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Once the survey data was transformed into UTM-WGS84, a portion of the mine
was generated and compared to the actual location of the truck to check for GPS/Map
consistency.
The next stage was to drive into the main pit in order to collect new survey data as
well as to confirm GPS positioning and map geometry.
The actual driver could see his own position with respect to the mine pit and he
was able to update the pit contour map on-demand, while the truck was moving within
the pit.
The real pit on the top and the equivalent virtual pit represented by a 3D contour
map version can be seen through the VirtualMine interface. Notice, in both pictures,
the mine truck CAT 793 driving at the ramp.
During this test, real time tracking was carried out as seen on Figure 72, checking
for consistency between truck location and pit geometry. Results were overwhelmingly
accurate.
Besides the real-time tracking test, an on-demand contouring test was also carried
out. Driving the CAT 793 truck down to the bottom of the pit, using the Topo command
and the Generate Contour command, produced a new contour map of the pit on the
truck’s on-board computer screen.
Once the contour map is generated on-demand by the driver, it can be loaded into
the computer screen, allowing so the driver to check position with respect to the new
updated map. Figure 71 and Figure 72, give a picture of the Morenci pit where the test
was carried out, as seen in real-life and in the VirtualMine program respectively.
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6.2.3 Proximity Warning System Tests
The next task at Morenci was to carry out proximity warning system tests between
two vehicles. As originally conceived this project considered the scenario of tracking
one moving vehicle with respect to a fixed point in the mine which could represent, for
example, a safety berm located next to the edge of a dump site.
VirtualMine was coded to perform such a task. The system was initially considered
a stand-alone system or a system that is independent of other vehicles and does not
transfer or receive information from them. At this point of development, however the
system was able to receive and translate GPS coordinates and to post in 3D the truck
position with respect to the mine map (also in 3D). The system is loaded with a
predefined virtual berm and is capable of keeping track of distance between the truck
and the berm.
At some predefined critical distance, the system warns the driver that the truck is
too close to the safety berm. (This Critical distance is represented as a safety 3D
bubble surrounding the truck). As soon the safety bubble touches the 3D plane
representing the safety berm, the alarm is triggered. As seen in Figure 73, the green
light at the bottom of the screen turns red as soon as the bubble collides with the
safety plane or safety berm.
■am
VirtuolMiae - Colorado Softool of Mines A VwftwlMme - Colorado School el Mine»
%= "-str
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Figure 73: Proximity warning into the virtual safety berm, as seen in VirtualMine
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However, within the scope of the proximity warning concept, another key aspect
had to be developed: proximity warning between vehicles.
In order to achieve proximity warning between the local truck and a remote vehicle,
VirtualMine was enabled with a new subroutine for TCP/IP communication in a peer-
to-peer scheme.
The two vehicles have a wireless link connection using the TCP/IP protocol, which
had to be compatible with the Trimble radio infrastructure as well as with IEEE
802.11 b radio technology.
The test was carried out mounting the VirtualMine system into a pickup truck and a
CAT 793 Truck. GPS signal for monitoring was activated over a previously loaded 3D
Morenci map.
The next step was to activate communication in the peer-to-peer interface in
VirtualMine, using predefined IPs corresponding to each truck as remote and as local.
At this point, both vehicles could see each other on the screen in both truck
computers. As both vehicles started to move into another section of the mine,
VirtualMine started to track their position in real time.
Figure 74 and Figure 75 show the screen’s computer immediately after TCP/IP
connection was established to display both vehicles moving from the test field into the
north section of the mine. VirtualMine keeps tracking their position and the distance
between them.
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As the truck wirelessly receives the terrain data, it can generate on-demand, an
updated 3D map containing the updated profile of the dump’s edge. The truck can
then proceed to the dump site using a newly updated terrain model, (see Figure 80).
Figure 80: The mine truck during the dumping process based on the new updated
map that was radio-transmitted from the dozer.
The objective is to make the truck driver base his/her bearings on the new updated
3D map that has been generated in his truck computer (as seen in Figure 81).
Based on tests carried out at the Holnam Cement Quarry, and the Morenci Mine,
VirtualMine proved to have the capacity to provide the operator with a self-view with
respect to the geometry of the mine and with respect to other GPS-equipped vehicles.
This extra view ability is given by the computer screen mounted in the tuck’s cabin.
It helps the operator monitor, in advance, nearby vehicles and previously fixed
dumping points, thus giving the driver an improved chance to react to a potential
collision or rollover.
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CHAPTER VII
7 CONCLUSION
7.1 Suitability of the system
The purpose of the project was to develop a GPS-based computer guidance
system that could be loaded into a panel computer which is mounted on cabin of
vehicles operating in open-pit mines to automatically warn the driver of the dumpsite’s
edge and the proximity of nearby vehicles.
Based on tests carried out at the Colorado School of Mines survey field, the
Holnam Cement Quarry, and the Morenci Mine, VirtualMine proved to have the
capacity to provide the operator with an extra digital “rear-view mirror” with respect to
the geometry of the mine and with respect to other GPS-equipped vehicles.
This digital “mirror” capability is given by the computer screen mounted in the tuck’s
cabin. It helps the operator monitor, in advance, nearby vehicles and previously fixed
dumping points, thus giving the driver an improved chance to react to a potential
collision or rollover.
The tests at Morenci have verified that VirtualMine can display truck position with
respect to the edge of the dump as well as with respect a remote vehicle. Together,
the technologies investigated in this dissertation are combined in this system to
improve safety of off-highway trucks in open-pit mines.
Thus, if the system described in this dissertation could be installed in mine vehicles,
it is believed that it would definitely reduce the number of accidents related to
proximity warnings and rollovers.
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VirtualMine in its current stage does not consider analysis of soil failure prediction
to assess the definition of the virtual safety berm with respect to the edge of the
dumping point. Further investigation related to this issue is recommended.
GPS Sub-meter accuracy was acquired using differential correction as shown
during CSM survey field tests. However, in order to assess the probability of acquiring
sub-meter accuracy 100% of the time, and to determine possible trends related to
DOR and atmospheric conditions, generation of accuracy tests should continue.
Even though VirtualMine was evaluated during testing at the CSM campus, Holnam
and Morenci, this system has to be rigorously tested to demonstrate its reliability and
effectiveness.
7.2 Original Contribution
This dissertation was based on a project to improve safety of off-highway trucks
through GPS, a project being developed at the CSM Mining Engineering Department,
in compliance with NIOSH requirements to improve safety conditions during dumping
and routine haul road operations.
There are four major contributions made in this dissertation:
• This dissertation follows a unique approach in developing a computer system
to improve safety in open-pit mines based on GPS, wireless networks, and 3D
graphics. The previously developed GPS systems have, so far, focused on
computer-aided excavation systems and on dispatching to improve productivity
of mining operations.
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• Another significant contribution is the integration of 3-D on-demand mapping
based on Virtual Reality Modeling Language (VRML), GPS, and wireless
communication networks to create a user-friendly, powerful yet inexpensive
system to monitor vehicle activity and mine geometry to improve safety in
open-pit mines. None of the GPS systems that are currently used in the mining
industry provide real-time relative vehicle positions with respect to updated
mine topography maps on-demand, nor with respect to any other vehicles.
• The proximity warning system developed in this dissertation uses a 3-D sphere
bubble concept to represent the safety zone around a given trench. The
adjustment of the safety sphere, relative to the operating conditions of the
vehicle and to the dump surface, is unique and has not been implemented
before in any other system based on GPS and wireless networks.
• The software implementation of 802.11b wireless networking protocol in the
mining environment is also new and provides, for the first time, an opportunity
to track mining equipment anywhere in the world as long as an internet
connection is available.
7.3 Conclusions
Fatal accidents related to dumping tasks occur at a significant rate, with over 20
deaths each year. Differential GPS-tracking applied as a proximity warning system,
can help to reduce these numbers.
The tests that were carried out on the Colorado School of Mines campus, as well
as in actual mining operations such as the Morenci Mine, indicate that VirtualMine can
successfully used as a tracking and collision warning system with respect to other
vehicles and to 3D mine geometry.
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The 3D-digital mapping generation using stored data collected from the GPS unit in
the truck and dozer was successful according to tests carried out and documented at
the Holnam Quarry and the Morenci Mine.
In addition, real-time 3D-digltal maps were successfully transferred wirelessly,
between truck and dozer using the TCP/IP protocol used in this system.
3D-digital mapping based on Virtual Reality Graphics, GPS, and wireless networks,
were combined to create a friendly, powerful and inexpensive system to monitor
vehicle activity and mine geometry.
7.4 Recommendations
New technologies, such as pseudo!ites and differential GPS based on
geostationary satellites, offer great potential in GPS applications to open-pit mining,
since they increase both the reliability of the GPS system and the overall accuracy at
a relatively low cost. Further investigation on the implementation of these technologies
is suggested.
Further research is also recommended to continue developing this system into a
mine expert system within the following scope: dispatching systems, optimizing shovel
operations, real-time mine planning, and control and interface for driverless systems.
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GLOSSARY
Anywhere fix: The ability of a receiver to start position calculations without being
given an approximate location and approximate time.
Bandwidth: The range of frequencies in a signal.
C/A code: The standard (Course/Acquisition) GPS code. A sequence of 1023
pseudo-random, binary, biphase modulations on the GPS carrier at a chip rate of
1.023 MHz. also known as the civilian code.
Carrier: A signal that can be varied from a known reference by modulation.
Carrier-aided tracking: A signal processing strategy that uses the GPS carrier
signal to achieve an exact lock on the pseudo-random code.
Carrier frequency: The frequency of the unmodulated fundamental output of a
radio transmitter.
Carrier phase GPS: GPS measurements based on the L1 or L2 carrier signal.
Channel: A channel of a GPS receiver consisting of the circuitry necessary to
receive the signal from a single GPS satellite.
Chip: The transition time for individual bits in the pseudo-random sequence.
Clock bias: The difference between the clocks indicated time and true universal
time.
Code phase GPS: GPS measurements based on the pseudo random code (C/A or
P) as opposed to the carrier of that code.
Control segment. A world-wide network of GPS monitors and control stations that
ensure the accuracy of satellite positions and their clocks.
Cycle slip: A discontinuity in the measured carrier beat phase resulting from a
temporary loss of lock in the carrier tracking loop of a GPS receiver.
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Data message: A message included in the GPS signal which reports the satellite's
location, clock corrections, and health. Included is rough information on the other
satellites in the constellation.
Differential positioning: Accurate measurement of the relative positions of two
receivers tracking the same GPS signals.
Dilution of Precision (DOR): The multiplicative factor that modifies ranging error. It
is caused solely by the geometry between the user and his set of satellites. Known as
DOP or OOP
Dithering: The introduction of digital noise. This is the process the DoD uses to
add inaccuracy to GPS signals to induce Selective Availability.
Doppler-aiding: A signal processing strategy that uses a measured Doppler shift
to help the receiver smoothly track the GPS signal. Allows more precise velocity and
position measurement.
Doppler shift: The apparent change in the frequency of a signal caused by the
relative motion of the transmitter and receiver.
Ephemeris: The predictions of current satellite position that are transmitted to the
user in the data message.
Fast switching channel: A single channel which rapidly samples a number of
satellite ranges. "Fast" means that the switching time is sufficiently fast (2 to 5
milliseconds) to recover the data message.
Frequency band: A particular range of frequencies.
Frequency spectrum: The distribution of signal amplitudes as a function of
frequency.
Ionosphere: The band of charged particles 80 to 120 miles above the Earth's
surface.
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Ionospheric refraction: The change in the propagation speed of a signal as it
passes through the ionosphere.
L-band: The group of radio frequencies extending from 390 MHz to 1550 MHz.
The GPS carrier frequencies (1227.6 MHz and 1575.42 MHz) are in the L band.
Multipath error: Errors caused by the interference of a signal that has reached the
receiver antenna by two or more different paths. Usually caused by one path being
bounced or reflected.
Multi-channel receiver: A GPS receiver that can simultaneously track more than
one satellite signal.
Multiplexing channel: A channel of a GPS receiver that can be sequenced
through a number of satellite signals.
P-code: The Precise code. A very long sequence of pseudo random binary
biphase modulations on the GPS carrier at a chip rate of 10.23 MHz which repeats
about every 267 days. Each one week segment of this code is unique to one GPS
satellite and is reset each week.
Precise Positioning Service (PPS): The most accurate dynamic positioning
possible with standard GPS, based on the dual frequency P-code and no SA.
Pseudolite: A ground-based differential GPS receiver which transmits a signal like
that of an actual GPS satellite, and can be used for ranging.
Pseudo random code: A signal with random noise-like properties. It is a very
complicated but repeating pattern of Ts and O's.
Pseudorange: A distance measurement based on the correlation of a satellite
transmitted code and the local receiver's reference code, that has not been corrected
for errors in synchronization between the transmitter's clock and the receiver's clock.
Satellite constellation: The arrangement in space of a set of satellites.
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Selective Availability (SA): A policy adopted by the Department of Defense to
introduce some intentional clock noise into the GPS satellite signals thereby
degrading their accuracy for civilian users.
Slow switching channel: A sequencing GPS receiver channel that switches too
slowly to allow the continuous recovery of the data message.
Space segment: The part of the whole GPS system that is in space, i.e. the
satellites.
Spread spectrum: A system in which the transmitted signal is spread over a
frequency band much wider than the minimum bandwidth needed to transmit the
information being sent. This is done by modulating with a pseudo random code, for
GPS.
Standard Positioning Service (SPS): The normal civilian positioning accuracy
obtained by using the single frequency C/A code.
Static positioning: Location determination when the receiver's antenna is
presumed to be stationary on the Earth. This allows the use of various averaging
techniques that improve accuracy by factors of over 1000.
User interface: The way a receiver conveys information to the person using it. The
controls and displays.
User segment: The part of the whole GPS system that includes the receivers of
GPS signals.
UTM: The Universal Transverse Mercator system uses a projection tangent to the
equator that divides the world into 60 zones spanning the equator. Within each zone a
Cartesian coordinate system is used.
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APPENDIX A:
SAFETY CONSIDERATIONS
According to a survey done by MSHA on “Haulage Fatalities at Surface Mines”
(See Krowczyk, 1998), there were 29 fatal accidents involving trucks rolling or driving
off bench, road or high wall in surface mining operations and 23 fatal accidents
involving vehicle collisions between 1992 and 1999.
In 1999 along, there were 4 fatal accidents involving trucks going over the edge of
waste dumps. I would like to mention that just last week (September 2001 ) as I was
reviewing this chapter it was reported another fatal accident in Chile during a dumping
task operation.
Following are six examples of MSHA reports related to truck rolling or driving
off bench in surface mining operations:
Report 1: COAL MINE FATALITY - On Monday, March 13, 1995, a truck driver
was fatally injured when the truck he was driving went backwards over the edge of the
spoil pile and down a 200 foot embankment. The truck contained a load of wet
material loaded from a parking area to be dumped at the edge of the spoil. During the
investigation the bulldozer operator at the site stated that he had advised the truck
drivers by radio to dump "short" because the edge of the spoil was soft.
The victim was making his eighth loaded trip to the dumping area, however he was
backing his truck to a different location from where his previous loads were dumped.
This was the only location on the edge where a rock pile, created by trucks dumping
"short", did not exist. An eyewitness observed the truck as it was rolling down the
slope, however he did not observe the victim being thrown from the truck near the
bottom of the slope. Some related factors are: the truck bed had not been raised; a
seat belt was not worn but the damage to the cab was so complete a fatal injury likely
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victim was dumping his truck at a 20-foot high waste dump when he backed over the
edge. The truck overturned onto the driver's side, pinning the victim under the left front
tire. He was not wearing a seat belt. Berm or bumper blocks were not provided at the
dump site.
Seat belts should be worn by all equipment operators. Berm or other impeding
devices are required at dumping locations where there is a danger of over travel.
Dumping locations should be visually inspected prior to work commencing.
This is the 24th fatality reported in calendar year 1998 in the metal and nonmetal
mining industries. As of this date in 1997, there were 29 fatalities reported in these
industries. This is the tenth fatality classified as Powered Haulage in 1998. There were
thirteen Powered Haulage fatalities in the same period in 1997.
Report 4: METAL/NONMETAL MINE FATALITY - On March 15, 1999, a 61 -year-
old truck driver with 40 years mining experience was fatally injured at a crushed stone
operation. The victim was backing a truck loaded with waste material to the edge of
the quarry preparatory to dumping. The rear wheels were backed onto a mound of
material which collapsed and the truck traveled over a 100-foot highwall. The victim
was not wearing a seat belt.
Dumping locations should be inspected and loads should be dumped at least one
truck length back from the edge where there is evidence the ground is unstable.
Haulage trucks should not back onto a berm which is provided to establish the
travel limit for their rear wheels.
Equipment operators should always wear seat belts.
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This is the 11th fatality reported in calendar year 1999 in the metal and nonmetal
mining industries. As of this date in 1998, there were 14 fatalities reported in these
industries. This is the fourth fatality classified as Powered Haulage in 1999. There
were four Powered Haulage fatalities in the same period in 1998.
Report 5: METAL/NON METAL MINE FATALITY - On November 9, 1998, a 51-
year old utility man with 18 months of mining experience was fatally injured at a
crushed stone operation. The victim was backing a loaded truck to the outer edge of a
stockpile, preparatory to dumping. Material had been loaded out from the base of the
stockpile. The pile collapsed and the truck overturned. The victim was not wearing a
seat belt.
Dumping locations should be visually inspected, and where there is evidence the
ground may fail to support the equipment, loads should be dumped at least one truck
length back from the edge.
The access ramp to stockpiles should be blocked once the load-out of material has
begun.
Equipment operators should always wear seat belts.
This is the 47th fatality reported in calendar year 1998 in the metal and nonmetal
mining industries. As of this date in 1997, there were 57 fatalities reported in these
industries. This is the 20th fatality classified as Powered Haulage in 1998. There were
22 Powered Haulage fatalities in the same period in 1997. (Please note the 1997 final
fatality count was changed from 60 to 61 after the close of the calendar year.)
Report 6: COAL MINE FATALITY - On Thursday July 31, 1997, in Perry County,
Kentucky, a truck driver was fatally injured while operating a large rock truck. The
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APPENDIX B:
VIRTUALMINE CODE DESCRIPTION
Main Terminal
Dim SafeDist As Double Dim LonMinusLonn As Double
Dim SafeBound As Double Dim ConvertLatPast As Double
Dim ConvertLonPast As Double
Dim AcumXYZ(5000) As String
Dim AcumXYZsp(5000) As String Dim Truckl_Pos2Base_File As String
Dim Acumlndex(5000) As String Dim Zoom As Integer
Dim CountXYZ, CountXYZprint As Integer
Private Sub Form_Load()
Dim DTM As VRMLNode
Dim DTM1 As VRMLNode 'for rotation algorithm eventually
Dim DTM2 As VRMLNode equalize to Zero
Dim DTM3 As VRMLNode ConvertLonPast = 481190
Dim DTM4 As VRMLNode ConvertLatPast = 4399900
Dim Contour As VRMLNode RotationPast = 0
Dim transformT As VRMLNode To enable scene editing
Dim transformT2 As VRMLNode Cortonal.Edit
Dim DTMDozl As VRMLNode ' The following instructs engine
Dim DTMDoz2 As VRMLNode to imnidiately apply
Dim SPlane As VRMLNode ' any changes in VRML nodes'
Dim SatNode As VRMLNode fields values
Dim RoadNode As VRMLNode Cortonal.Engine.AutoRefresh = True
Dim DtmTruckl As VRMLNode 'frmClient.Visible = False
Dim LatMinusLat As Double
Dim LonMinusLon As Double 'Cortonal.Scene = False
Dim ConvertLonDistDiff As Double Timerl.Enabled = False
Dim ConvertLatDistDiff As Double Timer2.Enabled = False
Dim DistTruck As Double Timer3.Enabled = False
Timer4.Enabled = False
Dim simcoordX(10000) As Double TimerS.Enabled = False
Dim simcoordY(10000) As Double
Dim simcoordZ(10000) As Double ' Set properties needed by MCI to
Dim tt As Integer open.
Dim zz As Integer MMControll.Notify = False
Dim LatMinusLatt As Double MMControll.Wait = True
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GPS Terminal: GPS Signal Handling Interface
® GPS I ermmul
file Çomm Port MSComm Call
O\0\*\+\ iffl Pi ilB
Enter Dec Races lor Degree and Minutes to be read from GPS data [Latitude and Longitude)
Deg Dec Races MinNumDigrts Deg Dec. Races MinNumDigits AltNumDig
F
(TT^
Longitude GPS Latitude GPS Altitude M GPS
{Degrees {Minutes jDegrees Minutes Meters
Projection System to be used
Geographic a UTM Longitud Latilud Gent Merid. UTM Zone
F33 FH ÎT- r™
Ellipsoid name Equatorial Radius Square of Eccentricity
{Ellipsoid jrj Refresh ^
Easting Northing Altitude
F---
E
Option Explicit
CenterMeridText.Enabled = True
Dim Ret As Integer Sph_a.Enabled = True
Dim Temp As String Sph_e2.Enabled = True
Dim hLogFile As Integer
Dim StartTime As Date 1 Stores MSComml.RThreshold =100
starting time for port timer MSComml.InputLen = 100
Dim Sph_a_num, Sph_e2_num, Sph_e4_num,
Sph_e6_num As Double On Error Resume Next
Dim UTMdm, UTMfx, UTMfy As Double
Dim UTM_XX, UTM_YY As Double * Set the default color for the
terminal
Dim LongDec As Double txtTerm.SelLength = Len(txtTerm)
Dim LatDec As Double txtTerm.SelText = ""
Dim LongDecRad As Double txtTerm.ForeColor = vbBlue
Dim LatDecRad As Double
Dim LatNl As Double ' Set Title
Dim LatN2 As Double T^p.Title = "GPS Terminal"
Dim CentMeridRad As Double 1 Set up status indicator light
Dim LatDiff As Double imgNotConnected.ZOrder
Dim UTMfl, UTMf2, UTMf3 As Double ’ Center Form
frmTerminal.Move (Screen.Width -
Private Sub Form_Load() Width) / 2, (Screen.Height - Height) / 2
Dim CommPort As String, Handshaking As * Load Registry Settings
String, Settings As String Settings = GetSetting(App.Title,
"Properties", "Settings", "") 1
frmTerminal.MSComml.Settings]\
'’' this terminal is a modification of If Settings <> "" Then
a modem terminal you may find some code MSComml.Settings = Settings
that does not have application in the If Err Then
process of reading the ASCII code from the MsgBox Error$, 4 8
GPS unit Exit Sub
''' definition of pi using Atn End If
PI = 4 * Atn(1) End If
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Open Temp For Binary Access Read As If Len(Temp) Then
hSend MSComml.SThreshold = Val(Temp)
If Err Then If Err Then MsgBox Error$, 48
MsgBox Error$, 48 End If
Else End Sub
' Display the Cancel dialog box.
CancelSend = False ' This procedure adds data to the Term
frmCancelSend.Labell.Caption = control's Text property.
"Transmitting Text File - " + Temp ' It also filters control characters,
frmCancelSend.Show such as BACKSPACE,
' carriage return, and line feeds, and
' Read the file in blocks the writes data to
size of the transmit buffer. ' an open log file.
' BACKSPACE characters delete the
'BSize = MSComml.InputLen character to the left,
BSize = MSComml.OutBufferSize ' either in the Text property, or the
passed string.
LF& = LOF(hSend) ' Line feed characters are appended to
Do Until EOF(hSend) Or CancelSend all carriage
' Don't read too much at the ' returns. The size of the Term
end. control's Text
If LF& - Loc(hSend) <= BSize ' property is also monitored so that it
Then never
BSize = LF& - Loc(hSend) + ' exceeds MAXTERMSIZE characters.
1 Private Static Sub ShowData(Term As
End If Control, Data As String)
' Read a block of data. Const MAXTERMSIZE = 1600
Temp = Space$(BSize) Dim TermSize As Long, i
Get hSend, , Temp Dim j As Long
Dim MinLongitud As String
' Transmit the block. Dim MinLatitud As String
MSComml.Output = Temp Dim Altitud As String
If Err Then Dim Lookat As String
MsgBox Error$, 4 8 Dim GradeLat As String
Exit Do Dim GradeLon As String
End If Dim pp As Long
Dim Comma As String
' Wait fo r a ll the data to be Dim CommaCount As Long
sent. Dim CheckLong, CheckEnd As String
Do Dim B20, C20, D20, E20, E20, G20,
Ret = DoEvents() H2 0 As Double
Loop Until Dim LatSouthCorr, TestError As
MSComml.OutBufferCount = 0 Or CancelSend Double
Loop
End If '' Make sure the existing text
doesn't get too large.
Close hSend TermSize = Len(Term.Text)
mnuSendText.Enabled = True If TermSize > MAXTERMSIZE Then
Term.Text = Mid$(Term.Text,
tbrToolBar.Buttons("TransmitTextFile").Ena 4097)
bled = True TermSize = Len(Term.Text)
CancelSend = True End If
frmCancelSend.Hide
End Sub ' Point to the end of Term ' s data.
Term.SelStart = TermSize
' This procedure sets the ^Threshold 'Add the filtered data to the
property, which determines SelText property.
' how many characters (at most) have to Term.SelText = Data
be waiting
' in the output buffer before the ''' following code is to detect GPGG
CommEvent property and extracts
' is set to comEvSend and the OnComm ''' the x y z values from de ASCII
event is triggered. stream comming
Private Sub mnuSThreshold Click() '1' from the GPS unit coming trough the
On Error Resume Next serial port of the PC
''1 here the coords are also process to
Temp = InputBox$("Enter New Geodetic or UTM
SThreshold Value", "SThreshold",
Str$(MSComml.SThreshold)) i = i
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ABSTRACT
The strategic open pit mine production scheduling problem is usually formulated as a large-
scale integer programming problem that is very difficult to solve. The life of mine production
scheduling problem is currently modeled on a block by block basis in order to decide which blocks
should be extracted, when they should be extracted, and what to do with the blocks once they are
extracted. Due to the nature of the problem, the decisions on whether or not to mine an individual
block should be addressed in a binary context. However, the large size of some real instances (3–
10 million blocks, 15–20 time periods) has made these models impossible to solve with currently
available optimization solvers. To overcome this challenge, many attempts have been made to
solve the problem with numerous heuristic and aggregation methods which cannot be proven to
converge to the true optimal solution. On the other hand, linear programming relaxation of the real
sized mine planning problems can be solved to a proven optimality by applying the existing exact
decomposition algorithms. However, the solution obtained from the LP relaxation problems may
result in fractional blocks being mined which cannot be implemented practically.
A novel integer solution algorithm is developed in this thesis which can solve the mine
production scheduling problems modeled with multi capacities, grade blending, grade uncertainty,
stockpiles, variable pit slopes, multi destinations and truck hours. It should be emphasized that the
blocks will not have any pre-determined destinations based on grades, cycle times, material type
or some other criteria since the best destination selection per block will be done automatically
during the optimization process to maximize the NPV, in other words the dynamic cutoff concept
is employed. Presently there is no known algorithm, either commercially available or presented in
the literature, that can provide an optimal integer solution to the open pit mine production
scheduling problem with capacity constraints together with lower and upper bound blending
constraints. Therefore, the solution algorithm that can generate an optimal integer solution to a
mine production scheduling problem that has never been solved will be a milestone in operations
research. Moreover, a new cone pattern generation scheme is developed in order to integrate the
variable pit slope angles based on complex geotechnical zones and multiple azimuths with any
size block dimensions to the new integer solution algorithm.
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ACKNOWLEDGMENTS
I would like to thank all of the significant people that made this PhD a success. First of all,
I wish to express my deepest appreciation to my thesis advisor Dr. Kadri Dagdelen who provided
continuous encouragement and support throughout this journey. He was not just an academic
advisor; he was also a life mentor and has been the biggest influence in my career. Without his
support, this endeavor would not have become a success. Next, I would like to thank Dr. Thys
Johnson whose brilliance was an inspiration for me to overcome the challenges that I frequently
encountered throughout this process. His friendship and his continuous support have been
priceless. I sincerely thank Dr. Marcelo Godoy, for making this research possible with his
contributions and his financial support. This research was funded by Newmont Mining
Corporation and I will be forever grateful for their support.
Special thanks are also given to Dr. Priscilla Nelson for her financial support in difficult
times. Many thanks to Dr. Christopher Painter Wakefield for being there for me whenever I needed
guidance with coding. I also wish to thank Dr. Hugh Miller for his time and effort in being a part
of the committee. I would like to also acknowledge the financial support provided by Larry Allen
in the latter semester.
I feel privileged to be surrounded by smart people at Mines. I would like to thank Onur
Golbasi, Marion Nicco and Marko Visnjic for their wonderful fellowship and always being there
to support me in times of need. I wish to thank my colleague Ady Van Dunem, who has always
been a brother to me. We have shared countless adventures during this period.
I would like to take the opportunity to express my gratitude to my parents in Turkey for
their motivation, friendship and financial support.
Lastly, I am forever grateful to my wife, Anna, for her love, constant encouragement,
support and understanding.
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CHAPTER 1.
INTRODUCTION
The strategic open pit mine production scheduling problem is a large-scale integer
programming problem which requires a solution on a block by block decision basis to determine
which blocks should be extracted, when they should be extracted, and what to do with the blocks
once they are extracted. Researchers have been trying to solve this problem since 1960`s but so
far, it has not been possible to find an optimal integer solution when it is modeled with mining and
mill capacity, blending and stockpile constraints. Since the size of a mining problem makes the
exact integer programming solution techniques inapplicable, many attempts have been made to
obtain a solution with numerous techniques such as pushback designs, heuristic and aggregation
methods which cannot be proven to converge to the true optimal solution. Also, the methods that
can find the closest integer optimal solutions to the models with a block by block basis can only
solve the problems constrained with upper and lower bound mining capacity constraints. Linear
programming (LP) relaxation of real sized mine planning problems can be solved to a proven
optimality by applying existing decomposition algorithms. However, the mining decisions
obtained from the LP relaxation are usually fractional and cannot be implemented practically. A
new integer solution algorithm will be introduced in this dissertation that can solve mine
production scheduling problems modeled with multi capacities, grade blending, grade uncertainty,
stockpiles, variable pit slopes and multi destinations. An integer solution to this complex
scheduling problem has never been achieved or at least never been reported in the literature.
1.1 Need for A Mine Production Scheduling Optimization
Mine production can be defined as a complex system formed by its elements which are the
operational mining system and the blocks to be mined and functions with a goal of maximizing
the discounted cash flows of the mining operations. The boundary within which the system
functions is expressed by an ultimate pit which contains all the blocks that are profitable to be
mined considering only pit slope constraints. This assumes all the resource capacities are
unlimited, the mills are free from any kind of blending requirements and the time value of money
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is ignored. The actual mine production is free of such unrealistic assumptions, therefore
understanding the interactions between the mining blocks within this system becomes significant
to achieve the objective of maximizing the profit. The factors that delineate these interactions
between the blocks and the operational mining system can be commodity price, mining cost,
processing cost, grade of the blocks, mineral resource classification categories, cycle times,
material type, tonnages mined and processed in a year, total available truck hours, certain blending
requirements depending on the material type, multiple processing destinations with different ore
recoveries, stockpiles, pit slope angles based on the rock formations, cutoff grades and time value
of money. Traditionally, some assumptions are made to reduce the complexity of the system.
One of the common assumptions is, designating the blocks to the most profitable
destination which neglects the factors influencing the block interactions. The decision based on
the value at the destination will underestimate the potential profit. It can be illustrated as follows.
Let us assume there are two possible processing destinations for an ore block, and that only one
more ore block can be sent to a destination where the highest recovery can be achieved. If there
are two ore blocks where the lower grade ore block is overlaying the higher-grade ore block,
sending the lower grade ore to a destination where less ounces will be recovered will create an
opportunity for a high-grade ore to be processed at a destination where more ounces could be
recovered. Another common assumption is made based on cycle times. If there are multiple waste
dumps to send a waste block, the operator makes a choice based on the lowest cycle time. The fact
is waste dumps have limited capacities, therefore if a waste block is required to be mined, the
optimizer should be able to send it to an available waste dump at the time of extraction. Otherwise,
the plan generated by an optimizer will be biased on the preselected destinations which will again
underestimate the potential profit.
These kinds of operational decisions always neglect the factors that play a role on the block
interactions which may end up costing millions of dollars. The complexity of the mine production
system also pushed the operators to come up with plans by aggregating the blocks into benches
and scheduling these benches within the pushbacks where the block level resolution is diminished.
But this approach will inherit all the disadvantages of the pushback design. Furthermore,
homogenizing the blocks on a single bench will lead to a loss of information on a block level.
Although some bench aggregation techniques may allow individual blocks to be mined from a
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given bench, restrictions on mining a block before mining the whole bench above reduces the
operational flexibility to meet with the blending requirements as well as other productional
requirements. All these assumptions are done with a goal of quickly generating mine production
plans by ignoring the critical relationship between the mentioned factors and the blocks; which
results with poorly developed mine plans that does not meet with the production and processing
requirements and may also end up with low profit margins. The only way to generate a mine
production plan that can incorporate all the factors characterizing the interactions between the
blocks simultaneously is mathematically modeling the mine production system as an optimization
problem shown by Johnson in 1968 and solving for an optimal integer solution. So far, nobody
was able to provide an optimal integer solution to a mine production scheduling problem on a
block by block level constrained by production and processing capacities, grade blending
constraints, risk blending constraints, multi destinations, variable pit slope angles and stockpiles.
1.2 Scope of Work
The linear programming (LP) relaxation of a mine production scheduling problem can be
solved fast to a mathematically proven optimality by decomposition algorithms. In this
dissertation, the Bienstock-Zuckerberg (BZ) decomposition algorithm is implemented to find the
optimal LP solution. Since the BZ algorithm generates fractional block mining decisions, the
results cannot be interpreted by any practical means, however the objective function value will
always remain as an upper bound to the mine production scheduling problem. If the optimal integer
solution is not known, the success of an integer feasible solution can always be determined by
solving the LP relaxation of the same problem and measuring the gap from the LP optimal solution.
The proposed integer solution algorithm will benefit from the mathematical theories behind
the decomposition algorithms to generate the closest optimal integer solution to the LP optimal
solution. Instead of using a linear combination of the extreme solution vectors obtained from the
consecutive sub-problem solutions in a decomposition algorithm, the proposed solution algorithm
will replace them with the convex combination of the integer feasible solution vectors to the
original problem that will narrow the span of a solution space from the sub-problem solution space
to the original problem solution space where the optimal integer solution exists. The proposed
solution algorithm should be able to provide an optimal integer solution very close to the LP
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CHAPTER 2.
LITERATURE REVIEW
In this chapter, an extensive literature review of the mathematical modeling methods
together with the solution techniques to the open pit mine production scheduling problems that
have been under investigation since the 1960s will be presented. The solution techniques proposed
by the researchers are presented under categories characterized by the variable types such as linear
programming, mixed integer linear programming and integer programming solution techniques.
The proposed solution techniques will be critiqued within each category. The reason for collecting
the assessment of the techniques under a separate section is to avoid repeating similar evaluations
since most of the techniques share resembling pitfalls. It is also worth mentioning that none of the
techniques proposed by the researchers were able to provide an optimal integer solution to the
mine production scheduling problems. This is the leading motivation behind the work in this thesis.
2.1 Ultimate Pit Limit Problem
Traditionally, the ultimate pit limit problem only investigates the most profitable blocks to
be mined as if all the resource capacities were unlimited and the mills were free from any kind of
blending requirements. In other words, the value of the pit is maximized subject only to the
extraction sequence of the blocks governed by the pit slopes. Since all the resource capacities are
assumed to be unlimited and the mills are assumed to have no blending requirements, the need for
a scheduling vanishes as well as the impact of time value of money on block extraction. It is usually
accepted as the maximum limits of mining or as the ultimate shape of the pit at the end of mine
life. There are various heuristic and exact algorithms that can solve the “ultimate pit problem” in
a reasonable amount of time.
The moving cone algorithm of Pana (1965) was a widely accepted and implemented
heuristic algorithm due to its fast solution time. Dagdelen (1985) shows that the algorithm
generates cones with a vertex positioned on a positive block and moving from one positive block
to another. The side walls of the cone obey the pit slope and the profitable cones are selected for
mining. Gauthier and Gray (1971), Barnes (1982) and Underwood (1996) mention that the moving
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cone method has shortcomings since the cones are only targeting a single positive block and
ignoring the combination of the other cones. Hence, if a single cone is not profitable, it will not be
mined although it would generate profit once combined with the other cones. Since the moving
cone method cannot promise the optimal answer, it falls into the heuristic category.
In 1965, an algorithm based on graph theory concepts was introduced by Lerchs and
Grossman which is currently known as the LG algorithm. The pit is initially represented with a
graph by converting each block to a node with a node mass equal to the block value and connecting
all the positive nodes (ore blocks) and the negative nodes (waste blocks) to the root node with the
arcs. In other words, an initial spanning tree is formed. Spanning trees will be partitioned into
strong and weak node groups. If the group of nodes represented with an arc coming from a root
node have a cumulative node value or mass that is positive, the group of nodes is called strong
otherwise they are called weak nodes. If the weak group of nodes overlies a strong group of nodes,
they are combined and reclassified into strong and weak group of nodes based on their cumulative
mass. The collection of strong nodes forms the maximum closure which indeed represents the set
of blocks that maximizes the profit and defines the ultimate pit limit.
Johnson (1968) is the first to show that when the ultimate pit limit problem is formulated
as a LP, the dual of the LP model can be transformed to a max flow network model. The author
also shows that the LP representation of an ultimate pit limit problem has a totally unimodular
structure where the entries of the coefficient matrix are integral, and the determinant of the
coefficient matrix is either 1 or -1, hence the problem can be converted to a bipartite graph. The
author became a pioneer in the field by showing that an ultimate pit limit problem is a maximum
flow problem which can be solved by any algorithm that can solve max flow network models.
Zhao and Kim (1992) modified the LG algorithm and extended its application to the large
block models. The pit is represented with a directed graph where the arcs are formed between the
positive (ore) and negative (waste) nodes (blocks) to imply the sequencing relationships. The
positive nodes are connected to negative nodes which is called as full support. Also, if an ore block
cannot support the waste block, then an arc is directed from waste block to the ore block called as
partial support. The value of the root node which is at the starting point of the last generated arc is
checked to determine if the tree should be categorized as strong or weak. If a weak group of nodes
overlies a strong group of nodes and only non-root nodes of the strong and weak trees have a
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precedence relation, the root node of strong tree is shifted to the non-root node of the strong tree
and the root node of weak tree is shifted to the non-root node of the weak tree since the arcs can
only be drawn between the root nodes. This approach solves the jointly support and reallocation
problems. Also, Zhao (1992) explains in his thesis that the proposed approach avoids the
normalization step of the LG algorithm which will significantly reduce the computation time.
However, a direct comparison with the original LG algorithm was not provided.
Yegulalp and Aries (1992) applied the excess-scaling algorithm of Ahuja and Orlin (1989)
to solve the ultimate pit limit problem as a max flow problem. However, the authors showed that
the LG algorithm implemented by Whittle Programming Pty. Ltd solves the ultimate pit limit
problem faster than their implementation of the excess-scaling algorithm.
Underwood and Tolwinski (1996) interpreted the graph theoretic methodology of LG
algorithm from a mathematic programming point of view by examining the similarity of the steps
of LG algorithm with the steps of dual simplex algorithm. The authors show that the dual simplex
algorithm maintains the same logic as the LG algorithm by removing the strong nodes and leaving
the weak nodes at every stage and once the optimality is achieved, the strong nodes will represent
the ultimate pit limit solution. The authors claim that the only difference between the two
algorithms is that the dual simplex algorithm updates the value of the arc between strong and weak
node by maximizing its value while maintaining the dual feasibility which improves the computing
performance. The authors show that dual simplex algorithm solves the ultimate pit problem a few
minutes faster than the LG algorithm for the cases they tested with up to 2.5 million blocks.
Hochbaum (2008) introduced the pseudoflow algorithm which is the fastest known
algorithm that can solve the ultimate pit limit problem or multi period sequencing problem as a
maximum flow or minimum s-t cut problem. The algorithm may start with a simple initialization
by saturating the sink adjacent and the source adjacent arcs and keeping the rest of the arcs with
zero flow (Chandran and Hochbaum, 2009). Hence, excesses are created at the source adjacent
nodes and deficits are created at the sink adjacent nodes. The method tries to reroute flow through
all the arcs going from a subset of nodes that has excess to a subset of nodes that has a deficit, so
that in the end a provable minimum cut in the graph will be achieved (Hochbaum, 2008).
According to the computational study of the pseudoflow and push-relabel algorithms for the
maximum flow problem conducted by Chandran and Hochbaum in 2009, the pseudoflow
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algorithm is faster than the push-relabel algorithm which was widely accepted as the fastest
algorithm to solve the maximum flow problem at that time. It has been also shown that the
pseudoflow algorithm solving the max flow formulation of the ultimate pit limit problem is faster
than the Lerchs and Grossman algorithm.
2.2 Pushback Design
The pushback design method determining a production schedule to an open pit mine should
start with determining the ultimate pit limits. Once the ultimate pit limit is determined, mine
planning continues with the goal of finding the optimal extraction sequence of the blocks, which
in the end, results in incremental pits called as pushbacks. The need for dividing the pit into sub
pits arises due to the scale of a block by block production scheduling problem hence, sub-pits or
pushbacks are used to schedule the blocks quarterly or yearly. Many heuristic techniques were
developed in order to sequence the pushbacks so that the schedules from the pushbacks will be in
compliance with the resource capacity and mill blending requirements.
Dagdelen and Francois-Bongarcon (1982) determined a series of pushbacks by varying the
price of the commodity, cutoff grade, mining or processing costs.
Gershon (1987) proposed an algorithm which generates cones with the shape of a pit
expanding towards the bottom of the pit with the vertex positioned on each block. The total quality
of the ore within the cone determines the positional weight of the block. The blocks are ranked
based on their positional weight and the highest rank block is scheduled first. Then, the positional
weights of the blocks are re-initialized with the remaining blocks. This allows scheduler to reach
the high-grade ore at the bottom of the pit quicker than the traditional approaches.
Whittle`s approach (1988) is a pit parametrization technique which varies the block values
incrementally and generates nested pits by implementing the LG algorithm.
Seymour (1995) proposed a parameterization algorithm that abandons the traditional pit
parametrization techniques which generate pits as a function of pit value. Instead, the method
selects both the pit volume and pit value as the parameters of this function. The algorithm is a
modified version of the LG algorithm by adding the parametrized variables and allowing to form
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sub trees that represent the small pits (Meagher et al., 2014). The branches are categorized as
strong or weak based on the threshold value of their strength which is calculated by dividing the
cumulative value of the nodes in the branch by the cumulative mass of the nodes in the branch.
Strong branches form the sub tress (small pits) which are sequenced based on their strength value.
Ramazan and Dagdelen (1998) developed a minimum stripping ratio pushback design
algorithm which is a modified version of the Seymour`s algorithm in 1995. The authors apply
break-even cutoff grade to differentiate ore and waste with an indicator value “1” assigned to ore
and indicator value “0“ assigned to waste. The authors replaced the cumulative value approach in
Seymour`s algorithm with the cumulative indicator value approach to calculate the strength of the
branch. The goal of the algorithm is to find pits with the minimum stripping ratio which will lead
to a schedule where the ore blocks are mined as soon as possible. Authors also demonstrated that
Whittle`s pit parametrization method may produce the same nested pits as the minimum stripping
ratio pushback design algorithm if all the ore blocks are assigned the value of the highest-grade
ore in the block model.
Somrit (2011) introduced a phase design algorithm which uses Lagrange parameters to
determine the size of the pits in compliance with the annual resource capacity requirements. The
usage of Lagrange parameters is similar to the Whittle`s pit parametrization technique. Although
the relationship between the Lagrange parameters and pit size is not linear, the author uses linear
interpolation to determine the parameters since it is impossible to predict the relationship with an
equation. This kind of approach may result in gap problems. The pits are determined in a reverse
order compared to the commonly adopted approaches. The algorithm first tries to find a pit that
meets the resource requirements of “t-1” periods. The unmined portion of the pit becomes a phase
for the period “t”. Then, the algorithm looks for the next pit that obeys the resource requirements
of “t-2” periods. The remaining blocks create the next phase for the period “t-1”. Hence, the
algorithm proceeds backwards in time till it determines the phase corresponding to the first period.
2.2.1 Shortcomings of the Pushback Design Methods
Due to the large scale of block by block mine production scheduling problems, the
pushback concept became attractive since it allows scheduling to be done from the sub-pits or
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nested pits which results in less variables in the optimization model. Dagdelen and Francois-
Bongarcon (1982), Gershon (1986), Whittle (1988), Seymour (1995), Ramazan and Dagdelen
(1998), Somrit (2011) proposed various techniques to obtain the pushbacks. The methods follow
the incremental fashion of obtaining pushbacks that fail to incorporate the blending requirements
as well as the uncertainty of the grade of ore. Moreover, the size of the pit has a non-linear
relationship with the value of the pit which makes it extremely hard to determine. But the common
approach attempts to find the pit size with an assumption of a linear relationship which in the end
results with a gap problem. The resulting gap problem will prevent pushbacks to meet with the
capacity requirements. So far, researchers have not found an approach to overcome the stated
problems which will in the end produce poorly designed pushbacks. Since many production
schedule plans highly depend on the design of pushbacks, poor designs will prevent the schedules
from achieving a maximum NPV or even obtaining a feasible solution.
2.3 Mathematical Programming and Integer Solution Techniques for Production
Scheduling
The mine production scheduling problem was originally formulated by Johnson (1968)
with a goal of finding the optimal block schedule that will maximize the NPV throughout the life
of a mine. This generic model incorporates the dynamic cutoff grade strategy by allowing the
blocks to be sent to the most attractive destination determined by the pit slope, capacity limitations
and average grade requirements at each destination and availability of the equipment. Johnson`s
model will generate a higher NPV since the destinations of the scheduled blocks will not be pre-
determined by the mine planner, instead the optimal destinations will be selected by the scheduler
to maximize the NPV. The best schedule would be obtained if the model could be solved with
integer decision variables. The complexity of the problem increases as the number of the blocks,
which are the decision variables in the model increases. A true integer optimal solution of the
proposed model still cannot be determined with the current solution techniques. The past and most
recent research in the area is geared towards developing solution techniques which can find such
feasible solutions that are close to the optimal integer solutions within a reasonable time frame.
Since the optimal integer solution to the proposed model is unknown, the solution techniques
developed cannot identify how far one is from the integer optimal solution, however the quality of
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the solutions are measured from the optimal linear programming solution since it provides an upper
bound (Dagdelen, 1985).
2.3.1 Linear Programming Solution Techniques
Linear Programming (LP) models are often called linear relaxation models, if one decides
to relax the integrality of the decision variables by presenting them as continuous variables in the
model. As mentioned before, the theoretical upper bound provided by the optimal LP solution is a
benchmark to measure the “optimality gap” of the integer solution to the same problem which
allows the evaluation of the success of the integer solutions.
G. Dantzig proposed the simplex algorithm in 1947, which is an exact optimization
technique to solve LP models. The simplex algorithm still remains one of the most popular
algorithm, adapted by commonly used optimization engines such as CPLEX and Gurobi. Once the
LP model is generated, the solution space bounded with the constraints creates a polytope where
the optimal solution may exist on one of its vertices which can be also called extreme points. Given
a mine production scheduling problem with an objective of maximizing NPV, the simplex
algorithm will solve the linear relaxation of the model by starting the search of an optimal solution
on an extreme point and moving to another extreme point until the objective function value cannot
be improved. This kind of a search is not practical for mining problems since the number of
variables and constraints may result with too many extreme points to be searched which results in
inefficient solution times.
Johnson (1965) proposed the Dantzig Wolfe (DW) decomposition algorithm to solve the
LP relaxation of the mine production scheduling model by decomposing the original model into
subproblems. The subproblem can be called a Lagrange relaxation of a model where the resource
capacity and blending constraints are moved to the objective function and penalized with the
associated dual values. The Lagrange relaxation problem is similar to the ultimate pit problem
except the blocks are sequenced in multi time period. Since the subproblem or Lagrange relaxation
problem has the totally unimodular structure, Dagdelen (1985) showed that the multi period
subproblem is a maximum flow network problem which is faster than solving the problem as a LP.
The DW algorithm uses a convex combination of the extreme points of the Lagrange relaxation
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problem to find the optimal solution to the original problem. The variables associated with the
extreme point solution vectors of the subproblem are used to solve the master problem subject to
the constraints of the original problem and a constraint that assures the convex combination of
extreme points. The optimal solution which may be in a fractional form can be used as a guidance
for scheduling the blocks since the optimal LP solution is an upper bound for the optimal integer
programming (IP) solution.
Bienstock and Zuckerberg (2009) improved the DW decomposition algorithm and the BZ
is presently recognized for being the fastest LP solving decomposition algorithm for the
precedence constrained production scheduling problems such as mine production scheduling
problems. The BZ algorithm can solve the LP relaxation of the mine production scheduling
problems constrained with upper and lower bound resource and blending constraints to a proven
optimality. The algorithm decomposes the original problem into a subproblem and master
problem. Each subproblem is solved likewise in the DW algorithm except the extreme point
solution vectors of the subproblem are orthogonalized in the BZ algorithm which will increase the
dimension of the solution space by adding more axes to the system which eventually reduces the
number of vectors linearly combined to represent the optimal solution in the master (original)
problem. In other words, since there is a variable associated with each one of the vectors generated
from the subproblem solution, a decrease in the number of vectors will result with less variables
to solve in the master problem. Detailed investigation of the BZ algorithm is presented in
Chapter 4.
Van Dunem (2016) integrated the uncertainty concept in the form of risk constraints to the
mine production scheduling problem. The author developed a model which allows the user to
reflect his/her risk tolerance within the production scheduling problem. The model consists of mill
capacity, mill blending and risk constraints where the mill blending, and risk constraints have both
minimum and maximum requirements. The resource modeling is done in order to estimate the
grade of a block with an associated level of uncertainty. Mineral resources are classified into the
categories as inferred, indicated and measured and defined as follows (CIM, 2006):
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(i) An Inferred mineral resource is that part of a Mineral Resource for which quantity and grade or
quality are estimated on the basis of limited geological evidence and sampling. Geological
evidence is sufficient to imply but not verify geological and grade or quality continuity.
(ii) An Indicated mineral resource is that part of a mineral resource for which quantity, grade or
quality, densities, shape and physical characteristics are estimated with sufficient confidence to
allow the application of modifying factors in sufficient detail to support mine planning and
evaluation of the economic viability of the deposit.
(iii) A Measured mineral resource is that part of a mineral resource for which quantity, grade or
quality, densities, shape, and physical characteristics are estimated with confidence sufficient to
allow the application of modifying factors to support detailed mine planning and final evaluation
of the economic viability of the deposit.
The risk constraints are developed in order to control the uncertainty by applying an upper bound
on the number of the inferred blocks sent to the mill and lower bounds on the amount of indicated
and measured blocks. With the incorporation of the risk constraints the author aims to minimize
the impact of uncertainty on meeting the operational or production targets. Traditional stochastic
production scheduling problems incorporate the grade uncertainty by varying the grade of the ore
blocks from one scenario to another which eventually increases the number variables with the
magnitude of the number of scenarios and results in a NP-hard problem with a huge variable space.
In contrary, the author proposed a novel approach by integrating the grade simulations once the
production scheduling is completed. In other words, the initial schedule is generated in order to
meet with the risk tolerance of the manager. Then, the simulated grades are integrated to check
whether the given schedule will meet the blending requirements. For example, if 9 out of 10
integrated simulations are within -15% of the lower bound and +15% of the upper bound blending
requirements, then the optimal production schedule is obtained under the grade uncertainty. If the
integrated simulations fail to meet with the blending requirements, the manager can lower the
number of the inferred blocks requested by the mill or increase the number of measured blocks
which will potentially lower the grade variability at the mill. Also, the management can drill more
drill holes to increase the level of confidence within the inferred group of blocks. The NPV
difference between the riskiest scenario and the considerably less risky scenario can give a hint to
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the management about the expected NPV that can be generated if more drilling is conducted on
the inferred group of blocks. Although modeling the production scheduling problem as a
deterministic model reduces the number variables significantly, the problem size is still
considerably large to solve as an integer programming model. Moreover, a solution for the LP
relaxation of the model might not be obtained by using CPLEX. Thus, the author proposed a
decomposition method developed by Bienstock and Zuckerberg in 2009 to solve the LP relaxation
of the model. The integration of the Pseudoflow algorithm to the BZ decomposition algorithm
made the LP relaxation of the large-scale open pit production scheduling problem under the metal
uncertainty solvable with fast solution times.
2.3.2 Mixed Integer Linear Programming Solution Techniques
Mixed integer linear programming techniques gained importance due to the size of the
mine production scheduling problems. Since the optimal integer solution of the problem has not
been achieved, a lot of research has moved towards solving the mining problems with the mix of
continuous and integer variables.
Gershon (1983) modified the mine production scheduling model proposed by Johnson
(1968) by adding continuous variables that will mine the blocks partially if all the preceding blocks
have been removed. The author claims that forcing the integrality of the decision variables will
not reflect the practical approach, hence by allowing partial mining, the schedules can meet with
the blending constraints. Osanloo et al. (2008) mentioned that Gershon`s approach will not be able
to handle large problems due to the large number of the binary variables and also since the size of
the problem is the main concern, dynamic cutoff grade strategy cannot be implemented.
Ramazan and Dimitrakopoulus (2004) improved Gershon`s (1983) model by setting the
ore blocks as binary variables and leaving the waste blocks as continuous variables. The decision
to partially mine the waste blocks still exist in the model. The authors claim that since the ore
blocks must be mined fully, the waste blocks will also be mined fully in order to satisfy the
sequencing requirements. Partially mined blocks may exist in the last period. The case study lead
to a conclusion that the number of binary variables is significantly decreased as well as the solution
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time by setting only the ore blocks as binary and also the partially mined blocks can be minimized
if the reserve constraints are equalities.
Kawahata (2006) took a unique approach to solve the MILP problems. The author divided
the pit into series of pushbacks and divided the pushbacks into increments (benches). The blocks
within the increments aggregated based on the similarities of the grades of the blocks and rock
properties. The original MILP formulation consists of binary variables that preserve the
sequencing between the pushbacks and increments and continuous variables which allow the
scheduler to pick the destination for the portion of the increment mined under dynamic cutoff grade
policy. The author reduces the solution space that the MILP model works within by splitting the
original problem into two Lagrange relaxation problems where the aggressive case relaxes the
process capacity constraints while keeping the mining capacity constraints and the conservative
case relaxes the mining capacity constraints while maintaining the process capacity constraints in
the model. Since the Lagrange relaxation model consists of only small number of pushback
variables achieved by aggregating the benches, the binary nature of the variables does not pose
any computational disadvantage. Since the solution of the subproblems will provide an upper and
lower bound to the original MILP problem, the variables which are not considered in the solution
set of the subproblems can be eliminated from the variable set of the MILP problem. This unique
approach will decrease the solution time of the MILP problem significantly.
Boland et al. (2008) formulated the MILP problem with the same aggregation technique
presented by Kawahata (2006). The author used binary variables to sequence the aggregated
blocks, however the model has two continuous variables, one for the fraction of the aggregate
excavated and the other variable represents the fraction of the aggregate processed. The author
claims that the blocks in any aggregate can have multiple attributes based on the different metal
content. The model does not have any blending constraints and the blocks in the aggregates are
already classified as ore and waste with a predetermined cutoff grade.
Goycoolea et al. (2015) proposed a MILP model which works with pre-defined pushbacks.
As in Kawahata`s approach (2006), the author aggregated the blocks within each pushback into
benches (increments). The model considers upper bound capacity on both production and
processing constraints. Also, a dynamic cutoff grade strategy was incorporated by allowing the
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scheduler to choose a destination for the mined blocks. Continuous variables were used to mine
the increments and the blocks from each increment partially. The author added additional volume-
flow constraints to limit the variance on the production from one period to another. Also, instead
of using binary variables to sequence the increments, the author achieved the same outcome with
a constraint that required all predecessor increments of any increment which is partially mined to
be mined completely.
King (2016) formulated the open pit to underground transition model with pre-determined
block destinations by varying crown and sill pillar locations. The author took a phase design
approach with the benches binned into blocks characterized by grades. The author combined the
surface mine production scheduling model with the underground mine production scheduling
model and called it an enhanced transition model. Binary variables were used to schedule the
underground activities such as extraction, backfilling and development as well as the sequencing
relationship between the benches for the surface mining. Also, continuous variables were used to
partially mine the benches and partially send the blocks to the mill. However, the partially bench
mining constraints were written in such a fashion that once the bench is mined it should be mined
completely so that the sequencing can take place. The author also allowed the stockpiles in the
model, however the stockpile rehandling costs are not considered. In order to solve the model, the
author used the BZ algorithm to solve the LP relaxation of the model and then a rounding heuristic
method is applied in order to get an integer solution. The author solved the model many times by
changing the locations of the sill and crown pillars in order to find the best location that will
maximize the NPV.
2.3.2.1 Shortcomings of the Mixed Integer Linear Programming Solution
Techniques
Mixed integer linear programming (MILP) techniques gained importance since the optimal
integer solution of the mine production scheduling problem has not been achieved due to the size
of the problems. Gershon (1983), Ramazan and Dimitrikapolus (2004), Kawahata (2006), Boland
(2008), Goycoolea et al. (2015), King (2016) solved the mine production scheduling problems
with the various forms of MILP techniques. Kawahata (2006) and Boland et al. (2008) further
aggregated the blocks into the benches in their MILP models in order to reduce the number of
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variables. MILP models are solved for binary variables and continuous variables. Most of the
authors use binary variables for the sequencing relations between the benches or blocks, and
continuous variables to partially mine the benches or the waste blocks. The main disadvantage of
aggregation techniques is the loss of resolution. In other words, once the blocks are aggregated
into benches, individual block grades are replaced with the average grade of the blocks like if the
block grades on a bench are distributed homogeneously. This assumption might generate schedules
which can easily mislead the operations in terms of meeting blending requirements. However, the
MILP model proposed by Goycoolea et al. (2015) may overcome this issue since the model allows
blocks to be scheduled individually from the benches but the sequencing relations are still
preserved between the benches which will create a dependency between the block and the bench
above. The plans generated with this approach might lead the scheduler to mine blocks which do
not have any spatial correlation with blocks on the next level. This might lead to extra stripping
for a particular period in certain situations. For the MILP models where the scheduling is done
block by block and no bench aggregation is considered, the number of binary variables may be too
large which will prevent the solver to find an optimal solution in a reasonable amount of time (Less
than 8 hours). Also, there is no real mining case study presented in the literature which can be
solved by modeling as MILP that takes into account mining and processing capacity and blending
requirements with scheduling on a block by block basis.
2.3.3 Integer Solution Techniques
True open pit mine planning schedules can only be achieved if one can solve Johnson`s
(1968) production scheduling model with integer variables.
Dagdelen (1985) was the first to show that the Lagrange relaxation technique can be
applied to solve Johnson`s model to generate integer solutions feasible to the original problem.
The Lagrange relaxation model has the same objective function as the original model with an
addition of side constraints penalized with the Lagrange multipliers and the model is subject to the
sequencing constraints. Hence, the Lagrange relaxation problem solution always provides an upper
bound to the optimal solution of the original problem. The goal is to find the best multipliers that
will generate solutions as close as possible to the original problem. Dagdelen proposed the
subgradient optimization method to generate Lagrange multipliers. If one cannot find such
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Lagrange multipliers that will produce feasible solution, to the original problem, it can be
concluded that the condition of gap exists (Everett, 1963). Dagdelen identified the totally
unimodular structure of the Lagrange relaxation problem and represented the multi-time period
sequencing problem on a network structure. The author solved the multi-time period sequencing
problem as a combination of a single time period maximum flow network problem.
Tachefine and Sumois (1996) also applied the Lagrange relaxation technique to solve the
open pit mine production scheduling problem which does not have any blending constraints. The
author uses the Bundle method to obtain the Lagrange multipliers. In order to make the solution
of the Lagrange relaxation feasible to the original problem, the author proposed a greedy heuristic
algorithm. The author solves the Lagrange relaxation problem as a maximum closure graph
problem. The greedy algorithm tries to find a set of frontier nodes which are candidates for
elimination. The nodes are selected for elimination in such a fashion that the violation of the
capacity constraints is minimized while keeping the impact on the value of the closure minimum.
Akaike (1999) extended Dagdelen`s Lagrange relaxation approach with the 4D network
relaxation method. The proposed method solves a Lagrange relaxation problem as a multi time
period maximum flow problem. Moreover, the method can handle the cutoff grade strategy and
the stockpiles can be incorporated in the model. Since the method tries to find the best Lagrange
multipliers with the subgradient method, the gap problem is inevitable if one exists.
Ramazan (2001) proposed the fundamental tree algorithm to solve the problem within the
pushbacks. The author removes the side constraints of the original model and represents the
problem as a network problem. The network model is formulated as a LP problem. At each
iteration, the arcs that have 0 flow are eliminated and a new tree(s) is formed. The iterations
terminate if the number of trees at the current iteration is equal to the previous iteration or if all
the trees have only 1 positive node. Each fundamental tree can be treated as aggregated blocks that
contain certain ore tonnage, waste tonnage and grade. Once the fundamental trees are obtained,
the production scheduling model for the pushback can be solved as an IP problem with the resource
capacity and blending constraints with a binary variable assigned for each fundamental tree.
Although the number of the binary variables in the model can be reduced significantly, large
deposits can still have large number of fundamental trees that will make model inefficient.
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Amaya et al. (2009) implemented a local search heuristic method to improve the initial
integer feasible solution of a production scheduling model constrained with an upper bound on
mining capacity constraint. Given an initial integer feasible solution, the method iteratively fixes
parts of the schedule and the remaining unfixed parts are solved within a binary context to find out
if a local improvement exists. If the solution provides an improvement, then the initial solution
space is updated. The author presented three strategies to search for an improvement. The cone
above strategy is used by first picking a block randomly from the set of blocks which are not
considered in the initial integer feasible solution set. Then, the target block and its predecessors
are solved as an integer while the rest of the initial feasible solution space is fixed. The periods
strategy can be applied by randomly selecting two-time intervals from a given integer feasible
solution space and solved for the best solution. The transversal strategy is applied by first defining
a distance and height and then searching for the blocks that are not considered in the initial integer
feasible solution.
Cullenbine (2011) proposed a sliding time window heuristic algorithm to approximately
solve the production scheduling problems with the minimum and maximum resource capacity
constraints. The algorithm initially fixes the initial integer feasible solutions for the T1 periods.
Then, the exact window T2 is determined where the integer solution will be obtained. Then, for
the remaining periods T3, Lagrange relaxation of the model is solved by penalizing the minimum
and the maximum resource capacity constraints with the duals obtained from the optimal solution
of the LP relaxation of the original IP model.
Chicoisne (2012) proposed a toposort heuristic algorithm which uses the optimal LP
solution as a guide to round the fractional solutions to an integer result. Expected extraction time
of a block is determined by multiplying the proportion of the block mined with its extraction
period. Once the integer feasible solution is found, Amaya`s local heuristic technique is applied in
order to improve the initial integer feasible solution. The algorithm can only be applied for models
with single or two capacity constraints in an upper bound form.
Lamghari and Dimitrakopoulos (2012) proposed the Tabu search meta-heuristic method
integrated with two diversification strategies that are long term memory search and variable
neighborhood search, to improve the feasible solution domain of the stochastic mine production
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scheduling problem. The authors modeled the two-stage stochastic problem by penalizing the
upper and lower bound mining capacity constraints in the objective function while preserving the
scenario dependent mill capacity and metal content constraints. In order to solve the modified
model, the initial integer feasible solution is obtained by randomly selecting the blocks from
a mineable set of blocks meaning that the blocks should not have any predecessors, or the
(cid:4666)(cid:1876)(cid:3036)(cid:4667)
predecessors are already mined. The random selection continues until the amount of the chosen
blocks for each period will be greater than or equal to the average of the upper and lower bounds
for the mining capacity constraint. The solution honors the non-stochastic constraints such as
mining capacity, reserve and sequencing constraints. The tabu search procedure begins once the
initial solution is obtained. For each block in the solution set, schedule times of the closest
successor and predecessor blocks are identified. Then, the latest schedule time of the predecessor
and the earliest schedule time of the successor is selected. For any block in , the
blocks can be shifted from one period to another between the time intervals of and at
(cid:1857)(cid:4666)(cid:1876)(cid:3036)(cid:4667) (cid:1864)(cid:4666)(cid:1876)(cid:3036)(cid:4667) (cid:1861) (cid:1876)(cid:3036)
each iteration. The block which is scheduled minimum amount of times is selected as a candidate
(cid:1857)(cid:4666)(cid:1876)(cid:3036)(cid:4667) (cid:1864)(cid:4666)(cid:1876)(cid:3036)(cid:4667)
in order to improve the searching activity in the less explored areas. If the solutions from the
consecutive iterations stop improving for a certain amount of pre-determined time, then the Tabu
search terminates. Then, the author applies diversification strategies to the Tabu search solution
space in order to explore the areas which are rarely considered. The first strategy is named as a
long-term memory diversification strategy. Since each block may be scheduled multiple times
during the Tabu search, the time period that a block has been scheduled for less frequently is stored
in a set. Then, a block randomly drawn from the set is scheduled to that time period. The second
proposed diversification strategy is the variable neighborhood diversification strategy. Given the
solution set obtained by Tabu search, a block which is scheduled the least number of times is
selected from the set and rescheduled to a time period between and which gives the
best NPV. The iterations terminate if the Tabu search solution cannot be improved or whenever
(cid:1857)(cid:4666)(cid:1876)(cid:3036)(cid:4667) (cid:1864)(cid:4666)(cid:1876)(cid:3036)(cid:4667)
the pre-defined number of iterations is reached.
Lambert (2012) proposed the Tailored Lagrange relaxation algorithm to solve the open pit
mine production scheduling problems with minimum and maximum resource capacity constraints.
The author proposed a maximum value feasible pit (MVFP) algorithm which will generate initial
integer feasible solutions to the Tailored Lagrange relaxation model. The MVFP is processed in 3
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phases. In Phase 1, a pit parametrization technique is used in order to find ore blocks that will
satisfy total processing requirements over the entire time horizon. If the Phase 1 cannot find any
pit that satisfies both the minimum and maximum ore production requirements over the time
horizon, the well-known gap problem exists. In Phase 2, if the minimum production requirements
are not satisfied, an integer program is solved to expand the pit. For the phase 3, an ultimate pit
limit approach can be conducted or sliding time window algorithm can be applied to generate an
initial integer feasible solution. Once the three phases are completed, the information collected
during the MVFP stage will determine the dualization strategy of the Tailored Lagrange relaxation
model.
Lamghari et al. (2014) proposed a method to generate an initial integer feasible solution
under mining and mill capacity constraints with an upper bound and to improve the generated
initial integer feasible solution. The author presented an exact approach and greedy heuristics to
get an initial integer feasible solution. For the exact approach, the production scheduling model
with mining and mill capacity constraints is solved within a binary context per period as a single
time period problem. The author also benefits from the variable elimination techniques by pre-
checking the mining capacity violations and generating a valid inequality to strengthen the
formulation. For the sequential greedy heuristics, a cone is generated with a vertex on each block
with a purpose of finding the set of blocks which will minimize the deviations from the mining
and mill capacity requirements while maximizing the NPV. One can make a similar analogy by
applying the floating cone algorithm to a multi time production scheduling problem to generate
initial integer feasible solutions. Then, the author implemented the variable neighborhood descent
(VND) search method of Hansen and Maladenovic (2001) to improve the initial integer feasible
solutions. The algorithm works in three steps. Given the initial integer feasible solution, the
algorithm first trades the ore and waste blocks between the consecutive time periods. Then, the
blocks together with the predecessors are shifted forward and backward in time until no more NPV
improvement is achieved.
Lamghari and Dimitrakopoulos (2015) proposed three heuristic algorithms, one for
generating an initial integer feasible solution called look-ahead heuristic (LAH) and the others for
improving the generated integer feasible solution to maximize the NPV of the two-stage stochastic
mine production scheduling problem named a network-flow based heuristic (NF) and a diversified
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local search heuristic (DLS). In order to apply the LAH, the multi-time period stochastic model is
re-written as a single time period stochastic model. LAH is an improved version of the greedy
heuristics presented in Lamghari (2014). LAH generates cones only with the blocks if the objective
function value can be improved while obeying the mining constraints. The iterations terminate
when there is no such cone that can improve the latest objective function value. Once the initial
integer feasible solution is obtained the author applied the NF heuristic to improve the generated
solution. The idea of the NF heuristic is to search the initially generated solution space extensively
by shifting the blocks from one period to another in order to delay and advance the extraction of
the blocks with a purpose of improving the NPV. Also, the NF heuristic has an advantage over the
variable neighborhood descent (VND) heuristic introduced by Lamghari (2014) by allowing a
search in the solution space instead of just considering the feasible solution space. Since the
neighborhoods generated with the forward and backward processes of the NF are large compared
to the Tabu Search (TS) presented by Lamghari (2012) and VND neighborhoods, the author
converted the problem to a network model and solved the longest path problem (LPP) to find the
solution that gives the best improvement in NPV. The author proposed the pulling algorithm of
Ahuja (1993) to solve the longest path problem as a network problem. If the solution obtained is
not better than the previous one, then the algorithm terminates. Secondly, the author also proposed
diversified the local search heuristic (DLS) to improve the initial integer feasible solution. The
heuristic DLS is basically the combination of the VND and NF heuristics. Instead of applying the
NF heuristic directly to the initial integer feasible solution obtained by LAH, the author first applies
VND to the solution so that the NF heuristic will begin with an improved version of the initial
integer feasible solution which will also avoid the local optima of the VND heuristic. If the
consecutive application of the VND and NF heuristics do not improve the solution, the algorithm
terminates, and the current solution is accepted as the final best solution to the problem.
2.3.3.1 Shortcomings of the Integer Solution Techniques
Mining operations can be truly optimized if an optimal integer solution can be found to the
production scheduling model proposed by Johnson in 1968. The generic model encompasses block
by block scheduling subject to mining and mill capacity and blending constraints while obeying
the pit slope requirements. The existing solution algorithms to solve the integer programming
models to a proven optimality such as branch and bound (A. H. Land and A. G. Doing, 1960) and
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branch and cut (M. Padberg and G.Rinaldi, 1983) are not efficient enough to solve large scale mine
planning production scheduling problems within a binary context.
2.3.3.1.1 Lagrange Relaxation Technique
Researchers investigated new solution techniques to solve the mine production scheduling
problems with the integer variables. Dagdelen (1985), Tachefine and Sumois (1996), Akaike
(1999), Kawahata 2006, Cullenbine (2011), Lambert (2013) implemented various forms of
Lagrange relaxation techniques with a goal of achieving an integer solution. If the optimal solution
of a Lagrange relaxation problem is feasible to the original problem, it provides an upper bound
solution to the original problem. On the other hand, there is no guarantee that a feasible solution
to the original problem by adjusting the Lagrange multipliers can be obtained. As shown by Everett
in 1963, Lagrange multipliers measure the change in the objective function value when the
resource constraints (mining and mill capacity or blending constraints) are expended. Since the
relationship between the resource expenditure and the objective function is non-linear, whenever
the function has a non-concave region, the hyperplane defined by the Lagrange multiplier cannot
be tangent to the accessible points on the region which constitutes the “gap” problem (Everett,
1963). The presence of a gap problem will prevent Lagrange relaxation solution from providing a
feasible solution to the original problem. So far, researchers have not identified a method to
overcome the gap problems. This will make mine production scheduling problems solved with
Lagrange relaxation techniques less attractive since gap problems are not avoidable which appears
to be due to the non-homogeneity of a mineral deposit.
2.3.3.1.2 Fundamental Tree
Ramazan (2001) aggregated the blocks into fundamental trees based on only considering
the values of the blocks and the precedence relationships. Then, the author treated each tree as an
integer variable and solved the mine production scheduling problem under mining and mill
capacity and blending constraints. Since, the complexity of the integer programming models is
directly related with the number of variables, the author first generated the pushbacks and then
scheduled the trees (grouped blocks) from the pushbacks. Unfortunately, since the author uses the
aggregation technique together with pushbacks, the weaknesses of those two approaches presented
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above will result in poor quality solutions. Moreover, for the large scale open pit mine production
scheduling problems, the model will have large number of fundamental trees which will increase
the complexity of the integer programming model which might prevent obtaining an integer
solution.
2.3.3.1.3 Heuristic Techniques
2.3.3.1.3.1 Heuristic Techniques to Obtain Initial Integer Feasible Solution
Since the scale of mine production scheduling problems restrict the usage of exact solution
algorithms, researchers implemented heuristic techniques to find the integer feasible solutions fast
and as close as possible to the optimal LP solution. Chicoisne (2012), Lamghari and
Dimitrakopoulos (2012, 2015), Lambert (2013), Lamghari et al. (2014) proposed methods to
obtain initial integer feasible solutions. Among these methods, Lamghari criticized the random
heuristic (RH) method proposed by Lamghari and Dimitrakopoulos in 2012 for its myopic
approach. Also, the solution times of the improvement heuristics applied to the initial integer
feasible solutions generated by the random heuristics are highest according to the cases presented.
The author also criticized the greedy heuristic proposed by Lamghari et al. (2014) for considering
all the unmined blocks as a candidate for the cones generated and selecting cones that will only
minimize the deviations from the upper bound mining capacity requirements even if the selected
cone does not improve the current solution. The author proposed a look ahead heuristic (LAH)
approach which is an improved version of the greedy heuristic where the candidate blocks for the
cone are selected if it leads to an improvement of the objective function value for the time period
t, while the upper bound mining capacity constraints are not violated. Although the author didn’t
present the actual solution times that it takes to obtain an integer feasible solution by applying RH
and LAH, it is mentioned that LAH is much faster. Also, since the quality of the solutions
generated by the improvement heuristics after obtaining an initial integer feasible solution are
highly dependent on the initial solution, the case studies show that regardless of which
improvement heuristic method is used, the initial integer feasible solutions generated by LAH
always results in a better solution compared to the RH and the combined solution times are faster.
Lambert proposed a maximum valued feasible pit (MVFP) method to generate an initial integer
feasible solution to the model which considers lower and upper bounds on the mining and mill
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capacity constraints. According to the case study presented by the author, it takes 5000 seconds (~
1hr 20 min) for the MVFP method to generate an initial integer feasible solution for the models
with 10000 blocks and a time horizon of 15 years. The solution time is quite high for a fairly small
size model. Chicoisne et al. (2012) proposed a TopoSort heuristic which uses LP relaxation
solutions to assign weights to each block based on the expected mining time and then schedules
the blocks by ranking the weights. The authors implemented the TopoSort heuristic to a case which
has more than 4 million blocks with a time horizon of 15 years. According to the results presented,
the initial feasible solution to the model which has an upper bound on mining and mill capacity
constraints is achieved in less than a second. So far, none of the heuristic methods that generate
initial integer feasible solutions are able to handle the models with blending constraints. Moreover,
MVFP is the only one that can handle lower bound mining and mill capacity constraints but
requires approximately 1h 20 minutes to generate a solution to a model with 10000 blocks which
is too long for such a small size model. Also, the comparison between MVFP, LAH and TopoSort
heuristic methods in terms of the quality of the solutions they generate has not presented in the
literature.
2.3.3.1.3.2 Solution Improvement Heuristic Techniques
An initial integer feasible solution serves as a guide to the improvement heuristic
techniques developed by Amaya et al. (2009), Cullenbine (2011), Lamghari and Dimitrakopoulos
(2012, 2015), Lambert (2013), Lamghari et al. (2014). The case study presented by Amaya et al.
(2009) shows that near optimal solutions were achieved in 4 hours by implementing Local search
heuristic (LS) to the Marvin (53668 blocks), AmaricaMine (19320 blocks), AsiaMine (772800
blocks) datasets where the models have only upper bound mining and mill capacity constraints.
Similar solutions are also presented by Chicoisne et al. (2012), when the authors ran the LS
heuristic in about 4 hours once they obtained an initial integer feasible solution with the TopoSort
heuristic. Moreover, the authors show that once the number of the resource capacity constraints
increase, the optimality gap achieved in 4 hours also increased. The applicability of LS to the real
mine models that include upper and lower bounds on resource and blending constraints still
remains elusive. The Sliding time window heuristic (STWH) proposed by Cullenbine (2011) can
solve the models consisting of upper and lower bounds on the mining and mill capacity constraints.
The results presented by the author shows that it may take up to 2hr 40 min to solve a model with
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25000 blocks and 15 time periods and produce a solution with an optimality gap of 4.3%. This is
so far the best-known solution obtained in the models with lower bounds on the resource
constraints. Moreover, since the approach utilizes Lagrange relaxation techniques, it will be
vulnerable to potential gap problems. The Tailored Lagrange relaxation method (TLR) proposed
by Lambert (2013) can be also applied to the models with upper and lower bounds on resource
constraints. The author applied the TLR to the same case studies presented by Cullenbine in 2011
which makes the comparison of the STWH with TLR easier. Although STWH was able to generate
a solution for the model with 25000 blocks, TLR could not obtain a solution with an optimality
gap less than 6.4% in 10 hours. By looking at the solution times of the remaining models where
the number of blocks vary from 10000 blocks to 18000 blocks, it can be concluded that the STWH
has a better performance than TLR. Like STWH, the TLR method may also suffer from the
weaknesses of the Lagrange relaxation approach since gap problems may exist. Lamghari and
Dimitrakopoulos (2015) presented a case study where the Tabu Search (TS), variable
neighborhood descent (VND), network flow (NF) and diversified local search (DLS) improvement
heuristic methods are compared in terms of the solution times and the optimality gap they produce.
The authors concluded that the TS has the worst performance and its solution time can increase
significantly depending on the quality of the initial integer feasible solution it starts with. Although
VND performs better than NF for the small size models, NF can outperform VND for the relatively
larger size problems but may take a longer time than VND to find a solution. The authors also
concluded that DLS which is the combination of NF and VND methods performs the best in terms
of the quality of the solution it generates and also it does not depend on the quality of the initial
integer feasible solution. These 4 methods are implemented on the stochastic models which consist
of at most 48000 blocks with upper bound mining and mill capacity constraints. A performance
comparison of these methods with the LS, STWH and TLR has not been presented. So far, it can
be concluded that among the existing improvement heuristic methods LS can solve the models that
has at most 4 million blocks in a block model and 15 time periods subject to upper bound mining
and mill capacity constraints within a 4% of optimality gap in 4 hours. Also, STWH remains the
only local improvement heuristic that can solve the model with 25000 blocks and 15 time periods
in 2hr 40min within an optimality gap of 4.3%. If one considers solving a large case mine
production scheduling problem that has 5 to 10 million blocks, multiple destinations subject to
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mining and mill capacity, blending and varying slope constraints within a binary context, the
currently developed heuristic algorithms will not be able to produce an integer optimal solution.
2.4 Summary of the Current State of Open Pit Mine Production Scheduling
The researchers have been trying to solve this problem since 1960`s unfortunately, the
proposed solution techniques are not capable of providing an optimal integer solution to the mine
production scheduling problems modeled with mining and mill capacity, blending and stockpile
constraints under grade uncertainty. There are numerous pushback design, heuristic and
aggregation methods investigated in the literature and the closest integer optimal solutions are
accomplished for the problems modeled with upper bound mining capacity constraints. Since, the
size of a mining problem makes the exact integer programming solution techniques inapplicable,
many attempts are made to solve the problem with pushback design, heuristic and aggregation
methods which cannot be proven to converge to the true optimal solution. So far, TopoSort
heuristic algorithm together with the Local search heuristic algorithm can generate an integer
solution for a block model consists of 4 million blocks with a 4% optimality gap in 4hr when
modeled with up to two upper bound capacity constraints and the sliding time window heuristic
can provide an integer solution within a 4.3% optimality gap in 2hr 40min when modeled with an
upper and lower bound capacity constraints for 25,000 blocks. Although these methods can
generate close integer optimal solutions, their limited applicability results with a preclusion of their
implementation on mine production scheduling problems. Contrarily, the new integer solution
algorithm proposed in this dissertation that can solve the mine production scheduling problems
modeled with multi capacities, grade blending, grade uncertainty, stockpiles, variable pit slopes
and multi destinations as an integer programming problem will realize the optimization process on
real life mining problems.
It is important to highlight the most influential developments closely related to the new
integer solution algorithm proposed in this thesis as shown in Figure 2.1. The major contributions
of each researcher are listed in the figure. Of course, there have been considerably more research
conducted on this topic as discussed throughout this Chapter, but the ones presented in Figure 2.1
set the stage for the takeoff platform of the developments presented in this thesis.
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CHAPTER 3.
MATHEMATICAL MODEL FOR OPEN PIT MINE PRODUCTION
SCHEDULING PROBLEM UNDER GRADE UNCERTAINITY
The mine production scheduling problem is an optimization problem which requires a
mathematical model that incorporates the factors delineating the interactions between the elements
of the complex mining system. The proposed integer solution algorithm in this thesis can solve the
new mathematical model presented in this chapter that incorporates production and processing
capacities, grade blending constraints, risk blending constraints, multi destinations, stockpiles and
variable pit slope angles on a block by block level. The new integer solution algorithm benefits
from the Bienstock-Zuckerberg decomposition algorithm. Thus, a new mathematical model will
be derived by introducing new variables in order to transform the model to a representative form
of the master problem of the Bienstock-Zuckerberg decomposition algorithm. Also, the original
mathematical model presented by Bienstock Zuckerberg (2009,2010 and 2015) will be derived
from the traditionally adapted mathematical formulation introduced by Johnson (1968). The
mathematical models presented in this chapter are in generic form that will make it easier for the
operator to adjust the models based on the needs of the operation. The proposed models differ from
the traditional mine production scheduling models with the integration of two important concepts.
The risk constraints which are first implemented by Van-Dunem (2016), are incorporated in order
to control the uncertainty by applying an upper bound on the number of the inferred blocks sent to
the mill and lower bounds on the amount of indicated and measured blocks. With the incorporation
of the risk constraints, the impact of uncertainty on meeting the operational or production targets
will be minimized. Moreover, stockpiles based on risk categories are also integrated to the
mathematical model. Stockpiles are utilized in mining operations for various purposes. In the event
of an equipment breakdown, instead of halting the production, the plant may continue to receive
materials from the stockpiles. Moreover, the upside potential of the commodity price is also
accounted by keeping the marginal ore in the stockpile for a potential recovery instead of sending
to the waste dump. Also, during the optimization process if the capacity of the plant can be filled
by mining a very valuable ore block, any marginal ore block preceding this nugget block will be
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sent to the waste dump if the stockpiles are excluded. Hence, the stockpiles can account for the
capacity limitations. Traditionally, the mine production scheduling problems are modeled with
stockpiles based on grade intervals or material types. The proposed model will further split the
stockpiles into risk categories, in order to prevent the blend of high risk ore with the low risk ore.
For example, if an inferred ore block is designated to a low-grade stockpile, it will be sent to a
low-grade stockpile that holds the inferred blocks. This approach will maximize the risk control in
the presence of stockpiles. Furthermore, the average grade of the material coming from the
stockpile was assumed to be equal to the lowest bound of the stockpile grade interval to prevent
overestimating the stockpile average grade. The average grade could also be used. The choice is
left to the user. The process flow representing the proposed mine production scheduling
mathematical model is shown in Figure 4.1.
Figure 3.1: Process Flow of the Mine Production Scheduling Problem
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requirement at process destination. Constraints (3.8) restrict the proportion of the inferred material
to be processed by enforcing an upper bound. Constraints (3.9) demand a minimum proportion of
the processed material to be indicated by enforcing a lower bound. Constraints (3.10) demand a
minimum proportion of the processed material to be measured by enforcing a lower bound.
Constraints (3.11) are stockpile material balance constraints. Constraints (3.12) ensure the tonnage
of the material sent from the stockpile is a continuous nonnegative variable. Constraints (3.13)
ensure the tonnage of the material remaining in the stockpile is a continuous nonnegative variable.
Constraints (3.14) ensure that all the decision variables are continuous between zero and one. There
could be other constraints such as minimum capacity and maximum blending constraints, but such
conditions seem unlikely therefore we specifically did not include those constraints to the
mathematical model.
3.2 Variable Substitution for Single Path “By” Variables
The new mathematical model adapted to the new integer solution algorithm will be derived
in 2 stages. The first stage of the variable substitution process will be demonstrated by initially
introducing the “by” variable. Transforming the “at” variables to single path “by” variables will
ensure interactions between the time periods as shown in Figure 3.2. All the rest of the indices,
sets, data and the stockpile variables will be kept the same as the ones used in the “at” formulation,
hence a new definition is not required. The variable substitution process will proceed with such
fashion that the ultimate goal is to maintain the “at” meaning for all the constraints.
Figure 3.2 : Node structure of the “at” variables and the “by” variables
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CHAPTER 4.
NEW INTEGER SOLUTION ALGORITHM TO SOLVE MULTI-
DESTINATION OPEN PIT MINE PRODUCTION SCHEDULING
PROBLEM
In this chapter, the new integer solution algorithm that can solve the mine production
scheduling problems modeled with multi capacities, grade blending, grade uncertainty, stockpiles,
variable pit slopes, multi destinations and truck hours will be covered extensively. It should be
stressed that the blocks will not have any pre-determined destinations based on grades, cycle times,
material type or some other criteria since the best destination selection per block will be done
automatically during the optimization process to maximize the NPV. The mathematical model for
the said problem is generated at the end of the previous chapter. A novel solution algorithm will
be introduced to solve this complex model. The new solution algorithm will exploit the
mathematical theories behind the Bienstock-Zuckerberg decomposition algorithm which is
presently the fastest solution algorithm for the precedence constrained production scheduling type
problems such as mine production scheduling problems. In order to set the groundwork, the BZ
algorithm will be covered in detail. Then, a comprehensive study on the new integer solution
algorithm will be presented. Discussions will be given to ensure the working mechanism of the
algorithm functions within the frame of mathematical theories.
The mine production scheduling problem is an integer programming problem solved for
mining decision variables in a binary form which constitutes the time period when the block is
mined and the destination to which the block is sent. So far there is no method that can generate a
theoretically proven optimal integer solution to the large-scale integer programming type
problems, including mine production scheduling problems due to the exponential computational
complexity of the existing exact algorithms. Nevertheless, a theoretical upper bound can be
achieved if the linear relaxation of the mine production scheduling problem can be solved to a
proven optimality. Henceforth, the knowledge of the optimal solution to a linear relaxation of the
mine production scheduling problem will facilitate the comparison of the integer feasible solution
to judge the success of the optimization process. In other words, if one can generate an integer
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feasible solution to the problem, the degree of success of the integer solution can be measured if
the optimal solution to the linear relaxation of the same problem is known. The difference between
the integer feasible solution and the optimal linear solution of the same problem is often called an
“optimality gap”. The goal of the optimization process is finding an integer feasible solution that
will minimize the optimality gap. The exhaustive search of the integer solutions attained by
heuristic methods cannot provide the optimality gap since the methods are not fundamentally
sophisticated with the mathematical theorems that will lead to a proven optimal solution. On the
other hand, the decomposition algorithms are developed based on intricate mathematical theorems
that can guarantee the optimality of the linear relaxation solutions for the mine production
scheduling problems. Henceforth, the integer solution algorithm developed in this thesis relies
heavily on the mathematical theories behind the Bienstock Zuckerberg decomposition algorithm.
4.1 The Bienstock-Zuckerberg Decomposition Algorithm
The Bienstock-Zuckerberg algorithm (BZ) is the most powerful decomposition algorithm
that can solve the LP relaxation of the large-scale mine production scheduling problems to a
proven optimality. The LP optimality is accomplished by iteratively solving the master and
subproblem until the convergence criteria is satisfied. It has been reported by Munoz et al. (2017)
that the BZ algorithm outperforms the counterpart Dantzig-Wolfe decomposition algorithm. The
algorithm derives its strength mainly by exploiting three important mathematical structures. First,
the subproblem formulation defines a totally unimodular system that allows the subproblem to be
formulated as a max flow problem as first shown by Johnson (1968). The subproblem is a multi-
time period sequencing problem which can be represented on a network structure as first shown
by Dagdelen (1985). Hochbaum (2008) proposed the Pseudoflow algorithm which is the fastest
known algorithm for solving the max flow problems. Henceforth, realizing the underlying network
structure of the subproblem and implementing the Pseudoflow algorithm to solve the subproblem
as a max flow problem will speed up the convergence process. Secondly, the orthogonalization
process that takes place between the subproblem solution and the former partitions increases the
dimension of the solution space at each iteration which will allow the LP optimal solution to be
captured very fast. The third strength is the contraction operation that leads to the conservation of
the original problem structure at the master problem while working with the significantly reduced
number of rows and columns.
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the single period version of the subproblem can be solved as a maximum flow network problem
which is faster than solving the problem as a LP. The idea is extended by Dagdelen (1985) by
exposing the mathematical relationship between the dual of the multi time period Lagrange
relaxation problem which is identical to this subproblem and the underlying network structure.
Currently Hochbaum` s Pseudoflow algorithm is the fastest known algorithm that can solve the
ultimate pit limit problem or multi time period sequencing problem as a maximum flow and
minimum s-t cut problem. The steps of the Pseudoflow algorithm will be outlined in the next
section.
The solution space of the BZ algorithm takes place in Euclidean n-space which
encompasses points with n coordinate values if the original problem contains n number of
variables. Since all the variables are positive, only the positive orthant of Euclidean n-space will
be considered. The constraints of the subproblem are the hyperplanes and the region bounded by
their intersections forms a polytope in n-space. Each subproblem solution is essentially a column
that represents the coordinate of an extreme point, located on the intersection of the hyperplanes
( ) on the polytope. The Figure 4.1 below represents a cross section taken from a polytope
and the blue colored points represent the extreme points where the subproblem solution could
(cid:1827)(cid:1874) (cid:3409) (cid:1854)
potentially take place.
Figure 4.1: Cross section of a polytope illustrating the subproblem solution space (outer
polytope) and the original problem solution space (inner polytope). Blue dots are the extreme
points of the subproblem solutions, the red dot is the optimum extreme point of the original
problem solution
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of the max flow solving algorithms. Nevertheless, due to the contraction operation, the number of
variables and the constraints in the master problem are still small enough to generate fast optimal
solutions to the master problem. The polytope corresponding to the feasible region of the original
(cid:2019)
problem takes place inside the subproblem solution space as shown in Figure 4.1. The optimal
solution generated for the master problem is a feasible solution to the original problem solution
space if not the optimal solution. Once the master problem solution is achieved; the dual vectors
corresponding to the side constrains (4.12) are also generated. The significance of the dual vectors
in the decomposition mechanism may be realized better if the relevance of the duality in the LP
concept is emphasized. Furthermore, the working mechanism of the dual vectors form a strong
connection between the simplex algorithm and the BZ decomposition algorithm which will be also
illustrated.
Given a LP problem, based on the strong duality theorem; if the feasible solutions of the
primal and the dual optimal solutions are equal, then the solution is proven to be the optimal LP
solution. Each constraint of the primal problem has an associated dual variable and the right-hand
side of the constraints are perceived as available resources. Obviously, the non-zero dual values
will only exist if the available resources are fully consumed, in other words if the constraints are
binding. The value of the associated dual variable in other words the shadow price will provide a
useful measure on the impact of a unit change in the available resource on the objective function
value. If the primal problem is a maximization problem, the positive dual variable can be
interpreted as such that expanding the available resource will have a positive contribution to the
objective function value and the opposite is true if the dual variable has a negative value. Let`s
also explore the significance of the duals in the simplex algorithm. In each iteration of the simplex
algorithm, the non-basic variable which has the highest contribution to the objective function value
is identified and included to the basic feasible solution. The variable selection process has an
underlying economic interpretation. For each non-basic variable, its net worth is determined by
the following equation where the original coefficient of the variable in the
objective function can be interpreted as revenue, dual variable or the reduce cost represents the
(cid:1855)(cid:3036)̅ = (cid:1855)(cid:3036) − (cid:1853)(cid:3038) (cid:1855)(cid:3036)
cost of consuming a unit of a resource and is the amount of the resources consumed. Thus,
can be interpreted as the modified value of the blocks.
(cid:1853)(cid:3038) (cid:1855)(cid:3036)̅
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The relationship between the master and subproblem of the BZ algorithm is apparent in the
light of the duality concept. The dual vector corresponding to the side constraints (4.12) will be
used similarly to characterize the net worth of the mining decision variables. To be more precise,
the block values will be modified with the corresponding dual values which may either make the
blocks favorable or not. In the simplex algorithm we include the decision variable with the highest
net worth, , into the basis. The pivoting process of the simplex algorithm is analogous to the
subproblem solution. The modified block values are the net worth of the blocks based on the duals,
(cid:1855)(cid:3036)̅
and the solution to the subproblem is essentially the best mining plan generated with the modified
block values. Henceforth, the variable corresponding to the plan enters the basis of the system.
If the master problem is constrained as a single destination problem, then the dual vector will
(cid:2019)
modify the discounted value of the blocks over the time periods which may make the blocks more
profitable for the later periods for the subproblem. If the blocks can be sent to multiple destinations
as in the original problem that is being solved in this research, then the value of the block at each
destination will be modified with the dual values per time period and the destination with the best
value will be selected for each time period in the subproblem. This was proven by Johnson (1968).
In a sense, the dual values will work similar to the cutoff grades to determine the best destination
for the blocks. The relationship between the destination variables across the time periods for a
single block was shown in the previous chapter. As it can be seen in Figure 4.2 below, the
subproblem variable is essentially represented by a single node, when the destination nodes of
(cid:3047)
the variable is collapsed by picking the best destination from the modified block values.
(cid:1878)(cid:3029)
(cid:3047)
(cid:1878)(cid:3029),(cid:3031)
Figure 4.2: Node structure of the original problem variables and the subproblem variables
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4.2 Pseudoflow Algorithm
The Pseudoflow algorithm is so far the fastest known solution algorithm for the network flow
problems modeled as max-flow or min-cut type problems. The strength of the Pseudoflow
algorithm is exploited in the BZ algorithm as well as the new integer solution algorithm. The
subproblem of the BZ algorithm is a multi-period sequencing problem which constitutes a totally
unimodular structure that allows the Pseudoflow algorithm to take the advantage of the underlying
network structure. Moreover, the new integer solution algorithm further splits these subproblem
max closures into single time period integer feasible sub-closures by iteratively solving the max
flow problems with the Pseudoflow algorithm. As this integerizing step may require many
iterations to satisfy the feasibility conditions, the fast working mechanism of the Pseudoflow
algorithm motivates the implementation of the new integer solution algorithm on large scale open
pit mining problems.
The theories behind the working mechanism of the Pseudoflow algorithm is extensively
discussed in the original paper by Hochbaum (2008), therefore only the summary of the steps of
the Pseudoflow algorithm will be presented. Moreover, the computational study of the Pseudoflow
algorithm and the push-relabel algorithm is presented by Chandran and Hochbaum (2009). Also,
the detailed review of the Pseudoflow algorithm is provided by Van Dunem (2016).
The specific steps of the single iteration of the Pseudoflow algorithm is summarized below
(Hochbaum 2008, Van Dunem 2016):
1. Initialize the algorithm by selecting a normalized tree. One simple normalized tree
corresponds to a Pseudoflow in (s, t graph that contains two distinguished nodes s and
t; source and sink nodes respectively) saturating all arcs adjacent to the source and all
(cid:1833)(cid:3046)(cid:3047)
the arcs adjacent to the sink , and zero on all other arcs.
(cid:1827)(cid:4666)(cid:1871)(cid:4667)
2. Find a residual arc from a strong set of nodes to a weak set of nodes called as merger arc.
(cid:1827)(cid:4666)(cid:1872)(cid:4667)
If such an arc does not exist, the current solution is optimal.
3. If residual arcs going from a strong node to a weak node exist then, the selected merger arc
is appended to the tree, the excess arc of the strong merger branch is removed, and the
strong branch is merged with the weak branch.
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4. Push the entire excess of the respective strong branch along the unique path from the root
of the strong branch to the root of the weak branch.
5. Split any arc encountered along the path described in 3) which does not have sufficient
residual capacity to accommodate the amount pushed. The tail node of that arc becomes a
root of a new strong branch with excess equal to the difference between the amount pushed
and the residual capacity.
6. The process of pushing excess and splitting is called normalization. The residual capacity
of the split arc is pushed further until it either reaches another arc to split or the deficit arc
adjacent to the root of the weak branch.
4.3 The New Integer Solution Algorithm
The new integer solution algorithm is developed mainly by exploiting the strengths of the
Bienstock-Zuckerberg algorithm outlined in the previous section. Importantly, the algorithm
works towards optimality in conjunction with the BZ algorithm. Although the BZ algorithm is
presently the most powerful decomposition algorithm to solve the LP relaxation of the mine
production scheduling problems towards proven optimality, the solutions are fractional which
results in impractical mine plans. However, the information that will lead solutions towards
integrality is also preserved in each iteration. The new integer solution algorithm will enjoy the
implicit use of this information together with all of the theories behind the working mechanism of
the BZ algorithm while unlocking new integer solution spaces for the master problem to satisfy
the integrality conditions. The algorithm ensures that every column solution generated by the
subproblem is partitioned into components that are integer feasible to the capacity constraints of
the original problem. This can be achieved simply by converting the original problem into a single
time period max flow problem (ultimate pit problem) and parametrizing the block values. While
it may take many iterations to be integer feasible to the capacity constraints, the Pseudoflow
algorithm will process the underlying network structure quite fast which may result with solution
times around 2-3 seconds per iteration. Since mine production scheduling problems also include
blending constraints, the duals generated by solving the master problem at each iteration will guide
the subproblem solutions towards feasibility for the blending constraints. At each iteration, the
orthogonalization process will further increase the dimension of the solution space where integer
feasible solutions for the master problem are also available. Furthermore, the contraction operation
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As the algorithm proceeds with splitting the original problem into master and subproblems,
the subproblem solutions play an important role in terms of defining the coordinates of the extreme
point of a polytope located on the intersection of the hyperplanes. It is important to mention that
the solution to the subproblem is not necessarily integer feasible to the capacity constraints of the
original problem. In the regular BZ algorithm, if the partitions defining the solution space of the
master problem are infeasible to the capacity constraints, the variables which are essentially the
weight of each partition may result with fractional values to honor the requirements enforced by
(cid:2019)
the capacity constraints. As this is very natural from a linear programming point of view, if the
variables are defined as binary variables, the solution to the master problem may end up being
(cid:2019)
infeasible. Henceforth, the subproblem solutions which are indeed max closures, must be
partitioned into sub-closures that are integer feasible to the capacity constraints of the original
problem. As it is illustrated in Figure 4.14, the outer polytope represents the subproblem solution
space where the blue dots are the extreme points which may not appear in the capacity feasible
region bounded with a green outline. The solution space to the original problem which is further
defined by the hyperplanes representing the additional side constraints will be inside the capacity
feasible region and is shown in Figure 4.14 outlined in black. While the yellow colored corner
point defines the LP optimal solution to the original problem, the red points are the set of integer
feasible points to the original problem space.
Figure 4.14: Cross section of a polytope illustrating the solution space of the new integer solution
algorithm
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values towards subproblem solution space is determined by the side constraints, the role of the
dual values will be analyzed for the capacity and blending type constraints separately. Capacity
type constraints limit the number of blocks handled or processed in some way such as total mining
production, processed tonnage, amount of material a crusher can handle, total available truck
hours, waste dump capacity or amount of material stored in stockpile. Given a mine planning
problem constrained with only upper bound capacity constraints, implementing the regular BZ
algorithm to solve the subproblem by penalizing the block values with the capacity duals is in fact
a well-known parametrization approach. The duals will modify the block values that exist in the
capacity constraint with the same penalty.
Axiom 4.3: As the penalty values increase, in the absence of gaps there may be an opportunity to
obtain max closure which is integer feasible to the original problem capacity constraints.
In the light of axiom 4.3, we can expect to generate an integer feasible solution towards capacity
(cid:1829)⃗(cid:3038)
constraint space at some iteration if the dual values constantly increase in consecutive iterations.
Proposition 4.4: The value of the duals does not constitute a monotonically non-decreasing
pattern on consecutive iterations.
Proof: The primal and dual formulation of the master problem is given in Figure 4.16. It has
already been proven that the master problem objective function values are monotonically non-
decreasing. Once the master problem is optimal for a particular iteration, the dual objective
function value must be equal to the primal objective function value due to the strong duality
theorem. This shows that since primal objective function values are monotonically non-decreasing,
the dual objective function values must be also monotonically non-decreasing.
Figure 4.16: Primal-dual relationship of the master problem
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It is clear that the objective function coefficients of the dual of a problem ( , ) are constant in
every iteration which means that any improvement on the objective function value must be
(cid:883) ℎ
justified by increasing either one or both of the value of the duals and . Also, the dual variable
does not have any contribution to the objective function value. We also know that the dual values
(cid:2020)
may be non-zero only if the primal constraints are binding. Let`s investigate the behavior of the
dual variables, first when a mine plan which is infeasible to the capacity constraint is obtained by
solving the subproblem and passed to the master problem solution space. If the partitions are
infeasible to the capacity constraint, the variables will be fractional which means some of the
reserve constraints and some of the sequencing constraints will be non-binding. This will result
(cid:2019)
with some of the variables having 0 values. Since the capacity constraints will be binding, all
the variables will be greater than or equal to 0. Hence, if consecutive iterations consist of
(cid:2020)
infeasible partitions, or must increase in order to justify the improvement of the objective
function value. If the subproblem solution generates a mine plan feasible to the capacity constraint,
(cid:2020)
we know that this is the most valuable integer feasible plan since in a max flow problem the
penalized block values will be the best ones to be mined. Therefore, once this plan is passed to the
master problem, any variable associated with this plan must become 1 in the master problem
solution space. Any resource constraint that contain this variable must be a binding constraint
(cid:2019)
which makes some of the variables become greater than or equal to 0. In this case, the
(cid:2019)
justification of the improvement in the objective function value can be made by increasing either
(cid:2020)
or . In either case, there is no guarantee that the variable will get a value greater than its
former value, in fact the value of may even decrease if the value of increases. Therefore, we
(cid:2020)
cannot claim a monotonically non-decreasing pattern for the capacity dual values.
(cid:2020)
In a sense, the proposition 4.4 is very important to justify the need for an integerizing
approach towards the subproblem solution space, since the regular BZ algorithm does not
guarantee a mining plan integer feasible to the original problem capacity constraint space.
Although an integer feasible region for the capacity constraints at each iteration can be secured,
the original problem may consist of blending constraints where mining plans generated by the
subproblem may need to be integer feasible for the blending constraints. Let`s investigate the
significance of the dual values generated for the blending type of constraints. First of all, the
blending type of constraints can be average grade, material type proportions or risk proportions
required at any type of processing destinations. In order to clarify the role of the duals for the
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of the columns may become infeasible. Let`s assume set where satisfies
the blending constraints. Let the orthogonal partitions deriv ℎ⃑⃑⃑e⃑(cid:2869) d = fro{(cid:1876)m(cid:2869) ,(cid:1876)(cid:2870) b …e .(cid:1876)(cid:3040)} ⃑ℎ⃑⃑⃑(cid:2869) and
where . Since is integ ℎ⃑e⃑⃑⃑(cid:2869)r fea ⃑(cid:1874)⃑s ⃑⃑(cid:2869) ib =le {f (cid:1876)o(cid:2869)r , (cid:1876)t(cid:2870)he … b .(cid:1876)le(cid:3041)n}ding
constraints, and have a mutually exclusive relationship in terms of being infeasible towards
(cid:1874)⃑⃑⃑⃑(cid:2870) = {(cid:1876)(cid:3041),(cid:1876)(cid:3041)+(cid:2869)….(cid:1876)(cid:3040)} |⃑(cid:1874)⃑⃑⃑(cid:2869) |+|⃑(cid:1874)⃑⃑⃑(cid:2870) | = (cid:1865) ⃑ℎ⃑⃑⃑(cid:2869)
blending co ⃑(cid:1874)n ⃑⃑⃑(cid:2869)s traint ⃑(cid:1874)s ⃑⃑⃑.(cid:2870) Because and are sub-partitions of , based on theorem 4.1, any
solution space governed by (cid:1874)⃑w ⃑⃑⃑(cid:2869) ill be c ⃑(cid:1874)⃑⃑a ⃑(cid:2870) ptured from the span of ℎ⃑⃑ ⃑⃑(cid:2869) and . In other words, the
blend of and still preserves the feasibility conditions for the blending constraints.
ℎ⃑⃑⃑⃑(cid:2869) (cid:1874)⃑⃑⃑⃑(cid:2869) (cid:1874)⃑⃑⃑⃑(cid:2870)
O⃑(cid:1874)⃑n⃑⃑(cid:2869) ce the(cid:1874)⃑⃑ ⃑⃑o(cid:2870) ptimality conditions mentioned in the BZ section are satisfied, the optimal LP
solution will be generated. The set of partitions generated in the final iteration will play a key role
in achieving an integer solution for the original problem. It is important to state that the final set
of partitions may be comprised of capacity feasible and combination of blending feasible and
infeasible columns. The relationship between the original problem constraint space and the span
of the final set of partitions is illustrated in Figure 4.17. The blue colored region represents the
solution space generated by the partitions whereas the outer polytope is the solution space of the
original problem. It should be pointed out that the linear combination of the partitions captures a
much smaller space where the optimal LP solution which is the extreme point colored with yellow
is included as well as some of the integer solutions shown with red color. This shows that once the
LP optimal solution is captured, the algorithm is ready to develop an optimal integer solution
within the span of the final set of partitions. This can be simply done by declaring the final set of
variables as binary variables and resolving the last iteration`s master problem. The reduced
number of rows and columns in the master problem allow the integer solution algorithms of
(cid:2019)
CPLEX to exploit the underlying mathematical structure and obtain an integer solution very fast.
Although the final integer solutions are optimal integer solutions within the solution space of the
final partitions, it cannot be proven that the solutions are true optimal integer solutions of the
original problem. Nevertheless, it was mentioned by the authors the BZ algorithm that the
integrality gap between the integer programming solution and the linear programming relaxation
is often small. The results indicate that there is a high potential for achieving an integrality gap of
less than 1% for the large-scale mining problems constrained with capacity and blending
constraints which will be presented in the next chapter.
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1 11 2(cid:2184)(cid:2202)
→
1
1(cid:2778)
→
0
0(cid:2779)
→
0
0(cid:2780)
→
0
0(cid:2781)
13 1 0 0 0
14 1 0 0 0
15 0 0 0 0
16 0 0 0 0
1
D 17 0 0 0 0
O
I 22 1 0 0 0
R
E 23 0 0 0 0
P
24 0 0 0 0
25 0 0 0 0
26 0 0 0 0
33 0 0 0 0
34 0 0 0 0
35 0 0 0 0
11 1 0 0 0
12 1 0 0 0
13 1 0 0 0
14 1 0 0 0
15 0 0 0 0
2 16 0 1 0 0
D
O 17 0 0 0 0
I R 22 1 0 0 0
E
P 23 0 1 0 0
24 0 0 0 0
25 0 0 0 0
26 0 0 0 0
33 0 0 0 0
34 0 0 0 0
35 0 0 0 0
11 1 0 0 0
12 1 0 0 0
13 1 0 0 0
14 1 0 0 0
15 0 0 1 0
16 0 1 0 0
3
D 17 0 0 0 1
O
I
R
22 1 0 0 0
E 23 0 1 0 0
P
24 0 0 1 0
25 0 0 1 0
26 0 0 0 1
33 0 0 1 0
34 0 0 1 0
35 0 0 0 1
Figure 4.26: Iteration 1- Initial set of integer feasible orthogonal partitions. Colors represent the
blocks considered in the partition
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1 11 2(cid:2184)(cid:2202) →(cid:2201)0
0(cid:2778)
1 11 2(cid:2184)(cid:2202) → 0
0(cid:2778)
→ 0
0(cid:2779)
→ 0
0(cid:2780)
→ 0
0(cid:2781)
13 0 13 0 0 0 0
14 0 14 0 0 0 0
15 0 15 0 0 0 0
16 0 16 0 0 0 0
1 1
D 17 0 D 17 0 0 0 0
O O
IR 22 0 IR 22 0 0 0 0
E 23 0 E 23 0 0 0 0
P P
24 0 24 0 0 0 0
25 0 25 0 0 0 0
26 0 26 0 0 0 0
33 0 33 0 0 0 0
34 0 34 0 0 0 0
35 0 35 0 0 0 0
11 1 11 1 0 0 0
12 1 12 1 0 0 0
13 1 13 1 0 0 0
14 1 14 1 0 0 0
15 1 15 0 0 1 0
2 16 1 2 16 0 1 0 0
D D
O 17 1 O 17 0 0 0 1
IR 22 1 IR 22 1 0 0 0
E E
P 23 1 P 23 0 1 0 0
24 1 24 0 0 1 0
25 1 25 0 0 1 0
26 1 26 0 0 0 1
33 1 33 0 0 1 0
34 1 34 0 0 1 0
35 1 35 0 0 0 1
11 1 11 1 0 0 0
12 1 12 1 0 0 0
13 1 13 1 0 0 0
14 1 14 1 0 0 0
15 1 15 0 0 1 0
16 1 16 0 1 0 0
3 3
D 17 1 D 17 0 0 0 1
O O
IR 22 1 IR 22 1 0 0 0
E 23 1 E 23 0 1 0 0
P P
24 1 24 0 0 1 0
25 1 25 0 0 1 0
26 1 26 0 0 0 1
33 1 33 0 0 1 0
34 1 34 0 0 1 0
35 1 35 0 0 0 1
Figure 4.29: Iteration 2- Subproblem solution column on the left is split into integer feasible
columns on the right
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11 21 (cid:2184)(cid:2202) → 11
(cid:2778)
→ 00
(cid:2779)
→ 00
(cid:2780)
→ 00
(cid:2781)
11 21 (cid:2184)(cid:2202) → 00
(cid:2778)
→ 00
(cid:2779)
→ 00
(cid:2780)
→ 00
(cid:2781)
13 1 0 0 0 13 0 0 0 0
14 1 0 0 0 14 0 0 0 0
15 0 0 0 0 15 0 0 0 0
16 0 0 0 0 16 0 0 0 0
1 1
D 17 0 0 0 0 D 17 0 0 0 0
O O
I 22 1 0 0 0 I 22 0 0 0 0
R R
E 23 0 0 0 0 E 23 0 0 0 0
P P
24 0 0 0 0 24 0 0 0 0
25 0 0 0 0 25 0 0 0 0
26 0 0 0 0 26 0 0 0 0
33 0 0 0 0 33 0 0 0 0
34 0 0 0 0 34 0 0 0 0
35 0 0 0 0 35 0 0 0 0
11 1 0 0 0 11 1 0 0 0
12 1 0 0 0 12 1 0 0 0
13 1 0 0 0 13 1 0 0 0
14 1 0 0 0 14 1 0 0 0
15 0 0 0 0 15 0 0 1 0
2 16 0 1 0 0 2 16 0 1 0 0
D D
O 17 0 0 0 0 O 17 0 0 0 1
I R 22 1 0 0 0 I R 22 1 0 0 0
E E
P 23 0 1 0 0 P 23 0 1 0 0
24 0 0 0 0 24 0 0 1 0
25 0 0 0 0 25 0 0 1 0
26 0 0 0 0 26 0 0 0 1
33 0 0 0 0 33 0 0 1 0
34 0 0 0 0 34 0 0 1 0
35 0 0 0 0 35 0 0 0 1
11 1 0 0 0 11 1 0 0 0
12 1 0 0 0 12 1 0 0 0
13 1 0 0 0 13 1 0 0 0
14 1 0 0 0 14 1 0 0 0
15 0 0 1 0 15 0 0 1 0
16 0 1 0 0 16 0 1 0 0
3 3
D 17 0 0 0 1 D 17 0 0 0 1
O O
I 22 1 0 0 0 I 22 1 0 0 0
R R
E 23 0 1 0 0 E 23 0 1 0 0
P P
24 0 0 1 0 24 0 0 1 0
25 0 0 1 0 25 0 0 1 0
26 0 0 0 1 26 0 0 0 1
33 0 0 1 0 33 0 0 1 0
34 0 0 1 0 34 0 0 1 0
35 0 0 0 1 35 0 0 0 1
Figure 4.30: Iteration 2- Current state of the partitions after appending the integer feasible
columns to the previous set of partitions . At this stage the columns are not
orthogonal to yet
ℎ⃑ (cid:1874) ℎ⃑
(cid:1874)
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11(cid:2202) 0 0 0 0 1 0 0
12(cid:2184) (cid:2778)(cid:1514) 0 (cid:2778) (cid:2779)(cid:1514)0 (cid:2779) (cid:2780)(cid:1514)0 (cid:2780) (cid:2781)(cid:1514)0 (cid:2781) (cid:2778)/1 ⃗ (cid:2780)/0 (cid:2781) 0/
13 0 0 0 0 1 0 0
14 0 0 0 0 1 0 0
15 0 0 0 0 0 0 0
16 0 0 0 0 0 0 0
1
D 17 0 0 0 0 0 0 0
O
I 22 0 0 0 0 1 0 0
R
E 23 0 0 0 0 0 0 0
P
24 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0
26 0 0 0 0 0 0 0
33 0 0 0 0 0 0 0
34 0 0 0 0 0 0 0
35 0 0 0 0 0 0 0
11 1 0 0 0 0 0 0
12 1 0 0 0 0 0 0
13 1 0 0 0 0 0 0
14 1 0 0 0 0 0 0
15 0 0 0 0 0 1 0
2 16 0 1 0 0 0 0 0
D
O 17 0 0 0 0 0 0 1
I R 22 1 0 0 0 0 0 0
E
P 23 0 1 0 0 0 0 0
24 0 0 0 0 0 1 0
25 0 0 0 0 0 1 0
26 0 0 0 0 0 0 1
33 0 0 0 0 0 1 0
34 0 0 0 0 0 1 0
35 0 0 0 0 0 0 1
11 1 0 0 0 0 0 0
12 1 0 0 0 0 0 0
13 1 0 0 0 0 0 0
14 1 0 0 0 0 0 0
15 0 0 1 0 0 0 0
16 0 1 0 0 0 0 0
3
D 17 0 0 0 1 0 0 0
O
I 22 1 0 0 0 0 0 0
R
E 23 0 1 0 0 0 0 0
P
24 0 0 1 0 0 0 0
25 0 0 1 0 0 0 0
26 0 0 0 1 0 0 0
33 0 0 1 0 0 0 0
34 0 0 1 0 0 0 0
35 0 0 0 1 0 0 0
Figure 4.31: Iteration 2- Integer feasible orthogonal partitions. Colors represent the blocks
considered in the partition
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11(cid:2202) 1 11(cid:2202) 1 0 0
12(cid:2184)
→
(cid:2201) 1(cid:2779) 12(cid:2184)
→
1(cid:2778)
→
0(cid:2779)
→
0(cid:2780)
13 1 13 1 0 0
14 1 14 1 0 0
15 0 15 0 0 0
16 0 16 0 0 0
1 1
D 17 0 D 17 0 0 0
O O
I 22 1 I 22 1 0 0
R R
E 23 0 E 23 0 0 0
P P
24 0 24 0 0 0
25 0 25 0 0 0
26 0 26 0 0 0
33 0 33 0 0 0
34 0 34 0 0 0
35 0 35 0 0 0
11 1 11 1 0 0
12 1 12 1 0 0
13 1 13 1 0 0
14 1 14 1 0 0
15 1 15 0 1 0
2 16 1 2 16 0 1 0
D D
O 17 1 O 17 0 0 1
I R 22 1 I R 22 1 0 0
E E
P 23 1 P 23 0 1 0
24 1 24 0 1 0
25 1 25 0 0 1
26 1 26 0 0 1
33 1 33 0 1 0
34 1 34 0 0 1
35 1 35 0 0 1
11 1 11 1 0 0
12 1 12 1 0 0
13 1 13 1 0 0
14 1 14 1 0 0
15 1 15 0 1 0
16 1 16 0 1 0
3 3
D 17 1 D 17 0 0 1
O O
I R 22 1 I R 22 1 0 0
E 23 1 E 23 0 1 0
P P
24 1 24 0 1 0
25 1 25 0 0 1
26 1 26 0 0 1
33 1 33 0 1 0
34 1 34 0 0 1
35 1 35 0 0 1
Figure 4.37: Iteration 3- Subproblem solution column on the left is split into integer feasible
columns on the right
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11 21 (cid:2184)(cid:2202) → 00
(cid:2778)
→ 00
(cid:2779)
→ 00
(cid:2780)
→ 00
(cid:2781)
→ 11 → 00 → 00 11 21 (cid:2184)(cid:2202) → 11
(cid:2778)
→ 00
(cid:2779)
→ 00
(cid:2780)
13 0 0 0 0 1 0 0 13 1 0 0
14 0 0 0 0 1 0 0 14 1 0 0
15 0 0 0 0 0 0 0 15 0 0 0
16 0 0 0 0 0 0 0 16 0 0 0
1 1
D 17 0 0 0 0 0 0 0 D 17 0 0 0
O O
IR 22 0 0 0 0 1 0 0 IR 22 1 0 0
E 23 0 0 0 0 0 0 0 E 23 0 0 0
P P
24 0 0 0 0 0 0 0 24 0 0 0
25 0 0 0 0 0 0 0 25 0 0 0
26 0 0 0 0 0 0 0 26 0 0 0
33 0 0 0 0 0 0 0 33 0 0 0
34 0 0 0 0 0 0 0 34 0 0 0
35 0 0 0 0 0 0 0 35 0 0 0
11 1 0 0 0 0 0 0 11 1 0 0
12 1 0 0 0 0 0 0 12 1 0 0
13 1 0 0 0 0 0 0 13 1 0 0
14 1 0 0 0 0 0 0 14 1 0 0
15 0 0 0 0 0 1 0 15 0 1 0
2 16 0 1 0 0 0 0 0 2 16 0 1 0
D D
O 17 0 0 0 0 0 0 1 O 17 0 0 1
IR 22 1 0 0 0 0 0 0 IR 22 1 0 0
E E
P 23 0 1 0 0 0 0 0 P 23 0 1 0
24 0 0 0 0 0 1 0 24 0 1 0
25 0 0 0 0 0 1 0 25 0 0 1
26 0 0 0 0 0 0 1 26 0 0 1
33 0 0 0 0 0 1 0 33 0 1 0
34 0 0 0 0 0 1 0 34 0 0 1
35 0 0 0 0 0 0 1 35 0 0 1
11 1 0 0 0 0 0 0 11 1 0 0
12 1 0 0 0 0 0 0 12 1 0 0
13 1 0 0 0 0 0 0 13 1 0 0
14 1 0 0 0 0 0 0 14 1 0 0
15 0 0 1 0 0 0 0 15 0 1 0
16 0 1 0 0 0 0 0 16 0 1 0
3 3
D 17 0 0 0 1 0 0 0 D 17 0 0 1
O O
IR 22 1 0 0 0 0 0 0 IR 22 1 0 0
E 23 0 1 0 0 0 0 0 E 23 0 1 0
P P
24 0 0 1 0 0 0 0 24 0 1 0
25 0 0 1 0 0 0 0 25 0 0 1
26 0 0 0 1 0 0 0 26 0 0 1
33 0 0 1 0 0 0 0 33 0 1 0
34 0 0 1 0 0 0 0 34 0 0 1
35 0 0 0 1 0 0 0 35 0 0 1
Figure 4.38: Iteration 3- Current state of the partitions after appending the integer feasible
columns to the previous set of partitions . At this stage the columns are not
orthogonal to yet
ℎ⃑ (cid:1874) ℎ⃑
(cid:1874)
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11(cid:2202) 0 1 0 0 0 0 0 0 0
12(cid:2184) (cid:2778)(cid:1514) 0 (cid:2778) (cid:1514) 1 (cid:2778) (cid:2779)(cid:1514) 0 (cid:2779) (cid:2780)(cid:1514)0 (cid:2779) (cid:1514)0 (cid:2779) (cid:2780)(cid:1514)0 (cid:2780) (cid:2781)(cid:1514) 0 (cid:2780) (cid:1514)0 (cid:2780) (cid:1514)0 (cid:2780)
13 0 1 0 0 0 0 0 0 0
14 0 1 0 0 0 0 0 0 0
15 0 0 0 0 0 0 0 0 0
16 0 0 0 0 0 0 0 0 0
1
D 17 0 0 0 0 0 0 0 0 0
O
I 22 0 1 0 0 0 0 0 0 0
R
E 23 0 0 0 0 0 0 0 0 0
P
24 0 0 0 0 0 0 0 0 0
25 0 0 0 0 0 0 0 0 0
26 0 0 0 0 0 0 0 0 0
33 0 0 0 0 0 0 0 0 0
34 0 0 0 0 0 0 0 0 0
35 0 0 0 0 0 0 0 0 0
11 1 0 0 0 0 0 0 0 0
12 1 0 0 0 0 0 0 0 0
13 1 0 0 0 0 0 0 0 0
14 1 0 0 0 0 0 0 0 0
15 0 0 0 0 1 0 0 0 0
2 16 0 0 1 0 0 0 0 0 0
D
O 17 0 0 0 0 0 0 0 0 1
I R 22 1 0 0 0 0 0 0 0 0
E
P 23 0 0 1 0 0 0 0 0 0
24 0 0 0 0 1 0 0 0 0
25 0 0 0 0 0 0 0 1 0
26 0 0 0 0 0 0 0 0 1
33 0 0 0 0 1 0 0 0 0
34 0 0 0 0 0 0 0 1 0
35 0 0 0 0 0 0 0 0 1
11 1 0 0 0 0 0 0 0 0
12 1 0 0 0 0 0 0 0 0
13 1 0 0 0 0 0 0 0 0
14 1 0 0 0 0 0 0 0 0
15 0 0 0 1 0 0 0 0 0
16 0 0 1 0 0 0 0 0 0
3
D 17 0 0 0 0 0 0 1 0 0
O
I 22 1 0 0 0 0 0 0 0 0
R
E 23 0 0 1 0 0 0 0 0 0
P
24 0 0 0 1 0 0 0 0 0
25 0 0 0 0 0 1 0 0 0
26 0 0 0 0 0 0 1 0 0
33 0 0 0 1 0 0 0 0 0
34 0 0 0 0 0 1 0 0 0
35 0 0 0 0 0 0 1 0 0
Figure 4.39: Iteration 3- Integer feasible orthogonal partitions. Colors represent the blocks
considered in the partition
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CHAPTER 5.
IMPLEMENTATION OF THE NEW INTEGER SOLUTION
ALGORITHM TO THE LARGE-SCALE OPEN PIT MINING
PROBLEMS
In this chapter, case studies will be presented to illustrate the implementation of the new
integer solution algorithm on large scale open pit mining problems. Some of these mining
problems may allow multiple processing options for a given block where the destination selection
will be a function of a dynamic cutoff grade optimization process defined by the state of the system
under capacity, average grade blending and risk blending constraints. Some of the mining
problems may have multiple sources to feed the blocks into the process stream such as multiple
pits and stockpiles. Again, it is important to emphasize that presently there is no known algorithm,
either commercially available or presented in the literature, that can provide an optimal integer
solution to the open pit mine production scheduling problem with capacity constraints together
with lower and upper bound blending constraints. The strength of the integer solution algorithm
developed in this thesis will be highlighted on the ability of solving problems that have more than
7 million variables as an integer problem with an optimality gap as small as 0.01 % within 5 hours
30 minutes.
5.1 Case Study 1 (McLaughlin Deposit)
The first case study will demonstrate the implementation of the new integer solution
algorithm on scheduling a data set referred to as the McLaughlin Deposit for 10 years. The
schematic description of the assumed mining complex is given in Figure 5.1. The economic
parameters that will be used to derive the block values are given in Table 5.1. The blocks will be
initially subjected to a break even cutoff grade 0.03 oz/t which will separate the invaluable waste
blocks from the ore blocks which carry a recoverable value once processed. Any ore block that
has a grade less than 0.05 oz/t will be treated at the leach pads or sent to the waste dump once it is
mined. Moreover, an ore block with a grade greater than or equal to 0.08 oz/t will be processed at
the mill or sent to the waste dump. If the grade of a mined ore block is between 0.05oz/t and
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5.1.1 The Ultimate Pit
Once the block model characteristics are outlined, the next step is to determine the ultimate
pit for the McLaughlin Deposit. The mathematical model for the ultimate pit problem which is
essentially a single period version of the subproblem shown in Chapter 4 requires a single value
for a mining decision variable; which means that the destination of a block will be pre-determined.
Hence, given a set of possible destinations for a single block, the destination where the highest
recoverable value can be achieved will be selected. Moreover, the cone pattern generation
technique is implemented to create arcs between the blocks to accomplish a uniform slope
(cid:2868)
angle. Then the solution to the ultimate pit problem is determined by implementing the pseudoflow
(cid:886)(cid:887)
algorithm. Table 5.5 illustrates the number of blocks and tonnage mined in the ultimate pit. It is
clear that while the block model consists of 1.9 million blocks, the ultimate pit has only 245.6
thousand blocks. There are about 2 thousand leach blocks that existed in the block model but not
mined in the ultimate pit since they are not economical. We can also say that leaving those leach
blocks on the ground lead to a decrease of the proportion of the inferred blocks about 1.4 % as
shown in Table 5.6. Also, there is no observable change on the average grades mined in the
ultimate pit. The value of the pit which is calculated with the undiscounted block values determined
by picking the most valuable destination was found to be $2.2 billion.
Table 5.5: Summary of blocks in the ultimate pit
MATERIAL TYPE BLOCKS TONNAGE AVG GRADE
TOTAL 245,617 122,808,500 -
WASTE 160,445 80,222,500 -
ORE 85,172 42,586,000 0.062 oz/t
MILL 18,140 9,070,000 0.133 oz/t
LEACH 42,737 21,368,500 0.034 oz/t
UNDECIDED 24,295 12,147,500 0.057 oz/t
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Table 5.7: Production requirements for the mine plan
PROCESS CAPACITY (TONNAGE) AVG GRADE (oz/t) RISK PROPORTIONS
PERIODS
MILL LEACH >= MILL >= LEACH <= INF >= IND >= MEA
1 1,500,000 1,500,000 0.062 0.035 20% 10% 35%
2 1,750,000 1,750,000 0.062 0.035 20% 10% 35%
3 2,000,000 2,000,000 0.062 0.035 20% 10% 35%
4 2,750,000 2,750,000 0.062 0.035 40% 10% 25%
5 3,000,000 3,000,000 0.062 0.035 40% 10% 25%
6 3,000,000 3,000,000 0.062 0.035 40% 10% 25%
7 2,750,000 2,750,000 0.062 0.035 50% 10% 25%
8 2,000,000 2,000,000 0.062 0.035 50% 10% 25%
9 1,750,000 1,750,000 0.062 0.035 50% 10% 25%
10 1,500,000 1,500,000 0.062 0.035 50% 10% 25%
The optimal mine plan is generated by implementing the new integer solution algorithm.
The results are shown in Table 5.8. It is clear that all of the yearly requirements are honored. In
Figure 5.5 it can be seen that the mill is working at full capacity for the first 7 years and after that
there is a shortage of mill blocks to fill the mill capacity. On the other hand, the Figure 5.6 shows
that the resulting mine plan is able to fill leach pad capacity every year. Also, the red dotted line
that appears in both figures shows the average grade of the ore blocks processed every year. It is
obvious that the optimizer prioritizes the high-grade zones in the earlier years of the production in
order to prevent the loss in value occurring naturally by the discount factor. Hence, both mill and
leach average grades are significantly higher in the earlier periods and gradually decrease until
they become equal to the minimum yearly average grade required by the process destination. The
Figure 5.7 shows the risk behavior of the mine plan over the years. It is apparent that the earlier
years of the production, the optimizer mines from the less riskier areas. This is indeed true in
practice where the approved business plan must deliver the amount of ounces promised to the
shareholders, therefore the confidence level on the processed ore tonnage plays a key role. The
riskier areas are postponed to the later years of the production since the confidence level on the
riskier areas can be always increased by adding more drill holes later on.
As it was mentioned before the new integer solution algorithm also provides the theoretical
upper bound on a given solution. The theoretical upper bound is calculated by solving the mining
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Table 5.9: Summary of the results
LP NPV @ 12.5 % $1,581,250,000
IP NPV @ 12.5 % $1,581,085,000
OPTIMALITY GAP % 0.01%
SOLUTION TIME 5 h 30 min
The yearly schedules are presented on a plan view in Figure 5.9, north-south and east west
cross sections are shown in Figure 5.8 and Figure 5.10 respectively. Once the pit outline is
established based on the yearly production schedules; it is worth making a comparison with the pit
outline generated by the traditional ultimate pit as shown in Figure 5.11 and Figure 5.12. First of
all, it can be clearly seen in Figure 5.11 that the undecided blocks colored with blue on the
traditional ultimate pit plan view received yellow or pink colors on the schedules which shows that
the ore blocks are not blindly designated to the mill just because there is more recoverable value;
instead the state of the system determined the best destination for each block. Secondly, the
ultimate pit is essentially an unconstrained max closure where the mine plan is a constrained max
closure. Hence, the true ultimate pit can be only determined once the production schedule is
generated.
Figure 5.8: Yearly production schedules on East-11075 cross section
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Figure 5.16: North- 43000 cross section taken from the ultimate pit
5.2.2 Mine Production Schedules
The optimal mine plan will be generated for a time horizon of 8 years with the
implementation of the new integer solution algorithm. The discount factor will be assumed as 7%.
The production requirements encompass crusher capacity, mill capacity and total mining capacity
as shown in Table 5.11. The leach pads have enough capacity to handle all the leach materials,
henceforth there is no need to include a leach capacity constraint into the model. There are 4
stockpiles at the site with initial capacities and the average grades shown in Table 5.12. Moreover,
the stockpile materials can be only processed at the mill. However, the stockpile materials will be
treated at the old mill for the first 6 months and after that a new mill will be available to treat the
stockpile materials with enhanced recoveries. Table 5.12 also shows that the processing costs of
the stockpile materials at the old mill are lower than the costs at the new mill.
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Table 5.11: Mine plan production scheduling requirements
CRUSHER MILL MINING
PERIOD
CAPACITY CAPACITY CAPACITY
1st 6months 11,000,000 906,660 21,500,000
2nd 6months 11,000,000 906,660 21,500,000
3 22,000,000 1,813,320 43,000,000
4 22,000,000 1,813,320 43,000,000
5 22,000,000 1,813,320 43,000,000
6 22,000,000 1,813,320 43,000,000
7 22,000,000 1,813,320 43,000,000
8 22,000,000 1,813,320 43,000,000
9 22,000,000 1,813,320 43,000,000
Table 5.12: Initial stockpile parameters
RECOVERIES COST ($/t)
STOCKPILES TONNAGE AVG. GRADE
OLD MILL NEW MILL HAULAGE OLD MILL NEW MILL
STK 1 436000 0.150 (oz/t) 60.0% 93.0% 0.5 16.1 19.2
STK 2 385000 0.070 (oz/t) 10.0% 90.0% 0.5 16.1 19.2
STK 3 614000 0.055 (oz/t) 52.4% 87.0% 0.5 16.1 19.2
STK 4 1338000 0.085 (oz/t) 65.0% 90.0% 0.5 16.1 19.2
Since the stockpile materials will be treated differently for the first 6 months, the
production requirements of the first year are split into 6 months intervals. After the first year, the
mine plan will be generated based on yearly intervals. The results of the mine production schedule
are illustrated in Table 5.13 where all the production requirements are honored. First of all, the
scheduler prioritized the stockpile 4 to feed the mill in the first 6 months since the stockpile 4 has
the highest recovery value. The second half of the first year, stockpile 1 is fully consumed since
the stockpile has the highest average grade which translates into a higher profit that needs to be
realized before reduced by a discount factor. Also, no mill material is mined from the pits during
the second half of the year which shows that the average grade of the ore available from stockpile
1 and stockpile 4 is higher than the available ore in the pit. Furthermore, the behavior of the tonnage
and the average grade of the material processed in the mill is shown in Figure 5.17 and the tonnage
and the average grade of the material treated in the leach pads is shown in Figure 5.18. It can be
observed in Figure 5.17 that the mill capacity is not filled in years 4,7,8 and 9. However, Figure
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The case studies demonstrate the behavior of the components of the mining system, their
interactions with each other, and how the targets of system production, grades and risks can be
realized with a true optimization technique. The strength of the new integer solution algorithm will
be further supported by the results obtained from the implementation on various types of mine
production scheduling problems. Table 5.15 presents the results generated by scheduling the
McLaughlin deposit with pre-determined destinations, on various ranges of time horizons and
different uniform slope angle requirements by enforcing process capacity, grade blending and risk
blending constraints. The optimality gaps range from 0.0003% to 1.3%. The next Table 5.16
illustrates the results obtained by scheduling the McLaughlin deposit for 3, 5 and 10 years using a
dynamic cutoff grade strategy technique which allows the optimizer to pick the best destination
for the blocks and with uniform slope angle requirements. The mine plan is constrained with
(cid:2868)
mill and leach capacity, mill and leach grade blending, and risk blending constraints. The total
(cid:886)(cid:887)
number of variables range from 2.2 million to 7.3 million and the resulting optimality gaps range
from 0.01% to 0.05%. Table 5.17 demonstrates the solutions obtained for the multi pit Gold deposit
that consists of 17 geotechnical zones. Both mine plans are scheduled for 9 years and subject to
the same total mining capacity, mill capacity and crusher constraints, however they are modeled
under different slope angle requirements which changes the schedules drastically. The optimality
gaps of 0.18% and 0.55% are again significantly small.
So far, 15 large scale open pit mining problems have been scheduled by implementing the
new integer solution algorithm. The problems were subject to various pit slope requirements, multi
destinations and multiple capacity and blending constraints with upper and lower bounds. The
results demonstrate that for 11 out of 15 problems, the optimality gaps are less than 0.2%, three
problems have a gap between 0.2% and 0.7% and only one problem ended up with an optimality
gap of 1.3%. Although it cannot be proven that the integer solutions are true optimal integer
solutions, we know that the optimality gap generated between the true optimal LP and the true
optimal IP solution of the small 2D integer example in the previous chapter was 3.75%. Moreover,
it was also mentioned a couple times by the developers of the BZ algorithm that the integrality
gaps between the linear relaxation of the problems and the integer programs are often small. Hence,
the fact that resulting gaps are so small may show that the integer solution to the problem may be
the true optimal integer solution, if not the tightness of the gaps proves the quality of the integer
solutions.
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CHAPTER 6.
CONCLUSIONS
The production scheduling problem is a critical component of open pit mine planning
whose solution impacts the cash flows of the operations. As stated in the introductory paragraph
of this thesis, an open pit mine production schedule that maximizes the profitability of the mining
operation subject to operational constraints is a very important issue for any mining problem. A
complex mine production scheduling system requires a mathematical model in order to integrate
all the factors characterizing the interactions between the blocks and the operational mining
system. So far there is no known algorithm, either commercially available or presented in the
literature, that can provide an optimal integer solution to the open pit mine production scheduling
problem with capacity constraints together with lower and upper bound blending constraints.
The major contributions from the research discussed in this thesis are listed below:
1. Developed a novel integer solution algorithm which can solve the block by block mine
production scheduling problems modeled with multi capacities, grade blending, grade
uncertainty, stockpiles, variable pit slopes, multi destinations and truck hours.
2. Developed a program in C++ platform to implement the new integer solution algorithm to
the large scale open pit mining problems.
3. The new integer solution algorithm achieved optimality gaps less than 1.3% in all the large
scale open pit mine production scheduling problems tested which is the best ever reached.
4. Developed algorithm can be used as a phase design tool that serves as a practical guide for
mine planning.
5. Developed a new cone pattern generation scheme which can handle variable pit slope
angles based on complex geotechnical zones or multiple azimuths with any size block
dimensions.
6. Introduced a new variable substitution methodology to formulate the open pit mine
production scheduling problem with significantly reduced number of sequencing
constraints.
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7. Developed new refining rules to orthogonalize the set of partitions between the consecutive
iterations of the new integer solution algorithm which is different than the
orthogonalization procedure given in Bienstock Zuckerberg (2009).
The major contributions are discussed further in detail in the following paragraphs. These
contributions should have a significant impact in the field of mine planning and a major
enhancement to industry in their desire to maximize profitability.
In this thesis, a novel integer solution algorithm which can solve the mine production
scheduling problems modeled with multi capacities, grade blending, grade uncertainty, stockpiles,
variable pit slopes, multi destinations and truck hours was developed. Since the new integer
solution algorithm also solves the LP relaxation of the mine production scheduling problem as part
of the solution process, the theoretical upper bound is provided together with the integer solution
which will allow the user to assess the quality of the integer solutions from the resulting optimality
gap.
A program was developed on a C++ platform to implement the new integer solution
algorithm. The program was tested on 15 large scale open pit mining problems subject to various
pit slope requirements, multi destinations and multiple capacity and blending constraints with
upper and lower bounds. 11 out of 15 problems, the optimality gaps were less than 0.2% and the
highest optimality gap encountered was 1.3%. Furthermore, the strength of the proposed integer
solution algorithm is highlighted by the ability of solving open pit mine production scheduling
problems that have more than 7 million variables as an integer problem with an optimality gap as
small as 0.01 % within 5 hours 30 minutes as an example.
The developed software in this thesis can also function as a phase design tool that serves
as a practical guide for mine planning. Since the underlying algorithm generates optimal plans; the
pushback generation with the new integer solution algorithm will honor all sorts of productional
requirements which makes the solution strategy far more advanced than the traditional phase
design methods with parametrization techniques.
A new cone pattern generation scheme was developed in this thesis which can handle
variable pit slope angles based on complex geotechnical zones or multiple azimuths with any size
block dimensions. The technique will minimize the number of arcs required to represent the actual
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pit slope angles on a block level, which will allow the new integer solution algorithm to work with
fast solution times.
A new mathematical model was developed in this thesis by introducing the dual path
double “by” variable “ ” by summing the single path “by” variable “ ” across the
(cid:3047) (cid:3047)
destinations. The mathematical model incorporates the production and processing capacities, grade
(cid:1873)(cid:3029)(cid:3031) (cid:1875)(cid:3029)(cid:3031)
blending constraints, risk blending constraints, multi destinations, stockpiles and variable pit slope
angles on a block by block level. The derived model represents a generic form which will make it
easier for the operator to adjust the model based on the needs of the operation.
As the new integer solution algorithm requires master problem to be formulated with the
set of orthogonal columns, new refining rules were developed in order to orthogonalize each
integer feasible “orthogonal” column from the current iteration`s partition set with each one of the
“orthogonal” partitions from the previous iterations `s partition set.
6.1 Recommended Future Work
The new integer solution algorithm developed in this thesis will contribute to the ongoing
research in mine production scheduling as well as to the general fields of operations research.
The traditional approach towards solving a mine planning problem is limited from a
deterministic point of view which neglects the level of uncertainty involved in a mining operation
at the expense of not meeting the production targets promised to the shareholders. As the
commodity price and the ore grade are the major uncertain components of the complex mining
system, instead of running a plan by incorporating their estimated value, a wide range of possible
values should be generated by simulations and the simulated values should be considered in the
mine plan. This requires a stochastic modeling which has a limited implementation in the mine
planning due to the massive number of variables attained by the simulations. The currently adapted
solution methods are heuristic methods which are not fundamentally mathematically sophisticated
so as to define the optimality gap. On the other hand, we know that the decomposition algorithms
can guarantee the optimality of the linear relaxation solutions. Therefore, research into the
implementation of the new integer solution algorithm to solve the large scale stochastic mine
production scheduling algorithm is suggested.
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Another important direction of future research relates to the development of a methodology
to generate practical mineable phases which is the widely accepted mine planning approach today.
Using the new integer solution algorithm provided in this thesis will greatly improve this approach
to mine planning. Incorporation of the minimum mining widths to the new integer solution
algorithm should be further investigated.
So far, the new integer solution algorithm has been tested on 15 large scale open pit mine
production scheduling problems subject to various type of constraints and the optimality gaps were
all less than 1.3%. More testing is suggested in order to demonstrate and confirm the tightness of
the optimality gap over a wide range of large-scale mine production scheduling problems and
operations.
The implementation of the new solution algorithm is not limited to the mining industry. In
fact, any precedence constrained production scheduling type problem can be solved with the new
solution algorithm. Therefore, there seems to be an advantage in testing the new integer solution
algorithm on production scheduling problems in general. We know that the totally unimodular
structure of the subproblems allows fast exploitation of the underlying network structure during
the integerizing process. However, there could be an efficient way to generate integer feasible
solution columns by solving the subproblems which do not constitute a totally unimodular
structure. This should be further investigated.
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APPENDIX A.
MODELING WITH VARIABLE PIT SLOPE ANGLES
The iterative nature of the proposed open pit mine production scheduling solution
algorithm requires the dependencies between the blocks based on the required pit slope angles to
be preprocessed. This is accomplished by generating arcs between the blocks and storing them in
map containers when programmed with a C++ coding language. Then the arcs between the blocks
are transformed into sequencing constraints at every iteration of the algorithm. Hence, there is a
strong correlation between the number of arcs generated versus the processing time of the
sequencing constraints and the memory allocated to store these arcs. In order to achieve fast
computing times, the number of arcs generated per block should be optimized. Therefore, a new
cone pattern generation scheme is needed to eliminate the redundant arcs while minimizing the
deviation from the required pit slope angle for block models with any size block dimensions.
Pit slope design is a fundamental element of the mining system where the slope stability is
ensured for the life of the mine, which may extend beyond closure. The safe slope angles are
determined to maximize the safety of the operating personnel and equipment. The pit may have
multiple slope angles depending on the presence of faults, joints, water and the strength of a rock
together with the slope governing boundary conditions and rock mass failure modes (Gilani and
Sattarvand, 2015). The economic contributions of the pit slope angles to the mining operation are
also significant. Pit slope angles form the domain that delineates the boundary of the mining
system. As the pit proceeds deeper, the magnitude of the pit outline expansion will be dictated by
the pit slopes. The operators tend to work with the safest steep slope angles to maximize the
economic benefits. The steepened slopes will enable the production to proceed with less stripping.
Moreover, there could be potential additional ore recovered because of the horizontal and vertical
expansion at the pit bottom due to steeper slopes. (Figure A.1)
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A.1 Background on Variable Pit Slope Angles
The arc generating concept that represents the pit slopes on a block level became a
compelling subject among researchers as the geotechnical properties impose variations on the pit
slopes from one azimuth direction to another also in vertical directions. Researchers have
investigated various methods that would generate an extraction pattern to minimize the deviations
from the required pit slope angles. The earliest research focused on generating a block extraction
pattern based on repetitive implementation of certain block configurations where the block size is
assumed to be cubical. Gilbert (1966) was first to show that given a cubic block model, 1:5:9
configuration gives the best approximation for a pit slope as shown in Figure A.3. Johnson
°
(1968) illustrated that if the pit is operating with a single slope angle different than , a block
(cid:886)(cid:887)
°
configuration pattern can still be implemented by adjusting the size of the blocks. Lipkewich and
(cid:886)(cid:887)
Borgman (1969) proposed the knight`s move pattern for a cubical block model to accomplish
°
slope angles. The method generates arcs from the target block to the 5 blocks on the level above
(cid:886)(cid:887)
and the 8 blocks on two levels above. The selection of the 8 blocks from the second level follows
the movement pattern of the knight in a chess game. This method imposes a minimum search
pattern of 13 blocks. Giannini (1990) showed that repeatedly applying a knight`s move pattern will
result with slope angles ranging from to . Unfortunately, the block configuration patterns
° °
are limited by hard assumptions on cubical block sizes and pit slope angles. It is practically
(cid:886)(cid:883).(cid:890) (cid:886)(cid:887)
°
infeasible to represent the multiple variable pit slope angles and non-cubical block models with
(cid:886)(cid:887)
any block configuration patterns.
The shortcomings of the block configuration patterns shifted the direction of the research
towards cone-based templates. The idea behind the cone-based templates is, once the cone is
constructed on a base block, it is projected to the surface. If the midpoint of any block appears
within the extraction cone, the block must be extracted before removing the base block. Moreover,
the cone-based search methods can handle variable pit slopes with non-cubical block models. Chen
(1976) implemented the cone-based search pattern to the predetermined pairs of azimuths and dips.
The author used linear interpolation technique to smooth the pit slopes between the azimuths.
Giannini (1990) proposed the idea of a minimum search pattern to remove the redundant arcs
generated during the cone construction process. The search pattern is generated as follows:
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Given a vertex “ ” representing the identification of the base block, the geometric pattern “ ”
that obeys the pit slope on various azimuths can be created by forming the set
(cid:1861) (cid:1842)
that hol (cid:1871)d (cid:3045)s the blocks to be remove (cid:1828)d(cid:3045) w hen the block with vertex “ ” is extra (cid:1827)ct(cid:3036)ed =.
F {o (cid:1854)r (cid:2869) ,e (cid:1854)a (cid:2870)c ,h (cid:1854) (cid:2871),… ,(cid:1854)(cid:3038) }there exists the preceding blocks selected with the same geometr i (cid:1861)c pattern “ ”
captured i (cid:1854)n(cid:3037) ∈set (cid:1827) (cid:3036) . In other words, for every block in set from , there exists a se t (cid:1842) o f
preceding block (cid:1827)s (cid:3037)from to determined by the geomet (cid:1827)ri(cid:3036)c patter (cid:1854)n(cid:2869) “
(cid:1872)(cid:1867) (cid:1854)
”(cid:3038). Again, there may be
more preceding sets for every block within the sets of to . The blocks from each one of the
(cid:1827)(cid:2869) (cid:1827)(cid:3038) (cid:1842)
preceding sets are identified until the top level is reache (cid:1827)d(cid:2869) and
(cid:1827)
s(cid:3038)tored in a set “ “. If all the blocks
extracted from the set “ “ generates a conical pit that has the slope angle equal to for each
(cid:1846)
on azimuth , then
(cid:1846)
is called a search pattern for the block “ ”. A pattern g (cid:1871)e(cid:3045)nerated by
(cid:1870)
minimum number of elements is called minimum search pattern (Giannini, 1990). If the pit
(cid:1828)(cid:3045) (cid:1827)(cid:3036) (cid:1861)
consists of multiple azimuths, the author implements the linear spline interpolation technique to
calculate the slope angles for different directions. Then, the search pattern is generated for different
tolerance level inputs by the user. The number of blocks included in the pattern depends on the
tolerance level. The author suggests that achieving 10% of the required wall slope angle may yield
efficient and practical results. Khalokakaie (1999) proposed a cone method that considers variable
slope angles on four principal directions (East, West, North, South). For a given base block, the
author calculates the radius of the cone on four principal directions per level. Then for each
direction for each level, the author calculates the number of blocks that the cone will mine by
simply dividing the radius of the cone by the width of the block. The next step is to figure out if
the remaining blocks will be included in the extraction cone. The author accomplishes it by using
the ellipse equation. Eventually, the upper area of the extraction cone is split into four regions
where each region falls between the consecutive principal directions and the regions are smoothed
into an ellipse shape. Shishvan and Sattarvand (2012) investigated the spline interpolation
techniques to smooth the pit walls for any number of slope angles in any direction. The authors
compared quadratic, cardinal and cubic splines and concluded that the cubic spline generates
smoother and more representative curves. The authors first identify the coordinates of the points
where the slope line for a given slope direction intersects the blocks on each level and name them
as corner points. Then, the authors generate spline interpolation matrices and solves them for the
corner points on each level. The resulting tangent vectors are inserted into a cubical polynomial
function which is solved by changing the value of “ ” from zero to one in order to calculate the
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coordinates of the points on each spline segment which the author refers to as border points. Then,
for each level, the distance from the center of mass of each block to the cone apex is compared to
the distance from the border points for the same direction in order to determine whether the block
should be included in the cone template. Gilani and Sattarvand (2015) proposed a non-linear
interpolation technique to construct a cone template for variable pit slope angles. The authors first
calculate the radius of the cone for the predefined azimuths in all levels. Then, the authors
implement an inverse distance law to interpolate the radius on azimuths from to with the
° °
increments of . Lastly, the authors calculate the distance from the center of the base block to
(cid:882) (cid:885)(cid:888)(cid:882)
°
the center of the intended block and compare it against the radius of the cone along the direction
(cid:883)(cid:882)
of the intended block in order to specify whether or not the intended block is inside the cone
template.
So far, the cone template generation techniques presented in the literature have been able
to address the shortcomings of the block configuration techniques but smoothing the pit walls with
a cone pattern generation technique that reduces the number of preceding blocks while minimizing
the deviation from the pit slope angles still remains to be perfected. Chen (1976), Giannini (1990)
and Khalokakaie (1999) used linear interpolation techniques to smooth the pit slopes from one
azimuth direction to another. The linear interpolation approach results with polygonal shaped pits
with sharp intersections (Figure A.4a) while nonlinear interpolation techniques achieve smoother
pit walls (Figure A.4b). Moreover, the cone template generated by Khalokakaie (1999) was
restricted with variable pit slopes for only principal directions that limits the practical
implementation. Gilani and Sattarvand (2015) emphasized the shortcomings of the spline
interpolation technique proposed by Shishvan and Sattarvand (2012) based on the modeling
difficulties of the cone template whenever the pit slope consists of few slope directions. Although
the technique proposed by Gilani and Sattarvand (2015) can handle any number of slope angles,
the authors did not mention any pattern that can reduce the number of preceding blocks included
in the cone of the base block.
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the slope angle. The maximum elevation of the search pattern is level 6, which means that the
°
first 5 benches will have even less than 80 arcs connecting the target blocks. The block size plays
(cid:886)(cid:882)
a significant role to determine the extent of the cone pattern. If the cone pattern is generated with
20x20x15ft block dimensions for the same slope angle, the cone template would look like the
°
one demonstrated in Figure A.8.
(cid:886)(cid:882)
Figure A.8: Final cone pattern template for pit slope angle, 20x20x15ft blocks
°
(cid:886)(cid:882)
Another important factor that governs the pattern stopping criteria is the minimum
deviation of the radius from the center of mass of the block closest to the side of the cone. Since
the blocks are discreate objects, the side of the cone will assume to pass from the center of the
block which may shorten or extend the radius of the cone representing the true slope angle. The
proposed algorithm will select the level at which the center of the block is closest to the side of the
cone. If the actual slope angle of the cone template for 50x50x35ft block size is measured for
principal and diagonal directions, it is found that on principal directions the actual slope is and
°
on diagonal directions the actual slope is For 20x20x15ft block size, the actual slope is
(cid:886)(cid:882)
° °
and on diagonal directions the actual slope is
(cid:885)(cid:891).(cid:887) . (cid:886)(cid:882).(cid:888)
A.2.1 Complex Slope Angles Based on
Geot°
echnical Zone
(cid:886)(cid:882).(cid:885) .
The pit may consist of a set of geotechnical domains where each domain inheres a unique
behavior which will lead to a complex slope design process. These domains can also be named
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zones. A medium size open pit mine can have couple dozen geotechnical zones. The zones may
be altered both horizontally and vertically. The geotechnical properties of each zone will determine
the steepest slope angle at which mining can occur without risking pit slope failure. Depending on
the behavior of the rock type, a medium scale open pit mine may have slope angles varying from
to from one zone to another. The schedules generated by mine production plans must
° °
honor these complex slope angles on a block level. If the complex slope angles are not reflected
(cid:885)(cid:882) (cid:887)(cid:890)
precisely in the production plans, the resulting NPVs may be either underestimated or
overestimated depending on how shallow or deep the pit slopes are generated. So far, it hasn’t
been shown in the published literature how to integrate the vertical alteration of the slope zones
into the cone generation techniques.
As was mentioned before, the dependencies between the blocks are formed by connecting
the target block by arcs to each one of the preceding blocks. The arcs must impose the variation of
the slope angles from one zone to another. It is important to keep the number of arcs formed as
low as possible in order to minimize the computing time and the memory usage. The proposed
cone pattern generation algorithm aims to minimize the number of the arcs generated as wells as
the deviations from the required slope angles.
The geotechnical properties of each zone may lead to a simplification of the possible slope
angles within a domain to a unique slope angle. In other words, slope angle will be constant for all
directions within the domain. This interpretation empowers the use of cone pattern generating
algorithm proposed for the constant pit slope angles. If an open pit mine consists of n number of
domains with n number of distinct slope angles, then n number of different cone pattern templates
will be generated which means that each domain will have its own cone pattern template. The arcs
will connect the base block to the dependent block addressed by the cone pattern template
developed for the domain of the base block. If the dependent block does not share the same domain
with the base block, then the arc generation stops. When the dependent block becomes the new
base block, the arc generation will continue by using a new cone pattern template reflecting the
domain of the new base block. The proposed technique can handle any number of geologic
domains.
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The final outline of the cone template derived for the first quadrant is shown in Figure A.16.
Figure A.16: Cone template generated for the 1st Quadrant
The proposed cone pattern generation algorithm can handle any number of azimuths with
any size of block dimensions. The pattern is aimed to stop on such a level that the difference
between the distance from the center of a mass of the base block to the side of the cone and the
cone radius on that level is minimum. This property will enable the algorithm to generate slope
angles closest to the actual slope angle on a given direction. If the slope angles change form one
direction to another, the non-linear interpolation technique will generate smoothed pit walls. The
size of the cone template brings another key advantage to the mine production scheduling problems
by preventing the generation of the redundant arcs. The proposed cone pattern algorithm adds a
strength to the goal of this thesis by enabling the proposed mine production solution algorithm to
work with fast solution times as well as reflecting the actual pit slopes on a block level.
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ABSTRACT
This thesis examines excavation chamber pressure behavior within a 17.5 m diameter
earth pressure balance tunnel boring machine (EPBM) used on the Alaskan Way viaduct
replacement tunnel project in Seattle, Washington, USA. The study examines behavior during
the first 150 rings of tunneling (10% of the project) through till and till-like deposits, granular
soils, and cohesive silts and clays.
A portable laboratory was established on the project site to characterize key properties of
the muck as it came through the screw conveyor and onto the belt conveyor, namely, vane shear
strength, density, slump, consistency and grain size distribution. Testing was performed on
representative muck samples from a series of rings. Machine data, including excavation chamber
pressures, screw conveyor pressures, soil conditioning inputs, and key operating parameters such
as thrust, cutterhead and screw conveyor torque, cutterhead and screw conveyor rotation speeds,
etc., were studied in great detail to determine what parameters influenced chamber pressures and
how. The results of detailed EPBM data analysis supported with field lab test results from muck
testing produced a number of key findings.
Excavation chamber pressures measured by 12 pressure sensors varied up to 3 to 3.5 bar
from crown to invert. Chamber pressures varied during ring mining and standstill, and the
responses from different heights in the chamber were synchronous. Chamber pressure variations
during excavation were influenced by changes in volumetric flow rates into the chamber via the
cutterhead and out of the chamber via the screw conveyor. Increases/decreases in net volumetric
inflow caused increases/decreases in chamber pressure. The magnitudes of pressure changes
were linearly correlated to the net volumetric flow rate changes. A quantitative analysis of these
data produced estimates of chamber material compressibility that could provide useful
information in assessing the effectiveness of soil conditioning.
An understanding of the role of cutterhead force on EPBM advance rate was developed.
By estimating the change in cutterhead force (thrust force minus chamber pressure force), a
relationship to advance rate was observed, i.e., increased cutterhead force increased the advance
rate. The same was not true between thrust force and advance rate. The increases/decreases in
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chamber pressure mentioned above were also related to decreases/increases in cutterhead force
through the mechanical concept of compressibility. Increases in chamber pressure resulting from
material compression and stiffening means that the chamber soil takes on more of the force at the
face (owing to relative stiffness increase). The cutterhead force therefore decreases. The behavior
also works in reverse. The vertical gradients of chamber pressure provided significant insight
into muck consistency and behavior. Magnitudes of gradients matched reasonably well with
muck densities. Changes in gradient both locally and globally provide information about muck
density under pressure and whether the chamber material is locally being compressed and
decompressed. Horizontal differences in chamber pressure were evident throughout mining and
standstill. When cutterhead rotation was clockwise, left side chamber pressures were higher, and
when cutterhead rotation was counterclockwise, right side chamber pressures were higher. The
fluctuation in these horizontal differences was influenced by many parameters including a
possible compressed air gap at the crown, steel/muck adhesion, and conditioning.
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CHAPTER 1 - INTRODUCTION
1.1. Overview
An earth pressure balance tunnel boring machine (EPB TBM or EPBM) is a mechanized
technique for tunneling in soil and soft ground in general. Shown in Figure 1.1, an EPBM
cutterhead with cutting tools rotates at a rate of 1-3 rpm to scrape the soil, guiding it into the
excavation chamber through openings in the cutterhead. The excavation chamber is kept full of
soil (muck) and is used to counteract the pressure of the in-situ soil at the cutterhead. The
excavation chamber soil is extracted in a controlled fashion by a screw conveyor onto a conveyor
belt for disposal.
Figure 1.1: An Earth Pressure Balance (EPB) Tunnel Boring Machine
Tunneling in complex urban areas requires full face support to counteract the geostatic
stresses of soil and groundwater (see figure 1.2). Proper face support minimizes ground
deformation as well as material and water inflow into the EPBM. Face support is provided via
the excavation chamber pressure that is applied by controlling excavation chamber material
inflow and outflow rates. Maintaining adequate face support via chamber pressure becomes more
challenging as the EPBM diameter increases. As shown in Fig. 2, geostatic stresses and
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Current soil conditioning practice is empirical and largely accomplished through trial and
error. It is often difficult to assess whether soil conditioning has been effective. Due to machine
complexity, geological heterogeneity, and ambiguities about conditioning and the mixing
phenomenon, there is no analytical model or process to guide the process of soil conditioning nor
is there a rational approach to gauge conditioning effectiveness based on EPBM performance
data.
The goal of this research is to improve the understanding of EPBM performance and how
it is influenced by soil conditioning. The particular focus of this study is on excavation chamber
pressure on a large diameter (17.5 m) EPBM being used in heterogeneous glacial soils in Seattle,
WA. Using extensive data collected during early tunneling (first 150 rings) combined with field
testing of the conditioned soil properties, this study aims to improve the understanding of
chamber pressure distribution and the influence that soil conditioning has on this pressure.
1.2. Literature Review
In the last 15 years most of the soil conditioning research being conducted has focused on
laboratory testing of conditioner properties, behavior of conditioned soil and foam as well as
modeling the screw conveyor, pressure chamber and last but not least, actual EPB micro models.
Each of these previous studies individually added great value to this field; however, very limited
research has been done on the evaluation of the EPBM’s performance considering actual
machine parameters and muck characteristics.
In EPBM related research, a limited number of papers has focused on the application of
muck engineering properties of soil conditioning in the field. Bezuijen A. and A.M. Talmon
measured the Soil pressures on both sides of the cutterhead of an EPB-shield. This study is
conducted on Botlek Rail railway Tunnel with 9.75m diameter. The ground geology is consists
of saturated Sand. Foam is used for soil conditioning purposes. The measured pressures are
compared with the pressures at the pressure bulkhead. In this paper it is suggested that the
pressures are not purely hydrostatic but the results are also influenced by the yield strength of the
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muck and the effective stresses. They also demonstrate that at the invert, sensors sometimes
measured effective stresses. Pressure on the bulkhead and in front of the cutterhead are measured
and found to be the same but the pressure gradients can vary remarkably (around 30% difference
in gradients). Gradients vary from around 10-15kPa/m at the tunnel face. In this study chamber
pressure shows a non-symmetrical distribution due to CH rotation.
In a research by Bezuijen et al. (2005), the excavation chamber pressure distribution was
analyzed through saturated Pleistocene Sand with high water permeability. Twenty muck
samples were obtained from the chamber employing a stopcock on the bulkhead in six different
locations of the Botlek rail tunnel alignment in Rotterdam, Netherland. Total pressure at the
bulkhead in nine locations and pore pressure in three locations from top to bottom of the plenum
were measured using pressure gauges.
Bezuijen et al’s measurements of chamber pressure show fluctuation during the mining of
different rings in comparable geology. In some cases, the pressure gradient exceeded the in-situ
soil density and sometimes the pressure gradient was lower than the water density. Chamber
pressure change is linked to adhesion between muck and steel surfaces which leads to higher or
lower pressure gradients due to flow direction in comparison to what density test results shows.
Cutterhead influence on chamber pressure is investigated with horizontal pressure gradient. It is
suggested that more mixture of air-Sand-water is available in one side. In their study, the
pressure gradients and their changes during standstill were not discussed.
Borghi and Mair (2008), through field data collection during the construction of the
Channel Tunnel Rail Link (CTRL) in London, demonstrate the effects of soil conditioning on the
performance of the EPBM. The study focused on soil conditioning treatment and modifications
therein while tunneling through different geology. Furthermore, chamber pressure was visualized
in time intervals to capture the effect of varying conditioning parameters on the chamber
distribution. Geology played a very important role in Borghi’s and Mair’s analysis.
Thewes and Budach (2010) classified the desired soil conditioning parameters of
successful EPBM tunneling. They suggest ranges for muck permeability, consistency, and
plasticity to enable better EPBM control. Additionally, in this paper the operational data was
assessed.
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1.3. Project Background
A 17.5 m diameter, 99.4 m long, 7000 ton EPBM manufactured by the Japanese firm
Hitachi Zosen Corp is used to excavate the Alaskan Way Viaduct tunnel. This machine is the
biggest EPB-shield TBM in the world at the time of this writing. This EPBM is used to tunnel
underneath the city of Seattle to replace the SR 99 Alaskan Way Viaduct, a double-decker
highway. This EPBM is equipped with hundreds of sensors. Both analogue and discrete signals
are recorded and interpreted in programmable logic controller (PLC) systems.
The SR99 tunnel alignment is located in the Puget Sound area of Seattle, Washington
(see figure 1.3). Puget Lowland geological condition is complex due to glacially influenced
sedimentary deposits. During several glacier advances, a new deposit of sedimentary layered
over previous material including glaciolacustrine Clays and silt, glacial outwash Sands and
Gravels, glacial till and till-like soils. Furthermore, soil setting complexity increased with erosion
and re-deposition of some soils, and the local deposition of fluvial and marine sediments. It is
also common for glacial deposits to contain cobbles and boulders. The hydrogeological regime in
the Seattle area is highly influenced by this complex stratification. Material permeability differs
in adjacent soil layers and local perched groundwater and multiple piezometric surfaces are
available.
Borehole
Figure 1.3: SR 99 tunnel alignment and geological sub-surface condition
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Table 1.1: Cutterhead tools on SR99 EPBM
No. Tool Qty.
1 Scraper bit 32
2 Double Disc Cutters 20
3 Cutter bits 260
4 pre-cutting bits 87
5 pre-cutting bits (Replaceable) 69
6 Trim bits 12
7 Fish tail 1
8 Copy cutters 2
9 Emergency bits 45
10 Wear detection bits 14
The cutterhead is 17.45m in diameter and 1.71m deep. Behind the cutterhead is a 17m in
diameter and 1.75m deep excavation chamber (also known as mixing chamber or plenum). Eight
pedestals connect the main bearing to the cutterhead (see figure 2.2). These pedestals help
mixing the material between the center and perimeter of the excavation chamber. A center
agitator (CA) is installed on the machine bulkhead in the center of the main bearing to agitate
muck that lies inside the pedestal circumference. The CA consists of six rounded 2m long mixing
bars and six blades facing back toward the bulkhead. Five static bars are installed on the
bulkhead to help mixing the muck available toward the perimeter of the excavation chamber.
The cutterhead rotates with 24 electric motors providing enough torque to achieve the
required rpm. Six hydraulic motors are installed on the bulkhead to rotate the center agitator. To
transfer the required soil conditioning to the front of the rotating cutterhead, a rotary union was
installed in the center of the cutterhead. All the fluids including water, polymer, and foam are
attached to the rotary union (see figure 2.3, page 9).
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(a) (b
Figure 2.2: Excavation chamber; (a) Cutterhead assembling process; (b) Mixing chamber with center
agitator in red, static bars and cutterhead legs all attached to bulkhead
The rotary union functions as a conduit that transfers fluid between stationary and
rotating equipment. Here, stationary equipment includes the soil conditioning pipes attached to
the rotary union and cutterhead is the rotating object. There is a seal between the stationary and
rotating part in the rotary union to prevent flow from high pressure to low pressure condition.
The rotary union is designed with a shaft and multiple fluid passages. Each of these fluid
passages is routed radially to the outside diameter of the shaft and a housing encircles the shaft
where the passage exits.
The EPBM can be operated in open and closed mode mining. Open mode operation is a
term for continuous mining in stable ground conditions with no to very low hydrostatic pressure
while the excavation chamber is partly filled with the muck. Closed mode operation is
continuous mining in projects with minimal ground loss requirement or in running ground with
high hydrostatic pressure. This requires the excavation chamber to be full of muck in order to
create appropriate pressure equilibrium. Vigilant control of the pressure is needed to perform the
mining operation with minimal ground loss and water inflow. 12 pressure sensors are installed
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on the bulkhead to continuously record total lateral muck pressure in the plenum (see figure 2.4).
The pressure sensors are fluid-filled pancake sensors (see figure 2.5).
Figure 2.3: Rotary union
The machine shield is 17.9m long including a front shield with a length of 7.7m with an
articulation joint, as well as a back shield with a 10.1m length with four seal brushes. Three types
of jacks are used in the shield of this machine. 56 thrust cylinders provide the required force to
displace the EPBM forward. Thrust cylinders push the machine against the precast concrete
lining. 28 articulation jacks are also available to articulate the machine through curves.
Cutterhead support jacks are also available for retracting the cutterhead in case of emergency or
maintenance. Three five-story back up gantries travel with the EPBM shield on rubber tires. All
the pumps, cables, transformers, compressors as well as air, water, soil conditioning agent,
polymer tanks and the scaffolding and stairs are pulled by the machine.
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2.2. Screw Conveyer
The screw conveyor (SC) in an EPB machine acts like a pump that controls the outflow
of material on the machine conveyor belt and dissipates the pressure from the excavation
chamber to atmospheric pressure. Two types of SC are normally used in EPB machines: (1) a
Archimedes screw SC with a shaft and a continuous auger of specific thickness, flight, and pitch
size, and (2) a ribbon type SC without a shaft and with a conduit in the center. Ribbon type SCs
accept higher boulder sizes, but because of the conduit in the center, the efficiency is lower than
the shaft type. In the SR99 tunnel, the SC being used is a ribbon type (see figure 2.6).
Figure 2.6: Ribbon SC of SR 99 tunnel project
In projects with higher hydrostatic pressure and for a better control over excavation
chamber (EC) pressure, two SCs are used (see figure 2.7). SC2 is usually not full with material
and SC1 is always packed with material. The rotation speed should always be higher than
in order to prevent cloggage in between the two SCs. SC θ1S Ca2nd SC2 are inclined 37 and 10
d θeSgC1rees from the horizontal, respectively. SC1 has 9 flights and SC2 has 17 flights. Four gates
are available along the SC1 and SC2. Gate G0 is manually controlled and is only operated in
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case of emergency or when maintenance is required. Gate G1 is a vertical sliding gate that is
controlled hydraulically by the operator and it is located by the end of SC1. Gate G3 is located
by the end of SC2, and it is a guillotine gate operated hydraulically by the operator. Gate G2 in
this machine is only for emergencies and is also controlled hydraulically. SC1 has one hatch on
top and SC2 has three hatches on the side for maintenance. Four earth pressure sensors were
installed on the SC1 casing and six earth pressure sensors were installed on the SC2 (see figure
2.7). The last sensor is installed between gate G2 and G3. A conveyer belt transfers the muck
from the SC chute to a continuous conveyor system which eventually dumps the material outside
the tunnel portal.
Figure 2.7: Location of gates and earth pressure sensors on the SC1 (left) and SC2 (right)
2.3. Soil Conditioning:
Soil conditioning is the process of preparation and injection of foam, polymer, bentonite,
water, and other additives to the ground material (see figure 2.8). Each of these additives is
injected and mixed with the soil for a specific requirement in the EPBM. Injections are
performed through ports in three different locations including cutterhead front, excavation
chamber (bulkhead), and screw conveyors.
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Figure 2.13: Air flow system
2.4. Delivery of Soil Conditioning
The SR99 EPBM has an extensive system of tanks, pumps and ports to deliver the soil
conditioning foam and solution through the cutterhead, in the excavation chamber and in the
screw conveyors. 39 ports are used for foam and polymer conditioning, including 22 ports on
the cutterhead, 8 ports on the bulkhead (rear face of the excavation chamber), 3 ports in screw
conveyor 1 and 6 ports in screw conveyor 2 (see figure 2.14). The majority of the 39 ports (24
ports) are used to deliver foam only. The remaining 15 of the 39 ports can be used to inject foam
or liquid (see figure 2.14). Five ports on the cutterhead, four ports in the chamber, 3 ports in
screw conveyor one, and two ports in the beginning of screw conveyor 2 are capable of injecting
foam or polymer). Among 22 cutterhead ports, 4 ports are injecting on the cutter outer rim. Two
set of port in on the cutterhead perimeter are set up to switch the flow in 30 seconds (see figure
2.14). These ports are located close to each other and one is closer to the cutterhead outer rim.
This suggests that with 1 rpm cutterhead rotation speed, in the first 30 seconds cutterhead travels
180 and during this period one port is injecting and in the second 30 seconds that cutterhead
travels back to the initial position ( ) the seconds port is injecting foam.
°
360°
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Cutterhead Rear View Bulkhead Rear View
P P
P
P P
P
P
P
P P
P
P
P
P Open Ports
Screw Conveyer #2
Ports Switching Flow
Every 30 Seconds
P
Closed Ports
P Polymer Port
Screw Conveyer #1
Figure 2.14: Foam and Polymer Ports Alignment and Orientation
Each port distributes foam or liquid that is supplied by 20 conditioning solution pumps
(see figure 2.12) and 20 foam generators (see figure 2.10). Each foam generator has a dedicated
pump and some ports have individual pumps and foam generators (these ports are identified in
figure 2.18). Other ports are grouped and operated by a pump and a foam generator.
Some foam generators provide foam to more than one port. Ports can be either on or off
(see figure 2.15). Two 10 (m3) tanks provide the system with conditioning solution. Each tank
has a specific recipe of diluted surfactant and polymer solution. The concentration of surfactant
and polymer is controlled by the soil conditioning operator. Polymer is provided from each of the
two 2.0 m3 tanks, and foaming agent is provided from a 5.0 m3 tank, all in the gantry backup of
the EPBM.
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Individually Pumpedto Cutter Head 2 Group Pump CH BH SC#1 SC#2 Individual Pump
4
5 1 F1 F23 - F35 2
55
7 3 F3 F25 - F34 4
Group Pumps
6 F6 F26 F33 - 5
8 F8 F28 F32 - 7
1
8 9 F9 & F20 - - - 14 13 3 1 10
8 8 10 F10 - - F39 17
6
10 6
9 11 F11 - - F38
9 6 3
12 F12 F27 - -
Cutter Head Bulk Head Screw #1
13 F13 F24 - -
15 F15 F29 - -
Individually Pumpedto Cutter Head 16 F16 F30 F31 -
1417
18 F18 & F22 - - -
19 F19 - - F37
Group Pumps
21 F21 - - F36
18
18
21 15 16 19 12 16 15 13 2119 11
12
13
11 16
Cutter Head Bulk Head Screw #1
Figure 2.15: Individually and group pumped foam ports and orientation
Five air compressors in the backup section of the EPBM supply air to the foam
generation guns. Either automatically or manually, the operator controls the solution flow or air
flow rates into the foam generators (see figure 2.11). Twenty pumps are installed to pump
foaming solution to foam generator units. A foam generator is basically a pipe with a matrix of
spongy-like restrains inside with materials such as wire wools and ceramic balls to provide a
station for solution and air to be mixed together and produce bubbles. Pressure sensors are
installed after the foam generators. 36 water injection ports are available, including 24 ports on
the bulkhead and 12 on the cutterhead. 26 Bentonite injection ports (mud slurry) are also
available in the plenum (see figure 2.16). On top of the bulkhead, two air supply and two air
exhaust pipes are also available for emergency to control pressure or release possible
accumulated air on top of the chamber.
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01
#-
1
#
pmuP
02
#-
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A specific concentration of the foaming agent (C ) and optional polymer concentration
f
(C ) are required to mix the foam liquid (also called foam solution). Air and solution flow rates
p
are measured behind the foam gun with the flow meters. The air flow rate divided by the solution
flow rate represents the foam expansion ratio (FER). Due to Boyle’s law gas is compressed and
decompressed with increase and decrease in pressure. The majority of the foam volume consists
of air, and according to Boyle’s law the foam expansion ratio will be dominated by the pressure
difference between the foam gun and the injection port location (ΔP) as well as the temperature
around the injection port. Foam is injected through ports in different locations. The configuration
and flow rates can be controlled automatically with soil conditioning system default setting or
manually by the operator during mining. The amount of foam injection during the mining
operation divided by the amount of soil excavated while the machine advances represents the
foam injection ratio (FIR) (see figure 2.18).
100; (2.1)
QF
FIR = QS ×
100; (2.2)
QB
BIR = QS ×
100; (2.3)
QP
PIR = QS ×
100; (2.4)
QW
WIR = QS ×
(2.5)
QA
FER = QL;
; (2.6)
Qfa
C f = Qw ×100
(2.7)
Qp
Cp = Qw×100
;
l l
QF = Foam Flow rate �m i;n � QS = Soil excavation volume permin�min�;
l l
QA = Air flow rate �min�
;
QL = Liquid flow rate �min�
l l
QB = Bentonite flow rate �m;i n � QP = Polymer f low rate �min�
l
QW = Water flow rate �min�; QL = qfa +qw ;
l l
qw = Water flow rate �min� qfa = Foam agent flow rate �min�
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