MANAGEMENT PLANNING PROBLEMS with conflicting

Transcription

1 Fndng Effcen Harves Schedules under Three Conflcng Objecves Sándor F. Tóh and Marc E. McDll Absrac: Publc foress have many conflcng uses. Desgnng fores managemen schemes ha provde he publc wh an opmal bundle of benefs s herefore a major challenge. Alhough a capably o quanfy and vsualze he radeoffs beween he compeng objecves can be very useful for decsonmakers, developng hs capably presens unque dffcules f hree or more conflcng objecves are presen and he soluon alernaves are dscree. Ths sudy exends four mulobjecve programmng mehods o generae spaally explc fores managemen alernaves ha are effcen (nondomnaed) wh respec o hree or more compeng objecves. The algorhms were appled o a hypohecal fores plannng problem wh hree mberand wldlfe-relaed objecves. Whereas he -Consranng and he proposed Alpha-Dela mehods found a larger number of effcen alernaves, he Modfed Weghed Objecve Funcon and he Tchebycheff mehods provded beer overall esmaon of he mber and nonmber radeoffs assocaed wh he es problem. In addon, he former wo mehods allowed a greaer degree of user conrol and are easer o generalze o n-objecve problems. FOR. SCI. 55(2): Keywords: mulobjecve fores plannng, spaal opmzaon, neger programmng, radeoffs MANAGEMENT PLANNING PROBLEMS wh conflcng objecves occur frequenly n foresry. The publc expecs more from fores resources han merely mber producon, ncludng waershed proecon, wldlfe haba managemen, aeshecs, recreaon, and carbon sequesraon. Sakeholder groups such as he mber ndusry and envronmenal organzaons ofen hold srongly conflcng values relaed o hese uses, and conflcs beween mber and nonmber objecves are common. Harvesng can fragmen sensve habas, obsruc he movemen of wldlfe, ncrease fre rsk, and reduce he aeshec value of he fores. On he oher hand, elmnang mber producon from publc foress n ndusralzed naons, as many sugges, would only ncrease he pressure on fores resources n less developed naons, where envronmenal conrols may be less effecve (Thomas 2000). Moreover, oher uses, such as recreaon, can also sress fores ecosysems. However, he radeoffs beween conflcng goals can ofen be balanced effecvely whn a landscape or fores hrough careful plannng (Rosenbaum 2000). In mos cases, quanfyng he radeoffs o deermne he degree of ncompably beween compeng fores uses can help decsonmakers (DMs) selec he bes compromse managemen alernaves. The spaal layou of foresry operaons such as harvesng or road consrucon can have a profound mpac on many nonmber objecves. Spaally explc fores managemen plannng models are useful for effcenly desgnng he locaon and mng of hese operaons whle also addressng wldlfe haba concerns (e.g., Reban and McDll 2003a, 2003b). These models, usually formulaed as neger programs (IPs), are used o deermne when specfc harves uns should be cu and when and where oher se-specfc managemen nervenons should be performed o balance varous fores uses. Mos ofen, hese models have been formulaed as sngle-objecve problems, n whch one fores use s opmzed subjec o a range of resrcons (e.g., Leuschner e al. 1975, Mealey and Horn 1981, Cox and Sullvan 1995, Benger e al. 1997, Reban and McDll 2003a, 2003b). Some of hese resrcons ensure ha mnmum requremens on boh mber and nonmber objecves are me. One example would be o maxmze mber oupu or he dscouned ne revenues from a fores subjec o consrans requrng a balanced endng age-class dsrbuon, a smooh flow of mber producon over me, and mananng a mnmum amoun of maure fores haba n large compac paches whle never exceedng a maxmum harves openng sze. Alernavely, he amoun of maure fores haba could be maxmzed subjec o mnmum ne presen value (NPV) or mnmum mber oupu consrans. In hese ypes of formulaons, he DM(s) s requred o specfy he mnmum requremens on some fores uses before he opmzaon process. Defnng harves arges mgh be relavely sraghforward, bu seng lms on he amoun of maure fores haba paches, for nsance, mgh no. Whou knowng whch haba requremens are feasble and whch are oo modes, specfyng such lms s ypcally guesswork and can lead o poor decsons. Furhermore, beer decsons can be made f he DM(s) undersand he radeoff srucure beween compeng objecves before seng requremens on varous objecves. Anoher frequenly used approach, goal programmng (Charnes e al. 1955), does no compleely overcome hs problem eher, as requres he DM o se up arges on he objecves ha, unlke consrans, do no have o be me. Goal programs (GPs) mnmze he devaons from he arges n eher an order of preference (preempve GP) or n lne wh a se of Sándor F. Tóh, Unversy of Washngon, College of Fores Resources, Bloedel Hall 358, Box , Seale, WA Phone: (206) ; Fax: (206) ; ohs@u.washngon.edu. Marc E. McDll, The Pennsylvana Sae Unversy, Unversy Park, PA mmcdll@psu.edu. Acknowledgmens: We hank he assocae edor and he hree anonymous revewers for her helpful commens. Thanks also o he Pennsylvana Bureau of Foresry for provdng fnancal suppor for hs research. Manuscrp receved March 13, 2008, acceped November 18, 2008 Copyrgh 2009 by he Socey of Amercan Foresers Fores Scence 55(2)

2 weghs assgned o each objecve (nonpreempve GP). Eher way, he DM has o specfy boh he arges and he weghs or a preference ls for he objecves before formulang he model. When possble, generang and vsualzng he complee, or an effecvely flered, se of effcen (.e., Pareoopmal) (Pareo 1909) soluons o fores plannng problems should help he DM(s) acqure a holsc vew of he problem and enable a more nformed decson when selecng a bes compromse managemen alernave. As opposed o domnaed soluons, effcen alernaves are hose whose objecve achevemens canno be furher mproved whou compromsng a leas one objecve. Ths unque se of soluons defnes he so-called effcen froner. Sudyng hs froner can be valuable o he DM(s) for wo reasons. Frs, he effcen froner separaes he regon where addonal soluons do no exs from he regon where domnaed soluons mgh exs (Tóh e al. 2006). Thus, he DM can assess he lms of smulaneously achevng several conflcng objecves. In oher words, he effcen froner answers he queson: wha s possble? Second, by movng along hs froner (.e., by movng beween Pareo-opmal soluons), one can also assess he amoun of one objecve ha mus be forgone o acheve a gven ncrease n he amoun of anoher objecve. Thus, he effcen froner denfes he srucure of he radeoffs beween he compeng objecves represened by he axes of he effcen froner space. Because n spaal fores plannng many managemen decsons are bnary, such as wheher or no a ceran fores un wh predefned boundares should be cu whn a gven me nerval, he se of feasble model soluons s dscree (Tóh e al. 2006). As a resul, he se of aanable objecve funcon values, and hence he effcen froner self, s also dscree and herefore nonconvex. Ths propery makes compuaonally challengng o generae he effcen froners for spaally explc fores managemen plannng problems. Tóh e al. (2006) evaluaed and esed four radonal mehods of generang he effcen froner for a bobjecve spaally explc fores managemen plannng problem. The four approaches, (1) he Weghed Objecve Funcon mehod (Geoffron 1968), (2) he -Consranng mehod (Hames e al. 1971), (3) he Decomposon mehod based on he Tchebycheff merc (Eswaran e al. 1989), and (4) he Trangles mehod (Chalme e al. 1986), were esed on a 50-managemen un hypohecal fores plannng problem. Tóh e al. (2006) also proposed a new approach called he Alpha-Dela mehod ha performed well compared wh he oher approaches. The wo objecves of neres n her es case were o maxmze he NPV of he fores and o maxmze he mnmum amoun of maure fores haba n large paches over he plannng horzon. Mos fores plannng problems nvolve more han wo compeng objecves. A lmed number of heorecal sudes on generang he se of effcen alernaves for hree or more objecve IPs have been documened. One prmary area of research has been he famly of he so-called reference pon mehods (Ehrgo and Wecek 2005). The concep s smple: an effcen soluon can be found by mnmzng he dsance beween a reference pon, whch can be any unaanable soluon n he objecve space, such as he deal soluon or goal programmng arges, and poenal nondomnaed soluons (he deal soluon s a vecor of objecve funcon values ha are ganed by opmzng he problem for each objecve, one a a me, whou regard o he res of he objecves). The dsance measure ha s mos ofen used s he Weghed (or no) Tchebycheff merc and s varans. Dfferen effcen soluons can be found by changng he weghs on he dsance merc (e.g., Eswaran e al. 1989). Moreover, he reference pons hemselves can be vared o denfy oher soluons (e.g., Alves and Clímaco 2001). However, due o he dscree naure of neger programmng, dfferen wegh combnaons and dfferen reference pons can also lead o dencal soluons. Thus, a smar decomposon of he wegh space (or he reference pon space) no regons ha lead o he same soluons s needed o reduce he me spen fndng redundan soluons. These regons are called ndfferen ses and are convex n he wegh space and nonconvex n he reference pon space (Alves and Clímaco 2001). The reference pon mehods fnd effcen soluons by deermnng or approxmang hese ndfference ses. An algorhm proposed by Chalme e al. (1986) fnds Z effcen soluons wh respec o n objecves by solvng a leas nz 1 IPs. The effcency of hs approach s quesonable, gven ha he -Consranng mehod (Sadagopan and Ravndran 1982), whch appeared o be slow n he bobjecve case (Tóh e al. 2006), fnds he same number of soluons by solvng only nz IPs. Case sudes assessng he numercal and compuaonal performance of he avalable mulobjecve mehods as appled o larger han llusrave problem nsances are no common. Ths fndng s parcularly rue for he area of fores resources and wldlfe managemen. Alhough effcen froners wh respec o wo objecves have been suded n (Rose e al. 1990, Holland e al. 1994, Cox and Sullvan 1995, Arhaud and Rose 1996, Church e al. 1996, Snyder and ReVelle 1997, Wllams 1998, Church e al. 2000, Rchards and Gunn 2000), he effcen froners of dscree fores managemen decson problems wh hree or more objecves have apparenly no been researched. Generalzng algorhms ha work well for b-objecve problems o problems wh hree or more objecves s challengng because of he added mahemacal and compuaonal complexy. Neverheless, hs ask s mporan as few fores-plannng problems nvolve only one or wo compeng managemen objecves. The presen sudy bulds on Tóh e al. (2006) by exendng hree of he four radonal approaches (he Weghed Objecve Funcon mehod, he -Consranng mehod, and he Decomposon mehod based on he Tchebycheff merc) and he proposed Alpha-Dela mehod o handle hree or more objecves. The same 50-managemen un hypohecal fores plannng problem used n Tóh e al. (2006) s used o demonsrae and evaluae he mechancs of he exended mehods n generang he effcen froner of a r-objecve spaal fores plannng problem. Two of he objecves are he same as hose n Tóh e al. (2006): maxmze dscouned ne mber revenues and maxmze he 118 Fores Scence 55(2) 2009

3 mnmum area of maure fores haba ha evolves n large paches over he plannng horzon. The hrd s o mnmze he oal permeer of he paches (summed over all perods). The formulaon of hs objecve was nroduced n Tóh and McDll (2008). Maxmzng he area of maure fores haba paches whle mnmzng her oal permeer promoes foresed landscapes wh several desrable spaal characerscs. Ths approach fosers he developmen of paches wh low permeer-area raos, ncreases her emporal overlap, and resuls n fewer and larger paches. Model Formulaon Ths secon descrbes a r-crera neger programmng formulaon ha maxmzes he ne presen value of he fores, maxmzes he mnmum amoun of maure fores haba n large paches over he plannng horzon, and mnmzes he oal lengh of he edges of hese paches over he plannng horzon. The model ncludes harves flow consrans, maxmum harves openng sze consrans, consrans ha defne he mnmum area of maure fores haba paches, and an average endng age consran. The formulaon of he maure fores pach creron s a slghly modfed verson of he one presened n Reban and McDll (2003b). Formulaon of he maxmum harves area consrans s a generalzaon of he formulaon presened n McDll e al. (2002): subjec o M Max Z m1 T X m0 X m 1 h m m T A c m0x m0 c m X h m m (1) Max (2) T Mn (3) 1 for m 1,2,...,M (4) mm h v m A m X m V 0 for 1,2,...,T (5) b l V V 1 0 for 1,2,...,T 1 (6) b l V V 1 0 for 1,2,...,T 1 (7) mm p X m n P 1 for all p P and h,...,t (8) jj m X mj O m 0 for m 1,2,...,M and 1,2,...,T (9) jj m X mj J m O m 0 for m 1,2,...,M and 1,2,...,T (10) mm c O m n c B c 0 for c C and 1,2,...,T (11) mm c O m B c n c 1 for c C and 1,2,...,T (12) cc m B c BO m 0 for m 1,2,...,M and 1,2,...,T (13) cc m B c C m BO m 0 for m 1,2,...,M and 1,2,...,T (14) M A m BO m m1 M m1 for 1,2,...,T (15) N P m BO m 2 CB pq pq pq1 for 1,2,...,T (16) BO p BO q 2 pq 0 for 1,2,...,T, pq 1,2,...,N (17) BO p BO q pq 1 for 1,2,...,T pq 1,2,...,N (18) M T A m Age m0 m1 X m 0, 1 T Age T X m0 Age T m Age T X m h m for m 1,2,...,M and B c 0,1 0 (19) 0, h m, h m 1,...,T (20) for c C, 1,2,...,T (21) O m, BO m 0, 1 for m 1,2,...,M and 0,1,...,T (22) Fores Scence 55(2)

4 pq 0, 1 for pq N (23) where he decson varable s X m a bnary varable whose value s 1 f managemen un m s o be harvesed n perod for h m, h m1,,t (when 0, he value of he bnary varable s 1 f managemen un m s no harvesed a all durng he plannng horzon [.e., X m0 represens he do-nohng alernave for managemen un m]). The auxlary and accounng varables are: O m a bnary varable whose value may equal 1 f managemen un m mees he mnmum age requremen for maure paches n perod,.e., he managemen un s old enough o be par of a maure pach; B c a bnary varable whose value s 1 f all of he managemen uns n cluser c mee he mnmum age requremen for maure paches n perod,.e., he cluser s par of a maure pach; BO m a bnary varable whose value s 1 f managemen un m s par of a cluser ha mees he mnmum age requremen for large maure paches,.e., he managemen un s par of a pach ha s bg enough and old enough o consue a large, maure pach; pq a bnary varable whose value s 1 f adjacen managemen uns p and q are boh par of a cluser ha mees he mnmum age requremen for large maure paches n perod ; V a connuous varable ndcang he oal volume of sawmber n m 3 harvesed n perod ; and he oal permeer of maure fores haba paches n perod. The parameers are: h m he frs perod n whch managemen un m s old enough o be harvesed; he mnmum area of maure fores haba pach over all perods; M he number of managemen uns n he fores; N he number of pars of managemen uns n he fores ha are adjacen; T he number of perods n he plannng horzon; c m he dscouned ne revenue per ha f managemen un m s harvesed n perod, plus he dscouned resdual fores value based on he projeced sae of he managemen un a he end of he plannng horzon; A m he area of managemen un m n ha; P m he permeer of managemen un m n meers; CB pq he lengh of he common boundary beween he wo adjacen managemen uns p wh q n meers; v m he volume of sawmber n m 3 /ha harvesed from managemen un m f s harvesed n perod ; M h he se of managemen uns ha are old enough o be harvesed n perod ; b l a lower bound on decreases n he harves level beween perods and 1 (where, for example, b l 1 requres nondeclnng harves; b l 0.9 would allow a decrease of up o 10%); b h an upper bound on ncreases n he harves level beween perods and 1 (where, for example, b h 1 allows no ncrease n he harves level; b l 1.1 would allow an ncrease of up o 10%); P he se of all pahs, or groups of conguous managemen uns, whose combned area s jus above he maxmum harves openng sze (he erm pah, as used n hs arcle, s defned n he followng dscusson); M p he se of managemen uns n pah p; n Mp he number of managemen uns n pah p; h he frs perod n whch a managemen un n pah can be harvesed; J m he se of all prescrpons under whch managemen un m mees he mnmum age requremen for maure paches n perod ; C he se of all clusers, or groups of conguous managemen uns whose combned area s jus above he mnmum large, maure pach sze (he erm cluser, as used n hs arcle, s defned n he followng dscusson); M c he se of managemen uns ha compose cluser c; n c he number of managemen uns n cluser c; C m he se of all clusers ha conan managemen un m; Age T m he age of managemen un m a he end of he plannng horzon f s harvesed n perod ; and Age T he arge average age of he fores a he end of he plannng horzon. Equaon 1 specfes one of he hree objecve funcons of he problem, namely o maxmze he dscouned ne revenue from he fores durng he plannng horzon plus he dscouned resdual value of he fores. Equaon 2 maxmzes he mnmum amoun of oal area n large, maure fores paches over he me perods n he plannng horzon. The raonale behnd hs objecve s o ensure ha he needs of sensve speces ha requre a mnmum area of conguous old fores haba a any parcular pon n me o survve and o dsperse wll be me by he soluon o he model. Equaon 3 mnmzes he sum of he permeers of hese paches over he enre plannng horzon wh he goal of promong pach shapes ha conan as much neror haba versus edge haba as possble. Our prmary goal wh hese objecve funcons was o provde a realsc example ha demonsraes he mechansms and he ules of he proposed mulcrera mehods. Consran se 4 consss of logcal consrans ha allow only one prescrpon o be assgned o a managemen un, ncludng a do-nohng prescrpon. To preven managemen uns from beng scheduled for harves before hey reach a mnmum harves age, harves varables (X m ) are creaed only for perods durng whch managemen un m s old enough o be harvesed. Consran se 5 consss of harves accounng consrans ha assgn he harves volume for each perod o he harves varables (V ). Consran ses 6 and 7 are flow consrans ha resrc he amoun by whch he harves level s allowed o change beween perods. In he example below, harvess were allowed o ncrease by up o 15% from one perod o he nex or o decrease by up o 3%. Consran se 8 consss of adjacency consrans generaed wh he Pah algorhm (McDll e al. 2002). These consrans lm he maxmum sze of a harves openng, ofen necessary for legal or polcy reasons, by prohbng he concurren harves of any conguous se of managemen uns whose combned area jus exceeds he maxmum harves openng sze. The excluson perod mposed by hese consrans equals one plannng perod. A pah s defned for he purposes of he algorhm as a group of conguous managemen uns whose combned area jus exceeds he maxmum harves openng sze. These pahs are enumeraed wh a recursve algorhm descrbed n McDll e al. (2002). A consran s wren for each pah o preven he concurren harves of all of he managemen uns n ha pah, because hs would volae he maxmum harves openng sze. Ths s done for each perod n whch s acually 120 Fores Scence 55(2) 2009

5 possble o harves all of he managemen uns n a pah. (In he nal perods of he plannng horzon, some of he managemen uns n a pah may no be maure enough o be harvesed.) Consran ses 9 15 are he maure pach sze consrans. Consran ses 9 and 10 deermne wheher or no managemen uns mee he mnmum age requremen for maure paches. These consrans sum over all of he prescrpon varables for a managemen un under whch he un would mee he age requremen for maure paches n a gven perod. O m s equal o 1 f and only f one of hese prescrpons has a value of 1, ndcang ha he managemen un wll be old enough n ha perod. One par of hese consrans s wren for each managemen un n each perod. Consran ses 11 and 12 deermne wheher or no a cluser of managemen uns mees he mnmum age requremen for maure paches. Clusers are defned here as conguous groups of managemen uns whose combned area jus exceeds he mnmum maure pach sze requremen. All possble clusers are enumeraed usng a recursve algorhm descrbed n Reban and McDll (2003b). A cluser mees he age requremen for maure paches n perod f all of he managemen uns ha compose ha cluser mee he age requremen, as ndcaed by he O m varables for he managemen uns n ha cluser. B c akes a value of 1 f and only f cluser c mees he age requremen n perod. These pars of consrans are wren for each cluser n each perod. Consran ses 13 and 14 deermne wheher or no ndvdual managemen uns are par of a cluser ha mees he mnmum age requremen,.e., wheher a managemen un s par of pach ha s bg enough and old enough. Because he clusers may overlap, hs consran se s necessary o properly accoun for he oal area of large, maure pach haba. These consrans say ha a managemen un s par of a pach ha mees he mnmum age and sze requremen for large, maure paches n perod (BO m 1) f and only f a leas one of he clusers belongs o mees he age requremen n ha perod. Consran se 15, workng n concer wh objecve funcon 2, assgns he smalles of he hree oal maure pach areas ha correspond o he hree plannng perods o an accounng varable:. Ths s done by specfyng hrough consran se 15 ha canno be larger han he maure fores pach area n any perod. Objecve funcon 2 maxmzes o ensure ha wll no ake a value ha s less han he smalles of he hree oal haba areas. Consran ses also work ogeher. Consran 16 calculaes he oal permeer of he maure fores paches ha arse n each perod and assgns hs value o accounng varable (denong he oal permeer of he paches n perod ). The sum of he oal permeers over he plannng horzon s mnmzed by objecve funcon 3. Consrans 17 and 18 defne a new bnary varable pq ha subsues for wha would oherwse be a nonlnear cross-produc erm ( pq BO p BO q ) n 16. Consran se 18 s no necessary f objecve funcon 3 s o mnmze he permeer. On he oher hand, f he objecve was o maxmze edge haba, hen consran se 18 would be necessary and 17 could be dropped. Consran 19 s an endng age consran. I requres he average age of he fores a he end of he plannng horzon o be a leas Age T years, prevenng he model from overharvesng he fores. In he example below, he mnmum average endng age was se a 40 years or one-half he opmal economc roaon. Consran ses denfy he managemen un prescrpon, maure pach sze, and he pq varables as bnary. Mehods Ths secon descrbes how he four b-objecvegenerang echnques esed n Tóh e al. (2006) can be generalzed o denfy he effcen se wh respec o hree or more objecves. These echnques are he Alpha-Dela, he -Consranng, he Modfed Tchebycheff, and he Weghed Objecve Funcon mehods. The Alpha-Dela Mehod The Alpha-Dela mehod fnds effcen soluons by progressvely movng from one end of he effcen froner o he oher (Tóh e al. 2006). A slghly sloped weghed objecve funcon s used wh weghs ha are consan hroughou he algorhm. The weghs are normalzed, and he crera values are scaled usng he deal soluon vecor. The relave dfference n he weghs assgned o he respecve objecves s conrolled by he parameer (slope). Ths parameer has o be small enough o no mss any soluons bu mus be greaer han zero o avod domnaed soluons. A each eraon a new effcen soluon s found, and he search space s consraned usng he achevemen vecor correspondng o he new soluon. Consranng he search space for problems wh hree or more objecves s no rval, however. Suppose ha he r-crera problem descrbed n he Model Formulaon secon s solved afer he hree objecves are scalarzed usng he slope parameer () ofhe Alpha-Dela algorhm. Le N 1, H 1, and E 1 denoe he achevemens on objecve funcons 1, 2, and 3, respecvely. As long as 0, he followng wll be rue for he res of he effcen soluons: f 1 (x) N 1 and eher f 2 (x) H 1 or f 3 (x) E 1, where f (x) denoes he value of objecve funcon. The laer wo of hese consrans can be used o resrc he search space for he remanng soluons. Inequaly f 1 (x) N 1 mus hold for any soluon n hs regon, because f f 1 (x) N 1 was rue for any one of he remanng effcen soluons hen ha soluon would have been found a he frs eraon. As long as s small enough, we can rule ou he regon where f 2 (x) H 1 and f 3 (x) E 1 boh hold because f here were a soluon n ha regon ha domnaes (N 1, H 1,orE 1 ), would have had a hgher f 1 (x) han N 1 and should have been found a he frs eraon. To ensure ha he search space s confned o objecve funcon values of f 2 (x) H 1 or f 3 (x) E 1 a he second eraon, he followng se of consrans are added o he orgnal problem. H 1 hab y 1 (24) Fores Scence 55(2)

6 E 1 edge y 2 (25) T y 1 y 2 1 (26) y 1, y 2 0, 1 (27) where he mnmum area of maure fores haba n paches over all perods; he oal permeer of maure fores haba paches n perod ; H 1 achevemen on from eraon one; E 1 achevemen on T from eraon one; hab, edge user-defned, suffcenly small consans; and y 1, y 2 bnary varables ha ensure ha only one of he consrans 24 and 25 s enforced. Consrans ensure ha eher he mnmum area of maure fores haba paches over all perods () s srcly greaer han H 1 or he oal permeer of he paches ( T ) s srcly smaller han E 1. The srcly greaer (or smaller) requremen s needed o avod repeaedly pckng up he same soluon. Ths requremen s acheved by addng suffcenly small consans, hab and edge,ohe bounds on haba area (H 1 ) and permeer (E 1 ). The eher-or relaonshp beween consrans 24 and 25 s acheved by usng consrans 26 and 27, whch requre ha eher y 1 1 and y 2 0 or vce versa. If y 1 1 hen only consran 24 s enforced, and f y 2 1 hen only consran 25 s enforced. The problem s hen solved agan wh hese addonal consrans. Afer each eraon, a new quadruple of consrans lke s added o he formulaon. The process s repeaed unl he problem becomes nfeasble. Fgure 1 llusraes he general mplemenaon of hs algorhm when appled o n-objecve problems. In he frs sep, he deal soluon s denfed, as s needed for scalng and normalzaon. In sep 2, he weghed objecve funcon s generaed wh slope. Ths objecve funcon s hen maxmzed subjec o he orgnal se of consrans (x X). If hs problem s nfeasble, hen he algorhm ermnaes; here are no soluons o he problem. Oherwse he problem s solved, and he aanmen values on he objecves wh he smaller weghs (F k, for P\{j}) are used o buld and add a new se of consrans o he orgnal problem (seps 3 and 4). These consrans wll be smlar o 24 27, excep now here are (n 1) resrced objecves, Fgure 1. The Alpha-Dela algorhm. 122 Fores Scence 55(2) 2009

7 and only one of he n consrans, f (x) F k (for 1,, n bu j), mus hold. Sep 5 checks wheher he newly consruced consran se from sep 4 domnaes any of he prevously consruced ses. If does,.e., f F m F k (for each P\{j} and for m k), hen he domnaed consran se (he one ha was generaed n eraon m) can be elmnaed from he problem. Afer oponally removng he redundan consrans (we found n es runs ha hs sep dd no mprove effcency and herefore we dd no use ), he new IP s solved and he process (seps 2 5) s repeaed unl he problem becomes nfeasble. The -Consranng Mehod The mplemenaon of he -Consranng mehod s very smlar o ha of he Alpha-Dela mehod. The key dfference s ha a each eraon n IPs are solved, as opposed o one (for an n-objecve problem), o guaranee an effcen soluon. Suppose he followng eraon s he kh eraon. Frs, one of he n objecves, say f k 1 (x) s maxmzed whou regard o he res of he objecves. If hs problem s nfeasble, he algorhm ermnaes; no more effcen soluons exs. Oherwse, he problem s solved and he resulng objecve funcon value, F k 1 (x), s recorded. Nex, anoher objecve s maxmzed, say f k 2 (x) subjec o f k 1 (x) F k 1. Ths problem s feasble because we know ha here exss a leas one soluon wh f k 1 (x) F k 1. Call he objecve funcon value of he resulng soluon F k 2. Now, a hrd objecve s maxmzed subjec o f k 1 (x) F k 1 and also o f k 2 (x) F k 2. The process s repeaed unl each of he objecves s maxmzed. When he las objecve, f k n (x), s maxmzed, he res of he objecves are consraned o f k 1 (x) F k 1, f k 2 (x) F k 2,, k k f n1 (x) F n1, where F k ( 1, 2,, n 1) s he objecve funcon value ha was obaned by maxmzng f k (x). The resulng objecve funcon value, F k n, ogeher wh he prevously obaned F k 1, F k k 2,,F n1 consue he aanmen values on he n objecves for effcen soluon k (sep 3). Seps 4 and 5, as well as he soppng rule, are exacly he same as n he Alpha-Dela algorhm. An mporan common characersc of he wo mehods s ha as he soluons are progressvely found along he effcen froner, he aanmen on one objecve gradually ges worse a each new soluon, whereas he aanmen on he oher objecves gradually, alhough no necessarly monooncally, mproves. Ths algorhmc propery can be benefcal n decsonmakng as enables one o fnd effcen managemen alernaves ha are smlar n achevemen values. The Weghed Objecve Funcon and he Tchebycheff Merc-Based Mehods Boh he Weghed and he Tchebycheff mehods make use of an effcen decomposon of weghs when appled o b-objecve problems (Tóh e al. 2006). In he case of he Weghed mehod, hese weghs are assgned o he compeng objecves and he sum of hese weghed objecves s maxmzed. In he case of he Tchebycheff approach, he weghs are assgned o he componens of he Tchebycheff merc, whch measures he maxmum dfference beween he aanmen values of a poenal soluon and ha of he deal soluon. The Tchebycheff merc s hen mnmzed o oban soluons ha are as close o deal as possble. One problem wh usng he Tchebycheff merc s ha may fnd weak Pareo-opma; soluons ha lead o objecve values ha le on he effcen froner bu are no corner pons. In oher words, a leas one of he objecves can sll be mproved. Ths problem s well documened n he leraure and can be overcome by usng he augmened (Seuer 1986, Seuer and Choo 1983) or he modfed (Kalszewsk 1987) verson of he merc. In hs sudy we used he laer approach. Insead of mnmzng he maxmum, we mnmzed he weghed dfferences beween he aanmen values wh a much hgher wegh pu on he dfference n NPV han on he dfference n mnmum haba area or edge lengh. Ths wegh allocaon, whch resuls n a slghly sloped Tchebycheff merc, s kep consan hroughou he decomposon process. Alhough varyng he relave weghs on he compeng objecves or on he componens of he Modfed Tchebycheff merc wll ofen yeld dfferen effcen soluons, s also possble ha wo dfferen combnaons of weghs resul n he same soluon. To mnmze he number of redundan soluons and he amoun of compuer me ha s needed o fnd hese soluons, Tóh e al. (2006) used an algorhm ha decomposes he se of possble normalzed wegh combnaons no secons (lne segmens n he bcrera case) ha correspond o he same effcen soluons (Eswaran e al. 1989). The decomposon for bobjecve problems s based on he fac ha f wo dfferen wegh combnaons yeld he same soluon hen any lnear combnaon of hese weghs wll do so as well. Thus, hese lnear combnaons can be elmnaed from furher consderaon. The decomposon of he wegh space s no as sraghforward wh hree or more objecves. For hree objecves, he se of possble normalzed wegh combnaons can be mapped as a rangle (Fgure 2). The apexes of he rangle represen he combnaons when a wegh of one s assgned o one objecve (or o one componen of he Tchebycheff merc) and zeros are assgned o he oher wo. Ths rangle s llusraed n Fgure 2 wh apexes (1, 0, 0), (0, 1, 0), and (0, 0, 1). The proposed procedure, he Trangles Algorhm, decomposes hs rangle no rangular secons (ndfference regons) ha correspond o he same effcen soluons. A each eraon one rangle s consdered. If he hree wegh combnaons ha represen he hree apexes of he rangle yeld he same soluon, hen no furher decomposon of ha rangle s necessary. Any pon whn he rangle (or, equvalenly, any lnear combnaon of apex weghs) wll yeld he same soluon. If, however, he weghs a he apexes yeld wo or hree dfferen soluons, he rangle mus be dvded no four smaller bu dencally shaped subrangles. These are (( 1 2, 1 2, 0), (0, 1 2, 1 2), ( 1 2,0, 1 2)), ((1, 0, 0), ( 1 2, 1 2, 0), ( 1 2, 0 1 2)), (( 1 2, 1 2, 0), (0, 1 2, 1 2), (0, 1, 0)), and ((0, 0, 1), (0, 1 2, 1 2), ( 1 2, 0, 1 2)) n Fgure 2. The apexes of he subrangles are eher dencal o one of he apexes of he paren rangle ((1, 0, 0), (0, 1, 0), and (0, 0, 1)) or are consruced as he mean of he wo of hose Fores Scence 55(2)

8 Fgure 2. The rangular decomposon of he wegh space for he Weghed and Tchebycheff mehods. Each pon on he rangle represens a wegh combnaon ha sums o 1. apexes. If wo of he hree soluons from he paren rangle was he same, e.g., weghs (1, 0, 0) and (0, 1, 0) boh yelded he same soluon, hen wegh combnaon ( 1 2, 1 2, 0), whch s a lnear combnaon of (1, 0, 0) and (0, 1, 0), can be assgned ha soluon as well. There s no need o solve he problem wh weghs ( 1 2, 1 2, 0). A he nex eraon, one of he four subrangles s seleced, and he same process as n he frs eraon s followed. The weghs ha defne he apexes of he subrangle are appled o he objecves of he problem (or o he componens of he Tchebycheff merc, dependng on whch mehod s used). I s enrely possble ha one of he wegh combnaons correspondng o one of he apexes of he subrangle has already been appled o he problem and solved a a prevous eraon, eher as par of he larger rangle or when an adjacen rangle was explored. In hs case, here s no need o solve he problem wh hese weghs agan. The soluon from he adjacen rangle can be used n he comparsons needed o deermne wheher he curren subrangle should be furher decomposed. Those of he hree problems ha have no been solved before or are no lnear combnaons of oher weghs ha yeld he same soluon are hen solved, and her soluons are compared wh he soluons a he oher apexes. The algorhm ermnaes eher when here are no more subrangles lef o decompose or he larges dfference beween he wegh combnaons ha correspond o he apexes of he remanng subrangles ha could poenally requre decomposon s smaller han hs predefned lm: a mnmum mesh or rangle sze. The followng noaon s used o llusrae he mechansm of he Trangles Algorhm. Le T be he se of acve (unexplored) rangles. Le w 11 W w 21 w 31 w 12 w 22 w 32 w 13 w 23 w 33 denoe he weghs assocaed wh he apexes of rangle T and Max(w 11 w 21, w 11 w 31 ) denoe he deph of. Ths laer merc, deph descrbes he sze of a gven rangle and s used n he algorhm o denfy he 124 Fores Scence 55(2) 2009

9 larges rangles. The greaer he value of, he larger rangle s. The algorhm decomposes he larges acve rangles frs. Lasly, le F k (for k 1, 2, 3) denoe he objecve funcon values ha correspond o he soluons of problems P k Max{w k1 f 1 (x) w k2 f 2 (x) w k3 f 3 (x) :x X} for k 1,, 3, respecvely, where f 1 (x), f 2 (x), and f 3 (x) are he objecve funcons. Seps 1 and 2 are he nalzaon phase of he algorhm. Sep 1 s o oban he deal soluon, whch s needed o scale and normalze he weghs for boh mehods. The mnmum mesh sze parameer,, s also defned (by he user) o lm he sze of he rangles o be decomposed. A hs pon, se T s empy. Sep 2 s o add he frs rangle o he ls of acve rangles (se T). Ths rangle s he so-called paren rangle whose apexes represen wegh combnaons (1, 0, 0), (0, 1, 0), and (0, 0, 1). The soluons o hese sngle-objecve problems have already been obaned n sep 1 when he deal soluon was denfed. A he begnnng of each eraon, se T s checked. If se T s empy, he algorhm ermnaes. If se T s nonempy, hen one of he larges rangles, say rangle, s seleced. If s smaller han he predefned or he soluons ha correspond o he apexes of are dencal, hen s removed from se T. Oherwse, four new rangles are creaed ( T1, T2, T3, and T4 ) wh he followng weghs on he apexes (sep 3): W T1 1 W T2 W T3 2w w w 11 w w 12 w w 13 w 23 w w 22 w w 23 w 33 w w 12 w w 13 w 33, 2w 11 w w 12 w w 13 w w 21 w w 22 w w 23 w 33, w 21 w 22 w 23 paren apexes ha led o dencal soluons. Fnally, he four new rangles are added o se T (sep 5), and he process sars all over agan by selecng anoher rangle. A Case Sudy To demonsrae he four mehods as hey generae he effcen se wh respec o hree objecves, a hypohecal fores plannng problem, he same as n Tóh e al. (2006), was used. Ths fores conssed of 50 managemen uns and could be consdered slghly overmaure, because approxmaely 40% of he area s beween 60 and 100 years old and he opmal roaon s 80 years (Fgure 3). The average managemen un sze was 18 ha, and he oal fores area was 900 ha. A 60-year plannng horzon was consdered, composed of hree 20-year perods. The four possble prescrpons for a gven managemen un were o harves he managemen un n perod 1, perod 2, or perod 3, or no a all. The mnmum roaon age was 60 years. A maxmum harves openng sze of 40 ha was mposed, and groups of conguous managemen uns were allowed o be harvesed concurrenly as long as her combned area was less han he maxmum openng sze. All managemen uns were smaller han he maxmum harves openng sze. The wldlfe speces under consderaon was assumed o need haba paches ha are a leas 50 ha n sze and a leas 60 years old. Because he mnmum pach sze was greaer han he maxmum harves sze, hese paches had o be composed of more han one un. We also specfed n he IP formulaon ha a leas one haba pach mus develop over he 60-year plannng horzon. Ths mean ha no effcen managemen alernaves were sough below 50 ha of maure fores pach producon. The r-crera IP descrbed n he Model Formulaon secon was bul for hs hypohecal fores, resulng n a model wh 4,794 consrans and 2,412 varables. Table 1 shows he dsrbuon of he consrans by consran ypes. The algorhms nroduced n he Mehods secon were mplemened usng a wo-hread CPLEX 11.1 (ILOG 2w 11 1 w 11 w w 11 w w 12 w 12 w w 12 w w 13 w 13 w w 13 w 33, W T4 w w 21 w w 11 w 31 w w 22 w w 12 w 32 w w 23 w w 13 w 33, The nex sep s o generae and solve 12 problems (P T1, P T2, P T3, and P T4 for 1, 2, 3) wh he wegh combnaons ha correspond o he apexes of he four rangles (sep 4). A mos, only 3 of he 12 problems would have o be solved, because he same wegh combnaons are assgned o more han one apex (Fgure 2). Furhermore, f any par of apexes n he paren rangle led o he same soluon, hen any lnear combnaon of hese apexes wll do so as well. There s no need o solve for a new apex f corresponds o he lnear combnaon of Fgure 3. The es fores. The fgures n each polygon denoe he harves un denfcaon number and he nal age class of he un. For example, 1 represens he 0 20 year age class, 2 represens 21 40, and so on. Darker polygons represen hgher nal age classes. Fores Scence 55(2)

10 Table 1. Tes problem sze parameers: he number of consrans s lsed for each consran ype Consran ype (equaon number) No. of consrans and 7 2 each and each 11 and 12 1,617 each 13 and each 15 and 16 3 each 17 and each 19 1 Toal 4,794 CPLEX 2008) on a dual processor Inel XEON CPU 3.00 GHz compuer wh 3.25 GB of RAM under a Wndows plaform (Mcrosof Wndows XP Professonal Verson 2002, Servce Pack 3). Programs o auomae he algorhms were wren n Mcrosof Vsual Basc 6 and.ne 2005 usng he ILOG CPLEX Callable Lbrares. The relave MIP gap olerance parameer (opmaly gap) was se o 0 and he negraly olerance parameer was se o 1.e07 ( %). These src sengs were needed o avod domnaed soluons and o make sure ha he numercally sensve algorhmc consrucs, such as he eher-or consrans n he Alpha-Dela and he -Consranng mehods and he compose weghed expressons n he oher wo echnques, would work properly. The workng memory lm was se o defaul 1 MB. The parameers of he Alpha-Dela and he -Consranng algorhms,, hab, and edge,were se o 1, 0.01 ha, and 0.47 m (1 pxel), respecvely ( only apples o he Alpha-Dela mehod). The laer wo sengs nsruc he algorhms no o look for soluons ha smulaneously lead o a less han 0.01-ha dfference n pach area producon and less han 0.47 m n edge producon. We red smaller values for hese parameers bu found no change n he effcen se. The deph parameer n he Trangles algorhm () was se o zero for boh he Weghed Objecve Funcon and he Modfed Tchebycheff merc-based approaches meanng ha runnng me was he only consran for hese algorhms o fnd effcen soluons. We se he parameers for each echnque so ha he hghes number of effcen soluons would be found gven a me lm. Alhough rgorous parameer unng was ousde of he scope of hs sudy, we made an aemp o opmze he performance of he algorhms. Our prmary goal was o provde an nsgh for he reader abou he pros and cons of he mechansms of he proposed mehods. The expermen addressed he followng quesons: (1) How many of he effcen soluons can each algorhm denfy whn 20 hours of compuer me? (2) How evenly are hese soluons dsrbued along he effcen froner? (3) How easly can a user fler he soluons n lne wh he DM s neress? and (4) How easly do he mehods generalze o he n-objecve case? Resuls and Dscusson Number and Dsrbuon of Effcen Soluons The -Consranng mehod found he hghes number of Pareo-opmal managemen alernaves (99) whn he prese me nerval of 20 hours. The Alpha-Dela mehod found 97, he Tchebycheff mehod found 76, and he Weghed mehod found 35 soluons. Fgures 4 and 5 graph he soluons n he objecve space: Fgure 4 n a hree- Fgure 4. Effcen alernaves found by he four echnques. Mulshaded markers ndcae soluons ha were found by more han one algorhm. 126 Fores Scence 55(2) 2009

11 Fgure 5. Effcen alernaves n wo-dmensonal projecons. Mulshaded markers ndcae soluons ha were found by more han one algorhm. dmensonal renderng and Fgure 5 n wo-dmensonal projecons. The soluon mes are summarzed n Fgure 6, n whch he man dagram dsplays he cumulave soluon mes for each mehod. The four smaller chars show he ndvdual soluon mes ha were requred o fnd each new effcen soluon. Noe ha because he -Consranng mehod fnds each new soluon n hree seps, he Alpha- Dela mehod n one sep, and he Tchebycheff and he Weghed mehods n many seps, he ndvdual soluon mes do no necessarly correspond o ndvdual IPs. One common rend ha can be seen from he dagrams s ha, afer a very producve nal phase, fndng new effcen soluons became ncreasngly me-consumng for each mehod. We provde an explanaon for hs rend n he dscusson ha follows. The Alpha-Dela and he -Consranng mehods found soluons mosly on one sde of he effcen froner, whereas he soluons denfed by he oher wo mehods were more evenly dsrbued. Even hough he Weghed and especally he Tchebycheff mehods provded a beer overall esmaon of he froner, he -Consranng and he Alpha-Dela mehods descrbed one par of he froner n more deal. The man reason for he dfference s ha whereas he Alpha-Dela and he -Consranng mehods worked gradually off of one sarng pon and found soluons sequenally, he oher wo mehods found more evenly dsrbued soluons as hey sar ou wh soluons ha are as conrasng wh respec o her assgned weghs as possble. I s mporan o pon ou, however, ha boh he Alpha-Dela and he -Consranng mehods can be Fores Scence 55(2)

12 Fgure 6. Soluon mes. The lne graph shows he cumulave and he bar graphs show he sequenal ndvdual soluon mes for each effcen soluon and for each of he four algorhms. nsruced o fnd soluons ha are more separaed from each oher along he froner by assgnng hgher values o parameers hab and edge. All mehods excep he Weghed Objecve Funcon mehod are capable of denfyng nonsuppored Pareoopma (nonsuppored Pareo-opma are effcen soluons ha are no corner pons of he convex hull of he effcen se) (Tóh e al. 2006). As a resul, he hree mehods can explore he effcen froner n more deal han he Weghed mehod. However, as he Alpha-Dela and he -Consranng mehods explore he effcen froner sarng from one end (from he hghes levels of NPVs) and hey requre a se of eher-or consrans wh assocaed bnary varables o be added o he problem a each eraon, he IP ha needs o be solved becomes ncreasngly hard as he algorhms proceed (see boom chars n Fgure 6). The srucure of he r-crera fores managemen plannng problem used n hs expermen mgh also accoun for hs ncreasng combnaoral complexy. Opmal soluons ha lead o greaer amouns of maure fores haba n large paches and o lesser amouns of mber revenues mgh be harder o denfy. Ths s because a hgher number of spaal arrangemens of maure fores haba exs f larger areas are allowed and hs larger se mus be evaluaed n he opmzaon process o fnd opmal soluons. Dependng on when he IP becomes oo me-consumng o solve and wha me consrans are mposed, he Alpha-Dela and he -Consranng mehods mgh or mgh no be able o scan he enre effcen froner. One way o mgae hs problem s o run hree separae algorhms, each sarng from a dfferen end of he froner. The Alpha-Dela mehod can be nsruced o work s way off he hghes possble level of mnmum haba area or from he lowes possble lengh of permeer. Ths way, only he cenral par of he effcen 128 Fores Scence 55(2) 2009

13 froner would have o be explored usng a larger burden of eher-or consrans, and he perpheral soluons mgh be found relavely easly. Ths combned algorhm can sop once he hree subalgorhms (n subalgorhms for an n- objecve problem) mee somewhere n he mddle of he effcen froner,.e., when hey fnd an dencal effcen soluon. Of course, here s no guaranee ha he IPs would no become nracable before hese subalgorhms mee. Ths combned approach, however, should ncrease he number and dversy of effcen soluons found. Alhough he Weghed mehod found only 35 soluons, mgh sll be a preferred alernave o solve mulobjecve IPs f he orgnal problem has a specal srucure (e.g., oal unmodulary) (Wolsey 1998) ha would be desroyed by usng oher mehods (e.g., ReVelle 1993). In hese cases, desroyng hs srucure o fnd he nonsuppored Pareoopma and hus beer descrbng he effcen froner mgh no be worhwhle. Toally unmodular or oher specal srucures are unlkely, however, n realsc fores plannng problems n whch n mos cases a large varey of complcang consrans need o be mposed n he model (Tóh e al. 2006). In concluson, as n he bcrera case, a combned approach can be recommended. The fores planner could use he Weghed or he Tchebycheff mehods o oban a welldsrbued effcen se and presen he resuls o he DM. Then, n lne wh he DM s neress, one area of he effcen froner could furher be explored usng he -Consranng or he Alpha-Dela mehod. Ths approach would ake advanage of he algorhmc dfferences n each mehod. Flerng he Effcen Soluons Besdes usng he Weghed or he Tchebycheff mehods as nal flers, here are several oher, ndrec ways o conrol he spacng of he effcen soluons along he effcen froner wh he proposed algorhms. Flerng he effcen se mgh be advanageous because: can sgnfcanly reduce compung me and a well-dsrbued subse of he effcen froner mgh provde a suffcen pool of alernaves for he DM o choose from or o gude hm or her o furher explore a parcular subregon of neres along he froner. Increasng he values of parameers, hab and edge n he Alpha-Dela algorhm and parameers hab and edge n he -Consranng algorhm wll ncrease he spacng of he effcen soluons n he objecve space. The sengs of hab and edge wll ensure ha no wo soluons wll be found whose achevemens n erms of mnmum haba area and edge lengh are boh whn hab and edge, respecvely. The spacng of he effcen soluons wh he Weghed Objecve Funcon and he Tchebycheff merc-based approaches can be conrolled o some exen by adjusng he mnmum mesh sze parameer n he Trangles algorhm. In pracce, mgh be useful o sar wh a large mesh sze and cover he enre wegh space and hen focus he search o a subregon of neres usng a smaller mesh sze n ha regon. How large should he nal mesh sze be? In our es case, boh he Weghed and he Tchebycheff mehods were processng rangles a he [ 1 128] level afer 20 hours of compung me. The Weghed mehod decomposed 88% and he Tchebycheff mehod 77% of hese rangles. Generalzaon o he n-objecve Case One addonal advanage of he Alpha-Dela and he -Consranng mehods over he oher wo approaches s ha her generalzaon o he n-objecve case s farly sraghforward, a leas from a echncal, neger programmng pon of vew (see Fgure 1). Generalzng he Trangles Algorhm o he n-objecve case s no as sraghforward as wh he above wo mehods. Insead of he rangles n he r-objecve case, n-dmensonal polyhedra (erahedra n he four-objecve case) would have o be decomposed no n-dmensonal subpolyhedra. Because each n-dmensonal polyhedron has n apexes, n new problems would have o be solved and compared (n! parwse comparsons) a each eraon. Ths ncreased compuaonal burden, however, mgh be offse by he fac ha he IP subproblems (he problems ha are solved a he apexes of he n-dmensonal polyhedra) are smpler han hose of he Alpha-Dela- or he -Consranng mehods. Unlke he feasble regon of he laer wo mehods, he feasble regon of he IP subproblems n he Trangles algorhm s consan, only he weghs on he objecves (or he componens of he Tchebycheff merc) change. Ecologcal and Managemen Implcaons The radeoff nformaon generaed for he hypohecal es problem demonsraes he uly ha one can expec from he proposed echnques n real applcaons. Lookng a Fgure 4 or he dagram n he cener of Fgure 5, one can conclude, for nsance, ha mnmum maure fores haba pach producon coss roughly $70,000 $100,000 n erms of forgone mber revenues for every 50-ha ncrease beween he 50-ha requred mnmum and he 170-ha poenal maxmum. An exra $100,000 $200,000 s needed f mnmum boundary paches are desred. The vercal cluserng of effcen soluons around some mnmum haba area hresholds, such as 135, 150, or 170 ha (Fgure 5, boom), mples ha he fores manager would have some flexbly (as much as $200,000 a he 170-ha level) (Fgure 5, op) o smulaneously generae maure fores paches as well as mber revenues. The poenal losses or gans assocaed wh edge producon are raded off agans poenal gans or losses n mber revenues when one swches beween managemen alernaves whn hese clusers. Ths resul demonsraes he benefs of he mulcrera echnques n ha hey offer a range of choces whenever possble and whou he DM s a pror bas ha would oherwse manfes n he form of arges or hard consrans f alernave approaches such as goal programmng were used. The vercal cluserng around some mnmum haba area hresholds mgh have landscape ecologcal mplcaons as well. Consder, for nsance, he cluser around he 170-ha level one more me (Fgure 5, boom). Sarng a Fores Scence 55(2)

14 he opon ha leads o he leas amoun of NPV, approxmaely $2 M (see he lowes pon n he cener dagram n Fgure 5), a far amoun of addonal harvesng s possble o acheve as much as $2.18 M n mber revenues whou havng o forgo any of he 170 ha of mnmum maure fores haba producon. The only loss s he ncrease n permeer-area raos of he paches. Once, however, anyhng beyond he $2.18 M s desred, he exra harvess ha would be needed would cause a sgnfcan drop n mnmum maure fores haba producon. One would have o swch o he 150-ha cluser o fnd more profable alernaves. Are hese dscree drops beween he clusers characersc of hs parcular problem? The fores planner would benef from knowng abou hresholds for whch small amouns of change n harves nensy can cause sgnfcan losses n haba srucure or coheson. The echnques proposed n hs arcle can help n denfyng hese hresholds. Fnally, any cluserng of soluons n he objecve space can carry sgnfcan value for he DMs. Clusers of soluons ha are smlar n erms of achevng he objecves ha are explcly ncorporaed n he model can be analyzed n erms of her conrbuon o a fourh or ffh objecve. Alhough here s no guaranee ha any of he pons n he cluser would be Pareo-opmal wh respec o an addonal creron, hs creron mgh help he DMs elmnae some soluons from he pool. Numercal Issues I s mporan o pon ou ha he algorhms proposed n hs sudy buld on numercally sensve consrucs such as he eher-or consrans n he -Consranng and he Alpha-Dela mehods or he compose objecve funcons n he oher wo mehods. Some of hese consrucs do no funcon properly f he opmaly and negraly olerance parameers n he IP solvers are no se gh enough. If a weghed objecve funcon s used, for example, he smalles amouns of subopmaly mgh lead o soluons ha are oally rrespecve of he assgned weghs. Consder he example n Fgure 7: alhough he weghed objecve funcon (dashed lne) n hs case n maxmzed a pon A, usng a small opmaly olerance gap could easly lead o Pon D. I s easy o see how hs loss of opmaly can make he rangular decomposon process dysfunconal. The assumpon ha he lnear combnaon of wo ses of weghs Fgure 7. Subopmaly and mulple domnance. ha lead o dencal soluons would also lead o he same soluon does no hold anymore. Anoher obvous consequence of subopmaly s ha he mulobjecve echnques mgh produce domnaed soluons. The only way o avod hs suaon s o se he opmaly olerance gap as small as possble, whch has he assocaed cos of ncreased compung me. Conclusons The prmary value of generang he se of effcen soluons o fores managemen problems s o help DMs acqure a more holsc undersandng of he problem by provdng nformaon abou he radeoffs, he producon possbles, and he degree of ncompably beween compeng objecves. Ths undersandng should faclae selecng he bes compromse managemen alernaves. We presened four ways o generae he se of Pareoopmal soluons o spaal fores plannng problems wh hree or more compeng objecves. Alhough generalzng he b-objecve algorhms o hree objecves s no rval, generalzng he proposed r-crera algorhms o handle problems wh four or more objecves s mehodologcally sraghforward. The resuls from one es of he four algorhms sugges ha here s no clear wnner n erms of compuaonal performance: each mehod has posves and negaves. Gven he algorhmc dfferences behnd he echnques and he snapsho of compuaonal resuls, we can conclude ha a combned ulzaon of he benefcal properes of eher he Weghed or he Modfed Tchebycheff mehods and eher he -Consranng or he Alpha- Dela mehods would probably work he bes n pracce. Use of one of he former wo mehods o generae an nal rough esmae of he radeoffs can be followed up by use of one of he laer wo echnques o fnd furher soluons n he DM regons of neres. Applyng he four mehods o larger problems mgh be compuaonally expensve f, as n hs sudy, ruly opmal soluons are sough. Reducng he opmaly olerance,.e., accepng subopmal soluons s ceranly an opon f compung me s a consran: he effcen froner could be approxmaed n a fracon of he me ha would be needed o provde an exac represenaon. Because of he numercally sensve naure of mulobjecve programmng, however, some of he proposed echnques mgh exhb adverse algorhmc behavors leadng o domnaed soluons or o oo few effcen soluons. How much opmaly can be forgone whle sll provdng a meanngful represenaon of he radeoff froner for he DMs? Ths queson can only be answered by comprehensve compuaonal expermens ha explore he radeoffs beween soluon mes and Pareo-opmaly on larger fores plannng problems. In he meanme, small-scale, plo applcaons of he echnques wh real DMs sll reman mporan as hey can answer quesons such as how mporan s o fnd nonsuppored Pareo-opma, how o vsualze and presen he alernaves o he DMs, and (3) o wha exen can hese mehods promoe consensus beween mulple sakeholders. By he me hese quesons are answered, our compuaonal capables may mprove o a degree ha 130 Fores Scence 55(2) 2009