(1, T) policy for a Two-echelon Inventory System with Perishableon-the-Shelf

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1 Journal of Optmzaton n Industral Engneerng 6 (24) 3-4 (, ) polcy for a wo-echelon Inventory ystem wth Pershableon-the-helf Items Anwar Mahmood a,, Alreza Ha b a PhD Canddate, Industral Engneerng, Department of Industral Engneerng, harf Unversty of echnology, ehran, Iran b Assocate Professor, Industral Engneerng, Department of Industral Engneerng, harf Unversty of echnology, ehran, Iran Receved 4 October, 23; Revsed 9 August, 24; Accepted 2 eptember, 24 Abstract hs paper deals wth an nventory system wth one central warehouse and a number of dentcal retalers We consder pershable-on-theshelf tems; that s, all tems have a fxed shelf lfe and start to age on ther arrval at the retalers Each retaler faces Posson demand and employs (, ) nventory polcy Although demand not met at a retaler s lost, the unsatsfed demand at the central warehouse s backordered In ths study, the long-run system total cost rate s derved Moreover, a proposton s proved to defne a doman for the optmal soluton Also, a search algorthm s presented to obtan ths soluton Further, we extend the exstent paper of the well-known (-, ) polcy to cope wth our consdered model Fnally, n a numercal study, we compare (, ) polcy wth (-, ) polcy n terms of system total cost he results reveal that when transportaton tme from the central warehouse to the retalers s long, the (, ) polcy outperforms the (-, ) polcy Keywords: (, ) polcy, (-, ) polcy, wo-echelon nventory system, Pershable tems Introducton In ths paper, a two-echelon nventory system consstng of one central warehouse and a number of dentcal retalers s consdered Each retaler faces Posson demand, and replenshes ts stock from the central warehouse he central warehouse, n turn, replenshes ts stock from an external suppler whch has an nfnte supply After onng the stock at a retaler, an tem has a constant shelflfe beyond whch t s no longer usable Demand that cannot be met mmedately at the retalers s lost he fxed orderng cost s neglgble from whch t s ustfed to employ (, ) or (-, ) polces he (, ) polcy s to order one unt at each fxed tme perod hs polcy was ntroduced by Ha and Ha (27) for a sngle nstallaton nventory system wth nonpershable tems When (, ) polcy s employed n the frst level of a supply chan, t prevents expandng the demand uncertanty for other levels and makes ther demand determnstc, one unt every unts of tme herefore, ths polcy enoys a number of advantages For example, the cost of holdng the safety stock n the upstream locatons s elmnated Also, ths polcy s very easy to apply and leads to smplfy the nventory control and producton plannng Followng Ha and Ha (27), Ha et al (29) appled (, ) polcy to a two-echelon Correspondng author Emal address: a_mahmood@esharfedu nventory system wth nonpershable tems Mahmood et al (24a) consdered (, ) polcy for a sngle stage nventory system wth pershable tems All of these papers, n ther numercal experments, compare (, ) polcy wth the well-known (-, ) polcy hey conclude that when the lead tme s hgh, (, ) polcy s better than (-, ) polcy n terms of total cost rate o our lmted knowledge, there are very few papers concernng pershable tems n mult echelon nventory systems Abdel-Malek and Zegler (988) analyzed the optmal replenshment polces for a sngle pershable tem, two-echelon nventory system assumng determnstc demand Under demand uncertanty, Fuwara et al (977) consdered the problem of perodc revew orderng and ssung polces n a two-echelon nventory system assumng separate lfetme for each echelon Kanchanasuntorn and echantsawad (26) nvestgated the effect of product pershablty on system total cost n a two-echelon nventory system hey assume Normal demand for retalers and develop an approxmate nventory model under perodc revew polces Olsson (2) developed an approxmate technque for evaluaton of base stock polces n a contnuous revew, two-echelon seral nventory system wth pershable tems He assumes that the downstream locaton faces Posson demand, and unsatsfed demand s backordered 3

2 Anwar Mahmoud & Alreza Ha/ (, ) polcy for a wo-echelon Mahmood et al (24b) consder a two-echelon seral nventory system wth pershable-on-the-shelf tems and lost sales hey employ (-, ) polcy to control the nventory of both echelons Usng MERIC approach (herbrooke 968), they approxmate the system total cost rate, and present two procedures to obtan the optmal or near optmal solutons In ths paper, we apply (, ) nventory polcy to a two-echelon dstrbuton system wth dentcal retalers All tems start to age as they arrve at a retaler Further, we extend the model of Mahmood et al (24b) to deal wth a dstrbuton system wth dentcal retalers Fnally, we compare (, ) and (-, ) polces n terms of system total cost rate n a numercal experment he results show that when both the shortage unt cost and the transportaton tme from the central warehouse to the retalers are hgh and the pershng unt cost s low, (,) polcy outperforms (-, ) polcy he proceedng parts of ths paper are organzed as follows In secton 2, the consdered model s descrbed In secton 3, consderng (, ) polcy the system total cost rate s derved Also a soluton procedure s presented hen n secton 4, we extend (-, ) polcy to deal wth a dstrbuton system wth dentcal retalers Furthermore, n secton 5, a numercal study s carred out to compare (, ) polcy wth (-, ) polcy Fnally, the concluson and future research are presented n secton 6 2 Model Descrpton We consder a sngle tem, two-echelon dstrbuton system consstng of a central warehouse and a number of dentcal retalers he tems have a constant shelflfe and start to age on ther arrval at a retaler All retalers face Posson demand he fxed orderng cost n all locatons s neglgble Although demand not met at a retaler s lost, the unsatsfed demand at the central warehouse s backordered All transportaton tmes are fxed All satsfed demands are met based on FIFO (Frst n frst out) polcy A fxed penalty cost per lost sale and a fxed penalty cost per pershed tem are ncurred at the retalers Items held n stock both at the central warehouse and at a retaler ncur holdng costs per unt per tme unt Also the central warehouse pays a fxed purchase cost per tem herefore, hortage, pershng and holdng costs are ncurred at the retalers, and purchasng and holdng costs are ncurred at the central warehouse wo scenaros are consdered to deal wth the nventory control of presented model he frst one s to apply (, ) polcy at the retalers and (N, ) polcy at the central warehouse, whereas the second s to employ (, ) polcy at the central warehouse and (, ) polcy at the retalers, n whch and represent nventory poston at ther correspondng locaton he (N, ) polcy s to order N unts at each fxed tme perod he obectve s to fnd optmal n the frst scenaro and optmal nventory postons n the second he followng notatons are used n subsequent parts of the paper N: he number of retalers μ : he customer demand rate at a retaler π: Cost of a lost sale h : Holdng cost per unt per tme unt at the central warehouse h : Holdng cost per unt per tme unt at a retaler p: Cost of a pershed tem c: Purchase cost per unt at the central warehouse α: he probablty of pershng for an tem α : he rate of pershng at a retaler τ : ransportaton tme from the external suppler to the central warehouse τ : ransportaton tme from the central warehouse to a retaler m: Items shelflfe at the retalers I : Average on-hand nventory at the central warehouse I : Average on-hand nventory at a retaler B :he average number of backorders at the central warehouse : Inventory poston at the central warehouse : Inventory poston at a retaler H : Average total holdng cost per tme unt at the central warehouse H : Average total holdng cost per tme unt at a retaler Π: Average total shortage cost per tme unt at a retaler OC: Average total pershng cost per tme unt at a retaler PC: Average total purchasng cost per tme unt at the central warehouse C : otal cost rate at the central warehouse C : otal cost rate at a retaler C : ystem total cost rate for (,) nventory system C : ystem total cost rate for (-,) nventory system 3 (, ) Polcy 3 Prelmnares ngle nstallaton model Mahmood et al (24a) consder a sngle pershable tem, sngle locaton nventory system operatng under (, ) polcy wth lost sales, Posson demand, pershng costs, purchase costs, and per unt per perod holdng costs he results of ther model are used to obtan the cost functon of the retalers n our model herefore, some key results of ther model are presented n follows Let m represent the shelflfe, and μ denote the Posson demand rate hey present the followng formula for the percent of pershed tems, α 32

3 Journal of Optmzaton n Industral Engneerng 6 (24) 3-4 N e ( ) e m [ m ]! m m Where N floor ( ) hey also show that the proporton of tme that the system s out of stock, P, could be obtaned from α P μ () (2) Furthermore, they obtan the expected on hand nventory as follow I ( mα Θ) (3) In whch Θ m yh( y) dy, (4) Where h( y ) s as n Eq (5) 32 Central Warehouse cost warehouse faces a determnstc demand of N unts per tme unts he orderng cost s neglgble, and all transportaton tmes are fxed herefore, the optmal polcy for the central warehouse s to order N unts at each fxed tme perod nce the transportaton tmes are fxed, the central warehouse can place an order for N unts to the external suppler n such a way that these tems arrve at the tme of whch the retalers orders are placed Consequently, the central warehouse receves ts orders from the external suppler and sends them to the retalers at the same tme Accordngly, the holdng cost at the central warehouse s zero Also the purchase cost rate s Nc PC Hence, the central warehouse total cost rate s: Let represent the optmal nce the retalers are dentcal, for all of them s the same hus the central ( ) e ( m y ) N ( m y )! e ; y m N N ( ) e ( m y ) ( )! ( ) e ( m y ) n ( m y )! h ( y ) e ; m ( n ) y m n ; n,, N n ( ) e ( m y ) ( )! ( m y ) e ; m y C Nc (6) 33 Retalers cost nce the central warehouse polcy s to dspatch an tem at the same tme as a retaler places ts order, the leadtme at the retaler s determnstc and equals to the fxed transportaton tme from the central warehouse to the retaler hus the retalers model s qute smlar to the sngle nstallaton model of Mahmood et al (24a) herefore, (5) 33

4 Anwar Mahmoud & Alreza Ha/ (, ) polcy for a wo-echelon snce the proporton of tems that s pershed s α, from () the pershed cost rate at a retaler s obtaned as: OC pα pe N ( μ ) e μ m μ [ m ]! Furthermore, snce P s the proporton of tme that the system s out of stock, the proporton of demand lost at a retaler s μ P herefore, from (2) the shortage cost rate at a retaler s: π( α) Π πμp πμ (8) In addton, the holdng cost rate at a retaler could be obtaned from (3) as follow h H hi ( mα Θ) (9) Fnally, one can obtan the total cost rate at a retaler from (7), (8) and (9) as n Eq () C OC Π H N ( μ ) e π ( α ) h πμ ( mα Θ ) pe μ m μ [ m ]! (7) () Consequently, from (6) and () the system total cost rate for (, ) polcy s c N μ ( ) μ e C C NC N pe μ m [ m ] π ( α ) h πμ ( mα Θ )! () From (), one can conclude that the system total cost rate s ndependent of lead tme whch s an nterestng characterstc of (, ) Polcy 34 oluton procedure Due to the complex form of (), we cannot prove the convexty of system total cost rate However, we can prove that f the optmal s not nfntve, t always would be n the nterval of (, m] hs s proved n the followng proposton Proposton : If the establshment of the nventory system s affordable, say, then would be n the nterval of (, m] Proof: see Appendx A nce proposton defnes a doman for the optmal soluton when the establshment of the nventory system s affordable ( ), one can use a smple search algorthm to obtan the optmal soluton If, then total lost sales s the optmal soluton and the system total cost rate s Nπμ herefore, by usng a search algorthm to fnd the best amount of n (, m] and compare ts total cost rate wth Nπμ, the optmal soluton of (, ) nventory system could be obtaned Let ε be a small value, the optmal soluton of (, ) could be found wth the accuracy of ε usng the followng procedure Procedure tep : et ε Calculate and C C C usng () et tep 2: If ε m, set ε, calculate C usng (), and go to step 3 Otherwse, go to step 4 tep 3: If C C, set Go to step 2 tep 4: If Nπμ Go to step 5 C, set and and C C C Nπμ tep 5: he algorthm s fnshed and (, C ) s the best soluton of (, ) nventory system 4 (-, ) Polcy In ths scenaro the central warehouse operates under (, ) polcy, and the retalers operate under (, ) polcy he obectve s to obtan and n such a way that the total cost rate s mnmzed Mahmood et al (24b) consder a two-echelon seral nventory system wth smlar assumptons hey approxmate the central warehouse demand wth a Posson process, and approxmate the retaler leadtme usng the well-known MERIC approach of herbrook (968) her model could easly be extended to a dstrbuton system wth dentcal retalers hus n ths paper, we modfy ther approach to cope wth our consdered nventory system he only requred modfcaton s that the demand of the central warehouse 34

5 Journal of Optmzaton n Industral Engneerng 6 (24) 3-4 s the summaton of N dentcal retalers demand herefore, the demand of the central warehouse, μ, s μ Nμ ( P ) Nα (2) Furthermore, adaptng Equaton (4) and (7) of Mahmood et al (24b), α and P at a retaler s obtaned as follows Ke ( τ m) α ( )! P μ Ke In whch τ μ ( τ m) τ! τ e K B μτ τ! τm τ x ( μx e dx )! (3) (4), and τ (5) μ ( μτ) μτ Where B ( ) e (6)! Consequently, adaptng Equaton () of Mahmood et al (24b), the total cost rate of a retaler s approxmated as μ( τ m ) K e ( τ m ) C ( p hτ ) ( )! K e τ ( π h τ ) μ h h τ μ μτ! (7) Also, by substtuton of μ n Equaton (6) of Mahmood et al (24b), the approxmated total cost rate of the central warehouse s ( μ τ ) C cμ h e! μτ ( ) (8) herefore, from (7) and (8) the system total cost rate s ( ) C C NC c h e ( )! K e N ( p h ) Nh Nh ( m ) ( m ) ( )! K e N ( h )! (9) Mahmood et al (24b) present two procedures to obtan the optmal soluton of the system In ther frst procedure, they present a heurstc to fnd the optmal soluton usng the approxmated total cost rate However, the optmal soluton of the approxmated model s not necessarly the optmal soluton of the consdered system herefore, they present the second procedure whch s a smulaton-based neghborhood search heurstc he obtaned soluton of the frst procedure s used as the startng pont of the second procedure In second procedure, for a gven pont, ), eght neghborhoods ncludng (, ),, ),, ), ( (, ) (, (, ( ( (, (,, ), ), ) and ) are consdered hen for each of them 3 smulatons wth tme unts s executed he average cost rate obtaned from these 3 smulatons s assgned to the correspondng neghborhood If at least one mprovng soluton has been determned, the neghborhood search s restarted wth respect to the best of them If no superor soluton was found, the procedure stops and returns the best found soluton Mahmood et al (24b), n a numercal experment show that ther second procedure obtans the optmal soluton of the system almost n all consdered problems We adapt ther procedures for our consdered model as follows Procedure 2 (Adapted from Procedure of Mahmood et al (24b)) EP : et :, mn opt C : Nπμ, : and opt : EP : et s :, :, τ : τ, C mn : Nπμ and C mn : πμ Calculate α from (3), and P from (4) EP 2: Whle mn ( C pα), do { μ Nμ ( P ) Nα, Calculate τ from (5), and C(, τ ) from (7) Calculate C, ) from (8) and set ( μ 35

6 Anwar Mahmoud & Alreza Ha/ (, ) polcy for a wo-echelon C If ) : C (, μ ) NC (, ) ( τ mn ( ) C mn : C( ) : μ C, then set C, μ and s : If mn ( ) C mn : C( ) : C, then set C et Calculate α from (3), and P from (4) } EP 3: Calculate C (, ) from (8), μ mn mn mn If C C then C : C opt and opt : s and let : mn If C μ) mn C (, then set opt opt (, ), return : and mn C, and stop the procedure Otherwse, set : and go to EP Procedure 3 (Adapted from Procedure 2 of Mahmood et al (24b)) tep : et : and J : 5 max Execute Procedure 2 Let be the best polcy found by Procedure 2 tep : Execute 3 smulatons wth tme unts for and set C to the average total cost rate obtaned by these smulatons tep 2: Execute 3 smulatons wth tme unts for all neghborhoods of whch are not smulated before et to the polcy wth the mnmum average total cost rate among all neghborhoods Also set C to ths total cost rate tep 3: If C C and Jmax, set :, and go to step 2 Otherwse, return as the best polcy and C as ts total cost rate top the Procedure he nterested reader s referred to Mahmood et al (24b) for more detals about convergence, effectveness and accuracy of these procedures 5 Numercal tudy hs secton s devoted to compare the performance of (, ) polcy (cenaro ) wth (-, ) polcy (cenaro 2) n terms of system total cost rate o do ths, for consdered problems, % ΔC C C C s calculated Where %ΔC shows how much n percent (, ) polcy performs better than (-, ) polcy Apart from comparson between two consdered polces, the effect of varyng the values of π, p, m and τ on the optmal soluton for both polces s studed In all consdered test problems, we have N 5, c 5, μ, h 2, h, τ 5, τ {, 2,3,4,5,6,7,8,9,} and ε o obtan the best soluton of cenaro and cenaro 2, we utlze Procedure and Procedure 3, respectvely Matlab software s used to codng these procedures In table, the effect of varyng tems shelflfe and transportaton tme from the central warehouse to the retalers on both polces s studed In ths table, the results for p, π 4 and m {5,,2} are presented As t could be seen, ncreases as m ncreases hs observaton s ntutvely expected, snce ncreasng of m leads to decreasng of pershng rate otal cost rate of (, ) polcy s ndependent of transportaton tmes, but total cost rate of (-, ) polcy ncreases as τ ncreases herefore, for small values of τ, %ΔC s negatve, and by ncreasng τ, %ΔC becomes postve he postve amount of %ΔC means that the (, ) polcy outperforms the (-, ) polcy hs behavor s llustrated n Fgure for m Accordng to able, the optmal nventory postons are not a monotonc functon of m, as one mght expect However, as expected, ncreasng n m leads to decreasng n total cost rate of both polces Although s an ncreasng functon of τ, decreases as τ ncreases hs appears due to that we consder a fxed transportaton tme from the external suppler to the central warehouse, then wth ncreasng of τ the delay n the central warehouse due to stockouts would be a smaller percent of retalers leadtme able 2 presents the results for m, π 4, p {5,, 2} As can be seen, when τ ncreases, %ΔC ncreases whch means that for hgh values of τ the (, ) polcy outperforms the (-, ) polcy hs result s smlar to that obtaned from able Moreover, as p ncreases the threshold value of τ, for whch (, ) s better than (-, ), ncreases Most of results are what one ntutvely expects For example, as p ncreases the total cost rate of both polces ncreases Furthermore, 36

7 Journal of Optmzaton n Industral Engneerng 6 (24) 3-4 ncreasng n p leads to decreasng of and ncreasng of, n order to decrease the total number of pershed tems he results for m, p and π {2,4,6} are presented n able 3 For the comparson of two polces, smlar results can be seen hat s, for enough ystem otal Cost ٠١ ٠٢ 2٠٣ 3٠۴ 4٠۵ 5 ٠۶6 ٠٧7 ٠٨8 ٠٩9 ١ τ long τ, (, ) polcy s better than (-, ) polcy Moreover, as π ncreases the threshold value of τ, for whch (, ) outperforms (-, ), decreases Other results of ths table are the same as what expected For example, as π ncreases, the total cost rate of both polces does (-, ) Polcy ncrease, ncreases, and decreases Fg ystem total cost of the polces wth respect to τ able Numercal results for p and π 4 m C ( ) τ C ( ) %ΔC C ( ) %ΔC C ( ) %ΔC % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 37

8 Anwar Mahmoud & Alreza Ha/ (, ) polcy for a wo-echelon able 2 Numercal results for m and π 4 P C ( ) τ C ( ) %ΔC C ( ) %ΔC C ( ) %ΔC % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % able 3 Numercal results for m and p π C ( ) τ C ( ) %ΔC C ( ) %ΔC C ( ) %ΔC % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Fnally, based on the observed results, when both shortage unt cost and transportaton tme from the central warehouse to the retalers are hgh and the pershng unt cost s low, the (,) polcy s preferred Moreover, for fxed values of system parameters, there s a fxed value of transportaton tme from the central warehouse to the retalers for whch the (,) polcy outperforms the (, ) polcy Further, as the lead tme ncreases, ths superorty s more pronounced 6 Conclusons and Future Research In ths paper, a two-echelon nventory system wth a sngle pershable-on-the-shelf tem was consdered he consdered system conssts of a central warehouse and a number of dentcal retalers wo scenaros were nvestgated to deal wth the nventory control of the system In the frst one (, ) polcy was employed, whereas n the second (-, ) polcy was appled When (, ) polcy s used for retalers, the central warehouse faces a determnstc demand herefore, the central warehouse can place ts orders to the external suppler n such a way that these orders arrve at the central warehouse and are dspatched to the retalers at the same tme Consequently, the holdng cost at the central warehouse s zero hs s an nterestng advantage of employng (, ) polcy for mult-echelon nventory systems Moreover, the system total cost rate of (, ) polcy s ndependent of the transportaton tme from the central warehouse to the retalers Furthermore, a numercal study was carred out to compare two consdered scenaros Accordngly, for fxed values of system parameters, there s a fxed value of transportaton tme from the central warehouse to the retalers for whch (, ) polcy s better than (, ) 38

9 Journal of Optmzaton n Industral Engneerng 6 (24) 3-4 polcy Further, as the lead tme ncreases ths superorty s more pronounced An attractve drecton for future research s to develop (, ) polcy to deal wth a two-echelon nventory system wth non-dentcal retalers Furthermore, consderng a model from whch the tems start to age at the central warehouse would be another challengng drecton uch a model s more complex than ours, snce the retalers concern the tems wth stochastc shelflfe Consderng the orderng cost s also mportant and would be a challenge for future research Appendx A Proof of Proposton We should obtan the total cost rate for m o do ths, we can nterpret the nventory problem at the retalers as a D/M/ queue wth mpatent customers, a sngle channel queueng system n whch the nter-arrval tmes are constant, equal to, the servce tmes have exponental dstrbuton wth mean /μ, and a customer (tems n nventory system) leaves the queue system whenever hs soourn tme s larger than m It s clear that the number of customers (unts) n ths queueng system s equal to the nventory on hand n our nventory system Let W denote the queue watng tme of an arrvng q customer (tem) n the queueng system n steady state, W denote a customer (tem) average watng tme n the s queueng system n steady state, denote the random varable of the requred servce tme of a customer (tem) n steady state, and represent the random varable of the occurred servce tme of a customer (tem) n steady state Consequently, usng ths queueng system, the total cost rate of the consdered nventory system s obtaned as follows It s clear that m f f W for m q m m hus we have, (A) From (A), t s clear that W s E( ) (A2) Usng Lttle s formula and (A2), the long-run average number of unts at a retaler s obtaned as I Hence, E( ) nce W, the probablty that an tem s pershed at q the retalers s obtaned as follows Pr( ) m e m herefore, the proporton of products that s pershed n a retaler s α hus p pe m OC (A4) Furthermore, the proporton of tme that the nventory system s out of stock, P, could be nterpreted as the proporton of tme for whch the server n queueng P system s dle Hence, effectve Where n our case and E( ) herefore, P E( ) effectve (A5) nce P s the proporton of tme that the system s out of stock, the proporton of demand that s lost s μ P Hence, E( ) P ( ) (A6) In addton, the amount of tems purchased per tme unt at the central warehouse s N, hus Nc PC (A7) Form (A3), (A4), (A6), and (A7), the total cost rate of the system for m can be wrtten as: C PC N OC Π H μm c E ( ) pe E ( ) N h πμ ( ) μ m c E ( ) pe C N ( h πμ ) πμ (A8) E( ) H h I h (A3) 39

10 Anwar Mahmoud & Alreza Ha/ (, ) polcy for a wo-echelon m c E( ) pe From (A8), f ( h ), It s clear that Moreover, f m c E( ) pe ( h ), evdently C s ncreasng wth respect to, also t s assumed that m, thus m herefore, f, wll be n the nterval of (,m] References [] Abdel-Malek, L and Zegler, H (988) Age dependent pershablty n two-echelon seral nventory systems Computers and Operatons Research, 5, [2] Fuwara, O, oewand, H and edarage, D (997) An optmal orderng and ssung polcy for a two stage nventory system for pershable products European Journal of Operatonal Research, 99, [3] Ha, R and Ha, A (27) One-for-one perod polcy and ts optmal soluton Journal of Industral and ystems Engneerng,, 2 27 [4] Ha, R, Prayesh Neghab, M and Babol, A (29) Introducng a new orderng polcy n a two-echelon nventory system wth Posson demand Internatonal Journal of Producton Economcs,7, [5] Kanchanasuntorn, K and echantsawad, A (26) An approxmate perodc model for fxed-lfe pershable products n a two-echelon nventory-dstrbuton system Internatonal Journal of Producton Economcs,, 5 [6] Mahmood, A, Ha, A and Ha, R (24a) One for One Perod Polcy for Pershable Inventory ubmtted to Computers & Industral Engneerng [7] Mahmood, A, Ha, A and Ha, R (24b) A twoechelon nventory model wth pershable tems and lost sales ubmtted to centa Iranca [8] Olsson, F (2) Modellng two-echelon seral nventory systems wth pershable tems IMA Journal of Management Mathematcs, 2, 7 [9] herbrooke, CC (968) MERIC: A mult-echelon technque for recoverable tem control Operatons Research, 6,