STOCHASTIC INVENTORY MODELS INVOLVING VARIABLE LEAD TIME WITH A SERVICE LEVEL CONSTRAINT * Liang-Yuh OUYANG, Bor-Ren CHUANG 1.

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1 Yugoslav Journal of Operatons Research 10 (000), Number 1, STOCHASTIC INVENTORY MODELS INVOLVING VARIABLE LEAD TIME WITH A SERVICE LEVEL CONSTRAINT Lang-Yuh OUYANG, Bor-Ren CHUANG Department of Management Scences, Tamkang Unversty, Tamsu, Tape, Tawan 5137, R.O.C. Abstract: The stochastc nventory models analyzed n ths study nvolve two models that are contnuous revew and perodc revew. Instead of havng a stockout cost term n the objectve functon, a servce level constrant s added to each model. For both these models wth a mxture of backorders and lost sales, we frst assume that the lead tme ( L )/protecton nterval ( T L ) demand follows a normal dstrbuton, and then relax ths assumpton by only assumng that the mean and varance are known. For each case, we develop a procedure to fnd the optmal soluton, and then an llustratve numercal example s gven. Keywords: Inventory, lead tme, servce level, mnmax dstrbuton free procedure. 1. INTRODUCTION In most of the early lterature dealng wth nventory problems, n both determnstc and probablstc models, lead tme s vewed as a prescrbed constant or a stochastc varable, whch therefore s not subject to control (see, e.g., Naddor [8] and Slver and Peterson [11]). In 1983, Monden [5] studed the Toyota producton system, and ponted out that shortenng lead tme s a crux of elevatng productvty. The successful Japanese experences usng Just-In-Tme (JIT) producton show that the advantages and benefts assocated wth efforts to control the lead tme can be clearly perceved. Ths research was support by the Natonal Scence Councl of Tawan under Grant NSC E The authors are also thankful to the anonymous referees for ther constructve suggestons to mprove the paper.

2 8 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme Recently, several contnuous revew nventory models have been developed to consder lead tme as a decson varable. Lao and Shyu [4] frst presented a contnuous revew nventory model n whch the order quantty was predetermned and lead tme was a unque decson varable. Later, Ben-Daya and Raouf [1] extended Lao and Shyu's [4] model by consderng both lead tme and order quantty as decson varables. Ouyang et al. [9] allowed both the lead tme and order quantty as decson varables and consdered a stockout case. Ouyang and Wu [10] utlzed the mnmax decson crteron to solve the dstrbuton free model. However, n the models prevously mentoned [1, 9, 10], reorder pont was not taken nto account, and they merely focused on the relatonshp between lead tme and order quantty. In other words, they neglected the possble mpact of the reorder pont on the economc orderng strategy. Such a phenomenon s usually not perfect n a real nventory stuaton. In a recent research artcle, Moon and Cho [7] revsed Ouyang et al.'s [9] model by consderng the reorder pont to be another decson varable. We note that the stockout cost n ther paper s an exact value. However, n many practces, t s dffcult to determne an exact value for the stockout cost, hence, we here replace the stockout cost term n the objectve functon by a servce level constrant. The objectve of ths paper s to extend Ouyang and Wu's [10] contnuous revew models to accommodate a more realstc stuaton. That s, our goal s to establsh a ( Q, r, nventory model wth a servce level constrant. From the numercal example provded, we can show that our new model s better than that of Ouyang and Wu [10]. On the other hand, we also propose a new ( T, R, nventory model for perodc revew. For both of these models, we frst assume that the lead tme/protecton nterval demand follows a normal dstrbuton, and then try to fnd the optmal orderng polcy. We next relax ths assumpton and merely assume that the frst and second moments of the probablty dstrbuton of lead tme/protecton nterval demand are known and fnte, and then solve ths nventory model by usng the mnmax dstrbuton free approach. An llustratve numercal example s provded n each case.. NOTATIONS AND ASSUMPTIONS To develop the mathematcal models, we utlze the followng notatons and assumptons n our presentaton. Notatons: D expected demand per year A orderng cost per order h holdng cost per unt per year σ varance of the demand per unt tme α proporton of demand that s not met from stock. Hence, servce level, α takes a sutable value 1 α s the

3 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 83 β the fracton of the demand durng the stockout perod that wll be backordered, 0 β 1 Q order quantty, a decson varable n the contnuous revew case r reorder pont, a decson varable n the contnuous revew case R target nventory level, a decson varable n the perodc revew case T length of a revew perod, a decson varable n the perodc revew case L length of lead tme, a decson varable X the lead tme/protecton nterval demand whch has a probablty densty functon (p.d.f.) f X E ( ) mathematcal expectaton Assumptons: x maxmum value of x and 0,.e., x max{ x, 0}. 1. The lead tme L conssts of n mutually ndependent components. The -th component has a mnmum duraton a, a normal duraton b, and a crashng cost per unt tme c. Further, for convenence, we rearrange c such that c1 c! c n. Then, t s clear that the reducton of lead tme should be frst on component 1 because t has the mnmum unt crashng cost, and then on component, and so on.. If we let L0 b j and L be the length of lead tme wth components 1,,..., n j 1 crashed to ther mnmum duraton, then L can be expressed as L n b j j 1 j 1 ( b a ), 1,,..., n ; and the lead tme crashng cost C ( per j j cycle for L [ L, L 1 ] s gven by C ( c ( L 1 c j ( b j a j ). j 1 3. The demand per unt tme has a probablty dstrbuton wth mean D and standard devaton σ, then the dstrbuton of the demand durng the length of tme t s t -fold, that s, the demand X follows p.d.f. f X (x) wth mean Dt and standard devaton σ t CONTINUOUS REVIEW MODEL In ths secton, we assume that nventory s contnuously revewed. An order quantty of sze Q s ordered whenever the nventory level drops to the reorder pont r. The reorder pont r s gven by r DL kσ L, where k s a safety factor. The

4 84 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme expected demand shortage at the end of the cycle s gven by E( X r) r ( x r) f ( x) dx, and thus the expected number of backorders per cycle s X β E( X r), and the expected number of lost sales per cycle s 0 β 1. ( 1 β ) E( X r), where We further suppose that the length of lead tme does not exceed an nventory cycle tme so that there s never more than a sngle order outstandng n any cycle. The expected net nventory level just before the order arrves s r DL ( 1 β ) E( X r), and the expected net nventory level at the begnnng of the cycle s Q r DL ( 1 β ) E( X r) (t can be verfed that ths quantty s greater than the reorder pont r ). Thus, the expected annual holdng cost s approxmately Q h[ r DL ( 1 β ) E( X r) ]. On the other hand, n many practces, the stockout cost often ncludes ntangble components such as loss of goodwll and potental delay to the other parts of the nventory system, and hence t s dffcult to determne an exact value for the stockout cost. Therefore, n ths study, we replace the stockout cost by a constrant on the servce level. That s, our objectve s to mnmze the expected total annual nventory cost whch s the sum of orderng cost, holdng cost, and lead tme crashng cost, subject to a constrant on servce level. Symbolcally, the mathematcal model of ths problem can be formulated as subject to mn EAC( Q, r, D Q Q A h r DL ( 1 β ) E( X r) D C( Q E ( X ) Q r α. (1) 3.1. Normal dstrbuton case In ths subsecton, we assume the lead tme demand X follows a normal dstrbuton wth mean DL and standard devaton σ L. Gven that r DL kσ L, we can also consder the safety factor k as a decson varable nstead of r. Thus, the expected shortage quantty E ( X r) at the end of the cycle can be expressed as a functon of k ; that s, E( X r) r X k z ( x r) f ( x) dx σ L( z k) f ( z) dz σ LG(k) > 0, () where f z (z) s the p.d.f. of the standard normal random varable Z and G(k) ( z k) f k z( z) dz. Therefore, model (1) can be rewrtten as

5 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 85 mn EAC N ( Q, D [ A C( ] hq h[ kσ L ( 1 β) σ LG( k)] Q subject to σ LG( k) αq, (3) where the subscrpt N n EAC denotes the normal dstrbuton case. The nequalty constrant n model (3) can be converted nto equalty by addng a nonnegatve slack varable, S. Thus, the Lagrangean functon s gven by EAC N ( Q, λ, S) EAC N ( Q, λ[ σ LG( k) S αq] D [ A C( ] hq hσ L[ k ( 1 β) G( k)] λ[ σ LG( k) S αq], (4) Q where λ s a Lagrange multpler. L For any gven ( Q, λ, S), EAC N ( Q, λ, S) s a concave functon n L, L ], because [ 1 EAC N ( Q, λ, S) 1 3 / 1 3 / hkσl σl G( k)[ h( 1 β) λ] < 0. L 4 4 Hence, for fxed ( Q, λ, S), the mnmum expected total annual cost wll occur at the end ponts of the nterval [ L, L 1]. On the other hand, we can further prove that, for any gven L [ L, L 1 ], by the Kuhn-Tucker necessary condtons for mnmzaton problem, we can obtan the slack varable S 0 (hence, the nequalty constrant n model (3) wll become an equalty). Therefore, for fxed L [ L, L 1 ], the mnmum value of Eq. (4) (n whch the varable S 0 ) wll occur at the pont ( Q, whch satsfes and EAC λ 0 N ( Q, ) D [ A C( ] h λα, (5) Q Q EAC N ( Q, λ 0 hσ L k 1 ( 1 β) Pz ( k) Pz ( k), (6) h EAC ( Q, 0 N σ LG( k) αq, (7) λ

6 86 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme where ( k) P ( Z k), and Z s the standard normal random varable. P z Solvng (5), (6), and (7) for Q, λ, and G (k), respectvely, leads to and where α < 1/ 4. 1 / D[ A C( ] Q, (8) h λα h λ [ 1 ( 1 β ) Pz ( k)], (9) P ( k) z αq G( k). (10) σ L Substtutng (9) nto (8) yelds D[ A C( ] h Q 1 ( 1 β ) Pz ( k) 1 α P ( k) z 1 /, (11) We cannot obtan the explct general solutons for Q and k by solvng Eqs. (10) and (11) because the evaluaton of each of the expressons requres knowledge of the value of the other. Therefore, we can establsh the followng teratve algorthm to fnd the optmal ( Q, k, L ). Algorthm 1 Step 1. For each L, 0,1,,..., n, perform () to (v). () Start wth k 1 0 and get P z( k 1 ) () Substtutng P ( 1 ) nto (11) evaluate Q 1. z k () Usng Q 1 determne G ( k ) from (10). (v) Check G ( k ) from Slver and Peterson [11, pp ] or Brown [, pp ] to fnd k, and then P z( k ). (v) Repeat () to (v) untl no change occurs n the values of Q and k. Step. For each ( Q, k, L ), compute the correspondng expected total annual cost EAC Q, k, L ), 0, 1,,..., n. N (

7 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 87 Step 3. Fnd EAC ( Q, k, L ). If EAC N ( Q, k, L ) EAC ( Q, k, L ), mn N 0, 1,,..., n mn N 0, 1,,..., n then ( Q, k, L ) s the optmal soluton. And hence, the optmal reorder pont s r DL k σ L. Example 1. In order to llustrate the above soluton procedure, let us consder an nventory system wth the data used n Ouyang and Wu [10]: D600 unts/year, A $ 00 per order, h $ 0/unt/year, σ 7unts/wee the servce level 1 α 0.985,.e., the proporton of demand that s not met from stock s α The lead tme has three components wth data as shown n Table 1. Table 1: Lead tme data Lead tme component Normal duraton Mnmum duraton Unt crashng cost 1 3 b (days) a (days) c ($/day) We assume that the lead tme demand follows a normal dstrbuton and consder the cases when β 0, 0.5, 0.8 and 1. Applyng the Algorthm 1 procedure yelds the results as shown n Table. From ths table, the optmal nventory polcy can be found by comparng EAC N ( Q, r, L ), for 0,1,,3, and thus we summarze these n Table 3. Table : Soluton procedure of Algorthm 1 ( L n week ) L C L ) ( β 0 β 0. 5 β 0. 8 β 1 r Q EAC Q, r, L ) r Q EAC Q, r, L ) r Q EAC Q, r, L ) r Q EAC Q, r, L ) N( N( N( N( $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $,640.9 Table 3: Summary of the optmal procedure soluton for Algorthm 1 ( L n week) β ( Q, r, L ) EAC N ( Q, r, L ) 0.0 (1, 55, 4) $, (13, 54, 4), (14, 54, 4), (14, 54, 4),54.05

8 88 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme Observng Table, shows that beneft can be acheved by crashng. For example, under β 1 comparng the dfference between the cases of L 8 weeks (no crashng case) and L 4 weeks (the mnmum cost case), beneft $,577.65,54.05 $ Remark 1: In Ouyang and Wu's [10] model, whch consdered a fxed reorder pont r (.e., they let P ( X > r) 0. ). Wth the normal dstrbuton demand case and β 1, they obtaned the optmal ( Q, ( 116, 4) and the mnmum expected total annual cost EAC N ( 116, 4) $, In our model, for the same case β 1, the optmal ( Q, r, L ) ( 14, 54, 4) and EAC N ( 14, 54, 4) $, Thus, we fnd that our model savngs EAC N ( Q, EAC N ( Q, r, L ) EAC N ( 116, 4) EAC N ( 14, 54, 4) $, , $.89, whch can be vewed as the reward due to controllng the reorder pont as a decson varable. Remark : As n Example 1, we further perform a senstvty analyss by consderng the change of values of h, D, A and σ whch range from 50% to -50%. For the case β 1, computed results are shown n Table 4. Table 4: Effect of parameters on the total cost and order strategy Change of the parameter (%) Order polcy ( Q, r, L ) Expected total annual cost EAC N ( Q, r, L ) Change of percentage of total cost h30 (50%) 5 (5%) 0 (0 %) 15 (-5%) 10 (-50%) D900 (50%) 750 (5%) 600 (0 %) 450 (-5%) 300 (-50%) A300 (50%) 50 (5%) 00 (0 %) 150 (-5%) 100 (-50%) σ 10.5 (50%) 8.75(5%) 7.00 (0 %) 5.5 (-5%) 3.50 (-50%) (10, 56, 4) (11, 55, 4) (14, 54, 4) (138, 80, 6) (168, 78, 6) (146, 79, 6) (134, 80, 6) (14, 54, 4) (108, 56, 4) ( 90, 57, 4) (148, 53, 4) (136, 54, 4) (14, 54, 4) (106, 8, 6) ( 89, 84, 6) (17, 64, 4) (16, 59, 4) (14, 54, 4) (119, 75, 6) (117, 70, 6) $ 3,196.70,871.77,54.05, , $,989.86,769.96,54.05, ,899.5 $,965.58,754.51,54.05, , $,71.78,60.06,54.05,411.06, % 13.78% 0% 15.45% 3.99% 18.46% 9.74% 0% 11.38% 4.74% 17.49% 9.13% 0% 10.00%.47% 7.83% 3.80% 0% 4.48% 8.63%

9 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 89 From Table 4, we can easly observe that the holdng cost s the most mportant parameter of cost savngs. 3.. Dstrbuton free case In many practcal stuatons, the dstrbutonal nformaton of lead tme demand s often qute lmted. Hence, n ths subsecton, we relax the assumpton about the normal dstrbuton of the lead tme demand by only assumng that the probablty dstrbuton of the lead tme demand X has gven fnte frst two moments (and hence, mean and varance are also known and fnte);.e., the p.d.f. f of X belongs to the class F. of p.d.f.'s wth fnte mean DL and varance σ L. Snce the form of the probablty dstrbuton of lead tme demand X s unknown, we cannot fnd the exact value of E ( X r). Hence, we use the mnmax dstrbuton free procedure to solve our problem. The mnmax dstrbuton free approach for ths problem s to fnd the most unfavorable p.d.f. f X n F for each ( Q, r, L ) and then mnmze over ( Q, r, L ); that s, our problem s to solve subject to mn Q,r,L f X F max EAC( Q,r, X E ( X ) Q r α. (1) For ths purpose, we need the followng proposton whch was asserted by Gallego and Moon [3]. Proposton. For any f F, X 1 E( X r) σ L ( r D ( r D. (13) Moreover, the upper bound (13) s tght. Because we have r DL kσ L, and demand X for any probablty dstrbuton of the lead tme, the above nequalty always holds. Then, usng model (1) and nequalty (13) and consderng the safety factor k as a decson varable nstead of r, model (1) s reduced to mn EAC U ( Q, D [ A C( ] Q hq 1 hσ L k ( 1 β)( 1 k k)

10 90 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme subject to σ L 1 k k αq, (14) where the subscrpt U n EAC denotes the dstrbuton free case. Therefore, the Lagrangan functon of model (14) can be formulated as EAC U ( Q, λ, S) EAC U ( Q, λ σ L( 1 k k) S αq D [ A C( ] hq 1 hσ L Q k ( 1 β )( 1 k k) λ σ L ( 1 k k) S αq, (15) where λ s a Lagrange multpler and S s a nonnegatve slack varable. By analogous arguments n the prevous normal dstrbuton demand case, t can be verfed that for fxed ( Q, λ, S), EAC U ( Q, λ, S) s a concave functon of L [ L, L 1 ]. Thus, for fxed ( Q, λ, S), the mnmum value of EAC U ( Q, λ, S) wll occur at the end ponts of the nterval [ L, L 1]. Furthermore, for fxed L [ L, L 1 ], by the Kuhn-Tucker necessary condtons for mnmzaton problem, we can get the slack varable S 0. Therefore, for fxed L [ L, L 1 ], the mnmum value of equaton (15) (n whch the varable S 0 ) wll occur at the pont ( Q, k, λ ) EAC whch satsfes U ( Q, EAC 0, U ( Q, EAC 0 and U ( Q, 0, Q k λ smultaneously. The resultng solutons are and 1 / D [ A C( ] Q, (16) h 4λα 1 k 1 λ h ( 1 β ), (17) 1 k k αq 1 k k. (18) σ L Combnng Eqs. (16), (17), and (18), the order quantty

11 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 91 1 / 4αD [ A C( ] hσ L Q, (19) αh( 1 αβ ) where α < 1/. Consequently, we can establsh the followng algorthm to fnd the optmal ( Q, k, L ). Algorthm Step 1. For each (18) to determne k. L, 0,1,,..., n, we use Eq. (19) to evaluate Q, and then use Eq. Step. For each ( Q, k, L ), compute the correspondng expected total annual cost EAC Q, k, L ), 0,1,,..., n. U ( Step 3. Fnd EAC ( Q, k, L ). If EAC U ( Q, k, L ) EAC ( Q, k, L ), mn U 0, 1,,..., n mn U 0, 1,,..., n then Q, k, ) s the optmal soluton. And hence, the optmal reorder pont s ( L r. DL k σ L Example. Usng the same data n Example 1, we assume that the dstrbuton of the lead tme demand s free except that ts frst and second moments are gven. Applyng Algorthm, the summarzed optmal values are presented n Table 5. Table 5: Summary of the optmal procedure soluton for Algorthm ( L n week ) β ( Q, r, L ) EAC U ( Q, r, L ) 0.0 (141, 65, 4) $, (14, 65, 4), (143, 65, 4), (143, 65, 4), The expected total annual cost EAC N Q, r, ) s obtaned by substtutng ( L ( Q, r, L ) nto model (3); and thus, EACN ( Q, r, L ) EACN ( Q, r, L ) s the largest amount that we would be wllng to pay for the knowledge of the probablty dstrbuton of demand. Ths quantty can be regarded as the expected value of addtonal nformaton (EVAI),.e., EVAI EAC ( Q, r, L ) EAC ( Q, r, ). N N L Moreover, t can be observed from Table 6 that the amount of EVAI ncreases as β ncreases.

12 9 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme Table 6: Calculaton of EVAI for the contnuous revew model β EAC N Q, r, ) EAC N ( Q, r, L ) EVAI ( L 0.0 $, $, $ ,781.1, ,779.6, ,777.55, Remark 3. Analogous to the argument n Remark 1, we dscuss the case that the dstrbuton of the lead tme demand s free and β 1. Ouyang and Wu [10] obtaned the optmal ( Q, ( 116, 4) and the expected total annual cost EAC U (116,4 ) $, In our model, the optmal ( Q, r, L ) ( 143, 65, 4) and EAC U ( 143, 65, 4) $, Thus, observe that our model savngs EAC U ( Q, EAC U ( Q, r, L ) EAC U ( 116, 4) EAC U ( 143, 65, 4) $, , $ 61.51, whch s the reward due to controllng the reorder pont. Remark 4: From Table 3 and Table 5, t s nterestng to observe that, regardless of the normal dstrbuton or dstrbuton free model n the contnuous revew case, ncreasng the backorder rate β results n a decrease n the mnmum expected total annual cost, but t results n an ncrease n the order quantty. On the other hand, there s a robustness property for the optmal reorder pont and lead tme as β vares. 4. PERIODIC REVIEW MODEL In ths secton, we consder a perodc revew nventory model. We frst assume that the nventory level s revewed every T unts of tme, a suffcent orderng quantty s ordered up to the target nventory level R, and the orderng quantty s arrved after L unts of tme, where L < T so that at most one order s outstandng n any cycle. Agan, we suppose that the protecton nterval s T L, demand X has a p.d.f. f X (x) wth mean D ( T L ) and standard devaton σ T L, and the target nventory level R s gven by R D( T δ σ T L, where δ s a safety factor. Therefore, the expected demand shortage at the end of the cycle s gven by E( X R) ( x R) f ( x) dx, and the expected number of lost sales per cycle s R X ( 1 β ) E( X R). For ths new model, we attempt to utlze some results n Montgomery et al. [6], and get the mathematcal expresson of ths problem as follows: mn EAC( T, R, A T DT C ( h R DL ( 1 β) E( X R) T

13 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 93 subject to E( X R) D( T α. (0) 4.1. Normal dstrbuton case In ths subsecton we assume that the protecton nterval demand for the ( T, R, perodc revew model follows a normal p.d.f. f X (x) wth mean D ( T and standard devaton σ T L. By applyng R D( T δ σ T L and further allowng the safety factor δ as a decson varable nstead of R, the expected shortage quantty E ( X R) at the end of the cycle can be expressed as a functon of δ as E( X R) R X x R) f ( x) dx ( σ T L( z δ ) f z) dz δ z ( σ T LG(δ ) > 0, where f z (z) and G (δ ) are defned n Secton 3.1. Hence, model (3) s reduced to mn EAC N ( T, δ, A C( DT h T DT δ σ T L ( 1 β) σ T LG( δ) subject to σ T LG( δ ) αd( T. (1) The Lagrangan functon s thus gven by EAC N ( T, λ, S) where λ s a Lagrange multpler and A C( hdt hσ T L[ δ ( 1 β ) G( δ )] T λ[ σ T LG( δ ) S αd( T ], () S s a nonnegatve slack varable. Smlar to the arguments n Secton 3, we obtan the slack varable S 0. And, for fxed ( T, δ,, EAC N ( T, has a mnmum value at the ponts of [ L, L 1]. On the other hand, for fxed L [ L, L 1 ], the mnmum value of () wll occur at the pont ( T, δ, whch satsfes EAC N ( T, T 0, EAC N ( T, δ 0, and EAC N ( T, λ 0. Thus, we obtan the followng equatons: A C( hd / hσδ σ λα D T 1 ( T G( δ )[ h( 1 β ) λ], (3)

14 94 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme h λ [ 1 ( 1 β ) Pz ( δ )], (4) P ( δ ) z and αd G ( δ ) T L, (5) σ where P z (δ ) s defned as n Secton 3.1. Substtutng (4) nto (3) yelds A C D αd ht Pz ( δ ) 1 σ G( δ ) [ 1 ( 1 β ) P ( δ )] ( T δ ( / z P ( δ ) z. (6) Snce t s not easy to solve Eqs. (5) and (6) for T and δ, an teratve algorthm can be employed to fnd the optmal ( T, δ,. Algorthm 3 Step 1. For each L, 0,1,,..., n, perform () to (v). () Start wth δ 1 0, and then get Pz ( δ 1 ) 0.5 and G ( δ 1) by checkng the table from Slver and Peterson [11] or Brown []. () Substtute δ 1, G ( δ 1 ), and Pz ( δ 1 ) nto (6), and use a numercal search method to obtan T 1. () Usng T 1, determne G( δ ) from (5). (v) Check G( δ ) from Slver and Peterson [11] or Brown [] to fnd δ, and then P δ ). z( (v) Repeat () to (v) untl no change occurs n the values of T and δ. Step. For each T, δ, L ), compute the correspondng expected total annual cost ( EAC T, δ, L ), 0,1,,..., n,. N( Step 3. Fnd mn EAC ( T, δ, L ). If EAC N ( T, δ, L ) mn EAC ( T, δ, L ), then 0, 1,,...,n N 0, 1,,...,n ( T, δ, L ) s the optmal soluton. And hence, the target nventory level s R D( T L ) δ σ T L. Example 3. Consder the same data as n Example 1. Usng Algorthm 3, the optmal values for ths example are shown n Table 7. N

15 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 95 Table 7: Summary of the optmal procedure soluton for Algorthm 3 ( T, L n week) β ( T, R, L ) EAC N ( T, R, L ) 0.0 (9.34, 17, 8) $, (9.41, 18, 8), (9.46, 18, 8), (9.48, 18, 8), By comparng the results n Table 3 and Table 7, t s obvous that the contnuous revew model s less expensve than the perodc revew, and the amount of cost dfference decreases as β ncreases. 4.. Dstrbuton free case In ths subsecton, snce the form of the probablty dstrbuton of protecton nterval demand X s unknown, we cannot determne the exact value of E ( X R). Therefore, usng the same proposton as presented n the contnuous revew case, we can obtan the least upper bound of the expected demand shortage at the end of the cycle as follows: For any f X F, let F denote the class of p.d.f. f X 's wth fnte mean D ( T L ) and varance σ ( T, then 1 E( X R) σ ( T [ R D( T ] [ R D( T ]. (7) The upper bound (7) s tght. Based upon the results of (7) and R D( T δ σ T L, the safety factor δ can be vewed as a decson varable nstead of R, and thus model (0) s reduced to mneac U ( T, subject to A C( T hdt 1 hσ T L δ ( 1 β) 1 δ δ σ T L ( 1 δ δ ) αd( T. (8) The Lagrangan functon of ths model s gven by EAC U ( T, δ, λ, S) A C( hdt 1 hσ T L T δ ( 1 β )( 1 δ δ ) λ σ T L ( 1 δ δ ) S αd( T, (9)

16 96 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme where λ s a Lagrange multpler and S s a nonnegatve slack varable. As mentoned n the above cases, t can be shown that the slack varable S 0. And, for fxed ( T, δ,, EAC U ( T, has a mnmum value at the end ponts of [ L, L 1]. Further, for fxed L [ L, L 1 ], the mnmum value of EAC U ( T, wll occur at the pont ( T, δ, whch satsfes EAC U ( T, T 0, EAC U ( T, δ 0, and EAC U ( T, λ 0. Smplfyng these equatons leads to A C( hd / hσδ σ λαd T 4 1 ( T ( 1 δ δ )[ h( 1 β ) λ], (30) 1 δ 1 λ h ( 1 β ), (31) 1 δ δ and 1 δ αd δ σ T L. (3) Substtutng (31) and (3) nto (30), we get the revew perod as [ A C( ] T, (33) hd ( 1 αβ ) where α < 1/. The dervng process s smlar to (), and hence we omt t. A smlar algorthm procedure as proposed n Secton 3. can be performed to obtan the optmal solutons. Example 4. Usng the same data n Example 1 and applyng a smlar procedure as n Algorthm yelds the optmal values gven n Table 8. Table 8: Summary of the optmal procedure soluton ( T, L n week ) β ( T, R, L ) EAC U ( T, R, L ) 0.0 (9.7, 63, 8) $ 3, (9.80, 63, 8) 3, (9.85, 64, 8) 3, (9.88, 64, 8) 3, From Table 9, we can see that the amount of EVAI ncreases as β ncreases.

17 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme 97 Table 9 : Calculaton of EVAI for the perodc revew model β EAC N T, R, ) EAC N ( T, R, L ) EVAI ( L 0.0 $ 3, $, $ ,49.95, ,494.9, ,495.08, Remark 5: Analogous to the argument n Remark 4 (contnuous revew case), from Table 7 and Table 8, t s obvous that, for the perodc revew case, ncreasng the backorder rate β results n a decrease n the mnmum expected total annual cost, but t results n an ncrease n the optmal revew perod. And the optmal target nventory level and lead tme as β vares s robust. 5. CONCLUSION In ths study, we presented a mxture nventory model wth backorders and lost sales, where the stockout cost term n the objectve functon s replaced by a servce level constrant. Frst, we extended Ouyang and Wu's [10] contnuous revew model by smultaneously optmzng order quantty, reorder pont, and lead tme. Next, we developed a perodc revew nventory model n whch revew perod, target nventory level, and lead tme are treated as decson varables. For these two models, we assumed that the lead tme/protecton nterval demand follows a normal dstrbuton, and found the optmal soluton. Then, we relaxed ths assumpton and appled the mnmax decson crteron to solve the dstrbuton free case. In future research on ths problem, t would be of nterest to consder an nventory model nvolvng the problem of net present value. Another possble extenson of ths work may be conducted by consderng the backorder rate β as a decson varable. REFERENCES [1] Ben-Daya, M., and Raouf, A., "Inventory models nvolvng lead tme as decson varable", Journal of the Operatonal Research Socety, 45 (1994) [] Brown, R.G., Decson Rules for Inventory Management, Holt, Rnehart, and Wnston, New Yor [3] Gallego, G., and Moon, I., "The dstrbuton free newsboy problem: revew and extensons", Journal of the Operatonal Research Socety, 44 (1993) [4] Lao, C.J., and Shyu, C.H., "An analytcal determnaton of lead tme wth normal demand", Internatonal Journal of Operatons & Producton Management, 11 (1991) [5] Monden, Y., Toyota Producton System, Insttute of Industral Engneers, Norcross, Georga, 1983.

18 98 L.-Y. Ouyang, B.-R. Chuang / Stochastc Inventory Models Involvng Varable Lead Tme [6] Montgomery, D.C., Bazaraa, M.S., and Keswan, A.K., "Inventory models wth a mxture of backorders and lost sales", Naval Research Logstcs, 0 (1973) [7] Moon, I., and Cho, S., "A note on lead tme and dstrbutonal assumptons n contnuous revew nventory models", Computers & Operatons Research, 5 (1998) [8] Naddor, E., Inventory System, John Wley, New Yor [9] Ouyang, L.Y., Yeh, N.C., and Wu, K.S., "Mxture nventory model wth backorders and lost sales for varable lead tme", Journal of the Operatonal Research Socety, 47 (1996) [10] Ouyang, L.Y., and Wu, K.S., "Mxture nventory model nvolvng varable lead tme wth a servce level constrant", Computers & Operatons Research, 4 (1997) [11] Slver, E.A., and Peterson, R., Decson Systems for Inventory Management and Producton Plannng, John Wley, New Yor 1985.