106 M.R. Akbar Jokar and M. Sefbarghy polcy, ndependent Posson demands n the retalers, a backordered demand durng stockouts n all nstallatons and cons

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1 Scenta Iranca, Vol. 13, No. 1, pp 105{11 c Sharf Unversty of Technology, January 006 Research Note Cost Evaluaton of a Two-Echelon Inventory System wth Lost Sales and Approxmately Normal Demand M.R. Akbar Jokar and M. Sefbarghy 1 The nventory system under consderaton conssts of one central warehouse and an arbtrary number of retalers controlled by a contnuous revew nventory polcy (R; ). Independent Posson demands are assumed wth constant transportaton tmes for all retalers and a constant lead tme for replenshng orders from an external suppler for the warehouse. Unsatsed demands are assumed to be lost n the retalers and unsatsed retaler orders are backordered n the warehouse. An approxmate cost functon s developed to nd optmal reorder ponts for gven batch szes n all nstallatons and the related accuracy s assessed through smulaton. INTRODUCTION A two-echelon nventory system s consdered consstng of one central warehouse and an arbtrary number of retalers wth dentcal orderng batch szes. The nventory control polcy s assumed to be a contnuous revew (R; ) polcy n all nstallatons, whch means that when the nventory poston reaches a predetermned value of R, an order of sze s placed. The demand processes for a consumable (not reparable) tem are assumed to be ndependent Posson and unsatsed demands to be lost n all retalers. The transportaton tme of each order placed by the retalers s assumed to be constant. A constant lead tme s assumed for replenshng the warehouse orders from an external suppler and unsatsed retaler orders to be backordered n the warehouse and all backordered orders are lled accordng to a FIFO-polcy. The reorder pont and batch sze of the warehouse are assumed to be nteger multples of the retalers dentcal batch sze. One of the oldest papers n the eld of contnuous revew mult-echelon nventory systems s a basc and famous one wrtten by Sherbrooke [1] n He assumed (S 1;S) polces n a Depot-Base system for reparable tems n the Amercan ar force and could *. Correspondng Author, Department of Industral Engneerng, Sharf Unversty of Technology, Tehran, I.R. Iran. 1. Department of Industral Engneerng, Sharf Unversty of Technology, Tehran, I.R. Iran. approxmate the average nventory and stockout level n the bases. The result of ths paper has been used by many subsequent researchers because t uses an ecent approxmaton for the lead tme of the bases (whch s usually one of the complextes of multechelon systems). However, other papers, lke [], studed Sherbrooke's model by changng some of ts crtcal assumptons and ganed some more nterestng results. Contnuous revew models of mult-echelon nventory systems n the 1980's concentrated more on reparable tems n a Depot-Base system than on consumable tems. For example, Graves [3] worked on the determnaton of the stockng levels n such a system, Monzadeh and Lee [4] consdered the ssue of determnng the optmal order batch sze and stockng levels at the stockng locatons by usng a power approxmaton and Lee and Monzadeh [5] generalzed prevous models on mult-echelon reparable nventory systems to cover the cases of batch orderng and batch shpment. On consumable tems, Deuermeyer and Schwarz [6] proposed a smple approxmaton for a complex mult-echelon system (one warehouse and multple retalers) assumng the backorderng of stockouts n all nstallatons wth a batch orderng polcy. Svoronos and Zpkn [7] proposed several renements by consderng second-moment nformaton (standard devaton as well as mean) n ther approxmatons. In the 1990's, Axsater [8] provded a smple recursve procedure for determnng the holdng and stockout costs of a system, consstng of one central warehouse and multple retalers wth an (S 1;S)

2 106 M.R. Akbar Jokar and M. Sefbarghy polcy, ndependent Posson demands n the retalers, a backordered demand durng stockouts n all nstallatons and constant lead tmes. Axsater [9] proposed exact and approxmate methods for evaluatng the prevous system for the case of a general batch sze n all nstallatons but wth dentcal retalers. For the case of non-dentcal retalers and a general batch sze, Axsater [10] proposed methods for the exact evaluaton of two retaler cases and an approxmate evaluaton for the case of more than two retalers. Forsberg [11] presented a method for exactly evaluatng the costs of a system wth one warehouse and a number of derent retalers usng another approach. The common assumpton of the above papers s that demands durng stockout n the retalers are backordered. However, on some condtons, demands may be lost. Andersson and Melchors [1] have proposed an approxmate method for the case of lost sales when the nventory control polcy s (S 1;S) n all nstallatons (one warehouse and multple retalers) and unsatsed demands are lost n the retalers. They also ntroduced the cost evaluaton of such a system n case of a general batch orderng polcy as a future eld of research. Ths s what s beng consdered n ths paper. The contents of ths paper are now outlned. Frst, a detaled problem formulaton s gven and the revew of two specal sngle echelon problems that are referred to later are presented. Then, t wll be explaned how to overcome the two mportant complextes of the model. After that, the approxmate total cost of the system s presented and dscussed for ndng reorder ponts. Fnally, numercal results and some conclusons and further research opportuntes are presented. PROBLEM FORMULATION It s assumed that (the dentcal batch sze of all retalers) s determned through a determnstc model wth a known replenshment cost at both warehouse and retalers, as many smlar papers such as [6,7,9,10] have done before, to smplfy the problem. The objectve s to nd the optmal reorder ponts by mnmzng the total holdng costs of the warehouse and retalers and the stockout costs of the retalers. Let the followng notaton be ntroduced: N number of retalers, demand rate at retaler ; =1; ; :::; N, o demand rate at the warehouse, L transportaton tme for delveres from the warehouse to retaler ; =1; ; :::; N, L o lead tme of the warehouse orders, dentcal batch sze of all retalers, batch sze of the warehouse, o R R o h h o C C o T C reorder pont of retaler (nteger value, snce demand s one at a tme), =1; ; :::; N, reorder pont of the warehouse (an nteger multple of ), holdng cost per unt tme at retaler ; =1; ; :::; N, holdng cost per unt tme at the warehouse, penalty cost per unt of lost sale at retaler ; =1; ; :::; N, cost per unt tme of retaler n steady state, =1; ; :::; N, cost per unt tme of the warehouse n steady state, total cost of the nventory system per unt tme n steady state. REVIEW OF TWO SPECIAL CASES Revew of Exact Soluton for Backorderng Problem wth Normal Demand Consderng a sngle-echelon nventory system wth a contnuous revew control polcy, a reorder pont of R and batch sze of, a constant lead tme for replenshng orders, demand (per unt tme) as a normal dstrbuton wth mean and standard devaton and backordered unsatsed demand, Axsater [13] presents formulae for the average stock level (D(; R)) and the average stockout level (B(; R)). Assumng the lnear unt costs of holdng and stockout, the correspondng annual costs can be obtaned. The results are brey revewed and the parameters are ntroduced snce they wll be used later n the authors approxmaton. B(; R)= 0 where: H(x)= and: (x) = Z 1 x H G()d()= 1 Z x 1 R 0 0 H R ; (1) (x +1)(1 (x)) x'(x) ; (a) 1 p e u du; '(x) = 1 p e x ; (b) D(; R) = + R 0 + B(; R): (3) The denton of the parameters n the above formulae s as follows: orderng batch sze of a contnuous revew polcy, R reorder pont of a contnuous revew polcy, L the constant lead tme of each order, 0 average of lead tme demand, 0 = :L, 0 standard devaton of lead tme demand, 0 =: p L.

3 Cost Evaluaton of a Two-Echelon Inventory System 107 Revew of Exact Soluton for Lost Sale Problem wth Posson Demand Consderng a sngle-echelon nventory system wth a contnuous revew control polcy, a reorder pont of R and batch sze of, a constant lead tme for replenshng orders, demand generated by a Posson process, a lost demand durng a stockout and R<(to make sure of not havng more than one order outstandng at a tme), Hadley and Whtn [14] developed formulae for the average stock level, (D), and for the average number of lost sales ncurred per unt tme, (E). Assumng the lnear unt costs of holdng and stockout, they obtaned the correspondng annual cost. Here, ther results are brey revewed and the parameters they have used n ther formulae are ntroduced, snce they wll be used n the presented approxmaton. E = : ^T ; (4) + ^T ( +1) D = + R + ^T L + E; (5) where: and: ^T = LP(R; L) R P (R +1;L); (6) T = + ^T : (7) The denton of the parameters n the above formulae s as follows: R L ^T T and: orderng batch sze of a contnuous revew polcy, reorder pont of a contnuous revew polcy, demand rate (mean of Posson demand dstrbuton), constant lead tme, the expected length of tme per cycle that the system s out of stock, tme per cycle. P (x; L) = 1X =x L (L) e! x =0; 1; ; 3; ::: COMPLEXITIES OF THE PROBLEM In many mult-echelon systems that whch makes the problem dcult s how to exactly, or approxmately, determne the type of demand n hgher echelons and, also, the real replenshment tme of orders, from the downstream echelons to hgher ones, because of possble stockouts n the hgher ones. In the authors problem, the same problem occurs and some approxmatons are used to tackle them. The approxmatons seem to be ecent and reasonable, but they wll be tested through some numercal problems n the next secton. Here, how to analyze the demand n the warehouse and, also, the lead tme of the retalers are explaned. Demand Analyss n the Warehouse The average number of cycles per unt tme n a contnuous revew nventory system when demand s lost durng a stockout s T 1 = [14], wthout +: ^T any specal assumpton concernng the nature of the stochastc processes generatng demands and lead tmes except to assume that they do not change wth tme and that unts are demanded one at a tme. Equaton 7 s just a specal case of ths relaton when the stochastc process generatng demand s Posson and the lead tme s constant. Snce a batch sze of s ordered n each cycle, the mean rate of demand (from ths nventory system to a hgher echelon) wll be T 1 n terms of the batch sze of. As Monzadeh and Lee [4] menton, when the stockout s backordered n the retalers and the demand process at each retaler s Posson, the arrval process of orders at the warehouse (hgher echelon) s a superposton of N arrval processes n the case of one warehouse and N retalers, so that each nter-arrval tme s Erlang dstrbuted wth shape parameter. When the number of retalers n the model s large, the arrval process can be well approxmated P by a Posson N process wth mean rate of. =1 Monzadeh and Lee also stress that such an approxmaton has been used or suggested by Muckstadt [15], Deuermeyer and Schwarz [6], Albn [16] and Zpkn [17]. However, ths classc Posson approxmaton has also been used n some recent papers, lke [18]. The man derence between the authors model and thers s that demand durng a stockout s lost nstead of backorderng n the retalers. As Axsater [13] ponted out, for tems wth hgh demand, t s usually more convenent to model the demand over a tme perod by a contnuous dstrbuton and the dscrete Posson demand wll become approxmately normally dstrbuted. Usng the sprt of the two mentoned approxmatons and extendng t for the case of lost sales, one can assume that when the number of retalers n the model s large, the arrval process can be well approxmated by a contnuous normal demand process wth a mean rate P of o = N per unt and the standard devaton + =1 ^T of such a normal demand process can be approxmated by p o (both n terms of the dentcal batch sze of

4 108 M.R. Akbar Jokar and M. Sefbarghy, snce the retalers' orderng batch sze was dentcal and equal to ). Approxmatng the Retalers Lead Tme As mentoned before, retalers at the rst echelon of the model experence ndependent Posson demand processes. Demand durng a stockout s assumed to be lost. Each order that s placed on the warehouse by each retaler wll have a mnmum lead tme equal to the transportaton tme. Snce some of the orders are placed when there s a stockout at the warehouse, the lead tme may be more than just the transportaton tme. The real lead tme of each retaler order conssts of two components: Frst, the transportaton tme of the orders from the warehouse nto the retaler, and second, an addtonal watng tme, whch results from a stockout n the warehouse. Ths watng tme does not have any clear dstrbuton and t s just known that t s zero when the orders do not ncur stockouts n the warehouse and has a postve value when they are backordered n the warehouse. Based on the approxmaton of demand at the warehouse descrbed n the prevous secton, the stock n the warehouse behaves just lke an nventory system of the type descrbed before. From Lttle's famous formula n the queung theory (as Andersson and Melchors [1] use t n ther approxmaton), one can use the expresson for the average stock level gven by Equaton 1 to obtan the average watng tme of each retaler order, as gven by Equaton 8. It should be notced that Equaton 1 s vald when customer demands occur one at a tme. Snce each retaler orders a batch sze,, Equaton 1 can stll be used f one makes the addtonal assumpton that the batch sze and reorder pont of the warehouse, o and R o, are nteger multples of the dentcal batch sze of the retalers,. W = B o( o ; Ro ) o ; (8) where: o = NX =1 + ^T : (9) In the above formula, W s the average watng tme of the orders placed by retalers. Based on an approxmaton, W s added to the transportaton tme of each retaler to make the approxmate constant lead tme of the orders. Ths can be used for evaluatng the retalers costs (holdng and stockout costs). o s the mean rate of demand n the warehouse. Equaton 9 follows drectly from the result n the prevous secton. APPROXIMATE TOTAL COST AND OPTIMIZATION METHOD Total Cost Functon of the System Based on the results of the prevous sectons, the total cost of holdng and shortage n the retalers and holdng n the warehouse s as follows: T C = C o + NX =1 C : (10) The warehouse cost conssts of just the holdng cost, as follows: C o = h o :D o ( o ; R o ):: (11) In the above formula, D o ( o ; Ro ) s the average stock level n the warehouse and s as follows, usng Equaton 3 and notng that o should be an nteger multple of : ) D o ( o ; R o ) = ( o + R o ol o + B o ( o ; R o ): (1) In the above equaton, o s obtaned through Equaton 13 (as explaned before) and B o ( o ; Ro ) s the average backorder level n the warehouse n terms of, whch s obtaned n Equaton 14 usng Equatons 1 and, notng, agan, that o should be an nteger multple of : o = NX =1 o B o ; R o where: + ^T ; (13) = (p o L o ) H Z1 H(x)= G()d = 1 and: (x) = x Z x 1 o " R0 1 p e u du; " + o H " Ro # ol o p o L o ol o p ##; (14) o L o (x +1)(1 (x)) x'(x) ; (15a) '(x) = 1 p e x : (15b)

5 Cost Evaluaton of a Two-Echelon Inventory System 109 Knowng o and B o ( o ; Ro ), one can determne W from the followng equaton, as explaned before: W = B o( o ; Ro ) o : (16) An mportant pont n our approxmaton s that W s dependent on o, whch, tself, s dependent on ^T. Snce the stochastc dstrbuton of lead tme for the retaler,, s not clear, ^T does not have any exact form n ths complcated model. It s, therefore, approxmated by the constant lead tme, L, as gven n Equaton 17. The authors experence shows that the eect of ths approxmaton s neglgble on o. However, the numercal tests wll show how accurate ths approxmaton s: ^T = L P (R ; L ) R P (R + 1; L ): (17) Each retaler cost conssts of shortage and holdng costs as follows: C = :E + h :D ; =1; ; ;N; 0 R : (18) In the above formulae, E s the average number of lost sales ncurred per unt tme n retaler and D s the average stock level n retaler. Based on the approxmaton shown n the secton of \Approxmatng the Retalers Lead Tme", L + W s assumed to be constant. Usng Equatons 4 to 6, for the case of a constant lead tme, one has the followng: E = D = 0 [ + ^T 0 ^T ]; (19) (+1) + R (L + ^T 0 +W ) + E ; (0) ^T 0 = (L + W )P (R ; (L + W )) R P (R + 1; (L + W )); (1) ^T 0 s the average length of tme per cycle, for whch retaler s out of stock when the lead tme s the constant value, L + W. Optmzaton Method It s clear that one should nd the optmal values of the reorder ponts of all nstallatons through mnmzng TC. Snce t s a complcated cost functon (even C by tself s very complcated as Hadley and Whtn menton [14]), t s not easy to nd the optmal reorder ponts of the warehouse and retalers. Here, by denng some new notaton, a method s presented for mnmzng the cost functon. It s necessary to state that the reorder pont of retaler (R ) s bounded by 0 and, 0 R, snce there should not be more than one order outstandng n each retaler at any tme and ths constrant satses ths condton for a contnuous revew nventory system wth lost sales [14]. Furthermore, snce there are N retalers n ths model and none of them can have more than one order outstandng, one has R o ( N). Ths s because, f R o N, then, the reorder pont s never reached n the warehouse. Snce, n the numercal problems, one only consders the dentcal retalers case (lke many other papers n the area of mult-echelon modelng), the optmzaton method s based on ths assumpton. In short, R o s ncreased from ts lower lmt and the optmal, R, for all retalers s found. Ths contnues untl a local mnmum s reached. Because of the complexty of the cost functon one s not able to prove ts convexty. However, by consderng a logcal upper lmt for the reorder pont of the warehouse (R o ), one can constran the soluton space. The warehouse reorder pont can be lmted by R o o + 3 p o wth the condence coecent of 99.7%. In the numercal tests that wll be presented n the next secton, both the optmzaton method has been used and the total soluton space has been searched (snce both the reorder ponts are lmted). The results have been the same and ths strongly suggests that the authors local mnmum s the global mnmum. The notatons used n the algorthm are as follows: R o (n) reorder pont of the warehouse n stage n, R o optmal reorder pont of the warehouse, R reorder pont of the dentcal retalers n each stage (0 R ), R (n) optmal reorder pont of the dentcal retalers n stage n, R optmal reorder pont of the dentcal retalers (through all stages), T C(n) total system cost n stage n (assumng R(0 R ) and R o (n)), T C (n) optmal total system cost n stage n, T C optmal total system cost. Here, the algorthm for ndng the optmal reorder ponts of the warehouse and the retalers s presented as follows: Step 0: Set n =0, Step 1: Set R o (n) =( N + n):, Step : Set R=0;TC (n)= a large enough number, Step 3: Determne TC(n) usng Equaton 10, assumng that n s a counter of stages,

6 110 M.R. Akbar Jokar and M. Sefbarghy Step 4: If TC(n) TC (n), then, R (n) =R and T C (n) =TC(n), Step 5: If R 1, then, R = R + 1 and go to Step 3, Step 6: If n = 0, then, n = n + 1 and go to Step 1, Else: If TC (n) TC (n 1), then, T C = TC (n 1), R o = R o (n 1), R = R (n 1) and stop, Else: If TC (n) TC (n 1), then, n = n + 1 and go to Step 1. NUMERICAL RESULTS In order to determne the power of the authors approxmaton, a set of 36 numercal problems have been desgned wth the assumpton of dentcal retalers. To the best of the authors knowledge and, as Andersson and Melchors [1] menton, no work has been done on the case of lost sales n retalers wth the polcy of batch orderng n all nstallatons. Snce there were no prevous numercal problems as a reference wth whch to compare our approxmaton, a problem set was developed, whch oered a reasonable range of model parameters. The optmal reorder ponts of all nstallatons were found usng the optmzaton method descrbed n the prevous secton. As already mentoned, the same optmal reorder ponts were found through searchng the total soluton space by settng an upper lmt on the warehouse reorder pont. Each numercal problem was, also, smulated 10 tmes (consderng 10 runs for each problem), for the optmal reorder ponts obtaned from the approxmate model, usng GPSS/H smulaton software. The smulaton tme length of each run s unt tmes wth unt tmes as a \run n" perod. Derent startng random number seeds were employed for each problem. All of the results show that ths length of tme s sucent for the system to reach a steady state. Ths s also clear from the standard devaton of the total system cost. The cost error s obtaned by the followng relaton: Cost Error = jsmulated total cost approxmated total costj : smulated total cost () The dentcal retalers' servce level s also reported as ther ready rate (the fracton of tme wth postve stock on hand). Ths can be obtaned through Hadley and Whtn [14] as: Servce Level = 1 average number of lost sales per unt tme average number of demands per unt tme : (3) The above relaton has been employed n the approxmaton and, also, n the smulaton model to nd the servce levels. The numercal problems are as n Table 1. The number of retalers s 0 (a large enough number to approxmate the demand dstrbuton as normal n the warehouse), the holdng costs of the warehouse and the dentcal retalers per unt per unt tme are assumed to be 1, h o = h = 1 and the transportaton tme for the dentcal retalers and, also, the lead tme of the warehouse are assumed to be 1, L o = L =1. The total cost and servce level results are shown n Table. As can be seen from Table, the errors n the approxmate total cost and approxmate servce levels are small n comparson wth smulated values. The error levels are consstent wth those obtaned for smlar approxmatons used by other researchers n ths area. CONCLUSION AND FUTURE RESEARCH In ths paper, an approxmate cost functon s developed for a two-echelon nventory system wth one warehouse and several retalers, where unsatsed demand n the retalers s lost and the control polcy s contnuous revew. The man pont of ths paper s to assume lost sales durng a stockout n the retalers, snce most of the prevous papers had assumed demand durng a stockout to be backordered. Only Andersson and Table 1. Desgned numercal problems. No o No o

7 Cost Evaluaton of a Two-Echelon Inventory System 111 No R o R Total Cost Table. Total cost and servce level results. Approxmaton Smulaton Approxmaton Smulaton Mean of Standard Cost Servce Servce Total Devaton of Error Level % Level % Cost Total Cost % Servce Level Error % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Mean.3% % Standard Devaton 1.4% %

8 11 M.R. Akbar Jokar and M. Sefbarghy Melchors [1] have developed an approxmate soluton for the case of lost sales usng an (S 1;S) polcy n all nstallatons. The warehouse arrval process has been approxmated by a contnuous normal demand process and each retaler lead tme by a constant lead tme, obtanng the average watng tme of the retalers orders usng Lttle's formula from the queung theory. The approxmatons were compared wth smulaton results for 36 numercal problems. The mean error of the cost s.3 %, whch seems to be good. In future research, the retalers lead tme could be approxmated by other dstrbutons and the mean error reduced. Another future research eld s to use a servce level objectve for determnng the optmal control polcy. ACKNOWLEDGMENT The authors wsh to thank Dr R.M. Hll and Dr D.K. Smth, two senor lecturers at the Unversty of Exeter n the UK, for ther helpful comments. REFERENCES 1. Sherbrooke, C.C. \METRIC: A mult-echelon technque for recoverable tem control", Operatons Research, 16, pp (1968).. Wang, Y. and Cohen, M.A. and Zheng, Y.-S. \A two-echelon reparable nventory system wth stockngcenter-dependent depot replenshment lead tme", Management Scence, 46, pp (000). 3. Graves, S.C. \A mult-echelon nventory model for a reparable tem wth one-for-one replenshment", Management Scence, 31, pp (1985). 4. Monzadeh, K. and Lee, H.L. \Batch sze and stockng levels n mult-echelon reparable systems", Management Scence, 3, pp (1986). 5. Lee, H.L. and Monzadeh, K. \Operatng Characterstcs of a two-echelon nventory system for reparable and consumable tems under batch orderng and shpment polcy", Naval Research Logstcs, 34, pp (1987). 6. Deuermeyer, B. and Schwarz, L.B. \A model for the analyss of system servce level n warehouse/retaler dstrbuton system: the dentcal retaler case", Multlevel Producton/Inventory Control Systems, Chap. 13 (TIMS Studes n Management Scence 16), L. Schwarz, Ed., Elsever, New York, USA (1981). 7. Svoronos, A.P. and Zpkn, P. \Estmatng the performance of mult-level nventory systems", Operatons Research, 36, pp 57-7 (1988). 8. Axsater, S. \Smple soluton procedures for a class of two-echelon nventory problems", Operatons Research, 38, pp (1990). 9. Axsater, S. \Exact and approxmate evaluaton of batch-orderng polces for two-level nventory systems", Operatons Research, 41, pp (1993). 10. Axsater, S. \Evaluaton of nstallaton stock-based (R; ) polces for two-level nventory systems wth Posson demand", Operatons Research, 46, pp (1998). 11. Forsberg, R. \Exact evaluaton of (R; ) polces for two-level nventory systems wth Posson demand", European Journal of Operatonal Research, 96, pp (1996). 1. Andersson, J. and Melchors, P. \A two echelon nventory model wth lost sales", Internatonal Journal of Producton Economcs, 69, pp (001). 13. Axsater, S., Inventory Control, Kluwer's Internatonal Seres (000). 14. Hadley, G. and Whtn, T.M. \Analyss of nventory systems", Prentce-Hall, Englewood Cls, N.J., (1963). 15. Muckstadt, J.A. \Analyss of a two-echelon nventory system n whch all locatons follow contnuous revew (s; S) polcy", Techncal Report, (337), School of Operatons Research and Industral Engneerng, Cornell Unversty, USA (1997). 16. Albn, S. \On Posson approxmaton for superposton arrval processes n queues", Management Scence, 8, pp (198). 17. Zpkn, P.H. \Models for desgn and control of stochastc, mult tem batch producton systems", Operatons Research, 34, pp (1986). 18. Cheung, K.L. and Hausman, W.H. \An exact performance evaluaton for the suppler n a two-echelon nventory system", Operatons Research, 48, pp (000).