Determining a production run time for an imperfect production-inventory system with scrap

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1 74 Journal of Scienific & Indusrial Researc J SCI IN RES VOL 66 SETEMBER 007 Vol. 66, Sepember 007, pp eermining a producion run ime for an imperfec producion-invenory sysem wi scrap Gary C Lin and Hong-ar Lin, * eparmen of Indusrial and Manufacuring Engineering and Tecnology, Bradley Universiy, 50 Wes Bradley Avenue, eoria, IL 665, USA eparmen of Indusrial Engineering and Managemen, Caoyang Universiy of Tecnology, 68 Jifong E. Rd., Wufong Townsip, Taicung Couny, 449, Taiwan Received 07 July 006; revised 0 April 007; acceped 8 April 007 A producion-invenory model is proposed for a manufacuring process used o produce a single produc in baces o mee a deerminisic demand. Te process is assumed o be in-conrol a e sar of a producion run. Ten, e process may sif o an ou-of-conrol sae a a random poin in ime, and a fixed porion of iems produced is defecive. Tese defecive iems canno be repaired or reworked and mus be scrapped wi an addiional cos. Models are proposed o deal wi siuaions regarding weer and wen a screening process is implemened. Te objecive is o deermine an opimal producion run ime a minimizes expeced (long-run) average cos per uni of ime including seup, producion, invenory olding, screening, and defecive coss. A numerical approac is developed for finding an opimal soluion. A closed-form soluion, wic is derived based on an approximaion o an exponenial funcion, is also given. Keywords: Invenory, Socasic processes Inroducion A producion process ends o deeriorae due o aging or oer facors a cause generaion of defecive unis during a producion run. efecive unis may be repaired, reworked, or scrapped depending on e caracerisics of produc and process. Tis paper considers e impac of an imperfec producion process (I) on e classical Economic Manufacuring Quaniy (EMQ) model, in wic e defecive unis mus be scrapped wi an addiional cos. EMQ Model EMQ model is probably one of e mos commonly applied invenory conrol models in indusry. Rosenbla & Lee,4 sudied an EMQ model wi an assumpion a afer e process sifs from an in-conrol sae o an ou-of-conrol sae, a fixed percenage of iems produced will be defecive unil enire lo is compleed. Random variable X represens elapsed ime from e beginning of a producion run o e ime poin wen e process sifs o e ou-of-conrol sae. I is assumed *Auor for correspondence dlin@cyu.edu.w o be exponenially disribued wi a mean value /. Based on is and e assumpions in e previous secion, number of defecive iems produced during a producion cycle can be expressed a funcion of producion run ime as 4 0 if X N() α( X) oerwise Expeced number of defecive iems produced in a producion cycle is given as E[N()] α (... () Lee & Rosenbla 5 considered a similar model wi periodic inspecion scedules. Lee & Rosenbla 6 considered a resoraion cos a is dependen on a delayed deecion process. Ceng 7 exended e model of Rosenbla & Lee 4 o deal wi a demand-dependen uni producion cos and an I, and formulaed an invenory decision problem as a geomeric program (G), wic allows a closed-form soluion o be derived. Lee & ark 8 sudied an I on EMQ model wi reworking

2 LIN & LIN: OTICAL ROUCTION RUN TIME FOR AN IMERFECT ROUCTION-INVENTORY SYSTEM 75 On-and Invenory epleion Rae /Y Fig. Sample pa of Model cos and warrany cos. efecive producs are reworked a some cos before being sipped, or if passed o e cusomer, incur a larger warrany cos. A model developed wi inspecion scedules 9 was looked ino e effecs of inspecion errors 0. Lee sudied an EMQ model for opimal lo sizing of a wafer probe macine widely uilized in semiconducor indusry. Kouja & Merez deal wi a model were producion rae can be deliberaely slowed down. A model,4 wi a general ime-o-sif disribuion was developed. Lin 5 considered inegraed EMQ models for a single produc sysem wi an I and a resource consrain on raw maerials. Ben-aya & Raim 6 invesigaed effecs of an I on muli-sage producion sysems. Ben-aya & Hariga 7 considered e economic lo sceduling problem subjec o an I. Inegraed producion-invenory models wi mainenance and qualiy conrol subjec o an imperfec process ave been reviewed 8. Lin e al 9 developed inegraed EMQ models a are subjec o random deerioraion wi periodic inspecion scedules. Based on EMQ model a deals wi I, is paper proposed ree models wi e objecive o deermine an opimal producion run ime a minimizes expeced (long-run) average coss per uni ime. Ten based on renewal reward eorem, expeced average coss per uni of ime can be obained by dividing e expeced oal coss per renewal cycle o e expeced duraion of a renewal cycle. roposed Models Model : No Screening rocess In is model (Fig. ), eac defecive uni is scrapped immediaely afer i is found a e momen of demand during a producion cycle. No screening aciviy is aken place. I is assumed a a defecive uni, once found, is discarded immediaely wi a cos, and during e producion run ime of a producion cycle, iems used o fulfill demand are all good unis. Tis could occur in a siuaion were producion rae is more an demand, and demand is filled on e basis of a firs-in-firs-ou principle. As a resul, ime-o-sif exceeds e ime required o produce enoug unis o mee e demand for producion run ime. Hence, defecive unis are deeced only a producion downime of a cycle. A e end of a producion run, maximum invenory is reaced a () unis. Le Y denoe yield rae of a producion lo. Ten Y [( ) N()]/[( )]. Since defecive unis are assumed o be evenly spread roug e invenory, e invenory during producion down ime is erefore depleed a e rae of /Y. roducion down ime (Fig. ) is given as T ( ) / Y [( ) N() ] T, e producion run ime. Hence, expeced cycle ime can be obained as

3 76 J SCI IN RES VOL 66 SETEMBER 007 E[N()] (α) α ( ) ( ) () E[T T ] e...() Funcion () is sricly increasing and concave in e inerval (0, ) of. Nex, i is considered a all coss are incurred during a producion cycle. Le g () denoe e expeced oal coss incurred during a producion cycle. roducion se-up cos is K and producion cos is c. efecive (scrap) cos is N(). Average invenory can be compued as I() ( ) T [( ) ] T ( ) ( )N() Te expeced oal cos of a cycle is composed of seup, producion, invenory olding, and defecive coss. I can be expressed as a funcion of as g () K c K c E[ I()] δe[n ()] ( α) ( ) ( δα ( α ( ) () Hereafer, i is assumed a α 0. Tis implies a enoug non-defecive iems will be produced o mee e demand roug e enire producion cycle. Noe a since e problem is modeled as a renewal process, objecive funcion can be wrien as Z () g ()/ (). Nex, following properies are used o esablis convexiy of e funcion Z (). ropery Consider e funcion is defined by f(x) g(x)/(x). If x S, were S is an open se and g and are differeniable, en f(x) is pseudoconvex if e following wo condiions old: i) g is convex on S, and g(x) 0 for eac x S; and ii) is concave on S, and (x) > 0 for eac x S. ropery Funcion Z () is a psuedoconvex funcion for > 0 if e following condiion is saisfied:. roof Te firs and e second derivaives of funcion g () wi respec o can be obained as dg () d α ( α) c ( ) ( ) ( e δα( d g() ( α) ( ) d α ( ) ( ) e δαe I can be seen a d g ()/d 0 provided. Tis implies a g () is a convex funcion of for > 0. From Eq. (), i is seen a e funcion () is a concave funcion in e inerval (0, ) of. Also, g () 0 and () > 0 for > 0. Hence, Z () is a psuedoconvex funcion for > 0. Since e opimal soluion * is no readily available, i can be obained by applying a numerical searc procedure o minimize Z ( ) and be subjec o 0 <. An opimal soluion * can be solved by using e bisecion meod. Since in mos cases, is very small, i is reasonable o use e Maclaurin series o approximae e (4) Using is approximaion in Eqs () and (), and ignoring e erms wi e ird or iger power of, one obains α () (5) ( α) g () K c () α α δα ( ) ()...(6) 4 Hence,

4 LIN & LIN: OTICAL ROUCTION RUN TIME FOR AN IMERFECT ROUCTION-INVENTORY SYSTEM 77 K c Z () α α [ ( ) δα ] α ( )α 4 α (7) Tis implies a wen a process is igly unrealiab (immediaely afer sar of a producion run e process will sif o ou-of-conorl sae), e bes policy is no o run any producion. In e remainder of is secion, a case is considered were all defecive unis found a e momen of demand are kep and scrapped as a bac a e end of a cycle. Enire bac of defecive unis is eld for a period of e producion downime. Hence, average invenory (AI) can be compued as Noe a Z () becomes a convex funcion of. Seing dz ()/d 0 yields I() ( ) T [( ) N ()] T ( ) ( ) 8 4 ( α ) α...(8) [ ( ) (c δ)α] αk K 0 If one ignores e firs and e second erms of Eq. (8), a quadraic equaion of is obained. Selecing e posive roo of is quadraic equaion yields an approximae opimal soluion of as αk αk K [ ( ) (c δ)α] appx- ( ) (c δ) α Remark appx (9) Wen α or approaces zero, Eq. (9) becomes K, wic is an opimal ( ) producion run ime for e classical EMQ model. Tus wen process is near perfec or penaly of a defecive uni is negligible, opimal soluion is idenical o a for EMQ model. On e oer and, wen becomes larger, Eq. (9) becomes αk αk K [ ( ) (c δ)α ] lim ( ) (c δ)α 0 NT ( ) N() To compue expeced AI of a cycle, obain second momen of random variable as E[N() ] ( α) 0 ( α) ( x) e x dx e Hence, e expeced AI of a cycle is given as E[I()] (α) ( α ) ( Expeced oal cos of a cycle can be expressed as ( α ) g () K c ( δα ( (α) (0) Noe a since problem is modeled as a renewal process, objecive funcion can be wrien as g() Z ()...() () ropery Te funcion Z () is a psedoconvex funcion for > 0.

5 78 J SCI IN RES VOL 66 SETEMBER 007 On-and Invenory Scrap Unis T T T Fig. A sample pa of Model Time roof Te second derivaive of e funcion respec o can be obained as ( α ) d g() c d (α) d g() d ( δα( g () wi (α) ( α ) e δαe Since ± ± 0 for 0 < α <, i can be seen a d g () /d 0. Tis implies a () is a convex funcion of for > 0. Hence, Z () is psuedoconvex for > 0. Again, opimal soluion of Eq. () is no readily available. Bu an opimal * can be obained by using a numerical searc procedure. A closed-form soluion can be obained by using e Maclaurin series o approximae exponenial erms found in e objecive funcion provided a is very small. Model : Afer-roducion Screening In is model, a 00% screening process akes place immediaely afer e end of producion run ime in a cycle (Fig. ). Since screening rae is more an e demand rae, screening will be done before all invenories g are depleed. Also, since defecive unis found in screening process are scapred as a bac, ey will be eld for a period of ime T. Afer screening process is compleed, all remaining unis are good unis and are used o mee e deamnd for a period of ime T. I is assumed a enire lo a e end of e producion run ime is screened for defecive unis a a rae of. Hence, ime required o screen all ese unis is obained as T (/)( ). Te producion run ime is T. Afer e end of e screening process of a cycle, all defecive unis found are scrapped as a bac. Hence, invenory level is reduced by a oal of N() unis. Te invenory level a is poin in ime is ( ) (/)( ) N(). Tus, T can be pu up as T ) ( ) ( ) N( Le (), a funcion of, denoe expeced duraion of a producion cycle, i can be derived as () E[T T T ] ( ) E[N()] ( ( α) α () Te funcion () is idenical o e funcion () given in e Firs Model. Le g () denoe e expeced oal cos incurred during a producion cycle, seup cos is K, producion cos is c, screening cos is v r( ) st (Tis cos includes a fixed cos, a per uni variable cos carged on ow many unis are screened, and a per ime uni variable cos carged on ow long e screening

6 LIN & LIN: OTICAL ROUCTION RUN TIME FOR AN IMERFECT ROUCTION-INVENTORY SYSTEM 79 process is run.), and scrap cos is ÀN(). AI can be compued as I() ( ) T [ ( ) N() ] ( ) ( ) (T T ) T N() N() N() Hence, expeced AI of a cycle can be compued as E[I()] ( ) ( ) [( α)( α) α ] ( ) ( ) ( ) α α e ( E[N() E[N()] α ( ) ( α ) ] () Now, expeced oal cos of a cycle can be expressed as g () K c v r K v c r α α π ( ) ( ) s [( α)( α) α] ( α) ( ) α ( ( ) ( s ( ) E[I()] πe[n()] α ( ) πα (4) Te objecive funcion can be wrien as Z () g ()/ (). ropery 4 Te funcion Z () is a psuedoconvex funcion for > 0 if e following condiions are saisfied: and α /. roof Te firs and e second derivaive of e funcion g () wi respec o can be obained as dg () c r d ( ) s ( [( α)( α) α] ( α) ) πα α α α π e ( ) α ( ) ( e d g () d ( α) ( [( α)( ) ] παe α ( ) e α ( ) ( ) Te second derivaive of g () is 0 for > 0 if and α /. Tis implies a g () is a convex funcion of > 0. Since () is idenical o (), i is sricly increasing and concave in e inerval (0, ) of. Hence, Z () is a psuedoconvex funcion for > 0. Since opimal soluion * is no readily available, a numerical searc procedure like bisecion meod can be developed o find *. Ignoring erms wi ird or iger power of, one obains s g () K v c r ( α)( ) α ( ) Z () ( (K v ) α α)( ( ) α ( ) ( ) πα s c r ( ) α α ) ( ) πα α (5) (6) Noe a Z () can be sown o be a convex funcion of. Hence, seing dz ()/d 0 yields α ( ) 4 ( )

7 T T Fig. A sample pa of Model 70 J SCI IN RES VOL 66 SETEMBER 007 On-and Invenory Scrap Unis Time Ignoring firs and second erms of above equaion, a quadraic equaion of is obained. Hence, approximae opimal soluion can be obained by aking posiive roo of resuling quadraic equaion as appx were α s c r ( ) α ( α)( ) ( ) πα α(k v) (K v) 0 α(k v) α A c r s α(k v) A ( ) α ( α)( ) ( ) πα 4A(K v) Tis soluion is similar o e classical EMQ model. Wen approaces zero, one obains lim appx 0 ( K v ) ( ) α α Wen becomes larger, appx approaces 0. Tis implies a wen a process sifs o e ou-of-conrol sae muc sooner during a producion run, e bes policy is no o run e process. Model : In-roducion Screening Te model considered in is secion is one a as a 00% screening process aking place a e beginning and rougou e enire producion run ime (Fig. ). Since enire lo is screened wile e producion is going on, all defecive unis will be found afer e process as sifed ino e ou-of-conrol sae. Afer e end of a producion run, defecive unis will be scraped as a bac. Hence, a e beginning of producion downime, ere are only ( ) N() good unis lef o fulfill e demand during e producion downime. Tis implies a T and T [( ) N()]/. Le () denoe e expeced duraion of a producion cycle, wic is a funcion of, en Remark Wen α approaces zero, one obains lim appx α 0 ( K v) ( ) () E[N()] E[T T ] (α) α ( ) ( Once again, e funcion () is idenical o e funcion () given in e Firs Model. Le g () denoe e

8 LIN & LIN: OTICAL ROUCTION RUN TIME FOR AN IMERFECT ROUCTION-INVENTORY SYSTEM 7 expeced oal coss incurred during a producion cycle, seup cos is K, producion cos is c, screening cos is v r s, and defecive cos is ÀN(). AI is derived as I() ( ) T [ ( ) N() ] ( ) [ ( ) N() ] ( ) ( ) N() T N() dg () (c r) s d ( α) [( α)( α) α] πα α α α π e ( ) ( e Hence, expeced AI of a cycle can be compued as E[I()] ( α ) ( ) ( ) [( α)( α) α ] ( α ) ( α ( ) ( E[N() ] E[N()] (7) Combining all cos componens and Eq. (7), expeced oal cos of a cycle can be expressed as g () K v (c r) s α K c v r s E[I()] πe[n()] ( α) [( α)( α) α] α π πα ( α ) ( e ( ) (8) Since e problem is modeled as a renewal process, objecive funcion can be wrien as Z () g ()/ (). ropery 5 Te funcion Z () is a psuedoconvex funcion for > 0 if e following condiions are saisfied: and α /. roof Since proof is similar o e one given in e proof of ropery, firs and second derivaives of e funcion g () wi respec o are presened as d g () d παe [( α)( ) ] α ( )e ( ) ( α) ( I can be seen a e second derivaive of g () is no less an 0 for > 0 provided a < and α < /. Tis implies a Z () is a psuedoconvex funcion for > 0. I can seen a e opimal soluion * is no readily available, bu a numerical searc procedure can be employed for is case. As an alernaive, an approximaion used in e derivaion of Model is applied o e exponenial erms in Eq. (8) and Z (). Ignoring erms wi e ird or iger power of m, one obains Z () [(c r ) s] g () K v ( α )( ) πα α ( ) [( c r ) s ] [ ( α)( ) πα ] α( ) α (K v) α α α