Introducing a non-stationary matrix model for stand-level optimization, an even-aged pine (Pinus sylvestris L.) stand in Finland

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.