A Glorious Literature on Linear Goal Programming Algorithms

Transcription

1 Amercan Journal of Operatons Research, 2014, 4, Publshed Onlne March 2014 n ScRes. A Glorous Lterature on Lnear Goal Programmng Algorthms Ukamaka Cyntha Orume, Danel Ebong Department of Mathematcs/Statstcs, Unversty of Port Harcourt, Port Harcourt, Ngera Emal: amakaorume@yahoo.com, danel.ebong@unport.edu.ng Receved 16 October 2013; revsed 16 November 2013; accepted 23 November 2013 Copyrght 2014 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract In the last several years, there has been a marked mprovement n the development of new algorthms for solvng Lnear Goal programmng (LGP). Ths paper presents a survey of current methods for LGP. Keywords Lnear Goal Programmng; Algorthms; Current Methods 1. Introducton Goal programmng (GP) has been a popular theoretcal method for dealng wth multple objectve decsonmakng problems. In recent years, a number of lnear goal programmng algorthms have evolved based on the Dantzg [1]. Charnes et al. [2], Lee [3], Ignzo [4] and many others have been nstrumental n the development of varous algorthms of lnear goal programmng. Ln [5] presents an mpressve lst of artcles whch propose or apply goal programmng. However, a major setback n applyng lnear goal programmng (LGP) method has been the lack of an algorthm capable of reachng optmalty n reasonable tme. Lterature recorded that the most commonly used goal programmng soluton methods were ntroduced by Lee [3] and Ignzo [4] based on Dantzg s smplex method. Both methods requre columns n the smplex tableau for postve and negatve devatonal varables. They also requre separate c j z j rows for each prorty level, all of whch ncrease the computatonal tme of the soluton method. Ths paper presents a survey of current goal programmng algorthms and ther lmtatons. The paper s organzed as follows: Introducton and/general structure of the Lnear Goal Programmng Model s presented n Secton Two. Goal Programmng Assumptons and General Prncples (Concepts) are presented n Secton Three, whereas current lterature on lnear goal programmng algorthm and ts lmtatons are presented n Secton Four. Secton fve presents the concludng remark. How to cte ths paper: Orume, U.C. and Ebong, D. (2014) A Glorous Lterature on Lnear Goal Programmng Algorthms. Amercan Journal of Operatons Research, 4,

2 2. Introducton and/general Structure of the Goal Programmng Model Goal programmng can be consdered as a branch of mult-objectve optmzaton whch tself s a part of multcrtera decson analyss. Goal programmng s one of the oldest mult crtera decson makng technques used n optmzaton of multple objectve goals by mnmzng the devaton for each of the objectves from the desred target. The basc concept of goal programmng s that whether goals are attanable or not, an objectve wll be stated n whch optmzaton gves a result whch come as close as possble to the desred goals (satsfed soluton). The objectve of goal programmng s to mnmze the achevement of each actual goal level. If non achevement s drven to zero, then t means that actual attanment of the goal has been accomplshed. For a sngle goal problem, the formulaton and soluton s smlar to lnear programmng wth the excepton that, f complete goal attanment s not possble, goal programmng wll provde a soluton and nformaton to the decson makers. In general, the dea of goal programmng s to convert orgnal multple objectves nto a sngle goal. The resultng model yelds a satsfcng soluton whch may not be optmum wth respect to all the conflctng objectves of the problem. That s GP yelds only an effcent and satsfactory result rather than optmum, soluton to the problem. Ths s because, t s uncommon to always satsfy every goal, so goal programmng attempts to reach a satsfactory level of the multple objectves under consderaton. To avod the possble bas effect of the soluton to dfferent measurement unt, goal normalzaton takes place. The procedures for structurng Goal programmng model are smlar to those for a Lnear Programmng. The man dfference between the LP and GP s that, LP maxmzes or mnmze (optmzes) a sngle objectve functons whereas, GP mnmzes the devatons between the target values of the objectves and the realzed results (satsfcng soluton). The basc steps for structurg goal programmng as stated n Rfa [6] are as follows; * Goals are dscovered and converted to constrants by ntroducg devatonal varables. * Examne the goals to determne the exact devatonal varables needed for them,.e., whether d, d +, or both as summarzed below n Table 1. In the second objectve goal (row 2 of Table 1), t mples that anythng below the target value b s acceptable, so the over-achevement of the target d + should be mnmzed to 0. In row three, the objectve goal s that anythng below the target value b should be drven to zero whle the over-achevement of the target d + should be accepted. The last objectve goal mples that anythng below or above the target value b s unacceptable, so the over-achevement of the target d + and under achevement of the goal d should be mnmzed to 0. * Goals are ranked n order of mportance and pre-emptve prorty factor, p assgned to each of them.. * In case of tes n prorty, assgn weghts to each of the devatonal varables n the prorty. Once the above steps are completed, the problem can be quantfed as a GP model. Schnederjans and Kwaks [7] referred to the most commonly appled type of goal programmng as pre-emptve weghted prorty goal programmng and a generalzed model for ths type of programmng s as follows: Mnmze: s.t n ( ) m (2.1) z= wp d + d ( ) ax j j + d d + = b = 1, 2,, m, (2.2) j x, d, d + 0, w > 0 (2.3) j ( = 1, 2,, m; j = 1, 2, 3,, n) (2.4) The above representaton can be found n some recent publcatons such as Rfa [6], Ken and Perushek [8], Ken and Perushek [9], Sharma and Sharma [10], Jafar et al [11], Orume and Ebong [12], Nabendu and Mansh [13] among others. For each of the objectves, a target value or goal would be gven ( b ), whch s needed to be acheved. Fnally, the undesred devatons d = ( d, d ) from the gven set of targets ( b ) are mnmzed by usng an achevement functon (z). In effect, a devatonal varable represents the dstance (devaton) between the aspra- 60

3 Table 1. General structure of goal programmng model. Goal devaton varable to be mnmzed(ncluded) n z ax b d + j j ax b d j j ax = b d + d j j ton level and the actual attanment of the goal. Hence, the devaton varable d s replaced by two varables: d = d d where d, d + 0. The precedng ensures that the devatonal varables never take on negatve values. The constrant ensures that one of the devaton varables wll always be zero. Fnally, the unwanted devatonal varables need to be brought together n the form of an achevement functon whose purpose s to mnmze them and thus ensure that a soluton that s as close as possble to the set of desred goals s found. Ths soluton s called a compromsed (harmonzed) soluton rather than optmal and that s why t s called a satsfcng technque. Weghted goal programmng and Lexcographc goal programmng are the most popular varants of the GP model as shown n Schnederjans and Kwak [7]), Crowder and Sposto [14], Tamze et al. [15], Ken and Perushek [8], Tamz and Jones [16], Charles et al [17], Alp and Erosy [18] and many others. The two methods do not generate the same soluton and nether s one method superor to the other because, varant s desgned to satsfy certan decson makers preferences. Rosenthal [19] represented Goal programmng problems formulatons n three categores. One s called weghted goal programmng where weghts are assgned to the goals that measure ther relatve mportance and then fnds a soluton that mnmzes the weghted sum of the devatons from the targets. The second approach s called preemptve goal programmng whch requres rankng the goals n order of mportance. The hgher prortzed goal s consdered frst before the next ranked goal. Fnally the thrd one s called prortzed GP whch s the combnaton of the weghted goal programmng and preemptve goal programmng above. These varants are dscussed below Pre-emptve Goal Programmng (Lexcographc Goal Programmng) In many stuatons, however, a decson maker may not be able to determne precsely the relatve mportance of the goals. I.e. apply pre-emptve goal programmng, the decson maker must rank hs or her goals from the most mportant (goal 1) to least mportant (goal m). Preemptve goal programmng procedure starts by concentratng on meetng the most mportant goal as closely as possble, before proceedng to the next hgher goal, and so on to the least goal.e. the objectve functons are prortzed such that attanment of frst goal s far more mportant than attanment of second goal whch s far more mportant than attanment of thrd goal, etc, such that lower order goals are only acheved as long as they do not degrade the soluton attaned by hgher prorty goal. When ths s the case, pre emptve goal programmng may prove to be a useful tool as ntroduced by Ijr [20], and developed by many others. The achevement functon for the general preemptve GP model s gven as such that Equaton (2.1)-(2.2) holds. The above s detaled n Orume and Ebong [21] Weghted Goal Programmng (WGP) k ( ) lex mn z = p d + d (2.5) Weghted Goal Programmng (WGP), weghts are attached to each of the objectves to measure the relatve mportance of devatons from ther target. WGP handles several objectves smultaneously by establshng a specfc numerc goal for each of the objectves and then fnd a soluton that comes close to each of these goals. Popular choces for normalzaton constants are the goal target value of the correspondng objectve (hence turnng all devatons nto percentages) or the range of the correspondng objectve (between the best and the worst possble values, hence mappng all devatons onto a zero-one range). The algebrac formulaton of a WGP as presented n Ken and Perushek [8] s gven as: 61

4 m + ( ) mn z= wd + wd (2.6) such that equaton (2.2)-(2.4) holds. Where w, w are numercal weghts assocated wth negatve and postve devatonal varable d and d + ( 0) respectvely whch denotes how far the decson s from the goal and how much the decson has exceeded the goal respectvely. The above s detaled n Orume and Ebong [21]. Tamz et al. [15] reported that artcles that used lexcographc and weghed goal programmng n pre-1990 applcaton papers s 75% and 25% respectvely. Jones and Tamz [22] reported that ther splt n the perod s 59% lexcographc and 41% weghted. The reasons for ths are due to the greater flexblty by the weghted goal programmng, and the am of the decson makers to do more trade-off analyss and drect comparson between goals Protzed Goal Programmng Ths s stuaton where both weghted and preemptve approaches are combned to form a model to a problem. Ths occurs when the goals can be categorsed nto groups where the goals wthn each group are of equal mportance, but there are slght dfferences between the groups n ther level of mportance. In ths knd of stuaton, weghted goal programmng can be used wthn each group n turn whle preemptve goal programmng s beng appled to deal wth each group n order of mportance. Each prorty level (each group) has a number of unwanted devatons to be mnmsed. Ths means that mnmsaton of devatonal varables placed n a hgher prorty level s assumed to be nfntely more mportant than that of devatonal varables placed n a lower prorty level (group). Ths s represented as n equatons (2.1) to (2.4). 3. Goal Programmng Assumptons and General Prncples (Concepts) Wnston [23] stated the axoms of lnear goal programmng models as follows: Addtvty: addtvty assumpton mples that the level of penalsaton for undesred devatonal varables from a target level does not depend on the levels of unwanted devatonal varables from the other goals. Proportonalty: Proportonalty assumpton n the goal programmng model requres that the penalsaton for an unwanted devatonal varable from a target level s drectly proportonal to the dstance away from the target level. Dvsblty: Ths assumpton mples that all the decson varables should be free to take any value wthn ther stated range,.e., a decson varable cannot be forced to take an nteger or a dscrete value. Certanty: Ths assumpton mples that all the data coeffcents are known wth certanty. However, the use of goal programmng s not necessarly mpossble f any of the above axoms s volated. A nonlnear goal programme could be formulated f the addtvty condton does not hold. In the case where the dvsblty axom does not hold, an nteger or bnary goal programmng could be formulated. When the certanty axom s not holdng, then the method to be used wll depend on the type of coeffcents over whch there exsts uncertanty. A certan amount of uncertanty over weghts and target values often exsts and ths can frequently be handled by good senstvty or weght analyss technques or an nteractve method. Another good alternatve s to use the fuzzy goal programmng varant. If there s uncertanty over the technologcal coeffcents then ether the fuzzy goal programmng varant or a combnaton wth a technque such as smulaton could be used. Nevertheless, t s necessary to see and understand the general prncples and concepts of goal programmng to ensure that the goal programmng varants are chosen correctly and the parameters set approprately. Satsfcng Goal programmng s prmarly a satsfcng technque. Smon [24] descrbes t as a behavour n whch decson makers am to reach a set of defned goals. If they reach those goals, then they are satsfed. Ths s dfferent from the concept of optmsng. He argued that human bengs are more nterested and able to reach goals than n the abstract concept of optmsng each outcome of the decson problem. Meetng goals as closely as possble s the man am of the goal programmng technque. Tamze and Jones [25] portrayed satsfcng as the prme underlyng phlosophy of goal programmng and that ts solutons should be judged solely on how well they meet the goals of the decson maker and whether they produce a practcal soluton to the decson problem. Although goal programmng can produce Pareto-neffcent solutons, ths s manly due to poor formulaton and model- 62

5 lng of the decson maker s preferences and target levels by the analyst buldng the goal programme. Optmsng Optmzaton mples lookng for the decson whch gves the best possble value of some measure of performance from amongst the set of possble decsons. The theory of optmsng n the presence of multple objectves s defned by adaptng a concept of Pareto optmalty n a mult-objectve model. Accordng to Tamze as Jones [25], optmsng phlosophy has mportance n Goal programmng n the stuatons where: 1) If Pareto optmalty detecton and restoraton take place then the goal programme has a mx of the satsfcng and optmsng phlosophes. 2) If the goals are two-sded (.e. a partcular value s optmal rather than a more s better or less s better stuaton) then the satsfcng and optmsng phlosophes can be thought of as concdng for those goals. Orderng or Rankng Orderng or rankng s mportant n the lexcographc goal programmng because, t s assumed that the rankng of the goals n order of mportance to the decson maker exsts and s known or able to be estmated by the decson maker. In real-lfe stuatons, goals does not take place lexcographcally, and n these cases the decson maker wll explore the trade-offs or the balance between the goals. In these cases lexcographc goal programmng should not be used and nstead another goal programmng varant should be chosen. 4. Lterature Revew on Lnear Goal Programmng Algorthms Goal programmng (GP) s a popular multobjectve optmzaton technque used n handlng problems wth multple objectves. There are several approaches avalable for solvng lnear GP problems n the lterature. Charnes and Cooper [2] developed goal programmng model for lnear system (lnear programmng problems) n whch conflctng goals were ncorporated as constrants. The model s represented as; lex mn z = p( d + d ) such that ax j j + d d = b, ( = 1, 2,, m). Ther model s lmted to those models that employ only a sngle objectve prorty. Because of the conflctng goals, t was mpossble to use the concept of lnear programmng. Ignzo [26] tred to apply ths method to solvng problems he faced n the U.S. Space program, but was faced wth the problem of ncommensurablty of the devatonal varables taken from dfferent goals whch made t mpossble to fnd weghtng factors for such varables so that a meanngful summaton s possble n the objectve functon. For nstance, most of the problems were nonlnear functons. The only beneft of ths model s that t uses any exstng lnear programmng packages for soluton. Ijr [20] appled the concept of generalzed nverse approach to develop a preemptve prorty levels to handle goals n ther order of mportance. He presented an analyss of problems n whch management can quantfy the goals that they wsh to obtan usng goal programmng approach. Ignzo [27] developed a GP soluton approach, known as sequental (lnear) GP method. Ths approach was establshed on the remark by Huss [28] that lnear goal programmng model (problem) could be solved through a sequence of conventonal L.P models. Ths method nvolves the soluton to an nterrelated sequence of conventonal lnear programmng problems. Based upon the soluton to the prevous lnear programmng (LP), an augmented constrant s added to the subsequent models. The purpose of ths augmented constrant s to ensure that all solutons to lower prorty (subsequent LP models) do not volate solutons prevously obtaned for hgher prorty goals. It utlzes any exstng lnear programmng package. Ths was letter coded (computersed) by Ignzo and Perls [29]. Lee [3] presented a method that nvolves a modfcaton of the standard smplex algorthm, capable of handlng pre-emptve prorty goals. It s an extenson of two-phase smplex algorthm known as multphase approach. Ths method approaches the entre goal programmng problem as a sngle model. The Lee s modfed smplex algorthm treats the full smplex tableau, expandng the evaluaton secton (z j c j ) row for every pre-emptve prorty and the selecton rules for ths algorthm follow conventonal prmal lnear programmng as descrbed by Olson [30]. Ignzo [31] descrbed ths method as an mprovement over the sequental smplex technque snce t requres fewer pvots (n general) and removes the need for the constructon of new constrants at each sequence. It has been shown that ths algorthm takes so much tme to converge. Schnederjans and Kwaks [7] descrbed Lee [3] and Ignzo [4] based on Dantzg [1] smplex method as the most commonly used GP soluton method; but have or requre columns n the smplex tableau for postve and negatve devatonal varables. The separate objectve functon rows for each prorty levels add to the computa- 63

6 tonal tme of the soluton method. Ignzo [4] developed a GP soluton approach known as multphase (lnear) GP method that handled the challenges he faced usng Lee s algorthm more effcently. He emphaszed the need for the constructon of new constrants for the accomplshed prorty level of the devaton varables for each lower level prorty model. At the frst stage of ths procedure, the only goals ncluded n the LP model are the frst-prorty goals, and the smplex method s appled n the usual way. If the resultng optmal soluton s unque, we adopt t mmedately wthout consderng any addtonal goals. However, f there are multple optmal solutons wth the same optmal value of Z (call t Z * ), move to the second stage and add the second-prorty goals to the model. If Z * = 0, all the varables representng the devatons from frst-prorty goals must equal zero (goals are fully acheved) for the solutons remanng under consderaton. On the other hand, f Z * 0, then, proceed to the second-prorty whch becomes the objectve functon but then, t adds the constrant that the frst soluton obtaned equals Z *. Apply the smplex method agan, f there exst multple optmal solutons, repeat the same process for any lower prorty goals. Ignzo [32] remarked that t s an effcent method but, there s a constructon of new constrant at each sequence. Authur and Ravndran [33] developed an effcent algorthm for solvng lnear goal programmng problems that adopts the concepts of herarchcal structure of pre-emptve models usng parttonng and elmnaton procedures. The algorthm took advantage of the defnton of ordnal pre-emptve factors n the objectve functon nherent n most goal programmng formulatons. Ths algorthm starts by handlng only those constrants that concerns the frst prorty goals. If there exsts multple optmal solutons to ths frst prorty model, constrants affectng the next hgher prorty are added, and the new model solved. Ths procedure contnues untl a sngle optmal soluton s obtaned. The authors solved problems of varous szes and complextes to test ts effcency wth the wdely used goal programmng algorthm by Lee [3]. In all tested problems, the parttonng algorthm consstently dd much better than the algorthm by Lee [3], takng as lttle as 12% of Lee s tme and never more than 60%. Hwang et al. [34] cted Arthur and Ravndran [33] method as beng an effcent soluton method. Olson [30] stated that n ths method, computatonal economes are ganed by consderng only rows and columns affectng the most mportant unsatsfed goal and that for models wth few prortes, or where all goals can be satsfed, lttle theoretcal computatonal advantage s expected wth ths method. But Schnederjans and Kwaks [7] portrayed ths computatonal procedure as beng restrcted to problems that are prortzed and/or do not have conflctng goal constrants that lose varables va the varable elmnaton process. The goal constrants that are later augumented to an already optmzed tableau as descrbed n ther procedure wthout an teratve adjustment run the rsk of volatng the orgnal G.P. problem as demonstrated by Schnederjans and Kwaks [7]. Ignzo [31] presented a general goal programmng mathematcal model that s dfferent from the tradtonal lnear programmng formulaton, but provded a real problem representaton n practce. Hs model s represented thus: Fnd x = x1, x2,, x j, so as to mnmze: a = g n, p, g n, p,, g n, p (4.1) such that: and { 1( ) 2( ) k ( )} ( ) for all 1,, f x + n p = b = m (4.2) xnp,, 0 (4.3) where: x j s the j th decson varable, a s denoted as the achevement functon; a row vector measure of the attanment of the objectves or constrants at each prorty level, gk ( n, p ) s a functon (normally lnear) of the devaton varables assocated wth the objectves or constrants at prorty level k, k s the total number of prorty levels n the model, b s the rght-hand sde constant for goal (or constrant), f ( x ) s the left-hand sde of the lnear or nonlnear goal or constrant. where n represents the negatve devaton from goal and p s the postve devaton from goal. Ignzo [35] developed another approach for solvng GPP that compresses tableau element wth the use of the condensed smplex tableau alongsde wth the concept of column droppng. He mantaned the negatve devatonal varables n the bass. The nteror rows of the tableau are assocated wth the present set of basc varables and the nteror columns wth the present set of non-basc varables. The bottom row of the tableau, under the 64

7 non-basc varables, gves the shadow prces for the prorty level under consderaton. Check marks ( ) are placed above the columns whch are no longer elgble for entry nto the bass. He also used the concept of reflected p -space to reduce storage. But, Schnederjans and Kwak [36] reported that, Ignzo [35] algorthm takes more computatonal tme manpulaton than the Schnederjans and Kwak [37] algorthm, and also fals to generate useful nformaton that s commonly found n more popular G.P. algorthms. In partcular, the Ignzo [35] algorthm does not provde any nformaton for subsequent parameter senstvty analyss. He concluded that, researchers who are nterested n usng a G.P. algorthm requrng the least number of tableau elements or readers who are nterested n maxmzng the nformatonal value of a G.P. soluton should not refer to Ignzo [35] method. Schnederjans and Kwaks [7] provded a new approach for goal programmng problems soluton based on Baumol s [37] smplex method that fully utlzed postve devatonal varable, together wth a step-by-step soluton of an llustratve example. Ther method yelds a substantal reducton n the number of tableau elements computatons. At that tme no computer program exsted usng the proposed goal programmng method and they recommended that such a computer program be developed by nterested researchers. Olson [30] stated that Schnederjans and Kwak s soluton procedure elmnates up to one half of the devatonal varable columns n the smplex bass relatve to Lee s full smplex approach, and does away wth the z j c j secton of the tableau, also, computatonal effcency s ganed by not mantanng the dentty matrx column elements n the smplex tableau, and f the optmal soluton ncludes a hgh proporton of postve devatonal varables, ths method can be expected to be relatvely faster than other approaches. He also stated that the algorthm however, does not follow a path of guaranteed soluton mprovement f the soluton contans a large proporton of negatve devatonal varables; ths method requres extra computatonal tme and cyclng often occurred n models where an unsatsfed goal was found n the fnal soluton (see Orume and Ebong [21]). But Ignzo [38] argued that Schnederjans and Kwaks paper s ncorrectly compared, snce the correct comparson ndcates that the new algorthm requres tableaux of the same szes as are requred of those for the older, elementary forms, and that the fact that there s no justfcaton that the Baumol method for Lnear Programmng s more effcent than the smplex method dsqualfes ther argument. Olson [30] compared four goal programmng methods; Schnederjans and kwaks [7], Lee [3], Authur and Ravndran [33], and Olson [30]. And also developed a revsed smplex algorthm (RSM) for solvng LGP problem that adopts Schederjans and Kwaks [7] dual smplex rules appled for the calculaton of new tableau elements. The algorthm utlzes effcency of not mantanng dentty columns whle ganng the systematc securty of searchng for optmal soluton utlzng only feasble soluton. Negatve devatonal varables are fully mantaned, but other varables columns are developed only as necessary. The RSM works well when negatve devatonal varable s n the soluton. In hs result tests, the dual smplex method appears to have superor computatonal tmes for models wth a large proporton of postve devatonal varables n the soluton, whereas the revsed smplex algorthm appears more consstent n tme and accuracy for general goal programmng models. He summarzed that the Lee algorthm proved accurate n models tested, although extra teratons were requred n some models, whereas the Arthur and Ravndran code was not tampered wth other than to ncrease dmensons owng to unfamlarty wth the specfc program (see Ebong and Orume [21]). Accordng to hm, accuracy was found to be a problem for the code n larger models tested. The Arthur and Ravndran code was compettve n tme for those models where t obtaned the correct soluton, but for larger models run, ncorrect solutons were obtaned. The computatonal effcency of the dual smplex code presented by Schnederjans and Kwak over the revsed smplex code can be substantal for models nvolvng solutons wth a hgh proporton of postve devatonal varables n the soluton as descrbed by the author. Ignzo [32] developed another approach for solvng GP problems called multdmensonal dual smplex algorthm (MDD). The dual of a lexcographc GP s smlar to that of a lnear programmng wth the excepton that the rght hand sdes of the dual are mult-dmensonal and ranked n order of mportant. Ths follows from the fact that the achevement functon n the prmal are ranked lexcographcally. However, snce t has been remarked that the mult-dmensonal dual s a lnear programmng problem wth multple and prortzed rghthand sdes, each model s dentcal to one another, wth the exceptons that: the rght-hand sde wll be changed at each sequence and some constrants dropped dependng on the soluton obtaned to the prevous lnear programmes. All non-bndng dual constrants correspondng to prmal varables are removed as llustrated by the author. Accordng to the author, one mght conjecture, that the soluton to a multdmensonal dual wth, say, fve rght-hand sdes would requre about fve tmes as long as that needed for the smplex soluton to the ntal 65

8 model and that an attractve feature of the dual smplex-based algorthm for obtanng the soluton of the LGP problem s that t can be mplemented easly wth any conventonal smplex software system (sngle-objectve functon). But Crowder and Sposto [14] dsagreed wth hs method and argued that removal of non-bndng constrants n the dual problem after obtanng the optmum for the dual problem assocated wth prorty level, concdes wth the removal of non-basc varables n the LGP prmal at prorty level. Ths mples that pre-emptve prorty condtons are volated whle solvng MDD problem, otherwse wrong soluton wll be obtaned. They supported ther clam by solvng a LGP problem as shown n Crowder and Sposto [14] whch yelds a 0 = ( 0,1, 2), x 1 = 6, x 2 = 0 when solved usng SLGP. But, f the same problem s solved usng the multdmensonal-dual by Ignzo [32], changng rght-hand sdes and removng slack rows at each prorty level as outlned, wll lead to ncorrect soluton: a 0 = ( 0,1, 0), x 1 = 4, x 2 = 0. Shm and Chun [39] showed that Resource Plannng and Management Systems (RPMS) network approach can be used to solve goal programmng (GP) problem. In RPMS-based GP, devatonal varables are represented by resource nodes. These resources, wth dfferent prortes and weghts, are used to attan maxmum goal achevement, whereas resources are consumed to produce products at process nodes (decson varables). The nodes to be placed n the ntal RPMS graph are process nodes, mnmum/maxmum nodes and resource nodes, except postve devatonal varables. The nterrelatons (technologcal coeffcent) are represented by arrows between process nodes and resource nodes, and between dfferent resource nodes. The nterrelatons between resource nodes and the maxmum node ndcate the prorty structure, whle the nterrelatons between resource nodes and the mnmum node stand for goal levels. Values below the x s at resource nodes mean remanng resources for negatve devatonal varables and resources consumed for postve devatonal varables, respectvely. The postve devatonal varables work as process nodes wth prorty assgned to t. But Ogryczak [40] argued that RPM defers from GP formulaton snce t utlzes negatve weght and addtonal regularzaton of the mn max aggregaton. He argued also that RPM s an teratve technque and ponted out serous flow snce practcal large problem usually have multple optmal soluton. Calvete and Mateo [41] used the deas of prmal dual algorthms for the mnmum cost generalzed network flow (MGNF) problem to develop a lexcographc optmzaton of mult objectve generalzed network flow (LGNF) problem. The author showed that ther algorthm s effcent n reachng optmalty, but dffcult n labelng process due to several nodes, arcs, paths whch result n multple solutons. Baykasoglu et al. [42] developed an algorthm for solvng lnear GP usng multple objectve tabu search (TS). Ther computatonal expermental result obtaned showed that ther method s effcent. Ths algorthm may recycle old solutons and become trapped n a loop as ndcated by the author, whch mples that t does not solvee all knds of goal functons and constrants. The procedure s very tedous as t works wth more than one soluton vector. Baykasoglu [43] used the multple objectve tabu search (MOTS) algorthm, whch was proposed prevously by Baykasoglu et al. [42] to solve GP models. In the proposed approach, GP models are frst converted to ther classcal Mult objectve optmzaton (MOO) equvalent by usng some smple converson procedures. Then the problem s solved usng the MOTS algorthm. The proposed approach also avods the problem of weghtng the goals n weghted GP and orderng the goals n pre-emptve GP. The results obtaned from the test problems showed that MOTS can successfully fnd many alternatve solutons to a gven GP. To enable the TS algorthm to work wth more than one objectve, selecton and updatng stages of the basc TS are redefned. Kasana [44] developed groupng algorthm for LGPP soluton. Ths method consders all goals and real constrants together as one group wth the objectve functon beng the sum of all the unwanted devatons, and solves a sequence of LP sub problems, each usng the optmal soluton of the prevous sub problems. Ths algorthm s beng domnated by the parttonng method as ndcated by the author. He ndcated that t s good and performs well only f a large number of goals are satsfed. In other word, f an unsatsfed goal s n the fnal tableau, t s neffcent. It utlzed sequental method (see Orume and Ebong [21]). Orume and Ebong [12] developed an effcent method of solvng a generalzed GPP that reduced the computatonal tme drastcally when compared wth the exstng ones. Ther method recognsed the fact that a goal n programmng model may nclude rgd constrants lke aj x j b.e. Goal programmng also allows for an addton of a set of lnear programmng style hard constrants whose volaton wll make the soluton nfeas- j 66

9 ble. Ther new goal programmng algorthm s formulated nto ntal tableau n the same format wth that of smplex method of solvng lnear programmng problems whether t s weghted, prortzed or generalzed model. The dfferent s that n the new algorthm, the devatonal varables that appeared n the achevement functons wth ther prorty or weght attached to them, together wth slacks from the rgd constrants f exsts forms v the bass. If both d (postve and negatve devatonal varables) from the same row are n the achevement functon z, then the one wth hghest prorty or weght wll be n the bass whereas the lesser one wll be placed at the non bases. The column of the decson varables, the slacks varables from the rgd constrants, and the devatonal varables from the achevement functons forms the non basc varables. The procedure consders the goal constrants (g ) as both objectve functons and the constrants. It starts by not ncludng the devatonal varables column that dd not appear n the achevement functon n the tableau (.e. whle searchng for the optmal soluton), but others can be augmented when necessary. Ths s because, postve devatonal varables columns coeffcent s the same as negatve of the negatve devatonal column coeffcent. Iskander [45] suggested an approach for Solvng Weghted Goal Programmng Problem that reformulated the weghted goal program as a lexcographc goal program wth two man goals. The frst goal, whch has the frst prorty, seeks to mnmze the maxmum weghted undesred normalzed devaton. The second goal, havng the second prorty, mnmzes the sum of the undesred normalzed devatons. The proposed approach seeks to provde a soluton n whch the goals achevements are proportonally related to the relatve weghts. Orume and Ebong [21] separated ther generalzed algorthm of 2011 nto lexcographc and weghted method. Ther algorthm s very effcent n reachng optmalty. However, debated weakness of LGP s that goal-programmng approach, regardless of the weghtng structures (pre-emptve or Archmedean) and regardless of the goals (one-sded or two-sded), can lead to nferor (domnated or neffcent), suboptmal solutons whch are not necessarly the best ones avalable to the decson-maker. But, n Ignzo [4], t was proved that the optmal soluton obtaned by the lexcographc problem s Pareto optmal. Thus, the lexcographc method s always adopted as an addtonal optmzaton approach n methods that can only guarantee weak optmalty by themselves. Evans [46] descrbed GP problem as a technque for fndng that soluton whch mnmzes the devaton over all feasble solutons; such a soluton s called a best compromse soluton and that under the assumpton that more of each objectve s preferred to less; a best compromse soluton. Mn and Storbeck [47] stated that GP s a technque not desgned to fnd an optmal pont, but to fnd an acceptable range, and advsed that the dspute of GP domnance wll contnue unless the management scentst can accept goal programmng s satsfcng prncple and not beng captvated by the prncple of optmalty. Mettnen [48] proved that GP technque yelds nondomnated solutons f the goal pont s chosen n the feasble doman. However, n Goal programmng, there s no method to determne f a soluton s better than other. Nabendu and Mansh [13] stated that the computatonal procedure n goal programmng s to select a set of solutons whch satsfes the constrants and provdng a satsfactory goal, ranked n prorty order snce GP approach seeks satsfcng solutons whch come as close to the desred aspraton levels as possble. Antono et al. [49] descrbed Pareto domnance relaton as the most commonly adopted method n mult objectve optmzaton to compare solutons whch, nstead of a sngle optmal soluton, leads to a set of alternatves wth dfferent trade-offs among the objectves. Ther solutons are called Pareto optmal solutons or non- domnated solutons. Although there are multple Pareto optmal solutons, n practce, only one soluton has to be selected for mplementaton. Hannan [50] and [51], and Romero [52] developed an approach for detecton of Pareto neffcency n a goal programmng soluton. These methods provde ways of restorng Pareto effcency by calculatng a Pareto-effcent soluton that domnates the goal programmng soluton. These algorthms carry out a maxmsaton of some functon of the wanted devatonal varables (that are not penalzed n the objectve functon) n the orgnal goal programmng, such that f the optmal soluton from that of the orgnal goal programme s altered ndcated that the orgnal goal programmng soluton was Pareto neffcent. Tamz and Jone [53] separated detecton and restoraton procedure, and showed that Pareto detecton can be performed wthout resortng to optmzaton n fve tests as descrbed n Tamz and Jone [25]. In all the fve steps, algorthm termnates wth only degenerate teratons havng taken place. Hence the soluton pont s stll the same wth the orgnal goal programme soluton. Tamz and Jones [25] descrbe ther approach as the most computatonally effcent means of detectng Pareto neffcency. Hannan [50] method presented dfferent possble ways of calculatng a Pareto-effcent soluton. These nclude; 1) A possble weghtng scheme to gve relatve mportance to the mprovement of the objectves. 2) A 67

10 prorty order n whch the objectves should be mproved. 3) A vector maxmsaton multple objectve programmng model to fnd a set of effcent solutons that domnate the orgnal goal programme s soluton. But Romero [52] ponted out that the Hannan [50] formulatons do not restrct the values of devatonal varables whch are not penalsed n the orgnal formulaton correctly, and thus generates worst soluton for such an objectve, and hence does not domnate the orgnal goal programme. Romero [52] modeled the orgnal goal programmng soluton and Pareto restoraton as a sngle lexcographc process. Tamz and Jones [53] ponted out a further complcaton that occurred n the case where one or more objectves are Pareto unbounded. In ths case the restoraton optmsaton wll become unbounded and an effcent soluton s not possble. In ths case the objectves that are unbounded wll be dscarded from the fnal prorty level and the optmsaton contnued from that pont. Ths step may contnue teratvely as not all the unbounded objectves may be detected at that pont. The method wll termnate wth all objectves classfed as ether Pareto effcent or unbounded. Ths s the closest to effcency that can be reached for a Pareto-unbounded model. Tamz and Jones [53] presented three ways of restorng Pareto effcency: Straght restoraton, Preference-based restoraton and Interactve restoraton. Another debated ssue on goal-programmng approach s unbounded soluton problem. But Ignzo [4], Markowsk and Ignzo [54], and Schnederjans [55], argued that unbounded problems dd not exst n GP because aspraton levels (target values or rght-hand-sde values) were assocated wth every objectve and GP orgnally sought a satsfcng soluton whch allowed for some flexblty n aspraton levels as stated n Mn and Storbeck [47]. Evans [46] showed that GP soluton cannot be bounded snce t has only mnmum values. Therefore, they concluded that GP problems could not generate unbounded solutons unlke n LP whch does not allow any flexblty n determnng rght-hand-sde values of feasblty constrants; hence an unbounded soluton can occur n LP formulatons. GP s much more flexble n determnng rght-hand-sde values than s LP. The major appeal of GP as explaned n Mn and Storbeck [47] s that t consders the cogntve nature of much human behavour whch can be rratonal or non-omnscent. Another ssue on goal programmng approach s feasblty. It has been proved that when the goals are feasble, the soluton gven by the GP s effcent, but the effcency of the provded soluton when the goals are not feasble remans an open problem n general as llustrated by Ontaro, et al. [56]. An effcent soluton s a feasble soluton, f there does not exst any other feasble soluton whch does at least as well on every sngle objectve, and better on at least one objectve. Weghted goal programmng and preemptve goal programmng provde pareto optmal solutons f the goals form a pareto optmal pont or f all devaton varables, d j + for functons beng ncreased and d j + for functons beng reduced, have postve values at the optmum as llustrated by ken and Perushek [9]. Ths was repeated n Marler and Arora [57]. Major advantage of goal programmng s that there always exsts a soluton to the problem, provded that t has feasble regon and ths s because of the ncluson of the devatonal varables. 5. Conclusons Ths paper presented a survey of current methods for LGP and lmtaton of LGP n general. The queston that one would ask at ths pont s: Whch procedure s the best? Orume and Ebong (2013) have compared varous LGP technques. Ther algorthm was compared n terms of accuracy and tme requrements wth exstng algorthms by Lee [3] and by Ignzo [4] and Ignzo [35]. Computatonal tmes for 10 goal programmng models of varous szes and complextes were compared. Number of teraton per problem, total entres per problems s used as benchmark for the comparsons. The new method by Orume and Ebong (2011) have better computatonal tmes n all the problem soluton and proved the best snce there s a reducton n computatonal tme n all the problems solved. The controverses surroundng LGP mostly came from msconceptons about the prncple of satsfcng whch underles GP theores. It s almost mpossble for the decson-maker to acheve deal goals wthout the expense of other goals n optmzaton of multple goals. In ths sense, the mult-objectve s an unfortunate name. Therefore, the effcency of GP solutons s problem-dependent and user-dependent. In the modellng and soluton processes of GP, so much freedom s gven to the decson-maker. If the decson-maker nadvertently sets unreasonable targets or assgns ncorrect weghts and/or prortes, a GP soluton cannot provde the best avalable or effcent soluton. Therefore, the lmtatons of GP, f any, are due manly to errors of ts users, not to the ratonale behnd GP theores. However, there s no doubt that further developments of GP theores are 68

11 no longer needed. Hence, the basc concept of goal programmng s that whether goals are attanable or not, an objectve wll be stated n whch optmzaton gves a result whch come as close as possble to the desred goals (satsfed soluton). GP technque yelds non-domnated solutons f the goal pont s chosen n the feasble doman. GP soluton cannot be bounded snce t has only mnmum values. It has been proved that when the goals are feasble, the soluton gven by the GP s effcent, but the effcency of the provded soluton when the goals are not feasble remans an open problem. Major advantage of goal programmng s that there always exsts a soluton to the problem, provded that t has feasble regon and ths s because of the ncluson of the devatonal varables. References [1] Dantzg, G.B. (1948) Programmng n a Lnear Structure. Comptroller, Unted States Ar Force, Washngton DC. [2] Charnes, A. and Cooper, W.W. (1961) Management Models and the Industral Applcatons of Lnear Programmng. John Wley, New York. [3] Lee, S.M. (1972) Goal Programmng for Decson Analyss. Auerbach, Phladelpha. [4] Ignzo, J.P. (1976) Goal Programmng and Extensons. D. C. Heath and Company, Lexngton. [5] Ln, W.T. (1980) A Survey of Goal Programmng Applcatons. Omega, 8, [6] Rfa, A.K. (1996) A Note on the Structure of the Goal-Programmng Model: Assessment and Evaluaton. Internatonal Journal of Operatons and Producton Management, 16, [7] Schnederjans, M.J. and Kwak, N.K. (1982) An Alternatve Method for Solvng Goal Programmng Problems: A Reply. The Journal of the Operatonal Research Socety, 33, [8] Ken, W. and Perushek, D.E. (1996) Lnear Goal Programmng for Academc Lbrary Acqustons Allocatons. [9] Ken, W. and Perushek, D.E. (2000) Goal Programmng as a Soluton. [10] Sharma, H.P. and Sharma, D.K. (2006) A Mult-Objectve Decson-Makng Approach For Mutual Fund Portfolo. Journal of Busness & Economcs Research, 4, [11] Jafar, H., Koshtel, R. and Khabr, B. (2008) An Optmal Model usng Goal Programmng for Rce Farm. Appled Mathematcal Scences, 2, [12] Orume, U.C. and Ebong, D.W. (2011) An Alternatve Method of Solvng Goal Programmng Problem. Ngeran Journal of Operatons Research, 2, [13] Nabendu, S. and Mansh, N. (2012) A Goal Programmng Approach to Rubber Plantaton Plannng n Trpura. Appled Mathematcal Scences, 6, [14] Crowder, L.J and Sposto, V.A. (1987) Comments on An Algorthm for Solvng the Lnear Goal-Programmng Problem by Solvng Its Dual. The Journal of the Operatonal Research Socety, 38, [15] Tamz, M., Jones, D.F. and El-darzn, E. (1995) A Revew of Goal Programmng and Its Applcatons. Annals of operaton Research, 58, [16] Tamz, M. and Jones, F. (1997) Interactve Frame Works for Investgatng of Goal Programmng Models. Theory and Practce. Journal of Mult-Crtera Decson Analyss, 6, [17] Charles, H.F., Boaz, G. and Hussen, N. (2005) Modellng Tradeoffs n Three-Dmensonal Concurrent Engneerng: A Goal Programmng Approach. Journal of Operatons Management, 23, [18] Alp, S., Yavuz, E. and Ersoy, N. (2011) Usng Lnear Goal Programmng n Surveyng Engneerng for Vertcal Network Adjustment. Internatonal Journal of the Physcal Scences, 6, [19] Rosenthal, R. E. (1983) Goal Programmng A Crtque. NZOR, 11, 8. [20] IJr, Y. (1965) Management Goals and Accountng for Control. Rand-McNally, Chcago. [21] Orume, U.C. and Ebong, D.W. (2013) An Effcent Method of Solvng Lexcographc Lnear Goal Programmng Problem. Internatonal Journal of Scentfc and Research Publcatons, 3, 1-8. [22] Tamz, M. and Jones, D.F. (2002) Goal Programmng n the Perod In: Ehrgott, M. and Gandbleux, X., Eds., Multple Crtera Optmzaton: State of the Art Annotated Bblographc Surveys, Kluwer, [23] Wnston, W. (2004) Operatons Research: Applcatons and Algorthms, Duxbury Press, Pacfc Grove. 69

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