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The economic impact of green-up constraints in the southeastern United States
Forest Ecology and Management 145 (2001) 191±202
The economic impact of green-up constraints in the southeastern United States Kevin Bostona,*, Pete Bettingerb a
Warnell School of Forest Resources, University of Georgia, Athens, GA 30602, USA Department of Forest Resources, Oregon State University, Corvallis, OR 97331, USA
b
Received 27 December 1999; accepted 21 March 2000
Abstract Green-up, or adjacency, requirements are a common constraint in forestry. The American Forest and Paper Association has developed a Sustainable Forestry Initiative that includes a green-up constraint which limits the average clearcut opening to 48 ha for 3 years or until the average height of the regenerated trees is >1.4 m. In addition to constraining the average clearcut size, many forestry companies in the southeastern USA voluntarily limit their maximum clearcut size to between 60 and 90 ha. In this research, a heuristic algorithm was used to develop tactical forest plans that consider both the maximum and average clearcut sizes. Economic effects of the green-up constraints were estimated for situations where intensive management can reduce the length of the green-up time from 3 to 2 years on a 21 600 ha ownership in Georgia (USA). For a 60-ha maximum opening size, this reduction in green-up time from 3 to 2 years resulted in an additional US$ 66 600 in present net worth (PNW) over a 10-year analysis period. This corresponds to a US$ 10 per harvested ha, or a 0.8% increase in PNW. The bene®t gained by reducing the length of the green-up period is less with a 90-ha maximum clearcut size, where PNW increases by US$ 45 600, or US$ 6.70 per harvested ha, a 0.5% increase. While the total volume per period was near the volume goal produced by a strategic forest plan, the spatial restrictions and the desire to maximize net present value resulted in lower volume of timber products (sawlogs and chip-and-saw logs) from older forest stands. A sensitivity analysis showed that an increase in price or yield further reduced the economic incentive for the reduction of the length of the green-up constraint. As price or volume decreased below expectations, however, the incentive to use intensive forest management practices to reduce the length of the green-up constraint became more attractive, since the differences between a 2-year and 3-year green-up time requirement may be large enough to pay for more intensive management practices. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Green-up constraints; Economic effects; Forest management
1. Introduction Increasing competition from international pine-producing regions is forcing forest industry organizations in the southeastern United States to adopt intensive *
Corresponding author. E-mail addresses: [email protected] (K. Boston), [email protected] (P. Bettinger).
forestry practices, including extensive site preparation, weed control, and fertilization. It is well accepted that these practices can accelerate forest growth rates (Knowe et al., 1985; Bacon and Zedaker, 1987), but they also add to the operational costs of growing trees. Forestry ®rms are under increasing pressure to produce higher returns for their shareholders and may only adopt practices that improve their overall pro®tability. Activities are often selected on their ability to
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improve pro®ts, not just improve growth rates and reach sustainability. In addition to increased competition from international producers, most forest products companies in the southeastern United States have adopted the American Forest and Paper Association (AF&PA) Sustainable Forestry Initiative (SFI). The initiative is a voluntary agreement among AF&PA members to promote sustainable forestry practices. One component of the SFI guidelines is a green-up constraint that restricts the average clearcut size to <48 ha for either 3 years or until the average height of the regenerated trees is >1.4 m. Additionally, many companies have developed an internal policy to voluntarily limit their maximum clearcut size to 60±90 ha. Estimating the effects of the combination of these land management constraints poses a challenging combinatorial problem. The ®nancial impact of various green-up constraints has generally been estimated by using relatively long periods of time for adjacent stands to `green-up'. For example, in Canada, three different green-up periods (10, 20, and 30 years), three different maximum opening sizes (20, 40, and 80 ha), and three different rotation lengths (80, 100, and 120 years) were used to compare the cost of various harvesting restrictions (Daust and Nelson, 1993). The smallest maximum size and the longest greenup time resulted in a 29% reduction in harvest volume compared with an optimal solution from a linear programming effort, which did not contain the spatial constraints. In Oregon (USA), increasing the greenup period from 20 to 30 years showed a 34±40% reduction in the present net worth (PNW) (Yoshimoto and Brodie, 1994). In California (USA), increasing the green-up period from 10 to 20 years reduced PNW by 13% for a 4-ha maximum opening size. As the minimum patch size increased to 32 ha, there was smaller difference for increased exclusion period (Barrett et al., 1998). A variety of heuristic techniques can be used to solve problems that consider management options with green-up size constraints, or spatially constrained harvest scheduling problems. Van Duesen (1999) was able to solve numerically-large spatially constrained problems by using a simulated annealing algorithm, but did not compare the quality of his solutions. Two problems based on a smaller spatial
scale showed no signi®cant difference in the objective function values for a random starting location hill-climbing algorithm, Monte-Carlo integer programming, simulated annealing, and tabu search (Murray and Church, 1995). Four problems of larger spatial scales were used to compare Monte-Carlo integer programming, simulated annealing, and tabu search. Simulated annealing found the best solution to three of the four problems while tabu search found the best solution to the fourth (Boston and Bettinger, 1999). Most of the tabu search algorithms used to solve harvest scheduling problems have a single search neighborhood. Bettinger et al. (1999) describe a technique that expands this by allowing two search neighborhoods to change their status simultaneously. This technique produces signi®cant improvements in the objective function value, but at the cost of longer processing times. Tabu search also tends to ®nd solutions that are concentrated in one portion of the solution space (Glover et al., 1995). The development of a technique that can combine several local optimal solutions into a single new solution may yield better results (Glover et al., 1995). This paper begins by describing a heuristic technique that combines tabu search and genetic algorithm techniques. Then, the heuristic technique is used to develop several tactical forest management plans and, subsequently, estimate the ®nancial effects of reducing the green-up time and maximum clearcut size constraint. The economic effects are estimated for situations where the green-up time is reduced from 3 to 2 years, and where the maximum clearcut size ranges between 60, 70, 80, and 90 ha. The change in revenue, if positive, could provide additional ®nancial incentive for companies to adopt intensive forest management practices. The goal is to determine whether variations in green-up parameters matter ®nancially in forest planning. Obviously, this can, and has been, shown to be true for long green-up times and small maximum clearcut sizes. However, it has not been shown for shorter differences in green-up times and larger maximum clearcut sizes. If there is an economic bene®t for reducing the length of the green-up time in areas where tree species can be more intensively managed, this will provide further incentive for using intensive management techniques.
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2. Methods We ®rst develop the problem formulation for evaluating green-up constraint parameters, then present the heuristic technique used to develop the tactical forest plans. Then, a database from a southeastern United States (Georgia) forest industry organization, representing 21 600 ha, is used to evaluate the economic effects of variations in green-up requirements. 2.1. Model formulation Two models were used to solve this problem, one for a strategic 30-year plan, and one for a tactical 10year plan. The strategic plan provides the target harvest goals for the tactical plan, and it can be considered a `relaxed' problem, since the spatial constraints for green-up are absent. The results from this plan provide a theoretical upper-bound on the solution. The objective function consists of maximizing the net present value of revenue from harvests less logging costs: maximize
J X N X T X
Revnt ÿ Lcnt Vjnt Xnt
(1)
j1 n1 t1
where J is the number of products, j the product type, N the number of harvest units, n the harvest unit; T the number of time periods, t the time period; Revnt the revenue per cubic meter for unit n harvested in time period t; Lcnt the logging cost per cubic meter for unit n harvested in time period t, Vjnt the volume per hectare of product j in unit n harvested during time period t, Xnt the continuous variable indicating whether unit n is harvested during time period t. A minimum harvest age of 19 years was assumed for each timber stand, and individual product volumes could not change by >5% per time period, except sawlog volumes, which were set to 10% per period. PN PN n1 Vjnt Xnt ÿ n1 Vjnt1 Xnt1 1 ÿ PN n1 Vjnt Xnt deviation j 8j; t 1; . . . ; t ÿ 1
(2)
The formulation for the tactical planning problem is similar to the strategic plan, and to the objective
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function found in Boston and Bettinger (1999). The objective function for the problem is: maximize
J X N X T X
ÿ
Revnt ÿ Lcnt Vjnt Xnt
j1 n1 t1 J X T X j1 t1
Vpjt dujt ÿ
J X T X Vpjt dljt
(3)
j1 t1
where Xnt0,1 is a variable indicating whether unit n is harvested during time period t; Vpjt the volume penalty per cubic meter of product j during time period t, dujt the positive deviation from volume goal of product j during time period t, and dljt the negative deviation from volume goal of product j during time period t. The volume goals we used were those that resulted from the strategic plan. Penalties were used to force the heuristic search process towards solutions which would emulate the strategic planning solution. A three-part step function was used to penalize deviations from the volume goal for each product. For deviations <10% from a volume goal, the penalty value is 50% of the product price (Table 1). For deviations between 10 and 25% from a volume goal, the penalty value is equal to the price. For deviations >25% from the volume goal, the penalty value is 150% of the price. Three main constraints were considered. The ®rst is a volume constraint, which is an accounting constraint used to sum the volume harvested in each period and determine the deviations from the goal for each product: N X
Vjnt Xnt ÿ dunt dlnt volume goal 8j; t
(4)
i1
The second constraint is a singularity constraint, which limits each unit to one treatment during the planning horizon: T X
Xnt 1 8n
(5)
t1
Table 1 Prices assumed for forest products Products
Price ($/m3)
Sawlogs Chip-and-saw logs Pulpwood
46.24 32.12 12.00
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The third constraint defines a maximum opening size for each logging unit n and its set of adjacent neighbors (Un). For a 3-year green-up constraints, we define a set (Tm) of near-time periods (mz, where m1tÿ3, m2tÿ2, m3tÿ1, m4t, m5t1, m6t2, and m7t3; all mz0, otherwise not in Tm; all mzT, otherwise not in Tm). We define a similar set for a 2year green-up constraint (mz, where m1tÿ2, m2tÿ1, m3t, m4t1, m5t2; all mz0, otherwise not in Tm; all mzT, otherwise not in Tm). Therefore, an opening is not just the harvests that occur in time period t, but also includes those around each unit n that have occurred during the near-time periods. The maximum opening size constraint is: " # Un X Tm X Ak Xkm An Xnt k1 m1
Maximum opening size 8n where Xnt 1; t (6) where Un is the set of adjacent neighbors to unit n, k the adjacent neighbors to unit n, Tm the set of neartime periods, m a near-time period, Ak the area of unit k; Xkm is 0,1, a variable indicating whether unit k is harvested during near-time period m; An the area of unit n; and Xnt is 0,1, a variable indicating whether unit n is harvested during time period t. A maximum average opening size can be de®ned for each time period. If each opening is centered around a focal unit (f) during a time period (t), we can de®ne the size of the opening (Oft) as: " # Un X Tm X Ak Xkm Af Xft Oft (7) k1 m1
where Un is the set of adjacent neighbors to focal unit f, k the adjacent neighbors to focal unit f, Tm the set of near-time periods, m a near-time period, Ak the area of unit k, Xkm the 0,1 variable indicating whether unit k is harvested during near-time period m, Af the area of focal unit f, and Xft is 0,1, a variable indicating whether focal unit f is harvested during time period t. As we are attempting to calculate the average opening size, we do not wish to count openings more than once. This mis-counting could occur if we allow each unit (n) in an opening (composed of multiple units n) to be considered the `center'. Therefore, only one unit (n) can be delineated as the focal center (f) of the
opening in any time period, and the total number of openings equals the number of focal centers of openings. Thus, X X X X f n and Xft Xnt (8) The average opening size for a set of openings in a time period (t) can then be constrained with the following equation: hP i F f 1 Oft average opening size 8t (9) F where F is the total number of openings, f the focal center of an opening, or an opening itself, and Oft an opening centered around a focal unit f during time period t. Unlike our strategic plan, the tactical forest plans include the spatial constraints noted in equations (6)± (9). For Eq. (6), we examined maximum opening size constraints of 60, 70, 80, and 90 ha. For Eq. (9), we used the AF&PA SFI maximum average clearcut size of 48 ha. The strategic plan was developed using linear programming techniques, and the tactical forest plans were developed using a heuristic technique. 2.2. Sensitivity analysis Two types of sensitivity analysis were completed when developing the tactical forest plans, one for price and one for volume. Prices for sawlogs and chip-andsaw logs were increased by 5%, then decreased by 5%. A similar approach was used to test the impact of volume sensitivity where a uniformly distributed random number generator was used to assign an adjustment factor for each unit to ®rst increase, then decrease the volume harvested in a unit between 0 and 5%. In each of these analyses, we examined maximum opening size constraints of 60, 70, 80, and 90 ha, and used the AF&PA SFI maximum average clearcut size of 48 ha. 2.3. Heuristic technique The heuristic technique is a hybrid algorithm consisting of ®ve components (Fig. 1). The ®rst is a Monte-Carlo integer programming algorithm that randomly develops an initial solution. This process selects a logging unit, determines if it meets the
K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
Fig. 1. Flow chart of the heuristic technique used to develop tactical forest plans.
minimum harvest age, and assures that its incorporation into the solution does not violate the green-up constraints. This continues until 10% of the sawlog volume has been achieved in all periods. Because each new run of the heuristic technique uses a new seed for a random number generator, this component will allow an increase in the proportion of the solution space explored when the program is executed repeatedly.
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The second component is the core tabu search routine, similar to the algorithms described in Murray and Church (1995), Bettinger et al. (1997), and Boston and Bettinger (1999), and is composed of two elements: (1) a tabu list that maintains a record of the recent moves; and (2) the aspiration criteria. After experimentation, 100 iterations were selected as the tabu list length. For this application, the aspiration criterion was assumed to be the overall best objective function value. The best move from the neighborhood of moves will be considered ®rst, whether or not it improves the current solution. If the move is not tabu, it will be accepted into the solution. If the move is tabu, yet exceeds the aspiration criteria, it too will be accepted into the solution. If a move is tabu, yet does not exceed the aspiration criteria, it is not accepted. The third component is the intensi®cation routine. The objective of an intensi®cation routine is to explore a portion of the neighborhood that has already yielded a good solution to search for better solutions. This intensi®cation routine begins by recalling the current best solution from the core tabu search routine. By using a 2-opt neighborhood search routine described in Bettinger et al. (1999), two units can simultaneously change their status. the intensi®cation routine has the same short-term memory features as the core tabu search routine, but the tabu list has been reduced from 100 to 20 iterations. Tabu search ®nds good solutions to large combinatorial problems, but they tend to be concentrated in a small portion of the solution space (Glover et al., 1995). Thus, the fourth and ®fth components of the heuristic technique have the goal of forcing the search to new parts of the solution space. The fourth component (Fig. 1) is a diversi®cation routine that schedules those units with the lowest frequency of entering the solution, while maintaining the minimum harvest age and not violating the green-up constraints. This diversi®cation will force the algorithm to the least explored portion of the solution space. The resulting solution becomes the starting point for the core tabu search routine. The ®fth component has the goal of combining two neighborhoods where good solutions were found with the hope of ®nding a superior solution. It is based on a crossover routine used in genetic algorithms. The genetic crossover routine treats the solution to the forest planning problems as if they were chromosomes, with each unit being a gene on a chromosome.
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The values for the genes, the alleles, become the periods when the unit is harvested. By using a random number generator that selects the crossover point, the two `chromosomes' are recombined into two new solutions. The solution with the highest objective value survives the crossover and becomes the starting solution for a continuation of the core tabu search routine. This heuristic technique has been shown to produce good results to similar scheduling problems which have 3000±5000 0±1 integer variables (companion paper regarding validation of the heuristic technique is in preparation), although the heuristic technique described here uses 2-opt moves. Extending the use of this solution method to larger scheduling problems therefore seems reasonable, although the performance of the solution method cannot be directly compared against known optimal solutions. 2.4. Data description The data set, from a forest products ®rm in Georgia (USA), represents a typical industrial ownership in the southeastern United States. The data set contains mostly pine plantations, as well as a mixture of hardwood areas, which for this analysis are not managed for timber production. It contains 1300 logging units, resulting in 10 000 0±1 decision variables. Yields were estimated with equations developed by
Harrison and Borders (1996) for three types of wood products: sawlogs, chip-and-saw logs, and pulpwood. The product prices that are assumed are shown in Table 1. We assumed a logging cost of US$ 7.41 per m3, and an 8% real discount rate. All penalty values were discounted by using an 8% real discount rate. 3. Results Production possibility curves for the 3- and 2-year green-up time when used in conjunction with the various maximum clearcut sizes are illustrated in Fig. 2. When applied to the 21 600 ha forest from the southeastern United States, the heuristic technique shows that reducing the green-up time from 3 to 2 years had the largest improvement in the objective function value when used in conjunction with the smallest maximum opening size, or an improvement of about US$ 66 600 compared to US$ 45 600 for the largest opening size. These improvements range from approximately US$ 10 per harvested ha for the 60-ha maximum clearcut size to US$ 6.70 per harvested ha for the 90-ha maximum clearcut size. Again, in all these analyses, the average opening size was constrained to the SFI guidelines of 48 ha. As expected, the larger maximum opening size allows the harvested volume to be closer to volume
Fig. 2. Production possibilities over a range of maximum clearcut sizes.
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Fig. 3. Chip-and-saw volume produced in the first 10 years with a 3-year green-up time requirement.
goals generated from our 30-year strategic plan (the relaxed LP problem formulation). Each maximum opening size (60, 70, 80, 90 ha) produces similar results for timber volume in the ®rst years of each
10-year tactical plan, but the larger openings allow for more volume to be harvested in the later periods. Figs. 3 and 4 show that the chip-and-saw volume is >90% of the target volumes in the ®rst few years of the
Fig. 4. Chip-and-saw volume produced in the first 10 years with a 2-year green-up time requirement.
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K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
Fig. 5. Sawlog volume produced in the first 10 years with a 3-year green-up time requirement.
strategic plan. The deviations from the sawlog volume targets are larger (Figs. 5 and 6), and often the tactical plan is able to only schedule about 75% of the target volume. The deviations are also consistently larger for
the 3-year green-up constraint than for the 2-year constraint. The opposite pattern is observed for pulpwood volume, where the tactical plan meets or exceeds the goals of the strategic plan in all time
Fig. 6. Sawlog volume produced in the first 10 years with a 2-year green-up time requirement.
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Fig. 7. Pulpwood volume produced in the first 10 years with a 3-year green-up time requirement.
periods for both the 3- and 2-year green-up constraints (Figs. 7 and 8). The results from the sensitivity analysis showed that as prices or volumes are decreased the impact of the
green-up constraint becomes more signi®cant (Figs. 9 and 10). As prices are reduced, the magnitude of the difference between the 3- and 2-year green-up constraint increases to US$ 106 per harvested ha for a
Fig. 8. Pulpwood volume produced in the first 10 years with a 2-year green-up time requirement.
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Fig. 9. Production possibilities when prices are modified.
Fig. 10. Production possibilities when volumes are modified.
60-ha maximum clearcut size and US$ 92 per harvested ha for 90 ha maximum clearcut size. A similar pattern is shown for a reduction in volume yield with US$ 157 per harvested ha for 60-ha maximum clearcut size. It is slightly reduced to US$ 153 per harvested ha for a 90 ha maximum clearcut size. When volume or price was increased there was no signi®cant difference between the 3- and 2-year green-up constraints.
4. Discussion and conclusions The goal of this analysis was to determine whether variations in green-up parameters matter ®nancially in forest planning efforts. Clearly, for relatively shortrotation pine plantations in the southeastern United States, shortening the green-up time assumption from 3 to 2 years does make a small difference in the PNW
K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
of the resulting solutions. While other researchers (Daust and Nelson, 1993; Yoshimoto and Brodie, 1994; Barrett et al., 1998) have also shown that the length of the green-up constraint is ®nancially important, they do so at much longer temporal scales than are required by the AF&PA SFI. To reduce the length of the AF&PA green-up constraint will require the adoption of intensive forest management techniques (chemical site preparation, herbaceous weed control, etc.). From this data set, the increase in PNW represents only 5±10% of the cost of the average herbicide treatment in the southern United States (Dubois et al., 1997). The use of intensive forest management techniques cannot be justi®ed solely based on the economic gain that results from a reduction in the green-up constraint. However, many ®rms in the SE region are currently using some intensive management techniques and may require only a marginal increase in expense to achieve the growth rate that would allow a reduction of the green-up constraint from 3 to 2 years. Others may have to adopt a full suite of intensive practices. We recommend that both forest and stand level analysis should be completed when evaluating the use of intensive forest management practices. We have also shown, at least for one forested area and ownership in the southeastern United States, that the maximum opening size assumed does matter. Variations in the distribution of forest conditions and the spatial arrangement of the age classes of other forests may result in different conclusions, but larger maximum opening sizes should allow more ¯exibility in the spatial arrangement and timing of harvests. If forest management organizations are in a position to develop ¯exible forest policies, such as the maximum allowable clearcut size, they should examine the costs and bene®ts of various green-up times as well as varying the maximum size of forest openings. The impact of the green-up constraint on the volume harvest was twofold. First, the products that are derived from longer rotations, such as sawlogs and chip-and-saw logs, did not meet the target goals that we developed in our 30-year strategic plan. Second, a surplus of pulpwood volume would be produced. Because the area of older forest stands is limited at the beginning of the planning horizon, the adjacency constraint prevents harvesting them at the time indicated by the strategic plan. Thus, stands younger than
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desirable will be harvested in order to meet the overall volume goals of the forest plan, yet they will not be old enough to produce the desired mix of sawlog or chipand-saw log volume. Sensitivity analysis shows that an increase in price or yield will further reduce the economic incentive for using intensive management to reduce green-up time requirements. However, as prices decrease or the yields do not meet their expected value, the incentive to use intensive forest management practices to reduce the greenup constraint can be justi®ed, as the PNW differences for this analysis come closer to the cost of herbaceous weed control. Organizations that attempt to match forest harvest scheduling to the needs of wood processing facilities may need to incorporate the required green-up constraints into their planning process if they are to accurately anticipate the amount and types of products that will be available. With regard to the heuristic technique we used to generate solutions to these forest planning problems, we feel that the hybrid tabu search-genetic algorithm method can produce better results than a traditional tabu search technique, because it allows for both an intensi®cation and a diversi®cation in the search process, and because it uses a genetic algorithm to combine local optimal solutions. We recommend that others developing tabu search applications should consider combining multiple search strategies into heuristic harvest scheduling models and move beyond the basic approaches heuristics offer. References Bacon, C.G., Zedaker, S.M., 1987. Third-year growth responses of loblolly pine to eight levels of competition control. South. J. Appl. For. 11, 91±95. Barrett, T., Gilless, J.K., Davis, L.S., 1998. Economic and fragmentation effects of clearcut restrictions. For. Sci. 44, 569±577. Bettinger, P., Boston, K., Sessions, J., 1999. Intensifying a heuristic forest harvest scheduling procedure with paired attribute decision choices. Can. J. For. Res. 29, 1784±1792. Bettinger, P., Sessions, J., Boston, K., 1997. Using tabu search to schedule timber harvests subject to spatial wildlife goals for big game. Ecol. Model. 42, 111±123. Boston, K., Bettinger, P., 1999. An analysis of Monte-Carlo integer programming, simulated annealing, and tabu search heuristics for solving spatial harvest scheduling problems. For. Sci. 45, 292±301.
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Daust, D.K., Nelson, J.D., 1993. Spatial reduction factors for stratabased harvest schedules. For. Sci. 39, 152±165. Dubois, M.R., McNabb, K., Straka, T.J., 1997. Costs and cost trends for forestry practices in the South. For. Farmer 32, 7±13. Glover, F., Kelly, J.P., Laguana, M., 1995. Genetic algorithm and tabu search: hybrids for optimization. Comput. Oper. Res. 22, 111±134. Harrison, W.M., Borders, B.E., 1996. Yield prediction and growth projection for site-prepared loblolly pine plantations in the Carolinas, Georgia, Alabama, and Florida. PMRC Technical Report 1996. Plantation Management Cooperative, D.B. Warnell School of Forest Resources, University of Georgia, Athens, GA.
Knowe, S.A., Nelson, L.R., Gjerstad, D.H., Zutter, B.R., Glover, G.R., Minogue, P.J., Dukes Jr., J.H., 1985. Four-year growth and development of planted loblolly pine on sites with competition control. South. J. Appl. For. 9, 11±15. Murray, A.T., Church, R.L., 1995. Heuristic solution approaches to operational forest planning problems. OR Spektrum 17, 193±203. Van Duesen, P.C., 1999. Multiple solution harvest scheduling. Silva Fenn. 33 (3), 1±9. Yoshimoto, A., Brodie, J.D., 1994. Short- and long-term impacts of spatial restrictions on harvest scheduling with reference to riparian zone planning. Can. J. For. Res. 24, 1617±1628.
The economic impact of green-up constraints in the southeastern United States Kevin Bostona,*, Pete Bettingerb a
Warnell School of Forest Resources, University of Georgia, Athens, GA 30602, USA Department of Forest Resources, Oregon State University, Corvallis, OR 97331, USA
b
Received 27 December 1999; accepted 21 March 2000
Abstract Green-up, or adjacency, requirements are a common constraint in forestry. The American Forest and Paper Association has developed a Sustainable Forestry Initiative that includes a green-up constraint which limits the average clearcut opening to 48 ha for 3 years or until the average height of the regenerated trees is >1.4 m. In addition to constraining the average clearcut size, many forestry companies in the southeastern USA voluntarily limit their maximum clearcut size to between 60 and 90 ha. In this research, a heuristic algorithm was used to develop tactical forest plans that consider both the maximum and average clearcut sizes. Economic effects of the green-up constraints were estimated for situations where intensive management can reduce the length of the green-up time from 3 to 2 years on a 21 600 ha ownership in Georgia (USA). For a 60-ha maximum opening size, this reduction in green-up time from 3 to 2 years resulted in an additional US$ 66 600 in present net worth (PNW) over a 10-year analysis period. This corresponds to a US$ 10 per harvested ha, or a 0.8% increase in PNW. The bene®t gained by reducing the length of the green-up period is less with a 90-ha maximum clearcut size, where PNW increases by US$ 45 600, or US$ 6.70 per harvested ha, a 0.5% increase. While the total volume per period was near the volume goal produced by a strategic forest plan, the spatial restrictions and the desire to maximize net present value resulted in lower volume of timber products (sawlogs and chip-and-saw logs) from older forest stands. A sensitivity analysis showed that an increase in price or yield further reduced the economic incentive for the reduction of the length of the green-up constraint. As price or volume decreased below expectations, however, the incentive to use intensive forest management practices to reduce the length of the green-up constraint became more attractive, since the differences between a 2-year and 3-year green-up time requirement may be large enough to pay for more intensive management practices. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Green-up constraints; Economic effects; Forest management
1. Introduction Increasing competition from international pine-producing regions is forcing forest industry organizations in the southeastern United States to adopt intensive *
Corresponding author. E-mail addresses: [email protected] (K. Boston), [email protected] (P. Bettinger).
forestry practices, including extensive site preparation, weed control, and fertilization. It is well accepted that these practices can accelerate forest growth rates (Knowe et al., 1985; Bacon and Zedaker, 1987), but they also add to the operational costs of growing trees. Forestry ®rms are under increasing pressure to produce higher returns for their shareholders and may only adopt practices that improve their overall pro®tability. Activities are often selected on their ability to
0378-1127/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 0 0 ) 0 0 4 1 7 - 5
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improve pro®ts, not just improve growth rates and reach sustainability. In addition to increased competition from international producers, most forest products companies in the southeastern United States have adopted the American Forest and Paper Association (AF&PA) Sustainable Forestry Initiative (SFI). The initiative is a voluntary agreement among AF&PA members to promote sustainable forestry practices. One component of the SFI guidelines is a green-up constraint that restricts the average clearcut size to <48 ha for either 3 years or until the average height of the regenerated trees is >1.4 m. Additionally, many companies have developed an internal policy to voluntarily limit their maximum clearcut size to 60±90 ha. Estimating the effects of the combination of these land management constraints poses a challenging combinatorial problem. The ®nancial impact of various green-up constraints has generally been estimated by using relatively long periods of time for adjacent stands to `green-up'. For example, in Canada, three different green-up periods (10, 20, and 30 years), three different maximum opening sizes (20, 40, and 80 ha), and three different rotation lengths (80, 100, and 120 years) were used to compare the cost of various harvesting restrictions (Daust and Nelson, 1993). The smallest maximum size and the longest greenup time resulted in a 29% reduction in harvest volume compared with an optimal solution from a linear programming effort, which did not contain the spatial constraints. In Oregon (USA), increasing the greenup period from 20 to 30 years showed a 34±40% reduction in the present net worth (PNW) (Yoshimoto and Brodie, 1994). In California (USA), increasing the green-up period from 10 to 20 years reduced PNW by 13% for a 4-ha maximum opening size. As the minimum patch size increased to 32 ha, there was smaller difference for increased exclusion period (Barrett et al., 1998). A variety of heuristic techniques can be used to solve problems that consider management options with green-up size constraints, or spatially constrained harvest scheduling problems. Van Duesen (1999) was able to solve numerically-large spatially constrained problems by using a simulated annealing algorithm, but did not compare the quality of his solutions. Two problems based on a smaller spatial
scale showed no signi®cant difference in the objective function values for a random starting location hill-climbing algorithm, Monte-Carlo integer programming, simulated annealing, and tabu search (Murray and Church, 1995). Four problems of larger spatial scales were used to compare Monte-Carlo integer programming, simulated annealing, and tabu search. Simulated annealing found the best solution to three of the four problems while tabu search found the best solution to the fourth (Boston and Bettinger, 1999). Most of the tabu search algorithms used to solve harvest scheduling problems have a single search neighborhood. Bettinger et al. (1999) describe a technique that expands this by allowing two search neighborhoods to change their status simultaneously. This technique produces signi®cant improvements in the objective function value, but at the cost of longer processing times. Tabu search also tends to ®nd solutions that are concentrated in one portion of the solution space (Glover et al., 1995). The development of a technique that can combine several local optimal solutions into a single new solution may yield better results (Glover et al., 1995). This paper begins by describing a heuristic technique that combines tabu search and genetic algorithm techniques. Then, the heuristic technique is used to develop several tactical forest management plans and, subsequently, estimate the ®nancial effects of reducing the green-up time and maximum clearcut size constraint. The economic effects are estimated for situations where the green-up time is reduced from 3 to 2 years, and where the maximum clearcut size ranges between 60, 70, 80, and 90 ha. The change in revenue, if positive, could provide additional ®nancial incentive for companies to adopt intensive forest management practices. The goal is to determine whether variations in green-up parameters matter ®nancially in forest planning. Obviously, this can, and has been, shown to be true for long green-up times and small maximum clearcut sizes. However, it has not been shown for shorter differences in green-up times and larger maximum clearcut sizes. If there is an economic bene®t for reducing the length of the green-up time in areas where tree species can be more intensively managed, this will provide further incentive for using intensive management techniques.
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2. Methods We ®rst develop the problem formulation for evaluating green-up constraint parameters, then present the heuristic technique used to develop the tactical forest plans. Then, a database from a southeastern United States (Georgia) forest industry organization, representing 21 600 ha, is used to evaluate the economic effects of variations in green-up requirements. 2.1. Model formulation Two models were used to solve this problem, one for a strategic 30-year plan, and one for a tactical 10year plan. The strategic plan provides the target harvest goals for the tactical plan, and it can be considered a `relaxed' problem, since the spatial constraints for green-up are absent. The results from this plan provide a theoretical upper-bound on the solution. The objective function consists of maximizing the net present value of revenue from harvests less logging costs: maximize
J X N X T X
Revnt ÿ Lcnt Vjnt Xnt
(1)
j1 n1 t1
where J is the number of products, j the product type, N the number of harvest units, n the harvest unit; T the number of time periods, t the time period; Revnt the revenue per cubic meter for unit n harvested in time period t; Lcnt the logging cost per cubic meter for unit n harvested in time period t, Vjnt the volume per hectare of product j in unit n harvested during time period t, Xnt the continuous variable indicating whether unit n is harvested during time period t. A minimum harvest age of 19 years was assumed for each timber stand, and individual product volumes could not change by >5% per time period, except sawlog volumes, which were set to 10% per period. PN PN n1 Vjnt Xnt ÿ n1 Vjnt1 Xnt1 1 ÿ PN n1 Vjnt Xnt deviation j 8j; t 1; . . . ; t ÿ 1
(2)
The formulation for the tactical planning problem is similar to the strategic plan, and to the objective
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function found in Boston and Bettinger (1999). The objective function for the problem is: maximize
J X N X T X
ÿ
Revnt ÿ Lcnt Vjnt Xnt
j1 n1 t1 J X T X j1 t1
Vpjt dujt ÿ
J X T X Vpjt dljt
(3)
j1 t1
where Xnt0,1 is a variable indicating whether unit n is harvested during time period t; Vpjt the volume penalty per cubic meter of product j during time period t, dujt the positive deviation from volume goal of product j during time period t, and dljt the negative deviation from volume goal of product j during time period t. The volume goals we used were those that resulted from the strategic plan. Penalties were used to force the heuristic search process towards solutions which would emulate the strategic planning solution. A three-part step function was used to penalize deviations from the volume goal for each product. For deviations <10% from a volume goal, the penalty value is 50% of the product price (Table 1). For deviations between 10 and 25% from a volume goal, the penalty value is equal to the price. For deviations >25% from the volume goal, the penalty value is 150% of the price. Three main constraints were considered. The ®rst is a volume constraint, which is an accounting constraint used to sum the volume harvested in each period and determine the deviations from the goal for each product: N X
Vjnt Xnt ÿ dunt dlnt volume goal 8j; t
(4)
i1
The second constraint is a singularity constraint, which limits each unit to one treatment during the planning horizon: T X
Xnt 1 8n
(5)
t1
Table 1 Prices assumed for forest products Products
Price ($/m3)
Sawlogs Chip-and-saw logs Pulpwood
46.24 32.12 12.00
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The third constraint defines a maximum opening size for each logging unit n and its set of adjacent neighbors (Un). For a 3-year green-up constraints, we define a set (Tm) of near-time periods (mz, where m1tÿ3, m2tÿ2, m3tÿ1, m4t, m5t1, m6t2, and m7t3; all mz0, otherwise not in Tm; all mzT, otherwise not in Tm). We define a similar set for a 2year green-up constraint (mz, where m1tÿ2, m2tÿ1, m3t, m4t1, m5t2; all mz0, otherwise not in Tm; all mzT, otherwise not in Tm). Therefore, an opening is not just the harvests that occur in time period t, but also includes those around each unit n that have occurred during the near-time periods. The maximum opening size constraint is: " # Un X Tm X Ak Xkm An Xnt k1 m1
Maximum opening size 8n where Xnt 1; t (6) where Un is the set of adjacent neighbors to unit n, k the adjacent neighbors to unit n, Tm the set of neartime periods, m a near-time period, Ak the area of unit k; Xkm is 0,1, a variable indicating whether unit k is harvested during near-time period m; An the area of unit n; and Xnt is 0,1, a variable indicating whether unit n is harvested during time period t. A maximum average opening size can be de®ned for each time period. If each opening is centered around a focal unit (f) during a time period (t), we can de®ne the size of the opening (Oft) as: " # Un X Tm X Ak Xkm Af Xft Oft (7) k1 m1
where Un is the set of adjacent neighbors to focal unit f, k the adjacent neighbors to focal unit f, Tm the set of near-time periods, m a near-time period, Ak the area of unit k, Xkm the 0,1 variable indicating whether unit k is harvested during near-time period m, Af the area of focal unit f, and Xft is 0,1, a variable indicating whether focal unit f is harvested during time period t. As we are attempting to calculate the average opening size, we do not wish to count openings more than once. This mis-counting could occur if we allow each unit (n) in an opening (composed of multiple units n) to be considered the `center'. Therefore, only one unit (n) can be delineated as the focal center (f) of the
opening in any time period, and the total number of openings equals the number of focal centers of openings. Thus, X X X X f n and Xft Xnt (8) The average opening size for a set of openings in a time period (t) can then be constrained with the following equation: hP i F f 1 Oft average opening size 8t (9) F where F is the total number of openings, f the focal center of an opening, or an opening itself, and Oft an opening centered around a focal unit f during time period t. Unlike our strategic plan, the tactical forest plans include the spatial constraints noted in equations (6)± (9). For Eq. (6), we examined maximum opening size constraints of 60, 70, 80, and 90 ha. For Eq. (9), we used the AF&PA SFI maximum average clearcut size of 48 ha. The strategic plan was developed using linear programming techniques, and the tactical forest plans were developed using a heuristic technique. 2.2. Sensitivity analysis Two types of sensitivity analysis were completed when developing the tactical forest plans, one for price and one for volume. Prices for sawlogs and chip-andsaw logs were increased by 5%, then decreased by 5%. A similar approach was used to test the impact of volume sensitivity where a uniformly distributed random number generator was used to assign an adjustment factor for each unit to ®rst increase, then decrease the volume harvested in a unit between 0 and 5%. In each of these analyses, we examined maximum opening size constraints of 60, 70, 80, and 90 ha, and used the AF&PA SFI maximum average clearcut size of 48 ha. 2.3. Heuristic technique The heuristic technique is a hybrid algorithm consisting of ®ve components (Fig. 1). The ®rst is a Monte-Carlo integer programming algorithm that randomly develops an initial solution. This process selects a logging unit, determines if it meets the
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Fig. 1. Flow chart of the heuristic technique used to develop tactical forest plans.
minimum harvest age, and assures that its incorporation into the solution does not violate the green-up constraints. This continues until 10% of the sawlog volume has been achieved in all periods. Because each new run of the heuristic technique uses a new seed for a random number generator, this component will allow an increase in the proportion of the solution space explored when the program is executed repeatedly.
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The second component is the core tabu search routine, similar to the algorithms described in Murray and Church (1995), Bettinger et al. (1997), and Boston and Bettinger (1999), and is composed of two elements: (1) a tabu list that maintains a record of the recent moves; and (2) the aspiration criteria. After experimentation, 100 iterations were selected as the tabu list length. For this application, the aspiration criterion was assumed to be the overall best objective function value. The best move from the neighborhood of moves will be considered ®rst, whether or not it improves the current solution. If the move is not tabu, it will be accepted into the solution. If the move is tabu, yet exceeds the aspiration criteria, it too will be accepted into the solution. If a move is tabu, yet does not exceed the aspiration criteria, it is not accepted. The third component is the intensi®cation routine. The objective of an intensi®cation routine is to explore a portion of the neighborhood that has already yielded a good solution to search for better solutions. This intensi®cation routine begins by recalling the current best solution from the core tabu search routine. By using a 2-opt neighborhood search routine described in Bettinger et al. (1999), two units can simultaneously change their status. the intensi®cation routine has the same short-term memory features as the core tabu search routine, but the tabu list has been reduced from 100 to 20 iterations. Tabu search ®nds good solutions to large combinatorial problems, but they tend to be concentrated in a small portion of the solution space (Glover et al., 1995). Thus, the fourth and ®fth components of the heuristic technique have the goal of forcing the search to new parts of the solution space. The fourth component (Fig. 1) is a diversi®cation routine that schedules those units with the lowest frequency of entering the solution, while maintaining the minimum harvest age and not violating the green-up constraints. This diversi®cation will force the algorithm to the least explored portion of the solution space. The resulting solution becomes the starting point for the core tabu search routine. The ®fth component has the goal of combining two neighborhoods where good solutions were found with the hope of ®nding a superior solution. It is based on a crossover routine used in genetic algorithms. The genetic crossover routine treats the solution to the forest planning problems as if they were chromosomes, with each unit being a gene on a chromosome.
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The values for the genes, the alleles, become the periods when the unit is harvested. By using a random number generator that selects the crossover point, the two `chromosomes' are recombined into two new solutions. The solution with the highest objective value survives the crossover and becomes the starting solution for a continuation of the core tabu search routine. This heuristic technique has been shown to produce good results to similar scheduling problems which have 3000±5000 0±1 integer variables (companion paper regarding validation of the heuristic technique is in preparation), although the heuristic technique described here uses 2-opt moves. Extending the use of this solution method to larger scheduling problems therefore seems reasonable, although the performance of the solution method cannot be directly compared against known optimal solutions. 2.4. Data description The data set, from a forest products ®rm in Georgia (USA), represents a typical industrial ownership in the southeastern United States. The data set contains mostly pine plantations, as well as a mixture of hardwood areas, which for this analysis are not managed for timber production. It contains 1300 logging units, resulting in 10 000 0±1 decision variables. Yields were estimated with equations developed by
Harrison and Borders (1996) for three types of wood products: sawlogs, chip-and-saw logs, and pulpwood. The product prices that are assumed are shown in Table 1. We assumed a logging cost of US$ 7.41 per m3, and an 8% real discount rate. All penalty values were discounted by using an 8% real discount rate. 3. Results Production possibility curves for the 3- and 2-year green-up time when used in conjunction with the various maximum clearcut sizes are illustrated in Fig. 2. When applied to the 21 600 ha forest from the southeastern United States, the heuristic technique shows that reducing the green-up time from 3 to 2 years had the largest improvement in the objective function value when used in conjunction with the smallest maximum opening size, or an improvement of about US$ 66 600 compared to US$ 45 600 for the largest opening size. These improvements range from approximately US$ 10 per harvested ha for the 60-ha maximum clearcut size to US$ 6.70 per harvested ha for the 90-ha maximum clearcut size. Again, in all these analyses, the average opening size was constrained to the SFI guidelines of 48 ha. As expected, the larger maximum opening size allows the harvested volume to be closer to volume
Fig. 2. Production possibilities over a range of maximum clearcut sizes.
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Fig. 3. Chip-and-saw volume produced in the first 10 years with a 3-year green-up time requirement.
goals generated from our 30-year strategic plan (the relaxed LP problem formulation). Each maximum opening size (60, 70, 80, 90 ha) produces similar results for timber volume in the ®rst years of each
10-year tactical plan, but the larger openings allow for more volume to be harvested in the later periods. Figs. 3 and 4 show that the chip-and-saw volume is >90% of the target volumes in the ®rst few years of the
Fig. 4. Chip-and-saw volume produced in the first 10 years with a 2-year green-up time requirement.
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Fig. 5. Sawlog volume produced in the first 10 years with a 3-year green-up time requirement.
strategic plan. The deviations from the sawlog volume targets are larger (Figs. 5 and 6), and often the tactical plan is able to only schedule about 75% of the target volume. The deviations are also consistently larger for
the 3-year green-up constraint than for the 2-year constraint. The opposite pattern is observed for pulpwood volume, where the tactical plan meets or exceeds the goals of the strategic plan in all time
Fig. 6. Sawlog volume produced in the first 10 years with a 2-year green-up time requirement.
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Fig. 7. Pulpwood volume produced in the first 10 years with a 3-year green-up time requirement.
periods for both the 3- and 2-year green-up constraints (Figs. 7 and 8). The results from the sensitivity analysis showed that as prices or volumes are decreased the impact of the
green-up constraint becomes more signi®cant (Figs. 9 and 10). As prices are reduced, the magnitude of the difference between the 3- and 2-year green-up constraint increases to US$ 106 per harvested ha for a
Fig. 8. Pulpwood volume produced in the first 10 years with a 2-year green-up time requirement.
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Fig. 9. Production possibilities when prices are modified.
Fig. 10. Production possibilities when volumes are modified.
60-ha maximum clearcut size and US$ 92 per harvested ha for 90 ha maximum clearcut size. A similar pattern is shown for a reduction in volume yield with US$ 157 per harvested ha for 60-ha maximum clearcut size. It is slightly reduced to US$ 153 per harvested ha for a 90 ha maximum clearcut size. When volume or price was increased there was no signi®cant difference between the 3- and 2-year green-up constraints.
4. Discussion and conclusions The goal of this analysis was to determine whether variations in green-up parameters matter ®nancially in forest planning efforts. Clearly, for relatively shortrotation pine plantations in the southeastern United States, shortening the green-up time assumption from 3 to 2 years does make a small difference in the PNW
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of the resulting solutions. While other researchers (Daust and Nelson, 1993; Yoshimoto and Brodie, 1994; Barrett et al., 1998) have also shown that the length of the green-up constraint is ®nancially important, they do so at much longer temporal scales than are required by the AF&PA SFI. To reduce the length of the AF&PA green-up constraint will require the adoption of intensive forest management techniques (chemical site preparation, herbaceous weed control, etc.). From this data set, the increase in PNW represents only 5±10% of the cost of the average herbicide treatment in the southern United States (Dubois et al., 1997). The use of intensive forest management techniques cannot be justi®ed solely based on the economic gain that results from a reduction in the green-up constraint. However, many ®rms in the SE region are currently using some intensive management techniques and may require only a marginal increase in expense to achieve the growth rate that would allow a reduction of the green-up constraint from 3 to 2 years. Others may have to adopt a full suite of intensive practices. We recommend that both forest and stand level analysis should be completed when evaluating the use of intensive forest management practices. We have also shown, at least for one forested area and ownership in the southeastern United States, that the maximum opening size assumed does matter. Variations in the distribution of forest conditions and the spatial arrangement of the age classes of other forests may result in different conclusions, but larger maximum opening sizes should allow more ¯exibility in the spatial arrangement and timing of harvests. If forest management organizations are in a position to develop ¯exible forest policies, such as the maximum allowable clearcut size, they should examine the costs and bene®ts of various green-up times as well as varying the maximum size of forest openings. The impact of the green-up constraint on the volume harvest was twofold. First, the products that are derived from longer rotations, such as sawlogs and chip-and-saw logs, did not meet the target goals that we developed in our 30-year strategic plan. Second, a surplus of pulpwood volume would be produced. Because the area of older forest stands is limited at the beginning of the planning horizon, the adjacency constraint prevents harvesting them at the time indicated by the strategic plan. Thus, stands younger than
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desirable will be harvested in order to meet the overall volume goals of the forest plan, yet they will not be old enough to produce the desired mix of sawlog or chipand-saw log volume. Sensitivity analysis shows that an increase in price or yield will further reduce the economic incentive for using intensive management to reduce green-up time requirements. However, as prices decrease or the yields do not meet their expected value, the incentive to use intensive forest management practices to reduce the greenup constraint can be justi®ed, as the PNW differences for this analysis come closer to the cost of herbaceous weed control. Organizations that attempt to match forest harvest scheduling to the needs of wood processing facilities may need to incorporate the required green-up constraints into their planning process if they are to accurately anticipate the amount and types of products that will be available. With regard to the heuristic technique we used to generate solutions to these forest planning problems, we feel that the hybrid tabu search-genetic algorithm method can produce better results than a traditional tabu search technique, because it allows for both an intensi®cation and a diversi®cation in the search process, and because it uses a genetic algorithm to combine local optimal solutions. We recommend that others developing tabu search applications should consider combining multiple search strategies into heuristic harvest scheduling models and move beyond the basic approaches heuristics offer. References Bacon, C.G., Zedaker, S.M., 1987. Third-year growth responses of loblolly pine to eight levels of competition control. South. J. Appl. For. 11, 91±95. Barrett, T., Gilless, J.K., Davis, L.S., 1998. Economic and fragmentation effects of clearcut restrictions. For. Sci. 44, 569±577. Bettinger, P., Boston, K., Sessions, J., 1999. Intensifying a heuristic forest harvest scheduling procedure with paired attribute decision choices. Can. J. For. Res. 29, 1784±1792. Bettinger, P., Sessions, J., Boston, K., 1997. Using tabu search to schedule timber harvests subject to spatial wildlife goals for big game. Ecol. Model. 42, 111±123. Boston, K., Bettinger, P., 1999. An analysis of Monte-Carlo integer programming, simulated annealing, and tabu search heuristics for solving spatial harvest scheduling problems. For. Sci. 45, 292±301.
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Daust, D.K., Nelson, J.D., 1993. Spatial reduction factors for stratabased harvest schedules. For. Sci. 39, 152±165. Dubois, M.R., McNabb, K., Straka, T.J., 1997. Costs and cost trends for forestry practices in the South. For. Farmer 32, 7±13. Glover, F., Kelly, J.P., Laguana, M., 1995. Genetic algorithm and tabu search: hybrids for optimization. Comput. Oper. Res. 22, 111±134. Harrison, W.M., Borders, B.E., 1996. Yield prediction and growth projection for site-prepared loblolly pine plantations in the Carolinas, Georgia, Alabama, and Florida. PMRC Technical Report 1996. Plantation Management Cooperative, D.B. Warnell School of Forest Resources, University of Georgia, Athens, GA.
Knowe, S.A., Nelson, L.R., Gjerstad, D.H., Zutter, B.R., Glover, G.R., Minogue, P.J., Dukes Jr., J.H., 1985. Four-year growth and development of planted loblolly pine on sites with competition control. South. J. Appl. For. 9, 11±15. Murray, A.T., Church, R.L., 1995. Heuristic solution approaches to operational forest planning problems. OR Spektrum 17, 193±203. Van Duesen, P.C., 1999. Multiple solution harvest scheduling. Silva Fenn. 33 (3), 1±9. Yoshimoto, A., Brodie, J.D., 1994. Short- and long-term impacts of spatial restrictions on harvest scheduling with reference to riparian zone planning. Can. J. For. Res. 24, 1617±1628.