 Seam Model Within 3D Block Model
Block values are the input parameters to run the optimization algorithm until the optimum
pit at a given price is reached. A series of nested pits is then required to generate a mining
sequence which results in maximum cash flow and NPV. Those can be created by determining
higher coal sale prices so that the optimum pit will be larger, as extra revenue makes a deeper pit
more economic.
Mining in the 1, 2, 3, 4 sequence maximizes NPV because the smallest pit is optimum at
the smallest sale price. Mining of pit 1 first maximizes early cash flow and thus maximizes NPV
(the least profitable pit will be mined last, where the price is discounted the most.)
4.2 Case Study: 3D Geologic and Economic Models
4.2.1 3D geological model
The mine has been operating for mor than 20 years. There are mining contracts currently
running with specified equipment specifications. Existing infrastructure is also built with certain
specifications, such as coal size requirements, that can be the input. This is directly related to the
heavy equipment used (digger, ripper and hauler) to be able to produce the required size of coal.
Therefore, the selection of SMU, in term of bench height criteria, is adjusted to the current
operation.
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Blasting and ripping design parameters are in a way as to produce a coal size of about 20 mm.
Considering the equipment used and coal seams’ average thicknesses, the average bench height
of 8m was used in the mining operation. Accordingly, the block model size was defined to be 8m
for the vertical size, and 25m x 25m in the lateral direction.
4.2.2 Production capacity
As stated in the introduction to mine section, the main bottleneck of mine production at
this operation is the coal transportation throughput capacity. There are four throughput
infrastructures (train loading station, or TLS) which are divided into two areas. TLS 1 and 2
serve Mine Area 1 with a total capacity (dedicated for Mine 1) of 8.3 million tonnes annually.
TLS 3 and 4 serve Mine Area 2 and 3 with total yearly capacity of 10.7 million tonnes. The
production target is set to allow infrastructure to operate at their optimal capacity. Waste volume
capacity will not be defined at the beginning of the simulation because the mine has a partnership
with mining contractors, so the annual production capacity of waste can be adjusted as long as it
is stable over the 4-5 years of the contract.
4.2.3 Economic block modeling
The input parameters for this case are as follows:
1. Specific gravity
a. Coal : 1.26 – 1.30 ton/m3
b. Waste : 2.20 ton/m3
2. Destination Economic
The coal’s calorific value in this deposit ranges from 4,700 to 7,000 kcal/kg gar, with the
majority price is based on the Indonesian Coal Index (ICI) – 3 for 5,000 kcal/kg and
calculated proportionally. The company uses a coal forecasting price, which has been
analyzed by Wood Mackenzie until 2050 (as presented in Figure 4.10). For the ultimate pit
limit generation, the coal price of $60/tonne was selected. The economic material used in
this study is CVAR (coal calorific value on as-received basis).
Since the control grade of the evaluation has been determined by, the price will be adjusted
by dividing the $60/tonne price by the 5,000 kcal/kg calorific value, so the unit price as the
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input parameter will be $0.012 /(tonne kcal/kg). The recovery is assumed to be 90%,
including geological and mining losses.
160
140
120
100
80
60
40
20
0
2009 2013 2017 2021 2025 2029 2033 2037 2041 2045 2049
FOB Newcastle @ 6,000 kcal/kg NAR, market FOB Newcastle @ 6,322 kcal/kg GAR, JPU contract
FOB HA Newcastle@ 5,500 kcal/kg NAR FOB Richards Bay @ 6,000 kcal/kg NAR
FOB Indonesia @ 5,000 kcal/kg GAR FOB Indonesia @ 4,200 kcal/kg GAR
FOB Richards Bay @ 6,000 kcal/kg NAR FOB Bolivar @ 6,300 kcal/kg GAR
S Id i t th A Mdi G (hit ) W d M k i C l M kt S i (F t)
Figure 4.10 Thermal Coal Price Forecast, FOB (US$/t, real 2021)
Source: the Argus Media Group and Indonesian govt. (history), Wood Mackenzie Coal
Market Service (forecasts)
3. Mining Costs
The cost structure in this case study only considers operating and sustained capital cost.
Both costs are already calculated in USD/tonne. Capital cost is not included in the
calculation because, for the currently operating mine, there is no capital cost assumed,
except if there will be increasing capacity for the coal production infrastructure. The cost
parameters assumed in the generation ultimate pit and production scheduling are presented
in Table 4.2.
Table 4.2 Cost Parameters (in USD/tonne)
Description Mine Area 1 Mine Area 2 Mine Area 3
Stripping Cost 1.31 1.31 1.15
Coal Mining Cost 2.04 2.14 2.24
Other Costs, include: 28.02 28.02 28.02
- Indirect production cost
- Coal transport to port
- G&A cost
- Marketing cost
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4.2.4 Ultimate pit analysis
The pit shell generation is limited to the elevation -250 m, about +400 m from the surface,
considering the life span of the slope based on geotechnical recommendations. There is also a
hard restriction on the surface area at Mine Area 1 and 3, which have been considered in this
study.
As for the slope parameters, Mine Area 1 and 2 use 15˚ as the overall slope angle, while
Mine Area 3 is divided into three parts based on the elevation representing different material
characteristics. Elevation above 78 m uses an overall slope of 15˚; between elevation -2 and 78
m uses an overall slope of 18˚; and below -2 m uses an overall slope of 20˚.
The upper part of material in Mine Area 3 is loose material, so the geotechnical study
recommended using a gentler slope with lower bench height at an elevation above -78 m. After
creating the ultimate pit limit from pit shell generation, the pit designs are made using more
detailed slope parameters shown by Table 4.3. Another restriction made is related to the area
constraints in Mine 1 and 3. The red area in the Figure 4.11 show the restricted parts, including a
residential area and an existing waste disposal.
Table 4.3 Pit Design Parameters
Mine Area 3
Mine Mine
Description Elv. Elv. Elv.
Area 1 Area 2
above 78 m -2 to 78 m below -2 m
Overall Slope 15˚ 15˚ 12˚ 18˚ 20˚
Face Slope Angle 27˚ 27˚ 18˚ 27˚ 45˚
Berm Width 12 m 12 m 15 m 10 m 13 m
Bench Height 8 m 8 m 6 m 8 m 8 m
Figure 4.11 Area Restrictions in Mine Area 1 (left) and Mine Area 3 (right)
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4.3 Reserve Comparison Results
4.3.1 Reserve estimation and model validation
Before establishing ultimate pit limits to determine the estimated coal reserves with the
parameters assumed above, validation is needed of the procedure using the previous study’s
assumption parameters. We can see in the below comparison between the ultimate pit limit in pit
shell generation in Project Evaluator with the one from the previous study (since the study was
conducted in 2018, the reserve had already subtracted the real production volume through mid-
2021). For validation purposes, only Figure 4.12 expresses the amount of reserves before being
factored.
Figure 4.12 Model Validation Using the Previous Study’s Assumption Parameters
The figure above confirms the model’s result is close to the previous study when the same
parameters are used. Then the new assumptions were run, resulting in an estimation of optimum
reserves considering all new adjusted constraints given in Table 4.4.
Table 4.4 Reserve Estimation Result with Adjusted Assumption Parameter
Description Mine Area 1 Mine Area 2 Mine Area 3
Coal tonnage 75.4 MT 63.5 MT 185.8 MT
Waste volume (BCM) 315.5 million 365.5 million 1,560.5 million
Strip Ratio 4.18 5.76 8.40
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4.3.2 The feasible effort to manage slope and area restrictions
To decide the level of worthwhile effort to manage mine restrictions, the benefits,
especially for coal conservation, need to be evaluated.
The assumption variations are made for slope parameters to be considered if there are
specific geotechnical modifications, dewatering plans or any other reinforcements that can be
implemented which cost less than the additional coal that can be conserved. Compared to the
previous study, there are also additional area restrictions. The comparison of reserves shown in
in Figure 4.13, Figure 4.14, and Figure 4.15 is presented to check whether it is economically
feasible to relocate them.
Mine Area 1’s challenge is to relocate residential housing and existing disposal area to
maximize its coal reserve. Managing those restriction is worth about 74 million tonnes of coal.
Moreover, if there are any geotechnical efforts to make the steeper slope work, up to 50 million
tonnes of coal can be conserved with a 25% lower stripping ratio. The comparison reserve for
Mine Area 1 is described in Figure 4.13.
OPTIMIZED RESERVE - MINE AREA 1
900 5.00
800 4.50
700 4.00
3.50 600
3.00
500
2.50
400
2.00
300
1.50
200 1.00
100 0.50
- -
Slope 15, area not Slope 15 Slope 20 Slope 25
restricted
Coal Waste Strip Ratio
Figure 4.13 Mine Area 1 - Reserve Comparison Based on Slope Variation and Area Restrictions
Meanwhile, Mine Area 2 do not have any additional area restrictions, so the variation is
only conducted to the slope design parameters as follows below. We can see that geotechnical
reinforcement to create a steeper overall slope can result in an additional 50 million tonnes of
coal. It also produces a lower stripping ratio by 15%.
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Figure 4.14 Mine Area 2 - Reserve Comparison Based on Slope Variation
A different result is obtained for Mine Area 3. A steeper slope can increase the coal reserve
up to 40 million tonnes and decrease the stripping ratio by up to 10%. The significant rise can be
realized if the existing disposal area can be mined again, which will give an additional 65 million
tonnes of coal with a 20% reduction in strip ratio. It should be noted that any plan to
relocate/remine the existing waste disposal is not an easy job; an engineer is needed to manage
mud material, and a new waste dump design would need to be developed to cover the wet, loose
material which may ultimately need a larger area and a more detailed study.
OPTIMIZED RESERVE - MINE AREA 3
2,500 9.00
8.00
2,000 7.00
6.00
1,500
5.00
4.00
1,000
3.00
2.00
500
1.00
- -
Slope 15-20, area not restricted Slope 15-20 Slope 20-25
Coal Waste Strip Ratio
Figure 4.15 Mine Area 3 - Reserve Comparison Based on Slope Variation and Area Restrictions
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4.4 Mining Pushbacks and Production Scheduling
In this case study, pushbacks are generated by varying coal prices to obtain a relatively
stable increment of coal and waste volume between each pushback. Pushbacks are then cut by
gridset plan (bench) using Auto Cut Generation Tools, to be set as mining phases to generate
production scheduling in the Mine Plan Schedule Optimizer (MPSO).
As explained in the previous chapter, the main constraint in scheduling production in this
case is coal production capacity. As such, in the first simulation, only the coal tonnage constraint
will be set. After that, waste volume constraints will be defined step-by-step to achieve a
balanced production profile over the life of the mine. Other general constraints set by the
optimization algorithm, as described in Chapter 3, are the level and phase precedence constraint,
reserve constraint and waste production constraint to balance the production volume.
Table 4.5 The Number of Levels/Benches in Each Phase of Each Mine Area
Mine Area (i) Phase (j) Number of Benches (K) Elevation
1 21 102 to -58
2 23 102 to -74
3 28 102 to -114
Mine Area 1
4 28 102 to -114
5 26 102 to -98
6 26 102 to -98
1 20 120 to -50
2 23 120 to -74
3 25 120 to -90
4 27 120 to -106
Mine Area 2
5 29 120 to -122
6 30 120 to -130
7 32 120 to -146
8 35 120 to -170
1 23 134 to -42
2 31 134 to -114
3 36 134 to -146
4 38 134 to -162
5 47 134 to -234
Mine Area 3
6 49 134 to -250
7 26 134 to -66
8 38 134 to -162
9 49 134 to -250
10 47 134 to -234
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By following Equations 3.6a, 3.6b, 3.7a and 3.7b to satisfy the precedence constraint
(mining sequence), Table 4.5 presents the number of benches as the upper limit for the
algorithm. It is important to note that, when fluctuating forecast commodity prices are set, and
there is no minimum production constraint, the optimizer program will have the potential to skip
the year which is not favored in production scheduling. It can happen in a year with a low price.
assumption or at a low maximum production constraint; therefore, it is important to set minimum
coal production constraint. Moreover, as long as the discount factor is set, it is better to use the
assumption of high commodity prices if the reserves in the ultimate pit are planned to be entirely
mined.
4.4.1 Mine Area 1
Since Mine Area 1 has a dedicated infrastructure in eastern area, the production schedule is
limited to a single mine. The process flow is simply illustrated in Figure 4.16. There are six
pushbacks defining six gradual coal prices. Pushbacks have been selected at the prices of $40,
$42.5, $43.5, $45, $47.5 and $60tonne USD. Setting a price lower than $40/tonne produces an
insignificant amount of coal.
Figure 4.16 Mine Area 1 Process Flow
As Clark and Dagdelen (2022) defined, in the best case, pushbacks generated by
commodity price give a continuous increase in and maximized cumulative net present value
(NPV), presented in Figure 4.17 and Figure 4.18.
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In this multi-mine schedule optimization, the NPV resulting from Mine 2 and 3 is $1.51
billion, $328 million higher than single-pit production scheduling. While evaluating the schedule
optimization results, Mine Areas 1 and 2 offered a relatively stable result between years.
Conversely, Mine Area 3 is a bit challenging. To maximize the coal mined, it requires a
pre-stripping period of about three years, which will have a negative impact on cashflow.
Negative cashflow in the early years of mining will significantly affect the NPV of the project.
However, failing to do pre-stripping in the early period will prevent the mine from having a life
span of more than five years, and an impact from not fulfilling the infrastructure minimum
requirements to send coal to port. Creating multi-mine scheduling can at least reduce the years
with a negative cashflow.
4.5 Financial Analysis
The yearly coal price assumptions used to generate cashflow are based on Wood
Mackenzie forecasts on nominal coal prices of Indonesia free-on-board FOB-5,000 kcal/kg.
However, this data is not used when optimizing the schedule in MineSight 3D because, to
maximize NPV, the program will tend to avoid mining in years of lower coal prices. Considering
we have already have the ultimate pit limit that we want, we can just set up the coal price
assumptions to be high so the only factor influencing the schedule is the discount factor.
Thermal coal price forecast, FOB Indonesia @ 5,000 kcal/kg GAR
120
100
80
60
40
20
0
2009 2013 2017 2021 2025 2029 2033 2037 2041 2045 2049
Historical & Forecasted Nominal Price Forecasted Nominal Discounted Price
Figure 4.36 Thermal Coal Price Forecast (FOB ICI-3, US$/t, nominal)
Source: the Argus Media Group and Indonesian govt. (history), Wood Mackenzie Coal Market
Service (forecasts)
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To analyze which scenario is better, the incremental rate of return analysis is generated by
varying the discount rate. Figure 4.41 below shows the incremental rate of return for the entire
project is 17.5%. It gives enough reason to decide on choosing the optimization scenario over the
current alternatives. For any discount rate below 17.5%, the optimization scenario is a better
choice. Otherwise, for any discount rate higher than 17.5%, it is better to keep on using the
current alternatives.
Figure 4.41 Incremental ROR Analysis
Considering the result of schedule optimization, the mine can extend its life. Moreover,
coal conservation for more sustainable mining and sales contracts to supply a national power
plant should be considered by the company to make the best decision.
4.6 The Effect of Increasing Production Capacity
The increase in production capacity is then analyzed to evaluate its impact on NPV. The
analysis is conducted by assuming that both existing infrastructures in the eastern area is
allocated only for Mine Area 1, which the capacity is about 10.7 million tonnes annually, and
adding one more train loading station to accommodate Mine Area 2 and 3 so the western area
will have 14.8 million tonnes of annual coal throughput capacity.
Once again, waste volume capacity will be adjusted by setting a mine contract following
this scheduling result, so long as it is stable during at least a four- to five-year period. The
investment needed to build additional infrastructure in the western area was not considered.
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CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1 Conclusion
Given the challenges faced by mining projects before running or during operation, it is
important to perform a continuous evaluation and renew mine planning efforts periodically. To
maintain the amount of coal reserves in the initial ultimate pit limit, updated production
scheduling is required.
The optimal schedule can be achieved by following three general steps: 1) aggregating
blocks into pushbacks; 2) expressing the scheduling problem into a linear system that can be
solved through linear programming using an iterative approach to finding an optimal schedule;
and 3) iterating semi-randomly through the solution space to ensure the schedule obtained is
close to the optimal solution.
Schedule optimization using Hexagon’s Mine Plan Schedule Optimizer (MPSO) needs to
be accompanied by an engineer’s manual evaluation and adjustments. From the study, we can
predict that MPSO looks to maximize profit in each period, but we cannot be sure if the program
doing the iteration is giving the most optimal NPV over the life of the mine. However,
adjustments can be made to the initial result generated by the program, which gives proper
understanding to the deposit’s economic behaviors to be able to set appropriate constraints and
generate a new production schedule that is more reliable and implementable.
By deciding to mine at a higher stripping ratio than the existing operation, this study results
in a higher coal reserve and a longer life of mine, which leads to a higher NPV for the project.
The optimization can produce 64% higher coal reserves than current operations for the mine
areas. It also results in an increase of $571 million net present value for the entire project.
By evaluating each mine area, we can conclude that Mine Area 3 has the highest coal
reserve, but with the highest risk, because of its pre-stripping requirement, which as noted before
will result in a negative cashflow for the first couple of years. However, by combining all areas,
the incremental rate of return (IRR) analysis shows 17.5%, which proves the optimization result
is a better project than current alternatives.
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Increasing production capacity is also a way to get a higher NPV. Depending on the
deposit characteristics, it can also solve the stripping problem, allowing the negation of pre-
stripping because the size of production can deal with the significant amount of waste above
coal.
The valuation generated in this study aimed to provide a more supportable basis for a
production schedule over the life of the mine, as a guidance and a consideration for the company
to make better decisions as well as more advantageous sales contracts and sustainable mining
considerations.
5.2 Future Work
T his study contributes to laying out a better understanding of the economic behavior of the
mine’s coal deposit and production schedule from the existing mine areas (and how to do that).
The author realizes the MPSO result may not be the most optimum schedule to maximize NPV,
but the mathematical model that builds on this study can be implemented to check whether there
are other schedule alternatives with better results.
An ancillary topic that may be explored in the future is improving scheduling algorithms to
plan alongside disposal capacity constraints, allowing potential problems to be overcome, such
as the need to do backfilling because of a lack of disposal locations within the concession area.
This, too, could be a thorough analysis that would be very useful for the company to evaluate the
entire project, not only by knowing the coal reserves left but also having a yearly production
breakdown over the mine’s life.
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APPENDIX D
PRODUCTION SCHEDULE CASHFLOW
Table D.1 Mine 1 Production Schedule Cashflow
Price Forecast
Year 2022 2023 2024 2025 2026 2027 2028 2029 2030
Coal Price USD/tonne 89.27 63.86 62.69 63.80 60.82 61.91 62.68 64.18 65.23
Cost Assumptions
MINE 1
Mining cost to dump USD/tonne 1.31
Mining cost to TLS USD/tonne 2.04
Costs to FOB USD/tonne 28.02
Production Volume
Period 1 2 3 4 5 6 7 8 9 Total
Waste Stripped Mil. BCM 38.85 38.84 38.95 41.22 41.12 41.60 41.73 25.14 9.63 317.07
Coal Mined Mtonnes 8.29 8.27 8.34 8.35 8.34 8.34 8.23 8.14 8.09 74.39
Stripping Ratio (BCM/tonnes) 4.68 4.70 4.67 4.94 4.93 4.99 5.07 3.09 1.19 4.26
Cashflow (in million USD)
Cash Flow Group Description 1 2 3 4 5 6 7 8 9 Grand Total
Revenue CVAR 7 40.48 5 28.26 5 23.11 5 32.40 5 07.00 5 16.13 5 16.05 5 22.19 5 27.63 4 ,913.24
Mining Cost Dump 1 11.99 1 11.97 1 12.27 1 18.81 1 18.52 1 19.93 1 20.29 7 2.47 2 7.76 9 14.01
TLS 1 6.94 1 6.90 1 7.04 1 7.05 1 7.03 1 7.03 1 6.82 1 6.62 1 6.52 1 51.94
Mining Cost Total 1 28.93 1 28.87 1 29.31 1 35.85 1 35.55 1 36.96 1 37.11 8 9.09 4 4.28 1 ,065.95
Costs to FOB 2 32.42 2 31.80 2 33.81 2 33.84 2 33.56 2 33.59 2 30.70 2 27.98 2 26.65 2 ,084.36
Net Cash Flow 3 79.13 1 67.59 1 59.98 1 62.71 1 37.89 1 45.58 1 48.24 2 05.13 2 56.69 1 ,762.93
Discount Factor 0 .95 0 .87 0 .79 0 .72 0 .65 0 .59 0 .54 0 .49 0 .45
Present Value 3 61.90 1 45.43 1 26.21 1 16.69 8 9.90 8 6.28 7 9.88 1 00.48 1 14.31 1 ,221.06
Cumulative Present Value 3 61.90 5 07.32 6 33.53 7 50.22 8 40.11 9 26.40 1 ,006.27 1 ,106.75 1 ,221.06
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Table D.2 Mine 2 Production Schedule Cashflow
Price Forecast
Year 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033
Coal Price USD/tonne 89.27 63.86 62.69 63.80 60.82 61.91 62.68 64.18 65.23 66.30 67.46 68.83
Cost Assumptions
MINE 2
Mining cost to dump USD/tonne 1.31
Mining cost to TLS USD/tonne 2.24 TLS 3
2.14 TLS 4
Costs to FOB USD/tonne 28.02
Production Volume
Period 1 2 3 4 5 6 7 8 9 10 11 12 Total
Waste Stripped Mil. BCM 33.63 46.46 43.46 10.91 44.18 29.98 26.41 47.58 55.35 10.41 11.61 8.11 368.07
Coal Mined Mtonnes 8.83 7.93 5.46 5.02 5.38 5.10 4.59 4.55 4.53 4.81 4.85 2.29 63.35
Stripping Ratio (BCM/tonnes) 3.81 5.86 7.96 2.17 8.21 5.88 5.76 10.45 12.22 2.16 2.39 3.53 5.81
Cashflow (in million USD)
Cash Flow Group Description 1 2 3 4 5 6 7 8 9 10 11 12 Grand Total
Revenue CVAR 7 56.77 4 85.83 3 28.45 3 07.68 3 14.16 3 03.23 2 76.04 2 80.64 283.54 3 06.40 3 14.36 1 51.57 4 ,108.67
Mining Cost Dump 9 6.96 1 33.92 1 25.27 3 1.45 1 27.34 8 6.42 7 6.12 1 37.14 159.55 3 0.00 3 3.47 2 3.38 1 ,061.02
TLS 1 8.92 1 6.98 1 1.69 1 0.77 1 1.53 1 0.93 9 .83 9.76 9 .70 1 0.32 1 0.40 4 .92 1 35.75
Mining Cost Total 1 15.88 1 50.90 1 36.97 4 2.21 1 38.87 9 7.35 8 5.95 1 46.90 169.26 4 0.31 4 3.87 2 8.29 1 ,196.77
Costs to FOB 2 47.43 2 22.06 1 52.92 1 40.77 1 50.76 1 42.95 1 28.55 1 27.62 1 26.88 134.90 1 36.01 6 4.28 1 ,775.13
Net Cash Flow 3 93.46 1 12.86 3 8.56 1 24.70 2 4.53 6 2.92 6 1.54 6 .11 - 1 2.59 131.20 1 34.48 5 9.00 1 ,136.77
Discount Factor 0 .95 0 .87 0 .79 0 .72 0 .65 0 .59 0 .54 0 .49 0 .45 0 .40 0 .37 0 .33
Present Value 3 75.57 9 7.94 3 0.42 8 9.43 1 5.99 3 7.29 3 3.16 2 .99 - 5 .61 5 3.11 4 9.49 1 9.74 7 99.53
Cumulative Present Value 3 75.57 4 73.51 503.93 5 93.36 6 09.35 6 46.65 6 79.81 6 82.80 6 77.19 730.30 7 79.80 7 99.53
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Table D.3 Mine 3 Production Schedule Cashflow
Price Forecast
Year 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034
Coal Price USD/tonne 89.27 63.86 62.69 63.80 60.82 61.91 62.68 64.18 65.23 66.30 67.46 68.83 70.30
Cost Assumptions
MINE 3
Mining cost to dump USD/tonne 1.15
Mining cost to TLS USD/tonne 2.24
Costs to FOB USD/tonne 28.02
Production Volume
Period 1 2 3 4 5 6 7 8 9 10 11 12 13
Waste Stripped Mil. BCM 81.24 68.54 71.43 146.12 112.85 127.05 130.81 109.51 101.88 146.63 146.36 25.50 33.26
Coal Mined Mtonnes 1.89 2.81 5.28 5.72 5.36 5.64 6.15 6.18 6.21 5.93 5.88 8.42 10.70
Stripping Ratio (BCM/tonnes) 42.96 24.38 13.52 25.55 21.04 22.52 21.26 17.71 16.41 24.74 24.88 3.03 3.11
Cashflow (in million USD)
Cash Flow Group Description 1 2 3 4 5 6 7 8 9 10 11 12 13
Revenue CVAR 184.03 1 95.67 3 61.10 3 97.73 3 55.54 3 80.67 420.39 432.50 4 41.52 4 28.23 4 32.55 6 31.73 819.84
Mining Cost Dump 2 04.92 1 72.87 1 80.16 3 68.55 2 84.65 320.47 329.96 276.22 2 56.98 3 69.85 3 69.16 6 4.31 8 3.90
TLS 4 .24 6 .31 1 1.85 1 2.83 1 2.03 1 2.65 1 3.80 1 3.87 13.93 1 3.29 1 3.19 1 8.89 24.00
Mining Cost Total 2 09.16 1 79.18 192.01 381.38 2 96.68 3 33.12 3 43.76 2 90.08 270.91 3 83.14 3 82.36 83.20 107.90
Costs to FOB 5 2.99 78.77 1 48.07 1 60.27 1 50.27 1 58.06 1 72.42 173.23 1 74.01 1 66.05 1 64.82 2 35.96 299.80
Net Cash Flow - 7 8.13 - 6 2.28 2 1.01 - 1 43.92 - 91.40 - 1 10.51 - 9 5.79 - 30.81 - 3 .39 - 1 20.96 - 1 14.62 3 12.57 4 12.14
Discount Factor 0 .95 0 .87 0 .79 0 .72 0 .65 0 .59 0 .54 0 .49 0 .45 0.40 0.37 0.33 0.30
Present Value - 7 4.58 - 5 4.05 1 6.58 - 1 03.22 - 5 9.59 - 65.50 - 51.61 - 15.09 - 1 .51 - 48.97 - 4 2.18 1 04.58 1 25.35
Cumulative Present Value - 7 4.58 - 1 28.62 - 1 12.04 - 215.26 - 2 74.85 - 340.35 - 3 91.97 - 4 07.06 - 408.57 - 4 57.53 - 4 99.72 - 3 95.14 - 2 69.79
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Table D.3 (continued)
Price Forecast
Year 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045
Coal Price USD/tonne 71.18 73.24 74.53 76.77 78.51 79.50 85.24 87.20 88.52 89.78 91.64
Cost Assumptions
MINE 3
Mining cost to dump USD/tonne 1.15
Mining cost to TLS USD/tonne 2.24
Costs to FOB USD/tonne 28.02
Production Volume
Period 14 15 16 17 18 19 20 21 22 23 24 Total
Waste Stripped Mil. BCM 33.91 33.45 34.47 33.23 34.09 18.43 8.91 16.15 15.05 20.26 12.46 1 ,561.58
Coal Mined Mtonnes 10.70 10.67 10.72 10.68 10.62 10.43 10.32 11.09 8.71 7.81 6.51 184.45
Stripping Ratio (BCM/tonnes) 3.17 3.13 3.22 3.11 3.21 1.77 0.86 1.46 1.73 2.59 1.91 8.47
Cashflow (in million USD)
Cash Flow Group Description 14 15 16 17 18 19 20 21 22 23 24 Grand Total
Revenue CVAR 8 30.10 8 52.13 8 70.68 8 93.65 9 09.10 9 04.03 9 58.52 1 ,054.06 8 40.13 7 64.13 6 50.75 1 5,008.79
Mining Cost Dump 8 5.53 8 4.37 8 6.96 8 3.82 8 5.97 4 6.49 2 2.49 4 0.73 3 7.97 5 1.10 3 1.42 3 ,938.83
TLS 2 4.00 2 3.94 2 4.04 2 3.96 2 3.83 2 3.40 2 3.14 2 4.88 1 9.53 1 7.52 1 4.61 4 13.75
Mining Cost Total 1 09.53 1 08.31 1 11.00 1 07.78 1 09.80 6 9.89 4 5.63 6 5.61 5 7.51 6 8.61 4 6.03 4 ,352.58
Costs to FOB 2 99.80 2 99.08 3 00.32 2 99.23 2 97.66 2 92.32 2 89.08 3 10.74 2 43.98 2 18.79 1 82.54 5 ,168.25
Net Cash Flow 4 20.77 4 44.73 4 59.36 4 86.64 5 01.64 5 41.82 6 23.81 6 77.71 5 38.65 4 76.73 4 22.18 5 ,487.96
Discount Factor 0 .28 0 .25 0 .23 0 .21 0 .19 0 .17 0 .16 0 .14 0 .13 0 .12 0 .11
Present Value 1 16.34 1 11.79 1 04.97 1 01.09 9 4.74 9 3.02 9 7.36 9 6.16 6 9.48 5 5.90 4 5.00 7 16.07
Cumulative Present Value - 1 53.45 - 4 1.66 6 3.31 1 64.40 2 59.14 3 52.16 4 49.52 5 45.68 6 15.16 6 71.06 7 16.07
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ABSTRACT
Mine production scheduling influences the economic outcome of the mining
project, maximizes equipment usage and effectively solves the demand requirements in
order to maximize the mine's profit. To maximize the mine's profit while adhering to the
operational constraints of the mine, the main objective of the schedule is to find answers
for the following questions:
• Which production blocks should be mined,
• If it is to be mined when it should be mined.
Since the early 1960's, operations research techniques have been applied to mine
production scheduling problems. Despite the effort that has been devoted to developing
optimal multi-period production schedules, to date, no such schedules have been made.
This dissertation uses the operation research technique of Mixed Integer
Programming (MIP) to produce a 3 to 5 year production schedule with monthly time
fidelity in a single run. A new production block database and a well-formulated MIP
model is developed for underground sublevel caving operation. The model and new block
database reduces the number of integer variables dramatically. The number of integer
variables is further reduced by assignment of an earliest and latest possible start time for a
production block. In order to determine the earliest and latest possible start times for
production blocks without violating optimality, two new algorithms are developed.
The model and algorithms are implemented for Kiruna Mine scheduling system.
Resulting production schedules exhibit significant cost savings over current manually
schedules by more closely satisfying required production targets and by reducing the
scheduling time.
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ACKNOWLEDGMENTS
I am very grateful to my thesis advisor, Dr. Mark Kuchta for his friendship,
guidance, financial and intellectual support at all times during this thesis study. Without his
support and guidance, it would not have been possible to complete this dissertation.
I would like to express also my thanks to Dr. Alexandra Newman. Her suggestions
and advice were very helpful on writing of my thesis and completion of this research.
I would like to thank to my other committee members Dr. Kadri Dagdelen, Dr.
Levent Ozdemir, and committee chair Dr. Christian Dubrunner for their time and
considerations on this research.
I would like to acknowledge The Ministry of The National Education of The
Republic of Turkey for providing financial support for six years for my studies.
I owe thanks to the LKAB Kiruna Mine for sponsoring this research and inviting
me to the mine.
I am very thankful my wife, Nadiye, for her understanding, friendship and
encouragemen. Without my wife’s encouragement, and support, I could not have
completed this dissertation. Also, I would like to thank to my brother, Ersan, for his
friendship, and encouragement during my study.
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1
CHAPTER 1
PRODUCTION SCHEDULING OF UNDERGROUND MINES
1.1. INTRODUCTION
Today, mining companies face the challenge of scheduling production in their
mines in a way that is economically optimal. The scheduling process should provide the
company with profit maximization, a high-level of equipment utilization, and high
quality products in every time period according to demand requirements. Production
schedules not only support the company’s planning and scheduling process in changing
market conditions, but should allow the company to examine various strategic scenarios
in advance. There are a number of different methods for mine scheduling. One of the
most common methods is manual scheduling using a spreadsheet. Using typical block
model information including geometry, grade, and tonnage, the time periods in which
each production block is mined manually determined. Manual schedules can take weeks
to develop depending on the mine size, and there is no way to know whether the schedule
is close to the best solution achievable.
The second method of mine scheduling is computer assisted scheduling. In the
last two decades, operations research techniques have been applied to production
scheduling in the mining industry. However, the available software and programs for
production planning are not sufficiently robust for solving the large models required for
multi-time period production scheduling.
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3
permits noninteger values for the decision variables. Fourth, certainty ensures that the
parameters are known and constant. Fifth, the solution is static or invariant for a given
scenario, i.e., a given set of parameters. Any LP model can be solved with an algorithm
known as the Simplex Method. Although this algorithm has exponential theoretical
complexity, the algorithm relies on convexity arguments, and, in practice, its
performance is good.
In certain situations, the decision variables must assume integer values. When this
restriction is added to the problem, the model is called an integer programming (IP)
model. Such a requirement arises naturally when the variables represent entities like
packages, people, and trucks that can’t be divided. Integers, specifically, binary variables
are used to capture yes/no decisions such as whether or not to mine the ore from a
particular block at a given time period. For example, if block i is mined in a given time
period, xi assumes the value of one and zero otherwise.
IP solution time depends on the number of integer variables and also on the
number of constraints, and increases exponentially with the number of integer variables.
In the worst case, all possible combinations of values of integer variables must be
examined in the solution process, which gives rise to a combinatorial problem of great
magnitude. The Branch and Bound algorithm is the most common technique for solving
IPs.
If some variable values are allowed to be continuous while others must assume
integer values, the resulting model is a mixed integer program (MIP). For all practical
purposes, these models are as difficult to solve as IPs.
1.2.1. Branch and Bound Algorithm
The Branch-and-Bound Algorithm is the most frequently used solution technique
for MIP problems. The algorithm fixes some variable values, thereby dividing the MIP
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4
problem into sub-problems containing mutually exclusive and collectively exhaustive
solution sets with respect to the original problem.
The Branch-and-bound algorithm for a binary minimization problem can be
summarized as follows;
Step 0. Set the objective function value Lower Bound (LB) = - oo and the Upper
Bound (UB) = + oo.
Step 1. Solve the noninteger version of the original problem (i.e., the LP
relaxation of the original problem) using normal linear programming procedures.
• If the resulting solution is integer, stop. The solution is optimal for the
integer problem also.
• If the solution is infeasible or unbounded, stop. The integer problem is also
infeasible or unbounded, respectively.
• Otherwise, let LB = the optimal value of the LP relaxation, (LP*). Then, go to
Step 2.
Step 2. Select an integer-constrained variable, which has a fractional value in the
LP relaxation solution. For example, choose the variable with the smallest objective
function coefficient.
Step 3. Create two subproblems to replace the current problem by adding a
constraint on the upper value and on the lower value of the variable selected in Step 2. In
other words, constrain the variable value to be equal to 1 in one subproblem and equal to
0 in the other subproblem. Note: these two subproblems constitute a mutually exclusive
and collectively exhaustive set of outcomes. Figure 1 represents a branch and bound tree
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Step 5. Fathom a branch if:
• The correspoding subproblem yields an infeasible solution.
• The subproblem yields an integer solution.
• The objective function value in this subproblem is greater than that
corresponding to the best known integer solution found.
Step 6. Repeat the process until no unexamined or unfathomed subproblems
remain or until the upper and lower bounds are "sufficiently" close.
Worst case performance for this algorithm involves evaluating each subproblem.
Therefore, solving an integer programming problem is exponentially complex.
For example, consider a problem with 100 binary variables, each of which has
two possible values, 0 or 1. At level 0, there are 2° nodes; at level 1, there are 21 nodes; at
level 2, there are 22 nodes, etc., and at the last level there are e 2100 nodes. Hence, the total
number of nodes that must theoretically be examined:
1 _ y.W+1
2° +21 +22 +23 + ....2100 =2101 -1. (Note 1 + x + x2+x3+... + *” =---------).
l-x
(Fortunately, in practice all nodes are usually not examined because some branches are
fathomed early due to the criteria listed in Step 5).
To put the solution time for this problem in perspective, consider that as many as
_ i = 2.5353 *1030 linear programs must be solved for this problem. If it is assumed
2100+1
that each LP requires one second of solution time, it will theoretically take
2.5353 * 1030 seconds or 9.7812*1023 months or 8.151*1022 years to solve the MIP
problem.
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The cutting plane algorithm follows:
Step 0. Set the objective function value Lower Bound (LB) = - oo and the Upper
Bound (UB) = + oo.
Step 1. Solve the noninteger version of the original problem (i.e., the LP
relaxation of the original problem) using normal linear programming procedures. Let X*
be an optimal solution.
• If the resulting solution is integer, stop. The solution is optimal for the integer
problem also.
• If the solution is infeasible or unbounded, stop. The integer problem is also
infeasible or unbounded, respectively.
• Otherwise, go to Step 2.
Step 2. Add a linear inequality constraint to the LP relaxation of the original
problem that all integer solutions satisfy, but X* does not.
Step 3. Go to Step 1.
Cutting plane algorithms are very effective for reducing the integrality gap i,e.,
the gap between the LP relaxation solution (used as an objective function value bound)
and the integer solution and for reducing the number of subproblems that must be
examined (the amount of enumeration required). However, just as branch-and-bound can
generate a huge number of subproblems, this technique can generate an enormous
number of constraints. Additionally, while branch and bound always (eventually) finds an
optimal solution (if one exists), this algorithm may not. Most of the time, branch-and-
bound and cutting plane algorithms are used in conjunction with each other. This
combination is called Branch-and-Cut. This combination first reduces the feasible region
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1.3. LITERATURE REVIEW
Mining projects start with the exploration stage. In this stage, the location and
definition of the orebody is determined by sampling processes. From these samples, such
as drill hole satiiples, the ore body is divided into certain sizes of blocks (Figure 4). The
size of a block can vary depending on the geology of the ore body and the mining
methods: A block model can be used to depict the economic value or mineral content of
the ore body. A geologic block model represents the grades of different minerals existing
in the ore body and their tonnages. A geologic block model can be converted into an
economic block model by using various costs of extraction and prices of the metals
contained in the deposits.
The next step of mining project is to answer the mining strategy questions:
• Should the mine be explored with surface or underground mining?
• If the mining method is surface mining, what is the break-even depth between
surface and underground mining (ultimate pit limits)?
• If the mining method is underground mining, what underground mining
method should be used?
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The computerized approach to calculate the break-even mining depth is called the
ultimate pit limit calculation. It is an area of mine planning that has received a great deal
of attention. There are many efficient algorithms to determine ultimate pit limits. Only
the material inside the pit boundary can be economically mined with open pit methods
under the existing economic conditions. Outside of the ultimate pit limits, underground
methods can be used, otherwise, it is not economical to mine under current condition,
such as the price of the commodity or cost of the mining operation. After the ultimate pit
limits have been defined, the next step in planning is to divide the ultimate pit limits into
smaller volumes to obtain the extraction sequence. Each of these smaller volumes is
called a “pushback”, “incremental cut”, or “phase”. Then, an extraction schedule is
developed for the life of the mine. Some important surface mine scheduling research
follows:
Lerchs and Grossmann’s algorithm is the most widely used graph theoretic
method to find the ultimate final pit limits. In this algorithm, all the blocks are
represented as nodes, with the economic block values as node values. The algorithm
includes the following steps:
• To initialize, connect all the blocks (nodes) to the root and label the arcs. Arcs
initially connected to positive nodes are classified as strong arcs; the others are
classified as weak arcs.
• Next, connect the nodes which are related to the root by a strong arc, to other nodes
that must be removed in order to mine this strong block. When the connection is
made, eliminate the strong arc going to the root for this node. Then relabel the arcs.
• If there is a node that overlays a node, which is connected to the root by a strong arc,
return to Step 1. Otherwise stop.
This algorithm is widely used to calculate ultimate pit limits because it guarantees
optimality. It is also used to obtain the best pit with respect to economic value by
changing the cutoff grades or the price of the commodity.
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The Moving Cone Algorithm has been one of the most widely used heuristic
algorithms for determining ultimate pit limits because of its rapid execution speed and
easy implementation. After finding the ultimate pit limits, the algorithm is used to find
the best pit with respect to economic value for different cutoff grades. This method
cannot guarantee that an optimal solution is obtained because it may miss the
combination of profitable blocks and extend the ultimate pit beyond the pit limits.
Considerable additional work has been done to determine the ultimate pit limits
and pushback design other than moving cone and Lerchs and Grossmann’s methods, e g,
Whittle (1988), Seymour (1995), Wang and Sevim (1993), and Underwood and
Tolwinski (1996,1998).
Mathematical programming methods used for mine planning consist of Linear
programming (LP), Dynamic Programming (DP), Mixed Integer (MIP) or Integer
Programming (IP). LP is most widely used. Danzig introduced Linear Programming
methods in 1947. Considerable amount of research has been done for surface mine
scheduling using LP.
Johnson (1969) uses LP for mine production scheduling in open pit mines. He
develops a mathematical model for long term scheduling to determine a feasible
extraction schedule in order to maximize the total profits over the planing horizon for an
open pit mine. He uses Dantzig-Wolfe decomposition to obtain smaller linear
programming problems, the master problem, and the sub-problem. His master problem
obtains a combination of mining plans which obey the given constraints and maximize
total profits. The sub-problem develops mining plans for each period based on the block
values (adjusted by master the problem).
Wilke and Reimer (1977) use a linear program to develop a short-term schedule
for the amount of ore and waste to extract from a given production block in an open pit
mine. They seek to maximize priorities associated with the extraction of the production
blocks; these priorities change from period to period contingent upon previous extraction
decisions. The model contains restrictions on ore quality, the capacity of the blending
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bed, the amount of ore contained, and therefore extractable, in a block, the ore-to-waste
ratio, and equipment capacity.
Winkler and Griffin (1998) develop a network-based linear programming model
for several applications. The network captures precedence relationships between mining
blocks of ore or coal, and sending products to a stockpile, to a blending facility, or to a
processing plant. Additional constraints are placed on equipment capacity, product
quality, and target production level. Deviations from constraint goals are minimized. The
resulting production schedule considers only a single time period; multi-period schedules
are generated by concatenating single-period schedules.
In addition to the above mentioned linear programming strategies, dynamic
programming and mixed integer programming has been applied to surface mine
scheduling.
Tolwinski and Underwood (1996) model an open pit mine scheduling problem to
maximize total discounted profit subject to constraints on slope angles, the use of mining
equipment, the available working space, and production requirements. They use a
heuristic dynamic programming solution technique in which a state corresponds to a
subcollection of production blocks and their (three-dimensional) position in the mine.
Their algorithm yields solutions that improve upon current operating procedures for their
test cases, but optimality is not guaranteed.
Dagdelen (1985) formulates the open pit-scheduling problem as a MIP. He
decomposes his multi-time period problem into smaller single time period problems
using Lagrangian relaxation. He develops an algorithm to solve these Lagrangian sub
problems. Overall, he uses an iterative approach to solve the production-scheduling
problem.
Barbaro and Romani (1986) use MIP to determine the mine production schedule
and processing plant types and location to satisfy multiple markets. In order to maximize
the profit, they try to optimize the amount of ore produce by each mine, the amount of
ore processed by each processing plant and satisfaction of market demand. They use
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binary integer variables for selection of the mine, processing site and processing type in a
given time period.
Smith (1998) uses mixed integer programming to determine whether or not a
block should be mined in a silver and gold surface mine. The objective is to minimize
positive and negative deviations from production targets subject to mine precedence
constraints. Smith declares that the use of binary variables precludes him from
developing a production schedule for more than a single time period. The number of
variables, and, hence, the solution time increases for a multi-time period model due to the
number of time periods under consideration, and the number of faces of each production
block that could potentially be mined.
Linear programming models have similarly been applied to underground mine
scheduling for a variety of mining methods.
Williams et al (1972) formulate a linear programming model to determine the
amount of ore to be mined from a particular face in a given month using sublevel stoping
in a copper mine. They seek to minimize monthly production fluctuations subject to
production targets, reserve limits, face profile constraints, and limits on the amount of ore
that can be extracted from a face in a given time period. They are able to solve their
model for a 12 time period scenario.
Jawed (1993) uses linear programming to minimize deviations from production
and goal targets for underground coal mining operations using room and pillar mining
techniques. His single period model determines the amount of coal to extract from a
particular position using a given technology while adhering to constraints on pillar
requirements, manpower, mine equipment capacity, fan capacity, and processing plant
capacity. Because these linear programming models can only capture the amount of ore
to be mined, they are not capable of making discrete decisions regarding whether to mine
a production block or not.
Winkler (1998) and Tang et al. (1993) combine linear programming with
simulation for a quality-oriented short term mine plan. In the former case, the author first
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uses a simulation model to make discrete decisions for each relevant time period
regarding which production blocks should be mined in a sublevel caving operation. The
simulation, while unable to produce "one" optimal set of production blocks for each time
period, yields an evaluation with respect to the objective of each set of production blocks
that could be mined. Based on a choice of these discrete decisions, a linear program then
determines the quantity to be mined from each production block to minimize a weighted
sum of deviations from goal production quantities subject to constraints on ore quality,
minimum demand requirements, block tonnages, and continuity of block mining
operations. In the latter case, Tang et al. create a model to determine the quantity of ore
to mine to minimize deviations from elasticized constraints on resource limits, mine
capacity, demand requirements, profit levels, and grade fluctuations. After these
aggregate decisions are determined, a simulation model evaluates these decisions at a
finer level of detail. The authors use other techniques as well, e.g., a binary program, to
make sequencing decisions.
In addition to the above mentioned linear programming strategies, mixed integer
programming has been applied to underground mine scheduling.
Trout (1995) actually formulates and attempts to solve a mixed integer multi
period production scheduling model for underground stoping operations for base metal
(e.g., copper sulphide). His model contains discrete decisions regarding not only when to
extract ore from a given block, but also whether a stope is void or backfilled in a given
time period. Continuous-valued variables account for the amount of ore extracted from
and the amount of backfill components used in each stope and time period. The model
seeks to maximize net present value of the operations subject to a myriad of constraints,
e.g., (i) extraction and backfill duration, quantity and capacity, (ii) block mining and
stope status sequencing, (iii) demand satisfaction, (iv) mine equipment capacity, and (v)
backfill supply. Although the model produces a solution that is considerably better than
what is currently realized in practice, solution time exceeds 200 hours, without a
guarantee of optimality.
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Winkler (1996) points out the importance of using mixed integer programming in
(mine) production scheduling to account for fixed costs, logical conditions, e.g., either
activity "x" or activity "y", sequencing constraints, and machine use restrictions. Indeed,
mixed integer programming cannot be ignored as an optimization tool if production block
mining sequence is to be accounted for. However, Winkler declares that the use of mixed
integer programs to address multi-period mine scheduling is infeasible due to the
theoretical complexity of solving such models. Instead, he uses mixed integer
programming to schedule underground coal mining operations for a single time period.
His model minimizes fixed and variable extraction costs subject to quality constraints,
and demonstrates that significant differences in solutions exist when fixed costs are
ignored.
Carlyle and Eaves (2001) use an integer programming model to plan a production
schedule for a sublevel stoping operation at Stillwater Mining Company. The decision
variables of the model represent the activity of the development, drilling, preparation, and
production. For a given time (quarter) period and given stope, development and drilling
decision variables take a value of 1 if the activity occurs in that time period or before and
0 otherwise. The model provides near-optimal solutions, for 10-quarter planning period,
to maximize revenue from mining platinum and palladium; however, the authors do not
describe any special techniques to expedite solution time. Also, there is not any
description for elimination of end effects of the solution.
Several attempts at multi-period scheduling using mixed integer programming
have been made at Kiruna Mine, yet none thusfar have produced acceptable multi-period
production schedules.
Almgren (1994) provides the first attempt to apply mixed integer programming to
production scheduling at Kiruna Mine. He divides the data into 100 m blocks
(Figure 5), where one to three of these blocks constitutes a machine placement. Several
additional sets of binary variables account for whether a block, or a portion of a block,
has been mined by a given time period. Various continuous-valued variables track the
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amount of various ore types mined in given locations at a given time period. In general,
because of the database from which model parameters are obtained, and of the resulting
unnecessarily complicated formulation, the model is too large to be solved in its
monolithic form. Hence, Almgren generates a five-year schedule by running the model
one month at a time for 60 monthly time periods. Schedules with acceptable deviations
from the production demands can be produced in a few hours, although there is no
guarantee of optimality for the complete five-year period, especially because “end
effects” between months are ignored. He also employs Lagrangian relaxation on the
monolithic multi-time period model, and empirical results indicate that the test model
works well, but only for small cases.
Topal (1998) and Dagdelen et al. (1999), propose a modeling approach in which
they derive the input data from the same database as Almgren. However, their model
contains fewer integer variables, specifically, only a binary variable representing the
mined state of a block in a given time period. Their model also contains continuous
valued variables to track the amount of the each ore type in a production block that has
been mined in a given time period. Because, similarly, this model cannot be solved in its
monolithic form, the authors reduce the number of integer variables by solving for yearly,
rather than 5 year, schedules. Each yearly model considers only those production blocks
that could feasibly be mined in the corresponding timeframe, and excludes those blocks
that would already have been mined in a prior year. The authors
subsequently obtain a monthly production schedule based on the yearly production
schedule. However, there is again no guarantee of an optimal schedule for the complete
five-year period, nor, by first solving for a yearly schedule, is there a guarantee that
production requirements can be met on a monthly basis.
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The resulting model is large enough that today’s hardware and software preclude
obtaining a schedule in a reasonable amount of time. Therefore, the number of integer
variables is reduced by carefully formulating the model and preprocessing the
production data. The number of integer variables is further reduced by the assignment of
an earliest and latest possible start time for a production block. Algorithms to determine
the earliest and latest possible start times for production blocks without eliminating the
optimal solution are developed. It is shown that through the application of the above-
mentioned techniques, the number of integer variables required can be significantly
reduced, thus enabling model solutions on today’s computers in a reasonable amount of
time.
An overview of the Kiruna Mine and the associated mining method is discussed
in Chapter 2. Along with the current manual mine planing system and the production
data. Chapter 3 covers the mathematical model description and mixed integer (MIP)
formulation for the Kiruna Mine. Several different optimization strategies are illustrated.
Methods of reducing the number of integer variables are discussed in Chapter 4,
including the logic of and an example for the early and late start algorithms. The effect
of the early and late start algorithms on the running time is also discussed. Chapter 5
presents the implementation of the model to the Kirune Mine. The scheduling outcomes
for different objectives are presented. Conclusions and recommendations for further
research appear in Chapter 6.
1.5. ORIGINAL CONTRIBUTION
This research has made the following major contributions:
1. This thesis presents a way to successfully to solve multi-period production
scheduling problems for a large-scale underground mine.
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2. MIP model running time depends exponentially on the number of integer
variables in the model. The number of integer variables is reduced
dramatically by preprocessing the production data and carefully formulating
the model. For the thirty six time periods, corresponding production blocks for
each machine placement for the Kiruna Mine data set would have required 36
(time periods) * 1,173 (blocks) = 42,228 integer variables in the model. The
proposed model has 36 (time periods) * 56 (machine placements) =2,016
integers.
3. It is possible to further reduce the number of integer variables by assigning an
earliest and a latest possible start date for each machine placement. Unique
algorithms have been developed to determine the earliest and latest possible
start time periods of the machine placements without violating optimality.
4. The MIP model formulation includes constraints that limit the number of
active production areas within each shaft group at a given time without
introducing any new binary integer variables.
5. The Kiruna Mine has adopted the optimization model developed in this thesis
as part of its production scheduling system.
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CHAPTER 2
THE KIRUNA MINE AND PRODUCTION SCHEDULING PROBLEM
2.1. THE KIRUNA MINE
The Kiruna Mine is situated in the far north of Sweden above the Arctic Circle
(Figure 6 a,b). The mine is the second largest and the most modem underground mine in
the world today, producing some 24 Mton of iron ore per year. The first documented
discovery of the Kiruna site was in 1696. In about 1890 the company Loussavaara-
Kiimnavaara Aktiebolag (LKAB) was established, and eight years later, mining
operations commenced at Kiruna. By 1900, mining operations were fully underway, and
rail had been laid from the Kiruna mine to Narvik, a major port city located in northern
Norway from which goods are exported to Europe and the Far East. For about the first
50 years of its life, the deposit was mined by surface methods. In 1952, underground
mining operations commenced, and by 1962, surface mining operations were completely
terminated, by which time 140 million tons of waste rock and 209 million tons of iron ore
had been mined. All mining since 1962 has been done from underground. At present,
approximately 1000 Mton of ore, only one half of the original orebody, has been
extracted. Today, the company employs about 3,000 workers and produces
approximately 24 million tons of iron ore per year (Figure 6,c).
Compared with other major iron ore producers in the world, Kiruna has two
unique features. It is an underground mine and the ore deposit is a very high-grade
magnetite ore. Most other iron ore operations around the world mine hematite ores by
surface mining methods.
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The orebody is a high-grade magnetite deposit, approximately 4 km long, and
about 80 m wide on average. The orebody lies roughly in the north-south direction, with
a dip of about 60 degrees and reaches a depth of 2 km. There are two main in-situ ore
types differentiated by their phosphorus content. About 20% of the orebody contains a
very high phosphorous, appetite-rich magnetite known as D ore, and the other 80%
contains a low-phosphorous, high-iron content magnetite known as B ore. The orebody is
characterized by clear-cut boundaries between the two ore types as well as between the
ore and surrounding waste rock at the hangingwall and footwall contacts. It has two main
contaminates, phosphorus (P) and potassium (K O). The phosphorus grade in the D ore
2
varies considerably with an average of 2%. The best B ore contains about 0.025% of
phosphorous with an iron (Fe) grade about 68%. In some areas the B ore is highly
brecciated, resulting in lower iron grades and higher potassium grades.
From the two major in-situ ore types, the mine delivers three raw ore products,
Bl, B2, and D3 (Table 1), used to supply four ore post-processing plants, or mills.
Phosphorous is the main contaminant and ore type classified according to the
phosphorous content.
• Bl quality ore contains the least amount of phosphorous (%P=0.06). It is used
to produce in high-quality fines (of the granularity of fine sand). Crushing,
grinding, and magnetic separation are the processing steps for removing the
contaminants from Bl ore. Bl ore accounts for 21% of total reserves.
• B2 quality ore includes medium percent phosphorus (%P=0.2). It is used to
produce ore pellets (approximately spherical in shape and 1 cm diameter).
Crushing and grinding the ore into finer consistency than Bl ore is the first
processing step. Second, binding agents and other minerals such as olivine
and dolomite are added. Finally, the resulting product is fired in large kilns to
form hard pellets. It accounts for 50% of total reserves. The B2 quality ore
generally cannot be mined directly from the in-situ ore. The blending of high
phosphorous D ore with low phosphorus B ore produce B2 ore during the
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2.2. MINING METHOD DESCRIPTION
The Kiruna mine currently produces about 65,000 tons of raw ore per day. The
mining method is large scale sublevel caving, a mass mining method based upon the
utilization of gravity flow of the blasted ore and the caved overlying waste rock mass.
The caving action is very complex and needs to be controlled. To apply the sublevel
caving method, the ore must be strong enough to stand without excessive support because
of the large amount of development work, and the waste must be weak enough to cave.
Sublevel caving is very suitable for low-grade ore and has had successful application
medium and high-grade metal sulfite ores. It is very flexible and adaptable for
mechanization and automation. One of the biggest disadvantages of the method is high
dilution of the ore by caved waste. A typical sublevel caving operation is shown in Figure
8
.
Transversal sublevel caving is used in the Kiruna Mine (Figure 9) with mining
proceeding from the hangingwall to the footwall. The spacing between sublevels is about
28.5 m and the spacing between crosscuts is about 25 m. The production drifts are
normally 7 m wide and 5 m high. The orebody is currently being mined from the 1045 m
main level (Figure 10). The mine is divided into ten production areas, each of which has
its own group of ore passes and ventilation shafts; this group is also known as a shaft
group. These production areas are about 400 to 500 m in length. The 1045 transportation
level and production areas are shown on Figure 10.
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term strategic planning in the undeveloped areas of the mine. The orebody is divided into
blocks 100 m in length, which extend from the hangingwall to the footwall. The height of
the blocks is the same as the proposed mining sublevel height. The quantities of B and D
ore are first estimated from the in-situ geologic block model. Where applicable, these
initial ore estimates consider blending raw ore to produce various ore types. Specifically,
the quantities of B and D ore from the in-situ block model are used to calculate the
expected run-of-mine quantities and grades of the three ore products Bl, B2, and D3. The
calculation procedure is based on principles of the gravity flow of broken rock. The 100
m block data within a machine placement is then reworked taking into account the
expected extraction sequence, haul distance, and LHD capacity to yield the expected
monthly quantity of the three ore. The production data for a one month time period is
called a production block.
The second method of calculating machine placement production data is applied
when more block information is available, specifically, when development and planning
of a production area is complete. The in-situ tons and grades, and hence, the expected
tons of Bl, B2, and D3 ore, can be calculated ring-by-ring. These calculations, in turn,
are used to generate monthly production data for each machine placement. The data are
considerably more accurate with the second method because the actual production blasts
are planned and expected quantities more precisely estimated (Kuchta, 1999).
Figure 14 shows the production block ore tonnage for a machine placement
located at the 820 m level from y-coordinate 29 to 30.
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Machine Placement Level 820 y 29-30
B1 82 D3 Total
25% 50% 75% 100%
Month I % kTon P % kTon P % kTon P kTon P
2000 1 100% 50 1 00 ---5-0-- 1.00
2000 2 18% 20 0.25 82% 90 1.00 110 0.86
2000 3 9% 10 0.08 18% 20 0.25 73% 80 1.00 110 0.78
2000 4 8% 10 0.08 23% 30 0.25 69% 90 1.00 130 0.76
2000 5 11% 10 0.08 44% 40 0.25 ' 44% 40 1.00 90 0.56
2000 6 8% 10 0.08 33% 40 0.25 58% 70 1.00 120 0.67
2000 7 9% 10 0.08 27% 30 0.25 64% 70 1.00 110 0.71
2000 8 25% 30 008 25% 30 0.25 50% 60 1.00 120 0.58
2000 9 20% 20 0.08 30% 30 0.25 50% 50 1.00 100 0.59
2000 10 25% 30 0.08 33% 40 0 25 42% 50 1.00 120 0.52
2000 11 14% 20 0.08 50% 70 025 36% 50 1.00 140 0.49
2000 12 15% 20 0.08 54% 70 025 31% 40 1.00 130 0.45
I 2001 1 21% 30 0.08 50% 70 0.25 29% 40 1.00 140 0.43
■ I 2001 2 21% 30 0.08 50% 70 0 25 29% 40 ,.00 140 0.43
2001 3 17% 20 0.08 50% 60 025 33% 40 1.00 120 0.47
Total 14% 250 0.08 36% 620 0.25 50% 860 1.00 1730 0.60
Figure 14. Production blocks ore tonnage for a machine placement located at the 820 m
level from Y-coordinate 29 to 30
The production data set has 56 machine placements for the 3 year schedule used
for model development (Figure 15). These 56 machine placements contain a total of 1173
monthly production blocks (see attached CD).
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The data file has two important properties with respect to scheduling. These are:
• Consideration of LHD Capacity. The LHD capacity is one of the most
important limitations for underground mine planning and design. The
mine production strictly depends on machine capacity. Normally, a
machine capacity constraint is required when the available reserves in a
production block exceed the machine capacity for a given time period. The
need for a machine capacity constraint is eliminated with this production
data because the reserves available for scheduling in each time period for a
given machine placement exactly equal the machine capacity. As seen in
Figure 14, one LHD can haul (transport) 50 Kton of D3 ore during the first
month, 20 Kton of B2 and 90 Kton of D3 during the second month, 10
Kton of Bl, 20 Kton of B2 and 80 Kton of D3 during the third month, and
so on.
• Elimination of the reserve constraint. This constraint ensures that the total
material mined for each product from a given production block during all
the scheduling periods has to be less than or equal to the available reserves
for the each product. Because once the production block is selected to be
mined in a time period, the amount of material that will be mined is
already defined.
2.4. THE CURRENT KIRUNA MINE PLANNING SYSTEM
At the Kiruna Mine, long-term (approximately 5 years) mine planning
(approximately 5 years) is done manually with the help of a computer. The program
developed for scheduling, called RULLPLAN97, a database application written in
Microsoft Access 97. The application includes a user interface consisting of various
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forms for data entry and program control. The computer is used to store the schedule as
the schedule progresses and to produce required reports.
The scheduling process can be characterized as a computer-assisted manual
heuristic approach. Before beginning the schedule, production targets for the three ore
products are determined for each month within the time horizon. Next, all the machine
placements where mining is currently taking place are added to the schedule. As active
machine placements are depleted, at some point the required production can no longer be
met. The mine planner then selects what appears to be the best machine placement to
start producing from given the list of available machine placements while adhering to
mine sequencing constraints. The operational constraint can be explained as follows.
• Vertical sequencing constraints that preclude mining a machine placement directly
beneath a given machine placement until at least 50% of the given machine
placement has been mined,
• Horizontal sequencing constraints that require adjacent machine placements on the
same sublevel to be mined after 50% of the given machine placement has been
mined, and
• Shaft group constraints that restrict the number of LHDs active within a shaft group
at any one time to a pre-determined maximum, usually two or three.
The machine placement is added to the schedule by entering the corresponding starting
date. The scheduling program then updates the production dates sequentially for all the
remaining monthly production blocks for that machine placement. After viewing the
schedule with the updated monthly planned quantities of the three raw ore products, the
scheduler repeats the process of assigning start dates for the available machine
placements on a month-by-month basis until a complete three to five-year schedule is
produced. Figure 16 shows the final production plan for March 1999 for the mine with
the current planning system.
A great deal of experience and time are needed to produce production schedules
using this method. Furthermore, these schedules are clearly obtained myopically, i.e.,
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CHAPTERS
MATHEMATICAL MODEL DESCRIPTION AND FORMULATION
This chapter presents the mixed integer-programming (MIP) model that is used to
develop a production schedule for the Kiruna Mine.
3.1. MODEL DESCRIPTION
3.1.1. Scheduling Objective
Because the mine can stockpile only small amounts of ore, the mine must meet its
production targets almost exactly so that the mills can meet their production demands.
The objective function is to minimize the deviation from the Bl, B2 and D3 raw ore
production targets. Moreover, the mine must observe various operational constraints.
3.1.2. Production Requirements
The mine has to meet the production requirement for each ore type in every time
period. The targeted ore tonnage for Bl, B2, and D3 ore quality month-by-month are 297,
977, and 728 Kton respectively.
3.1.3. Continuity of Mining
Once a machine placement has started to be mined, mining that machine
placement must progress continuously for each production block within the machine
placement and in a specific order until all available ore has been removed. This constraint
is a requirement for efficient sublevel caving.
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3.1.4. Vertical Sequencing Between Levels
To prevent undermining between levels, a machine placement cannot be started
until 50% of the machine placement on the level above has been mined.
3.1.5. Horizontal Sequencing
To prevent blast damage in adjacent machine placements, machine placements
located to the left and right on the same level must begin to be mined after no more than
50% of the given machine placement has been mined.
3.1.6. Currently Active Machine Placements
Some of the machine placements are currently being mined at the start of the
schedule. This constraint places those machine placements that are being mined into the
production schedule.
3.1.7. Machine Limitation for Each Shaft Group
Each shaft group has 3 or 4 ore passes, one of which is used for mine
development. Only 2 or 3 Load Haul Dump units can simultaneously be active in each
shaft group. For quality control reasons, one shaft is required for each LHD; therefore,
the total number of LHD's operating with in a shaft group is limited to 2 or 3.
3.1.8. Cutting Plane Constraints
In this group, two constraints are used in the model.
• Mining must start if the start date for a machine placement is within the
production time frame. This constraint is called a set covering constraint.
• Mining may start in one and only one time period for a given machine
placement if late start date occurs after the end of the scheduling horizon. This
constraint is called a set partitioning constraint.
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du^dd^iO Mt,k
yht binary Vb,t
3.2.1. Objective Function
The objective function minimizes the deviation from the Bl, B2 and D3
production targets so that the mills can meet their respective production demands. The
variables dutk and ddtk represent deviations above and below the production target,
respectively, for each ore type at each time period. Either dutk or ddtk must be zero in a
given time period.
Different strategies can be applied to realize a preferred pattern of deviations.
• The deviation terms in the objective can be weighted by time period and/or
ore type to emphasize the importance of meeting demands as closely as
possible in a specific time period and/or for a specific ore type. This weight
could be further differentiated by positive and negative deviations. Then, if
and Wfr are defined as the objective function weight for positive and
negative deviations, respectively, for ore type k in time period t, the objective
follows:
Min E WZduJ + 2 (W^ddJ
k,t k,t
• The structure of the objective can be changed to smooth out the deviation
across time periods as follows:
Min
X E t Z (r* * y* - r* * A m )]
k b fEQ6|t>l
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MinZ
st.
=> z yk,t
=> Z >ddkt \fk,t
3.2.2. Constraints
Constraints (1) calculate the tons of Bl, B2, and D3 ore mined per time period
and the corresponding deviations from the specified production levels. Implicitly, this
constraint assumes continuous mining of each production block within each machine
placement. The first term on the left hand side calculates the total tons of ore type k
mined in time period t and the second and third terms account for the excess and
shortage, respectively of the ore type in that time period. The righthand side represents
the demand for ore type k in time period t.
This constraint also accounts for the requirement that mining of a machine
placement, (explained in Section 3.1.3). Because each binary variable is multiplied by the
corresponding amount of ore generated in each time period by mining the given machine
placement, once the variable assumes a value of , the result of mining the ore from all
1
the production blocks in the machine placement is accounted for accordingly.
Constraints ( ) comprise the vertical sequencing constraints between mining
2
sublevels.
Constraints (3) and (4) enforce horizontal sequencing constraints between
adjacent production blocks.
Constraints (5) ensure that no more than the allowable number of LHDs is active
within a shaft group.
Constraints ( ), set packing constraints, ensure that a production block starts to be
6
mined no more than once during the time horizon if its late start date occurs beyond the
length of the time horizon.
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CHAPTER 4
REDUCING THE NUMBER OF INTEGER VARIABLES
4.1. INTRODUCTION
MEP model solution times depend on the number of integer variables and also on the
number of constraints in the model. MIP solution time increases exponentially as the
number of integer variables increases. The practical application of MIP is limited by the
number of integer variables (decision variables), because the large number of integer
variables required for model formulation results in solution times that may be
unacceptable for practical planning purposes. For a three year schedule, the approximate
number of integer variables for the Kiruna Mine production schedule is 36 (time periods)
* 1173 (blocks) = 42,228. The corresponding solution time for the production scheduling
problem is unreasonable with today’s computers. Therefore, the number of integer
variables is reduced by preprocessing the production data and carefully formulating the
model. It is possible to further reduce the number of integer variables by assigning an
earliest and a latest possible start date for each machine placement. In this chapter,
integer variable reduction methods are explained.
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4.2. PREPROCESSING THE PRODUCTION DATA AND FORMULATING THE
MODEL
The production data includes the available tonnage of each ore type for each
production block. Therefore, given the production blocks that are being mined, the
amount of each ore type that is extracted can be determined. It may initially appear as if a
single binary variable needs to be defined indicating whether production block b is mined
in each time period t, i.e., ybt- However, because all production blocks within a machine
placement must be mined continuously in a specific order, it suffices to define a binary
variable indicating whether machine placement a starts to be mined in time period t, i.e.,
yat. From the production data and mining start date the amount of any ore type available
(mined) in a given time period can be determined. Using the variables that depend on
machine placements, rather than production blocks, requires nontrivial accounting to
consider the amount of time required to mine each production block, and, given the
number of production blocks in each machine placement, the amount of time to mine
(some portion of) the machine placement. However, because there are far fewer machine
placements than individual production blocks, usings, rather thanytt, as the binary
variable reduces the number of binary variables from 42,228 to 2,016 for the 36 time
period example case (56 machine placements * 36 time periods = 2016).
Moreover, some of the machine placements are already being mined, i.e., some
machine placements are active; hence the corresponding binary variable can be fixed to a
value of 1. In this case, the variable need not be considered in the optimization model.
Currently, 15 machine placements are active, so the number of integer variables can be
reduced from 2,016 to 1,440 (2,016 - 16 machine placements * 36 time periods = 1,440).
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4.2.1. Example Application of the MIP Model on a Small Problem
A small example is created to explain how the binary variable indicating
whether machine placement a starts to be mined in time period t is used. The following
are the assumptions.
• There are four machine placements (MP), each containing different production
blocks (PB). For example, machine placement 1 (MPI) has 4 production blocks
(PB), machine placement 2 (MP2) has production blocks (PB), etc. (Figure 17).
6
• A maximum of two LHDs can be operable within a shaft group at a given time
and there is only one shaft group.
• Constraints are illustrated for four to seven time periods depending on the
complexity of the constraint.
• The machine placement 1 (MPI) is currently active.
• Available tonnages are given in Table 2.
• The targeted ore tonnage for Bl, B2, and D3 ore quality month-by-month are 170,
150, and 155 Kton respectively.
• Earlyup, = l>£œ yjtfP = 1, = '> a n d = 4.
7 2
• = 1, La/eMP2= 10, Late^ = 10, and LaleilPA= 10
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ba. Parameter representing the number of production blocks contained in machine
placement a
mv: Dynamic array containing the max{zv} greatest completion times for each shaft
2
group v for machine placements for which the early start dates have been set, ordered
from greatest to least
The function member(X,y) is used to denote the yth (greatest) member of set X.
Step . Initialize by setting the earliest start date (ES) to 1 for all machine
0
placements a and correspondingly initialize all elements of the dynamic array m
to .
1
Loop through the sorted list of sublevels starting with the uppermost sublevel
For each sublevel do
Set the early start date for all machine placements on the current sublevel
according to the vertical sequencing constraints
For each machine placement on current sublevel do
Step 1. Set the early start date for a affected by the vertical sequencing
constraint owing to a0 and the corresponding number of production blocks
within machine placement a0\
Set ES(a) = max (ES(a), ES(ac) + 50%*(^)).
Next machine placement
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Repeat for each machine placement a to which a late start date is wished to assign.
Step 0. Initialize by setting the late start date (LS) to T (the maximum number of time
periods in the planning horizon) for all machine placements.
Step 1. Define the boundaries of influence for machine placement a. That is, determine
all production blocks b within adjacent machine placements a on the same sublevel that
can never be mined as a result of not mining a. Denote this set »9a.
Step 2. Given machine placement a is never mined during the time horizon, determine
the production blocks b within machine placements adjacent to a that can be mined in
each time period t while adhering to vertical and horizontal sequencing constraints.
Denote this set .
Step 3. Given the production blocks b that can be mined (from Step 2), calculate the
t
cumulative available tonnage of each ore type k for every time period t.
U=1
Step 4. Determine the cumulative required production tonnage of each ore type k
u-1
for each time period t. Then, for each time period t=\...T, check whether: <
U=1
U=1
If the condition is true, stop. Time period t defines an upper bound on the latest
start date for a.
To determine whether this upper bound is tight, check whether:
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Table 7. Monthly cumulative required production for ore type B1 {r± )
Time Period (Months) Required B1 Production
(Tons)
1 250
2 500
3 750
4 1000
5 1250
A comparison between the monthly required ore tonnage and the available tonnage for
each ore type is provided a in Table 8.
Table 8. Comparison between cumulative available ore and required ore for B1 ore type
Time Period (Months) Required B1 Production Available B1 Ore type
(Tons) (Tons)
1 250 260*
2 500 515
3 750 735
4 1000 990
5 1250 1115
As seen in Table 8, by comparing column 2 with column 3, if MP3 is never mined, the
total required B1 production by time period 3 is 750 tons and the total available B1
tonnage is 735 tons. Because required B1 production is greater than available B1
tonnage, time period 3 is an upper bound on the time at which MP3 must start to be
mined. To ensure that this is a tight upper bound, we must calculate the available ore
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given that machine placement 3 starts to be mined in time period 3. The result of these
calculations appears in Table 9.
Table 9. Comparison between cumulative available ore and required ore for B1 ore type
assuming MP3 starts to be mined in time period 3
Time Period (Months) Required B1 Production Available B1 Ore type
(Tons) (Tons)
1 250 260
2 500 515
3 750 950+
4 1000 1445
5 1250 1950
For example, the amount of ore available until time period 3 is calculated as the
sum of the following quantities: (i) the amount of ore available from Table 6 in time
period 3, (ii) the additional ore available from mining production block 3 of machine
placements adjacent to MP3, and (iii) the amount of ore in production block 1 of MP3.
These quantities are, respectively, 735 + (85+50) + 80 = 950 tons, and are denoted by a +
in Table 9. Following this logic, starting to mine MP3 in time period 3 will make 1445
tons of B1 ore available until time period 4, 1950 tons of B1 ore available until time
period 5. By comparing column 2 with column 3, it can be seen that the cumulative
available ore tonnage given MP3 starts to be mined in time period 3 exceeds the required
amount of B1 ore from time period 3 until the end of the horizon. Hence, time period 3 is
a tight (exact) upper bound on the late start date for MP3.
Additionally, each machine placement defines a late start time for its adjacent
machine placements because of the sequencing constraints, which require adjacent
machine placements to be mined after 50% of the given machine placement has been
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mined. In our example, MP2 and MP4 must start to be mined after 50% of MP3 has been
mined. If the late start of MP3 is time period 3, and 3 time periods are required until 50%
of MP3 has been mined, then MP2 and MP4 must start being mined no later than time
period 6.
4.5. NUMERICAL RESULTS
The efficiency gained from using the early and late start algorithms is illustrated
with five scenarios. The first scenario, or base case, contains actual data from LKAB’s
Kiruna Mine (see attached CD). The other four scenarios were generated by modifying
demand data and the available number of LHDs from the base case to reflect realistic
changes given the availability of each ore type in each machine placement. For all
scenarios, the original objective is used, minimizing total deviation from planned
production quantities for all ore types in all time periods, and a three-year planning
horizon is assumed.
Several points regarding the late start dates should be noted before numerical
results are presented. In theory, the late start algorithm for each machine placement
could be run. Practically speaking, this would be computationally feasible. However,
late start dates for machine placements adjacent to a machine placement for which a late
start date has been determined can also be set using left and right sequencing constraints.
Initially, a late start date only for the machine placement in the geometric center of each
sublevel is assigned, because using sequencing rules often results in a “tighter” late start
date than using the late start algorithms on all machine placements. This does not
preclude the possibility that other mine structures might suggest determining late start
dates for all machine placements to realize the maximum variable reduction possible.
Second, although the late start algorithm is presented as an optimal procedure, it
cannot, in actuality, be implement it as such. Recall that the objective is to minimize
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deviations from planned production quantities. Without knowing, a priori, that the
solution to the original optimization problem results in meeting demand for each ore type
in each time period, (which may, in general, not be the case!) late start dates cannot be set
based on satisfying such a constraint. To approximate the extent to which we can
reasonably require that demand be satisfied, for each ore type and cumulative time,
deviation from the demand is based on the ratio of the amount of available ore to the
amount of ore demanded. So, for example, if this ratio were .95 for ore type k until time
period /, then for the late start date for the relevant machine placement to be time t or
greater, we would only require that the demand for ore type k until time period /be ~ .95,
say, .95 *(1- s). The question now arises as to an appropriate value for s. There is no
robust answer, but empirically and intuitively it has been found that a value of 2% works
well for ratios less than .95; for ratios greater than this value, s need not be used.
Admittedly, such an ad hoc procedure could result in an overly-restrictive
solution, i.e., one in which a late start date is actually set earlier than it should be. This,
in turn, would result in a suboptimal solution. One can argue that for large enough
problems, the loss of optimality resulting from “early” late starts, with, of course, a
judicious choice of s, is preferable to the loss of optimality resulting from the inability to
solve the monolith, i.e., the original model without reducing the number of integer
variables using a late start date.
The difference between the best solution and the bound is called gap-, this gap
provides a sense of the quality of a sub-optimal. Finding a solution within 5% or 10% of
optimality may suffice, especially given the accuracy of the data.
The numerical result is presented in Table 10 as follows: The first column gives
the scenario number; the second column provides the number of integer variables
contained in the scenario without the use of either the early or the late start algorithm.
The following two columns give the reduction in the number of integer variables
resulting from the use of the early and late start algorithms, respectively. The fifth
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Given the structure of the problem, the original number of integer variables - over
1400 - precludes us from obtaining a solution in a reasonable amount of time. In fact, it
is not shown the solution times for this version of the problem. By implementing early
start, on average, the number of integer variables is reduced by over 53%. Implementing
the late start algorithm results in, on average, an additional 24% reduction in the number
of variables. Correspondingly, near-optimal solutions are obtained in less than half hour
for three of the five scenarios by implementing only the early start algorithm. However,
in several cases, the algorithm cannot find a solution better than as much as 6-16% from
optimality within imposed time limit of three hours. A user may choose to implement
only the (optimal) early start algorithm, and still realize solutions within an acceptable
amount of run time. Larger and/or more complicated scenarios may suggest that both the
early start and the (heuristic) late start algorithms be used. The ratios of total deviation to
total demand demonstrate that the scenarios for which are obtained a solution within 5%
of optimality, yield reasonable results.
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CHAPTER 5
RESULTS AND COMPARATIVE ANALYSIS
5.1. PRODUCTION SCHEDULING OF THE KIRUNA MINE
In this study, the mixed integer programming model is implemented by using the
AMPL programming language, and the CPLEX solver version 7.0.
The AMPL programming language is one of several optimization modeling
systems that is designed around specialized algebraic modeling languages. AMPL has its
own syntax for set and indexing expressions. Sets and indexing expressions make the
model size smaller and more efficient than normal programming codes. Optimal solutions
can be verified and reported easily and AMPL provides numerous alternatives for
presenting the data and results. The CPLEX software contains a state-of-the-art integer
programming solver based on the standard branch-and-bound algorithm (for more
information see Chapter 1, Section 1.2.1), embellished with heuristics that aid in finding
feasible solutions, and methods for developing “cuts” that tighten the search space. The
algorithm finds increasingly better (lower, for a minimization problem) feasible solutions,
and provides a (lower) bound on the best solution attainable. Also, users can direct the
branching process or select specialized techniques that take advantage of structures in
their specific problems. One of the major problems with the AMPL programming system
is the weakness of the system in locating errors. An error report simply indicates the line
where the error occurs. However, there is no information provided about why the error
has occurred, where exactly it has occurred, and how it can be corrected. A complete
listing of AMPL code for the model used for the Kiruna Mine scheduling system is given
in Appendix - A.
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In addition to good software, state-of-the-art hardware is necessary to solve large
mixed integer programming problems. The model instances discussed in this thesis were
solved on a Sun Ultra 10 machine with 256 MB RAM.
5.2. PRODUCTION SCHEDULING OF THE KIRUNA MINE WITH DIFFERENT
STRATEGIES
5.2.1. Production Scheduling by Minimizing Deviation from "Planned Production
Demands"
Figures 20, 21 and 22 show the complete three-year schedule obtained when
using the objective function which minimizes the sum of the production deviations for
the three ore products Bl, B2, and D3, ( Section 3.2.1.). The optimal solution is obtained
in 96 seconds for three-year monthly time horizon.
The ratio of the total tons of deviation to the total tons demanded is 6.32 %. This
number gives a sense of the quality of objective. 6.32 % is produced by summing all the
deviations dukt and ddkt then dividing this number by total tons demanded and
multiplying by 100. Table 11 presents the output for dukt and ddkt for this scenario. The
first and fourth columns give the time periods (month), the second and fifth columns give
the ore type, and the third and sixth columns provide the over and under production
tonnages, dukt and ddkt. The summation of the deviation from the third and sixth
columns is 4,554 Kton and total production target is 72,072 Kton for 36 months. Dividing
4,554 by 72,072 gives 6.32 %.
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achine Placement Level 201 | 202 | 203 | 204 205 206 207 208 209 210 211 | 212 I Total
■ [ 1 765 42- 43 45 i I ....... 690
!■ ■N* 765 44- 46 45 2 : ■ ■ ! ■ ■ i p l L 1680
in in |n in n mm nkn te tu be
792 14- 15 16 i 1630
792 19-21 19 2 I n I I I 130
792 33- 34 33 2 n □ n □ 540
792 35- 37 37 i 420
792 39- 42 40 a j E j e i i n E l eL i l o i n I n I n *==: | a 1760
792 43- 46 45 2 ■ 750
818 16-17 16 2 p p □ b B B T I I esc
820 16- 17 16 2 M i l t p ] | 150
820 18- 19 19 i j d n ■=□ ■ZJ IZJ ILJ IZJ lEZI p 1160
820 23-26 25 i MU e u m u MU tu LI) tu % u 1120
n
820 27-28 28 i n n 1 1 n i i i n m m n 1250
820 29- 30 28 2 I ■ 1 1 I..... 260
820 31-32 33 i i n n m m m i n m mm |n m I n i n 1650
|n |n |m |n n pi pm
820 33-34 33 2 ■ m ■ Km cm | leoo
820 35-37 37 1 ■ ■ ! ■ ! ■ I n i n | n n | n i n | n p e l 1690
n ft i m ni m
820 38- 41 40 1 s n 770
ii pi iD^|m_g^jc* _n fzm
849 27- 28 28 930
849 29- 30 28 I lai |om lorn Inn lea *=a I n |[=m 1210
pi
849 31- 34 33 ja IT Im loi Q : I d 820
849 35- 38 37 140
mm urn
Development Ore « n n n |n u ■■ n n n 1560
Development Waste 996
Total Kton/Month 2053 1983 1973 1913 1883 1913 2063 2143 1983 1853 1823 1973| 23556
Total Kton/Day 66.2 70.8 63.6 63.8 60.7 63.8 66.5 69.1 66.1 59.8 60.8 63.6
B1 Kton/Day 11.9 11.4 11.0 11.3 10.3 11.0 11.6 11.9 11.8 11.3 11.0 10.0
B2 Kton/Day 29.8 31.5 27.2 28.8 27.2 30.1 30.7 31.7 29.6 24.6 24.4 27.2
D3 Kton/Day 24.5 27.9 25.5 23.7 23.2 22.7 24.2 25.5 24.7 23.9 25.3 26.5
Month 201 202 203 204 205 206 207 208 209 210 211 212
Figure 20. Illustration of a first year production schedule for the Kiruna Mine with the
original objective function
1
ft
I
1
ft
I
1
ft
1
1
I
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Machine Placement Level 301 302 303 304 305 3— 06 307 308 309 310 311 312 Total
765 44- 46 45 2 ■ 140
792 14- 15 16 1 I n 340
792 39- 42 40 1 1 1 r a l a i a i u |i__e | a 970
mm mugims
792 43- 46 45 2 ■ ■ B E 1920
I ma m
820 14- 15 16 1 am am ■ 620
2 c |d id tzi qzi t=i d
820 16- 17 16 d |d |d |d |d 1560
820 33- 34 33 2 ■m i i i l i i I 1 1 1 LJ 950
am
820 35- 37 37 1 m m m m e r n m 1200
i [ni in in |p in |azi ! c jc t j
820 38- 41 40 \m u 1790
820 42- 46 45 ’ I i___________ » r« w * ^ m* 680
849 23- 26 25 i id id d i d i d d d d 1290
849 27- 28 28 i Id l^ jd iliZ lld lld llC II^ Z l^ Z ] Ç 3 Ç3 1 1640
849 29- 30 28 2 | a | a l a j a jca |cm |c * |c mt=a [m |m 1760
849 31-34 33 1 jeu l u [ i i | a |b j |d i j u |h j |d i |ci3 |d i|r .;"il 1930
849 35- 38 37 Lij m ._ M 2010
849 39- 41 40 1 1 1 ...1 I I 1 1 1 # ti t=i 210
878 27- 30 28 i ■ |n n « i m i m i m i « ■■ ■ 1480
n n n n n n : "
Development Ore I— I— I— | i I— |mm 1560
Development Waste 996
Total Kton/Month 1843 1733 1933 1913 1933 1953 2053 1973 1883 1923 1983 1923j 23046
Total Kton/Day 59.5 61.9 62.4 63.8 62.4 65.1 66.2 63.6 62.8 62.0 66.1 62.0 j
B1 Kton/Day 9.0 9.6 10.0 10.8 10.3 10.7 9.7 9.0 9.7 9.4 9.3 8.7
B2 Kton/Day 25.6 29.0 29.5 30.6 30.1 32.1 31.4 27.8 28.8 28.8 32.4 29.8
D3 Kton/Day 24.8 23.2 22.9 22.3 21.9 22.3 25.2 26.8 24.3 23.9 24.3 23.5
301 302 303 304 305 306 307 308 309 310 | 311 312
Figure 21. Illustration of a second year production schedule for the Kiruna Mine with the
original objective function
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