Periodic inventory model with unstable lead-time and setup cost with backorder discount. Chandra K. Jaggi* and Neetu Arneja

Transcription

1 Int. J. Appled Decson Scences, Vol. 3, No., Perodc nventory model th unstable lead-tme and setup cost th backorder dscount Chandra K. Jagg* and Neetu Arnea Department of Operatonal Research, Faculty of Mathematcal Scences, Unversty of Delh, Delh-0007, Inda Fa: E-mal: E-mal: E-mal: *Correspondng author Abstract: he man purpose of ths study s to nvestgate the perodc nventory model th backorder prce dscounts, here shortages are partally backlogged. he applcaton of ust-n-tme (JI) phlosophy.e., crashng of the lead-tme and setup cost has been carred component se. o cases have been dscussed vz protecton nterval demand dstrbuton s knon (normal dstrbuton approach) protecton level demand dstrbuton s unknon (mnma dstrbuton approach). An algorthm has been developed hch ontly optmses the reve perod, the backorder prce dscount, the setup cost and lead-tme for a knon servce level. Numercal results clearly ndcate that sgnfcant savngs could be acheved. Keyords: nventory; setup cost; lead-tme; crashng cost; backorder prce dscount. Reference to ths paper should be made as follos: Jagg, C.K. and Arnea, N. (00) Perodc nventory model th unstable lead-tme and setup cost th backorder dscount, Int. J. Appled Decson Scences, Vol. 3, No., pp Bographcal notes: Chandra K. Jagg s an Assocate Professor, Department of Operatonal Research, Faculty of Mathematcal Scences, Unversty of Delh, Inda. He receved hs PhD and MPhl n Inventory Management and Masters (O.R.) from the Unversty of Delh. Hs teachng ncludes nventory and fnancal management. Hs research nterest les n modellng nventory systems. He has to hs credt 34 publcatons n Internatonal Journal of Producton Economcs, Journal of Operatonal Research Socety, European Journal of Operatonal Research, Internatonal Journal of Systems Scences, Internatonal Journal of Canadan Journal of Pure and Appled Scences, OP, OPSEARCH, Advanced Modellng and Optmsaton, Journal of Informaton and Optmsaton Scences, and many more. Copyrght 00 Inderscence Enterprses Ltd.

2 54 C.K. Jagg and N. Arnea Neetu Arnea s a Research Scholar n the Department of Operatonal Research, Faculty of Mathematcal Scences, Unversty of Delh, Inda. She receved her MSc (Operatonal Research) from the Unversty of Delh. She has cleared Natonal Elgblty est (NE) conducted by UGC-CSIR, Inda. At present, she s pursung PhD n Inventory Management. Introducton In nventory management, the lead-tme has been consdered as a prescrbed constant or a stochastc varable. In realty, lead-tme usually conssts of the follong components as suggested by ersne (98): order preparaton, order transt, suppler lead tmes, delvery lead tme and setup tme. o gan the compettve success n busness, each frm knos the mportance of tme n the market place. herefore, the obectve of each frm s to reduce the lead-tme n a practcal stuaton to satsfy the customer s demand on tme. he applcaton of ust-n-tme (JI) phlosophy ncludes the crashng of lead-tme to run the system proftably. he lead-tme can be decomposed nto several crashng perods for makng the orkng system effectve. In many practcal stuatons, lead-tme s controllable or can be reduced at an etra-added cost. Generally, the etra cost of reducng lead-tme nvolves admnstratve, transportaton and suppler s speed-up costs. As a result, the clent servce and manufacturng schedule can be mproved even th reduced safety stocks. Recently, several contnuous reve nventory models have been developed to consder lead-tme as a decson varable (Lao and Shyu, 99; Ben-daya and Raouf, 994; Ouyang et al., 999; Moon and Cho, 998; Ouyang and Wu, 997; Ouyang and Chuang, 998; Wu and sa, 00). One fnds that these papers manly lay emphass on the benefts of reducng lead-tme. So, e have consdered supposton of reducng lead-tme at an etra cost n ths paper. It s generally observed that hle shortage occurs, demand can be captured partally. Some customers may prefer ther demands to be backordered.e., some customers hose needs are not urgent can at for the tem to be satsfed, hle others ho cannot at have to fll ther demand from another source hch s lost sale case. But there are some factors that motvate the customer for the backorders out of hch prce dscount from the suppler s the maor factor. By offerng suffcent prce dscounts, the suppler can secure more backorders through negotaton. Wth hgher prce dscount, the suppler could fetch a large number of back order rato. Km and Park (985) nvestgated an nventory model th a mture of sales and tme eghed backorders. Subsequently, Pan and Hsao (00) presented contnuous nventory models th backorder dscounts and varable lead-tme. Later on, Pan and Hsao (005) epanded the model by consderng the case here lead-tme crashng cost s gven as the functon of reduced lead-tme and ordered quanttes. In ths paper, the backorder dscount has been taken as one of the decson varable. he consderaton s as unsatsfed demand durng the shortages can lead to optmal backorder rato by controllng the prce dscount and the suppler s to mnmse the relevant total nventory cost. Further, Porteus (985) eamned the mpact of captal nvestment n reducng setup costs n the classcal EOQ model for the frst tme. Many researchers have reported several relatonshps beteen the amount of captal nvestment and setup cost level (Nor

3 Perodc nventory model th unstable lead-tme and setup cost 55 and Sarker, 996; Km et al., 99; revno et al., 993; Hall, 983, Sarker and Coates, 997). herefore, ths artcle deals th to mportant aspects of JI phlosophy.e., reducton of lead tme and setup costs here the lead tme and setup costs vary as a functon of captal epense. Setup costs can be controlled and reduced through varous efforts such as orker tranng, procedural changes and specal equpment acquston. Earler, Chuang et al. (004) presented a perodc reve nventory model th varable lead-tme and reducton of setup cost here the setup cost has been consdered as the logarthmc functon of nvestment thout backorder dscount. Alternatvely, Cheng et al. (004) have analysed the contnuous nventory model th crashng of lead-tme and setup cost component se here the lead-tme demand follos the normal dstrbuton only. But, certan questons reman to be ansered e.g., hat ll be the effect of backorder dscount n perodc nventory model or hether the component se reducton of lead-tme and setup costs are benefcal n perodc nventory model or not? If yes, hat s the ntensty of benefts? If the demand durng lead-tme does not follo normal dstrbuton, hat ll be the probable soluton and so on? o anser these questons, ths paper suggests a perodc reve nventory model th backorder dscount here lead tme and setup costs have been reduced component se, hch can be taken as the mture and etenson of Cheng et al. (004) and Chuang et al. (004) ork. In ths artcle, perodc reve nventory model has been consdered here demand durng the protecton nterval s partally backordered. Both cases of protecton nterval demand knon dstrbuton unknon dstrbuton, here frst and second fnte moments are knon, have been revealed. he man purpose of our study s to optmse the reve perod, the backorder dscount, the setup cost and lead-tme th knon servce level. he lead-tme and setup costs both are controllable and have shon that the sgnfcant savng could be obtaned by backorder dscount. Notaton and assumpton o develop the proposed model, e adopt the follong notaton and assumptons used n Cheng et al. (004) n ths paper.. Notaton D K h R average demand per year fed orderng cost per nventory cycle nventory holdng cost per unt per year target level β fracton of the demand back ordered durng stock out perod such as 0 β

4 56 C.K. Jagg and N. Arnea β 0 π 0 π L X upper bound of the backorder rato margnal proft (.e., cost of lost demand) per unt back order prce dscount offered by the suppler per unt length of lead-tme protecton nterval demand hch has a p.d.f. f th fnte mean D( L) and standard devaton ( L)( 0) σ > for the protecton nterval ( L) here σ denotes the standard devaton of the demand per unt Ω the class of p.d.f. f of the protecton nterval demand th fnte mean D( L) and standard devaton σ L S A E(.) fed shortage cost, $ per unt short safety factor length of a reve perod mathematcal epectaton mamum value of and 0.e., Ma {, 0} EAC epected annual cost EAC least upper bound of epected annual cost.. Assumptons he nventory level s reveed every unts of tme. A suffcent quantty s ordered up to the target level R, and the orderng quantty s arrved after L unts of tme. he length of the lead-tme L does not eceed an nventory cycle tme so that there s never more than a sngle order outstandng n any cycle. 3 he target level R Epected demand durng the protecton nterval safety stock (SS) and SS A * (standard devaton of protecton nterval demand),.e., R D( L) Aσ L here A s the safety factor and satsfes P[ > R] q, q represents the alloable stock out probablty (.e., defned servce level) durng the protecton nterval and s gven. 4 he lead-tme L conssts of n mutually ndependent components. he th component has a mnmum duraton a and normal duraton b, and a crashng cost per unt tme c. Arrangng c such that c c c 3 c n for the convenence. Snce t s clear that the reducton of lead-tme should be frst on component because t has the mnmum unt crashng cost, and then component, and so on.

5 Perodc nventory model th unstable lead-tme and setup cost 57 n 5 Let L0 b and L be the length of lead tme th components,,, crashed to ther mnmum duraton, then L can be epressed as L L0 ( b a),, n and the lead tme crashng cost per cycle C(L) s gven as C( L) c( L L) c ( b a). (Refer Chuang et al., 004). 6 he setup cost K conssts of m mutually ndependent components. he th component has a normal cost e and mnmum cost d and a crashng cost f hen the normal cost reduces to mnmum cost. So, there s a dscontnuous relatonshp beteen the crashng cost and setup cost reducton. We have arranged f such that f f f3... fn for the sake of convenence. Snce t s clear that the reducton of setup cost should be frst on component because t has the mnmum unt crashng cost, and then component, and so on. m 7 Let K0 e and K be the set up cost th components,,, crashed to ther mnmum cost, then K can be epressed as K K ( e d ) 0 and setup crashng cost per cycle C(K ) s gven as C( K ) al., 004).,,, m f (refer Cheng et 8 Assumng that a fracton β (0 β < ) of the demand durng the stock out perod can be backordered so the remanng fracton β s lost. he backorder rato β s varable and s n proporton to the prce dscount π offered by the suppler per unt. β0π hus β here 0 β 0 < and 0 π π 0 (Pan and Hsao, 00). π 0, 3 Basc model We have assumed that the protecton nterval demand X has a p.d.f. f th fnte mean D( L) and standard devaton σ ( L) th the target level R D L Aσ L here A s already defned. As Montgomery et al. (973) proposed the perodc reve model, the epected net nventory at the begnnng of the perod s R DL ( β ) E( X R), and the epected net nventory at the end of the

6 58 C.K. Jagg and N. Arnea perod s RDL D ( β ) E( X R). So, the epected holdng cost per year s D appromately hrdl ( β ) E( X R) and the epected stock out cost per πβe( X R) ( S π0 )( β) E( X R) year s, here E( X R ) s the epected E X R R f d (refer Chuang et al., demand shortage at the end of cycle.e., ( ) ( ) 004). When the lead-tme L s reduced to L then the annual lead-tmecrashng cost s. Smlarly, the setup cost K reduced to K, then annual setup crashngcost s f. c No, the obectve s to mnmse the total epected annual cost ( EAC) hch s the sum of the orderng cost, setup crashng cost, stock out cost, holdng cost and lead tme EAC, π, L here crashng cost. Symbolcally, our problem s to mnmse R EAC (, π, L) K f ( ) ( )( ) ( ) 0 πβe X R S π β E X R D hrdl ( β ) E( X R) c () Also, e have assumed that the backorder rato β depends on the backorder prce dscount π.e. βπ β and R D( L) Aσ L π 0 here A s safety factor. he equaton () can be rtten as

7 Perodc nventory model th unstable lead-tme and setup cost 59 EAC (, π, L) K c f βπ E( X R) S π 0 βπ π0 π0 D βπ h A σ L E( X R) π 0 () In the above model, the crashng of setup cost s ndependent of the lead-tme, hch means the reducton of setup costs can be carred out separately. Consder a case, here a frm conducts a specal tranng course to ts employees for mplementng the better qualty of the product nstead of makng ne appontments, as qualty mplementaton s one of the mportant concepts of JI. Hoever, the lead-tme and orderng cost reductons may be related closely n some cases. Here, to cases arse for dstrbuton of lead-tme demand.e., a normal dstrbuton b unknon dstrbuton 3. Lead tme demand th normal dstrbuton In ths secton, e have assumed that the probablty dstrbuton of protecton nterval demand X has a normal dstrbuton th mean D( L) and standard devaton σ ( L). So, the epected shortages occurrng at the end of the cycle s gven by ( ) ( ) σ ( ) ψ > 0 E X R R f d L A here ψ ( A) ϕ( A) A Φ( A) R respectvely. herefore, equaton () s reduced to, φ and Φ are the standard normal p.d.f. and d.f., EAC (, π, L) K c f βπ σ ( L) ψ ( A) S π 0 β0π π0 π0 D βπ h A σ L σ ( L) ψ ( A) π 0 (3)

8 60 C.K. Jagg and N. Arnea It can be checked that for fed and π, EAC (,, L) EAC (, π, L) L [, ], because < 0. L L L π s a concave functon of So, for fed (, π, L), the mnmum total epected annual cost ll occur at the end ponts of the nterval [, ], L, t can be shon that EAC (,, L) hand, for a gven value of L, L (, π ). hus for fed L ( L, L ), pont (, π ) that satsfy the mnmum value of EAC (,, L) EAC (, π, L) EAC (, π, L) 0 and π L L On the other π s conve n π ll occur at the 0. No, K c f EAC (, π, L) 0 βπ S π0 β0π σ Lψ ( A) π0 π0 βπ σ S π0 β0π ψ π0 π0 ( L) σ ψ ( A) ( L) D Aσ βπ σ h ( L) π 0 ( L) ψ ( A) ( A) (4) hs can be rtten as K c f βπ S π0 β0π σ Lψ ( A ) π0 π0 βπ S π0 β0π σ Lψ ( A) π0 π0 D Aσ βπ σ h ψ ( A) ( L) π 0 ( L) (5)

9 Perodc nventory model th unstable lead-tme and setup cost 6 here as ( π ) π 0 EAC,, L h ( S ) 0 π π (6) Snce t s dffcult to obtan the solutons for and π eplctly as the evaluaton of equaton (5) and equaton (6) need the value of each other. As a result, e must establsh the follong teratve algorthm to fnd the optmal (, π ). Algorthm Step Step Step 3 For each L, 0,, n, eecute (a) (d). (a) (b) For each K, 0,,, m, eecute (b) (d). Start th fed servce level 0.8 that s, A and ψ(a ) by checkng the table from Slver and Peterson (985, pp ). (c) By puttng the value of ψ(a ) nto equaton (5), usng numercal search technque, evaluate. If L then go to (d) otherse let L, go to (d). (d) By usng, calculate the value of π usng the equaton (6). Compare π and π 0. If π π 0, then π s feasble. Go to step () Otherse set π π 0, go to step () For each ( L ) ( L ) (,, L ), π,, compute the correspondng epected annual cost EAC, π,, 0,,, m from equaton (3) and fnd mnmum EAC π for 0,,, m. Go to step 3. Fnd ( π ) mn,,. EAC L 0,,,... n (, π, L ) R D ( L) Aσ ( L) * * * If EAC ( π L ) EAC ( π L ),, mn,,. Hence 0,,,... n s the optmal soluton. And hence, the optmal target level s * * * * *. heoretcally, for gven K, D, h, β 0, π 0, σ and each L ( 0,, n), from equatons (5) and (6), e can obtan optmal values of and π, then the correspondng total epected annual cost can be found. hus, the mnmum total epected annual cost could be obtaned hen the lead-tme demand s normally dstrbuted. 3. Lead tme demand th unknon dstrbuton If the lead-tme demand does not follo normal dstrbuton or the probablty dstrbuton s unknon th frst to moments, then the soluton can be obtaned by mnma approach (see Ouyang et al., 996). Snce the probablty dstrbuton of X s unknon, e cannot fnd the eact value of E(X R).

10 6 C.K. Jagg and N. Arnea 3.. Soluton by mnma approach In ths secton, e rela the constrant over the form of the probablty dstrbuton of lead-tme demand. Here, e assume that the lead-tme demand has an unknon dstrbuton th knon fnte mean D( L) and standard devaton ( L) the target level R D( L) A L σ here σ. No, e try to use a mnma dstrbuton free procedure to solve ths problem. So our problem s to solve: mn ma EAC, π, L, We use the follong proposton (Gallego and > 0, π > 0, L> 0 F Ω Moon, 993) to shorten the problem. Proposton : For any F Ω, E( Xr) σ ( L) ( R D ( L) ) R D ( L) σ ( L)( A A) (7) Moreover, the upper bound (7) s tght. hen the equaton () can be reduced to (, π, ) K c f EAC L βπ S π0 β0π σ ( L)( A A) π0 π0 D βπ h A σ L σ ( L)( A A) π 0 (8) here EAC (, π, L) s the least upper bound of EAC (, π, L). As notfed n the precedng secton, t can be shon that EAC (,, L) concave functon of L [, ] π s a L L for fed and π (Append ). herefore, the mnmum upper bound of the epected total annual cost ll occur at the end pont of the nterval L [ L, L ] for fed value of (, π ). Moreover, t can be shon that EAC, π, L s conve functon of and π for fed L (Append ). herefore, the frst order condtons are necessary and suffcent condtons for optmalty. Usng the frst condton of dervatves, e get

11 Perodc nventory model th unstable lead-tme and setup cost 63 and K c f βπ S π 0 β0π σ L A A π0 π0 βπ S π 0 β0π σ L A A π0 π0 4 D Aσ βπ σ ( h A A ) ( L) π 0 4 ( L) h ( S π 0) π. (0) Snce, t s dffcult to obtan the eact value of servce factor A hch depends upon the requred servce level on the bass of alloable stock out probablty q, because the p.d.f. f () s unknon. So, the follong proposton has been used to fnd accurate value of A. herefore, the algorthm to fnd the optmal reve perod, lead-tme and backorder dscount can be establshed by usng the proposton gven belo: Proposton: (Ouyang and Wu, 997) Let X represent the protecton nterval demand that has p.d.f. f () th fnte mean D( L) and standard devaton ( L) 0, σ L c > P[ X > c]. σ L crl If e take R nstead of c, then P[ X R] (9) σ then for any real number > σ σ ( L) ( L) ( RDL) P[ X > R]. A Furthermore, snce t s assumed that the alloable stock out probablty q durng the q P X > R. Hence q. A A herefore A 0,. q lead tme s knon so [ ] or

12 64 C.K. Jagg and N. Arnea Algorthm Step For Each q, dvde the nterval 0, q nto N equal subntervals. Let 0, q A0 A N AN A0 Al Al, N l,,... N Step For each L ( 0,,,, n) and K ( 0,,,, m) perform steps (3) and (4). Step 3 For gven A l {A 0, A,, A N }, l 0,,,, N, usng numercal search technque, evaluate from equaton (9) smultaneously. If L then go to Step (4) otherse Set L and go to Step (4). Step 4 By usng, calculate the value of π usng the equaton (0). Compare π and π 0. If π π 0, then π s feasble. Go to net step. Otherse set π π 0,, go to Step (5) Step 5 For each (, π, L ), compute the correspondng epected annual cost EAC, π, L, 0,,, m Step 6 Step 7 Fnd ' ' ' hen, (,, L) ( ) Mn EAC (, π, L ). Al { A0, A,... AN} If (, π,, ) A A A π Al { A0, A,... AN} EAC,, L Mn EAC (,, L ). ' ' ' Fnd EAC ( π L ) Mn EAC ( A, π,, ) A L A,,,,. 0,,,... n π s the requred optmal soluton. 4 Numercal eample In order to llustrate the soluton algorthms, e have consdered an nventory system havng data used by Chuang et al. (004) and Cheng et al. (004): D 600 unts per year, K $00 per order, S $50 per short out, σ 7 unts per eek, π 0 $ 50 per unt, h $0 per unt per year, q 0. herea 0 0 and A N, N 00. he lead-tme and setup cost has three components, hch have been shon n able and able. able Lead tme data Lead tme component Normal duraton (days) b Mnmum duraton (days) a Unt Crashng cost, c

13 Perodc nventory model th unstable lead-tme and setup cost 65 able Setup cost data Setup cost component Normal cost, e Mnmum cost, d otal crashng cost as reduced to mnmum cost, f We have solved the cases for dfferent upper bounds of the backorder rato β 0, 0.5 and hch represents the cases of no backorder, partally backorder and no lost sale. At frst, able 3 provdes the soluton thout the crashng of lead-tme and setup cost th normal dstrbuton. able 3 Optmal soluton of numercal eample (*, L* n eeks) thout crashng of lead-tme and fed setup cost (K 00) for normal dstrbuton β 0 L* * R* π EAC(, π, L) hen, by applyng the algorthm, crashng has been carred out for lead-tme and setup costs for dfferent backorder rato and llustrated n able 4(a) and able 4(b). It s observed that by reducng the lead tme and setup cost, the total epected cost decreases. able 4(a) Crashng of lead-tme and setup costs for dfferent backorder rato. L C(L ) C(K ) β 0 0 β EAC R π EAC R π

14 66 C.K. Jagg and N. Arnea able 4(b) Crashng of lead-tme and setup costs for dfferent backorder rato L β 0 C(L ) C(K ) EAC R π he optmal nventory results th relevant savngs here lead-tme and setup cost have been crashed gven n able 5. From able 5, e also observe that the total annual epected cost decreases as the backorder rato ncreases snce suppler can fetch a large number of backorders by offerng the prce dscount th no loss although th less cost. able 5 Optmal soluton of numercal eample (*, L* n eeks) th crashng of lead-tme and setup cost K for normal dstrbuton β 0 L* * R* K* π EAC(*, π, L*) Savng (%) { EAC L EAC K L EAC ( π L)} Note: * * * ( π ) savng %,,,, /,, 00% Furthermore, able 6 lsted the optmal results for controllable lead-tme and setup cost th unknon dstrbuton. able 6 Optmal soluton of numercal eample [(*, π, L*) n eeks] th crashng of lead-tme and setup cost for unknon dstrbuton β 0 L* * R* K* π EAC(*,, L*) π

15 Perodc nventory model th unstable lead-tme and setup cost 67 able 7 provdes the comparson of unknon dstrbuton model th that of normal dstrbuton model to obtan epected value of addtonal nformaton (EVAI), hch s the largest amount that the suppler ould be llng to pay for knong the dstrbuton of protecton nterval demand. It s the dfference of total epected annual cost that s obtaned by substtutng the optmal soluton of unknon dstrbuton and normal N ' ' ' N N N N EAC, π, L EAC, π, L, hch dstrbuton cases n equaton (3).e., ( ) ( ) could be referred to as the etra cost for utlsng the unknon dstrbuton nstead of normal. able 7 Calculaton of EVAI N ' ' ' N N N N β 0 EAC (, π, L ) EAC (,, L ) π EVAI It s nterestng to observe that an ncrement n backorder rato based on prce dscount offered by the suppler reduces the total epected annual cost by 9.6% n normal dstrbuton case and 8.63% n unknon dstrbuton. Moreover, the etra cost for utlsng the unknon dstrbuton nstead of normal, that s EVAI has been gven above for dfferent value of backorder rato. 5 Conclusons he man purpose of ths paper s to nvestgate the effect of controllng the lead-tme and setup cost n perodc nventory model th backorder prce dscount as a decson varable, and to solve the cases of knon (normal) as ell as of unknon dstrbuton of protecton nterval demand. When unsatsfed demand occurs, a suppler offers prce dscount to the customers for the stock out tems to secure more backorders. hat s hy e have consdered the dependency of backorder rato on the amount of prce dscount. he crashng of lead-tme and setup cost s a very natural phenomenon n realstc nventory models, hch have been carred out ndependently, under ths study. Usually, n crashng one tres to reduce the tme, to acheve ths one requres addtonal resources, hch costs money. herefore, one looks for a trade off beteen the tme and the cost. Moreover, safety stock protecton s requred for lead tme plus the order nterval.e., ( L), t s hgher for the larger lead tme and loer for the smaller lead tme. Wth ths very fact, crashng of lead tme has been carred out n ths paper. It has been shon from the results that the reve perod () and the mamum nventory level (R) decrease th the reducton of lead tme. Further, hen the reducton of lead-tme accompanes a decrease of setup cost th backorder dscount to customers then the total epected cost decreases sgnfcantly.

16 68 C.K. Jagg and N. Arnea Acknoledgements he authors are grateful to the anonymous referees for ther useful suggestons and comments. he frst author ould lke to acknoledge the support of Research Grant No. Dean (R)/R&D/009/487, provded by Unversty of Delh (Inda) for conductng ths research. he second author ould lke to thank Unversty Grant Commsson (UGC) for provdng the felloshp to accomplsh the research. References Ben-Daya, M. and Raouf A. (994) Inventory models nvolvng lead-tme as decson varable, Journal of the Operatonal Research Socety, Vol. 45, pp Cheng,.L., Huang, C.K. and Chen, K.C. (004) Inventory model nvolvng lead-tme and setup cost as decson varables, Journal of Statstcs & Management Systems, Vol. 7, pp.3 4. Chuang, B.R., Ouyang, L.y. and Chuang, K.W. (004) A note on perodc reve nventory model th controllable setup cost and lead tme, Computers and Operatons Research, Vol. 3, pp Gallego, G. and Moon, I. (993) he dstrbuton free nesboy problem: reve and etensons, Journal of the Operatonal Research Socety, Vol. 44, pp Hall, R. (983) Zero Inventores, Do Jones Irn, Homeood, Illnos. Km, D.H. and Park, K.S. (985) (Q, r) nventory model th a mture of sales and tme eghted backorders, Journal of the Operatonal Research Socety, Vol. 36, pp Km, K.L., Hayya, J.C. and Hong, J.D. (99) Setup reducton n economc producton quantty model, Decson Scences, Vol. 3, pp Lao, C.J. and Shyu, C.H. (99) An analytcal determnaton of lead-tme th normal demand, Internatonal Journal of Operatons & Producton Management, Vol., pp Montgomery, D.C., Bazaraa, M.S. and Kesan, A.I. (973) Inventory models th a mture of backorders and lost sales, Naval Research Logstcs, Vol. 0, pp Moon, I. and Cho, S. (998) A note on lead-tme and dstrbutonal assumptons n contnuous reve nventory reve models, Computers and Operatons Research, Vol. 5, pp Nor, V.S. and Sarker, B.R. (996) Cyclc schedulng for a mult-product, sngle faclty producton system operatng under a ust-n-tme producton systems, Journal of Operatonal Research Socety, Vol. 47, pp Ouyang, L.Y. and Chuang, B.R. (998) A mnma dstrbuton free procedure for perodc reve nventory model nvolvng varable lead-tme, Internatonal Journal of Informaton and Management Scences, Vol. 9, No. 4, pp Ouyang, L.Y. and Wu, K.S. (997) Mture nventory model nvolvng varable lead-tme th a servce level constrant, Computers and Operatons Research, Vol. 4, pp Ouyang, L.Y. Yeh, N.C. and Wu, W.S. (996) Mture nventory model th backorders and lost sales for varable lead-tme, Journal of the Operatonal Research Socety, Vol. 47, pp Ouyang, L.Y., Chen, C.K. and Chang, H.C. (999) Lead-tme and orderng cost reductons n contnuous reve nventory systems th partal backorders, Journal of the Operatonal Research Socety, Vol. 50, pp Pan, C.H. and Hsao, Y.C. (00) Inventory models th back-order dscounts and varable lead-tme, Internatonal Journal of System Scence, Vol. 3, pp

17 Perodc nventory model th unstable lead-tme and setup cost 69 Pan, C.H. and Hsao, Y.C. (005) Integrated nventory models th controllable lead tme and backorder dscount consderatons, Internatonal Journal of Producton Economcs, pp.93 94, pp Porteus, E.L. (985) Investng n reduced setups n the EOQ model, Management Scence, Vol. 3, pp Sarker, B.R. and Coates, E.R. (997) Manufacturng setup cost reducton under varable lead tmes and fnte opportuntes for nvestment, Internatonal Journal of Producton Economcs, Vol. 49, pp Slver, E.A. and Peterson, R. (985) Decson Systems for Inventory Management and Producton Plannng, Wley, Ne York. ersne, R.J. (98) Prncples of Inventory and Materals Management, North Holland, Ne York. revno, J., Hurley, B.J. and Fredrch, W. (993) A mathematcal model for the economc ustfcaton of setup tme reducton, Internatonal Journal of Producton Research, Vol. 3, pp.9 0. Wu, J.W. and sa, H.Y. (00) Mture nventory model th back orders and lost sales for varable lead tme demand th the mtures of normal dstrbuton, Internatonal Journal of Systems Scence, Vol. 3, pp Append Proof of EAC (, π, L) s concave n L L, L ) for fed, π. ( EAC (, π, L) L S π 0 β0π π0 π0 4 βπ βπ σ S ( L) A A βπ σ ( L) ( L) ( A A) haσ π 0 4 EAC (, π, L) L 3 βπ S π0 β0π σ( L) ( A A) π0 π0 8 3 haσ βπ 3 σ ( L) ( L) ( A A) 4 herefore EAC s concave n L (, ) < 0 π 0 8 for fed (, π ). L L (A)

18 70 C.K. Jagg and N. Arnea Append Proof of EAC (, L) For a gven value of L ( L L ),π s a conve functon of ( π,, )., e frst obtan the Hessan Matr H as follos: EAC (, π, L) EAC (, π, L) H EAC (, π, L) EAC (, π, L) π π (A) hen, evaluatng prncpal mnor of H, the frst prncpal mnor of H that s H > 0, here K c f EAC (, π, L) H βπ S π0 β0π π0 π0 σ L A A βπ S π0 β0π π0 π0 σ 4 L ( ) ( A A ) ( ) D Aσ h A A 4 hs can be rtten as βπ σ ( L) π 0 ( L) (A3) K c f L ( ) hd hσ βπ A ( A A) ( L) L π βπ βπ 3 π0 β0π σ π0 π0 π0 β0π σ π0 π0 3 S L A A 4 S L A A ( ) ( )

19 Perodc nventory model th unstable lead-tme and setup cost 7 or K c f hd hσ βπ A A A L L 4 L 4 π 0 3 ( π ) σ π σ N L A A N L A A 4 3 here βπ N( π) S π0 β0π, π0 π0 (A4) We have (, π, ) EAC L ψ ( ) hσ βπ A A A L 4 π 0 ( ) 3 (A5) here K c f N( π ) σ ( L) ψ A A N π σ L N π σ L A A A A 4 No, EAC (, π, L) H ψ hσ βπ A A A L 4 π 0 Let 3

20 7 C.K. Jagg and N. Arnea δ ( ) K c f N L A A L 4 ( ) ( π ) σ ( ) ( ) N L A A 3 ( π ) σ ( ) ( ) βπ δ > A A A L 4 π 0 So, from e have hσ hen 3 Snce H > ψ δ > 0 K c f L ψ δ 3 L ( ) ( 3 4 ) ( π )( 4 ) σ ( ) ( ) N L L A A > 0 3 L ( ) he second prncpal mnor of H s EAC (, π, L) EAC (, π, L) π H EAC (, π, L) EAC (, π, L) π π herefore, here EAC (, π, L) EAC (, π, L) EAC (, π, L) H π π

21 Perodc nventory model th unstable lead-tme and setup cost 73 EAC (, π, ) 0 L π βπ ( L) ( A A) 0 β0 EAC (, π, L) π0 π0 π βσ π 0 and Sβ σ ( L) ( A A) βπ Sβ π π βσ 0 β 0 σ L ( A A) 0 0 h 0 ( L) π 0 ( A A) Also, EAC (, π, L) EAC (, π, L) H H π π K c f ( 3 4L) N( π )( 4 ) L σ L ( A A ) 3 3 > ( L) ( L) EAC (, π, L) EAC (, π, L) π π After smplfcaton, e have H β 0 0 σ 4 K c f L A A 3 4L A A π 0 π L ( π ) h σβ ( ) βπ Sβ0 βπ Sβ0 4N( π) β0 45N( π) β0 L π0 π0 π0 π L 4 L 44N βπ Sβ0 β0 π0 π0 4 ( ) 4 L L > 0 Because 0 B0 < π 0 ( 4 B0 ) B0S > 3π 0 S > 0 here π 0 s the margnal proft and S s the shortage cost. EAC s a conve functon n (, π ) for a gven value of L L, L. hs completes the proof.