Inventory Management Under Random Supply Disruptions and Partial Backorders

Transcription

1 Inventory Mngement Under Rndom Supply Disruptions nd Prtil Bckorders Antonio Arreol-Ris, 1 Gregory A. DeCroix 2 1 Deprtment of Informtion & Opertions Mngement, Lowry Mys College & Grdute School of Business, Texs A&M University, College Sttion, Texs Fuqu School of Business Duke University, Durhm, North Crolin Received June 1997; revised Mrch 1998; ccepted 30 Mrch 1998 Abstrct: We explore the mngement of inventory for stochstic-demnd systems, where the product s supply is rndomly disrupted for periods of rndom durtion, nd demnds tht rrive when the inventory system is temporrily out of stock become mix of bckorders nd lost sles. The stock is mnged ccording to the following modified (s, S) policy: If the inventory level is t or below s nd the supply is vilble, plce n order to bring the inventory level up to S. Our nlysis yields the optiml vlues of the policy prmeters, nd provides insight into the optiml inventory strtegy when there re chnges in the severity of supply disruptions or in the behvior of unfilled demnds John Wiley & Sons, Inc. Nvl Reserch Logistics 45: , INTRODUCTION We consider stochstic-demnd inventory system where the product s supply is rndomly disrupted for periods of rndom durtion. The source of supply disruptions could be process-relted or mrket-relted. In the first cse, the supply my become unvilble due to brekdowns, trnsporttion disruptions, or strike. In the second cse, even though the goods my be physiclly vilble, mrket conditions my be such tht prticulr compny does not hve ccess to the product. One exmple of such conditions my rise s result of price fluctutions, where, during certin periods of time, the price t which supply equls demnd is so high tht purchsing the product becomes prohibitive for some compnies. Another exmple is scrcity of goods where the supplier gives preference to mjor plyers t the expense of minor plyers. The uthors re fmilir with compny tht fces this sitution. For excellent motivtions of inventory systems with rndom supply disruptions, see Prlr nd Berkin [18] nd Gupt [8]. Customers tht encounter stockout my decide to bckorder or my become lost sle. Most work in inventory mngement hs considered one of the two extremes of customers behvior during stockouts: All customers bckorder or ll customers re lost. In this pper we consider more generl stockout behvior, commonly clled prtil bckorders, where Correspondence to: T. Arreol-Ris 1998 by John Wiley & Sons, Inc. CCC X/98/ / 8m27$$ :11:55 nr W: Nv Res 994

2 688 Nvl Reserch Logistics, Vol. 45 ( 1998) frction of demnds tht rrive during stockout is bckordered nd the remining frction is lost. The costs of mnging the system re sttionry, nd include ordering, holding, lost sles, nd bckordering costs. The system is controlled ccording to the following modified (s, S) policy: When the inventory level is t or below s nd the supply is vilble, procure the necessry mount to bring the inventory level up to S. This modified (s, S) policy ws dopted for severl resons. The optiml policy for the inventory system under study is n open reserch question. In the bsence of supply disruptions, the proposed policy becomes the stndrd (s, S) policy which is known to be optiml for this sitution. Plcing orders when the inventory level is t or below the reorder point nd the supply is vilble nturlly modifies the stndrd ( s, S) policy to ccommodte supply disruptions. For the inventory system under considertion, we first derive the cost function nd use it to determine the optiml vlues of the policy prmeters (s, S). Then we explore the impct on the optiml vlues of the policy prmeters of vritions in the verge frequency nd durtion of supply disruptions, nd of vritions in the frction of stockouts tht re bckordered. We use the results of this nlysis to obtin insights into the optiml strtegy for using the policy prmeters s nd S to protect the system ginst supply disruptions with different levels of severity, nd lso to exmine the effect of different stockout behviors on this optiml strtegy. Due to the difficulty of hndling prtil bckorders, the inventory literture in this re is limited. Moinzdeh [ 12] considers bse-stock level inventory system with Poisson demnd, constnt resupply times, nd prtil bckorders. Other inventory models with prtil bckorders cn be found in Montgomery et l. [15], Kim nd Prk [10], nd Posner nd Ynsouni [22]. Tretment of supply uncertinty in production-inventory system cn be trced to Meyer et l. [ 11], which studied single-stge production system fcing constnt demnd where the supply source is subject to rndom filure. The uthors ssumed zero setup cost nd pure lost sles, nd developed expressions for the operting chrcteristics of the system. Other work on production-inventory systems with deterministic demnd nd supply disruptions includes [1], [4], [5], [6], [7], nd [14]. Prlr nd Berkin [ 18] study the clssic EOQ problem with supply disruptions. Prlr nd Perry [19] extend this nlysis to system where orders my be plced before the inventory level reches zero, nd where there is fixed cost for determining the stte of the supplier. Weiss nd Rosenthl [ 25] determine the optiml inventory policy when the timing (but not the durtion) of supply disruptions is known in dvnce. Prlr nd Perry [ 20] consider system with two suppliers subject to independent disruptions. To our knowledge, the only ppers deling with supply disruptions nd rndom demnd re [2], [3], [8], [17], [21], nd [23]. Posner nd Berg [21] extend Meyer et l. [11] to the cse of compound Poisson demnd. Cho [2] presents model in which the rte of inventory ccumultion or reduction cn be continuously djusted. Under liner cost structure the uthor chrcterizes the optiml control policy using single inventory trget. Cho et l. [3] present very generl model for mnging fuel inventories t electric utilities, nd describe solution procedures involving dynmic progrmming nd simultion. Relted work cn be found in [13] nd [16]. The work most similr to tht presented here is contined in Gupt [8], Prlr [17], nd Song nd Zipkin [23]. One of the models presented in Song nd Zipkin is essentilly periodic-review version of our model, with the dditionl ssumption of complete bckordering. In tht setting, the uthors show tht policy equivlent to our modified (s, S) policy / 8m27$$ :11:55 nr W: Nv Res 994

3 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions 689 is optiml, thus providing further motivtion for our policy choice. Prlr presents model similr to ours, with positive delivery led times nd more generl ssumptions regrding the frequency nd durtion of disruptions, but with the more restrictive ssumption of full bckordering. The uthor derives n pproximte cost function nd demonstrtes solution procedure for the specil cse when order quntities re lrge. Gupt studies model with demnd nd disruption ssumptions similr to those considered here. His model is more restrictive in tht it ssumes tht ll shortges result in lost sles, but more generl since it llows for positive delivery led times. In ddition, Gupt proposes using stndrd ( r, Q) policy, which my result in plcing multiple orders during supply disruption. The unnecessry ordering costs resulting from such policy mke it generlly inferior to the modified ( s, S) policy proposed here. In light of the previous work on inventory systems with supply disruptions, this pper mkes number of primry contributions. First, it is the only work to ddress the problem of mnging stochstic-demnd system with both supply disruptions nd prtil bckorders. Since the qulittive behvior of the model presented here remins intct for the extreme cses of full bckordering nd pure lost sles, our work subsumes those versions of previous models tht gree with our ssumptions. It thus provides unified tretment of some of the models considered by Song nd Zipkin [23], Prlr [17], nd Gupt [8]. In ddition, we obtin exct closed-form expressions for the system costs. Bsed on these expressions, we re ble to develop relible solution procedures nd extrct vluble insights into the behvior of the system. In prticulr, our nlysis provides guidnce to mngers regrding the best strtegy to use for protection ginst chnges in the severity of supply disruptions or chnges in the behvior of unfilled demnds. The contents of the pper re orgnized s follows. Section 2 dels with model development nd nlysis. The section includes derivtion of the cost function nd determintion of the optiml vlues of the policy prmeters. Sections 3 nd 4 respectively ddress the optiml inventory strtegy when there re chnges in the severity of supply disruptions nd in the stockout behvior. The lst section summrizes our reserch findings nd offers directions for future reserch. 2. MODEL DEVELOPMENT AND ANALYSIS Let D denote demnd per unit time, I denote the interrrivl time of supply disruptions, nd L denote the length of supply disruptions. We ssume tht D is Poisson process with prmeter, nd tht I nd L re exponentilly distributed with prmeters l nd m, respectively. Thus represents verge demnd per unit time, 1/ l represents verge time between supply disruptions, nd 1/ m represents verge durtion of supply disruptions. In our experience with systems fcing supply disruptions, the delivery led time fter n order is plced tends to be smll compred to the verge length of supply disruption. To reflect this, nd to simplify the nlysis, our model ssumes tht delivery led times re zero. Since zero led times result in never hving orders outstnding, the modified (s, S) inventory policy cn be implemented bsed on the inventory level rther thn the more trditionl inventory position. We will use the following nottion nd definitions. N Å long-run verge number of orders plced per unit time, OH Å long-run verge on-hnd inventory level, q Å probbility tht stockout becomes bckorder, / 8m27$$ :11:55 nr W: Nv Res 994

4 690 Nvl Reserch Logistics, Vol. 45 ( 1998) 1 0 q Å probbility tht stockout becomes lost sle, LS Å long-run verge number of lost sles per unit time, BR Å long-run verge number of units bckordered per unit time, BL Å long-run verge bckorder level, k Å ordering cost, h Å holding cost rte, p Å unit lost sles cost, p 0 Å unit bckordering cost, p 1 Å bckordering cost rte, K Å long-run verge totl cost per unit time. In nlyzing our modified (s, S) inventory policy, we will find it convenient to work with s nd D å S 0 s insted of s nd S. The policy prmeter D could be interpreted s the mount ordered if the supply is vilble when the inventory levels hits s or, lterntively, s the minimum quntity ordered Cost Function Since demnd for the product is Poisson process, we restrict s nd D to be integers. Nturlly D 1, nd following previous reserch in inventory theory, we ssume tht s 0. Let b denote the probbility of finding the supply unvilble when the inventory level hits s. The following proposition estblishes the fundmentl result of this section. PROPOSITION 1: In the inventory system under study K(s, D) Å N k / h D D / 1 2 / s / b m s 0 (1 0 r s ) m / r s bg m, N Å m md / b, nd b Å l D l / m 1 0 / l / m where r Å /( / m) nd g Å (1 0 q)p / q(p 0 / p 1 /m). All proofs re included in the Appendix. When no confusion rises, K will be used in lieu of K(s, D). Notice tht, in the limiting cse q r 0, the inventory system becomes one with pure lost sles nd g r p. In the limiting cse q r 1, the inventory system becomes one with full bckorders nd g r p 0 / p 1 /m which represents the expected cost per bckorder. Consequently, g cn be interpreted s the expected cost per demnd rriving during stockout. / 8m27$$ :11:55 nr W: Nv Res 994

5 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions Optimiztion Results Let s* nd D* denote the optiml vlues of the policy prmeters. In this section we focus on finding s* nd D*. Define nd s*(d) Å rg min s 0 K(s, D) D*(s) Å rg min D 1 K(s, D). For ese of exposition, the following development ssumes tht s*(d) nd D*(s) re unique. If this is not the cse, the rguments involving unique s*(d) nd D*(s) cn be dpted by focusing on prticulr vlues of s*(d) nd D*(s), sy the smllest or the lrgest. PROPOSITION 2: For fixed vlue of D, K is convex in s. Moreover, if D b m 0ln(r) ú r 1 0 r h / gm h 0 1, then s*(d) Å 0; otherwise s*(d) Å ln(rb) ln(r) or s*(d) Å ln(rb) ln(r) where Å m2 D / bm 0b 2 ln(r) nd b Å h h / g. Proposition 2 estblishes tht K is convex in s for fixed D. However, lthough visul nlysis of numericl trils suggests tht K is in fct jointly convex in s nd D, we hve not been ble to verify this nlyticlly, or even to show tht K is convex in D for fixed s. As result, closed-form expression for D*(s) does not seem possible, nd thus we cnnot rule out potentil difficulties in computing s* nd D* if K is not well behved. Despite this possibility, we were ble to obtin some qulittive informtion regrding the behvior of s*(d) nd D*(s), s indicted in the following proposition. PROPOSITION 3: s*(d) is nonincresing in D nd D*(s) is nonincresing in s. / 8m27$$ :11:55 nr W: Nv Res 994

6 692 Nvl Reserch Logistics, Vol. 45 ( 1998) In ddition to providing insight into the behvior of the optiml policy prmeters, Proposition 3 llows us to estblish the finite convergence of Algorithm A below to the optiml pir (s*, D*). ALGORITHM A: 1. Perform line serch on D to compute D*(0). 2. For ech D {1,...,D*(0)}, compute s*(d) using Proposition Among the D*(0) pirs (s*(d), D) computed in Step 2, the one tht yields the smllest vlue of K is the optiml pir (s*, D*). 3. OPTIMAL INVENTORY STRATEGY FOR CHANGES IN THE SEVERITY OF SUPPLY DISRUPTIONS Given the complexity of s*, D*, nd K(s*, D*), our chrcteriztion of the optiml inventory strtegy s m or l chnges will be bsed on nlyticl results for limiting vlues of m nd l, nd on results of extensive numericl experiments for their nonlimiting vlues. In preprtion for the forthcoming nlysis, we will tke smll digression to discuss how the protection of the inventory system ginst supply disruptions cn be djusted by vrying the vlues of s, D, nd S. While it is obvious tht incresing s would increse the protection level of the system, nd decresing s would decrese the protection level of the system, it is less cler wht would be the effect on the protection level when D increses or decreses. Specificlly, incresing D reduces the number of cycles per yer nd hence decreses the number of times tht potentil supply disruption my be fced, which in turn increses the protection level. On the other hnd, it is esy to see from Proposition 1 tht incresing D increses b nd thus increses the probbility of encountering supply disruption when the inventory level hits s, which in turn decreses the protection level. The following proposition settles this dilemm. PROPOSITION 4: The frction of time tht the inventory system is incurring stockouts (lost sles or bckorders) is decresing in D. While the bove discussion mkes it cler tht incresing either s or D increses the system s protection ginst supply disruptions, it is not obvious to wht extent ech of these prmeters should be used to chieve such protection. For exmple, if n increse in the protection level is needed, should s be incresed, should D be incresed, should both be incresed, or should one be incresed nd one be decresed? In ddition, if n increse in protection level is needed, will the djustments in s nd D lwys result in lrger vlues of S? Given Proposition 3, which sttes tht ll else being equl, increses in s tend to yield decreses in D nd vice vers, one should not be surprised to lern tht the nswers to the bove questions re complex. The following two subsections go long wy towrd nswering these questions, nd provide significnt insight into the impct of supply disruptions on the inventory system being studied Chnges in m Consider the limiting cse m r 0. Using Eq. (13) in the Appendix, it is not hrd to demonstrte tht lim mr0 0 iå0 P(i, 1)Å 1. Thus for ny vlues of (s, D) ll demnds will / 8m27$$ :11:55 nr W: Nv Res 994

7 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions 693 Figure 1. Optiml behvior of the inventory system s m vries: ( ) s*; ( ) D*; ( ) S*. be lost or bckordered. This indictes tht when the verge durtion of supply disruptions grows without bound, the product should not be inventoried. Consider now the limiting cse m r. Using Proposition 1, it is not difficult to show tht Hence, if m Å, then nd lim K Å k mr D / h(d / 1) 2 / hs. (1) s* Å 0 (2) D* Å 2k h or D* Å 2k h. (3) Therefore, s the verge durtion of supply disruptions goes to zero, the optiml behvior of the inventory system pproches tht of the bsic EOQ model. For nonlimiting vlues of m, to study the behvior of s*, D*, nd S* sm chnges, we formulted 128 test problems. These problems were generted by fixing h Å 1, nd then tking ll combintions of the following prmeter vlues: Å 100, 500; q Å 0.3, 0.7; k Å 50, 200; p Å 10, 50; p 0 Å 2, 5; p 1 Å 50, 200; l Å 1, 12. Bsed on the optimiztion results of Section 2.2, we solved ech of the 128 test problems for lrge number of vlues of m, nd produced grphs of s* vs. m, D* vs. m, nd S* vs. m. The behvior of s*, D*, nd S* sm chnges ws consistent cross ll test problems, nd this behvior is cptured by the exmple in Figure 1, where Å 100, q Å 0.3, k Å 50, h Å 1, p Å 10, p 0 Å 2, p 1 Å 50, l Å 1, nd EOQ Å 100. Note the following: / 8m27$$ :11:55 nr W: Nv Res 994

8 694 Nvl Reserch Logistics, Vol. 45 ( 1998) S*, s*, nd D* re decresing in m. As m r, s* r 0 nd D* r EOQ. The mngeril implictions of these reserch findings re severl. When the severity of the supply disruptions diminishes becuse m is incresing, less protection is needed, nd, s result, the optiml vlues of the policy prmeters should decrese. Conversely, when the severity of the supply disruptions increses becuse m is decresing, more protection is needed, nd the optiml vlues of the policy prmeters should increse. However, wht is not necessrily intuitive is tht, even though, by Proposition 3, s*(d) nd D*(s) tend to move in opposite directions, n increse in the system s protection level is best chieved by incresing both s* nd D*. With regrd to the second finding, we know from the nlyticl results in Eqs. (1) (3) tht, s m r, the limiting behvior of the system is tht of bsic EOQ system. Nevertheless, it is interesting to see tht in some circumstnces even for moderte vlues of m, the inventory system strts to exhibit EOQ behvior Chnges in l Consider the limiting cse l r 0. Using Proposition 1, it is not difficult to show tht lim lr0 K Å k D / h(d / 1) 2 / hs. (4) Compring (1) to (4), we conclude tht the results in (2) nd (3) pply here s well, nd hence when the verge time between supply disruptions goes to infinity, the optiml behvior of the inventory system lso becomes tht of the bsic EOQ model. Consider now the limiting cse l r. It is not hrd to demonstrte tht lim K lr Å m D(D / 1) / md / k h 2 / sd / m s 1 0 (1 0 r s ) m / r s g m. (5) As l r, the inventory system being studied pproches one where the supply is lwys unvilble t the time the inventory level hits s. As result, every ordering cycle experiences n exponentil inventory-replenishment time with men 1/m. However, since n order is not plced until the supply becomes vilble ( i.e., t the end of the inventory-replenishment time), this inventory system is distinctly different from clssicl inventory systems with exponentil inventory-replenishment times (see, for exmple, Hdley nd Whitin [9]). In prticulr, ech order lwys returns the on-hnd inventory to S, successive inventoryreplenishment times re clerly independent, nd orders cnnot cross. Using Eq. (5), it is not difficult to show tht, s l r, if D ú m 0ln(r) r 1 0 r h / gm h 0 1, then s*(d) Å 0; otherwise, / 8m27$$ :11:55 nr W: Nv Res 994

9 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions 695 s*(d) Å ln(rb) ln(r) or s*(d) Å ln(rb) ln(r), (6) where Å m2 D / m 0 2 ln(r) (6b) nd b Å h h / g. (6c) Therefore, s the verge interrrivl time of supply disruptions goes to zero, s* nd D* clerly converge to finite vlues. These vlues cn be obtined using n obvious vrition of Algorithm A. We conducted second numericl experiment with two objectives. First, we sought to gin dditionl insight into the limiting behvior of s*, D*, nd S* sl r. Second, we wished to study, for nonlimiting vlues of l, the behvior of s*, D*, nd S* sl chnges. We formulted 128 test problems by fixing h Å 1, nd then tking ll combintions of the following prmeter vlues: Å 100, 500; q Å 0.3, 0.7; k Å 50, 200; p Å 10, 50; p 0 Å 2, 5; p 1 Å 50, 200; m Å 1, 12. Using the optimiztion results of Section 2.2, we then solved ech of the 128 test problems for lrge number of vlues of l, nd produced grphs of s* vs. l, D* vs. l, nd S* vs. l. Once more, the behvior of s*, D*, nd S* sl chnges ws consistent cross ll test problems, nd this behvior is cptured by the three exmples in Figure 2, where Å 100, q Å 0.3, k Å 50, h Å 1, p Å 10, p 0 Å 2, p 1 Å 50, m Å 1, 6, 12, nd EOQ Å 100. Note the following: S* is incresing in l. There exists lo ú 0 such tht, on l õ lo, s* Å 0 nd D* is incresing in l, nd, on l lo, s* is incresing in l nd D* is decresing in l. lo is incresing in m e.g., lo (m Å 1) Å 0.2, lo (m Å 6) Å 1.0, nd lo (m Å 12) Å 2.0. As l r 0, s* r 0 nd D* r EOQ. As l r, D* r EOQ nd s* r s* (D Å EOQ). Severl mngeril implictions cn be derived from the bove reserch findings. Not surprisingly, when the severity of the supply disruptions diminishes becuse l is decresing, S* should decrese; nd conversely, when the severity of the supply disruptions increses becuse l is incresing, S* should increse. A second impliction is definitely less intuitive: Depending on the vlue of l, the system follows two different strtegies to increse the protection level s l increses. When l [ 0, lo ), the increses in S* should be the result of incresing D* lone. When l [lo, ), the increses in S* should be the net result of incresing s* nd decresing D*. Note lso tht lo, the vlue of l tht triggers the chnge of strtegy from increse S* / 8m27$$ :11:55 nr W: Nv Res 994

10 696 Nvl Reserch Logistics, Vol. 45 ( 1998) Figure 2. Optiml behvior of the inventory system s l vries: ( ) s*; ( ) D*; ( ) S*. / 8m27$$ :11:55 nr W: Nv Res 994

11 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions 697 solely by incresing D* to increse S* by incresing s* nd decresing D*, is ffected by m. The smller the vlue of m the sooner the strtegy shift. The finl finding bove is surprising nd, even with hindsight, difficult to explin: The finite vlue to which D* converges s l r is EOQ, nd, s result, the finite vlue to which s* converges is s* (D Å EOQ). We confess tht, from theoreticl stndpoint, we were even more surprised when ttempts to justify the behvior D* r EOQ s l r by using Eqs. (5) nd (6) were fruitless. 4. OPTIMAL INVENTORY STRATEGY FOR CHANGES IN STOCKOUT BEHAVIOR In this section we chrcterize the optiml inventory strtegy s stockout behvior chnges. The pproch used prllels tht of Section 3. To explore the response of the system to vrying moderte vlues of q, third extensive numericl experiment ws performed on 128 test problems. The test problems were formulted by fixing h Å 1, nd then tking ll combintions of the following prmeter vlues: Å 100, 500; k Å 50, 200; p Å 10, 50; p 0 Å 2, 5; p 1 Å 50, 200; l Å 1, 12; nd m Å 1, 12. Using the optimiztion results of Section 2.2, we solved ech of the 128 test problems for lrge number of vlues of q with 0 q 1, nd produced grphs of s*, D*, nd S* vs. q. The behvior of s*, D*, nd S* sq chnges ws consistent cross ll test problems, nd this behvior is cptured by the two exmples in Figure 3, where Å 500, k Å 50, h Å 1, p Å 50, p 0 Å 2, p 1 Å 200, l Å 1, nd m Å 1 (Exmple 1) nd 12 (Exmple 2). Hence for Exmple 1 we hve tht g Å 50 / 152q, nd for Exmple 2 we hve tht g Å q. Note the following: D* is nerly constnt in q. s* nds* re incresing/decresing in q when g is incresing/decresing in q. A number of mngeril implictions re suggested by these numericl results. First, since D* seems to be quite insensitive to the frction of stockouts tht re bckordered, the inventory mnger cn choose good minimum order quntity D without obtining precise estimte for q. This my be quite useful in prctice, since estimting q my be difficult. On relted vein, this result lso implies tht the vrying levels of protection ginst supply disruptions required by different stockout behviors re chieved lmost entirely by chnging the optiml reorder point s*. Finlly, the second reserch finding confirms our intuition tht, s stockout behvior shifts towrd more bckorders, more protection ginst supply disruptions is desirble when the expected cost per bckorder is greter thn the expected cost per lost sle, nd less protection ginst supply disruptions is desirble when the expected cost per bckorder is less thn the expected cost per lost sle. 5. CONCLUSION Our reserch interest in this pper hs been to explore the optiml mngement of inventory for stochstic-demnd systems with rndom supply disruptions nd prtil bckorders. Our results suggest the following mngeril insights: / 8m27$$ :11:55 nr W: Nv Res 994

12 698 Nvl Reserch Logistics, Vol. 45 ( 1998) Figure 3. Optiml behvior of the inventory system s q vries: ( ) s*; ( ) D*; ( ) S*. The optiml inventory strtegy to mnge chnge in the severity of supply disruptions should be to move the order-up-to level S* in the sme direction s the chnge. However, the specific strtegy to ccomplish such move in S* through the vribles s* nd D* depends on whether the chnge in the severity of supply disruptions is due to chnge in m or is due to chnge in l. The optiml inventory strtegy to mnge chnge in stockout behvior should be to move the order-up-to level S* in the sme direction s the chnge in expected cost per demnd rriving during stockout g. The djustment in the vlue of S* should be chieved lmost exclusively through the optiml reorder point s*. / 8m27$$ :11:55 nr W: Nv Res 994

13 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions 699 Due to the lck of previous reserch on the subject of rndom supply disruptions in stochstic-demnd inventory systems with prtil bckorders, the work presented here must be regrded s just one step forwrd in understnding such systems. Some obvious directions for extending this work prllel erlier dvnces in inventory mngement without supply disruptions or prtil bckorders. Such directions include serching for the optiml inventory policy or llowing for nonsttionry costs perhps depending on the stte of the supply. Although we expect tht the bsic flvor of our results will hold for much more generl systems thn the one considered here, significnt dditionl insights my be gined by explicitly ddressing some of these issues. APPENDIX PROOF OF PROPOSITION 1: Let us first derive the expression for b. Modeling the evolution of supply disruptions s two-stte continuous-time Mrkov chin, it is not difficult to demonstrte tht Pr{supply is unvilble t time t} Å l l / m [1 0 e 0(l/m)t ]. Define X s the time when the inventory level hits s nd let f X (r) denote its probbility density function. Then b Å l 0 e l / m [1 0(l/m)x ] f X ( x) dx. 0 Becuse D is Poisson process, X is distributed Erlng with prmeters nd D, where We thus hve b Å l l / m G(D) 0 f X ( x) Å (x)d01 e 0x G(D), x ú 0. [1 0 e 0(l/m)x ](x) D01 e 0x dx Å l l / m 1 0 / l / m D. Let i denote the inventory level, j Å 0 if the supply is vilble nd j Å 1 otherwise, nd P(i, j) be the limiting or stedy-stte probbility of being in stte (i, j). Using stochstic blnce rguments, we constructed the trnsitionrte digrm depicted in Figure 4. Bsic procedures plus tedious lgebric mnipultions led to P(S, 0)Å P(S, 1)Å m( / m) (md / b)( / l / m), (7) ml (md / b)( / l / m), (8) m P(i, 0)Å md / b l S/10i / l / m / l / m m, i Å s / 1,...,S 0 1, (9) l / m ml P(i, 1)Å (md / b)(l / m) 1 0 / l / m S/10i, i Å s / 1,...,S 0 1, (10) P(i, 0)Å 0, i s, (11) / 8m27$$ :11:55 nr W: Nv Res 994

14 / 8m27$$ :11:55 nr W: Nv Res 700 Nvl Reserch Logistics, Vol. 45 ( 1998) Figure 4. Stedy-stte probbility nd trnsition-rte digrm for the inventory system under study.

15 Arreol-Ris nd DeCroix: Inventory Mngement with Supply Disruptions 701 mb P(i, 1)Å md / b s/10i, i Å 1,...,s, (12) / m P(i, 1)Å m q / m b md / b / m s q q / m 0i, i 0. (13) From bsic principles K Å kn/ hoh/ pls/ p 0 BR / p 1 BL. Define cycle s the time intervl between order receipts nd denote its length by T. Let T 1 be the cycle length if the supply is vilble when the inventory level hits s, let T 2 be the cycle length if the supply is unvilble when the inventory level hits s, nd let A be n indictor rndom vrible defined by 1 if the supply is vilble when the inventory level hits s, A Å 0 otherwise. Also define X 1 to be n Erlng rndom vrible with prmeters nd D, nd define X 2 to be n exponentil rndom vrible with prmeter m. Then T Å T 1 / T 2, T 1 Å AX 1, nd T 2 Å (1 0 A 1 )(X 1 / X 2 ). Therefore, E(T 1 ) Å E(E(T 1 ÉA)) Å Pr{A Å 1}E(T 1 ÉA Å 1) / Pr{A Å 0}E(T 1 ÉA Å 0) Å (1 0 b)e(x 1 ) / br0 Å (1 0 b) D nd E(T 2 ) Å E(E(T 2 ÉA)) Å Pr{A Å 1}E(T 2 ÉA Å 1) / Pr{A Å 0}E(T 2 ÉA Å 0) Å (1 0 b)r0 / b(e(x 1 ) / E(X 2 )) Å b D / 1 m. Therefore, E(T ) Å (1 0 b) D / bd / b m Å D / b m Å md / b m. Hence, since N Å 1/E(T ), we hve N Å m md / b. The expressions for OH, LS, BR, nd BL follow from Eqs. (7) (13) nd lgebr. PROOF OF PROPOSITION 2: It is esy to show tht Ì 2 K/Ìs 2 0 for ny D, thus estblishing convexity in s. The reminder of the result follows from lgebric mnipultion of the condition / 8m27$$ :11:55 nr W: Nv Res 994

16 702 Nvl Reserch Logistics, Vol. 45 ( 1998) ÌK ú 0 Ìs så0 (which would imply s* Å 0) nd the condition (which yields the optiml s* if(ìk/ìs)é så0 0). ÌK Ìs Å 0 PROOF OF PROPOSITION 3: We will first show tht the rtio D/b is incresing in D. From Proposition 1 D b Å D l ( l / m)[1 0 ( / l / m) D ]. Let g(d) Å D/[1 0 (/( / l / m)) D ]. Then 1 / ( / l / m) D [D ln( / l / m) 0 1] g (D) Å. [1 0 ( / l / m) D ] 2 If 0 õ õ 1 nd x 0, clerly x 1 / x ln(). Hence D 1 / D / l / m ln, / l / m nd thus 1 / D D / l / m ln 0 / l / m 1 1 / 1 / D ln / l / m D ln 0 / l / m 1 2 ln 2 Å D 0. / l / m Therefore, D/b is incresing in D. The result in Proposition 3 follows from the fct tht K is superdditive on the sublttice {0, 1, rrr} 1 {1, 2, rrr}. (For summry of the necessry lttice progrmming results, see Topkis [24].) Superdditivity of K is equivlent to the condition Ì 2 K ÌsÌD 0. This condition on K holds term by term becuse D/b is incresing in D. PROOF OF PROPOSITION 4: Let e denote the frction of time tht the inventory system is incurring stockouts (lost sles or bckorders). Using bsic principles, it is not hrd to show tht e Å r s (md/b) / 1. Combining the bove eqution with the result tht D/b is incresing in D (demonstrted in the proof of Proposition 3) yields Proposition 4. / 8m27$$ :11:55 nr W: Nv Res 994

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