Hindawi Publishing Corporation International Journal of Forestry Research Volume 2015, Article ID 173042, 15 pages http://dx.doi.org/10.1155/2015/173042 Research Article An Evaluation of the Use of Simulated Annealing to Optimize Thinning Rates for Single Even-Aged Stands KaiMoriguchi,TatsuhitoUeki,andMasashiSaito InterdisciplinaryGraduateSchoolofScienceandTechnology,ShinshuUniversity,8304Minamiminowa, Kami-ina-gun,Nagano399-4511,Japan CorrespondenceshouldbeaddressedtoKaiMoriguchi;14st403f@shinshu-u.ac.jp Received18June2015;Revised27October2015;Accepted10November2015 AcademicEditor:IgnacioGarc´ıa-Gonza´lez Copyright©2015KaiMoriguchietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited. Weevaluatedthepotentialofsimulatedannealingasareliablemethodforoptimizingthinningratesforsingleeven-agedstands. Four types of yield models were used as benchmark models to examine the algorithm’s versatility. Thinning rate, which was constrainedto0–50%every5yearsatstandagesof10–45years,wasoptimizedtomaximizethenetpresentvalueforonefixed 3 rotationterm(50years).Thebestparametersforthesimulatedannealingwerechosenfrom11 patterns,usingthemeanofthenet presentvaluefrom39runstoensurethebestperformance.Wecomparedthesolutionswiththosefromcoarsefullenumerationto evaluatethemethod’sreliabilityandwith39runsofrandomsearchtoevaluateitsefficiency.Incontrasttorandomsearch,thebest runofsimulatedannealingforeachofthefouryieldmodelsresultedinabettersolutionthancoarsefullenumeration.However, variationsintheobjectivefunctionfortwoyieldmodelsobtainedwithsimulatedannealingweresignificantlylargerthanthoseof randomsearch.Inconclusion,simulatedannealingwithoptimizedparametersismoreefficientforoptimizingthinningratesthan randomsearch.However,itisnecessarytoexecutemultiplerunstoobtainreliablesolutions. 1.Introduction programming(DP)[2–6];nonlinearprogramming(NP)[7, 8]; and heuristic methods such as random search (RS) [9], Optimizingthethinningrateforsinglestandsisafundamen- tabusearch,orgreedyalgorithms[10].Ofthese,DPandNP talprobleminimprovingtheeconomyofforestmanagement. havebeenwidelyusedandhavebeenfurtherdevelopedbased In Japan, the proportion of small-scale forest holders is oncomparisonwitheachother. large.Forexample,65.4%ofprivateforestownersinNagano Roise [7] reported that two NP approaches, the meth- Prefectureownareas<1hainextent,and90.6%ownareas ods of Hooke and Jeeves [8] and Powell [11], were more smaller than 5ha [1]. They therefore have little leeway in efficient than the discrete DP algorithm which served as performingspatialoptimization,anditisimportanttoopti- a baseline comparison. On the other hand, Paredes et al. mize the thinning rates applied to single stands to improve [4]proposedanewDPalgorithmnamedPATH(projection profitability.Fortunately,recentdevelopmentsincomputing alternativetechnique)thatwasmoreefficientthantraditional enablerapidquantitativeoptimizationwithnoexperimental DP algorithms. Yoshimoto et al. [3] developed an extended knowledge. PATHalgorithmnamedMSPATH(multistagePATH),which The problem may be optimized in a straightforward can reflect long-term effects such as an increase in log manner using full enumeration (FE), that is, enumerating pricewithdiameterandshowedthatitcouldprovidebetter andexaminingallpatterns.Becausecalculationcostincreases solutionsthanPATH[6,12].Yoshimotoetal.[6]developed exponentiallywiththenumberofcontrolvariables(referred an improved PATH algorithm named RLS-PATH (region- toas“thecurseofdimensionality”),thishowevertendstobe limitingstrategiesPATH)todealwithyieldmodelsincluding impractical.Approachesthatreducethecomputingcostsare manyvariablesandreportedthatinmanycasesitprovided thereforerequired.Manyalgorithmsforsolvingthisproblem better solutions than Hooke and Jeeves’ method. Each of havebeenproposedandcanbegroupedasfollows:dynamic thesemethodshasbeenusedtooptimizethinningratesfor 2 InternationalJournalofForestryResearch single stands (e.g., NP: [13–21], DP: [22–32]). In particular, [10,12,29,32]orinsoilexpectationvalue(SEV)[14,36]),orin the MSPATH algorithm is more appropriate than PATH othercriteria[26,27].Itisdifficulttodealwiththenecessary whenlong-termeffectscausedbycompetition-densityeffects complexityusingNPalgorithms,sincethevarioustrade-off or fluctuations of log price with diameter are taken into relationships,forexample,betweenyieldvolumeatthinning account[12]. Yoshimoto[12]developedawaytoapplyitto andthatatfinalcutting,mayrequiremultimodalityofyield yield models based on stand density management diagram models, and the incorporation of logging processes makes (SDMD) [33], a type of forest growth model incorporating them discontinuous. On the other hand, theoretical proofs competition-densityeffectsthatiscommonlyusedinJapan, are required to ensure the reliability of the DP algorithm; whichhasbeenappliedinseveralstudies[27–32].However, however it may be difficult to obtain sufficient proof to itisnecessarytoconsidercarefullywhethersuchoptimized support complex yield models. We therefore suggest that resultsrepresenta“betterchoice”or“thebestchoice.” simulated annealing (SA) [37, 38], a stochastic algorithm With respect to the spatial planning of forests, it has inspired by the annealing process of metals, may satisfy been argued that it is important to validate optimization the requirements. It is worth noting that SA has been algorithms[34].Thisalsoappliestoalgorithmsforoptimizing theoretically proven to provide the global optimum for a thinningratesforsinglestands,butexistingalgorithmshave widevarietyofproblems[38],suchasthetravelingsalesman asyetonlybeencomparedwitheachother,notagainstexact problem[37,38]andtheknapsackproblem[39],aslongas optimal solutions as provided by FE. Producing a solution the“temperature”parameterisdecreasedgraduallyenough. thatisvalidatedasclosetotheoptimalsolutionispotentially Inthefieldofforestry,SAhasusuallybeenusedforforest notalwaysessential,iftheuserisonlyusingtheoptimization planningunderspatialconstraints[40–42]orwithaspecial algorithm as a way to improve the current situation; in objectivefunctionsuchasthatdefinedbasedon“forestspatial other words, for them “optimized” means “better choice.” value”[43,44].LatelytheperformanceofSAitselfhasbeen However, in some cases a solution close to the optimum is analyzed,forexample,itssensitivitytotheparametersused needed.Forexample,itisdifficulttomanageforestswithout [45]anditsapplicationinneighborhoodmethods[46].Thus, subsidy in Japan. In this case, the result of a stand-level SA is commonly used in the field and its performance is optimization analysis can determine whether a stand can evaluated. However, this approach has never been used to be managed successfully under optimized conditions and optimizethinningratesforsinglestandsanditsperformance whetheritisworthsubsidizing.Suchananalysiscanprovide inthisareahasnotbeenevaluated.Itisthereforenecessary importantinformationforregionalzoning,andinthiscasea firsttoevaluatewhetherSAcanbeusedreliablytooptimize highdegreeofreliability,thatis,confidencethatthemethod thinningratesforsinglestands. providessolutionscloseto“thebestchoice,”isnecessaryto In this study, we evaluate the potential performance of satisfyaccountabilityrequirements,especiallytowardsforest SAinoptimizingthinningratesforsingleeven-agedstands, ownerswhohaveappliedforrezoningandwhoseforestisnot using plural yield models based on SDMD. We set the subsidized.Thesameconsiderationsmayapplytocountries thinningintervalineachmodelatfiveyearsandtheobjective orregionsinsituationssimilartoJapan,thatis,littleleeway function as the NPV before planting for one rotation term. in performing spatial optimization and a limited supply of Theageoftherotationisoneofthemostimportantvariables funding. for improving the financial viability of stand-level forest In a previous study [35], RS with 106 iterations and management.However,wefixedthisvariableinourmodel, MSPATHwerecomparedwithcoarseFE(CFE),bysplitting because a change in rotation age leads to a change in variables into a lattice, comparing them using two simple the number of dimensions of the model (the number of SDMD-basesmall-scaleyieldmodels(differingonlyintheir variables, in this case the number of thinning ages). To pricefunctions)asbenchmark,andsettingnetpresentvalue optimizebothrotationageandthinningrates,itwouldthus (NPV) for 50 years as the objective function. The NPV of benecessarytodefineseveralmodelswithdifferentrotation thenonmonotonicpricingyieldmodelprovidedbyMSPATH ages, optimize them, and then choose the best one. The was60,419yen/halessthanthatderivedusingCFE,andthe searchingperformanceofthemodelwithafixedrotationage monotonic pricing yield model provided by MSPATH was thereforedeterminesperformanceinoptimizingamodelthat 30,920yen/ha less. In contrast, those provided by RS were treatsrotationageasavariable. 3,755yen/ha and 5,487yen/ha less, respectively, than that Whereas the “performance” of an optimization method includesbothreliabilityandefficiency,wemustfirstimprove derived using CFE. On the other hand, MSPATH reduced thecalculationcostby10−5.2∼10−5.1timesrelativetoCFEand its reliability and only then enhance its efficiency while 10−2.6∼10−2.5timesrelativetoRS.MSPATHthereforeconfers maintainingreliability.Theemphasisisthereforeonsecuring reliableoptimawithproven,orthodoxSAmethods.However, a great advantage in terms of calculation cost, but it is the combination of parameters used has strong effects on not possible to rely on it providing “the best choice.” Thus the performance of SA approaches [45]. In addition, SA is we need to establish a more reliable method of optimizing astochasticalgorithm,andasaresultthesolutionobtained thinningratesforsinglestands. fromeachsearchingprocessvaries.Wethereforesearchfor Therearediversethinningratemodelsforsinglestands “the best” combination of SA parameters using CFE, to asaresultofdifferencesinthecomplexityofthemodels(e.g., ensurethebestpossibleperformance,usingthemeanNPV whethertheyconsidervariationintrees[26,27]orinlogging from 39 runs of the SA as the objective function. We also andlumbering[31]),intheobjectivefunctionused(e.g.,NPV compare the 39 NPVs from the solutions obtained using InternationalJournalofForestryResearch 3 optimizedSAparameterswiththoseobtainedusingCFEto (trees/ha),𝑉𝑅𝑓isthestandvolumeperareaonthefulldensity evaluatewhetherSAcanprovidesolutionsclosetotheglobal curve(m3/ha),andRyistherelativeyieldindex(thestandard optimum.Finally,westatisticallycomparedthesevalueswith density index for the SDMD). 𝐴 and 𝐵 are parameters that thoseobtainedusingRSwithfivetimesiterations,toevaluate changewiththemeanheightofthedominanttreesandare thecomparativemeritofSA. definedasfollows: 𝐴=0.095669𝐻−1.274434, 2.MaterialsandMethods (10) 𝐵=8833.4𝐻−3.054618. 2.1. Benchmark Model. A hypothetical even-aged stand of Japanese larch (Larix kaempferi) in Nagano Prefecture with Notethattheseformulaearecommontotheaforemen- medium site quality and a density of 2500 planted trees/ha tioned previous study [35], but the aspects of the model wasusedasabenchmarkstand.Minimumthinningagewas described in the rest of this section and in the next one fixedin10yearsandthethinningintervalinfiveyears.The havebeenmodified.Thepresentmodelismoreappropriate rotationagewasfixedin50years,sincethisvaluewouldstill becauseitsimulatesself-thinningafterthefirstthinning. allowapplicationofCFE. We modeled self-thinning using Tadaki’s model [49], calculatedasfollows[50]: 2.1.1. Objective Function. We defined the objective function asNPVbeforeplantingforonerotationterm: 𝑠 𝑠2 𝑁𝐵 𝑁=− +√ + 𝑠 𝑛 2 4 𝐴 NPV=∑𝐼𝑖(1+0.01𝑟)−𝑡𝑖 −𝐶𝑟, (1) (11) 𝑖=1 𝑠 𝑠2 𝑁𝐵 where𝐼𝑖 istheincomefromthe𝑖thcutting(yen/ha),𝑟isthe if 0.417216𝑁𝑠 <−2 +√ 4 + 𝐴𝑠 , annual interest (%), 𝑡𝑖 is the stand age at the 𝑖th cut (year), wanadss𝐶e𝑟taist0th.9e%co[4st7]o.fArsegmeennetriaotnioenda(ybeonv/eh,a𝑡𝑖).vAalnunesuwaleirnetfierxeesdt 𝑁=10𝑐11𝐻𝑐12 if − 2𝑠 +√𝑠42 + 𝑁𝐴𝑠𝐵 ≤0.417216𝑁𝑠, (12) toincrementsoffiveyearsoverarangeof10–50years. where 𝑁𝑠 is “initial number of trees per area” (trees/ha), a parameteroftheself-thinningcurve.𝑠isgivenasfollows: 2.1.2. Growth Model. We used SDMD to define the growth model. The formulae for larch in Nagano Prefecture are as 𝐵 𝑁1.7159002 follows[48]: 𝑠= + 𝑠 −𝑁. (13) 𝐴 239631.3𝐴 𝑠 1 𝐵 𝑉 =𝐴+ 𝑁, (2) Equation(11)istheself-thinningcurvebeforethestand reaches the full-density condition. After that, self-thinning 0.04259𝐻 𝐻𝐹=0.578096+0.460651𝐻+ √𝑁, (3) proceedsaccordingto(12),thefull-densitycurve,whichtakes 100 themeanheightofthedominanttreesasavariable.𝑁𝑠iswhat 𝑉 the number of trees per area at age 0 would be if the stand 𝐺= , (4) hadonlybeensubjecttoself-thinningbutnottoadditional 𝐻𝐹 thinning by humans. 𝑁𝑠 is therefore only ever equal to the 𝐺 plantednumberoftreesperareaiftheforestisneverthinned. 𝑑𝑔=200√𝜋𝑁, (5) We call this parameter 𝑁𝑠 to avoid confusion with planted numberoftreesperareaandinitialvaluesofoptimization. 𝑑=−0.155598+0.982606𝑑𝑔, (6) SDMDisafunctionofthemeanheightofthedominant trees and the number of trees per area. It hypothesizes that log10𝑁𝑅𝑓 =5.529749−1.780184log10𝐻, (7) only the number of trees per area, not the mean height of the dominant trees, is affected by thinning. In other words, 1 𝐵 𝑉 =𝐴+ 𝑁 , (8) SDMD simulates thinning by translating the self-thinning 𝑅𝑓 𝑅𝑓 curve under a specific mean height of dominant trees. The 𝑉 valueof𝑁𝑠correspondstotheself-thinningcurvein(11).As Ry= 𝑉 , (9) therelationshipbetweenthenumberoftreesperareaundera 𝑅𝑓 specificmeanheightofdominanttreesand𝑁𝑠ismonotonic, where𝑉isthestandstemvolumeperarea(m3/ha),𝐻isthe wecansimulatethinningbydecreasing𝑁𝑠. Itiseasytocalculatethenumberoftreesperareabased mofetarneehsepigerhtaroefad(otmreiensa/hnat)t,r𝐻ee𝐹s(imst)h,e𝑁stiasnthdefoprrmesehnetignhutm(mbe)r, on 𝑁𝑠 under an arbitrary mean height of dominant trees using (11) and (12). However, if number of trees per area at 𝐺isthebasalarea(m2/ha),𝑑𝑔isthediameteratbreastheight anarbitrarymeanheightofdominanttreesisgiven,wefirst (DBH)ofthemeanbasalarea(cm),𝑑isthemeanDBH(cm), need to identify 𝑁𝑠 with a numerical approximation with 𝑁𝑅𝑓isthenumberoftreesperareaonthefull-densitycurve (11) to simulate self-thinning after that age. Moreover, after 4 InternationalJournalofForestryResearch the stand has reached full-density condition, we cannot Table1:Relationshipbetweenenddiameterandlogprice. compute 𝑁𝑠 using (12) and given the number of trees per 3 Enddiameter(cm) Logprice(yen/m ) area,becausethefull-densitycurveas(12)isindependentof 𝑁𝑠.Toaddresstheseproblems,wedefinethinningrateasthe 0–5 0 ratioofthedifferencebetween𝑁𝑠 beforeandafterthinning 6–11 10,500 asfollows: 12–14 7,000 16–18 11,000 𝑁 −𝑁 𝑅=100 𝑠all 𝑠main, (14) 20–22 12,300 𝑁 𝑠all 24+ 12,500 where 𝑅 is the thinning rate (%), 𝑁𝑠all is 𝑁𝑠 before the Source:Hokushinlogmarket[53]. thinning,and𝑁𝑠mainis𝑁𝑠afterthethinning. TosimulatethegrowthofthestandwithageusingSDMD, 2.1.5.StemProfile. Weusedastemprofilecurvetocalculate the mean height growth curve of the dominant trees is thesmallenddiameterofthelogsinacomplexyieldmodel required. The growth curve of Japanese larch at an average (definedinSection2.1.8(2)).TheBehreequationgivenbelow siteinNaganoPrefecturecanbepredictedusingthefollowing isawidelyusedrelativestemprofilecurve: formula[51]: 𝑑 2𝑥 = , 𝐻=25.81{1−1.182exp(−0.05𝑡)}, (15) 𝑑 𝑎+𝑏𝑥 (18) 0.9 where𝑡isthestandage(years).SinceSDMDassumeslower where𝑥istheheightforwhichstemwidthisbeingcalculated thinning,themeanheightofthedominanttreeswillnotbe relative to total tree height (scaled from 0 to 1), 𝑑 is the affectedbythinning. diameter at relative height 𝑥, and 𝑑0.9 is the diameter at relative height 0.9. Parameters 𝑎 and 𝑏 are calculated as follows[52]: 2.1.3. Volume and Diameter of Yield Trees. The total stem volumeofcuttreesperareaiscalculatedasfollows: 𝑎=0.9(2−𝑏), 𝑉cut(𝑁cut,𝑡)=𝑉(𝑁all,𝑡)−𝑉(𝑁main,𝑡), (16) 𝑏= 180𝜑−126𝛿 , 20𝜑−63𝛿+70𝛿𝜑 (19) where 𝑉 (𝑁 ,𝑡) is the cut stem volume per area when cut cut 𝑁 treesper areaare cutat time𝑡, 𝑉(𝑁,𝑡) is thestanding 7 𝜋𝐻󸀠 cut 𝛿=𝑑 √ , treevolumeperareacalculatedusing(2)and(15),wherethe bh 10 4V󸀠 numberoftreesperareais𝑁attime𝑡,𝑁 isthenumberof cut cuttreesperarea,𝑁main isthenumberofstandingtreesper whereV󸀠 isstemvolume,𝐻󸀠 isthetotalheightofthetree,𝜑 area, and 𝑁all is the total number of standing and cut trees istherelativeheightatbreastheight,and𝑑 istheDBH. bh perarea.Thisequationcanbeappliedtoboththinningand Equation (18) can be transformed as follows to provide finalcutting.Meanstemvolumepertreecanbecalculatedby therelationshipbetweenthediameteratanyheightandthe dividingtheleftsideof(16)by𝑁 .ThemeanDBHofthecut cut DBH: treeswascalculatedusingtheformula: 𝑥 𝑎+𝑏𝜑 𝑑=𝑑 ( ). (20) 𝑑 𝑁 −𝑑 𝑁 bh𝜑 𝑎+𝑏𝑥 𝑑 = all all main main, (17) cut 𝑁 cut 2.1.6.LogPrice. Therelationshipbetweentheenddiameter wherethesubscriptsof𝑑arethesameasthoseof𝑁in(16). of4mlongJapaneselarchlogsandthevalueinarealmarket SDMD does not describe variation in stem volume, so we inNaganoPrefecture[53]isshowninTable1.Althoughthere assumedaconstantsizeforallcuttreesforagivenstandage. isabriskmarketfor6–11cmlogsforlogpiles,thevalueof12– 14cmlogsislowerasthereislessdemandforthissize.This pricemodeladdsmultimodalitytotheNPV. 2.1.4.ConstraintsofThinningRatesandtheNumberofTrees perArea. BecausetheparametersofSDMDwereestimated 2.1.7.Cost ofRegenerationandYield. Thethinningcostwas usingdatafromrealforestssubjectedtostandardsilvicultural 3 processes,unusualconditionssuchasextremelysmallnum- setto4588yen/m ,basedonthemeantotalcostoflogging beroftreesperareashouldbeavoidedwhenusingthismodel. and transportation of Japanese larch [54], assuming a 50% Accordingly,thelowerboundof𝑁𝑠 wassetto200trees/ha. subsidy is provided, and the cost of final cutting was set 3 In addition, it is recommended that a high thinning rate to 5987yen/m , assuming no subsidies. Usually no subsidy be avoided when using SDMD; however intensive thinning for final cutting is provided; however for the purpose of is often performed to reduce thinning costs. For this study, examining a range of patterns, we also tested the model we placed emphasis on actual practice and restricted the on a scenario in which the final cost was similarly reduced 3 thinningratestotherange0–50%. to 2994yen/m . Hereafter, we refer to the two scenarios as InternationalJournalofForestryResearch 5 F (full) and H (half), respectively. The difference between numbersotherwise),and𝑘isthenumberofbuckedlogsfrom scenarios affects the flatness of the NPV. The present costs aharvestedtree,calculatedasfollows: forregenerationarecalculatedas1,078,590yen/haaccording 𝐻−0.5 tostandardgovernmentsources[55],whichincludesground 𝑘=floor( ), (24) clearanceofatypicalmeadowatthebeginningoftheplanting 4.1 year,planting2500trees/ha,preventionofmammaldamage byspreadingZiramsolutioninthreeyears,bushcuttingevery where“floor”roundsdowntothenextlowerinteger. yearfromonetofiveyears,treetrimmingandcrosscuttingin Becausethenumberofbuckedlogsfromatreeandthe 10 and 15 years, 8% consumption tax, 0.9% annual interest, end diameters were restricted to integers, the NPV was a and50%subsidy.Thistotalwasappliedto𝐶𝑟in(1). discontinuous, multimodal function. Hereafter we refer to this model as L and in combination with the half and full 2.1.8.YieldModel. Wedefinedtwotypesofyieldmodelsthat scenariosdescribedinSection2.1.7,L-HandL-Fmodelare differinthedegreeofdifficultyinvolvedinoptimizingthem. defined. (1)SimpleYieldModel.Thismodelcalculatestheyieldvolume 2.2. Optimization Method. This section details the imple- by simply multiplying the yield rate by the stem volume of mentation of SA, its application to the thinning rate opti- theharvestedtrees.Wecalculatedincomebasedontheprice mization problem, and the method we used to evaluate its inTable1correspondingtotheDBHoftheharvestedtrees. performance. Thethinningandfinalcuttingyieldratesweresetto58%and 65%,respectively.GapsintheclassesinTable1werelinearly 2.2.1. Simulated Annealing. SA was developed based on interpolated.Theincomefromacutwascalculatedasfollows: the metal annealing process, which finds the global mini- 𝑦 𝐼= 𝑉 (𝑃 −𝐶 ), mum“energy”requiredbygraduallydecreasingaparameter (21) 100 cut cut cut named “temperature.” At the beginning, the temperature is high, so frequent transitions occur for both low-energy where 𝐼 is the income from a cutting per area (yen/ha), 𝑦 and high-energy states. This makes it possible to find the is the yield rate (%), 𝑉 is the total volume of cut trees cut global optimum by searching a wide solution space. As the per area (m3/ha), 𝑃 is the price of a cut tree per volume (yen/m3),and𝐶 icsutthecuttingcostpervolume(yen/m3). temperature cools, the frequency of transitionsto a higher- cut energy state decreases, and the system tends to transition Hereafter, we refer to this model as S and in combination to a lower-energy state more frequently. Using these fea- withthehalfandfullscenariosdescribedinSection2.1.7,S- tures,approximateglobalminimacanbefoundheuristically. H and S-F model are defined. The models are identical to The flowchart is shown in Figure1. The procedure can be the nonmonotonic pricing yield models used in a previous described as a Metropolis algorithm [57] with changes in study [35], except for the self-thinning model and volume temperature. SA is an adaptable method that requires the calculationused. following four components: (1) a cooling function, which controls the rate of decrease in temperature; (2) a proposal (2)LoggingYieldModel.Thismodeltakestheloggingprocess densityfunction,whichisaprobabilitydistributionfunction intoconsideration.Asmany4mlogs(with0.1mmargin)as (PDF) that generates candidate variables, that is, thinning possiblewerebucked,fromstumpheight(0.5m)tothetopof rates; (3) an energy function, which is the metaobjective theharvestedtree.Diameterswerecalculatedateveryheight functiontobeminimized;and(4)anacceptanceprobability, from4.6mtothetopofthetreein4.1mincrements,using a temperature-dependent PDF that decides whether or not thestemprofilecurve,andwereassumedtobeenddiameters. theproposedstateisaccepted. TheJapaneseAgriculturalStandardforlogs[56]wasapplied Thecoolingfunctionisdefinedasanexponentialdecay tothecalculationsofdiameterandvolume.Thethinningand function;thatis,adecreasingratedeterminedbytheinitial finalcuttingyieldratesweresetto80%and90%,respectively. andfinaltemperaturesismultipliedwithtemperatureattime Incomefromacutwascalculatedasfollows: 𝑡toobtaintemperatureattime𝑡+1.Theproposaldensityis 𝑦 𝑘 definedbyanormaldistributionwithameanofthepresent 𝐼= 𝑁 ∑V (𝑃 −𝐶 ), 100 cut 𝑗 𝑗 cut (22) value of the target variable and a standard deviation that is 𝑗=1 optimizedasdescribedlater.Theenergyfunctionisdefined asfollows: where𝑗istheindexnumberofalogfromitsstump,𝑃𝑗isthe priceofthe𝑗thlog(yen/m3),V𝑗 isthevolumeofthe𝑗thlog 𝐸= NPVmax󸀠 −NPV , (m3)calculatedas NPVmax󸀠 −NPVmin󸀠 (25) V =4( 𝑑𝑗 )2, (23) where 𝐸 is the energy function, NPVmin󸀠 is the tentative 𝑗 100 minimumNPV,andNPVmax󸀠isthetentativemaximumNPV. Since our aim was to maximize the NPV, the problem was where𝑑𝑗istheenddiameterofthe𝑗thlog(cm)(omittedfor to minimize the negative NPV. This definition limits the multiplesoftwofordiameters>14cmandomittedfornatural differential of the energy function to the range 0-1. We set 6 InternationalJournalofForestryResearch Start Set initial temperature, final temperature (or rate of cooling), and number of iterations per temperature Initialize state variables and temperature Seti=0 Generate candidate state variables Calculate energy value and acceptance probability Random (0,1)

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