barber revisited: aggregate analysis in harvest scheduling models

Barber Revisited: Aggregate Analysis in Harvest Schedule Models

Steven D. Millsa, Bruce L. Carroll

b, and Karl R. Walters

c

a FORSight Resources, 8761 Dorchester Road, Suite 102, North Charleston, SC 29420, USA, Email:

[email protected]

b FORSight Resources, 8761 Dorchester Road, Suite 102, North Charleston, SC 29420, USA, Email:

[email protected]

c FORSight Resources, 3813 H Street, Vancouver, WA, 98663, USA, Email:

[email protected]

ABSTRACT. A landmark paper by Richard Barber (1985) focused on yield bias

introduced by aggregate analysis in forest planning. Although this paper is widely cited

by forest planners, the applicability of the results to linear programming harvest schedule

models is unclear. This study examines the relationship between assumptions used in

aggregate analysis and harvest volume, area, and average age bias. Results from periodic

linear programming models are compared against annual models with annual age classes.

Results indicate that constrained harvest schedule models with aggregated age classes

consistently exhibit positive volume bias relative to models with annual age classes,

regardless of initial age class distribution. Feasibility of the periodic harvest schedules is

examined and implications for strategic and operational implementation are discussed.

Key Words: Harvest scheduling, timber yields, periodic analysis, aggregate analysis,

forest planning.

1. Introduction

One of the most important yet commonly overlooked issues in harvest schedule modeling

is age class aggregation. Planners often take for granted the inherent assumptions

associated with aggregated age classes. In doing so, they fail to recognize bias that may

be introduced into results. A clear understanding of this bias provides important insights

into harvest projection errors and their potential impacts on operational planning.

Bias caused by aggregated age classes was first examined by Barber (1985). Since that

time, Barber’s work has been widely cited as justification for assumptions made during

model formulation. Many planners fail to recognize that Barber carried out his analysis

using area and volume control models, techniques used by few if any organizations

today. The appropriateness of expanding Barber’s results to modern mathematical

modeling techniques remains unclear. This paper revisits the topic of bias in aggregate

analysis by updating Barber’s methodology to reflect current planning practices.

The need for aggregated age classes arises for several reasons. First, inventory data

resolution may be poor; exact ages may be unavailable or unreliable, leaving the planner

with age class data only. A model conforming to the aggregated age classes is thus the

only reasonable option. Another issue that would cause one to consider aggregated age

classes is model size. Despite advances in hardware and software technology, forest

planning models often push against the upper limits of solver capacity. Aggregating

stands into age classes reduces strata numbers, effectively reducing model size.

A definition of age class width and planning period width are required before moving

further into a discussion of age class aggregation. Age class width describes the number

of ages combined into a single age class. Planning period width describes the length of

time each planning period represents. Although the convention in much of the harvest

scheduling literature is age classes and planning periods of equal width, in practice this is

often not the case. As a result, this article generalizes the work of Barber (1985) who

dealt with the specific case of equal age class and planning period widths to include the

cases of both age class width wider and narrower than planning period width.

Bias is not directly caused by aggregated age classes or the use of planning periods, but

rather assumptions drawn around them. Periodic analysis implicitly assumes that a single

harvest activity occurs each multiyear period (Barber 1985). Although it is understood

that some acreage will be harvested annually, from a modeling perspective each period

must be treated as having a single discrete harvest. As a result, a uniform harvest age is

required for yield calculations. The assumed harvest age is a function of assumptions

about harvest timing within a period and the initial age of acres in a class. It is these

assumptions that this study addresses.

Initial age class age generally follows one of two assumptions; class mid-point or class

end-point. A reasonable assumption is that the area is randomly distributed according to a

uniform distribution over the age class interval (Barber 1985). The resulting average age

is thus equivalent to the class mid-point. Alternately, some planners view the average age

as being equal to the upper end of the class. To simplify further discussion, let these

differing assumptions be referred to as age class mid-point and age class end-point.

Like initial age class age, two alternate assumptions exist regarding harvest timing within

a period. Common practice is to assume that annual harvest is equal to periodic harvest

divided by period width. By assuming equal annual harvest, one can see that on average,

harvest will occur at period mid-point. Alternately, some planners extend the assumption

of period end-point harvest used in annual models to periodic models. To simplify further

discussion, let these assumptions be referred to as period mid-point and period end-point.

Given the assumptions outlined above, the assumed harvest age for existing stands can be

expressed as:

(1) 1 2

2 2

w wu w

c pharvest age a p t= + ,

where: au = age equal to the upper end of the age class cw = age class width (years) pw = planning period width (years) t = harvest period

1 = 1 if assumed initial age is class mid-point, 0 otherwise 2 = 1 if assumed harvest timing is period mid-point, 0 otherwise

The first term describes the initial age of the class while the second term describes the

number of years the class has aged prior to harvest. It can be seen that the underlying

assumptions can lead to assumed harvest age differences as large as (cw /2) + (pw /2) years. Note that regardless of assumptions about initial age and harvest timing,

regenerated stands use the same harvest age calculation; pwtr where tr is the number of

planning periods since regeneration. Furthermore, age class width transitions from cw to

pw following the first harvest.

The remainder of this paper examines bias in linear programming based harvest schedule

models caused by aggregate/periodic analysis. Methods are outlined in §2. Methods

parallel those employed by Barber (1985), with changes reflecting the use of

mathematical modeling as well as updated management practices. There is a particular

emphasis on mimicking typical private, institutional investor and industrial planning

practices in an effort to maximize utility for planning foresters. Section 3 outlines results,

followed by conclusions and a discussion of management implications in §4.

2. Methods

Test forest – A hypothetical 1,000-acre test forest was used for the study. All stands were

Douglas fir (Pseudotsuga menziesii) plantations with a planting density of 360 trees per

acre and a Douglas fir site index of 140. Three initial age class distributions were used,

uniform, negative skew, and positive skew. In each distribution, one-twelfth of the area

was assigned to each five year interval from 1 to 60 years. The distributions differed in

their assignment of acres within each five year interval as follows:

Uniform – area uniformly distributed across all ages,

Negative skew – entire area placed at the lower limit of the interval, and

Positive skew – entire area placed at the upper limit of the interval.

The skewed distributions, while unlikely, represent the extremes that could be

encountered (Barber 1985). The resulting annual stratifications had either 60 (uniform

distribution) or 12 (skewed distributions) strata. Note that once aggregated into age

classes, all three initial age class distributions appear identical, with 1/12th (5-year age

class width) or 1/6th (10-year age class width) of the area in each age class.

Harvest schedules – Strategic harvest schedule models were developed utilizing a model

II linear programming formulation (Johnson and Scheurman 1977). The models

consisted of an objective function maximizing net present value (6% real discount rate)

over a 150-year planning horizon. Following common practice, cash flows were

discounted from period mid-point. Stumpage prices were developed from Oregon Region

1 average delivered prices (Oregon Department of Forestry 2008). Real trend prices less

logging and hauling costs were calculated based on Q1 1985 to Q1 2008 time series.

General modeling assumptions include: 1) all stands harvested must be replanted, 2) all

harvested stands are regenerated to the same forest type and planting density, and 3)

harvest levels cannot increase or decrease between planning periods. Although some

authors warn against the utility of a strict ‘even flow’ constraint (e.g., McQuillan 1986,

Pickens et al. 1990), these are often implemented in practice, pointing towards the

appropriateness for this analysis. Clearcut harvests were allowed on strata greater than 20

years in age. Clearcuts were the only removals modeled.

Yield data – Yield tables were constructed for existing and regenerated stands utilizing

the FORSim Pacific Northwest Growth Simulator v2008.7 (FORSight Resources 2008),

implementing the Stand Management Cooperative (SMC) variant of the ORGANON

growth and yield model (Hann 2006). ORGANON is a distance independent, individual

tree model based on empirical relationships and validated for conifer and some mixed

species stands. FORSim PNW was used to generate merchandized yields for No. 1, No.

2, No. 3, No. 4, and utility grade sawlogs (see Northwest Log Rules Advisory Group

2006 for product specifications). A third degree polynomial was fit to ORGANON 5-

year periodic yields for each product to allow annual interpolation. This follows Barber

(1985) who used this procedure to interpolate total cubic volume. Note that although

fitted functions allow volume calculations in partial year increments, all fractional harvest

ages were rounded up to the nearest next whole age.

Test cases – Test cases are outlined in Table 1. All combinations of the common initial

age and harvest timing assumptions were tested within the four general cases.

Comparisons are against annual models with one-year age classes developed for each

initial age class distribution. Because periodic/aggregate analysis is used to approximate

annual models, these serve as appropriate standards. The annual models will be termed

the ‘standard models’ throughout the remainder of the paper.

Table 1. Test cases run, including abbreviations for each case. Planning Period

Width

Age Class

Width

Initial Age

(Within Age Class)

Harvest Timing

(Within Planning Period) Abbreviation

1-year planning period width, 5-year age class width (PP1AC5##)

1 5 Mid-point End-point PP1AC5ME

1 5 End-point End-point PP1AC5EE


5 5 Mid-point Mid-point PP5AC5MM


5 5 End-point Mid-point PP5AC5EM


5-year planning period width, 10-year age class wdith (PP5AC10##)










3. Results

Results are reported for a 100-year subset and a 1-period subset of the 150-year planning

horizon. Using a subset of the full planning horizon ensures that results are not influenced

by end of horizon effects common to harvest schedule models. The 100-year subset

provides a comparison of long term averages, while the 1-period subset provides a

measure of the implications on short-term scheduling. The latter is important as it is often

only the first one to two years that are implemented, with the remaining periods included

to ensure long term feasibility.

In all cases, bias is measured as:

( )c i

i

o s

s,

where oc refers to the observed case value and si refers to standard model value for initial

age class distribution i. This yields the percent (as a decimal) over- or under-estimate for

each parameter. Negative values indicate under-estimates while positive values indicate

over-estimates.

Table 2 shows the volume bias results. Notice that results for cases PP5AC5ME and

PP5AC5EM are identical. Returning to equation (1), it can be seen that when cw = pw and 1 2 harvest ages are equivalent. Because the assumed harvest ages are identical, the

resulting harvest schedules are also identical. The case PP5AC5EM will be omitted from further tables with the understanding that its bias is equivalent to case PP5AC5ME. Additionally, volume bias does not vary with the planning horizon subsets. The even

volume flow constraint prevents volume from deviating between period, meaning

average annual harvest volume is the same throughout the planning horizon. As a result,

volume bias is the same at all time resolution levels, leading to the omission of time

resolution from Table 2.

Table 2. Volume bias values by test case and initial age class distribution. Bold values

indicate the minimal bias for each planning period/age class width combination.

Case Uniform Neg. Skew Pos. Skew

PP1AC5ME 0.009 0.028 -0.008

PP1AC5EE 0.025 0.044 0.008

PP5AC5MM 0.030 0.050 0.013

PP5AC5ME 0.056 0.076 0.038

PP5AC5EM 0.056 0.076 0.038

PP5AC5EE 0.071 0.091 0.053

PP5AC10MM 0.035 0.054 0.017

PP5AC10ME 0.051 0.071 0.034

PP5AC10EM 0.075 0.095 0.057

PP5AC10EE 0.090 0.110 0.071

PP10AC5MM 0.032 0.051 0.015

PP10AC5ME 0.072 0.093 0.055

PP10AC5EM 0.049 0.069 0.031

PP10AC5EE 0.087 0.108 0.069

In all periodic models volume bias is minimized with the age class mid-point/planning

period mid-point (MM) assumption. It is interesting to note that within this assumption

set, volume bias is of a similar magnitude for each initial age class distribution regardless

of age class width and planning period width (e.g., Figure 1). In the case of the annual

model, volume bias is minimized with the age class mid-point assumption (ME) for both

the uniform and negative skewed distributions. With the positive skewed distribution, the

ME and EE cases produce bias of the same magnitude, one positive and one negative.

-0.025

0.000

0.025

0.050

0.075

PP1AC5 PP5AC5 PP5AC10 PP10AC5

Planning Period (PP) and Age Class (AC) Width (Years)

Bia

s (D

ecim

al P

erce

nt)

.

Positive Skewed ACD

Uniform ACD

Negative Skewed ACD

Figure 1. Volume bias for the age class midpoint/period midpoint (age class

midpoint/planning period endpoint in the PP1AC5 case) assumption set by planning

period and age class width.

The trend shown in Figure 1 carries through to all cases; volume bias is highest with the

negative skew distribution and lowest with the positive skew distribution. The negative

skew distribution causes a reduction in the average harvest age with the standard models

when compared to the uniform distribution. As a result, the harvest volume with the

standard model goes down, increasing the positive volume bias. Alternately, the positive

skew distribution results in average harvest ages which are higher; harvest volume goes

up, and volume bias is reduced.

In all but one test case positive volume bias was observed, indicating that predicted

harvest volumes are higher than the true values. This is counter to observations by Barber

(1985), who noted negative bias in all test cases but one. The method used to calculate

assumed harvest age in Barber’s work is expressed as:

(2) ( ) 3 12

wB u w

pharvest age a p t= + ,

where au, pw, and t are defined as in (1) and 3 = {1 if yields are calculated at age class mid-point, 0 otherwise}. Note that this equation is valid only for the case cw = pw. By comparing this against equation (1), it can been seen that harvest age > harvest ageB except when cw = pw and 1 = 2 = 1 3, in which case the two are equivalent. This is evident when examining average harvest age bias (see Table 3). While Barber (1985) reports negative average harvest age bias, results from this study indicate that positive bias is more often the case. Differences between annual and aggregate/periodic models will thus be larger, contributing to the positive volume bias. Table 3. Harvest age bias values by test case and initial age class distribution. Results

are shown for 100-year and 1-period subsets of the 150-year planning horizon. Bold

values indicate the minimal bias for each planning period/age class width combination.

100-year Subset 1-period Subset

Case Uniform ACD Neg. Skew Pos. Skew Uniform ACD Neg. Skew Pos. Skew

PP1AC5ME 0.016 0.063 -0.025 -0.024 0.045 -0.025

PP1AC5EE 0.064 0.113 0.021 0.017 0.089 0.017

PP5AC5MM -0.031 0.014 -0.071 -0.006 0.064 -0.007

PP5AC5ME 0.029 0.077 -0.013 0.038 0.111 0.037

PP5AC5EE 0.076 0.126 0.032 0.081 0.157 0.080

PP5AC10MM -0.027 0.018 -0.067 -0.041 0.027 -0.042

PP5AC10ME 0.021 0.068 -0.021 0.043 0.116 0.042

PP5AC10EM 0.079 0.129 0.035 0.001 0.071 0.000

PP5AC10EE 0.126 0.178 0.080 0.084 0.161 0.083

PP10AC5MM -0.025 0.020 -0.065 -0.009 0.061 -0.010

PP10AC5ME 0.079 0.129 0.035 0.035 0.108 0.034

PP10AC5EM 0.021 0.069 -0.020 0.080 0.156 0.079

PP10AC5EE 0.127 0.179 0.081 0.124 0.203 0.123

Given the calculations outlined above, it could be expected that the case PP5AC5MM (cw = pw and 1 = 2 = 1) would exhibit negative volume bias for the uniform and positive skew distributions (see Barber 1985, Figures 5B and 5C, yield calculated at the top of the interval). Failure to observe negative bias is explained by differences in methodology between the studies. Consider the example age class distribution shown in Figure 2. On the top is a subset of the uniform initial age class distribution using five-year period and age class widths. The age class distribution after harvest of an area equivalent to the acres in a single class is shown at the bottom. The dark bars indicate the ending distribution if an oldest first selection strategy is used; the checked bars indicate the ending distribution if an alternate strategy is used.

Figure 2. Illustration of uniform initial age class distribution using 5-year age classes

(top). The age class distribution after five years of growth assuming harvest of an area

equal to the acres in each class is shown at the bottom. The dark bar indicates the ending

distribution if an oldest first selection strategy is used, the checked bar shows the ending

distribution using an alternate selection strategy. Barber enforced an oldest first selection strategy, the same strategy illustrated by the dark bars in Figure 2 (bottom). A linear programming model will exhibit similar behavior if it is unconstrained or the oldest acres are younger than the economically optimal rotation age. The models in this study conform to neither case. Over time the forest will shift to a

younger age class distribution as the model schedules over-mature acres. An average

harvest age reduction follows, along with a corresponding harvest volume decrease. This

harvest volume reduction limits the achievable even flow volume. The model counteracts

this phenomenon by retaining and harvesting some acres later (illustrated by the checked

bars in Figure 2). Because volume per unit area increases with age (during the pertinent

yield curve portion), retaining and harvesting older acres allows an even flow volume

increase. Age class aggregation exaggerates the increase to levels unattainable with

annual age classes, giving rise to the positive volume bias discussed earlier. Examining

the test case harvest schedules confirms this behavior, with a mix of different aged stands

scheduled for harvest and some older acres retained. An even flow volume shift is not possible using volume control models due to the implicit constraint requiring annual harvest to be equal to the long term sustained yield (LTSY). In contrast, the even flow constraints of a linear programming model simply require an even

harvest level sustainable throughout the planning horizon. Indeed, it is widely recognized that even flow constraints produce annual harvest volumes in excess of LTSY with unregulated, mature forests (e.g., McQuillan 1986, Pickens et al. 1990). This holds true on

the hypothetical forest used here, with annual harvest levels 41% to 56% higher than

LTSY. Sustaining the increased harvest levels requires harvest of additional young acres later in the planning horizon. This is evident by the negative bias in average harvest age and the positive bias in average harvest area for case PP5AC5MM (see Tables 3 and 4). Table 4. Harvest area bias values by test case and initial age class distribution. Results

are shown for 100-year and 1-period subsets of the 150-year planning horizon. Bold

values indicate the minimal bias for each planning period/age class width combination. 100-year Subset 1-period Subset

Case Uniform ACD Neg. Skew Pos. Skew Uniform ACD Neg. Skew Pos. Skew

PP1AC5ME -0.014 -0.044 0.014 0.025 -0.027 0.009

PP1AC5EE -0.040 -0.069 -0.013 0.007 -0.044 -0.008

PP5AC5MM 0.068 0.035 0.097 0.037 -0.016 0.021

PP5AC5ME 0.020 -0.011 0.049 0.009 -0.042 -0.007

PP5AC5EE -0.007 -0.037 0.021 -0.008 -0.059 -0.024

PP5AC10MM 0.062 0.029 0.091 0.070 0.016 0.053

PP5AC10ME 0.030 -0.002 0.058 0.051 -0.003 0.034

PP5AC10EM -0.011 -0.042 0.016 0.023 -0.029 0.007

PP5AC10EE -0.035 -0.065 -0.009 0.006 -0.046 -0.010

PP10AC5MM 0.058 0.025 0.087 0.034 -0.019 0.017

PP10AC5ME -0.013 -0.043 0.015 -0.013 -0.064 -0.029

PP10AC5EM 0.028 -0.004 0.056 0.014 -0.038 -0.002

PP10AC5EE -0.038 -0.067 -0.011 -0.030 -0.080 -0.046

Returning to Table 3, it can be seen that a trend across all cases does not exist as it does

with volume bias. At the 100-year resolution, the MM assumption set minimizes bias

with the negative skew initial age class distribution. For the uniform or positive skew

distributions, bias is minimized with either the ME or EM assumption sets. Moving to the

1-period subset, the cases with five-year age class width reach a minimum using the MM

assumption set. The 10-year age class width reaches the minimum for the negative

distribution using the MM assumption set, but for the other two distributions EM

produces minimal bias. Age bias is minimized for the PP1AC5## case using the ME

assumption for the negative skew and the EE assumption for the positive skew across

both time resolutions. Bias for the uniform distribution is minimizes using the ME

assumptions at the 100-year resolution but the ME for the 1-period resolution.

The average harvest age bias for each of the four general cases is lower moving from the

100-year to 1-period resolution. The linear programming model cannot take advantage of

the age shifts previously discussed until at least the second period. As a result, the age

makeup of the scheduled acres tends to be more consistent between the annual and

periodic models for the first period. Increasing harvest throughout the planning horizon

does, however, require immediate harvest of additional mature acres relative to the

standard models. Differences between the 100-year and 1-period subsets also highlight

the cumulative nature of bias. As outlined earlier, periodic models harvest more young

stands late in the planning horizon to allow higher harvest volumes. Volume bias for the

1-period subset does not include the impact of this shift, evident by the generally lower

age bias in the 1-period subset.

Average annual harvest area bias is shown in Table 4. As with average harvest age, there

is not a clear trend across all cases. Examining the correlation between area and age bias

(see Table 5) gives a better understanding of area bias results. In all test cases the two

exhibit a strong negative correlation (r -0.745). The negative correlation indicates an

opposing relationship, clearly visible in the example shown in Figure 3. Throughout the

pertinent portions of the yield curve, volume increases with age. This means that negative

age bias forces additional acres to be scheduled to achieve similar harvest volumes, while

positive age bias requires less harvest area. The relationship weakens at the 1-period

resolution since long term average conditions are no longer being examined. Instead, the

relationship reflects a single point along the yield curve. A final yet important

observation is that the assumption set minimizing volume bias (MM) often results in the

largest average annual harvest area bias.

Table 5. Correlation coefficients for average annual harvest area and average harvest

age bias. (Note: Cases PP5AC5ME and PP5AC5EM are both included in calculations).

100-year Subset 1-period Subset

Case Uniform ACD Pos. Skew Neg. Skew Uniform ACD Pos. Skew Neg. Skew

PP1AC5 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000

PP5AC5 -0.995 -0.995 -0.995 -0.988 -0.988 -0.988

PP5AC10 -0.998 -0.998 -0.998 -0.748 -0.748 -0.748

PP10AC5 -0.999 -0.999 -0.999 -0.745 -0.745 -0.745

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

PP5AC10MM PP5AC10ME PP5AC10EM PP5AC10EE

Test Case

Bia

s (

De

cim

al

Pe

rce

nt)

Area Bias

Age Bias

Figure 3. Average annual harvest area and average harvest age bias for the 5-year

planning period width and 10-year age class width cases for the uniform initial age class

distribution and 100-year planning period subset.

To test the implications of implementing the plans generated with the test cases, target

values were applied to the standard model with the uniform age class distribution.

Common practice is to use the strategic plan to set volume or area targets for on the

ground management. In general, only the first year is implemented, with re-planning

efforts in subsequent years to include updated information from inventory and

silvicultural treatments. This was mimicked by setting either harvest area or volume

targets for the first planning period. Targets were derived from the test case that

minimized area or volume bias, depending on the targets used, for each of the four age

class/planning period width combinations.

Results for the area targets are outlined in Table 6. In three of the cases, a fall down in

harvest volume of at least 2.5% was recorded. In the fourth case, PP10AC5, the solution

was infeasible. Only when the even flow volume constraint was relaxed did these area

targets lead to a feasible solution.

Table 6. Annual harvest volumes when area targets from the test case are used as

constraints in the uniform initial age class distribution standard model compared to

annual harvest volume with the test case.

Case Test Case Volume Annual Volume Fall Down

PP1AC5EE 1,485,147 1,448,591 2.5%

PP5AC5EE 1,551,975 1,432,616 8.3%

PP5AC10EE 1,578,577 1,448,589 9.0%

PP10AC5ME 1,553,557 Infeasible -----

No test case produced feasible solutions when harvest volume targets were set. This is an

expected result given the positive volume bias discussed earlier. The volume generated

with the test cases cannot be sustained throughout the planning horizon. Relaxing the

even flow constraint to first apply in the second period produced feasible solutions, but

there was a notable decline between first and second period harvest volumes. This

illustrates the declining even flow effect outlined by McQuillan (1986). The even flow

volume cannot be sustained long term, so in subsequent rounds of planning there is a

decline in even flow volume.

4. Discussion

The results indicate that constrained harvest schedule models with aggregated age classes

consistently exhibit positive volume bias relative to models with annual age classes,

regardless of initial age class distribution. This differs from observations made by Barber

(1985), who noted a bias toward underestimating harvest volume. Differences can be

attributed in part to assumed harvest age calculations as well as differences in

methodology. The latter is important because it indicates that extending Barber’s earlier

work to constrained mathematical programming models should be approached with

caution. Whether his results can be extended to unconstrained mathematical models

remains unclear, but this has little practical importance as few forest planning models fit

the unconstrained case.

Some planners have suggested initially using annual periods, then compressing time in

subsequent periods (e.g., Barber 1985). They believe that this may generate more

accurate harvest volume and area estimates in the short term. Short term accuracy is

important since the first few periods are often the one implemented. It should be noted,

however, that the implications of this technique are unclear. This methodology requires

further examination with a similar study to provide a full understanding of the problems

and bias that may arise.

Recent years has seen an increase in the popularity of annual models with aggregated age

classes. The general case PP1AC5## represents this scenario. As with the method

outlined above, the goal here is to better capture annual harvest. The volume bias results

indicate that this technique does in fact produce more accurate annual harvest volume

estimates than the other test cases. The benefit to harvest area estimates is inconsistent,

however, with larger bias for the negative skewed distribution relative to the other

general cases.

An alternate approach is the case of annual age classes with five-year planning periods, a

case absent from this analysis. The usefulness of this method is somewhat questionable

given that a primary reason for aggregate/periodic analysis is model size reduction.

Although five-year planning periods would reduce model size over an annual model, the

failure to aggregate stands into age classes would result in a large number of strata. In

general, a model with annual periods and five-year age classes will have fewer strata,

resulting in smaller model size.

The significance of the results outlined in §3 became apparent when strategic plan

implementation was considered. As described earlier, feasibility issues became a concern.

Even when plans were feasible, a decline in harvest volume from projected levels was

observed (see Table 6). This presents additional feasibility concerns in the presence of

minimum volume constraints, often implemented for wood or mill supply requirements.

If these constraints are binding, there will be a shortfall when the plan is implemented.

Additional acreage can be harvested to reach the needed volume, but this cannot be

sustained into the future. In addition, increasing harvest acreage can present

complications if area-based goals or constraints are included in the model, such as

wildlife habitat or a target ending age class distribution. Increasing harvest acreage will

directly impact these goals, possibly causing additional feasibility concerns.

These observations fall in line with authors studying uncertainty in model yields.

Weintraub and Abramovich (1995) discuss feasibility problems arising from yield

uncertainty in-depth. They recognize that feasibility issues that can arise when area

targets produce volume levels below those projected. Kangas (1999) notes that ignoring

uncertainty in model yields can lead to selection of non-optimal solutions which may be

infeasible. Although these authors dealt with the stochastic nature of yields, their

observations fall closely in line with those made here.

Planners are left with the question of whether it is better to minimize bias in harvest

volume or harvest area. In general, it is felt that is more important from a planning

perspective to implement the strategy defined by the model (both harvest and silviculture)

rather than to meet the volume objectives (get the wood out). Regardless of how a

planner chooses to proceed, there are signification implications for feasibility, both long

and short term. Perhaps the best approach is to minimize area bias when area-based goals

are present and volume bias when volume-based goals are present. This may not be

possible given that forest planning problems often include both. The decision of which

bias to minimize is beyond the scope of this research, but it is a critical topic for planners

and it requires further study.

This study also does not address yield bias at the tactical planning level. It is unclear if

similar results could be expected. Also unclear are the implications of strategic volume

and area bias when linking strategic and tactical planning. This too is a critical area for

further examination.

Regardless of the assumption set a planner chooses to follow, it is important to

understand the issues those assumptions may cause. As with any modeling exercise, the

harvest schedule model is an abstraction of reality, and, as such, some biases are

inevitable. A thorough understanding of bias sources is critical for planners to maximize

the utility of their models and understand issues that may arise during implementation.

References Barber, R.L. 1985. The aggregation of age classes in timber resource schedule models: Its effects and bias. Forest Science 31(1): 73-82. FORSight Resources. 2008. FORSim Pacific Northwest Growth Simulator (PNW) v2008-5 User Documentation. Hann, D.W. 2006. ORGANON User’s Manual, Edition 8.2. Johnson, K.N. and H.L. Scheurman. 1977. Techniques for prescribing optimal timber harvest and investment under different objectives – Discussion and synthesis. Forest Science Monograph 18 23(1). 38p. Kangas, S.A. 1999. Methods for assessing uncertainty of growth and yield predictions. Can. J. For. Res. 29: 1357-1364. McQuillan, A.G. 1986. The declining even flow effect – Non sequitur of national forest planning. Forest Science 32(4): 960-972. Northwest Log Rules Advisory Group. 2006. Northwest Log Rules Eastside and Westside Log Scaling Handbook. 12th ed., 76p.

Oregon Department of Forestry. 2008. Log price information: Domestically processed logs (delivered to a mill), Region 1. Available online: oregon.gov/ODF/STATE_FORESTS/TIMBER_SALES/logpage.shtml. Pickens, J.B., B.M. Kent, and P.G. Ashton. 1990. The declining even flow effect and the process of national forest planning. Forest Science 36(3): 665-679. Weintraub, A. and A. Abromovich. 1995. Analysis of uncertainty of future timber yields in forest management. Forest Science. 41(2): 217-234.

barber revisited: aggregate analysis in harvest scheduling models

Business