Users Guide for FRCS: 

Fuel Reduction Cost 

Simulator Software

Roger D. Fight, Bruce R. Hartsough, and Peter Noordijk

United States 

Department of 

Agriculture

Forest Service

Pacific Northwest 

Research Station

General Technical 

Report

PNW-GTR-668

January 2006
Authors

Roger D. Fight is a research forest economist and Peter Noordijk is a social 

science technician, U.S. Department of Agriculture, Forest Service, Pacific 

Northwest Research Station, P.O. Box 3890, Portland, OR 97208; Bruce R. 

Hartsough is a professor and Chair, University of California, Department of 

Biological and Agricultural Engineering, Davis, CA 95616-5294.

The Forest Service of the U.S. Department of Agriculture is dedicated to the principle of 

multiple use management of the Nation’s forest resources for sustained yields of wood, 

water, forage, wildlife, and recreation. Through forestry research, cooperation with the 

States and private forest owners, and management of the National Forests and National 

Grasslands, it strives—as directed by Congress—to provide increasingly greater service 

to a growing Nation.

The U.S. Department of Agriculture (USDA) prohibits discrimination in all its programs and 

activities on the basis of race, color, national origin, age, disability, and where applicable, sex, 

marital status, familial status, parental status, religion, sexual orientation, genetic information, 

political beliefs, reprisal, or because all or part of an individual’s income is derived from any 

public assistance program. (Not all prohibited bases apply to all programs.) Persons with 

disabilities who require alternative means for communication of program information (Braille, 

large print, audiotape, etc.) should contact USDA’s TARGET Center at (202) 720-2600 (voice 

and TDD). To file a complaint of discrimination, write USDA, Director, Office of Civil Rights, 

1400 Independence Avenue, SW, Washington, DC 20250-9410 or call (800) 795-3272 (voice) 

or (202) 720-6382 (TDD). USDA is an equal opportunity provider and employer.
Abstract

Fight, Roger D.; Hartsough, Bruce R.; Noordijk, Peter. 2006. Users guide for 

FRCS: fuel reduction cost simulator software. Gen. Tech. Rep. PNW-GTR668. Portland, OR: U.S. Department of Agriculture, Forest Service, Pacific 

Northwest Research Station. 23 p.

The Fuel Reduction Cost Simulator (FRCS) spreadsheet application is publicdomain software used to estimate costs for fuel reduction treatments involving 

removal of trees of mixed sizes in the form of whole trees, logs, or chips from 

a forest. Equipment production rates were developed from existing studies. 

Equipment operating cost rates are from December 2002 prices for new equipment and wage rates for the Pacific Northwest. These cost assumptions can be 

modified by the user. There are four ground-based systems, four cable systems, 

and two helicopter systems. Cost estimates are in U.S. dollars per 100 cubic 

feet, per green ton, and per acre.

Keywords: Cost (fuel treatment), harvesting economics, fuel treatment 

planning, software, simulation.
Contents

1 Introduction

3 Overview of FRCS Simulation

3 Harvesting Systems and Conditions

4 Ground-Based (Tractive) Systems

5 Cable-Yarding Systems

6 Helicopter-Yarding Systems

7 How to Get Started with FRCS

7 Entering Data in FRCS

10 Simulated Harvest Cost Results

11 Constraints and Error Messages

12 Changing Default Machine Cost Values

13 For Advanced Users

13 Description of Calculation Sheets

18 Use of FRCS in Batch Mode

18 Glossary

19 Acknowledgments

19 Metric Equivalents

19 References

21 Appendix
1

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

Introduction

The Fuel Reduction Cost Simulator (FRCS) can be used to estimate the cost of 

operations involved in cutting trees and delivering logs or trees to a landing for 

further processing or handling. It was designed to provide cost estimates for four 

ground-based systems, four cable systems, and two helicopter systems. It uses 

an engineering cost approach to estimate costs for individual machines (Miyata 

1980) and combines machines into systems by using the approach described in 

Hartsough and others (2001). In that regard it is similar to the STHARVEST 

model (Fight and others 2003). The major additions beyond the STHARVEST 

model include the consideration of multiple products (logs and chips) and multiproduct systems including chipping tops, limbs, and small trees. Four additional 

systems have been added: two additional cable systems and two helicopter 

systems. Production equations from additional studies are included, and equipment costs have been updated to December 2002. The FRCS model was designed 

to focus on the kinds of systems and thinnings that are designed to address the 

buildup of fuels in forests that contribute to risk of uncontrollable wildfire. 

Information is included to provide an understanding of system configuration 

and the situations being simulated; however, it was not our intent to provide the 

information needed to determine whether a particular system is appropriate 

for a particular situation. It is the responsibility of the user to get the technical 

and policy information needed to make that decision. The FRCS software, this 

document, and related documents can be found at http://www.fs.fed.us/pnw/data/

soft.htm. 

The kinds of treatments that are contemplated and being implemented to 

reduce fire risk typically include thinning small trees from stands that are judged 

too dense. These small trees occur either in dense single-canopy stands or in a 

lower canopy level in multistoried stands. The prevailing attitude among public 

land managers is that some proportion of these stands needs to be thinned to 

create a desirable mix of future stand conditions to meet forest health and habitat 

objectives and to reduce fire hazard. This generally involves removing part or 

the entire small-tree component of a stand and may include trees across the full 

diameter range of the stand.

The costs of harvesting and utilizing small trees can be prohibitive because 

the unit costs of harvesting smaller diameter trees are generally higher than those 

for larger diameter trees, and the unit costs of harvesting low volumes per acre 

are generally higher than those for higher volumes per acre. The cost penalties 

for harvesting low volumes per acre and small trees differ depending on the type 

of logging system used. For example, harvesting with a cable system is typically 
2

GENERAL TECHNICAL REPORT PNW-gtr-668

more expensive than with a ground-based system and harvesting with a helicopter system is typically more expensive than with a cable system. In addition, 

the incremental cost increases are greater for cable systems than for groundbased systems when lower volumes per acre or smaller trees are harvested. It 

is therefore appropriate to consider at the time the silvicultural prescription is 

developed, the type of harvesting system likely to be used in a stand. With that 

system in mind, harvest costs can be estimated that will indicate the effect of 

different management strategies on harvesting costs and net return. If harvesting 

costs are not considered, prescriptions may be developed that result in costs of 

harvest that exceed the value of the timber or exceed the perceived fuel reduction 

benefits. To accomplish a thinning as a timber sale, there needs to be enough 

value in the trees to be removed to attract bidders. Harvesting costs are often the 

primary issue in whether or not a stand treatment will pay for itself. It is therefore 

important to have reasonable harvesting cost estimates for planning purposes 

in advance of preparing a timber sale or designing a treatment that must be paid 

for with the available budget. The FRCS software was developed for that specific 

purpose. It is intended to give reasonable cost estimates for a general type of 

harvesting equipment and conditions. Because it estimates an average cost for 

several alternative machines in a configuration, it is not intended to provide 

estimates for a specific harvest unit with a specific set of machines.

The FRCS software uses more than 100 productivity equations drawn from 

the literature for machines doing various operations. Machines can be excluded 

from the model by setting their relevance to zero, a process described below. 

Capital and operating costs were developed with replacement purchase prices 

for new equivalent equipment for December 2002. The productivities and costs 

are used to develop stump-to-truck harvesting costs for the 10 system choices 

in FRCS. The resulting costs are an average of costs for each type of machine 

that might be included in a configuration weighted by the appropriateness of 

each machine for the conditions. These relevance weights vary from 1.0 (where 

the study is considered to be highly relevant) to zero in portions of the range 

where the relationships are not likely to be valid. Those interested in the detailed 

assumptions have access to them through the software and can override the 

hourly machine cost rates if they wish. When calculating costs for a harvest 

system, FRCS does not explicitly consider the differences in calculated production rates between individual activities such as felling and yarding. Harvesting 

contractors use a variety of methods to maintain a reasonable balance between 

the activities. The FRCS software does take average system imbalances into 

account because the assumed utilization rates include interactive delays caused 

It is important to have 

reasonable harvesting 

cost estimates for 

planning purposes in 

advance of preparing 

a timber sale or 

designing a treatment.
3

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

by the imbalances. Therefore, the costs reported by FRCS are expected to be 

reasonable on average.

A glossary near the end of this document contains definitions of some terms 

and variables that are important in the proper use of the FRCS software.

Overview of FRCS Simulation

An FRCS simulation typically involves specifying a logging system, partial cut 

or clearcut, average yarding distance, slope, move-in distance, number of acres 

being harvested at that location, whether or not residues will be chipped, and, 

for helicopter systems, the elevation. Results can be made more relevant to local 

conditions by changing default values for machine costs, wood density, the volume 

of tops and limbs removed with the bole wood (residue fraction), and the proportion of hardwood volume. Logging cost in U.S. dollars per hundred cubic feet 

(US$/ccf), and dollars per green ton ($/gt), and dollars per acre ($/ac) are shown 

for user-specified combinations of average tree size in cubic feet and number of 

harvested trees per acre by categories of trees to be chipped (chip trees), trees to 

be made into logs that are small enough to fell and process into logs by machine 

(small log trees), and trees to be made into logs that require chainsaw felling (large 

log trees). The cost table provides an estimate of cost that might be applied to a 

single harvest, a series of harvests that might occur in a stand, or harvests applied 

to multiple stands with similar harvesting conditions. 

Harvesting Systems and Conditions

Both whole-tree (WT) and log-length systems are included in FRCS. In a WT 

system, trees are felled either mechanically or by hand and delivered to the landing 

where the trees are processed into logs. (There are several terms for “delivered” 

to the landing: skidded for a skidder system, forwarded for a cut-to-length [CTL] 

harvester-forwarder system, and yarded for a cable or helicopter system. We use 

the generic term “delivered” where it may pertain to multiple systems to avoid 

long awkward sentences.) Tops and limbs left in the harvest unit may be left to 

decompose or may receive additional treatment not addressed by FRCS. Tops and 

limbs delivered to the landing may be chipped for fuel, left there to decompose, or 

may receive additional treatment not addressed by FRCS. In a log-length system, 

trees are felled, limbed, and bucked either mechanically (CTL) or by hand into 

logs at the stump. Logs are delivered to the landing by skidders, forwarders, cable 

yarders, or helicopters. Tops and limbs may remain in the woods to decompose or 

they may receive additional treatment not covered by FRCS. Those removed by a 

harvester on flat terrain may be bundled and delivered to the landing.

Results can be made 

more relevant to 

local conditions by 

changing default 

values for machine 

costs, wood density, 

the volume of tops 

and limbs removed 

with the bole wood, 

and the proportion of 

hardwood volume. 
4

GENERAL TECHNICAL REPORT PNW-gtr-668

Ground-Based (Tractive) Systems

Ground-based systems are used where management conditions allow because 

they are typically less expensive and cause less damage to reserve trees than do 

cable-yarding systems. The potential to damage soils on steep or wet ground, 

however, limits tractive equipment to sites where the soil is relatively dry and 

where slopes are less than 30 to 40 percent. For slopes greater than about 10 

percent, landings and road access for ground-based systems should normally 

be located on the downhill edge of the harvest unit. Skidding uphill on steeper 

slopes can cause excessive soil disturbance, and skidding or forwarding uphill 

is more costly.

With a ground-based manual-felling log-length system, trees are chainsawfelled, limbed, and bucked into logs at the stump. Rubber-tired skidders (choker 

and grapple) collect the logs and transport them to the landing. Logs to be 

hauled in log form are loaded onto log trucks, and logs to be chipped for board 

products or fuel are processed through a disk chipper and blown into chip vans. 

A ground-based manual-felling log-length system is normally used where trees 

are large enough that they must be bucked into two or more pieces to remove 

them from the woods. It also may be used when managers wish to retain tops, 

limbs, and their associated nutrients on site.

With a ground-based manual-felling WT system, trees are felled with 

chainsaws but not limbed or bucked. Rubber-tired skidders (choker and grapple) 

collect and transport whole trees. Trees are chipped or processed mechanically 

with stroke or single-grip processors and loaded onto trucks. A ground-based 

manual-felling WT system would typically be used for smaller trees than would 

the manual-felling log-length method and where feller bunchers are unavailable 

or where managers wish to confine machine traffic to a sparse network of skid 

trails. It often will be the most economical system where few trees per acre are to 

be removed. It is appropriate where managers wish to remove residues from the 

site to reduce fuel loading. 

With a ground-based mechanized-felling WT system, trees are felled and 

bunched; drive-to-tree machines are assumed for flat ground, whereas swingboom and self-leveling versions are included for steeper terrain. Rubber-tired 

grapple skidders transport bunches to the landing. Trees are chipped or 

processed mechanically with stroke or single-grip processors and loaded onto 

trucks. A ground-based mechanized-felling WT system is normally used when 

most or all of the trees to be removed are small enough to be handled by a feller 

buncher. It is useful where fuel loading is high because it removes tops and limbs 
5

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

from the stand. Because all operators are in machines, this system is safer than 

either of the manual-felling systems, where fallers and choker setters are exposed 

to the dangers of falling trees and rolling logs. 

With a ground-based CTL system, mechanized single-grip harvesters fell, 

limb, and buck the trees at the stump and pile the logs at trailside. Logs are transported to the landing by forwarders. Logs to be hauled in log form are loaded 

onto log trucks, and logs to be chipped are processed through a disk chipper 

and blown into chip vans. Because of the forwarder’s high center of gravity, the 

ground-based CTL system is limited to gentler slopes than are the other tractive 

systems, and at the upper limits of slope, to terrain that is fairly uniform rather 

than dissected. In addition, forwarder trails must run close to the fall line and 

must be laid out on a parallel, uniformly spaced network so the harvester can 

access the whole area. The physical constraints of the harvester and forwarder 

limit this system to trees under about 20 inches diameter at breast height (d.b.h.). 

Mills that prefer logs longer than the forwarder can carry may pay less for short 

CTL logs. Forwarders carry rather than drag logs and can travel on mats of the 

tops and limbs left by harvesters. Managers may therefore prefer CTL systems 

when it is critical to leave residues onsite, minimize soil disturbance, begin operations earlier in the year and continue longer, and minimize aesthetic impact. Cutto-length systems have safety advantages and can operate with small landings, 

but they are typically more expensive than mechanized-felling WT systems.

Cable-Yarding Systems

Cable-yarding systems are used where terrain is too steep or wet for groundbased systems or to reach across streams to areas not accessible by road. Yarding, however, requires deflection, that is, concave terrain profiles, in order to lift 

logs and avoid soil disturbance. Experience and careful planning are needed to 

ensure adequate deflection. In partial cutting, road access and landings usually 

should be located along the uphill edge of a harvest unit because yarding downhill can cause excessive damage to residual trees. Stand damage may be less of a 

problem when the trees being removed are smaller than the residual trees. The 

types of yarders include Idaho jammers (for clearcuts only), live skylines, and 

running skylines. All the cable-yarding studies included in FRCS were for uphill 

yarding, so the results should be applied to downhill yarding with caution.

With a manual-felling log-length cable-yarding system, trees are chainsawfelled, limbed, and bucked at the stump. Cable yarders transport the logs to the 

landing. Logs to be hauled in log form are loaded onto trucks, and logs to be 

chipped are processed through a disk chipper and blown into chip vans. Manual 
6

GENERAL TECHNICAL REPORT PNW-gtr-668

felling is the most common means of preparing trees for yarding because it c

an be used on essentially any type of terrain.

With a manual-felling WT cable-yarding system, trees are felled with chainsaws, but not limbed or bucked. (Trees too large to be yarded in one piece or too 

large to be mechanically processed at the landing are limbed and bucked in the 

woods.) Cable yarders transport the trees to the landing for chipping or mechanical processing and loading onto trucks. This system is appropriate where managers wish to remove residues from the site to reduce fuel loading.

Cable yarding typically is used on steeper terrain, and landings may not 

have enough space to accommodate a processor as well as a loader and chipper. 

In these situations a manual WT/log-length system may be appropriate for fuel 

reduction operations. Trees to be chipped can be felled and then yarded as whole 

trees, and those to be processed into sawlogs can be felled, limbed, and bucked 

prior to yarding. This combination reduces the amount of tops and limbs added 

to surface fuels as compared to using a log-length system exclusively, but removes 

less fuel than would the pure WT system. 

With a CTL cable-yarding system, mechanized single-grip harvesters fell, 

limb, and buck the trees and bunch the logs along predesignated yarding corridors and along harvester trails between the corridors. A cable yarder (a standing 

skyline with motorized slackpulling carriage was the only machine for which 

data were available) transports the bunched logs to the landing. Logs to be 

hauled in log form are loaded onto trucks, and logs to be chipped are processed 

through a disk chipper and blown into chip vans. The CTL system is applicable 

where the terrain is gentle enough and trees are small enough to allow the use of 

a harvester. It is not very common but shows promise for reducing the costs and 

residual stand damage associated with manual-felling log-length cable yarding. 

Cable yarders do not have the log-length constraints that forwarders have.

Helicopter-Yarding Systems

Because helicopters are generally used on steeper terrain, the most common helicopter system (manual-felling log-length) uses chainsaws to fell, limb, and buck 

trees at the stump. The helicopters then transport the logs out of the stand. Large 

landings are required because of the high production rates of helicopters and to 

provide adequate space for safe operation of equipment outside the load drop 

zone. Because large landings are typically few and far between, yarding distances 

for helicopters are generally longer than they would be for ground-based or cable 

operations. Logs to be hauled in log form are loaded onto trucks, and those to 

be chipped are processed through a disk chipper and blown into chip vans. 
7

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

With a CTL helicopter-yarding system a mechanical harvester fells, limbs, 

and bucks trees at the stump. Subsequent activities are the same as for the 

manual-felling log-length system. The CTL system is only applicable where 

terrain conditions allow access and operation of a mechanical harvester. If 

use of a harvester is possible, planners should consider carefully before using 

a helicopter for yarding rather than a much less costly forwarder. 

How to Get Started with FRCS

The FRCS application is an Excel1 spreadsheet program. It creates estimates 

of harvesting cost in dollars per hundred cubic feet, dollars per green ton, and 

dollars per acre for user-specified conditions and harvesting systems. Software 

that will run an Excel spreadsheet is required. The FRCS application was developed on Excel 2002. We have not tested the application with earlier versions or 

with spreadsheet programs other than Excel. A shortcut to the file can also be 

created and placed on the desktop or another convenient location. 

Entering Data in FRCS

Figure 1 shows the screen where data are entered and results are displayed. A 

check-box in the upper left corner can be used to display or hide explanatory 

information on making cost simulations. There are two sections to the screen. 

The upper section provides for data entry for those variables that will be constant 

for all scenarios entered in the lower section of the screen. The upper portion 

deals with harvest systems and attributes of the site. The lower section deals with 

the size and volume of trees or logs being removed and presents the estimated 

costs in dollars per cubic foot, dollars per green ton, and dollars per acre. 

The left side of the upper section requires that four or five items be 

addressed; elevation is required only for helicopter systems.

1. Select from a list of 10 types of harvesting systems in the combo box at the 

top: ground-based mech WT, ground-based CTL, ground-based manual 

WT, ground-based manual log, cable manual WT/log, cable manual WT, 

cable manual log, cable CTL, helicopter manual log, or helicopter CTL.

2. Select the type of harvest: clearcut or partial cut.

3. Enter the average distance that logs or trees are delivered to the landing. 

For ground-based and cable systems this is measured along the slope. 

4. Enter the average slope for the harvest unit in percent.

5. For helicopter systems enter the elevation. 

1 The use of trade or firm names in this publication is for reader information and does 

not imply endorsement by the U.S. Department of Agriculture of any product or service. 
8

GENERAL TECHNICAL REPORT PNW-gtr-668

In the middle of the upper section two areas are addressed: changing 

machine costs, which will be discussed later, and move-in costs. 

To include the costs of moving equipment to the harvesting unit, check 

the box labeled “Include move-in costs.” If this is the only unit in a treatment 

contract or timber sale, these costs should be based on how far equipment would 

likely be brought in and the area of this unit. If there are multiple units that 

would likely be harvested together that are close enough to drive equipment 

between them, the area of the combined units would be the more appropriate 

value to use. Area is entered in acres, one-way move-in distance in miles.

On the right side of the upper section, check the box to include the cost of 

collecting and chipping residues. This option is not available for some systems. 

For the three WT systems, the tops and limbs are transported to the landing 

as part of the tree. If this option is selected, a chipper will be included in the 

equipment configuration and the tops, limbs, and small trees will be converted 

to dirty chips at the landing. This is accomplished at a small addition to overall 

cost because the tops and limbs are at the landing and only incur the added 

cost of chipping. The only other system for which this option is available is 

the ground-based CTL system. In this case, a bundler is used to gather up and 

bundle the tops and limbs that have been left in windrows by the CTL operation. 

These bundles will be transported to the landing on a forwarder for subsequent 

chipping. The cost of recovering and chipping CTL residues is considerably more 

than for residues from a WT system.

Figure 1—User interface for Fuel Reduction Cost Simulator where data are entered and results viewed.
9

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

The section labeled “Tree Characteristics” requires some explanation. Three 

types of trees are mentioned: chipped, small log, and large log. The chipped 

category includes the trees that will be WT chipped at the landing. The small 

log trees are those that will be converted to logs and that are small enough to 

be handled with mechanical equipment. The large log trees are those that must 

be felled, limbed, and bucked with chainsaws. 

Because the weight of different species of trees differs widely and weight is 

an important determinant of machine capacity and therefore productivity, we 

gain precision in cost estimates by adjusting the default values for wood density 

to reflect the situation. If the cell is left blank, a default value of 60 pounds per 

green cubic foot will be used. The densities provided are green weights of bole 

wood and bark per cubic foot of bole wood (gross scale). Appendix table 2 

presents some estimates of weights per cubic foot by species provided by regional 

offices of the Forest Service. In general, they are from loads of logs that have 

been weighed and scaled in cubic feet. In some cases where the weights for two 

species are the same in a Forest Service region, it is because the data were for 

the two species combined and the weight was assigned to both species.

The residue fraction is the ratio of the weight of tops and limbs to the weight 

of the bole. The log weights are based on converting the volumes that will be 

entered in the lower section of the screen to weight. For all classes of trees, if the 

cell is blank a default value of zero will be used. Appendix table 3 presents estimates of residue ratios for commercially important species for a range of average 

tree size. These values are based on a 4-inch minimum top and only include trees 

that are large enough to have at least a full 16-foot log. Comparable values for a 

6-inch minimum top are very close to these and are therefore omitted. 

The hardwood fraction is the volume of hardwood removed as a proportion of the total. The FRCS includes no information from production studies 

on hardwoods, so for the hardwood fraction, estimated costs for conifers are 

increased by 20 percent to adjust for the generally lower productivity associated 

with hardwoods owing to their lean, tree form, and branchiness.

In the lower section of the screen, enter the number of trees per acre and the 

average volume for each category of trees. It is acceptable to leave blanks for 

categories that do not have any cut trees. The screen will accommodate a large 

number of alternative combinations. These values can be copied and pasted from 

a spreadsheet. After entering any number of rows, click the calculate button to 

calculate results for all rows. The results are presented in the three columns on 

the right. All calculations are based on the values entered in the top section of 
10

GENERAL TECHNICAL REPORT PNW-gtr-668

the screen. Appendix table 4 presents volumes to a 4-inch top for commercially 

important species for a range of average tree diameters and heights. These 

volumes are calculated with the Inland Northwest Growth and Yield Cooperative profile equations (Flewelling and Ernst 1996). Volumes are only reported 

for trees that are large enough to have at least a full 16-foot log. 

To recalculate all of the combinations entered in the lower part of the screen 

with a different system or other variables in the upper part of the screen, just 

change the system selection or other variable and click the calculation button 

again. It is especially important when using the model to make comparisons of 

systems to make sure that the combination of variables is appropriate for both 

systems. For example, it would seldom be appropriate to make a comparison 

between a helicopter system and another system with the same delivery distance 

from stump to landing. The requirements for landings for ground or cable 

systems are such that they are usually at the edge or within the unit. The requirements for landings for a helicopter system are such that they are usually some 

distance beyond the unit boundary. So a direct comparison of a helicopter system with another system would ordinarily involve changing the average delivery 

distance. 

Simulated Harvest Cost Results

Cost comparisons for a system over a range of conditions should be quite robust 

and in most cases have smoothly changing cost curves. The method of averaging 

and weighting described in the “Introduction” and “Glossary” strongly promotes 

that outcome. In some cases, there can be an abrupt shift when the relevance for 

a machine goes to zero. Abrupt shifts are more likely if this machine has a cost 

that is considerably above or below the average and if there are a small number 

of machines in that average. 

Cost comparisons between systems may sometimes appear to give anomalous results, especially near the boundaries of the conditions where they are 

considered relevant. For example, some cable systems may have lower costs than 

some ground-based systems on slopes in the range of 35 to 40 percent owing to 

extrapolation of study results that showed increasing costs with slope for groundbased systems. These kinds of anomalies can only be avoided by artificially 

constraining the model to ensure that they do not occur. Doing this, however, 

would distort in unpredictable ways the comparisons for a system over a range 

of conditions. The primary purposes for which we see FRCS being used (evaluating prescriptions and getting broad average costs for a set of prescriptions 

Cost comparison 

anomalies can be 

avoided by artificially 

constraining the model.
11

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

applied to a set of conditions) make smooth cost curves for systems more important than artificially constrained comparisons between systems, so we have not 

added these constraints. 

There can be anomalies between the cost of partial cutting and clearcutting 

for cable-yarding systems. On average the clearcutting studies are older than the 

partial-cutting studies. Many of the clearcutting studies include equipment that 

is older, less efficient, and less commonly used now. We do not see FRCS being 

used to make decisions about whether to clearcut or partial cut so we do not see 

this as a serious problem. Again artificial constraints could be added to ensure 

that clearcutting is always less expensive than partial cutting under the same 

conditions, but to do so would distort the costs of some systems relative to other 

systems so we have not added these constraints. 

Constraints and Error Messages

There are constraints on the values that are permitted to be entered in FRCS. 

There are two general reasons for these constraints. Firstly, we try to prevent 

circumstances that are outside what is reasonable, given equipment limitations or 

safety considerations. We also try to prevent unreasonable extrapolation beyond 

the range of conditions that were represented in the studies on which the model 

was based. The latter constraints are internal to the application and are dealt 

with by use of the relevance factors that were discussed in an earlier section.

Table 1 shows the upper limits on average tree volume and slope; values 

greater than these will produce error messages. Cut-to-length harvesters cannot 

handle large log trees, but CTL systems can accommodate a small number of 

such trees by using chainsaws to fell and buck as needed to reduce piece size to 

that which can be subsequently handled by the harvester or forwarder. For CTL 

systems, we limit the number of large log trees removed to 10 per acre and the 

number of large log trees to 10 percent of the total number of log trees removed. 

Systems that involve mechanical felling or processing are limited to slopes less 

than 40 percent for safety or environmental damage considerations. Cable 

systems are limited to average yarding distances of no more than 1,300 feet 

owing to the typical drum capacities on cable yarders. 

Error messages will appear in the results cells for trees that violate the 

constraints listed above. The messages will include an indication of the error 

type and the tree group that is in violation. Error messages that appear when 

limits are exceeded are provided here: “Small Tree” for small tree average 

volume limits, “Chip Tree” appears if chip trees exceed limits, and “Large Log 

Tree” appears if large log trees exceed average volume limits. The word “Slope” 
12

GENERAL TECHNICAL REPORT PNW-gtr-668

appears if slope limits are exceeded for the equipment selected, and “Yard Dist” 

will appear if yarding distance is too great. There are also limits on the average 

d.b.h. of all trees and all log trees that may be constraining in some cases. These 

will be reported as “Avg Vol” or “All Log.” If there are multiple constraints that 

are violated, only one will be reported. In those cases, it may require more than 

one adjustment of the numbers to clear all of the constraint violations. 

Changing Default Machine Cost Values

The machine operating costs should include the full cost of ownership, labor, 

and fuel. 

The machine costs used by FRCS can be changed in order to make the perhour costs of equipment operation more closely match the costs in your area. To 

change the default machine costs, select the machine that needs to be changed 

from the drop-down list in the “Change Machine Costs” area on the user interface. Once a piece of equipment has been selected, a text box will appear prompting the user to enter the new per-hour machine costs. The text box will provide 

the current per-hour value and an entry field for the user to set the new costs. 

Once a new value has been entered, that value will be stored by FRCS until the 

user either resets default costs by selecting the “reset defaults” button, or changes 

the cost again.

Table 1—Constraints on values that average volume and slope can take under different 

harvest systems

Maximum average Maximum average Maximum average

volume for volume for volume for Maximum

System chip trees small log trees large trees slope

- - - - - - - - - - - - - - Cubic feet - - - - - - - - - - - - - - Percent

Ground-based: 

Mechanical whole tree 80 80 250 40

Cut-to-length 80 80 100 40

Manual whole tree 80 80 250 40

Manual log length 80 None None 40

Cable:

Manual whole tree/log 80 None None 100

Manual whole tree 80 80 250 100

Manual log length 80 None None 100

Cut-to-length 80 80 100 40

Helicopter:

Manual log length 80 None None 100

Cut-to-length 80 80 100 40
13

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

For Advanced Users

Description of Calculation Sheets

The following descriptions of the calculation sheets within FRCS are provided 

for those who want to understand in more detail how the model works and where 

the calculations are to be found.

The FRCS workbook includes 25 sheets. Twenty-one are calculation sheets; 

of the others, one is a user interface sheet, one is a treatment report, one is a 

data input sheet for batch processing, and the last is an error report sheet for 

batch processing. Tables of simulated harvesting costs can be run from the user 

interface sheet without ever accessing other sheets. The additional sheets are 

accessible for those who want to understand the derivation of results or change 

some of the model assumptions. Equations can be viewed on the formula bar by 

checking “Show Formula Bar” under the Options menu. If the user selects “no” 

in the message box asking if the user would like to format tool bars, the user will 

be able to use the sheet tabs for the sheets described in this section. Or the user 

can select the “Sheet tabs” box in the Options menu of Excel Tools.

User interface—

The name of the main user interface sheet is Interface_page. This sheet contains 

the results of one or more cost simulations. It also contains the values that have 

been used in making the simulations so that simulation assumptions can be 

verified and retained when the spreadsheet is saved. This sheet also contains the 

input features to allow the user to describe the harvest that they wish to simulate. 

The sheet will retain these values if the spreadsheet is saved. 

Batch Error Report—

The Batch_Error_Report sheet displays cell errors for users who have run a batch 

file, but have made errors in system selection or have received error warnings in 

the outputs of some cut profiles processed.

Bundle & Forward Residue—

The Bundle&ForwardResidue sheet calculates costs per green ton for CTL residues by using three productivity equations drawn from the literature and displays 

the relevances assigned for the last cost calculation shown on the Interface_page, 

i.e., the one at the bottom of the list.

Cable yarding—

The cable yarding sheet calculates cost per ccf based on 18 cable yarder productivity equations drawn from the literature and displays the relevances assigned 

for the harvest cost calculation shown on the Interface_page.
14

GENERAL TECHNICAL REPORT PNW-gtr-668

Chipping—

The chipping sheet calculates costs per ccf for chip trees and costs per green ton 

of residues by using eight chipping productivity equations drawn from the literature and displays the relevance assigned for the harvest cost calculation shown 

on the Interface_page.

Data—

The data sheet is provided for analysts who want to use the FRCS spreadsheet 

with large batches of data. Procedures have been developed to allow the spreadsheet to be called by other programs to simulate harvesting costs for specific 

conditions as part of a broad analysis of fuels treatment. That use is not documented here, but the ability to process batch data is facilitated with the “Batch 

Mode” tool bar that appears when the batch mode button is clicked in 

the Interface_page. This is included so that one spreadsheet will serve both 

purposes. Values entered on this sheet will not affect the results shown in the 

table on the Interface_page, as the data are only entered when the “Process 

Batch” button is selected.

Data_management—

The Data_management sheet serves to facilitate updates of the calculation sheets 

without affecting the user input sheet. Its format is a combination of the Inputs

and Outputs sheets. The Data_management sheet shows intermediate results 

calculated for the last harvest cost calculation executed. Values entered for the 

treatments on the interface sheet are recorded on the Data_management sheet. 

The values on the Data_management sheet are sent to the Inputs sheet. Results 

from the Outputs sheet are collected with errors interpreted into the error codes 

for display on the Interface_page. The “Stand variables” table collects the data 

from each row of the input table on the Interface_page. The “Unit variables” 

table collects the variable inputs (yarding distance, slope, treatment area, movein distance, and elevation) that are common to all stands in the simulation. The 

following table collects the inputs from each of the binary inputs (clearcut or 

thinning, collecting residues or not, and including move-in costs or not) and 

translates them to be read into the calculation for each stand. To the right of 

the binary values is a small table that presents the values for the harvest system 

selected by the user, to be read into the results display on the Interface_page. The 

Data_management sheet also includes the table of machine cost values that allow 

the user to customize the machine cost values through the Interface_page. The 

next table down, “System cost summaries,” reads the system outputs from the 

Outputs sheet for display on the Interface_page. Below that is a table that checks 
15

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

for possible errors in the system summaries and interprets the errors for output 

to the user. The final section of the sheet contains a table of system components 

and the default average cost-per-hour values along with any user modifications. 

Modifications to machine costs should only be made through the Interface_page.

Fell & Bunch—

The Fell&Bunch sheet calculates the costs per ccf for 11 feller-buncher productivity equations drawn from the literature and displays the relevance assigned for 

the harvest cost calculation shown on the Interface_page.

Felling—

The felling sheets calculate the costs per ccf for four felling productivity equations drawn from the literature for chainsaw felling and three equations for 

chainsaw felling, limbing, and bucking. The sheet also displays the relevance 

assigned for the harvest cost calculation shown on the Interface_page. This and 

the other three felling sheets share common productivity equations, but receive 

different inputs from the Interface_page as indicated below.

(a) Felling (all trees)—

This sheet applies to the systems where all trees are cut with chainsaws.

(b) Felling (large log trees)—

This sheet applies to the systems where only the large log trees are cut with 

chainsaws.

(c) Felling (WT chip, log other)—

This sheet applies to the systems where chainsaws are used to fell the trees to be 

chipped, and to fell, limb, and buck all the other trees.

(d) Felling (WT small, log other)—

This sheet applies to the systems where chainsaws are used to fell the trees to be 

chipped or processed at the landing, and to fell, limb, and buck all the large log 

trees.

Forwarding—

The forwarding sheet calculates costs per ccf for six forwarder productivity 

equations drawn from the literature and displays the relevance assigned for the 

harvest cost calculation shown on the Interface_page.

Harvesting—

The harvesting sheet calculates costs per ccf for 17 harvester productivity 

equations drawn from the literature and displays the relevance assigned for the 

harvest cost calculation shown on the Interface_page.
16

GENERAL TECHNICAL REPORT PNW-gtr-668

Helicopter yarding—

The HelicopterYarding sheet calculates the daily costs, capacities, productivities, 

and costs per ccf for three helicopters. It also displays the relevance assigned for 

the harvest cost calculation shown on the Interface_page.

Inputs—

The Inputs sheet displays user inputs, model assumptions, and some intermediate 

calculations to facilitate validating the model and to provide a convenient place 

to pass variables between sheets within the model.

Loading—

The Loading sheet calculates costs per ccf for six loading productivity equations 

drawn from the literature and displays the relevance assigned for the harvest cost 

calculation shown on the Interface_page.

Machine costs—

The MachineCosts sheet calculates the cost per hour for each type of machine. 

This is where the default machine costs are calculated. These costs can be overridden by use of the “Change Machine Cost” pull-down menu found on the 

Interface_page user interface.

Move-in costs—

The MoveInCosts sheet calculates the move-in costs for each of the harvesting 

systems.

Outputs—

The Outputs sheet collects the outputs for all the system combinations. The 

products recovered per acre are presented in the top section. The costs of the 

elements of each system are displayed. These are added together and presented 

in the “system cost summaries” section of the sheet in three forms: $/acre, $/bole 

ccf and $/gt. Below, in the “Limits” section, the cut profile variables are checked 

against the constraints of each system, and any errors are reported. The Data_

management sheet collects the system cost outputs, and the limit error outputs 

for presentation on the Interface_page user interface sheet.

Processing—

The Processing sheet calculates costs per ccf for seven processor productivity 

equations drawn from the literature and displays the relevance assigned for the 

harvest cost calculation shown on the Interface_page.
17

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

Relevance Weight Inputs—

The RelevanceWeightInputs sheet calculates the relevance weights for each 

machine or study included in a system. The cost for a system is an average of 

the costs of all machines weighted by their relevance weights. A machine can be 

excluded from the average cost by putting in a weight of zero in column B by the 

machines to be excluded. User-supplied weights that are within the valid range of 

zero to 1 will be used except when the calculated weight is zero. If the calculated 

weight is zero, the user-supplied weight will be ignored to prevent extrapolation 

beyond what is deemed a reasonable use of the machine. Note that the current 

model weight shown in column C is for the last calculation made by the model, 

that is, the one reported at the bottom of the list, and depends on the circumstances input to FRCS. Column K shows the weights used in calculations. They 

will be identical to the model-calculated weights unless user-supplied weights 

are used.

Report—

The Report sheet displays the inputs and results for a group of harvests that 

differ only in the size and volume of cut trees. The first section displays the 

stand profile variables, and the second section provides the cut tree list and 

the estimated cost for the stand described in each row, and the third section 

displays the costs per hour used for all machines. The report sheet allows the 

user to select the “Print” button to have the report formatted and printed.

Skidding—

The Skidding sheet calculates costs per ccf for seven productivity equations for 

skidding unbunched logs and seven productivity equations for skidding bunched 

logs drawn from the literature. It also displays the relevance assigned for the 

harvest cost calculation shown on the Interface_page.

Variable list—

The VariableList(just for info) sheet is used by the spreadsheet developers to 

keep a complete list of the named ranges used in the spreadsheet and identify 

their source locations. Model users should not need to use the information on 

this sheet.
18

GENERAL TECHNICAL REPORT PNW-gtr-668

Use of FRCS in Batch Mode

The Interface_page also contains a hidden bar of menu buttons titled “Batch 

Mode” that facilitate the process of importing and processing large amounts of 

data. Detailed documentation of batch mode formats and operations are available in the program. To activate the batch mode, the user must click the batch 

mode button. A group of buttons will appear that allow the user to access batch 

mode functions (fig. 2).

Figure 2—User interface for Fuel Reduction Cost Simulator with batch mode button bar displayed.

The most important button for the new user is the “Documentation” button, as it will walk the user through the batch mode process. Briefly, to operate 

batch mode, the user must load a data file by using the “Load Batch Data” 

button. To process the data the user must select the “Process Batch” button, 

which will scan the data for input errors, then process the file. The user can 

then examine the error report to fix problems with the data. The results can be 

saved as a separate file by using the “Save Results” button. The data page can 

be cleared with the “Clear Data” button. The batch mode settings will process 

files with up to 60,000 rows.

Glossary

area harvested—The total area in acres to be harvested by one harvesting 

system during one entry.

dirty chips—Chipping of whole trees or boles that have not been debarked 

resulting in chips that are generally used for hog fuel.

green density—The weight of green wood and bark per cubic foot of bole 

wood measured in pounds per cubic foot of bole wood.

The most important 

button for the new user 

is the “Documentation” 

button, as it will walk 

the user through the 

batch mode process. 
19

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

one-way move-in distance—Distance in miles that equipment is transported to 

reach the harvest unit.

skidding/forwarding/yarding distance—This is the skidding distance for the 

ground-based skidder systems or the forwarding distance for the CTL system 

or the yarding distance for the cable and helicopter yarding systems. It refers 

to the average (not external) one-way distance. For ground-based and cable 

systems, distance is measured along the slope rather than horizontally.

slope—The average fall-line slope for the harvest unit, measured as a percentage.

tree volume—Average gross volume in cubic feet to the merchantable top 

diameter of trees being harvested. 

trees per acre—Number of harvested trees per acre.

utilization rate—Productive hours divided by scheduled hours.

Acknowledgments

Our thanks to Professor Glen Murphy of Oregon State University for identifying 

and evaluating the results of a number of recent field studies for inclusion in the 

model and for updating the equipment costs.

Metric Equivalents

When you know: Multiply by: To find:

Inches (in) 2.54 Centimeters

Feet (ft) .3048 Meters

Yards (yd) .914 Meters

Miles (mi) 1.609 Kilometers

Acres (ac) .405 Hectares

Cubic feet (ft3) .0283 Cubic meters

Dollars per hundred cubic feet ($/ccf) .353 Dollars per cubic meter

Dollars per green ton ($/gt) 1.102 Dollars per green metric ton

Dollars per acre ($/ac) 2.471 Dollars per hectare

Pounds per cubic foot (lb/ft3) 16.0185 Kilograms per cubic meter

References

Fight, R.D.; Zhang, X.; Hartsough, B.R. 2003. Users guide for STHARVEST: 

software to estimate the cost of harvesting small timber. Gen. Tech. Rep. 

PNW-GTR-582. Portland, OR: U.S. Department of Agriculture, Forest 

Service, Pacific Northwest Research Station. 12 p.
20

GENERAL TECHNICAL REPORT PNW-gtr-668

Flewelling, J.W.; Ernst, R.L. 1996. Stem profile estimation—east-side species. 

280 p. Unpublished report for the Inland Northwest Growth and Yield 

Cooperative. On file with: U.S. Department of Agriculture, Forest Service, 

Forest Management Service Center, 2150A Centre Avenue, Suite 341A, Fort 

Collins, CO 80526-1891. 

Hartsough, B.R.; Zhang, X.; Fight, R.D. 2001. Harvesting cost model for 

small trees in natural stands in the interior Northwest. Forest Products Journal. 

51(4): 54–61.

Miyata, E.S. 1980. Determining fixed and operating costs of logging equipment. 

Gen. Tech. Rep. NC-55. St. Paul, MN: U.S. Department of Agriculture, Forest 

Service, North Central Forest Experiment Station. 16 p.
21

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

Table 2—Green weight of logs to a merchantable top by species

Pacific

Rocky Pacific Northwest

Northern Mountain Southwestern Intermountain Southwest Region (6)

Species Scientific name Region (1) Region (2) Region (3) Region (4) Region (5) (east side)

- - - - - - - - - - Pounds per cubic foot of log volume (gross scale) - - - - - - - - - -

Aspen Populus tremuloides Michx. 58 58 54

Douglas-fir Pseudotsuga menziesii 60 50 52 64 61

(Mirb.) Franco

Engelmann spruce Picea engelmannii Parry 56 50 51

ex Engelm.

Grand fir/white fir Abies grandis (Dougl. ex 65 62 57

D. Don) Lindl./concolor

(Gord. & Glend.) Lindl. 

ex Hildebr.

Lodgepole pine Pinus contorta Dougl. 56 50 51 55 63

ex Loud.

Mountain hemlock Tsuga mertensiana (Bong.) 68

Carr.

Ponderosa pine Pinus ponderosa Dougl. 66 68 68 60 69 69

ex Laws.

Subalpine fir Abies lasiocarpa (Hook.) 53 46 51

Nutt.

Western larch Larix occidentalis Nutt. 63 52 55

Western redcedar Thuja plicata Donn ex 43

D. Don

Western white pine Pinus monticola ex D. Don 52

Table 3—Ratio of the weight of tops above a 4-inch minimum and limbs, to the 

weight of logs, by species and tree diameter

Diameter at breast height (inches)

Species 6 7 8 9 10 11 12 13+

Alpine fir 1.05 0.89 0.80 0.76 0.73 0.72 0.70

Douglas-fir .95 .73 .61 .54 .49 .46 .41

Engelmann spruce 1.08 .77 .61 .51 .45 .41 .32

Grand fir 1.26 .81 .61 .51 .44 .40 .38 .33

Lodgepole pine 1.00 .64 .47 .39 .33 .30 .28 .24

Mountain hemlock 1.05 .73 .58 .50 .46 .43 .41 .38

Ponderosa pine .90 .78 .70 .64 .60 .46

Western larch 1.22 .76 .55 .44 .37 .33 .30 .25

Western redcedar .53 .32 .23 .19 .16 .14 .13 .12

Western white pine .76 .56 .46 .39 .36 .34 .31

See table 2 for species scientific name.

Appendix
22

GENERAL TECHNICAL REPORT PNW-gtr-668

Table 4—Volume of logs to a 4-inch minimum top by species, tree diameter, and height

Diameter at 

breast height

Alpine fir Douglas-fir Engelmann spruce Grand fir Lodgepole pine

Height Log volume Height Log volume Height Log volume Height Log volume Height Log volume

Inches Feet Cubic feet Feet Cubic feet Feet Cubic feet Feet Cubic feet Feet Cubic feet

6.0

6.0 41 3 43 3

6.0 46 3 48 3 45 3 53 4 54 4

7.0 35 3 36 4

7.0 42 4 42 4 41 4 48 5 48 5

7.0 52 5 54 5 51 5 60 6 60 7

8.0 36 4 34 4 35 5 41 5 41 6

8.0 47 6 47 6 46 6 54 7 53 8

8.0 57 8 60 7 57 8 67 10 65 10

9.0 40 6 38 5 39 6 45 7 44 8

9.0 51 9 52 8 50 9 59 10 56 11

9.0 62 11 66 10 62 11 73 14 69 13

10.0 44 9 41 7 42 9 49 10 47 10

10.0 56 12 56 11 55 11 64 14 60 14

10.0 67 15 71 14 67 15 80 19 73 18

11.0 47 11 45 9 45 11 53 13 49 13

11.0 59 15 61 14 58 15 69 18 62 18

11.0 71 19 76 18 71 19 84 24 75 22

12.0 50 14 48 12 48 14 57 17 51 16

12.0 63 19 65 18 62 19 73 23 65 22

12.0 75 24 82 23 75 24 89 30 78 27

13.0 52 17 51 15 51 18 61 21 52 20

13.0 66 23 69 22 65 23 77 29 67 26

13.0 79 29 87 29 79 29 93 37 81 33

14.0 55 21 53 18 54 22 65 26 54 24

14.0 68 27 73 27 69 28 81 35 69 31

14.0 82 35 92 35 83 35 97 45 83 39

15.0 57 25 56 22 57 26 68 30 54 27

15.0 71 32 76 32 72 33 84 42 70 36

15.0 85 41 96 42 87 42 101 53 85 45

16.0 60 30 59 26 59 31 70 36 55 31

16.0 74 38 80 38 75 40 87 49 70 41

16.0 88 48 100 49 91 49 104 63 86 51

17.0 61 34 61 30 62 36 72 41 56 36

17.0 76 43 83 44 78 46 90 56 72 47

17.0 91 55 104 57 94 57 107 72 87 58

18.0 62 38 63 34 64 42 74 46 57 41

18.0 78 49 86 51 81 53 92 63 73 53

18.0 94 62 108 66 97 66 110 82 89 66

19.0 63 43 66 39 66 47 77 53 58 46

19.0 79 55 89 58 83 61 95 72 74 60

19.0 96 70 112 76 101 76 112 92 90 74

20.0 64 48 68 44 68 54 80 60 58 51

20.0 81 62 92 66 86 69 97 81 74 66

20.0 98 78 116 87 104 86 114 103 90 82

21.0 67 55 70 50 70 61 83 68 58 56

21.0 83 69 95 75 89 78 100 91 74 72

21.0 100 87 120 98 107 97 117 116 89 88

22.0 69 62 72 56 73 68 85 76 58 61

22.0 85 77 98 84 91 88 103 102 73 77

22.0 102 96 123 111 110 109 120 129 88 95

23.0 70 68 73 62 75 76 87 83 58 67

23.0 87 85 100 94 94 98 104 112 74 86

23.0 104 105 127 124 113 121 122 142 90 105

24.0 72 75 75 68 77 85 88 91 59 74

24.0 89 94 103 104 96 109 106 122 75 94

24.0 105 115 131 138 116 135 124 155 91 116
23

Users Guide for FRCS: Fuel Reduction Cost Simulator Software

Table 4—Volume of logs to a 4-inch minimum top by species, tree diameter, and height (continued)

Diameter at 

breast height

Mountain hemlock Ponderosa pine Western redcedar Western larch Western white pine

Height Log volume Height Log volume Height Log volume Height Log volume Height Log volume

Inches Feet Cubic feet Feet Cubic feet Feet Cubic feet Feet Cubic feet Feet Cubic feet

6.0 41 2

6.0 46 3 39 3 53 3

6.0 59 3 49 3 65 4 48 3

7.0 38 3 33 3 48 4

7.0 52 5 44 4 60 5 44 4

7.0 66 6 43 3 55 6 72 7 58 6

8.0 43 5 38 5 55 7 34 4

8.0 58 7 35 4 50 7 67 8 51 7

8.0 73 9 48 5 61 8 80 10 69 9

9.0 48 8 42 7 60 9 44 7

9.0 63 10 38 5 54 9 73 12 61 10

9.0 79 13 52 8 67 12 85 14 77 14

10.0 53 11 46 9 65 13 54 11

10.0 69 14 41 7 59 12 78 15 70 15

10.0 85 18 56 10 72 15 91 18 86 19

11.0 57 14 27 6 49 12 68 16 56 14

11.0 73 19 44 9 62 15 82 20 74 19

11.0 90 24 60 13 76 19 95 23 91 24

12.0 61 18 28 7 51 15 72 20 59 17

12.0 78 24 46 12 66 19 85 24 78 24

12.0 95 31 64 17 80 24 99 28 96 30

13.0 65 23 29 9 55 18 75 24 63 21

13.0 82 30 49 15 69 23 89 29 83 29

13.0 100 38 68 22 84 29 103 34 102 37

14.0 69 28 31 10 58 22 79 29 68 26

14.0 87 37 52 18 73 28 93 35 88 36

14.0 105 47 72 27 88 35 106 41 107 45

15.0 72 34 33 12 60 26 82 35 71 31

15.0 91 44 55 22 75 33 96 42 91 43

15.0 109 56 76 33 91 41 110 48 111 53

16.0 75 40 35 15 63 30 85 41 73 37

16.0 94 53 58 26 78 38 99 49 94 49

16.0 114 67 80 39 94 48 113 56 115 63

17.0 79 48 37 18 65 34 88 47 69 38

17.0 98 62 61 32 81 44 102 56 91 53

17.0 117 77 84 47 96 55 117 65 114 70

18.0 83 56 39 21 68 39 91 54 65 40

18.0 101 72 64 37 83 50 106 65 89 56

18.0 120 88 89 55 99 62 121 74 112 76

19.0 85 64 44 26 70 44 92 60 66 45

19.0 104 82 68 45 86 56 107 72 93 64

19.0 123 101 92 64 102 71 122 83 120 91

20.0 87 72 48 33 71 50 93 67 67 50

20.0 107 92 72 53 89 63 109 81 97 74

20.0 126 115 96 75 106 81 124 93 127 107

21.0 89 82 53 40 73 55 96 76 75 62

21.0 109 104 76 62 90 70 112 92 101 86

21.0 128 128 99 86 107 89 129 106 127 117

22.0 92 93 58 48 74 61 99 85 84 74

22.0 111 116 80 73 92 77 116 103 105 99

22.0 130 142 102 98 109 98 134 120 127 127

23.0 95 104 62 57 76 67 102 96 73 72

23.0 114 130 84 83 93 84 118 114 102 101

23.0 134 159 106 111 110 106 135 131 130 142

24.0 98 117 65 66 78 74 105 107 63 67

24.0 117 145 87 95 95 91 120 126 98 102

24.0 137 177 109 126 111 114 136 143 134 158

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