Difference between revisions of "Estimation of the potential production rate"

Latest revision as of 00:59, 8 February 2014

Article written by Sci '14: Travis Dominski, Brett Kolankowski, Andrew Marck, Michal Pasternak, Steve Shamba

Production rate and mine life play a large role in determining the project economics. A higher production rate typically allows for lower operating costs, while the subsequent shorter mine life maximizes the Net Present Value of ore extraction. However, a higher production rate requires a greater capital cost as larger equipment and infrastructure is required. Estimation of production rate is a problem that has been looked at by many scholars. The most well-known scholar to look at the problem was H. K. Taylor who developed the empirical Taylor’s Rule, which is a rule of thumb that is commonly taught to Mining Engineering students. While the most popular, Taylor’s Rule is not the only method that can be used when estimating production rate. Taylor’s Rule only considers tonnage, while other methods use the deposit's grade and financial factors.

Taylor’s Rule

Taylor's Rule of Thumb for Mine Life

H. K. Taylor a mining engineer working with Placer Development Limited, proposed "Taylor's Law" at a mine valuation and feasibility study seminar in Spokane, Washington in 1976^[1]. This rule was then published in 1977. Taylor realized the need for such a rule since the existing "supposedly optimum mining rate have long been estimated by elementary economic theory, usually by present-value methods, but it has been observed that many such exercises show a bias towards high rates of working that are unachievable or undesirable in practice."^[1] The previous methods had led to inexperienced companies proposing mines with wildly unrealistic rates. Taylor based the emperical rule off of nearly 30 mining projects, mostly young mines. Taylor's rule was tested in 1984 by McSpadden and Schaap, who checked it against 45 open-pit copper deposits.^[2] McSpadden and Schaap found the rule needed to be tweaked slightly, however this finding was due to the narrow range of their mine types compared to Taylor's wide range of mine types.

Equation for Mine Life and Production Rate

The empirical equation for mine life that Taylor developed is:

$Life=0.2{\sqrt[{4}]{Tonnage}}$

The equation can be used to find the production rate by

$Production(mt/day)={Tonnage \over MineLife\times OperatingDays}$

Assuming a mine operating 350 days a year, Taylor's rule gives the equation

$Production(mt/day)=0.0143\times Tonnage^{(0.75)}$

Taylor's rule was originally developed for metric tonnes, but it can be applied to short tons. This is because the difference between metric tonnes and short tons is lessened by the 4th root, so the resulting error in mine life would only be +2.5%.

Limitations

Taylor identified several senarios where his rule of thumb does not work well: ^[1]

Old mines far into the stages of operation (can work faster)
Unusually large deposits (>200Mt as suggested mining rate would be unobtainable)
Very deep, flat ore bodies (production limited by hoisting limits of shaft)
Steeply dipping tabular or massive deposits that are worked in steps towards great depths (limited to rate of deepening of the working, use #Tatman (2001))
Erratically mineralized multi-vein systems (production rate limited to discovery rate)

USBM/USGS Modifications

Modifications of Taylor's Rule

Taylor's rule has been modified and tweaked by the United States Bureau of Mines (USBM) and its successor the United States Geological Survey (USGS) to a large and more modern set of data.^[3] All the modifications of Taylor's rule use the same general relationship, and just revise the variables.

D.A. Singer, W.D. Menzie and K.R. Long revised Taylor's rule in 1998, based on a data set of 41 open pit gold-silver mines^[4]. Their model found that appropriate rates for open pit gold-silver mines should be significantly higher than what Taylor's Rule suggests. Their resulting equation was:

$Production(mt/day)=0.416\times Tonnage^{(0.5874)}$

D.A. Singer, W.D. Menzie and K.R. Long also adapted Taylor's rule for underground massive sulfide deposits in 2000.^[5] With this modification it is clear that the Taylor's rule overestimates the underground mining rate since it was calibrated to mostly open pit mines. The data used for this calibration included mines using mining methods such as room and pillar, cut and fill, crater retreat, shrinkage slope, or sublevel longhole mining methods. The resulting equation for underground massive sulfide deposits is:

$Production(mt/day)=0.0248\times Tonnage^{(0.704)}$

Long and Singer further studied Taylor's rule in 2001 and calibrated it to 45 open pit copper mines.^[6] The open pit copper model proved to have a curve halfway between the 1998 gold-silver curve and Taylor's Rule. This is due to base metal mines having less aggressive production rates than precious metal mines.

$Production(mt/day)=0.0236\times Tonnage^{(0.74)}$

The latest study on Taylor's rule was completed by Long in 2009.^[7] Long's study is the most extensive of all studies looking at the relationship between production rate capacity and reserves. The study looked at 342 open pit and 197 underground mines located in the Americas and Australia. Long found that there was a significant difference between the production rate of underground vs. open pit and block caving. The equation found for underground deposits was found to be:

$Production(mt/day)=0.297\times Tonnage^{(0.562)}$

The equation for open pit and block caving deposits was found to be:

$Production(mt/day)=0.123\times Tonnage^{(0.649)}$

Long's 2009 study also found that introducing the variables grade and capital cost played a factor in estimating production rate, however expected tonnage was the primary factor. Long did generate equations involving grade and capital cost for open pit, however the inputs for these equations were not clarified.

Applicability

Many scoping studies use the original Taylor's Rule as a starting point for production rate regardless of the type of mine. It is clear from the USBM/USGS modifications of Taylor's Rule that a better estimate is possible adding the variable of mine type as well as expected tonnage. The table below acts as a guide to selecting an appropriate version of Taylor's rule. The general equation for Taylor's rule is:

$Production(mt/day)=a\times Tonnage^{(b)}$

Mine Type	a	b	Source	# Mines
Unknown	0.0143	0.75	Taylor (1986)	~30
Open Pit - Gold/Silver	0.416	0.5874	Singer, Menzie, Long (1998)	41
Open Pit - Copper	0.0236	0.74	Singer, Long (2001)	45
Open Pit/Block Caving - Other	0.123	0.649	Long (2009)	342
Underground - Massive Sulfide	0.0248	0.704	Singer, Menzie, Long (2000)	28
Underground - Other	0.297	0.562	Long (2009)	197

Sample Scenario

For this senario, a theoretical block caving iron mine will be considered. This mine has an expected reserve of 50 million tonnes.

First, an appropriate version of Taylor's rule must be picked. Since this mine is a room and pillar mine, the most appropriate version would be Long's 2009 open pit and block caving modification. The a factor will be 0.123 and the b factor will be 0.649. The resulting equation will be:

$Production=0.123\times 50,000,000^{(0.649)}=12,205tonnes/day$

Assuming the mine operates 350 days a year, the mine life would be:

$MineLife={Tonnage \over Production(mt/day)\times OperatingDays}={50,000,000 \over 12,205\times 350}=11.7years$

Other Methods

Wells (1978)

In 1978, H.M. Wells published the paper “Optimization of mining engineering design in mineral valuation”, which proposed maximizing the present value ratio (PVR) in order to find the optimal production rate.^[8] PVR is the ratio of PVOUT (the present value of positive cash flows) to PVIN (the present value of negative cash flows). A PVR greater than 1 represented a profitable production rate while a PVR less than 1 was an unprofitable production rate. The optimal production rate is the rate that causes the PVR to be at its maximum.

Lizotte and Elbrond (1982)

Y. Lizotte and J. Elbrond researched optimization of production rates in 1982. They approached the problem using open-ended dynamic programming and created a model for the problem. However they concluded that there were vast differences between their model and realistic mining.^[8]

Cavender (1992)

B. Cavender took a theoretical approach to determining appropriate mine life by looking at the finance side of mining.^[8] He developed three techniques for finding the mine life that optimized the NPV of the project. Cavender looked at cash flow, stochastic risk modeling, and option pricing. Since the model deals with a hypothetical mining and does not include realistic mining constraints, it has little application to real mine design.

Smith (1997)

L.D. Smith in 1997 found that a mine's production rate was better determined by a range than a specific point.^[8] Smith's paper proposed a range of production rates with an upper limit as the rate that resulted in the highest NPV. The lower limit of this range was found to be the rate that best repaid capital costs.

Tatman (2001)

C. R. Tatman found that 35% of underground mines with steeply dipping tabular deposits did not manage to meet their estimated production rate.^[9] This indicated that there was a need for a new rule of thumb other than Taylor's rule. By studying 60 mines Tatman found that the production rate of steeply dipping deposits was based on the geometry of the ore body rather than the size. Tatman proposed the following empirical formula to estimate production rate:

${ProductionRate(tonnes/year)=RateFactor\times RateMultiplier}$

The rate factor is equal to tonnes/vertical meter, while the rate multiplier (found using table below) is an empirical value based on the deposit thickness and the risk the mine designer is willing to take. The level of risk is the probability that the project will not meet the resulting production.

Deposit thickness	Rate multiplier
(metres)	Low risk (<10%)	Moderate risk (10-30%)	High risk (>30%)
< 5	< 20	20 to 50	> 50
5 to 10	< 50	50 to 70	> 70
> 10	< 30	30 to 70	> 70

Abdel Sabour (2002)

Effects of various parameters on optimal production rate (Abdel Sabour)^[8]

Using a mathematical model, S.A. Abdel Sabour looked at the effect of various physical, economic and financial factors on the optimal production rate.^[8] As seen in the graphs on the right, it was found that the optimal production rate increases with both the tonnage and grade of the deposit. A higher gold price resulted in a higher production rate. The production rate was also dependent on the expected growth rate of gold prices, with a higher growth rate resulting in a lower production rate. If mining cost growth rate is expected to be high, the optimal production rate is should also be high to avoid higher mining costs in later years. The final factor considered was the cost of capital (discount rate). It was found that for a low (~5%) and high (~35%) cost of capital, the production rate should be low, however between these points of cost and capital, the production rate should be higher.

Abdel Sabour's definition of the optimal production rate is the rate that generates the highest NPV. To obtain the optimal rate this microeconomic theory was applied. Since these optimal production rates look at economic theory rather than engineering design constraints, his work is not useful on it's own to estimate production rate. It can however can be used to tweak the results of other production rate estimates, such as Taylor's rule.

Half Vertical Tonnage

The half vertical tonnage rule of thumb is used in steeply dipping mechanized vertical deposits. The rule is similar to Taylor's rule as it is based of tonnage, however it uses the average tonnage divided by depth instead. The rule is as follows:

$ProductionRate={1 \over 2}\times {Tonnage \over Depth(ft)}$ ^[10]

Alternatively a factor of 1/3 may be used instead of 1/2.

Summary

Taylor's rule is the best way get a preliminary estimate of the production rate and the mine life during mine design. This is due to its simplicity of calculation since it involves only one variable, as well as the real world applicability of the rule since it is built upon real world data. Modifications by the USBM/USGS should be considered when using Taylor's rule, as they tweak it to better suit the type of mine.

For the types of mines where Taylor's rule is not applicable, other methods such as Tatman's rule of thumb should be used.

References

↑ ^1.0 ^1.1 ^1.2 Taylor, H.K., 1986, Rates of working mines; a simple rule of thumb: Transactions of the Institution of Mining and Metallurgy, v. 95, section A, p. 203-204.
↑ McSpadden, G.M. & Schaap, W. 1984. Technical note – A test and comment on Taylor’s rule of mine life. The AusIMM Bulletin and Proceedings, 289(6):217-220.
↑ Abdeljalil, M. "A Critical Review Of The Inputs To Long Range Mine Planning Of Open Pit Porphyry Type Copper Deposits." (2013)
↑ Singer, D.A., Menzie, W.D., and Long, K.R., (1998). A simplified economic filter for open-pit gold-silver mining in the United States: U. S. Geological Survey Open-File Report 98-207, 10 p., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of98-207/OFR98-207.pdf
↑ Singer, D.A., Menzie, W.D., and Long, K.R., (2000), A simplified economic filter for underground mining of massive sulfide deposits. accessed February 02, 2014, at http://pubs.usgs.gov/of/2000/0349/report.pdf
↑ Long, K.R., and D.A. Singer, (2001), A Simplified Economic Filter for Open-Pit Mining and Heap-Leach., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of01-218/of01-218.pdf
↑ Long, K.R., (2009), A Test and Re-Estimation of Taylor's Empirical Capacity–Reserve Relationship. accessed February 02, 2014, at http://link.springer.com/article/10.1007%2Fs11053-009-9088-y
↑ ^8.0 ^8.1 ^8.2 ^8.3 ^8.4 ^8.5 Abdel Sabour, S. A. "Mine size optimization using marginal analysis." Resources Policy 28, no. 3 (2002): 145-151.
↑ Tatman, C.R., Production-rate selection for steeply dipping tabular deposits. Mining Engineering, Vol. 53, No. 10, October:62-64, 2001.
↑ Morin M., personal correspondence February 04, 2014

Annotated Bibliography

Taylor, H.K., (1986) Article hard to very hard to find, available in Douglas Library under reference number TN1 I51

In this paper, Taylor explains the origins of Taylor's rule and discusses the applications and limitations of the rule.

McSpadden, G.M. & Schaap, W., (1984) Article hard to very hard to find, available in Douglas Library under reference number TN1 A936

McSpadden and Schaap evaluate Taylor's rule in this paper to see if it is valid. They find the rule is valid but required adjustment

Abdeljalil, M., (2013)

Abdeljalil's master's thesis partly discusses the modification to Taylor's rule done by the USGS/USBM

Singer, D.A., Menzie, W.D., and Long, K.R., (1998)

The authors of this USBM/USGS publication fitted Taylor's rule to open pit gold and silver mines.

Singer, D.A., Menzie, W.D., and Long, K.R., (2000)

This USBM/USGS paper fitted Taylor's rule to underground massive sulfide mines.

Long, K.R., and D.A. Singer, (2001)

This USBM/USGS paper fitted Taylor's rule to open pit copper mines.

Long, K.R., (2009)

This USBM/USGS paper is the most recent paper on Taylor's rule and used the most extensive set of data to calibrate the rule for both open pit and underground mines.

Abdel Sabour, S. A., (2002)

The author wrote a brief history on previous methods of estimating production rate. The author also used his own mathematical models to investigate the effect on production rate of various deposit and economic factors.

Tatman, C.R., (2001) Article hard to very hard to find, available in Douglas Library under reference number TN1 A525

The author identifies the need for a rule of thumb for steeply dipping deposits. By studying 60 mines, the author develops a new empirical rule of thumb based on deposit geometry and risk.

[taylor-1] 1.0 ^1.1 ^1.2 Taylor, H.K., 1986, Rates of working mines; a simple rule of thumb: Transactions of the Institution of Mining and Metallurgy, v. 95, section A, p. 203-204.

[mcsp-2] McSpadden, G.M. & Schaap, W. 1984. Technical note – A test and comment on Taylor’s rule of mine life. The AusIMM Bulletin and Proceedings, 289(6):217-220.

[num1-3] Abdeljalil, M. "A Critical Review Of The Inputs To Long Range Mine Planning Of Open Pit Porphyry Type Copper Deposits." (2013)

[usbm1998-4] Singer, D.A., Menzie, W.D., and Long, K.R., (1998). A simplified economic filter for open-pit gold-silver mining in the United States: U. S. Geological Survey Open-File Report 98-207, 10 p., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of98-207/OFR98-207.pdf

[usbm2000-5] Singer, D.A., Menzie, W.D., and Long, K.R., (2000), A simplified economic filter for underground mining of massive sulfide deposits. accessed February 02, 2014, at http://pubs.usgs.gov/of/2000/0349/report.pdf

[usbm2001-6] Long, K.R., and D.A. Singer, (2001), A Simplified Economic Filter for Open-Pit Mining and Heap-Leach., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of01-218/of01-218.pdf

[usbm2009-7] Long, K.R., (2009), A Test and Re-Estimation of Taylor's Empirical Capacity–Reserve Relationship. accessed February 02, 2014, at http://link.springer.com/article/10.1007%2Fs11053-009-9088-y

[wells-8] 8.0 ^8.1 ^8.2 ^8.3 ^8.4 ^8.5 Abdel Sabour, S. A. "Mine size optimization using marginal analysis." Resources Policy 28, no. 3 (2002): 145-151.

[tatman-9] Tatman, C.R., Production-rate selection for steeply dipping tabular deposits. Mining Engineering, Vol. 53, No. 10, October:62-64, 2001.

[Morin-10] Morin M., personal correspondence February 04, 2014

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@@ Line 2: / Line 2: @@
-Production rate and mine life, play a large role in determining the [[project economics]].  A higher production rate typically allows for lower operating costs, while the subsequent shorter mine life maximizes the [[Project_economics#Present_Value_Calculation|Net Present Value]] of ore extraction.  However, a higher production rate requires a greater capital cost, as larger equipment and infrastructure is required.
+Production rate and mine life play a large role in determining the [[project economics]].  A higher production rate typically allows for lower operating costs, while the subsequent shorter mine life maximizes the [[Project_economics#Present_Value_Calculation|Net Present Value]] of ore extraction.  However, a higher production rate requires a greater capital cost as larger equipment and infrastructure is required.
-Estimation of production rate is a problem that has been looked at by many scholars.  The most well-known scholar to look at the problem was H. K. Taylor who developed the empirical Taylor’s Rule, a rule of thumb that is commonly taught to Mining Engineering students.  While the most popular, Taylor’s Rule is not the only method that can be used when estimating production rate.  Taylor’s Rule only takes into account tonnage, while other methods use the grade of the ore and financial factors.
+Estimation of production rate is a problem that has been looked at by many scholars.  The most well-known scholar to look at the problem was H. K. Taylor who developed the empirical Taylor’s Rule, which is a rule of thumb that is commonly taught to Mining Engineering students.  While the most popular, Taylor’s Rule is not the only method that can be used when estimating production rate.  Taylor’s Rule only considers tonnage, while other methods use the deposit's grade and financial factors.
 ==Taylor’s Rule==
 [[Image:Taylor%27s_Rule_Mine_Life.jpg|500px|right|thumb|Taylor's Rule of Thumb for Mine Life]]
-H. K. Taylor a mining engineering working with Placer Development Limited, proposed "Taylor's Law" at mine valuation and feasibility study seminar in Spokane, Washington in 1976<ref name="taylor">Taylor, H.K., 1986, Rates of working mines; a simple rule of thumb: Transactions of the Institution of Mining and Metallurgy, v. 95, section A, p. 203-204. </ref>.  This rule was then published in 1977.  Taylor realized the need for such a rule as the existing ''"supposedly optimum mining rate have long been estimated by elementary economic theory, usually by present-value methods, but it has been observed that many such exercises show a bias towards high rates of working that are unachievable or undesirable in practice."''<ref name="taylor"/>  The previous methods had lead to inexperienced companies proposing mines with wildly unrealistic rates. Taylor based the emperical rule off of nearly 30 mining projects, mostly young mines.<!--<ref name="num1">Abdeljalil, M. "A Critical Review Of The Inputs To Long Range Mine Planning Of Open Pit Porphyry Type Copper Deposits." (2013)</ref>-->
+H. K. Taylor a mining engineer working with Placer Development Limited, proposed "Taylor's Law" at a mine valuation and feasibility study seminar in Spokane, Washington in 1976<ref name="taylor">Taylor, H.K., 1986, Rates of working mines; a simple rule of thumb: Transactions of the Institution of Mining and Metallurgy, v. 95, section A, p. 203-204. </ref>.  This rule was then published in 1977.  Taylor realized the need for such a rule since the existing ''"supposedly optimum mining rate have long been estimated by elementary economic theory, usually by present-value methods, but it has been observed that many such exercises show a bias towards high rates of working that are unachievable or undesirable in practice."''<ref name="taylor"/>  The previous methods had led to inexperienced companies proposing mines with wildly unrealistic rates. Taylor based the emperical rule off of nearly 30 mining projects, mostly young mines.  Taylor's rule was tested in 1984 by McSpadden and Schaap, who checked it against 45 open-pit copper deposits.<ref name="mcsp">McSpadden, G.M. & Schaap, W. 1984. Technical note – A test and comment on
-  Taylor's rule was test in 1984 by McSpadden and Schaap, who checked it against 45 open-pit copper deposits.  McSpadden and Schaap found the rule needed to be tweaked slightly, however this finding was due to the specivity of their mine types compared to Taylor's wide range of mine types.
+Taylor’s rule of mine life. The AusIMM Bulletin and Proceedings, 289(6):217-220.
+</ref>  McSpadden and Schaap found the rule needed to be tweaked slightly, however this finding was due to the narrow range of their mine types compared to Taylor's wide range of mine types.
 ===Equation for Mine Life and Production Rate===
@@ Line 24: / Line 24: @@
 <math>Production(mt/day)= 0.0143 \times Tonnage^{(0.75)}</math>
-Taylor's rule was originally developed for metric tonnes, but it can be applied to short tons.  As the difference between metric tonnes and short tons is lessened by the 4th root, the resulting error in mine life would only be +2.5%.
+Taylor's rule was originally developed for metric tonnes, but it can be applied to short tons.  This is because the difference between metric tonnes and short tons is lessened by the 4th root, so the resulting error in mine life would only be +2.5%.
 ===Limitations===
-Taylor identified several senarios where his rule of thumb does not work well:
+Taylor identified several senarios where his rule of thumb does not work well: <ref name="taylor"/>
 *Old mines far into the stages of operation (can work faster)
 *Unusually large deposits (>200Mt as suggested mining rate would be unobtainable)
 *Very deep, flat ore bodies (production limited by hoisting limits of shaft)
-*Steeply dipping tabular or massive deposits that are worked in steps towards great depths (limited to rate of deepening of the working)
+*Steeply dipping tabular or massive deposits that are worked in steps towards great depths (limited to rate of deepening of the working, use [[#Tatman (2001)]])
 *Erratically mineralized multi-vein systems (production rate limited to discovery rate)
 ===USBM/USGS Modifications===
 [[Image:Taylor%27s_Rule_Modified.jpg|650px|right|thumb|Modifications of Taylor's Rule]]
-Taylor's rule has been modified and tweaked by the United States Bureau of Mines (USBM) and its successor the United States Geological Survey (USGS) to a large and more modern set of data.<ref name="num1" />  All the modicications of Taylor's rule use the same general relationship, and just revise the variables.
+Taylor's rule has been modified and tweaked by the United States Bureau of Mines (USBM) and its successor the United States Geological Survey (USGS) to a large and more modern set of data.<ref name="num1">Abdeljalil, M. "A Critical Review Of The Inputs To Long Range Mine Planning Of Open Pit Porphyry Type Copper Deposits." (2013)</ref>  All the modifications of Taylor's rule use the same general relationship, and just revise the variables.
-D.A. Singer, W.D. Menzie and K.R. Long revised Taylor's rule in 1998, based on a data set of 41 open pit gold-silver mines<ref name="usbm1998">Singer, D.A., Menzie, W.D., and Long, K.R., (1998). A simplified economic filter for open-pit gold-silver mining in the United States: U. S. Geological Survey Open-File Report 98-207, 10 p., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of98-207/OFR98-207.pdf</ref>.  Their model found that appropriate rates should for open pit gold-silver mines should be significantly higher than Taylor's Rule suggests.  Their resulting equation was:
+D.A. Singer, W.D. Menzie and K.R. Long revised Taylor's rule in 1998, based on a data set of 41 open pit gold-silver mines<ref name="usbm1998">Singer, D.A., Menzie, W.D., and Long, K.R., (1998). A simplified economic filter for open-pit gold-silver mining in the United States: U. S. Geological Survey Open-File Report 98-207, 10 p., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of98-207/OFR98-207.pdf</ref>.  Their model found that appropriate rates for open pit gold-silver mines should be significantly higher than what Taylor's Rule suggests.  Their resulting equation was:
 <math>Production(mt/day)= 0.416 \times Tonnage^{(0.5874)}</math>
-D.A. Singer, W.D. Menzie and K.R. Long also adapted Taylor's rule for underground massive sulfide deposits in 2000.<ref name="usbm2000">Singer, D.A., Menzie, W.D., and Long, K.R., (2000), A simplified economic filter for underground mining of massive sulfide deposits. accessed February 02, 2014, at http://pubs.usgs.gov/of/2000/0349/report.pdf</ref>  In this modification it clear that the Taylor's rule overestimates the underground mining rate as it was calibrated to open pit mines.  The resulting equation for underground massive sulfide deposits is:
+D.A. Singer, W.D. Menzie and K.R. Long also adapted Taylor's rule for underground massive sulfide deposits in 2000.<ref name="usbm2000">Singer, D.A., Menzie, W.D., and Long, K.R., (2000), A simplified economic filter for underground mining of massive sulfide deposits. accessed February 02, 2014, at http://pubs.usgs.gov/of/2000/0349/report.pdf</ref>  With this modification it is clear that the Taylor's rule overestimates the underground mining rate since it was calibrated to mostly open pit mines.  The data used for this calibration included mines using mining methods such as [[Hard-rock_room_and_pillar|room and pillar]], [[Cut_and_fill|cut and fill]], [[Vertical_crater_retreat|crater retreat]], [[Shrinkage_stoping|shrinkage slope]], or [[Sub-level_open_stoping|sublevel longhole]] mining methods.  The resulting equation for underground massive sulfide deposits is:
 <math>Production(mt/day)= 0.0248 \times Tonnage^{(0.704)}</math>
-Long and Singer further studies Taylor's rule in 2001 and calibrated it to 45 open pit copper mines.<ref name="usbm2001">Long, K.R., and D.A. Singer, (2001), A Simplified Economic Filter for Open-Pit Mining and Heap-Leach., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of01-218/of01-218.pdf</ref>  The open pit copper model proved to have a curve halfway between the 1998 gold-silver curve and Taylor's Rule.  Since the mines are the same type as those used for Taylor's rule, it can be seen that the realistic production rate has increased in the decades since Taylor's rule was first developed.
+Long and Singer further studied Taylor's rule in 2001 and calibrated it to 45 open pit copper mines.<ref name="usbm2001">Long, K.R., and D.A. Singer, (2001), A Simplified Economic Filter for Open-Pit Mining and Heap-Leach., accessed February 02, 2014, at http://geopubs.wr.usgs.gov/open-file/of01-218/of01-218.pdf</ref>  The open pit copper model proved to have a curve halfway between the 1998 gold-silver curve and Taylor's Rule.  This is due to base metal mines having less aggressive production rates than precious metal mines.
 <math>Production(mt/day)= 0.0236 \times Tonnage^{(0.74)}</math>
@@ Line 56: / Line 54: @@
 The latest study on Taylor's rule was completed by Long in 2009.<ref name="usbm2009">Long, K.R., (2009), A Test and Re-Estimation of Taylor's Empirical
-Capacity–Reserve Relationship. accessed February 02, 2014, at http://link.springer.com/article/10.1007%2Fs11053-009-9088-y</ref>  Long's study is the most extensive of all studies looking at the relationship between Capacity and Reserve.  The study looked at 342 open pit and 197 underground mines located in the Americas and Australia.  Long found that there was a significant difference between the production rate of underground vs. open pit and block caving.  The equation found for underground deposits was found to be:
+Capacity–Reserve Relationship. accessed February 02, 2014, at http://link.springer.com/article/10.1007%2Fs11053-009-9088-y</ref>  Long's study is the most extensive of all studies looking at the relationship between production rate capacity and reserves.  The study looked at 342 open pit and 197 underground mines located in the Americas and Australia.  Long found that there was a significant difference between the production rate of underground vs. open pit and block caving.  The equation found for underground deposits was found to be:
 <math>Production(mt/day)= 0.297 \times Tonnage^{(0.562)}</math>
 The equation for open pit and [[block caving]] deposits was found to be:
 <math>Production(mt/day)= 0.123 \times Tonnage^{(0.649)}</math>
 Long's 2009 study also found that introducing the variables grade and capital cost played a factor in estimating production rate, however expected tonnage was the primary factor.  Long did generate equations involving grade and capital cost for open pit, however the inputs for these equations were not clarified.
@@ Line 93: / Line 93: @@
 |}
+===Sample Scenario===
+For this senario, a theoretical block caving iron mine will be considered.  This mine has an expected reserve of 50 million tonnes.
+First, an appropriate version of Taylor's rule must be picked.  Since this mine is a room and pillar mine, the most appropriate version would be Long's 2009 open pit and block caving modification.  The a factor will be 0.123 and the b factor will be 0.649.  The resulting equation will be:
+<math>Production = 0.123 \times 50,000,000^{(0.649)} =  12,205 tonnes/day</math>
+Assuming the mine operates 350 days a year, the mine life would be:
+<math>Mine Life = {Tonnage \over Production(mt/day) \times OperatingDays}= {50,000,000 \over 12,205 \times 350} = 11.7years </math>
 ==Other Methods==
 ===Wells (1978)===
-In 1978, H.M. Wells published the paper “Optimization of mining engineering design in mineral valuation”, which proposed maximizing the present value ratio (PVR) in order to find the optimal production rate.<ref name="wells">Abdel Sabour, S. A. "Mine size optimization using marginal analysis." Resources Policy 28, no. 3 (2002): 145-151.</ref>  PVR was the ratio of PVOUT (the present value of positive cash flows) to PVIN (the present value of negative cash flows).  A PVR greater than 1 represented a profitable production rate while a PVR less than 1 was an unprofitable production rate.  The optimal production rate is the rate that causes the PVR to be at its maximum.
+In 1978, H.M. Wells published the paper “Optimization of mining engineering design in mineral valuation”, which proposed maximizing the present value ratio (PVR) in order to find the optimal production rate.<ref name="wells">Abdel Sabour, S. A. "Mine size optimization using marginal analysis." Resources Policy 28, no. 3 (2002): 145-151.</ref>  PVR is the ratio of PVOUT (the present value of positive cash flows) to PVIN (the present value of negative cash flows).  A PVR greater than 1 represented a profitable production rate while a PVR less than 1 was an unprofitable production rate.  The optimal production rate is the rate that causes the PVR to be at its maximum.
 ===Lizotte and Elbrond (1982)===
-Y. Lizotte and J. Elbrond researched optimization of production rates in 1982.  They approached the problem using open-ended dynamic programming and created a model for the problem.  However they concluded that there were vast difference between their model and realistic mining.<ref name="wells"/>
+Y. Lizotte and J. Elbrond researched optimization of production rates in 1982.  They approached the problem using open-ended dynamic programming and created a model for the problem.  However they concluded that there were vast differences between their model and realistic mining.<ref name="wells"/>
 ===Cavender (1992)===
-B. Cavender took a theoretical approach to determining appropriate mine life looking at the finance side of mining.  He developed three techniques for finding the mine life that optimized the NPV of the project.  Cavender looked at cash flow, stochastic risk modeling, and option pricing.  Since the model deals with a hypothetical mining and does not include realistic mining constraints, it has little application to real mine design.
+B. Cavender took a theoretical approach to determining appropriate mine life by looking at the finance side of mining.<ref name="wells"/>  He developed three techniques for finding the mine life that optimized the NPV of the project.  Cavender looked at cash flow, stochastic risk modeling, and option pricing.  Since the model deals with a hypothetical mining and does not include realistic mining constraints, it has little application to real mine design.
 ===Smith (1997)===
-L.D. Smith in 1997 found that estimating a mines production rate was better determined by a range than a specific point.  Smith's paper proposed that an appropriate range of production rates with a upper limit as the rate that resulted in the highest NPV.  The lower limit of this range was found to be the rate that best repaid capital costs.
+L.D. Smith in 1997 found that a mine's production rate was better determined by a range than a specific point.<ref name="wells"/>  Smith's paper proposed a range of production rates with an upper limit as the rate that resulted in the highest NPV.  The lower limit of this range was found to be the rate that best repaid capital costs.
+===Tatman (2001)===
+C. R. Tatman found that 35% of underground mines with steeply dipping tabular deposits did not manage to meet their estimated production rate.<ref name="tatman">Tatman, C.R., Production-rate selection for steeply dipping tabular deposits.
+Mining Engineering, Vol. 53, No. 10, October:62-64, 2001.</ref>  This indicated that there was a need for a new rule of thumb other than Taylor's rule.  By studying 60 mines Tatman found that the production rate of steeply dipping deposits was based on the geometry of the ore body rather than the size.  Tatman proposed the following empirical formula to estimate production rate:
+<math>{Production Rate(tonnes/year) = Rate Factor \times  Rate Multiplier}</math>
+The rate factor is equal to tonnes/vertical meter, while the rate multiplier (found using table below) is an empirical value based on the deposit thickness and the risk the mine designer is willing to take.  The level of risk is the probability that the project will not meet the resulting production.
+{| class="wikitable"
+|-
+! Deposit thickness !!Rate multiplier!!!!
+|-
+! (metres) !!Low risk (<10%) !!Moderate risk (10-30%)!!High risk (>30%)
+|-
+| < 5 ||< 20 ||20 to 50 ||> 50
+|-
+| 5 to 10 ||< 50 ||50 to 70 ||> 70
+|-
+| > 10 ||< 30 ||30 to 70 ||> 70
+|-
+|}
 ===Abdel Sabour (2002)===
 [[Image:Optimum_Production_Rate.jpg|700x500px|right|thumb|Effects of various parameters on optimal production rate (Abdel Sabour)<ref name="wells"/>]]
-Using a mathematical model, S.A. Abdel Sabour looked at the effect of various physical, economic and financial factors on the optimal production rate.<ref name="wells"/>  For the physical factors it was found that the optimal production rate increases with both the tonnage and grade of the deposit.  A higher gold price resulted in a higher production rate.  The production rate also depended on the expected growth rate of gold prices, with a higher growth rate resulting in a lower production rate.  If mining cost growth rate is expected to be high, the optimal production rate is should also be high to avoid higher mining costs in later years.  The final factor considered was the cost of capital (discount rate).  It was found that for a low (~5%) and high(~35%) cost of capital, the production rate should be low, however between these points of cost and capital, the production rate should be higher.
+Using a mathematical model, S.A. Abdel Sabour looked at the effect of various physical, economic and financial factors on the optimal production rate.<ref name="wells"/>  As seen in the graphs on the right, it was found that the optimal production rate increases with both the tonnage and grade of the deposit.  A higher gold price resulted in a higher production rate.  The production rate was also dependent on the expected growth rate of gold prices, with a higher growth rate resulting in a lower production rate.  If mining cost growth rate is expected to be high, the optimal production rate is should also be high to avoid higher mining costs in later years.  The final factor considered was the cost of capital (discount rate).  It was found that for a low (~5%) and high (~35%) cost of capital, the production rate should be low, however between these points of cost and capital, the production rate should be higher.
-Abdel Sabour's definition of the optimal production rate is the rate that generates the highest NPV.  To obtain the optimal rate this microeconomic theory was applied.  Since these optimal production rates look at economic theory rather than engineering design constrains, his work is not useful on it's own to estimate production rate.  It can however can be used to tweak the results of other production rate estimates, such as Taylor's rule.
+Abdel Sabour's definition of the optimal production rate is the rate that generates the highest NPV.  To obtain the optimal rate this microeconomic theory was applied.  Since these optimal production rates look at economic theory rather than engineering design constraints, his work is not useful on it's own to estimate production rate.  It can however can be used to tweak the results of other production rate estimates, such as Taylor's rule.
+===Half Vertical Tonnage===
+The  half vertical tonnage rule of thumb is used in steeply dipping mechanized vertical deposits.  The rule is similar to Taylor's rule as it is based of tonnage, however it uses the average tonnage divided by depth instead.  The rule is as follows:
+<math>Production Rate = {1 \over 2} \times {Tonnage \over Depth (ft)}</math><ref name="Morin">Morin M., personal correspondence February 04, 2014</ref>
+Alternatively a factor of 1/3 may be used instead of 1/2.
 ==Summary==
 Taylor's rule is the best way get a preliminary estimate of the production rate and the mine life during mine design.  This is due to its simplicity of calculation since it involves only one variable, as well as the real world applicability of the rule since it is built upon real world data.  Modifications by the USBM/USGS should be considered when using Taylor's rule, as they tweak it to better suit the type of mine.
+For the types of mines where Taylor's rule is not applicable, other methods such as Tatman's rule of thumb should be used.
 ==References==
 <references/>
+==Annotated Bibliography==
+'''Taylor, H.K., (1986)'''
+''Article hard to very hard to find, available in Douglas Library under reference number TN1 I51''
+In this paper, Taylor explains the origins of Taylor's rule and discusses the applications and limitations of the rule.
+'''McSpadden, G.M. & Schaap, W., (1984)'''
+''Article hard to very hard to find, available in Douglas Library under reference number TN1 A936''
+McSpadden and Schaap evaluate Taylor's rule in this paper to see if it is valid.  They find the rule is valid but required adjustment
+'''Abdeljalil, M., (2013)'''
+Abdeljalil's master's thesis partly discusses the modification to Taylor's rule done by the USGS/USBM
+'''Singer, D.A., Menzie, W.D., and Long, K.R., (1998)'''
+The authors of this USBM/USGS publication fitted Taylor's rule to open pit gold and silver mines.
+'''Singer, D.A., Menzie, W.D., and Long, K.R., (2000)'''
+This USBM/USGS paper fitted Taylor's rule to underground massive sulfide mines.
+'''Long, K.R., and D.A. Singer, (2001)'''
+This USBM/USGS paper fitted Taylor's rule to open pit copper mines.
+'''Long, K.R., (2009)'''
+This USBM/USGS paper is the most recent paper on Taylor's rule and used the most extensive set of data to calibrate the rule for both open pit and underground mines.
+'''Abdel Sabour, S. A., (2002)'''
+The author wrote a brief history on previous methods of estimating production rate.  The author also used his own mathematical models to investigate the effect on production rate of various deposit and economic factors.
+'''Tatman, C.R., (2001)'''
+''Article hard to very hard to find, available in Douglas Library under reference number TN1 A525''
+The author identifies the need for a rule of thumb for steeply dipping deposits.  By studying 60 mines, the author develops a new empirical rule of thumb based on deposit geometry and risk.