Calculating growth response
The models of Kukkola and Saramäki (1983) were used to predict the growth response obtained by fertilization:
$$ \begin{array}{c}\hfill \varDelta {I}_t = {\mathrm{A}}_t \times \mathrm{B}\hfill \\ {}\hfill {\mathrm{A}}_t = \mathrm{f}\left( t, N, P, T\right)\hfill \\ {}\hfill \mathrm{B} = \mathrm{f}\left( Fre,{H}_{\mathrm{dom}}\right) \times \mathrm{f}(SI)\hfill \end{array} $$
(1)
where ΔI
_{
t
} is the increase in volume growth t years after fertilization (m^{3} ha^{1}), A_{
t
} is the effect of time and the amount and type of fertilizer and B is the effect of site and growing stock characteristics on the response. N is the amount of added nitrogen (kg ha^{1}), T is the type of the nitrogen component of the fertilizer, P is indicator variable telling whether the fertilizer also includes phosphorus. Fre is number of trees per hectare, H
_{dom} is dominant height of the stand at the moment of fertilization (m) and SI is site index (dominant height at 100 years, m). The models were developed separately for pine and spruce forests. The two nitrogen types (T) with slightly different responses are: (1) urea, and (2) ammonium sulfate or ammonium nitrate with lime (Kukkola and Saramäki 1983). Examples of calculated responses are shown in Fig. 1.
The fertilizer used in this study is a commercial product containing 25% of N and 2% of P. 50% of N is in ammonium sulfate and the other 50% is in nitrate. The dose of 150 kg N ha^{1} means that 600 kg of fertilizer was used per hectare. The price of the fertilizer was 0.6 € kg^{1} and a “fixed” cost of fertilization was assumed to be 10 € ha^{1}. The total fertilization cost was 10 € ha^{1} + 600 kg ha^{1} × 0.6 € kg^{1} = 370 € ha^{1}, which corresponds to the actual cost of N fertilization in upland forests.
To find the relationship between additional volume increment (obtained from the response model) and additional diameter increment (required in simulation and optimization) the growth of several pine and spruce stands growing on different sites was simulated using individualtree models (Pukkala et al. 2013). The obtained results for volume and diameter increment were used to calculate the relationship between volume growth multipliers and diameter growth multipliers. The multipliers were obtained when comparing the growths of similar growing stocks on different sites. The following relationship was found:
$$ \mathrm{Diameter}\ \mathrm{growth}\ \mathrm{multiplier} = {\left(\mathrm{Volume}\ \mathrm{growth}\ \mathrm{multiplier}\right)}^{0.9} $$
After finding this relationship, the growth response was simulated as follows. First, the volume increment of the stand without fertilization was calculated using the model set of Pukkala et al. (2013). Second, the additional volume increment obtained from fertilization was predicted using the model of Kukkola and Saramäki (1983). Then, the volume increment multiplier was calculated as follows:
$$ \mathrm{Multiplier} = \left(\mathrm{Increment}\ \mathrm{without}\ \mathrm{fertilization} + \mathrm{Additional}\ \mathrm{growth}\right)\ /\ \mathrm{Increment}\ \mathrm{without}\ \mathrm{fertilization} $$
This multiplier was raised to power 0.9 to obtain diameter increment multiplier. For example, if the predicted annual volume increment without fertilization was 6 m^{3}ha^{1} and the additional increment due to fertilization was 2 m^{3}ha^{1}, the volume growth multiplier was (6 + 2)/6 = 1.333, resulting in a diameter increment multiplier of 1.333^{0.9} = 1.295.
The models of Kukkola and Saramäki (1983) give the response separately for different years. In the current study, growth was simulated in 5years steps because the used models (Pukkala et al. 2013) predict 5year growths. When calculating the response, 3^{rd} year since fertilization was used for the first 5year period and 8^{th} year for the second 5yeafr period. It was verified that this simplification gives almost the same total response as calculating the response separately for each year (Fig. 1).
Optimizations
The growth and yield simulator was linked with a nonlinear optimization algorithm (Hooke and Jeeves 1961). Three different management alternatives regarding fertilization were optimized:

Management without fertilization; this provided a reference.

Management schedules which included the possibility to fertilize once with 150 kg N per hectare; this aimed at identifying stands in which profitability can be increased most by using the currently recommended amount of N.

Management schedules in which the amount of fertilizer was also optimized and more than one fertilization treatments were allowed; these optimizations aimed at finding out how much the profitability of forestry can be increased by fertilization and what is the impact of optimal fertilization on timber production. The maximum number of fertilizations was equal to the number of optimized cuttings (i.e., three since three cuttings were optimized). The maximum allowed N dose was 300 kg ha^{1} in one fertilization treatment.
The simulationoptimization system used in this study is the same as in Pukkala et al. (2014a), with the exception that the response function of fertilization was added to the simulator and the year of fertilization as well as the amount of added N were added as optional decision variables. In addition, a decision variable telling whether or not a fertilization treatment is conducted was included in the problem formulation.
The simulation begins with an initial stand, and any number of future cuttings can be optimized. The value of the ending growing stock, after the last optimized cutting, is calculated with a model and added to the net present value (NPV) of the simulated cuttings (Pukkala 2005, 2016). The higher is the number of optimized cuttings, the smaller is the influence of the ending growing stock on NPV and optimization result.
The variables optimized for each cutting were: number of years since the beginning of simulation or previous cutting and two parameters of a model for the harvest percentage in different diameter classes (Fig. 2). Three cuttings were optimized in this study, which means that the number of optimized decision variables was 3 × 3 = 9 when fertilization was not used, 3 × 3 + 2 = 11 when the schedule included one fertilization with a fixed dose (150 kg N ha^{1}) and 3 × 3 + 3 × 3 = 18 when also the N dose was optimized with a maximum of three fertilization treatments.
The model for harvest intensity was as follows (Pukkala 2015):
$$ {p}_{\mathrm{remove}}=1/\left(1+ \exp \left({\mathrm{a}}_1\left({\mathrm{a}}_2\hbox{} d\right)\right)\right) $$
(2)
where p
_{remove} is the proportion of harvested trees and d is diameter at breast height (cm).
The stem quality was assumed to be normal (Q2 in the list of quality classes shown below) in all optimizations except when the effect of stem quality was analyzed. Normal quality means that the volume of saw log was calculated with the taper models of Laasasenaho (1982), taking into account the minimum top diameters (15 cm for pine, 16 cm for spruce) and minimum piece lengths of saw logs (4.3 m for both species). Then, quality deductions were made according to the models of Mehtätalo (2002), the predictions of which were corrected based on the empirical results of Malinen et al. (2007). The following four cases were compared when the effect of stem quality on fertilization benefit was analyzed:

Q1: No quality deductions

Q2: Normal quality as described above

Q3: Saw log volume was reduced by 50% compared to Q2

Q4. The quality was so poor that no saw logs were obtained
Quality deduction was simulated by transferring a part of the theoretical saw log volume (based on the taper model and minimum log dimensions) to pulpwood volume. The incomes from cuttings were calculated using roadside timber prices and harvesting cost functions (Rummukainen et al. 1995). Roadside timber price was 55 € m^{3} for saw log and 31 € m^{3} for pulpwood.
The used software allows evenaged management, continuous cover forestry (CCF) and socalled anyaged forestry (AAF) in which the silvicultural system is not specified beforehand. Always when the postcutting stand basal area falls below the Finnish legal limits (see Pukkala et al. 2014a), there is artificial regeneration if the amount of existing advance regeneration is insufficient (less than another legal limit).
When Pukkala et al. (2014a) used the software to optimize the AAF management of 200 real stands, final felling followed by artificial regeneration was included in few optimal management schedules. Therefore, most of the optimal schedules represented CCF management. To exclude management schedules where fertilization is made in a distant future, during the next rotation, all management schedules optimized in this study included only thinning treatments, i.e. they were forced to represent CCF management. However, the type of thinning, as specified by parameters a_{1} and a_{2} of Eq. 2 was not restricted. The CCF constraint was implemented by preventing the postthinning stand basal area from falling below the legal limit (8–9 m^{2}ha^{1} depending on site fertility). The objective variable was NPV, calculated to infinity with a 3% discount rate.
Two most common Scots pine and Norway spruce sites were included in the analyses. For Scots pine, the sites were subxeric (VT, Vaccinium type) and mesic (MT, Myrtillus type) and for Norway spruce they were mesic (MT) and herbrich (OMT, OxalisMyrtillus type). Twentyfive initial stands were included for each site and species. The basal area of the stands varied from 10 to 26 m^{2}ha^{1} at 4m^{2} intervals (10, 14, 18, 22 and 26 m^{2}ha^{1}) and with each stand basal area the mean dbh of trees varied from 12 to 24 cm at 3cm intervals (12, 15, 18, 21 and 24 cm). Tree height and stand age varied with mean dbh, differently on different growing sites. All these stands were in such a stage that fertilization would soon be a realistic option. All stands were assumed to represent the growing conditions of the southern part of Finland (temperature sum was 1200 degreedays >5 °C).