A Production-Delivery Inventory System under Continuous Price Decrease and Finite Planning Horizon

Transcription

1 Prceedigs f the 008 Idustrial Egieerig esearch Cferece J. Fwler ad S. as, eds. A Prducti-elivery Ivetry System uder Ctiuus Price ecrease ad Fiite Plaig Hriz Jufag Yu epartmet f Egieerig aagemet, Ifrmati & Systems Suther ethdist Uiversity, allas, exas , USA Bhaba. Sarker ad eiz uga epartmet f Idustrial Egieerig Luisiaa State Uiversity, Bat uge, Luisiaa 70803, USA Abstract I tday s cmpetitive wrld, the cst f high-tech prducts declies sigificatly ver their life cycle. A ivetry mdel fr such prducts is develped fr a rderig plicy f raw materials, ad fiished gds which is delivered t the custmer at a fixed iterval f time i a fixed quatity. his mdel itegrates the raw material ivetry with the fiished gds ivetry alg the supply chai. he gal f this research is t miimize the ttal cst f the supply chai while the price f the high-tech prduct is liearly decreasig. Keywrds Prducti-delivery system, just-i-time plicy ad ctiuus price decrease. 1. Itrducti I recet years, e f the appraches that have had a majr impact supply chais is just-i-time (JI) ivetry maagemet. JI plicy frces maufacturer t prduce required quatities f gds at the required time. I a JI evirmet, a supplier eeds t adjust the prducti schedule simultaeusly with the buyer s demad. At each cycle the supplier eeds t rder the exact required quatities f raw materials t prduce the expected demad f buyers while reducig the ivetry cst. A defiig characteristic f high-tech idustries is shrt prduct cycle with decreasig cmpet prices. he decreasig price leads buyers t place their rder at the last miute just fr small lts i rder t t icrease their ivetry cst. By dig this, they save mey frm bth payig less fr the purchasig cst ad less fr carryig cst. herefre, shrter prduct life cycles ad buyers rapid demads i small lts frce maufacturers t respd quicker ad t shrte the cycle times i tday s cmpetitive market. his explais the heavy emphasis the JI plicy practiced by the techlgy-related cmpaies. Glhar ad Sarker [1], Jamal ad Sarker [], ad Sarker ad Parija [3] develped varius sluti methdlgies fr the itegrated mdel ad determied a ptimal r ear-ptimal rderig plicy fr prcuremet f raw materials ad the maufacturig batch size t miimize the ttal cst while csiderig equal shipmets f the fiished prducts, at fixed itervals, t the buyers. hey csidered the JI philsphy i maufacturig sectrs ad frmulated the mdel frm the pit f view f the beefit t the maufacturig firm. he urealistic assumpti here is that the uit cst is cstat ver the plaig hriz. Hwever, especially fr the successful cmpaies i high-tech idustries, this is t a reasable assumpti. he price f the cmpets ad fiished gds decreases ctiuusly durig their life cycle. ecetly, Khuja et al. [4] develped a efficiet algrithm fr slvig the jit repleishmet prblem fr prducts that may be experiecig uit cst icrease r decrease. hey tested their prpsed algrithm ad shwed that it idetifies the glbal ptimal slutis i mst cases. he mai purpse f this research is t develp a itegrated ivetry mdel fr high-tech idustries i a JI evirmet uder ctiuus price decrease ver fiite plaig hriz while effectively ad successfully accmplishig supply chai itegrati s that the ttal cst f the system is miimal. 1580

2 Yu, Sarker ad uga. Prducti-elivery del uder Fiite Plaig Hriz A sigle-stage prducti-delivery ivetry mdel uder a fiite plaig hriz is develped t shw the sigificace f icrpratig the price chage it the supply chai system. he mdel i a JI evirmet fr high-tech idustries demstrates the relaxati f cstat uit cst i the previusly develped ivetry mdels. It is assumed that a prducti facility purchases raw materials frm utside suppliers i a fixed size ad cverts them it fiished prducts that are t be delivered t a buyer at a fixed iterval f time. Ulike the icreasig ivetry built up i a traditial ecmic maufacturig mdel with a ctiuus demad, a saw-tth fashi ivetry mdel is built up here durig the prducti perid. his is because a fixed demad f x uits f fiished gds is istataeusly csumed at the ed f every successive shipmet perid, L, due t the fixed-iterval batch supply. the ther had, prducts experiecig ctiuus decrease i price are csidered i this mdel. We assume that the uit cst f the fiished prduct is equal t the ttal cst f the required raw materials plus ther fixed maufacturig csts. A sigle-stage prducti-delivery system uder a fiite plaig hriz is develped t miimize the ttal ivetry csts f raw materials ad fiished gds. his ivetry mdel ca be applied t may techlgy-related cmpaies whse prducts are experiecig shrt life cycles with decreasig uit csts..1 del Frmulati hree types f ivetry csts are csidered fr bth raw materials ad fiished prducts. he csts csidered fr raw materials are: the raw material rderig cst, the raw material purchasig cst ad the raw material carryig cst, whereas the csts fr fiished gds are: the maufacturig setup fr each batch, the fiished gds maufacturig cst ad the fiished gd carryig cst. A cst fucti is develped herei t determie a ptimum umber f cycles durig the fiite plaig hriz by miimizig the ttal ivetry cst. Ntati: A rderig cst f raw materials, (dllars/rder) AS aufacturig set-up cst, (dllars/batch) b ecrease i uit cst per uit time, (dllars/year) C aw material cst at time t 0, (dllars/uit) C F (t) Uit fiished gds cst at time t, C F ( t) C ( t) f C, (dllars/uit) C (t) Uit raw material cst at time t, C ( t) C bt, (dllars/uit) C Uit maufacturig cst i.e., margial cst, (dllars/uit) emad fr fiished gds per uit time, (uits/year) emad fr raw materials per uit time, (uits/year) f Cversi factr f the raw materials, f Q Q i Fracti hldig cst f ivetry value per uit time L ime betwee successive shipmets, (i years) m Number f full shipmets f fiished gds, m L Q x x ' m ' Number f pssible shipmets f fiished gds, m L Q x x Number f cycles f equal-legths durig the plaig hriz, P Prducti rate per uit time, (uits/year) Q Average ivetry f fiished gds maufactured per cycle, (uits/cycle) avg Q Quatity f fiished gds maufactured per set up ver perid, (uits/batch) Q Quatity f raw materials required i each batch, Q Q f, (uits/rder) Cycle time durig the plaig hriz, Q ml, (i years) aufacturig perid (uptime), Q P, (i years) P P wtime, Q ( 1 1 P), (i years) P x Fixed quatity f fiished gds per shipmet, x Q m L, (uits/shipmet) Y Quatity prduced durig time L, Y LP Px Y x Fiished gds ivetry built up at the ed f each shipmet durig the uptime, Y x P 1x. Legth f the plaig hriz fr umber f fixed cycles,, (i years) 1581

3 Yu, Sarker ad uga. Geeral Cst Fucti JI purchasig shuld be embraced t miimize the ttal cst by csiderig frequet deliveries i small lts. he itegrated ttal cst fr cycles ver a fiite plaig hriz csists f the ttal cst f the raw materials EQ () ad the ttal cst f the fiished prducts EPQ () as shw belw. ( ) ( ) ( ) (1) EQ..1 aw aterial Csts he ttal cst f the raw materials ver the plaig hriz icludes the rderig cst, the purchasig cst ad the hldig cst. he rderig cst fr rders ver the plaig hriz is A. he uit cst at time j( ) is EPQ p / Q C bj( ). he average ivetry f raw materials ver e cycle is / where P / is a scalig factr fr the hldig cst distributed ver cycle time. Hece, the ttal ivetry hldig cst fr raw material per cycle is ( PQ / ) C bj( ) ( i / ). herefre, the ttal cst fr cycles ver the plaig hriz is EQ ( Q ) = A + 1 j 0 Q C bj + 1 j 0 Q p C he rescalig factr, P, csidered i the raw material hldig cst ca als be represeted by P. Als, raw material ca be trasfrmed t a fiished prduct thrugh the maufacturig prcess at a cversi rate, f Q Q frm which the ttal cst i equati () may be writte as EQ ( Q ) A + 1 Q f j0 Q C bj + C bj i (3) f P j0 I rder t express the cst fucti i terms f (umber f cycles), Q is replaced with i equati (3): EQ ( ) A Nw, it ca be easily shw that f 1 j0 1 C bj j 0 fp i 1 1 j 0 C bj bj i b( 1) C bj C (5) which yields the ttal cst f raw material ver the fiite plaig hriz, EQ () i equati (4) as ( ) A C b( 1) i f fp EQ (6).. aufacturig Csts he ttal cst f maufacturig the fiished gds ver the plaig hriz icludes the setup cst, the prducti cst ad the hldig cst f the fiished gds. he A value represets the ttal setup cst fr all the cycles ver the plaig hriz. he maufacturig cst f e uit f fiished gds is the ttal ivetry cst f fiished prducts ver the plaig hriz is EPQ 1 S 1 () (4) ( C bj( )) f C. herefre, 1 1 ( Q ) AS Q C bj C Qavg C bj C i f (7) f j0 Equati (7) ca be expressed i terms f umber f cycles,, as EPQ As i equati (5), ( ) A S 1 j0 1 j0 1 C f 1 C f bj bj C C C f Q j0 avg C i 1 j0 1 C f b( 1) f bj C (8) (9) 158

4 Hece, equati (8) reduces t Yu, Sarker ad uga C b( 1) EPQ ( ) AS C Qavgi (10) f f Usig the ifrmati f -had ivetry f fiished gds at ay time ad shipmet size x at the ed f every L time uits, fr m shipmets/cycle, Sarker ad Parija [3] develped a expressi fr the average ivetry f fiished gds per cycle, Q as: avg m( m 1) x Qavg Q 1 mx P (11) Q Assumig Q =, equati (11) is further expressed as Q m( m 1) x mx avg P (1) Cmbiig this result with equati (10), i C b( 1) i mxi m( m 1) x i EPQ( m, ) AS C (13) f f P Fially, the itegrated ttal cst fucti, ( m, ) i terms f m shipmets per cycle ad cycles ver the fiite plaig hriz ca be writte as b ( 1) i ( m, ) = EQ () + EPQ ( m, ) = ( A AS ) C f fp C f C b( 1) i i mxi m( m 1) x f P he first term i equati (14) represets the rderig ad the setup cst fr all the cycles, the ext term represet the purchasig ad hldig cst f the raw materials ad the last term represets the maufacturig ad hldig cst f the fiished prducts fr all the cycles ver the fiite plaig hriz..3 ptimal Sluti he ttal cst fucti ( m, ) i equati (14) is a fucti f ly tw decisi variables m ad. It may be ted that there are m shipmets ver the whle plaig hriz. Sice m ad are bth itegers ad m Q x, fidig the ptimal sluti t equati (14) is t trivial, ad thus a clsed-frm sluti fr cat be btaied directly. Glhar ad Sarker [1] applied a efficiet algrithm which was prpsed by izadeh ad Aggarwal (1990) t slve a similar prblem with the help f a discrete ptimizati techique. Subsequetly, Jamal ad Sarker [] imprved the prpsed algrithm t btai a ptimal r ear-ptimal sluti iteratively. his imprved algrithm is exteded i this paper t slve the sigle-stage ivetry mdel uder ctiuusly decreasig uit cst. Algrithm 1: dified Algrithm fr Ecmic aufacturig Quatity Step 1: Iitialize A 0, A S, C 0, b,,, f, P, C, i ad x. Set m = 0, 0, Q 0, Px, Q Q, Y ad ( m, ). Nte: ad Q are umber f cycles ad batch quatity f fiished gds, respectively, at a lcal miimum ttal cst where umber f shipmets, m, is fixed ad kw. Step : Q Q Y, ad m Q. x Step 3: Cmpute ( m, ). If ( m, ) ( m, ), Set, ( m, ) ( m, ) ad g t Step. i (14) 1583

5 Yu, Sarker ad uga therwise, Set Q Q ad the lcal miimum ( m, ) is btaied. Step 4: Set Q m x, ad slve ( m, ) t btai ptimal umber f cycles,. Jamal ad Sarker (199) prved that, where x differetiable. herefre, m is replaced with m x is determied at which ( m, ) is miimum at the discuted pit m is a kw ad fixed iteger. hey claim t fix m at m s that the ttal cst equati is i equati (14). he ttal cst fucti i (14) is a cubic equati i. his equati ca be easily slved prgrammatically. ce the ptimum miimum ttal cst ( m, ) is btaied, the ther variables ca be cmputed as well. t fid the 3. Numerical Studies ad Aalysis As a illustrati, a umerical example is preseted fr a sigle prduct by cmbig tw umerical prblems frm Jamal ad Sarker [] ad Khuja ad Park [5]. I additi t the give variables i thse examples, it is assumed that C = $4.00/uit, = 400 uits/year, P = 3600 uits/year, A 0 = $00/rder, A S = $300/setup, f = 1, x = 100 uits/shipmet, i = 8% aually, = 1 year, the iitial price per uit f raw material is C 0 = $8.00 ad it is decreasig at a rate f 1% per week. After applyig Algrithm 1, the results are reprted i able 1. he results shw i the table are calculated by addig the quatity prduced durig the shipmet perid, Y, t the quatity prduced per setup, Q, at every iterati ad determiig the umber f cycles,, ad the umber f shipmets, m, fr each iterati t fid the miimum ttal cst. he umber f shipmets, m, which gives the miimum cst, is selected as a ptimal result. herefre, the ptimum umber f shipmets, m, is fud at the third iterati where cycles/year, Q 450 uits/batch with ( m, ) $4,731. /year frm which m = Q x = 4 shipmets/cycle. able 1: esults f the mdified algrithm t btai the ptimum m Iterati Q m (m, ) , , , ,740 After fixig m = 4, the ptimal umber f cycles, cycles/year, is btaied by miimizig the ttal cst fucti, ad this is cverted t 5 cycles/year sice 5 cycles give smaller ivetry cst. herefre, the ptimal slutis are 5 cycles/year, Q 480 uits/batch ad ( 4,5) $ /year. he ttal cst f the sigle-stage ivetry mdel is cmputed fr all the pssible cmbiatis f m ad t check whether the result btaied frm the mdified algrithm is accurate r stable. All these cmbiatis f m ad were etered it equati (14) t fid the ttal cst values. his cmparis prves the accuracy f the mdified algrithm develped abve. A few mre prblems are examied t test the mdel uder a fiite plaig hriz. he effect f the rderig ad setup cst reducti has bee examied usig the umerical examples. It ca be ccluded frm the results that the ptimum lt size f the fiished gds, Q, is decreasig while the ttal umber f cycles ver the fiite plaig hriz,, is icreasig because f the effect f reduced rderig ad setup csts. his result shws the sesitivity f the mdel t the chages i rderig cst ad setup cst. 4. Special Case ad Alterate Sluti he ivetry mdel develped here is a mdel whe matchig is imperfect. If Q = mx, the it is called perfect matchig case, therwise it is imperfect matchig. I perfect matchig case, m = x, equati (1) yields 1584

6 he ttal cst fucti i (14) becmes ( ) ( A Yu, Sarker ad uga 1 1 P Q avg x A S ) C b ( 1) f fp C i b( 1) i xi C (16) f f P I additi t the algrithm discussed abve, a alterative methdlgy is studied fr the ivetry mdel. I this methdlgy, umber f shipmets, m, is expressed i terms f umber f cycles,, ad the ttal cst fucti f the ivetry mdel becmes a fucti f ly e decisi variable ver the fiite plaig hriz. I rder t represet m i terms f, set ( m, ) / m = 0 which leads t ( m, ) C b ( 1) imx i(1 m) x ixz C = 0 (17) m f f herefre, the ptimal umber f shipmets, m, ca be writte as: m ( x) x (18) I this alterative sluti methdlgy, istead f fidig the lcal miimum ttal cst ad fixig m t a value fud by a iterative prcedure, m is updated t a fucti f umber f cycles,, ad the ttal cst fucti i equati (14) is updated t () by settig m equals t ( x) x. he ttal cst values btaied frm Algrithm 1 ad the iteractive prcedure are really clse ad the differece is isigificat. he advatage f this methd is that the iteractive sluti prcedure is faster ad easier tha the iterative sluti prcedure. 5. Ccludig emarks I this study, a sigle-stage prducti-delivery mdel uder a fiite plaig hriz fr techlgy-related cmpaies whse prducts are experiecig ctiuus price decrease durig the life cycle is develped t emphasize the imprtace f csiderig price decrease it the supply chai system. Sice the price is ctiuusly decreasig, a maufacturig firm delivers the fiished gds i small quatities frequetly this prves the imprtace f the JI plicy i ivetry ctrl system. Frequet deliveries i small lts are really effective t reduce the ttal cst f the supply chai. Cmpaies i high-tech idustries ca be successful by fllwig the JI plicy ad a super-effective supply chai maagemet sice the price f their prducts is decreasig ctiuusly. Fially, sme suggestis fr future research ca be give sice the ivetry mdel develped i this paper is limited t certai cditis. he scpe f the develped ivetry mdel is limited t a fixed cycle time, ad this makes it harder fr the cmpaies t take the advatage f the decreasig price fucti. S, future research may be directed t relaxig this limitati by csiderig varyig cycle legth, which wuld prvide mre prfit. efereces 1. Glhar,.Y. ad Sarker, B.., 199, Ecmic maufacturig quatity i a just-i-time delivery system, Iteratial Jural f Prducti esearch, 30(5), Jamal, A... ad Sarker, B.., 1993, A ptimal batch size fr a prducti system peratig uder a just-i-time delivery system, Iteratial Jural f Prducti Ecmics, 3(), Sarker, B.. ad Parija, G.., 1994, A ptimal batch size fr a prducti system peratig uder a fixed-quatity, peridic delivery plicy, Jural f the peratial esearch Sciety, 45(8), Khuja,., Park, S. ad Saydam, C., 005, Jit repleishmet prblem uder ctiuus uit cst chage, Iteratial Jural f Prducti esearch, 43(), Khuja,. ad Park, S., 003, ptimal lt sizig uder ctiuus price decrease, he Iteratial Jural f aagemet Sciece, 31(6), i (15) 1585