RESOLUTION OF THE LINEAR FRACTIONAL GOAL PROGRAMMING PROBLEM

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1 Revsa Elecrónca de Comuncacones y Trabajos de ASEPUMA. Volumen Págnas 7 a 40. RESOLUTION OF THE LINEAR FRACTIONAL GOAL PROGRAMMING PROBLEM RAFAEL CABALLERO Unversdad de Málaga / Deparameno de Economía Aplcada Maemácas Campus El ejdo s/n Málaga MÓNICA HERNÁNDEZ Unversdad de Málaga / Deparameno de Economía Aplcada Maemácas Campus El ejdo s/n Málaga Recbdo 8/0/00 Revsado 8/04/00 Acepado 0/05/00 RESUMEN: En ese rabajo se resuelve el problema de Programacón por Meas cuando dchas meas adopan una formulacón fracconal lneal. La gran dfculad de ese po de problemas radca en las resrccones no lneales que aparecen al resolver el correspondene problema de programacón asocado al problema de mea Cuando esen solucones alcanzando odas las meas, el problema es fácl de resolver a ravés de la resolucón de un problema lneal asocado. En ese rabajo nos cenramos en el caso en el que no odas las meas se consguen sasfacer. En dcho caso, enconramos la solucón que más se acerca a los nveles de aspracón esablecdos por el decsor desde odas las apromacones del problema de meas que se conemplan en la leraura: Ponderadas, Mnma y Lecográfca Cada seccón se ermna con un análss de sensbldad de dchos nveles de aspracón. Palabras claves: Programacón Maemáca No Lneal, Programacón Fracconal, Programacón Mulobjevo, Programacón por Mea ABSTRACT: Ths work deals wh he resoluon of he goal programmng problem wh lnear fraconal crera. The man dffculy of hese problems s he non-lnear consrans of he mahemacal programmng models ha have o be solved. When here es soluons sasfyng all arge values, he problem s easy o solve by solvng a lnear problem. So, n hs paper we deal wh hose nsances where here s no guaranee such soluons es, and herefore we look for hose pons n he opporuny se closes o he arge value Ths sudy has been done akng no accoun all he dfferen approaches avalable for solvng a goal programmng problem, creang soluon-search algorhms based on hese approaches, and performng a sensvy analyss of he arge value Keywords: Non Lnear Programmng, Fraconal Programmng, Mul-objecve Programmng, Goal programmng. 7

2 8 Rafael Caballero, Mónca Hernández. Inroducon The man neres n fraconal programmng was generaed by he fac ha a lo of opmzaon problems from engneerng, naural resources and economcs requre he opmzaon of a rao beween physcal and/or economc funcon Such problems, where he objecve funcons appear as a rao or quoen of oher funcons, consue a fraconal programmng problem. When hese quoens have o verfy ceran arge values, we would have a se of fraconal goal In goal programmng GP problems s assumed ha he decson-maker gves up opmzaon whle he/she esablshes arge values for each objecve. When he levels are verfed, he epecaons or desres of he decson-maker are sasfed. In addon, he decson-maker usually ses a seres of preferences regardng he objecves and orders hem accordng o her relave mporance. In GP problems, when he objecves are lnear fraconal funcons, he formulaon of he correspondng goal problem o be solved s raher comple due o s nonlnear consran The resoluon of hs ype of problems by drec lnearzaon s no feasble []. In [], Hannan presens a characerzaon of he condons for he lnear problem such ha s equvalen o he orgnal fraconal problem. However, Soyser and Lev [3] argued ha Hannan's resul was erroneous by usng a counereample. These auhors developed a es problem whch, once solved, could be used o fnd ou wheher we are dealng wh equvalen problems or no. See [4] for furher nformaon and references abou hs ssue. Afer hese sudes, and wh he ecepon of he work of Kornbluh and Seuer [5], here have been very few references of goal programmng wh fraconal goal Fnally, Aude e al [6] have used global opmzaon echnques o solve hs problem. In hs paper, we propose soluon algorhms for he goal programmng problem where he goals ake a lnear fraconal form under all goal programmng formulaon Praccal eamples of fraconal goal programmng problems occur n many decson problems where he crera are epressed as a rao of wo gven funcons such as prof/capal, cos/me, cos/volume, oupu/npu, ec. Thus, hs model has mporan applcaons n areas such as fnance, ransporaon, nformaon heory, foresry managemen, agrculural economcs, educaon, resource allocaon and ohers see [7] and [8]. Recenly, he fraconal problem have appeared n envronmenal and naural resources problems [9], [0], [], among ohers. Prevous work by Caballero and Hernández [] shows ha by usng a smple lnear es s possble o deermne beforehand he esence or absence of soluons ha sasfy all he goals of he problem under sudy. Here we assume ha no all goals can be sasfed and solve he lnear fraconal goal programmng problem drecly. Then we carry ou a sensvy analyss for he arge values se by he decson-maker, ha have been oo consrcve. As sad, n GP problems he decson-maker esablshes arge values for each objecve. In general, he purpose of GP s o reach hese arge values as closes as

3 Resoluon of he Lnear Fraconal Goal Programmng Problem 9 possble, mnmzng he devaon beween he arge value and he level acheved for he correspondng arbue. The decson-maker s preferences regardng he achevemen of he arges may be ncorporaed n hs scheme followng dfferen way Thus, here are several GP varans such as weghed GP, mnma GP and lecographc GP, among oher Weghed GP mples a mahemacal cardnal specfcaon of weghs for each goal and seeks for a soluon ha mnmzes he weghed sum of all he unwaned devaon varable The mnma GP approach mnmzes he mamum devaon from he arge Fnally, Lecographc GP nvolves a preempve, ordnal weghng of goals ha allocaes hem no prory levels and mnmzes he unwaned devaon varables n a lecographc order. Formally, weghed GP and mnma GP can be vewed as a parcular case of lecographc GP wh a sngle prory level. Therefore, whou loss of generaly, we assume ha we are dealng wh he laer approach. A lecographc GP problem can have a sngle goal or more whn each prory level. Ths paper sudes and solves he model generaed n boh nsance When here s more han one goal a a sngle prory level, we wll ake no accoun whch approach.e. weghed or mnma has been used o aggregae he goals of he level. When approprae, a worked eample s ncluded n he secon n order o asss he undersandng of he algorhm. Thus, ne secon deals wh he case of a sngle goal for a ceran prory level. Followng secons descrbe nsances of more han one goal whn a gven prory level. In Secon 3, we assume ha goals were added usng he weghed approach, and n Secon 4 s assumed ha he mnma approach was he mehod chosen for addng goals n he level. Fnally, we summarze he mos sgnfcan conclusons of hs work followed by he reference. One goal per prory level We assume ha our problem has p lnear fraconal objecves and a consran se ha s a conve polyhedron. Whou loss of generaly, we assume ha he decsonmaker mposes a mnmum arge value for each objecve. Thus, he unwaned devaon varables are he negave ones, ha s, n for,, p. In he presen secon, we assume ha, usng he Lecographc GP approach, he prory levels are mposed n such a way ha n a gven level s here s only one goal, correspondng o he -h objecve. In hs case, n level s, where he nde se of he goals n s wll be denoed by N s he problem o be solved s as follows: mn n A b j f j u j g f n p u g, n, p 0 j N,..., N s

4 30 Rafael Caballero, Mónca Hernández where A M mn R and b R m. Le X s { R n / A b, 0, f j g j u j j N,..., N }, whch s ncludes he consrans lnear ha mpose he sasfacon of goals n he prevous levels; and f j c j α j, g j d j β j where c j, d j R n, α j, β j R, j,..., p. In addon, we assume ha g j, j,..., p are srcly posve for every X. Le us call f g ϕ j,...,p. j j j I s clear ha problem s no easy o solve due o he non-lnear consran correspondng o he goal we are aempng o sasfy. Takng no accoun he followng assocaed problem, mn n ' X f g u n ', p ' 0 s n ' p ' 0 Awerbuch e al n [] showed ha he soluon of he lnear problem * s no necessarly he soluon of. However, n [] s shown ha f he soluon of n he opmum s zero, hen hs pon wll be also he pon soluon of. In he oher case, where he opmum of s n * > 0, he followng heorem deermnes he soluon pon of. Theorem.. Le ** be he soluon of he sngle-objecve lnear fraconal problem ma ϕ If when solvng he soluon s such ha n * > 0, hen **, n **, 0 s he soluon of where n ** u ϕ **. Proof. On he one hand, he consrans of are verfed by pon **, u ϕ **, 0 because: ** X s ϕ ** u ϕ ** 0 u. By hypohess, he goal s no verfed n any pon of X s. Therefore s no verfed n **, whch means ϕ ** < u. Tha s, n ** u ϕ ** > 0. Obvously p ** 0 0. On he oher hand, hs pon also mnmzes he value of n among hose ha verfy he consran By reduco ad absurdum, suppose *, n *, p * s a soluon of. Accordng o our hypohess and Theorem n Caballero and Hernández [], gven ha n * > 0, hen also n * > 0, ha s, u > ϕ *. Gven ha hs pon has o verfy he X s

5 Resoluon of he Lnear Fraconal Goal Programmng Problem 3 consrans of he problem, hen necessarly ϕ * n * p * u, where n * > 0 and p * 0, and herefore, p * 0. In oher words ϕ * n * u, whch means ha n * u ϕ *. However, assumng ha *, n *, 0 s a soluon of would mean ha n * < n **. Tha s, u ϕ * < u ϕ **. Ths would mean ha ϕ * > ϕ **, where ** s he mamum of he funcon ϕ n X s. Ths s a conradcon and hus pon **, n **, 0 s he soluon of. As sad n he nroducon, Soyser and Lev [3] desgned a es problem ha can be used o esablsh he equvalence beween problems and. If he opmal value of such a problem s zero, he equvalence beween soluons for and s guaraneed. Usng our noaon, hs es problem s as follows: Le *, n *, p * be he soluon of wh n * > 0. Then, f he opmal soluon of he problem ma f ϕ * g X s s zero, hen * also solves. So, Theorem s no more han a generalzaon of Soyser and Lev resul because all hs es problem does s o use he mehod of Dnkelbach [3] o verfy wheher * s or s no he soluon of ma ϕ Once he goal programmng problem has been solved, makes sense o perform a sensvy analyss of he arge values whch have been se. In hs sense, he soluon pon of problem self offers he sensvy analyss we are lookng for and provdes us wh an nerval whn whch we can be sure ha here are pons whch sasfy he - h goal. Indeed, f ** s he soluon of, wh n ** > 0, ** s he pon of X s ha mnmzes he unachevemen of he goal and hs mnmum unachevemen s precsely n **. Therefore, he sensvy nerval s, u n **]. Thus, gven a arge value for he goal whch s whn he sad nerval, we can guaranee he esence of a leas one pon ha wll sasfy such a goal sasfyng prevous levels. On he oher hand, f he arge value s ou of he nerval, we guaranee ha he goal s no gong o be sasfed. X s 3. More han one goal per prory level. Weghed GP approach. In hs secon we wll solve he followng problem:

6 3 Rafael Caballero, Mónca Hernández mn X f n p u g n, p k λ n s 0,..., k,..., k 3 Where parameer λ represens he wegh of each goal n he weghed GP k approach. So hese weghs mus verfy ha λ, λ 0,, k. In order o solve problem 3, we consder s assocaed lnear problem mn X λ n' f g u n' p' 0 n', p' 0 k s,..., k,..., k 4 Smlarly o he case of he prevous secon, f when solvng 4 we oban a k soluon such ha λ n ' * > 0, n such a case here s no soluon ha sasfes all goals k n such prory level and he value λ n' * does no necessarly provde us wh he mnmum unachevemen n X s of he goals n he curren prory level. Le us solve problem 3 drecly, ha s, we seek a feasble soluon X s whch mnmzes he weghed sum of devaons of he fraconal crera from her arge value Gven an X s, for every {,..., k}, when consran f g u n p 0 s verfed, we have: f c u d α u β > 0, hen n c u d α u β f c u d α u β 0, hen n 0 Gven ha n n / g, hen we can esablsh he followng epressons of he varables n n relaon o, for a gven whn and k: f c u d α u β > 0, n c u d α u β / d β 5 f c u d α u β 0, n 0 In hs way, o fnd he soluon pon of 3, we jus have o mnmze n X s he weghed sum of epressons 5. However, n order o do hs, we frs have o esablsh he goals ha can gve rse o wo dfferen epressons of n whn X s. These goals are he ones ha verfy { / ϕ u } X s and spl he se X s no wo subess: for one of hem, n 0; whle for he oher, n s gven by epresson 5. To locae hose goals we propose solvng for each wh k a GP problem wh a sngle goal,

7 Resoluon of he Lnear Fraconal Goal Programmng Problem 33 where he achevemen funcon s hn, p n p, and hen check wheher he value of funcon h n he opmal soluon s zero. Le us assume ha ou of he k goals n he curren level, he frs m goals fulfl such requremen, where m k. We consder a paron of X s ha ncludes all possble combnaons of he consrans ϕ u on he one hand, and of ϕ u on he oher for,..., m. The consrans ha have o be added o hose esng n X s o creae he subses of he paron are showed n Table : Table. Consrans o be added o X s Subse Consrans o be added o he consrans of X s Y ϕ u, ϕ u, ϕ 3 u 3,..., ϕ m u m Y ϕ u, ϕ u, ϕ 3 u 3,..., ϕ m u m Y m ϕ u, ϕ u, ϕ 3 u 3,..., ϕ m u m ϕ u, ϕ u, ϕ 3 u 3,..., ϕ m u m Y m Y m ϕ u, ϕ u, ϕ 3 u 3,..., ϕ m u m Ths paron of X s s made up of, a mos, m polyhedral subses, where some of hese subses mgh be he empy se. To fnd he soluon of 3, we propose o mnmze he weghed sum of he correspondng epressons of n for each of hese subse Therefore, he followng m problems have o be solved, for each r from o m : c ud α uβ mn λ d β I r Y r where I r {,..., k such ha ϕ u n Y r } and λ s he wegh correspondng o he -h goal whch we assume has already been normalzed. Le us consder he opmal soluon r *, wh a value n he objecve funcon O r *, for r,..., m, and le mn O r * O r0 *.Then r0 * s he opmal soluon o he r,..., m orgnal goal programmng problem 3. Thus, for solvng 3, he followng algorhm s suggesed: Algorhm 3.. Sep. Le {,..., k / {ϕ u } X s } {,, m}. Sep. Make a paron of he se X s no m subses Y r as was descrbed earler. Le r.

8 34 Rafael Caballero, Mónca Hernández Sep 3. Gven r, f Y r, go o sep 4. If Y r, usng he algorhm n [4], solve he problem mn I r c λ Y r u d d α u β β where I r {,..., k such ha ϕ u n Y r }. Le r * be s soluon wh a value of he objecve funcon O r *. Sep 4. If r m, go o sep 5. If r m, le r r and go back o Sep 3. Sep 5. Le J {r,..., m / Y r }. Calculae mn O r * O*. r J If O * mn O r *, STOP: * s he soluon of 3. r J Before carryng ou he sensvy analyss, noe ha f here was only one goal a hs level, he prevous process would gve he same resuls as hose descrbed n Secon. Indeed, n he case of a sngle goal per level, gven ha he goal has no beng sasfed, he number m of possble goals o be sasfed s m 0. Ths means ha he paron for X s s made by a sngle subse whch s he acual X s. Gven ha n X s we have ϕ u, he epresson of n s n c u d α u β / d β, ha s, n f ug g ϕ u. Gven ha u s a consan, he pon n X s ha mnmzes hs epresson of n also mamzes he funcon ϕ n X s. Regardng he sensvy analyss, le us assume ha he soluon of problem 3, found wh he suggesed algorhm, s *, n *,..., n k *, p *,..., p k *. Thus, and usng smlar reasonng o ha used n he prevous secon, we reach he concluson ha, a leas, we have o lower he arge values o he correspondng value of ϕ * for each,..., k. Ths asseron s based on he fac ha pon *, obaned as a soluon o problem 3, s he pon ha mnmzes he weghed sum of he unachevemens of he goals esablshed n hs prory level. Le us conclude hs secon wh a worked eample ha res o asss n he undersandng of he soluon algorhm. Eample 3. Assume a problem whose opporuny se s X {, R / 4, 4, 5 } and suppose ha n he frs prory level we have he followng goals, all wh he same wegh:

9 Resoluon of he Lnear Fraconal Goal Programmng Problem ; 5 8 ; 5 5 ; 4 3 ϕ ϕ ϕ ϕ So ha he problem ha has o be solved s he followng:,...,4 0, mn p n p n p n p n p n s n 6 Afer solvng he lnear problem assocaed wh 6, he soluon n he opmum s greaer han zero, so we conclude ha here are no pons ha sasfy all he goal Fgure. Fgure of Eample 3. problem So we have o apply he algorhm proposed n order o solve 6 drecly. The frs sep s o denfy whch of he four goals could be sasfed n hs level. Fgure

10 36 Rafael Caballero, Mónca Hernández clearly shows ha only goals ϕ 3 - and ϕ 4 sasfy hs requremen, and herefore, m. Thus, we have o paron he se X no 4 subses, such as shown n Fgure : Y { X / ϕ 3 -, ϕ 4 }, Y { X / ϕ 3 -, ϕ 4 } Y 3 { X / ϕ 3 -, ϕ 4 }, Y 4 { X / ϕ 3 -, ϕ 4 }. The followng s o solve hree opmsaon problems, one for each non-empy subse of X. Because he wo frs goals canno be sasfed n eher of hese four subses, n all of hem we have o consder ha ϕ and ϕ. Therefore, he frs problem o solve s he followng: mn 5 Y Afer applyng he algorhm n [4], we fnd ha s soluon s * 3, 4 where he value of he objecve funcon s O *.5. The second problem o solve s 7 mn 5 7 Y 3 whose soluon s 3 *, 4 wh a value for he objecve funcon of O 3 * The hrd problem o solve s 7 3 mn Y4 whose soluon s 4 *, 3 wh a value for he objecve funcon of O 4 *. Therefore, akng O* mn {O *, O 3 *, O 4 *} O 3 *.9667, he pon ha solves 6 s pon 3 *, 4, snce he weghed sum of devaons of he fraconal crera from her arge values o he se X are mnmzed n hs pon. The values of he goals n hs soluon pon are ϕ * 0.75, ϕ * 0, ϕ 3 * , ϕ 4 * /3 and herefore, he sensvy analyss esablshes ha he arge values of he frs wo goals have o be dropped a leas o values u 0.75 and u 0 and n he fourh goal o value u 4 /3 n order o oban pons sasfyng he goals n hs level. We even can ncrease he arge value for he hrd goal up o value u We wan o pon ou ha hose are he values ha mnmzed he sum of he unachevemen In hs case, hs mnmum sum of unachevemen s O* Alhough here are oher ways of lowerng he arge values ha lead o soluons sasfyng he goals, hese oher ways mply a sum of devaons from he orgnal arge values greaer han O*.

11 Resoluon of he Lnear Fraconal Goal Programmng Problem More han one goal per prory level. Mnma GP approach. In hs secon we suppose ha he k goals n he curren level s are added under he mnma approach. In hs case he problem o solve s as follows: mn d X f n p u g λ n d n, p s 0,..., k,..., k,..., k As n prevous cases, we assocae a lnear problem wh he orgnal problem whch, once solved, wll ell us wheher here are any soluons ha sasfy all he goals n ha level or no. The lnear problem s epressed as follows: mn d' X f g u n' p' 0 λ n' d' n', p' 0 s,..., k,..., k,..., k However, o solve 7 drecly we should bear n mnd ha hs problem s equvalen o mn ma λ u ϕ 9 X s,..., k Gven ha for,, k ϕ are lnear fraconal funcons and he u values are prevously gven, he funcons of hs mnma problem are also lnear fraconal funcons and herefore we are dealng wh Generalzed Fraconal Programmng. Ths problem has been wdely suded n he leraure and dfferen algorhms are proposed o solve. See, for eample, he Newmodm algorhm [5]. Ne, we wll carry ou he sensvy analys Once 7 has been solved, where * s s soluon, f * s he unque soluon, hen he arge values of he goals whch have no been sasfed have o decrease o he value he goals ake a hs specfc pon. However, f * s no he unque soluon of 7, we have o connue wh he analyss because here mgh be some objecve funcons for whch mgh be possble o esablsh arge values hgher han hose reached n * and sll have pons sasfyng all he goal For hs case we propose an erave algorhm ha would lead us o he pon where he unachevemen of he goals a hs level s mnmzed and so, provde us wh he sensvy analyss sough. In each eraon of hs algorhm we ake, from he goals no sasfed, hose where he devaon varable d has reached s hghes value d*. For hese goals, we assume ha are he frs m goals we consder he followng consrans: ϕ u d* ϕ *,,m 7 8

12 38 Rafael Caballero, Mónca Hernández We add hese o problem 7 as hard consrans, and elmnae he sof consrans correspondng o hose goal We connue wh he process n hs way unl we eher reach a unque soluon for some of hese problems or have fnshed mprovng all possble mamum values for he arge values of he goals ha have no been sasfed. The algorhm suggesed s as follows: Algorhm 4. Sep. Le k ; X X s and I {,..., k}. Sep. Solve he lnear fraconal mnma problem usng he Newmodm algorhm mn ma λ u ϕ X k I k Le k * be he soluon pon and θ k * he value of he objecve n k *. Sep 3. Esablsh he followng nde se J k, whn I k : J k { I k / θ k * u ϕ k *}. Le I k I k \ J k. Sep 4. If I k, STOP: k * * s he pon ha mnmzes he unachevemen of he goal If I k, hen le X k X k { / ϕ u θ k * / J k }. Le k k, and reurn o sep. Ths s a fully convergen algorhm because n each eraon he se I k becomes smaller because, gven he acual consrucon of he value θ k *, he nde se J k can never be an empy se. So, f * s he pon obaned afer applyng he algorhm, for,, k, le be u ϕ * f u > ϕ *, and u u f u ϕ *. Wh hese new arge values, we make sure ha he mnma GP problem correspondng o he curren prory level s has a soluon ha sasfes all he goals, gven ha a leas pon * s one of hem. However, wh arge values hgher han u for hose goals where u > ϕ * we guaranee he non-esence of pons sasfyng all he goals of he problem. Ths s ensured because * s he pon ha mnmzes he mamum unachevemen of he goals esablshed. Neverheless, for arge values such ha u ϕ *, we can asser ha hese arge values could sll be ncreased up o he value of ϕ * and sll be pons ha sasfy all he goal

13 Resoluon of he Lnear Fraconal Goal Programmng Problem Conclusons In hs work we have eamned he goal programmng GP problem from all possble perspecves when he objecves assocaed wh he goals are lnear fraconal funcon We have carred ou a comprehensve sudy and have derved algorhms capable of fndng he pons ha solve he problem n a gven prory level. We have eplored all he possble cases, one goal per prory level and several goals per prory level. In hs laer case, he problem was also separaely analysed for goals grouped accordng o he weghed GP approach or he mnma GP approach. In all he cases he non-lnear problem derved has been solved. Once he lnear fraconal goal programmng problem was solved for each case, we also carred ou a sensvy analyss of he arge values se by he decson-maker. Thus, each secon of he work ncludes relaaon values for he arges whch guaranee he esence of soluons ha sasfy all he goals n he curren prory level. References. S. Awerbuch and J.G. Ecker, W.A. Wallace, A noe: Hdden nonlneares n he applcaon of goal programmng, Manage Sc E.L. Hannan, An nerpreaon of fraconal objecves n goal programmng as relaed o papers by Awerbuch e al. and Hannan, Manage Sc. 7, A.L. Soyser and B. Lev, An nerpreaon of fraconal objecves n goal programmng as relaed o papers by Awerbuch e al. and Hannan, Manage Sc C. Romero, Handbook of Crcal Issues n Goal Programmng, Pergamon Press, Oford, J.S.H. Kornbluh and R.E. Seuer, Goal programmng wh lnear fraconal crera, Eur J Oper Res, C. Aude, E. Carrzosa and P. Hansen, An eac mehod for fraconal goal programmng, J Global Opm, S. Schable, Fraconal programmng, n Handbook of Global Opmzaon, ed R. Hors and P. Pardalos Kluwer Academc Publshers, Dordrech, Neherlands, I. M. Sancu-Mnasan, A sh bblography of fraconal programmng. Opmzaon. 55, C. Casrodeza, P. Lara and T. Peña, Mulcrera fraconal model for feed formulaon: economc, nuronal and envronmenal crera, Agr Sys, 86, T. Gómez, M. Hernández, M. A. León and R. Caballero, A fores plannng problem solved va a lnear fraconal goal programmng model, For Ecol Manage, T. Gómez, M. Hernández, J. Molna, M. A. León and R. Caballero, Aspecos medoambenales denro de la planfcacón foresal con objevos múlples: Capura de

14 40 Rafael Caballero, Mónca Hernández carbono y resrccones de adyacenca, Revsa Elecrónca de Comuncacones y Trabajos de ASEPUMA, R. Caballero and M. Hernández, Resoraon of effcency n a goal programmng problem wh lnear fraconal crera, Eur J Oper Res, W. Dnkelbach, On nonlnear fraconal programmng, Manage Sc, 3, J.E. Falk and S.W. Palocsay, Opmzng he sum of lnear fraconal funcons, n Recen advances n global opmzaon, ed Ch. A. Floudas and P.M. Pardalos Prnceon Unversy Press, Prnceon, New Jersey, J.A.Ferland and J.Y. Povn, Generalzed fraconal programmng: Algorhms and numercal epermenaon, Eur J Oper Res, 0,

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