Preemptive Possibilistic Linear Programming: Application to Aggregate Production Planning

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1 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Preeptve Pssblstc Lnear Prgrang: Applcatn t Aggregate n Plannng Phruksaphanrat B. Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 Abstract Ths research prpses a Preeptve Pssblstc Lnear Prgrang (PPLP) apprach fr slvng ultbjectve Aggregate n Plannng (APP) prble wth nterval deand and precse unt prce and related peratng csts. The prpsed apprach attepts t axze prft and nze changes f wrkfrce. It transfrs the ttal prft bjectve that has precse nfratn t three crsp bjectve functns, whch are axzng the st pssble value f prft, nzng the rsk f btanng the lwer prft and axzng the pprtunty f btanng the hgher prft. The change f wrkfrce level bjectve s als cnverted. Then, the prble s slved accrdng t bjectve prrtes. It s easer than sultaneusly slve the ultbjectve prble as perfred n exstng apprach. Pssble range f nterval deand s als used t ncrease flexblty f btanng the better prductn plan. A practcal applcatn f an electrnc cpany s llustrated t shw the effectveness f the prpsed del. Keywrds Aggregate prductn plannng, Fuzzy sets thery, Pssblstc lnear prgrang, Preeptve prrty A I. ITRODUCTIO GGREGATE n Plannng (APP) s a edu range capacty plannng that typcally encpasses a te hrzn fr 3 t 8 nths. A prductn planner ust ake decsns regardng utput rates, eplyent levels, nventry levels, backrderng level as well as subcntractng t ptze the prductn plan. Ang the nuerus ethds capable f develpng atheatcal ptzatn dels nclude APP prbles []-[4], Lnear Prgrang (LP) s a cnventnally used technque. Hwever, LP dels assue that all requred data nput can be unquely deterned. These dels cannt be appled t real APP prbles snce a Decsn Maker (DM) frequently has nsuffcent nfratn n hw t specfy deand, related peratn csts ceffcents and unt prce by crsp real nubers. These data are typcally fuzzy n nature. Mrever, LP can deal wth nly a sngle bjectve prble. In general, an APP prble nvlves ultple bjectves, whch are ften cnflctng n nature [4], such as nze cst, axze prft, nze nventry level r nze change f wrkfrce level etc. The nput data r paraeters n real-wrld, such as deands, resurces, csts and unt prce are ften precse r Ths wrk was supprted n part f atnal Research Unversty Prject f Thaland Offce f Hgher Educatn Cssn, Faculty f Engneerng, Thaasat Unversty, THAILAD. B. Phruksaphanrat s an Assstant Prfessr, ISO-RU, Industral Engneerng Departent, Faculty f Engneerng, Thaasat Unversty, Rangst capus, Klngluang, Pathu-than, 220, Thaland (e-al: lbusaba@engr.tu.ac.th). fuzzy because se nfratn s ncplete r unbtanable. Cnventnal atheatcal prgrang cannt slve all prbles thse have precsn. In dealng wth precse data, se researchers ay apply stchastc prgrang t slve. Hwever, the an prble s the lack f cputatnal effcency and nflexble prbablstc dctrnes n whch the real precse eanng f the DM ght be pssble t del [5]. In 976, Zerann frst ntrduced fuzzy set thery nt cnventnal LP prbles [6]. That study cnsdered LP prbles wth a fuzzy bjectve and cnstrants, whch ultple bjectves prbles can be slved. Fllwng the fuzzy decsn-akng ethd prpsed by [7],[8] any f Fuzzy Lnear Prgrang (FLP) dels have been develped fr slvng ndustral prbles [9]-[]. Mrever, Zadeh (978) presented the prnence f the thery f pssblty, whch s related t the thery f fuzzy sets by defnng the cncept f a pssblty dstrbutn as a fuzzy restrctn, whch acts as an elastc cnstrant n the values that can be assgned t a varable [5]. He denstrated the sgnfcance f the thery f pssblty stes fr the fact that uch f the nfratn n whch huan decsns s based n s pssblstc rather than prbablstc n nature [2]. In 992, La and Hwang prpsed a new apprach t se f Pssblstc Lnear Prgrang (PLP) prbles. Wang and La (2005) prpsed a PLP del fr slvng a sngle bjectve APP prble wth precse deand, paraeters and capacty [5]. The fuzzy bjectve s cnverted t a Multple Objectve Lnear Prgrang (MOLP) del usng the ethd f La and Hwang [5]. Afterward, pssblstc ptzatn ethds have been appled n se f practcal applcatns [5],[3]-[20].FLP s based n the subjectve preferred cncept fr establshng ebershp functns wth fuzzy data, whle the PLP s based n the bjectve degree f event ccurrence requred t btan pssblstc dstrbutns wth precse data. FLP technques ay nt be applcable fr PLP [5]. PLP prvdes cputatnal effcency and flexblty. It als supprts pssblstc decsn akng n an uncertan envrnent [5],[6],[8]. PLP apprach sultaneusly nzes the st pssble value f the precse ttal csts, axzes the pssblty f btanng lwer ttal csts, and nzes the rsk f btanng hgher ttal csts. Iprecse frecast deands are cnverted t crsp deands by adptng weght average ethd. Other studes f PLP prbles als use ths strategy t slve ther applcatns [5],[6]-[8]. Anther way t deal wth uncertan nfratn s usng f nterval nubers [2]-[22]. Recently, L and Huang (200) prpsed Interval based Pssblstc Prgrang (IPP) by slvng sub-prbles t fnd nterval slutns because any f uncertan paraeters are expressed as fuzzy sets, nteractns ang these uncertantes ay lead t serus cplextes [2].Sultaneusly slve MOLP by adjustng ther ebershp functns are dffcult and take te because Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

2 c ~ j Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 each bjectve has dfferent range r scale t adjust. Mrever, ne bjectve ay extreely prtant than the thers. Especally n an APP prble, ttal prft s extreely prtant than the thers. S, ths bjectve shuld be set as the frst prrty. Fr frecast deand, DM ay feel uncfrtable t estate the deand n each perd because pprtunty lss f sales and prft wll ccur f deand s under-estated r hldng cst wll ncrease f deand s ver-estated. Weghts whch are used t transfr fuzzy deand n PLP del ay nt apprprate because after ultplyng weghts wth ther deands, the crsp deand s btaned. If the DM can estate the pssble range f frecast deand, gvng ths pssble range t fnd the ptal slutn f a prductn plan s better than gvng a crsp deand because reducng r ncreasng a few unts f deand ay extreely ncrease r decrease prductn cst due t change f the nuber f prductn lne. S, ths wrk prpses a ult-bjectve APP prble usng a Preeptve Pssblstc Lnear Prgrang (PPLP) apprach, whch can effectvely fnd the cprse slutn. The prpsed ethd slves the prble accrdng t bjectve prrtes. Paraeters related t the bjectve ceffcents are precse. These data are del by trangular pssblstc dstrbutns. Deand n each perd s cnsdered as an nterval deand fr the pssble range f deand. Tw an bjectve functns are cnsdered n the del. They are t axze prft and t nze change f wrkfrce level.the structure f the paper s as fllws: Sectn 2 descrbes the ntatns and frulatn f an APP prble. Sectn 3 prpses the PPLP del and the slutn prcedure. Sectn 4 llustrates the APP prble f a case study t shw the effectveness f the prpsed del. Fnally, sectn 4 delneates cnclusn and scpe f future wrks. II. OTATIOS AD FORMULATIO In rder t descrbe the ult-prduct ult-perd APP prble atheatcally, ntatns belw are ntrduced. Assue that a cpany anufactures types f prducts t satsfy the pssble arket deand ver a plannng hrzn T. Multple bjectve functns wth precse nfratn and pssble nterval deand s deterned n ths research. Indces: nuber f prduct types, =,2,,. t nuber f perds (nth) n the plannng hrzn, t =, 2,,T. Paraeters: Dt upper lt f frecast deand f prduct n perd t, (unts). Dt lwer lt f frecast deand f prduct n perd t, (unts). p unt prce f prduct, ($/unt). r regular prductn cst per unt f prduct, ($/unt). verte prductn cst per unt f prduct,($/unt). l t layng ff cst per wrker n perd t, ($/wrker). h t hrng cst per wrker n perd t, ($/wrker). s hldng cst per unt, ($/unt). b backrderng cst per unt f prduct n perd t, ($/unt). nuber f peratrs per prductn lne, (an/lne). d nuber f days n perd t, (days). a prcessng te per unt f prduct, (hrs/unt). n prductn capacty per prductn lne per perd f prduct, (unts/ lne/perd) regular wrkng hur per wrker per day, (hrs/an-day) verte wrkng hur per wrker per day, (hrs/an-day) A t nu prductn quanttes (regular prductn and verte prductn) n each perd t, (unts). I t ax axu nventry level n perd t, (unts). Wt ax axu labr level n perd t, (an). Decsn varables: Dt satsfed deand n an nterval [ D t, D t ] f prduct n perd t, (unts). X t regular te prductn f prduct n perd t, (unts). Yt verte prductn f prduct n perd t, (unts). Bt backrderng level f prduct n perd t, (unts). I t nventry level f prduct n perd t, (unts). Wt nuber f wrkers n perd t, (an). H t hred wrker n perd t, (an). Lt lad ff wrker n perd t, (an). t nuber f prductn lnes f prduct n perd t, (lnes). dentes uncertan nfratn. A. Objectve functns There are tw bjectve functns; t axze prft (O ) and t nze the changes f wrkfrce level (O 2 ). The frst bjectve functn s t axze the ttal prft Prft s the an bjectve functn fr every cpany. It ces fr revenue nus csts. Revenue generates fr quanttes f custer deands nus backrder parts ultply by unt prce f all prducts. In ths del, backrderng s unacceptable because st f custers wll nt wat fr delay parts. Csts cpse f ttal prductn cst fr regular and verte prductns, cst f changng wrkfrce level, nventry cst and penalty f backrderng cst. Fuzzy prft functn s represented by Max O p *( D B) ( rx Y ) T T t t t t t t T T T ( ll hh ) ( si ) ( bb ). t t t t t t t t t () The related unt prce ( p ) and cst ceffcents ( r,, l t, h t, s, b ) are precse. Trangular pssblty dstrbutns are used t represent these data. These are dscussed n the next sectn. The secnd bjectve functn s t nze the changes Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

3 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 f wrkfrce level Changng wrkfrce level effects rale and stablty fr wrkers. Ths level shuld be nzed. The ttal changes f wrkfrce level are the nubers f lad ff and hred wrkers. Ths bjectve can be shwn as T Mn O 2 = (L t +H t ). (2) t= B. Cnstrants The labr level cnstrants: Fr each perd, the fllwng cnstrants are appled: W W H L t. (3) t t t t, Wt Wtax, t. (4) * t dt * Wt, t. (5) The wrkfrce level n each perd ( W t ) shuld equal the wrkfrce level n the prevus perd ( Wt ) plus the new hred wrkers (H t ) nus the lad ff wrkers (L t ) as shwn n (3). The wrkfrce level n each perd shuld nt be greater than the axu avalable wrkfrce level as shwn n (4). Each prductn lne needs a specfc nuber f wrkers. uber f wrkers n all prductn lnes n each perd shuld be equal t the avalable wrkfrce level n perd t. The nventry level cnstrant: The suatn f nventry level f all prduct types shuld nt greater than the axu avalable nventry capacty n each perd. It It ax, t. (6) The prductn cnstrants: Xt Yt It Bt It Dt, t. (7) D D D, t. (8) t t t t t t X Y A, t. (9) Regular prductn, verte prductn, nventry level n prevus perd, backrdered unts f the current perd nus nventry level f the current perd f prduct equal t deand f the current perd f prduct as shwn n (7). Cnventnally, crsp deand s assued n the APP del. In ths prpsed del, a pssble nterval deand s used due t uncertanty n deand. It can ncrease flexblty n prductn because the apprprate level f deand can be selected fr the pssble range f deand fr prductn. Equatn (8) shwn that satsfed deand s n an nterval f a pssble nterval deand range. The suatn f regular prductn and verte prductn quanttes fr all prducts n each perd shuld nt lwer than the nu requreent n each perd as shwn n (9). The prductn capacty cnstrants: ax t dw t t, t. (0) ay t dw t t, t. () n t Xt, t. (2) n t Yt, t. (3) The regular and verte prductn hurs shuld nt be greater than the avalable labr hur n each perd as shwn n (0),(). Equatns (2) and (3) represent the nuber f prduct prduce n perd t fr regular te and verte. III. MODEL DEVELOPMET A. Mdelng the precse data wth trangular pssblty dstrbutn Ths wrk assues that a trangular pssblty dstrbutn can be stated as the degree f ccurrence f an event wth precse data [5]. Fg. presents the trangular pssblty dstrbutns f precse unt prce, p ( p,, p p p ) and cst ceffcents, A =(A,A,A ). These knds f precse nfratn exst n the frst bjectve functn ( O ). In practce, a DM can ake trangular pssblty dstrbutns f p and A based n the three prnent data, as fllws. p p The st pessstc values ( p, A ) that defntely belngs t the set f avalable values (pssblty degree = 0 f nralzed).the st pssble values ( p, A ) that defntely belngs t the set f avalable values (pssblty degree = f nralzed). p p p p p a) Pssblty dstrbutn f p A A A b) Pssblty dstrbutn f Fg. The trangular pssblty dstrbutn f unt prce, and cst ceffcents, A The st ptstc values ( p, A ) that has a very lw lkelhd f belngng t the set f avalable values (pssblty degree = 0 f nralzed). Cst ceffcents, A are cnsdered as precse data ( r,, l t, h t, s, b ). Trangular dstrbutns f these data can be wrtten as fllws: p A p A Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

4 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 r ( r, r, r ),. (,, ),. l t ( lt, lt, lt ), t. h ( h, h, h ), t. t t t t s ( s, s, s ). b ( b, b, b ),. B. An addtnal MOLP del La and Hwang (992) referred t prtfl thery and cnverted the fuzzy bjectve wth a trangular pssblty dstrbutn nt three crsp bjectves. Accrdng t ther ethd, the frst bjectve functn ( O ) can be fully defned by three prnent pnts (Z p, 0), (Z, ) and (Z, 0) as shwn n Fg. 2. The precse bjectve can be axzed by pushng the three prnent pnts twards the rght. Because f the vertcal crdnates f the prnent pnts beng fxed at ether r 0, the three hrzntal crdnates are the nly cnsderatns [5]. Cnsequently, slvng the precse bjectve requres axze Z, axze Z -Z but nze Z -Z, sultaneusly. These nvlve axze the st pssble value f the precse prft, Z, nze the rsk f btanng lwer prft, (Z -Z p ), and axze the pssble f btanng hgher prft, (Z -Z ). Three new crsp bjectve functns, (Z, Z 2, Z 3 ) are presented as fllws. Pssblty degree, Z 0 p Z Z Z Fg. 2 The trangular pssblty dstrbutn f fuzzy bjectve, axz Z I T I T p *( Dt Bt) ( r Xt Y t) t t T I T I T ll t t hh t t si tbb t t t t ( ). Z O. (4) n p Z2 Z Z I T p ( p p )*( Dt Bt ) t I T p p r r Xt Yt t T p p [( lt lt ) Lt ( ht ht ) Ht] t I T I T p p ( s s ) It ( b b ) Bt. t t (5) [( ) ( ) ] ax Z3 Z Z I T ( p p )*( Dt Bt ) t I T [( r r ) Xt ( ) Yt ] t T t t I T I T ( s s ) It ( b b ) Bt. t t (6) [( l l ) L ( h h ) H ] The reanng bjectve functn (O 2 ) can be rewrtten as Z 4 as the fllwng equatn. T n Z4 ( Lt Ht). (7) t C.Slvng the addtnal MOLP prble The addtnal MOLP prble can be changed nt an equvalent sngle gal LP prble usng the fuzzy decsn akng f Bellan and Zadeh [8] and Zerann s fuzzy prgrang ethd [6],[7]. The pstve deal slutn (PIS) and negatve deal slutn (IS) [23] f these bjectve functns can be used t defne a ebershp functn f each bjectve as fllws. PIS IS ax, n PIS p IS p 2 2 PIS IS 3 3 PIS IS 4 n, ax Z Z Z Z. (8) Z n( Z Z ), Z ax( Z Z ). (9) Z ax( Z Z ), Z n( Z Z ). (20) Z Z Z Z. (2) Fr each crsp bjectve functn, the crrespndng lnear ebershp functn s defned by PIS f Z Z IS Z Z IS PIS ( Z) f Z PIS IS Z Z, (22) Z Z IS 0 f Z Z t Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

5 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 f ( Z Z ) Z p PIS 2 IS p Z2 ( Z Z ) PIS p IS 2 IS PIS 2 2 Z2 Z2 ( Z ) f Z ( Z Z ) Z, (23) p IS 0 f ( Z Z ) Z2 f ( Z Z ) Z PIS 3 IS ( Z Z ) Z3 IS PIS 3 PIS IS 3 3 Z3 Z3 ( Z ) f Z ( Z Z ) Z, (24) IS 0 f ( Z Z ) Z3 f Lt Ht Z IS Z4 ( Lt Ht) PIS IS 4 IS PIS 4 t t 4 Z4 Z4 ( Z ) f Z L H Z IS 0 f Lt Ht Z4 4, (25) Then, the cplete equvalent ult-bjectve del fr slvng the APP prble can be frulated. rally Zeran s equvalent sngle-bjectve lnear prgrang del s used t btan the verall satsfactn cprse slutn [5][6]-[8]. Hwever, n rder t btan the satsfactry slutn, DM needs t dfy ebershp functn f each bjectve untl the cprse slutn s fund. It s dffcult t adjust these ebershp functns due t the dfference n range r scale. Mrever, DM ay need hgher degree f satsfactn level f ne bjectve functn than the thers. The better way t adjust the level f satsfactn fr dfferent prrty f bjectves can be dne by preeptve prrty, whch s usually apply t gal prgrang [23]. Ths type f weght can als be appled t PLP del. It s easer t set the desre level accrdng t ther prrtes and slve the rderly than adjustng ebershp functns f all bjectves sultaneusly. In the preeptve prrty fuzzy gal prgrang, the kth prrty, P k s preferred t the next prrty, P k +. The relatnshps ang the prrtes are P >>>P 2 >>> >>>P k >>> >>>P K. (26) Expressn (26) ndcates that the gals at the hghest prrty level (P ) have been acheved t the nu extent pssble, befre the set f gals at the secnd prrty level (P 2 ) are taken nt cnsderatn and the prcess ges n untl the last prrty level P K s cnsdered [24]. After applyng the preeptve prrty t the PLP fr APP, the Preeptve Pssblstc Lnear Prgrang (PPLP) fr APP wth pssble nterval deand can be transfred t the equvalent preeptve LP del as fllws: lex ax[,..., K ] (27) Subject t * k k, k,..., K. (3)-(3), (22)-(25). are, the ebershp functns f the bjectves that are k rank t be the kth prrty. * k s the desrable acheveent degrees fr the ebershp functns f kth prty. The DM cannt nly frulate preeptve prrty structure fr the prble, but als can requre nu acheveent degrees fr se bjectve belngng t the sae prrty level. D. Algrth The algrth f the prpsed PPLP apprach fr slvng an APP prble s as fllws.. Frulate the PPLP del fr the APP prble. 2. Deterne pssble range f nterval deand. Mdel the precse ceffcents ( p, r,, l t, h t, s, b ) usng trangular pssblty dstrbutns. 3. Develp crsp bjectve functns f the auxlary MOLP prble that are axzng the st pssble ttal prft, nzng the rsk f btanng lwer ttal prft, axzng the pprtunty t btan the hgher ttal prft and nzng the change f wrkfrce level. 4. Order prrty f all crsp bjectve functns. 5. Specfy lnear ebershp functns f crsp bjectve functns, and then cnvert the auxlary MOLP prble nt an equvalent preeptve LP del. 6. Slve and dfy the del nteractvely accrdng the bjectve prrty by settng the desrable acheveent degrees fr the ebershp functn f the kth * bjectve prrty ( k ) untl a satsfactry slutn s fund. IV. A CASE STUDY A. Case descrptn A real case study f a cpany wh prduces electrnc cpnent s llustrated. The plannng hrzntal s 6 nths. There are 6 dels. Pssble range f a frecast deand s estated fr a crsp frecast deand and ts errr fr the actual deand f hstrcal data. The unt prce, regular prductn csts, verte prductn csts and backrder cst are precse nubers wth trangular pssblty dstrbutns. Intal labr level s 84 wrkers wth 6 prductn lnes (6 wrkers per lne). Hrng and frng csts are $64.84 and $78.67 per wrker. Labr hur per day s 6 hurs fr regular te and 5.5 hurs fr verte prductn. Inventry cst per unt s $0.00. Maxze nventry level fr each perd s 304,050 unts. Capacty per prductn lne s 5,500 unts fr all prducts except prduct 5 and 6 capacty are 2,750 unts. Prcessng te s 0. hur per unt fr all prducts except prduct 5 and 6 are 0.22 hur per unt. Wrkng day fr perd t 6 are 25, 24, 26, 25, 24, respectvely and the axze labr level fr each perd are 96, 97, 98, 20, 20 and 20, respectvely. Begnnng nventry fr prduct -6 are 750, 2,00, 0, 0, 3,657, 0, 43, 2,39, 0, 0, 0,,384, 3,358, 7,360,,325 and 680 unts. Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

6 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 B. Slvng prcedures The slutn prcedures usng the PPLP del fr the case study s descrbed as fllws:. Frulate the PPLP del fr the APP prble usng ()-(3). 2. Deterne pssble range f nterval deand. Mdel the precse ceffcents ( p, r,, b ) usng trangular pssblty dstrbutn. Pssble range f frecast deand data s 2% f crsp frecast deand as shwn n Table I. The unt prce, regular prductn cst, verte prductn cst and backrder cst are precse nubers wth trangular pssblty dstrbutns shwn n Table II-III. 3. Develp crsp bjectve functns f the auxlary MOLP prble accrdng t (4)-(7). 4. Order prrty f bjectve functns. Fr the DM s vewpnt fr APP prble f the case study cpany, the bjectve functns can be ranked as fllws: The st prtant bjectve s the frst bjectve (ttal prft) that needs t be axzed. It shuld be satsfed at ne level f satsfactn. Secndly, the rsk t btan the lwer level f prft shuld be nzed. Mst f cpanes try t reduce ther rsks as uch as pssble. They ay allw drppng se f ther prft. TABLE I CRISP FORECAST DEMAD DATA Prd. Crsp frecast deand data n each perd (unts) 95,000 70, ,396 65,000 2,000 60, ,000 20, , , ,000 55, ,000 29,063 30,000 35,000 43,000 45, ,000 5,24 33,000 8, ,000 70,000 60,000 70,000 35,000 26, , , ,000 9, , ,000 40,000 44, ,000 40, , ,55 8,000 5, ,000 40,000 30,000 9,006 27,98 40,000 20, ,000 43,000 35,000 28,0 2 8,000 0,000,640 3,789 8,000 26, ,000 58,356 4,600 80,000 70, , , , ,400 40, , , ,000 35, ,000 20,000 0,000 Ttal,703,564,526,355,673,363,437,559,453,98,646,7 TABLE II IMPRECISE UIT PRICE Prce ($) Prce ($).420,.440, ,2.475, ,.440, ,.843, ,2.360, ,2.289, ,.989, ,450, ,.680, ,2.480, ,.520, ,.424, ,.85, ,.444, , ,.523,.543 TABLE III IMPRECISE REGULAR PRODUCTIO COST, OVERTIME PRODUCTIO COST AD BACKORDER COST Regular prductn cst ($) Overte prductn cst ($) Backrderng cst ($).047,.044, ,.048, ,0.26, ,.044, ,.048, ,0.26, ,.75, ,.754, ,0.354, ,.35, ,.358, ,0.298, ,0.660, ,0.669, ,0.252, ,0.660, ,0.664, ,0.227, ,.752, ,.756, ,0.272, , ,2.034, ,0.368, ,2.22, ,2.26, ,0.37, ,.432, ,.436, ,0.276, ,.750, ,.754, ,0.343, ,.700, ,.704, ,0.368, ,.750, ,.754, , ,.326, ,.330, ,0.28, ,.326, ,.330, ,0.27, ,.749, ,.753, ,0.228,0.226 ext, the change f wrkfrce level shuld be nzed, f t s pssble. Fnally, the pprtunty t btan the hgher prft shuld be ncreased, f there are se chances t prve the plan. S, bjectve prrty f ths case study s Z, Z 2, Z 4 and Z Specfy lnear ebershp functns f these crsp bjectve functns usng PIS and IS. The PIS f Z - Z 4 are ($3,2,90.33, $27,754.86, $478,550.87, 0) and the IS f Z -Z 4 are ($-2,550,249.26, $480,740.55, $246,908.84, 48). The crrespndng lnear ebershp functns f the fur bjectve functns can be defned accrdng t (22)-(25). ext, cnvert the auxlary MOLP prble nt equvalent preeptve LP del (27). 6. Slve and dfy the del accrdng the bjectve prrty by settng the desrable acheveent degrees * fr the ebershp functn f the kth prrty ( k ). Satsfactn levels f selected cprse slutn fr bjectve t 4 are.00, 0.602, 0.59 and 0.625, respectvely. The results f the sngle bjectve ptzatn wth crsp deand and nterval deand, the cprse slutn by PPLP del are shwn n Table IV. It can be seen that nterval deand can btan the better prft than crsp deand due t the flexblty f selectng the apprprate deand fr prductn. In ths case study, range f Z s 5,762,59.56, PIS IS IS PIS ( Z Z ), but range f Z 4 s 48, ( Z4 Z4 ). These range are extreely dfferent, t s dffcult t adjust f PLP apprach s appled. PPLP ethd des nt requre adjustng f ebershp functns. DM can select the apprprate level f satsfactn f each bjectve accrdng t ts prrty that DM desres. PPLP can btan the cprse slutn, whch has the hghest prft and has addtnal nfratn abut Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

7 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 pessstc case and ptstc case f the ttal prft. It s better than the sngle bjectve ptzatn. Mrever, uncertanty s als cnsdered. DM can adjust the desre level f satsfactn f each bjectve t get the ther satsfactry slutns. TABLE IV OPTIMIZATIO RESULTS Maxze/Mnze Prft ($) Maxze ttal prft wth crsp deand and precse nfratn Mnze change f wrkfrce level wth crsp deand and precse nfratn Maxze ttal prft wth nterval deand and precse nfratn Cprse slutn wth nterval deand and precse slutn Change f wrkfrce (Man) 3,50, ,2, ,889, ,2, ,595, TABLE V SATISFIED DEMADS Satsfed deand n each perd (unts) 98,900 73, ,684 66,300 2,240 6, ,400 22, ,27 459, ,000 58,00 3 0,200 29,644 30,600 35,700 43,860 45, ,240 5,226 33,660 8, ,300 7,400 6,200 7,400 35,700 26, , , ,600 94, , ,400 42,800 45, ,000 42, , ,602 8,60 5, ,800 40,800 30,600 9,386 27,742 4,600 20, ,600 43,860 35,700 28,57 2 8,60 0,200,673 3,865 8,360 26, ,000 59,523 4,892 8,600 7, , ,300 62, ,08 48, ,200 86, ,800 34, ,800 9,600 07,800 TTL,735,235,555,467,706,830,463,90,48,462,675,439 TABLE VI REGULAR PRODUCTIO Regular prductn n each perd (unts) 92,372 7, ,047 65,550,490 60, ,628 06,49 270,40 446, ,990 46, ,86 28,65 28,65 35,700 43,860 45, ,240 5,226 33,660 8,60 5 6,395 67,535 57,302 67,743 32,043 22, ,326 88,279 22,837 94, , ,907 39,63 40, ,587 42, , ,602 8,60 5, ,837 36,837 28,65 9,386 27,742 40,930 6, ,65 43,860 35,700 28,57 2 6,776 8, ,48 6,976 24, ,674 49,6 7,532 74,240 64,040 96, ,605 5, ,605 46,875 67,875 84, ,20 8,920 98,233 8 Table V shws the satsfed deands that were btaned fr the prpsed PPLP del, these deands are a lttle bt greater than the crsp frecast deand presented n Table I but t s n the pssble range f frecast deand. Usng pssble range f deand has advantages ver crsp deand because the del can select the best slutn fr DM fr the pssble range f deand that can satsfy the axu utlzatn f capacty. It s re flexble than exstng dels. The nuber f prductn quanttes by regular and verte prductn are shwn n Table VI-VII. TABLE VII OVERTIME PRODUCTIO Overte prductn n each perd (unts) 5, , ,762 3,97 87, ,04 993, , ,674 62,64 72, ,080 3,224 4, ,963 3,963, ,028 0, ,966 3, ,370 99,047 78, TABLE VIII UMBER OF PRODUCTIO LIE uber f prductn lne n each perd (lnes) TTL TABLE IX LABOR LEVEL,HIRIG LEVEL, AD FIRIG LEVEL Perd Ite. f wrker (en) Hred wrker (en) Internatnal Schlarly and Scentfc Research & Innvatn 5(8)

8 Wrld Acadey f Scence, Engneerng and Technlgy Internatnal Jurnal f Industral and Manufacturng Engneerng Internatnal Scence Index, Industral and Manufacturng Engneerng waset.rg/publcatn/7630 TABLE X IVETORY LEVEL Inventry level n each perd (unts) ,86 0 2,39 2,39 2,39 2,39 3 3,358 3,358 3,358 3,358 3,358 3,358 Table VIII shws the nuber f prductn lne n each nth. The nuber f prductn lne per day fr perd t perd 3 s 6 prductn lnes and fr perd 4 t perd 6 s 7 prductn lnes. Currently the nuber f prductn lne per day s 4 lnes. It wll be ncreased 2 lnes at the begnnng f perd wth addtnal 2 wrkers. At the end f perd 3, 6 addtnal wrkers wll be hred fr ne re prductn lne a day. Ttal nuber f wrker n each perd s shwn n Table IX. Backrderng unts fr perd t perd 6 ccur nly fr prduct 6, they are 58,20, 33,997, 0, 0, 0, 8,887 unts. Inventry levels are nt necessary fr alst all prducts n each perd except prduct 7, 8 and 3 as shwn n Table X. V.COCLUSIO Ths research presents a Preeptve Pssblstc Lnear Prgrang (PPLP) ethd fr slvng ultple bjectves f an APP prble wth tw bjectve functns; t axze prft and nze change f wrkfrce level. The prpsed PPLP apprach attepts t axze the st pssble ttal prft, nze the rsk f btanng the lwer ttal prft, nze the change f wrkfrce level and axze the pprtunty t btan the hgher prft respectvely by settng the satsfactn level f each bjectve. Ths ethd can reduce the prble f adjustng the ebershp functns f exstng PLP apprach. DM can anpulate the cprse slutn based n wn preferences t get the satsfactry slutn. Pssble frecast deand nterval s used n the del. It can ncrease flexblty t the del t btan the better slutn. Mrever, utlzatn f prductn lnes can be apprprately planed fr each perd. Ths ethd can be appled t thers case studes n ndustral applcatns. Iprecse capacty can als be ncluded. REFERECES [] F. Hanssann and S.W. 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