Optimizing continuous cover management of boreal forest when timber prices and tree growth are stochastic
 Timo Pukkala^{1}Email author
Received: 3 November 2014
Accepted: 28 January 2015
Published: 14 March 2015
Abstract
Background
Decisions on forest management are made under risk and uncertainty because the stand development cannot be predicted exactly and future timber prices are unknown. Deterministic calculations may lead to biased advice on optimal forest management. The study optimized continuous cover management of boreal forest in a situation where tree growth, regeneration, and timber prices include uncertainty.
Methods
Both anticipatory and adaptive optimization approaches were used. The adaptive approach optimized the reservation price function instead of fixed cutting years. The future prices of different timber assortments were described by crosscorrelated autoregressive models. The high variation around ingrowth model was simulated using a model that describes the cross and autocorrelations of the regeneration results of different species and years. Tree growth was predicted with individual tree models, the predictions of which were adjusted on the basis of a climateinduced growth trend, which was stochastic. Residuals of the deterministic diameter growth model were also simulated. They consisted of random tree factors and cross and autocorrelated temporal terms.
Results
Of the analyzed factors, timber price caused most uncertainty in the calculation of the net present value of a certain management schedule. Ingrowth and climate trend were less significant sources of risk and uncertainty than tree growth. Stochastic anticipatory optimization led to more diverse postcutting stand structures than obtained in deterministic optimization. Cutting interval was shorter when risk and uncertainty were included in the analyses.
Conclusions
Adaptive optimization and management led to 6%–14% higher net present values than obtained in management that was based on anticipatory optimization. Increasing risk aversion of the forest landowner led to earlier cuttings in a mature stand. The effect of risk attitude on optimization results was small.
Keywords
Adaptive optimization Anticipatory optimization Stochastic optimization Risk preferences Risk Uncertainty Reservation priceBackground
Maximizing the economic benefits from timber production is equal to maximizing the net present value of future net incomes. Unfortunately, the future net incomes are unknown at the moment when management decision should be made. Future net incomes depend on future timber prices, which show substantial temporal variation (Leskinen and Kangas 1998).
Also the growth and development of trees and stands are poorly known. Deterministic models explain only a part of the growth variation between years, stands and trees. Measurements of past growth show that there are periods of good growth while in other years or during longer periods trees grow less than the longterm average (e.g. Pasanen 1998). In addition to these weatherrelated seasonal variations in annual growth, there are also betweentree growth differences which cannot be explained by deterministic models. Another factor causing uncertainty in growth prediction is climate change. It is usually assumed that the growth rate will increase in the boreal forests of North Europe (e.g. Pukkala and Kellomäki 2012), but the estimated growth trends represent very uncertain knowledge.
Flowering, pollination, seed production and germination are subprocesses of the regeneration process of trees and stands. All these subprocesses are very sensitive to weather conditions such as temperature and rainfall. In addition, the eventual size of the seed crop depends on the fluctuations of seed predators and seed diseases. Since many subprocesses critical to regeneration success depend on weather conditions, it is impossible to predict the exact amount of regeneration in a certain year in the future, even when there are plenty of empirical regeneration data to fit models. The best that can be done is to predict the distribution of regeneration results or the probability of successful regeneration. Mortality is also hard to predict exactly. However, the socalled regular mortality (competitionrelated mortality) is very low in regularly thinned managed boreal forest. Therefore, if catastrophic events are excluded from the analysis (like in this study) uncertainty in mortality does not add much to the total degree of uncertainty in the prediction of stand development. For an attempt to include catastrophic events see Zhou and Buongiorno (2006).
The above discussion shows that decisions on future forest management must be made under risk and uncertainty. Risk is usually understood to be a situation in which the probabilities of different states of nature are known, which makes it possible to calculate the distribution of outcomes for a certain decision alternative. Uncertainty refers to situations in which the probabilities are unknown. The prevailing situation is uncertainty. However, to make analyses easier, the situation is transformed from uncertainty to risk, by assuming some distributions for the uncertain factors. This allows the analyst to calculate the probabilities of different outcomes of decision alternatives.
Forest landowners have different attitudes toward risk and uncertainty. Most people are risk avoiders, especially in “big” decisions with a major potential impact on their livelihood. A riskaverse person seeks decision alternatives, which are at least reasonable when the states of nature develop in an unfavorable way. Risk avoiders tend to select decision alternatives for which the lower end of the distribution of outcomes is as good as possible (Pukkala and Kangas 1996). They may also minimize the “regret”, i.e. the maximum loss compared to the best decision alternative under certain states on nature. On the contrary, risk takers are optimistic and favor decision alternatives that are good under favorable states of nature, even though the probability of such an outcome may be low.
There are two basic approaches to the optimization of stand management in a risk situation: anticipatory and adaptive optimization. Anticipatory optimization seeks a single management schedule, which produces the most favorable distribution of net present values or some other objective function (Valsta 1992). Risk neutral decision makers select management schedules which produce high average net present values. Risk takers often select management schedules for which the best outcomes are good whereas risk avoiders tend to maximize the worst outcomes of alternative management schedules.
Adaptive optimization does not try to find a single management prescription for the stand. Instead, it aims at finding rules that help the landowner to make right decisions in changing environment (Lohmander 2007). A wellknown rule is the reservation price function indicating the minimum price that the seller should obtain from timber (Brazee and Mendelsohn 1988; Lohmander 1995; Gong and Yin 2004). A more general approach is the Markov decision process model (Lembersky and Johnson 1975; Kaya and Buongiorno 1987).
It can be assumed that reservation price decreases with increasing financial maturity of the stand: the lower the relative value increment of the stand, the lower is the minimum selling price of a certain timber assortment. Since the relative value increment decreases with increasing stand density and mean tree size, it can be assumed that reservation price is negatively correlated with stand basal area and mean tree diameter (Lohmander 1995; Gong 1998; Lu and Gong 2003).
The aim of this study was to describe a system for stochastic optimization of the management of boreal forests in a situation where future timber prices, tree growth and regeneration are not known exactly. The developed simulation–optimization system was used to compare deterministic and stochastic optima, as well as the results of anticipatory and adaptive optimization approaches. Pukkala and Kellomäki (2012) compared anticipatory and adaptive management in evenaged forestry and Zhou et al. (2008) compared adaptive and anticipatory policies in unevenaged forests. In this study, continuous cover management of both evenand unevenaged initial stands was optimized. Continuous cover management refers to any sequence of cuttings that keep a minimum postcutting residual stand basal area. Regeneration by planting or sowing is not used.
Based on previous studies, it was hypothesized that in a risk situation it is optimal to grow more diverse stands than under certainty (Rollin et al. 2005). Risk avoiders were assumed to maintain more diverse stand structures than risk seekers (Pukkala and Kellomäki 2012). The third hypothesis was that adaptive optimization and management results in higher average net present value than anticipatory optimization (Gong 1998; Lu and Gong 2003).
Methods
Growth and yield model
The diameter increments obtained from the diameter increment model were multiplied with a multiplier that describes the effect of climate change on tree growth (Pukkala and Kellomäki 2012). The climateinduced growth trend is based on a processbased model (Kellomäki and Väisänen 1997; Ge et al. 2010) and corresponds to climate change scenario A1B. The effect of changing climate on diameter increment depends on tree species and growing site. The trends are linear and growth will improve approximately 20% in 50 years. In this study it was assumed that the influence of climate change on diameter increment is not known with certainty. Therefore, the slope of the trend line was assumed to be stochastic, with standard deviation equal to 0.1 times the slope coefficient.
Case study stands
Case study stands
Stand  Site  Strata  BA  Height  D _{ min }  D _{ mean }  D _{ max } 

Uneven spruce  MT  Spruce  18  21  17  22  28 
Spruce  7.6  6  1  8  16  
Mature mixed  MT  Pine  6  21  13  22  28 
Spruce  6  14  3  16  22  
Birch  6  20  13  21  27  
Young mixed  MT  Pine  4  17  13  18  22 
Spruce  3  14  3  16  22  
Birch  4  16  13  17  20  
Young spruce  OMT  Spruce  15  11  5  12  18 
Mature spruce  OMT  Spruce  28  21  15  23  29 
Young pine  VT  Pine  15  11  5  12  18 
Mature pine  VT  Pine  25  20  15  21  27 
Each species and canopy layer was initially described by basal area, mean diameter, mean height and minimum and maximum of the diameter distribution. Stand basal area and the three diameters were used to predict the diameter distribution of each stratum (species or canopy layer) present in the stand. The predicted diameter distribution was divided into 10 classes of equal width, and 5 trees were taken to represent each class. The random tree factors of the residuals of the diameter increment model were generated at this point (a _{ i } of Equation 1). As a result, each stratum of the stand was represented by 50 “representative trees” varying in size and inherent growth potential.
Growth, survival and ingrowth were simulated using 5year time steps. If there was ingrowth, a new representative tree was generated for every 10 new conifers or 50 hardwoods (each new tree represented 10 or 50 trees per hectare). The random tree factors of the residuals of diameter growth models were drawn from normal distribution for each new representative tree. Mortality was simulated by multiplying the frequency of the representative tree by its survival probability.
Optimization

Cutting year (exactly: number of years since the start or since previous cutting)

Parameters of the thinning intensity curve, which was defined separately for each species present in the initial stand
Parameter a _{2} gives the diameter at which thinning intensity is 0.5, and a _{1} defines the type of thinning. If a _{1} is negative, small trees are thinned more than large ones, resulting in low thinning. When a _{1} is positive, the thinning represents high thinning while a _{1}equal to 0 results in uniform thinning. As a result, the number of optimized variables was 3(1 + 2) = 9 for onespecies stand and 3(1 + 3 × 2) = 21 for a mixture of pine, spruce and birch.
The intensity and type of cutting were defined with the same logistic function that was used in anticipatory optimization. However, in adaptive optimization cutting may be postponed if timber price is not good enough. Using the same thinning intensity curve with varying cutting years may lead to situations in which the thinning is too heavy or too light, depending on how much and to which direction the cutting year is moved. To avoid this from happening, the problem formulation was changed so that parameter a _{2} (location of thinning intensity curve) was calculated with a model, and only parameter a _{1} (thinning type) was optimized. This resulted in problem formulations containing 3 + 3 × 1 = 6 decision variables in onespecies stands, and 3 + 3 × 3 × 1 = 12 decision variables in the mixture on pine, spruce and birch (the type of thinning was optimized separately for each species).
The current forestry legislation of Finland does not allow the landowner to thin the stand below a certain minimum residual basal area (typically around 10 m^{2} · ha^{−1}). If the minimum basal area requirement is not met, the landowner is obliged to regenerate the stand within a certain time frame. In this study, any solution in which the minimum basal area was not met was penalized with the consequence that the selected schedules were better in line with the current forestry legislation.
Each management schedule evaluated during an optimisation run was simulated 600 times, and the mean NPV of the 600 stochastic outcomes was passed to the optimization algorithm. The results therefore represent the optimal management for risk neutral decision makers. When the effect of risk attitude was analysed the 10% accumulation point of the distribution of outcomes was used as the objective variable for a risk avoider, leading to the selection of such a management schedule for which the worst outcomes are as good as possible (Pukkala and Kangas 1996). The corresponding accumulation point for a risk seeker was 90%. The used optimization method was the direct search algorithm of Hooke and Jeeves (1961). Afterwards, all optimal solutions – also the deterministic ones – were simulated 1000 times with stochastic variation in tree growth, growth trend, in growth and timber price. The reported results on NPV, removals etc. are based on these simulations.
Results and discussion
Effect of risk factors
The total removal of the three cuttings was 6%–23% lower in the stochastic anticipatory optima than in the deterministic optima. The interval between the 1^{st} and the 3^{rd} cutting was 5–20 years shorter in the stochastic optima. These are indications of risk sharing behavior: in a risky situation it is optimal to cut more often but remove a smaller volume at a time.
Results calculated from 1000 stochastic simulations with the optimal values of decision variables for a risk neutral decision maker in different problem formulations when tree growth, ingrowth and timber price are stochastic (Det = deterministic optimization, Anti = stochastic anticipatory optimization, Ada = stochastic adaptive optimization)
Number of the cutting  Young spruce  Mature spruce  Young pine  Mature pine  

Det  Anti  Ada  Det  Anti  Ada  Det  Anti  Ada  Det  Anti  Ada  
Cutting year  
1^{st}  20  20  21.9  0  0  4.6  20  20  25.3  0  0  5.6 
2^{nd}  35  35  42.1  20  15  19.6  35  35  47.4  15  15  21.2 
3^{rd}  55  50  62.9  55  45  30.8  55  50  70.3  45  25  39.2 
Diameter before cutting (cm)  
1^{st}  19.1  19.1  19.6  23.0  23.0  24.4  17.5  17.6  18.6  21.0  21.0  22.2 
2^{nd}  19.9  19.7  21.9  28.3  26.6  27.7  17.8  18.9  20.0  23.5  23.5  23.8 
3^{rd}  23.3  22.3  25.3  31.5  30.5  29.9  19.9  19.1  21.6  24.8  24.3  23.7 
Basal area before cutting (m^{2} · ha^{−1})  
1^{st}  35.7  35.7  37.1  28.1  28.1  31.3  32.1  32.2  35.8  25.1  25.1  28.1 
2^{nd}  29.5  27.1  33.6  21.6  19.8  19.7  26.7  26.6  33.2  16.4  19.4  19.5 
3^{rd}  27.6  25.7  29.0  26.7  21.6  14.9  27.6  26.0  32.0  23.9  14.1  19.6 
Basal area after cutting (m^{2} · ha^{−1})  
1^{st}  14.6  12.4  14.5  9.8  10.8  11.1  13.5  13.4  14.4  9.3  11.8  11.5 
2^{nd}  10.6  12.7  12.5  6.4  5.5  9.2  10.5  12.8  12.5  7.1  9.4  8.7 
3^{rd}  10.5  12.6  10.1  11.6  7.2  5.1  11.3  12.1  11.2  9.2  6.2  9.3 
Removed volume (m^{3} · ha^{−1})  
1^{st}  176  192  192  197  187  219  140  141  167  150  126  162 
2^{nd}  163  126  190  164  153  113  129  111  175  93  99  109 
3^{rd}  160  122  182  167  157  109  141  118  181  153  78  93 
Total  499  440  564  528  497  441  410  370  523  396  303  364 
Average roadside saw log price obtained (€ · m^{−3})  
1^{st}  56.8  55.8  64.6  56.1  56.3  61.9  56.7  56.4  66.8  55.7  56.3  62.9 
2^{nd}  56.2  56.5  65.0  56.5  56.4  66.2  56.1  55.8  67.0  56.2  56.5  67.5 
3^{rd}  56.5  55.6  65.1  54.5  55.1  67.4  55.6  55.8  65.2  54.5  55.7  68.1 
Effect of risk attitude
The effect of risk attitude on optimal management was analyzed in the mixed stands with the hypothesis that a risk avoider maintains a more diverse stand structure than a risk seeker. However, the thinning intensity curves were very similar for both risk attitudes suggesting that the postcutting diameter distributions were also similar for both attitudes. The same difference as in pure stands was observed between deterministic and stochastic anticipatory optima: the deterministic optima proposed diameterlimit cutting with a narrower postcutting diameter distribution than obtained in stochastic anticipatory optimization.
Optimal cutting years in anticipatory stochastic optima for different risk attitudes
No. of the cutting  Young mixed stand  Mature mixed stand  

Avoider  Neutral  Seeker  Avoider  Neutral  Seeker  
Cutting year  
1^{st}  20  20  20  0  5  10 
2^{nd}  35  35  35  15  20  25 
3^{rd}  65  65  55  45  50  55 
Removed volume (m^{3} · ha^{−1})  
1^{st}  142  145  162  77  107  139 
2^{nd}  110  107  83  102  130  115 
3^{rd}  193  215  127  229  210  243 
Total  446  467  372  408  448  497 
Adaptive optima
In adaptive optimization, cutting years were replaced by the reservation price function, resulting in cutting years that may be different in repeated stochastic simulations, depending on the realized stand development and timber price. To make the thinning intensity curve sensitive to changes in cutting year, the “location” parameter of the curve (a _{2}, dbh at which thinning intensity is 0.5) was calculated with a model (Equation 8) and only the type of thinning (low, uniform or high depending on parameter a _{1} of Equation 6) was optimized.
The solutions of the adaptive optimization problems were also simulated so that the optimized value of parameter a _{1} (thinning type) of the thinning intensity curve (Equation 6) was replaced by 1, corresponding to high thinning. The average NPVs of 1000 simulations were nearly the same as obtained with the optimized values of parameter a _{2}, except for the mature mixed stand. The result indicates that nearly optimal adaptive management can be found when optimizing only the reservation price function and calculating the thinning intensity curve with model, fixing parameter a _{1} to 1. The whole management schedule can be defined and optimized only by three decision variables, namely the parameters of the reservation price function. In the anticipatory optima for mixed stands there are 21 decision variables and yet the expected NPV is clearly better for the adaptive solution defined by only 3 decision variables.
The average roadside price obtained from saw log was about 20% higher in adaptive optima than in deterministic or stochastic anticipatory optima (Table 2). The difference was smaller in the first cutting of the mature stands, due to the high opportunity cost of the growing stock (high financial maturity of the initial stand). The results are in agreement with the assumptions made about the shape of the reservation price function.
Conclusions
All the hypotheses of the study were supported by the results. However, the effect of risk attitude on optimal management was very small, which may be related to the current forestry legislation which ruled out a part on the management options. Another reason may be the size differences of the species of mixed stands, which had a greater impact on the results than risk attitude. Positive correlation between timber prices of different tree species (Figure 3) also decreases the possibilities to reduce financial risk by increased species diversity. Roessiger et al. (2011) concluded that the optimal management for a cautious riskavoiding forest landowner uses tree species diversification, avoiding clearcutting and monospecies forest composition.
All thinnings of all solutions were high thinnings. The very high stochastic variation of ingrowth did not affect the expected NPV of the management schedule and it did not bring much uncertainty in decisionmaking. This is because the removals and incomes of the first three cuttings were obtained from trees that already existed in the initial stands. Ingrowth affects the incomes of distant cuttings whose effect on NPV is very small when the discount rate is 3% or higher. In addition, infrequent regeneration and ingrowth, combined with uneven growth rate of the ingrowth trees may provide a continuous enough supply of trees to larger diameter classes. Timber price was by far the most significant source of risk and uncertainty.
By looking at the average NPVs of 1000 stochastic simulations conducted with different optimal solutions (Figure 11) it can be concluded that there is also some uncertainty related to the optimality of the found solutions. Theoretically, stochastic anticipatory optima should produce better results than 1000 stochastic simulations with the deterministic optima, but this was not always the case. Correspondingly, fixing parameter a _{1} to 1 should decrease the simulated NPVs, compared to adaptive solutions where a _{1} was optimized, but this did not happen always. The results suggest that stochastic problems are more difficult to solve than the deterministic ones. Simulating each schedule clearly more than 600 times in optimization (600 realizations were used in optimization runs) would most probably partially solve the problem, but with a high computational cost. An alternative approach, namely the Markovian decision process model, would be better from the computational point of view (Kaya and Buongiorno 1987).
Corresponding to the hypotheses and previous studies (Gong 1998; Lu and Gong 2003; Pukkala and Kellomäki 2012), adaptive optimization led to higher NPVs than anticipatory optima. However, the differences were smaller than what could be expected on the basis of some earlier studies (Gong and Yin 2004; Pukkala and Kellomäki 2012). This was partly because the growth interval was always 5 years although the truly optimal cutting year might be one of the years within the 5year time step used in simulation. This most probably decreased the NPVs more in adaptive optimization since it was not possible to pick the year of the 5year period that had the highest timber price. Therefore, the results of this study can be interpreted so that the benefit of adaptive optimization is at least 6%–14% but it can be also higher. Zhou et al. (2008) found a 17% higher NPV for adaptive strategy compared to fixed strategy, with little difference in length of cutting cycle.
The adaptive approach facilitates very simple management rules. The optimal future management can be described with only 3 parameters, namely the coefficients of the reservation price function. A thinning treatment should be conducted when the actual price is higher than the reservation price. The thinning intensity of different diameter classes is calculated with Equation 6. Parameter a _{1} of the equation can be taken as 1, and parameter a _{2} is calculated with Equation 8. If the use of equations is difficult to the forest manager, the equations can be converted to diagrams that show the optimal management in a changing environment.
Declarations
Authors’ Affiliations
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