A icle
In oducing a Non-S a iona y Ma ix Model o
S and-Le el Op imiza ion, an E en-Aged Pine
(Pinus Syl es is L.) S and in Finland
Johanna Pyy 1,*, Anssi Ah ikoski 2, E kki Lai inen 1and Jouni Siipileh o 3
1Facul y o Sciences, Uni e si y o Oulu, P.O. Box 8000, FI-90014 Oulu, Finland; [email p o ec ed]
2Na u al Resou ces Ins i u e Finland (Luke) Oulu, Paa o Ha aksen ie 3, FI-90014 Oulu, Finland;
[email p o ec ed]
3Na u al Resou ces Ins i u e Finland (Luke), La oka anonkaa i 9, FI-00790 Helsinki, Finland;
[email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +35-844-545-0982
Academic Edi o s: Maa en Nieuwenhuis and Timo hy A. Ma in
Recei ed: 10 Ap il 2017; Accep ed: 9 May 2017; Published: 11 May 2017
Abs ac :
In gene al, ma ix models a e commonly applied o p edic ee g ow h o size-s uc u ed
ee popula ions, whe eas empi ical–s a is ical models a e designed o p edic ee g ow h based on a
as amoun o ield obse a ions. F om he heo e ical poin o iew, ma ix models can be conside ed
o be mo e gene ic since hei dependency on ad hoc g ow h condi ions is a less p e alen han ha
o empi ical–s a is ical models. On he o he hand, he main pi all o ma ix models is hei inabili y
o include a ia ion among he indi iduals wi hin a size class, occasionally esul ing in less accu a e
p edic ions o ee g ow h compa ed o empi ical–s a is ical models. Thus, he ele an ques ion
is whe he a ma ix model can cap u e essen ial ee-g ow h dynamics/cha ac e is ics so ha he
model p oduces accu a e s and p ojec ions which can u he be applied in p ac ical decision-making.
Such a dynamic cha ac e is ic in ou model is he basal a ea o ees, which causes nonlinea i y in
ime. The e o e, ou ma ix model is a nonlinea model. The empi ical da a o models was based on
20 sample plo s ep esen ing 8360 ee eco ds. Fu he , acco ding o he model, s and p ojec ions
we e p oduced o h ee Sco s pine (Pinus syl es is L.) sapling s ands (age o 25 yea s, s and densi y
luc ua ing om 850 o 1400 s ems ha
−1
). Then, (e en-aged) s and managemen was op imized by
applying sequen ial quad a ic p og amming (SQP) among hose g ow h p edic ions. The objec i e
unc ion o he op imiza ion ask was o maximize he ne p esen alue (NPV) o he ongoing
o a ion. The s ands we e loca ed in No he n Os obo hnia, Finland, on nu ien -poo soil ype.
The esul s indica ed ha ini ial s and densi y had an e ec on op imal solu ions—op imal s and
managemen a ied wi h espec o hinnings ( iming and in ensi y) as well as o op imal o a ion.
Fu he , an inc easing discoun a e sho ened conside ably he op imal o a ion pe iod, and elaxing
he minimum hinning emo al o 30 m
3
ha
−1
esul ed in an inc ease bo h in numbe o hinnings
and in he maximum ne p esen alue.
Keywo ds: o es managemen ; nonlinea ma ix model; op imiza ion; disc e e op imal con ol
1. In oduc ion
The ma ix popula ion models (in e changeably ansi ion ma ix models o Ushe ma ix models;
see [
1
] and [
2
], espec i ely) a e widely used o s udy he dynamics o o es ypes a ound he
wo ld [1,3]
. In gene al, ma ix popula ion models ely on a di ision o he diame e dis ibu ions in o
o de ed classes [
4
,
5
]. Wi h espec o classi ying o es dynamics models (depending on hei le el o
desc ip ion o he o es ), ma ix popula ion models all be ween s and models and indi idual ee
Fo es s 2017,8, 163; doi:10.3390/ 8050163 www.mdpi.com/jou nal/ o es s
Fo es s 2017,8, 163 2 o 13
models (e.g., [
6
,
7
]). Ma ix popula ion models a e g ounded in dis ibu ion-based popula ion models
whe e indi idual a ibu es a e summa ized by hei popula ion-le el dis ibu ion (e.g., [8]).
In ma ix popula ion models, ee g ow h is modelled as a ansi ion om a class o uppe
classes (upg ow h a e), su i al as he cumula ed ansi ions om one class o ano he (mo ali y
a e), and ec ui men as a ansi ion in o he i s class ( ec ui men a e) [
2
]. In p incipal, ma ix
models a e based on ou assump ions: Ma ko p ope y, Ushe p ope y, s a iona i y and geospa ial
independence [
1
]. Th oughou he his o y o de eloping ma ix models, hey ha e been c i icized e.g.,
o hei inabili y o include a ia ion among he indi iduals wi hin a size class [
9
,
10
], o he Ushe
p ope y (an indi idual canno mo e up by mo e han one class o mo e backwa ds), and o he
a bi a iness o he class di ision [
1
,
9
,
11
,
12
]). Se e al solu ions o hese abo e-men ioned d awbacks
ha e been de eloped and applied (e.g., [
4
,
13
,
14
]), and a ew mo e app oaches a e unde way [
11
,
15
].
Despi e he ew sho comings associa ed wi h ma ix popula ion models, hey ha e been applied o
almos all he subjec a eas o o es y [
1
]. The ad an ages o popula ion ma ix models, compa ed
o indi idual-based models (in e changeably empi ical–s a is ical indi idual ee models, see [
16
])
a e abundan , bu dependen on he applica ion; he bes app oach o a pa icula case should be
he one ha is he mos consis en wi h modelling pu poses while making he ewes assump ions
acco ding o he law o pa simony, i.e., Occam’s azo [
1
]. S a ed di e en ly, when he wo app oaches
(indi idual-based models and popula ion ma ix models) make p edic ions o simila quali y, Occam’s
azo a ou s pa simonious popula ion ma ix models (see, e.g., [
17
]). Gi en he complexi y o
indi idual-based models, he la ge amoun o in o ma ion (and ield measu emen s) hey equi e and
he long p ocessing ime s ill make hem di icul and labo ious o apply o o es managemen , hus
simple and mo e compac models dealing wi h e.g., size classes a e mo e e icien and p ac ical o
he majo i y o pu poses [
1
,
18
]. Especially when applying g ow h models wi hin an economic analysis
o o es managemen , popula ion ma ix models ha e been shown o be ad an ageous and e icien
o apply since a ious op imiza ion algo i hms can s aigh - o wa dly be combined, esul ing in a
sound and solid assessmen amewo k (e.g., [
19
,
20
]). In such a amewo k, he popula ion ma ix
model p oduces g ow h p edic ions using he op imiza ion algo i hm, which maximizes he objec i e
unc ion (e.g., ne e enues; see [20]) by dynamic compu ing.
In Ushe ma ix models, i is assumed ha he g ow h speed o a ee is s a iona y, i.e., i depends
only on he size o he ee [
1
]. In eali y, he g ow h speed depends also implici ly on ime. The e o e,
we elaxed he s a iona i y assump ion and used a non-s a iona y ma ix model whe e he g ow h
speed o he ee depends no only on he size class bu also on he basal a ea o he o es s and (basal
a ea e ol es in ime).
The objec i e o his s udy was i s o in oduce a nonlinea ma ix model o Sco s pine
ee-g ow h in an e en-aged s and, and hen o maximize he ne p esen alue (NPV) o he ongoing
o a ion by op imizing s and managemen wi h sequen ial quad a ic p og amming (SQP). We applied
h ee ad anced seedling s ands wi h a ying in ensi y (i.e., numbe o s ems pe hec a e) as an ini ial
poin o s and p ojec ions. Timing and in ensi y o hinnings and op imal o a ion pe iod associa ed
wi h he op imal s and-le el managemen we e analyzed in de ail o e eal po en ial pa e ns in esul s
wi h espec o ini ial s and cha ac e is ics.
2. Ma e ials and Me hods
2.1. The G ow h Model
We suppose ha he o es s and consis s o a single species o ees, which has di e en diame e s.
Mo eo e , we assume ha he diame e dis ibu ion o ees depends on he basal a ea.
Le us assume ha he ime is di ided in o subin e als
[ k
,
k+1]
,
k=
0,
. . .
,
M
, and he diame e is
di ided in o
N
disc e e and non-o e lapping size classes, which a e deno ed by subindices
i=
1,
. . .
,
N
.
We deno e,
yk= [yk
1
,
yk
2
,
. . .
,
yk
N]T
and
hk= [hk
1
,
hk
2
,
. . .
,
hk
N]T
a e he disc e e diame e dis ibu ion and
ha es ec o o he o es s and a ime e en
k
, espec i ely. Mo eo e , hei elemen s
yk
i
and
hk
i
Fo es s 2017,8, 163 3 o 13
a e he numbe o ees pe uni a ea and he numbe o emo ed ees pe uni a ea o size class
i
,
espec i ely.
Le us deno e by
bk
i
and
ak
i
he p obabili y ha a ee om diame e class
i
g ows o he nex
diame e class
i+
1 and he p obabili y ha a ee om diame e class
i
emains in he same diame e
class be ween
k
and
k+1
, espec i ely. Deno e also by
mk
i
he p obabili y ha a ee om diame e
class
i
dies be ween
k
and
k+1
. The a e
ak
i
is called s asis a e,
bk
i
upg ow h ansi ion a e and
mk
i
mo ali y a e [
1
]. We assume ha he upg ow h ansi ion a e
bk
i
and mo ali y a e
mk
i
depend
linea ly on he basal a ea o he o es s and [21], i.e.,
bk
i=c0i+c1iΠ(yk),i=1, . . . , N−1, k=0, . . . , M−1, (1)
mk
i=d0i+d1iΠ(yk),i=1, . . . , N,k=0, . . . , M−1, (2)
whe e
Π(yk) =
N
∑
j=1
yk
jπ(xj/
2
)2
is he basal a ea o he s and and
xj
is he cen e o he diame e class
j
.
The s asis a e ak
ican be calcula ed om he upg ow h ansi ion a e bk
iand mo ali y a e mk
i
ak
i=1−bk
i−mk
i,i=1, . . . , N,k=0, . . . , M−1, (3)
whe e bk
N=0 o all k=1, . . . , M.
We assume he ollowing explici ma ix equa ion o he size class dis ibu ion a ime k+1:
yk+1=U(yk)yk−hk(4)
whe e U(yk)is he o es g ow h ma ix, which has he ollowing s uc u e a ime e en k
U(yk) =









ak
10 . . . 0 0
bk
1ak
2. . . 0 0
0bk
2
...0 0
.
.
..
.
........
.
.
0 0 . . . bk
N−1ak
N









. (5)
The ime dependency in oduced in his pape con ibu es o he cu en li e a u e on ma ix
g ow h models (e.g., [1,11,18,22]).
Model Pa ame e Es ima ion
The da a used o es ima e he model pa ame e s o he size-s uc u ed ansi ion ma ix we e
de i ed om wo long- e m expe imen s (HARKAS se ies; see, e.g., [
23
]). The wo expe imen s we e
es ablished in 1978 and 1984 in e en-aged, pu e comme cial Sco s pine (Pinus syl es is L.) s ands
loca ed on mine al soil in Os obo hnia egion, Finland. The biological ages o he expe imen al
s ands a he ime o es ablishmen we e 43 and 58 yea s, espec i ely. The si es we e classi ied as he
Vaccinium o es si e ype, a sub-xe ic o es , a nu ien -poo soil ype [
24
], which p esen s app. 25% o
o es land a ea on mine al soils acco ding o he 11 h na ional o es in en o y (NFI) in Finland [25].
The s ands we e es ablished by sowing wi h seed o local o igin (i.e., unimp o ed seed ma e ial).
The expe imen s loca ed in Muhos municipali y, 26
°
06
0
05
00
E and 64
°
46
0
02
00
N, asl 60–70 m. (Figu e 1).
A e age g owing season alls in o a ange o 100–140 days ( h eshold +5
°
C, see [
26
]). The da a
o upg ow h and mo ali y in he ma ix model we e based on 8360 ee eco ds om 20 sample
plo s. The s and managemen among he sample plo s luc ua ed conside ably: om con ol plo s
(no hinnings) o e y in ensi e hinnings (60% o he basal a ea emo ed). The sample plo s o one
expe imen (al oge he 12 sample plo s) we e measu ed i e imes du ing 1978 and 2014. Acco dingly,
in ano he expe imen , he sample plo s (eigh sample plo s) we e measu ed ou imes. The age ange
Fo es s 2017,8, 163 4 o 13
o measu emen s in expe imen s co e s oge he a ime span om 43 o 88 yea s which p ac ically
p esen s all comme cial hinnings o Sco s pine du ing a o a ion in he Os obo hnia egion.
Figu e 1. Loca ions o expe imen s.
Acco ding o he da a desc ibed abo e, we es ima ed he coe icien s
c0i
,
c1i
,
i=
1,
. . .
,
N−
1
and
d0i
,
d1i
,
i=
1,
. . .
,
N
in Equa ions (1) and (2), espec i ely, by using he leas squa es me hod.
Technically, pa ame e es ima ions we e calcula ed by using Ma lab (R2012a 7.14.0.739, Ma hWo ks
Inc, Na ick, MA, USA) .
2.2. The Op imiza ion P oblem
The aim o he op imiza ion was o maximize he e enues om he hinnings. The op imiza ion
p oblem was o mula ed as ollows:
max
h∈RMN
M−1
∑
k=0 ∑N
i=1(cp p
i+cs s
i)hk
i
(1+ ) k−p(hk)!, (6)
subjec o Equa ion (4)
y0=y0, (7)
y≥0, h≥0 (8)
whe e
is he in e es a e;
cp
and
cs
a e he p ice o pulpwood and saw log, espec i ely;
p
i
and
s
i
a e he olume o pulpwood and saw log o a ee in diame e class
i
, espec i ely; and
y0
is he
ini ial alue o size class dis ibu ion. The unc ion
p(hk)
is a penal y e m, which ensu es ha i any
hinning is done a ime k, a leas Bcubic me e s pe hec a e ha e o be emo ed. I is in he o m
p(hk) = 






N
∑
j=1
( p
j+ s
j)hk
j(B−
N
∑
i=1
( p
i+ s
i)hk
i), i 0 <
N
∑
i=1
( p
i+ s
i)hk
i<B,
0, else.
(9)
Fo es s 2017,8, 163 5 o 13
The pulpwood and sawlog olumes o a Sco s pine we e abula ed a di e en diame e s a
in e als o 5 cm s a ing om 7.5 cm (c . [
27
]). By using able (o iginal alues acco ding o pine,
H100 = 20 m
) and cubic spline wi h no -a-kno end condi ion [
28
], we calcula ed he pulpwood and
sawlog olumes o a ee in diame e classes
i
,
i=
1,
. . .
,
N
. The sawlog and pulpwood olumes o
a ee a e shown in Figu e 2.
6 8 10 12 14 16 18 20 22 24 26 28 30 32
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
diame e o a ee
sawn imbe and pulpwood olumes, m3
pulpwood
saw log
Figu e 2. Pulpwood and sawlog olumes o a ee as a unc ion o diame e a b eas heigh , cm.
We applied FMINCON sol e (Ma lab Op imiza ion Toolbox) o sol ing he op imiza ion
p oblem
(6)
–
(9)
. Technical de ails and o mal subp oblems associa ed wi h SQP a e p esen ed in
Appendix A.
Ini ial Da a o S and-Le el Op imiza ion
Da a o s and-le el op imiza ion we e sough om a pe manen s and plo da abase o young
s ands (TINKA). We sea ched o Sco s pine-domina ed ad anced seedling s ands (dominan heigh o
app oxima ely 8 m) om Os obo hnia. We calcula ed he ounded a e age s and cha ac e is ics om
he a i icially egene a ed Sco s pine s and on a d yish si e. S and cha ac e is ics we e needed as inpu
a iables o c ea ing he indi idual ee dimensions, s em diame e s a b eas heigh (dbh) and ee
heigh s (h). These s and cha ac e is ics ep esen ed he a e age s and densi y ha ing
1000 s ems ha−1
a he age o 25 yea s. The s and basal a ea was 9 m ha
−1
, basal a ea-weigh ed mean diame e
(
∑dbh3/∑dbh2
) 12 cm and Lo ey’s heigh (
∑h·dbh2/∑dbh2
) 7.6 m. We c ea ed a ia ion in he
numbe o s ems while ixing he o he s and cha ac e is ics in o de o de elop he op ions o
one mo e spa se s and and also o one highe s and densi y (Table 1). The wo-pa ame e Weibull
dis ibu ion was sol ed om he s and cha ac e is ics using pa ame e eco e y [
29
]. The samples
o 20 andomly selec ed diame e s we e gene a ed o each op ion. We sampled ees om he
cumula i e p obabili y dis ibu ion by andomizing he pe cen ile (P) om he uni o m 0–1 dis ibu ion.
The cumula i e Weibull dis ibu ion unc ion is o o m
F(dbh) =
1
−e−(dbh/b)c
and ee diame e is
sol ed as
dbh =b{−ln(
1
−P)}(
1
/c)
whe e
b
and
c
a e he scale and shape pa ame e o he Weibull
dis ibu ion [
30
]. F om he sol ed pa ame e s (see Table 1), we no iced ha he a e age densi y
esul ed in almos no mal dis ibu ion (
c=
3.61) while he highes densi y (1400 ha
−1
) wi h pa ame e
c
o 2.01 is s ongly skewed o he igh and inally he spa se s and (850 ha
−1
) esul ed in a peaked
dis ibu ion wi h pa ame e co 7.53 (Figu e 3).

Fo es s 2017,8, 163 6 o 13
The pa ame e s
b0=
1.319 and
b1=
0.296 o he Näslund’s heigh cu e [
31
] we e p edic ed
om s and basal a ea, basal a ea-weigh ed mean dbh and Lo ey’s heigh oge he wi h he e ec i e
empe a u e sum (993
°
C d) using models by [
32
]. Models also included p edic ion o he s anda d
de ia ion o he esidual e o as a a iance unc ion (see [
32
]). Random de ia ion was added o he
p edic ed ee heigh in o de o esul in ealis ic a ia ion in s em dimensions.
Table 1. The s anda d s and cha ac e is ics and he eco e ed Weibull pa ame e s band c.
S and
Age, Yea s
Basal A ea,
m2ha−1
Numbe o
S ems ha−1
Weigh ed Mean
dbh, cm
Lo ey’s
Heigh , m
Dominan
Heigh , m
Scale
Pa ame e b
Shape
Pa ame e c
25 9 850 12 7.6 8.5 12.216 7.535
25 9 1000 12 7.6 8.5 11.352 3.612
25 9 1400 12 7.6 8.5 9.057 2.011
Fo es s 2017,8, 163 6 o 13
The pa ame e s
b0=
1.319 and
b1=
0.296 o he Näslund’s heigh cu e [
31
] we e p edic ed
om s and basal a ea, basal a ea-weigh ed mean dbh and Lo ey’s heigh oge he wi h he e ec i e
empe a u e sum (993
°
C d) using models by [
32
]. Models also included p edic ion o he s anda d
de ia ion o he esidual e o as a a iance unc ion (see [
32
]). Random de ia ion was added o he
p edic ed ee heigh in o de o esul in ealis ic a ia ion in s em dimensions.
Table 1. The s anda d s and cha ac e is ics and he eco e ed Weibull pa ame e s band c.
S and
Age, Yea s
Basal A ea,
m2ha−1
Numbe o
S ems ha−1
Weigh ed Mean
dbh, cm
Lo ey’s
Heigh , m
Dominan
Heigh , m
Scale
Pa ame e b
Shape
Pa ame e c
25 9 850 12 7.6 8.5 12.216 7.535
25 9 1000 12 7.6 8.5 11.352 3.612
25 9 1400 12 7.6 8.5 9.057 2.011
Figu e 3.
The shapes o he Weibull dbh dis ibu ions o he a e age s and (solid line) which ollowed
app oxima ely no mal dis ibu ion (
c=
3.61) and o spa se (b oken line) and dense s and (do ed line).
3. Resul s
In ou calcula ions, we used he ollowing alues o model pa ame e s: in e es a e
=
3%,
p ice o pulpwood
cp=
16.56
em−3
and sawlog
cs=
58.44
em−3
, and minimum hinning emo al
B=50 m3ha−1.
The esul s show ha wo o h ee in e media e hinnings ook place du ing o a ion (Table 2).
Fu he , i seems ha he hinning pa e n (wi h espec o ha es ed and emaining ees in di e en
size classes) was mo e o less simila , ega dless o he ini ial s and densi y: la ge ees we e always
emo ed (Figu e 4). Wi h ega d o he hinning in ensi y o he i s hinning, app. 43–45% o basal
a ea was emo ed. In he second hinning, om 45% (no mal s and) up o 68% (dense s and) o
basal a ea was emo ed. In he hi d hinning (spa se and no mal s ands), he emo al pe cen age
(exp essed as ela i e o basal a ea) was on a e age 64%. The emo al pe cen ages all in o he o iginal
ange o hinning in ensi ies in he modelling da a. The maximized e enues luc ua ed be ween 3713
and
4198 eha−1,
indica ing ha ini ial s and densi y has an e ec on he maximum ne p esen alue.
On he o he hand, wi h ega d o MAI, ini ial s and densi y has only a mino ole (Table 2).
Figu e 3.
The shapes o he Weibull dbh dis ibu ions o he a e age s and (solid line) which ollowed
app oxima ely no mal dis ibu ion (
c=
3.61) and o spa se (b oken line) and dense s and (do ed line).
3. Resul s
In ou calcula ions, we used he ollowing alues o model pa ame e s: in e es a e
=
3%,
p ice o pulpwood
cp=
16.56
em−3
and sawlog
cs=
58.44
em−3
, and minimum hinning emo al
B=50 m3ha−1.
The esul s show ha wo o h ee in e media e hinnings ook place du ing o a ion (Table 2).
Fu he , i seems ha he hinning pa e n (wi h espec o ha es ed and emaining ees in di e en
size classes) was mo e o less simila , ega dless o he ini ial s and densi y: la ge ees we e always
emo ed (Figu e 4). Wi h ega d o he hinning in ensi y o he i s hinning, app. 43–45% o basal
a ea was emo ed. In he second hinning, om 45% (no mal s and) up o 68% (dense s and) o
basal a ea was emo ed. In he hi d hinning (spa se and no mal s ands), he emo al pe cen age
(exp essed as ela i e o basal a ea) was on a e age 64%. The emo al pe cen ages all in o he o iginal
ange o hinning in ensi ies in he modelling da a. The maximized e enues luc ua ed be ween 3713
and
4198 eha−1,
indica ing ha ini ial s and densi y has an e ec on he maximum ne p esen alue.
On he o he hand, wi h ega d o MAI, ini ial s and densi y has only a mino ole (Table 2).
Fo es s 2017,8, 163 7 o 13
Table 2. Op imal s and-le el managemen s gene a ed by he ma ix model. Discoun a e 3%.
Thinning Age o he
S and (a)
Volume o Remo ed
T ees (m3ha−1)
P opo ion o
Saw log (%)
NPV
(eha−1)
MAI
(m3ha−1yea −1
)
Spa se (Numbe o s ems 850)
Fi s hinning 45 63.7 75
Second hinning 55 50.0 80
Thi d hinning 65 52.3 81
Final hinning 95 73.7 90
To al 239.6 82 4042 2.52
No mal (Numbe o s ems 1000)
Fi s hinning 45 66.3 78
Second hinning 55 50.0 76
Thi d hinning 65 56.3 75
Final hinning 90 73.7 84
To al 246.3 79 4198 2.74
Dense (Numbe o s ems 1400)
Fi s hinning 45 54.9 78
Second hinning 55 71.4 72
Final hinning 85 95.1 85
To al 221.4 79 3713 2.60
5−8 11−1417−2023−26 29−
0
100
200
300
Fi s hinning
850
5−8 11−1417−2023−26 29−
0
100
200
300
Second hinning
5−8 11−1417−2023−26 29−
0
100
200
300
Thi d hinning
5−8 11−1417−2023−26 29−
0
100
200
300
Numbe o ees / ha
1000
5−8 11−1417−2023−26 29−
0
100
200
300
5−8 11−1417−2023−26 29−
0
100
200
300
5−8 11−1417−2023−26 29−
0
100
200
300
1400
5−8 11−1417−2023−26 29−
0
100
200
300
Size class
T ees a e hinning
Remo ed
Figu e 4.
Diame e dis ibu ions associa ed wi h op imal hinnings. Numbe s (e.g., 5–8, 11–14)
ep esen diame e in cen ime es.
We es ed he sensi i i y o he esul s wi h espec o wo c i ical aspec s. Fi s , he e ec o
discoun a e on op imum s and managemen was analysed by changing he o iginal 3% in o 4%
and 5%. Then, we loosened he penal y e m (Equa ion
(9)
) so ha he minimum hinning emo al
would be
30 m3ha−1
, which can be conside ed o be he “decisi e limi ” o con ac o s o execu e
a hinning in Finland (e.g., [
33
]). Fo simplici y, we conduc ed bo h sensi i i y analyses only o he
no mal s and densi y op ion (numbe o s ems is 1000 ha
−1
). The esul s o he sensi i i y analysis a e
shown in Table 3. In he sensi i i y analyses, he hinning emo als anged om 31% ( hi d hinning
ela ed o 3% discoun ing and 30 m
3
minimum emo al c i e ion) up o 66% (six h hinning ela ed o
3% discoun ing and 30 m3minimum emo al c i e ion), exp essed as a pe cen age o basal a ea.
Fo es s 2017,8, 163 8 o 13
Op imal o a ion sho ened wi h inc easing discoun a e— o ins ance, wi h 5% discoun ing, he
op imal o a ion was 65 yea s whe eas wi h 3% discoun ing i was as long as 90 yea s (Tables 2and 3).
Ano he in e es ing esul wi h discoun a es was ha wi h 4% and 5% discoun ing, he i s hinning
occu ed ea lie han wi h 3%, and he numbe o in e media e hinnings d opped o wo wi h 4% and
5% discoun ing (Table 3). The di e ences in MAI be ween 3%, 4% and 5% discoun ing we e, howe e ,
mino . Relaxing he minimum hinning emo al c i e ion o 30 m
3
ha
−1
esul ed in a sligh ly highe
maximum ne p esen alue as well as mo e in e media e hinnings o be conduc ed—compa ed o
he baseline op imal solu ion (Table 3). Thus, om he o es owne ’s poin o iew, milde hinnings
(wi h espec o hinning emo als) a e a ou able: applying mo e equen hinnings wi h ela i ely
small hinning emo als inc eases he maximum ne p esen alue (Table 3).
Table 3.
Op imal s and-le el managemen s gene a ed by he ma ix model wi h di e en in e es a es
and minimum hinning emo als. “Baseline op imal solu ion” indica es s and managemen wi h 3%
in e es a e and no mal ini ial densi y (numbe o s ems 1000 ha−1).
Thinning Age o he
S and (a)
Volume o Remo ed
T ees (m3ha−1)
P opo ion o
Saw log (%)
NPV
(eha−1)
MAI
(m3ha−1yea −1
)
In e es a e 4% and minimum hinning emo al 50 m3ha−1
Fi s hinning 40 61.4 70
Second hinning 50 50.2 67
Final hinning 75 109.3 78
To al 220.9 73 3168 2.95
In e es a e 5% and minimum hinning emo al 50 m3ha−1
Fi s hinning 35 50.2 58
Second hinning 45 50.0 57
Final hinning 65 77.0 68
To al 177.2 62 2513 2.73
In e es a e 3% and minimum hinning emo al 30 m3ha−1
Fi s hinning 35 36.9 66
Second hinning 45 38.6 68
Thi d hinning 55 30.0 80
Fou h hinning 65 36.5 86
Fi h hinning 75 30.2 88
Six h hinning 85 43.1 88
Final hinning 105 41.2 88
To al 256.6 81 4355 2.44
Baseline op imal solu ion
Fi s hinning 45 66.3 78
Second hinning 55 50.0 76
Thi d hinning 65 56.3 75
Final hinning 90 73.7 84
To al 246.3 79 4198 2.74
4. Discussion
When dealing wi h modelling, one should bea in mind ha i is he end use s, no he modele s
hemsel es, who inally de e mine he alue o a model (e.g., [
1
]). Thus, om he end use s’ poin o
iew, simplici y and accu acy (in p edic ion) a e he key wo ds o emphasize. F om he compu a ional
and analy ical iewpoin , ma ix models a e ac ually simple by s uc u e han empi ical–s a is ical
indi idual ee models [
1
], and hey essen ially equi e handling o less in o ma ion [
8
]. Wi h espec
o accu acy, ma ix models ha e been shown o p edic ee g ow h accu a ely—bo h in he sho
e m [
34
] and long e m [
18
,
22
]. This pape in oduces a new ime-dependen ansi ion ma ix
model o p edic ing pine g ow h in no he n Finland, in which he ime-dependency o igina es om
modelling he basal a ea e ol ing in ime. To ou knowledge, such a model p esen s a no el app oach
in he li e a u e o ma ix popula ion models (c . [
1
]). P io o concluding, we need o compa e ou
esul s o exis ing li e a u e on simila g ow h condi ions and ee species in o de o disco e whe he
ou model is applicable o end use s (who a e esponsible o ac ual decision-making).
Fo es s 2017,8, 163 9 o 13
Fo s and-le el op imiza ion, he e a e a ious di e en algo i hms o choose om [
35
,
36
],
e.g., he de i a i e- ee
di ec sea ch me hod such as he Hooke and Jee es me hod, di e en ial
e olu ion, pa icle swa m op imiza ion [
37
], hyb id op imiza ion s a egies which combine sepa a e
algo i hms (e.g., [
16
]) o dep h- i s sea ch algo i hms which apply a sea ch ee consis ing o a
back acking mechanism [
37
]. In his s udy, we applied a new algo i hm which has ecen ly been
in oduced o o es applica ions: sequen ial quad a ic p og amming (SQP) [
38
]. Ten a i ely, he SQP
(as ep esen a i e o g adien - ype me hods) has been p o en o be obus and much mo e e icien
han he de i a i e- ee me hods [38].
Wi h espec o inancial pe o mance, ou esul s a e in line wi h exis ing li e a u e on he same
ee species (pine) and g ow h condi ions in Finland (e.g., [
36
,
39
,
40
])—gi en he ac ha we assessed
he maximum ne p esen alue (NPV) o he ongoing o a ion, whe eas he exis ing li e a u e ocuses
on assessing he maximum ba e land alue (BLV). Howe e , hese wo measu es (MaxNPV and
MaxBLV) can be echnically commensu a ed o compa ison (e.g., [
41
]). Op imal numbe o hinnings
a ied in his s udy, depending on he penal y e m (Equa ion
(9)
) and pa icula ly on he discoun a e.
This esul is cohe en wi h ea lie s udies [
39
,
40
]) sugges ing, e.g., ha highe in e es a es dec ease
he op imal numbe o hinnings. Mean annual inc emen s (MAIs) associa ed wi h he op imal s and
managemen we e he e sligh ly lowe han p esen ed in exis ing li e a u e [
40
,
42
], a ying om app.
2.52–3.0 m
3
ha
−1
yea
−1
o 3.6 m
3
ha
−1
yea
−1
. The easons o his mino disc epancy in olume
ou pu a e easy o depic : i s , he exac loca ions (in e ms o empe a u e sum and mic o-clima ic
condi ions) o s ands a e sligh ly di e en be ween his s udy and he o he s [
40
,
42
]). The o he eason
is ela ed o s udy amewo ks, mo e p ecisely o g ow h models which a e, o cou se, di e en , and
hus esul in sligh ly di e en ou comes. Howe e , one can a gue ha he MAIs unde lying op imal
managemen p oduced by al e na i e g ow h models can be conside ed o be simila . In addi ion,
di e en op imiza ion algo i hms we e applied in he s udies, which also has an e ec on ou comes
(see, e.g., [35]).
Finally, he unde lying a ionale (and mo i e) o cons uc ing a ime-dependen ma ix model is
o la e on be able o inco po a e gene ic gains in o ha pa icula model. The idea o inco po a ing
gene ic gains in o he ime-dependen ma ix model s ems om he ac ha gene ic gain in g ow h
e ol es wi h ime (e.g., [
43
]), indica ing ha gene ic gain once assessed a ju enile s age could change
owa ds ma u i y [
44
,
45
]. Thus, in he nea u u e, we need g ow h models which ake in o accoun his
phenomenon. Fu he , we could also include a ia ion in s em quali y (due o ee b eeding; see [
46
,
47
]),
and u he inco po a e i in o he ime-dependen ma ix model. Fo ins ance, his can echnically be
done wi h a subdi ision o he ca ego ies, i.e., size classes (see [
4
,
48
]). Ha ing inco po a ed he e ec
o ime-e ol ing gene ic gain and a ia ion in s em quali y in o he ime-dependen ma ix model,
his c ea es a new se o analysis ools o assessmen s. In his new assessmen amewo k, i would
inally be possible in o es y o alue he ele an ai s (such as enhanced g ow h and s em quali y)
in mone a y e ms, and o cons uc genuine ade-o s be ween b eedable ai s. This u he enables
e icien deploymen o imp o ed ma e ial in di e en clima ic condi ions (see [49]).
Acknowledgmen s: We wan o acknowledge he Jenny and An i Wihu i ounda ion o inancial suppo .
Au ho Con ibu ions:
Jouni Siipileh o p o ided he da a o op imiza ions; Johanna Pyy de eloped he
ma ix model and op imiza ion p oblem and conduc ed he op imiza ions unde supe ision o E kki Lai inen;
Anssi Ah ikoski analyzed he esul s; all au ho s con ibu ed o he w i ing o he manusc ip .
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The ounding sponso s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip , and in he
decision o publish he esul s.