A NEWSVENDOR MODEL WITH UNRELIABLE SUPPLIERS

Transcription

1 A NEWSVENDOR MODEL WIH UNRELIABLE SUPPLIERS MABOOL DADA NICHOLAS C. PERUZZI LEROY B. SCHWARZ 3 AUGUS 003 Abstract We consder the problem of a newsvendor that s served by multple supplers, where any gven suppler may be unrelable. By unrelable we smply mean that the margnal amount receved from a suppler s no more than, and typcally s less than, the margnal amount ordered from the suppler. In ths settng, the newsvendor needs to determne () whether or not to place an order wth a gven suppler, and () f so, then for how much? o address these questons, we develop a general framework n whch the newsvendor can dversfy ts rsk of nadequate delvery amounts by spreadng ts orders among any number and combnaton of avalable supplers that dffer n terms of cost and (delvery) relablty. Ultmately, we fnd that the newsvendor model wth unrelable supplers has the same structural propertes as a newsvendor model n whch all supplers are relable but have lmted capacty. Our resultng contrbuton s two-fold: Frst, we establsh propertes of the optmal soluton and develop correspondng nsghts nto the trade-off between cost and relablty. Second, we perform comparatve statcs on the optmal soluton, wth a partcular emphass on nvestgatng how changes n suppler cost or relablty affect the newsvendor s orderng decsons and customer servce level. Krannert Graduate School of Management, Purdue Unversty, 30 Krannert Bldg., West Lafayette, IN Phone: emal: dada@mgmt.purdue.edu. Department of Busness Admnstraton, Unversty of Illnos at Urbana-Champagn, 06 S. 6 th St., Champagn, IL 680. Phone: e-mal: petruzz@uuc.edu. 3 Krannert Graduate School of Management, Purdue Unversty, 30 Krannert Bldg., West Lafayette, IN Phone: emal: lee@mgmt.purdue.edu.

2 INRODUCION By defnton, a newsvendor faces the challenge of havng to decde how much of ts product to order from ts suppler for ts sngle sellng season, pror to observng the random demand for ts product. Correspondngly, the newsvendor s sales for the season s constraned both by the demand that materalzes and by the quantty that the newsvendor chooses to supply. he economc mplcatons are twofold: On the one hand, f realzed demand exceeds the supply, then the newsvendor wll sell ts entre stock, but at the expense of havng excess demand go unsatsfed; on the other hand, f the supply exceeds the realzed demand, then the newsvendor wll satsfy demand completely, but at the expense of havng leftovers. he famlar soluton to ths trade off ndcates that the newsvendor, to maxmze expected proft for the sellng season, should order ust enough supply so that the probablty of meetng demand s equal to the rato of the margnal overstockng cost to the sum of the margnal overstockng plus understockng cost. In ths classc settng, the newsvendor s suppler s perfectly relable n the sense that the quantty provded (by the suppler) s exactly equal to the amount ordered (by the newsvendor). Consequently, a newsvendor needs only to determne how much to order from a sngle suppler. In ths paper, we explore mplcatons f the newsvendor faced the addtonal challenge of havng to deal wth a supply source that dd not necessarly delver the quantty ordered. Such could be the case f a suppler faced, for example, random capacty or a random yeld process. In effect, a suppler s delvery quantty n ths varaton s constraned by ts own producton output functon. Consequently, f the suppler s producton output s nsuffcent to meet the newsvendor s order quantty, then the newsvendor wll receve less than what was ordered. In ths settng, the newsvendor needs to determne () whether or not to place an order wth a gven suppler, and () f so, then for how much?

3 o address these questons, we develop a modelng framework bult around a very general noton of unrelablty. Specfcally, n our model, we defne unrelable suppler to mean smply that there exsts a postve probablty that the margnal quantty delvered by the suppler s less than a margnal unt ordered from the suppler. hs framework hnges on the constructon of a suppler output functon that can be ether determnstc or random, as well as ether endogenous (n the sense that ts parameters can depend on the newsvendor s order quantty) or exogenous. Gven ths framework, our focus s two-fold: Frst, we establsh propertes of the optmal soluton and develop correspondng nsghts nto the trade-off between cost and relablty. Second, we perform comparatve statcs on the optmal soluton, wth a partcular emphass on nvestgatng how changes n suppler cost or relablty affect the newsvendor s orderng decsons and customer servce level. One contrbuton of our model s that t unfes a dsparate range of relablty constructs nterspersed throughout both the newsvendor lterature and the random yeld lterature. (Authortatve revews of the newsvendor and random yeld lteratures are provded by Porteus (990) and by Yano and Lee (995), respectvely. More recent updates nclude those by Khoua (999) and Mnner (00), respectvely.) Examples of relablty constructs appearng n these lteratures nclude, among others, the case of all-or-nothng delvery (Anupnd and Akella (993); Gerchak (996)), the case of random capacty (Carallo et al. (994)), the case of bnomal yeld (Chen et al. (00)), the case of stochastc proportonal yeld (Heng and Gerchak (990)), and combnatons thereof (Wang and Gerchak (996)). ypcally n these papers, the prmary focus s on characterzng the structure of the optmal polcy, gven that each of the supply sources are utlzed, so that t may serve as a buldng block for dynamc nventory models. In contrast, our focus (lke, to some degree, the focus of Chen et al. (00) and Gerchak

4 (996)) s on characterzng optmal suppler selecton and ts mplcatons for the newsvendor, for ts supplers, and for ts customer market. Ultmately, we fnd that the newsvendor model wth unrelable supplers has the same structural propertes as a newsvendor model n whch all supplers are relable but have lmted capacty. Another contrbuton of our model s the correspondng nsghts that t yelds. In partcular, we fnd that cost generally takes prorty over relablty when t comes to suppler selecton. Consequently, supplers wth costs that are hgh relatve to other supplers could be left wthout a share of the newsvendor s aggregate order, regardless of ther relatve relablty levels. For these supplers, relablty mprovements would be to no aval; only through mprovng costs could they break through the threshold to gan a share of the newsvendor s aggregate order. In addton, we fnd that, when compared to a newsvendor that does not have to deal wth an unrelable supply source, a newsvendor that does have unrelable supplers wll order more (n aggregate), but provde a lower customer servce level. Moreover, we complement these results wth useful comparatve statcs. he remander of ths paper s organzed as follows. In Secton, we develop our formal model and demonstrate that, n a reduced form, t yelds a structure amenable to a standard newsvendor nterpretaton. In Secton 3, we establsh propertes of the optmal soluton, dscuss mplcatons for prortzng supplers, and develop correspondng nsghts. hen, n Secton 4 we perform a comparatve statcs analyss on a two-suppler soluton to nvestgate how the newsvendor should react to changes n fundamental problem parameters. We conclude the paper wth Secton 5. 3

5 HE MODEL AND IS REDUCED FORM Pror to the begnnng of a sngle sellng season, a retaler places orders for ts product from among any of N ndependent supplers, some or all of whch are unrelable. In ths context, unrelable means that the margnal amount suppled (.e., delvered) by a gven suppler s less than or equal to the margnal amount ordered from the suppler. Accordngly, let denote the quantty ordered from suppler, and let R represent the relablty of suppler, where R s an exogenous construct characterzed as ether fxed or random. hen, S (,R ), suppler s supply functon, s such that S (,R ). Prevalent supply functons n the lterature nclude the followng: S(,R ) mn{,r }; S (,R ) R ; Pr{S } Pr{S 0} ρ. (a) (b) (c) In general, (a) characterzes lmted-capacty models; (b) characterzes proportonal yeld models ; and (c) characterzes all-or-nothng models. Note, however, that (c) also can be nterpreted as a specal case of ether (a) or (b). In partcular, f R s specfed as a Bernoull random varable that takes on a realzed value of ether or 0, then (a) reduces to (c); smlarly, f R s specfed as a Bernoull random varable that takes on a realzed value of ether or 0, then (b) reduces to (c). Gven (a) (c) as well as the applcablty of alternatve constructs, we model suppler s supply functon more generally by defnng the noton of a producton output functon: (,R ) mn{,k (, R )} S, () where K (.) represents the output functon. hus, a gven suppler s producton output functon explctly constrans the amount that the suppler can delver (.e., supply) to the retaler. hs 4

6 specfcaton s useful because t accounts for the possblty that a gven suppler s producton capablty could be ether endogenous (n the sense that t depends on the retaler s order-quantty decson) or exogenous, as well as ether determnstc or random. Notce, for example, f R n (a) s determnstc, then t corresponds to the case n whch K (,R ) n () s exogenous and determnstc. Lkewse, f R n (a) s random, then t corresponds to the case n whch K (,R ) n () s exogenous and random. Smlarly, f R n (b) s random (or, determnstc), then t corresponds to one partcular specfcaton of an endogenous and random (or, determnstc) output functon. Let s (,R ) S (,R )/ represent the margnal quantty suppled by suppler per margnal unt ordered from that suppler. We assume that s (,R ) for all and realzed values of R (or, equvalently, we assume that K (,R )/ ). If suppler s such that s (,R ) for all and realzed values of R, then we say that suppler s perfectly relable. In contrast, f s (,R ) < for some value of and realzed value of R, then we say that suppler s unrelable. Supplers are ndexed from least to most expensve so that c < < c N, where c denotes the per-unt purchase cost of nventory obtaned from suppler. We assume that the retaler pays suppler only for the amount actually suppled. hat s, we assume that the retaler s procurement cost assocated wth suppler s c S (,R ) c. One practcal mplcaton of ths assumpton s that our model s most applcable to a procurement envronment, where t s reasonable to expect that a retaler pays for what s receved rather than for what s ordered. In contrast, ths assumpton would be less applcable to a producton envronment n whch t s reasonable to expect that a manufacturer must pay not only for the yeld of a producton run (S (,R )), but also for the total number of defects that result from the producton run ( 5

7 S (,R )). In such an envronment, the total cost assocated wth orderng unts would be c rather than c S (,R ). In Secton 5, we dscuss some mplcatons f t s assumed that the decson maker pays for what s ordered rather than for what s suppled. Let D denote the random demand for the retaler s product; and let F and f represent the cumulatve dstrbuton functon (cdf) and probablty densty functon (pdf) assocated wth D. Note however, that the assumpton of random demand s not necessary. he random demand process D can be replaced by the determnstc process d wthout affectng the results of our analyss. hs s because our analyss centers on the ype-i servce level, whch s defned as the uncondtonal probablty that demand for the sngle perod does not exceed the supply avalable for the perod, and ths s a concept that apples as long as ether demand or supply (or both) s random. Because of the central role that servce level plays n the nterpretaton of our results, let SL() denote the retaler s servce level, gven that { } s the vector of order quanttes placed among the supplers. hen, by defnton, SL() Pr{D S ()} E[F(S ())], where S () S (,R ) S N ( N,R N ) represents the total supply avalable to the retaler for sale durng the perod. Analogously, let N denote the total quantty ordered by the retaler. o complete the specfcaton of our model, assume that the retaler sells ts product for the per-unt sellng prce p; that leftovers are salvaged for the per-unt value v (where a negatve v denotes a dsposal cost); and that shortages are assessed a per-unt penalty cost π (to sgnfy the cost of lost goodwll). Fnally, let φ (p π c)/(p π v) denote the standard newsvendor fractle (.e., rato of underage costs to the sum of underage and overage costs) assocated wth suppler. hs notaton, as well as the notaton ntroduced above, s summarzed n able. 6

8 N able. Summary of Notaton total number of supplers p, v, π retaler s per-unt sellng prce, salvage value, and penalty cost of loss goodwll, respectvely c retaler s per-unt purchase cost of supply procured from suppler φ (pπ c )/(pπ v) newsvendor fractle assocated wth suppler R relablty of suppler, { } quantty ordered from suppler, and order-quantty vector, respectvely K (,R ) suppler s output functon S (,R ) mn{, K (,R )} quantty suppled (.e., delvered) by suppler s (,R ) S / margnal quantty suppled by suppler (per unt ordered) N total quantty ordered from all supplers S () S S N total quantty suppled by all supplers D demand for retaler s product (random or determnstc) F, f cdf and pdf (or pmf) characterzng D SL() E[F(S ())] ype-i servce level provded by retaler he retaler s obectve s to maxmze Π(), ts expected proft for the sellng season, where ( ) E p sales( ) v Eleftovers( ) π shortages( ) c S (, R ) Π E pd v Eleftovers( ) ( p π) shortages( ) cs (, R ) E pd v S ( ) ( π) ( ),R D p D S,R cs (, R ). (3) Consder, then, the queston of how much to order from suppler (whether that be zero or otherwse). o that end, defne (, R ) X D S. (4) Note that X s a random varable representng suppler s apportoned, or rsk-adusted, demand. In other words, X ndcates the total (random) demand for the retaler s product, less the total (random) supply provded by all supplers other than suppler. As such, X corresponds to the random demand for the unts suppled by suppler. Accordngly, let 7

9 Pr { X S (,R ) R } Pr{ D S ( ) R } F ( S ( ) ). hen F (S ()) denotes the condtonal probablty that demand s less than or equal to total supply, gven a realzed value of R. Moreover, the retaler s servce level, whch, by defnton, s the uncondtonal probablty that demand s less than or equal to supply, can be expressed as SL ( ) E[ F( S ( ) )] E [ F ( S ( ) )]. (5) R Snce X s ndependent of, one convenent way to express (3) s as follows: where Π ( ) Π ( ) E ( p c ) S (, R ) Π, (6) [ ] ( ) E px v[ S (,R ) X ] ( p π) [ X S (,R )] c S (, R ). (7) he sgnfcance of (6) and (7) s as follows: From the perspectve of determnng how much, f any, to order from suppler, the second term of (6) can be gnored because t s ndependent of. hus, only Π () needs to be consdered. Notce from (7), however, that for any realzed value of R, Π () can be nterpreted as a standard newsvendor proft functon, where S (,R ) denotes the resource supply decson whle the exogenous X denotes the random demand for that resource. he one varaton here s that the supply decson (S (,R )) tself mght be random. Interestngly, ths smple varaton creates enough complcaton that, unlke ts standard newsvendor counterpart, (6) s not guaranteed to be concave n general. Nevertheless, Π () stll can be nterpreted as an expectaton over two random varables: X, whch, n general, actually s a convoluton of random varables, and R, whch, n general, s a random varable representng the relablty of suppler. o make ths double expectaton explct, we rewrte (7) as follows: 8

10 Π [ [ ] R ] ( ) E E px v[ S (,R ) X ] ( p π) [ X S (,R )] c S (,R ) R X. hus, the condtonal expected proft assocated wth orderng from suppler, gven a realzed value of R, s precsely a newsvendor proft functon. hs mples that the dervatve of Π() taken wth respect to s ( ) ( ) Π ( ) Π, (8) ( p π v) E [ s (,R )( φ F ( S ( ) ))] M R where, recall, φ (p π c)/(p π v) and s (,R ) S (,R )/. Correspondngly, M () 0, whch represents the necessary condton for, the optmal quantty to order from suppler, to be an nteror pont soluton, can be wrtten as E [ (,R ) F ( S ( ))] φ λ, (9) where λ (,R ) s E (,R ) [ s (,R )] and, for notatonal smplcty, we now use E [.] n leu of E R [.] to ndcate an expectaton over the random varable R. As n the classc newsvendor problem, the rght hand sde (RHS) of (9) s the famlar crtcal fractle assocated wth suppler, and the left hand sde (LHS) has a servce level nterpretaton. In partcular, the LHS can be nterpreted as the weghted condtonal servce level gven that matters, where λ (,R ) denotes the weght for any gven value of R. In ths context, matters f, for a gven value of R, the supply functon depends on. hat s, matters for realzed values of R that are such that s (,R ) > 0. 9

11 o better llustrate ths noton, consder the applcaton of (9) to the case n whch suppler s output functon s exogenous and random (note that, for the sngle-suppler verson of ths case, Carallo et al. (994) demonstrate that the retaler s proft functon s unmodal, but not concave): Illustraton. Let S (,R ) mn{,r }, where R s a random varable characterzed by a known probablty dstrbuton. hen, S (,R ) depends on f and only f the realzed value of R meets or exceeds. Hence, for any gven realzed value of R r, matters f and only f r ; and does not matter f r <. Moreover, for realzed values of R n whch does matter, s (,R ). Consequently, E [s (,R )] Pr{R }. Correspondngly, the condtonal weghted servce level gven that matters s E [ λ (, R ) F ( S ( ))] E 0 E [ Pr{ D k Sk ( k, R k )}] Pr{ R } Pr{ R } [ F ( S ( ) ) R < ] E[ F ( S ( ) ) R ] Pr{ R } { D S (, R )} Pr k k k k. herefore, from (9), the optmalty condton for to be an nteror pont soluton when suppler s output functon s exogenous and random s Pr { D k Sk ( k, R k )} φ, (0) whch s consstent wth the optmalty condton establshed n Carallo et al. (994). o conclude ths secton, we note that the model developed here does not requre the supply functon of each suppler to be of the same functonal form. Consequently, gven (), what dstngushes suppler from other supplers s ts specfc combnaton of cost (c ), relablty (R ), and output functon (K (,R )). 3 SUPPLIER SELECION: COS VERSUS RELIABILIY Snce supplers dffer on cost and relablty, n ths secton we examne how these two attrbutes affect the outcome of the retaler s suppler selecton decson, whch refers to the process of 0

12 choosng supplers wth whch to place orders. o characterze these results, we defne a suppler to be actve f t s optmal for the retaler to place an order wth that suppler. Conversely, we defne a suppler to be nactve f t s optmal for the retaler not to place an order wth that suppler. Gven these defntons, we fnd that, n general, cost takes precedence over relablty when t comes to choosng supplers. In partcular, we establsh and dscuss the followng propertes of optmal selecton:. If a gven suppler s nactve, then all more expensve supplers wll be nactve.. If a gven suppler s perfectly relable, then all supplers more expensve than the perfectly relable suppler wll be nactve. Hence, no more than one perfectly relable suppler wll be actve. Moreover, f a perfectly relable suppler s actve, then the retaler s optmal servce level wll be dentcally equal to the newsvendor fractle assocated wth that suppler. 3. Unless the margnal beneft assocated wth sellng a unt purchased from a gven suppler (p π c ), s strctly less than the expected margnal beneft assocated wth sellng a unt ordered from each less expensve suppler, the suppler wll be nactve. 4. For problem specfcatons n whch the Karush-Kuhn-ucker (KK) condtons are suffcent for determnng an optmal soluton, f orderng optmally from the n least expensve supplers would yeld a servce level that s greater than the newsvendor fractle assocated wth suppler n, then suppler n and all more expensve supplers wll be nactve. 5. he optmal total quantty ordered from all actve supplers s at least as large as the total quantty ordered n an otherwse equvalent problem n whch all supplers are perfectly relable. However, the resultng optmal servce level s no greater than the resultng servce level n the problem wth only perfectly relable supplers. o develop these results, frst note that φ > > φ N snce φ (p π c )/(p π v) and supplers are ndexed such that c < < c N. Next, note that any optmal soluton must satsfy the followng KK optmalty condtons for all :

13 ( ) 0 M ; () M ( ) 0, () where M () s gven by (8). Fnally, consder the followng 3 lemmas, whch serve as buldng blocks for the techncal analyss of ths secton (proofs of these lemmas are provded n the appendx): Lemma. For any gven, M ( ) ( p v) E [ s (,R )][ φ SL( ) ] Lemma. If 0 Lemma 3. If 0, then SL( ) φ. >, then SL( ) E[ s (,R )] φ. π. We now formally establsh and dscuss the results of ths secton. Proposton. For,,N-, f 0, then 0. Proof. Assume that, n an optmal soluton, 0. hen, from Lemma and the ndexng of supplers, SL ( ) φ > φ. hus, from Lemma, M But, from (), M ( ) 0 ( ) ( p π v) E [ s (,R )][ φ SL( )] 0 < <. mples that 0. In other words, f 0, then 0. Proposton, n effect, establshes a precedence rankng among the supplers based only on each suppler s cost. Bascally, n terms of selectng supplers, t s optmal for the retaler to start by choosng the least expensve suppler, and then to add supplers to ts selecton set one by one, accordng to how nexpensve the suppler s. Consequently, the optmal number of actve supplers, say N, wll be such that suppler s actve f and only f c c N.

14 Proposton. For,,N-, f suppler s perfectly relable, then 0 for,,n. Proof. Proposton mples that f 0, then 0 for all >. herefore, to establsh ths proposton, t suffces to show that that 0 f suppler s perfectly relable. Assume, then, that suppler s perfectly relable (that s, s (,R ) for all and realzed values of R ). here are two possble cases. On the one hand, f 0, then 0 drectly from Proposton. On the other hand, f > 0, then () mples that M ( ) 0. hus, from (8), [ s (,R )( φ F ( SL( )] φ SL( ) hat s, SL ( ) φ > φ. But, from Lemma, f φ < SL( ) 0 E., then M ( ) < 0. And, from (), f M ( ) < 0, then 0. herefore, f suppler s perfectly relable, then 0. Proposton ndcates that there can be at most one perfectly relable suppler that becomes actve. Accordngly, gven Proposton, f there exst two or more perfectly relable supplers, the only one that s even elgble to become actve s the least expensve one. An ntutve argument for why more expensve supplers wll not become actve when a perfectly relable suppler exsts s as follows: Suppose the retaler s consderng orderng a unt from a more expensve suppler when a less expensve suppler that s perfectly relable exsts. If the retaler nstead dverts the unt to the less expensve, perfectly relable suppler, then the retaler can save on procurement cost gven that the unt s receved. Moreover, snce dvertng the unt to the perfectly relable suppler also reduces uncertanty, the retaler can order less unts n the aggregate, thereby further reducng expected procurement cost. Even though a perfectly relable suppler can render all more expensve supplers nactve, the followng proposton ndcates that perfect relablty s hardly suffcent for a gven suppler to become actve tself. 3

15 Proposton 3. If, n an optmal soluton, E[ s (, R )] φ < φ for any <, then 0. Proof. Assume that, n an optmal soluton, > 0. hen, from (), M ( ) 0. Consequently, gven the proposton assumpton that there exsts an < that s such that φ < E[ s (,R )] φ, Lemma mples that SL( ) φ < E[ s (,R )] φ Lemma 3, f SL( ) < E[ s (,R )] φ, then 0. But, accordng to. And, from Proposton, f 0 for <, then 0, whch contradcts the orgnal assumpton that > 0. herefore, f there exsts an < that s such < E[ s (,R )] φ φ, > 0 s an nvald assumpton, whch mples that 0. o nterpret Proposton 3, note that the condton < E[ s (, R )] φ ( p π c ) < E[ s (,R )( p π c )] φ s equvalent to. In ths expresson, note that (p π c ) represents the condtonal margnal beneft assocated wth sellng a unt ordered from suppler (.e., the [ ] underage cost), gven that suppler actually delvers the unt. And, E s (, R )( p π c ) represents the expected margnal beneft assocated wth sellng a unt ordered from suppler, whch s an expectaton condtoned over the lkelhood that suppler actually delvers the unt. hus, Proposton 3 ndcates that the expected margnal beneft assocated wth orderng from less expensve supplers s pvotal for determnng whether or not to place orders wth more expensve supplers, regardless of the more expensve suppler s relablty. As a result, any less expensve suppler can effectvely render a more expensve suppler nactve, even f the more expensve suppler s perfectly relable. From a techncal standpont, Proposton 3 s useful because many typcal supply functons are such that s (,R ) actually s ndependent of. Indeed, s (,R ) reduces to s (R ) for each of the three prevalent supply functons noted n Secton. As a result, for these supplers, 4

16 [ s (, R )] E reduces to the constant E [s (R )]; thus, the elgblty test provded by Proposton 3 can be appled mmedately to reduce the set of elgble supplers before attemptng to solve the assocated optmzaton problem. Although Propostons and 3 establsh that perfect relablty s no guarantee for becomng actve, t s nterestng to note the followng corollary, whch ndcates that f a perfectly relable suppler becomes actve, then the retaler s optmal soluton can be nterpreted as a base servce level polcy. SL φ. Corollary. If suppler s perfectly relable and actve, then ( ) Proof. If suppler s perfectly relable, then, by defnton, s (,R ) for all and realzed values of R ). Moreover, f suppler s actve, then, also by defnton, > 0. herefore, from () and (9), [ s (,R )( φ F ( SL( )] φ SL( ) 0 E. hat s, SL( ) φ. Accordng to ths corollary, f t s optmal for the retaler to order from a perfectly relable suppler, then the retaler s optmal servce level s ndependent of any other actve suppler s cost and relablty. Consequently, f a less expensve suppler s cost or relablty were to change, for whatever reason, then the change would not affect the retaler s optmal servce level. he change would only affect the allocaton scheme that the retaler uses to acheve the optmal servce level. hus, f a perfectly relable suppler s actve, then that suppler s role s to complement the combned orders placed wth all less expensve supplers n a very specal way: regardless of how the retaler spreads ts orders among less expensve supplers, the purpose of the perfectly relable suppler s to delver the remanng quantty requred to brng the retaler s 5

17 servce level up to the base level φ. For ths reason, we refer to a perfectly relable suppler that becomes actve as an anchor suppler. Generally speakng, Propostons 3 establsh cost as a prorty over relablty when t comes to selectng supplers wth whch to place orders. In partcular, Propostons 3 all establsh condtons under whch more expensve supplers wll be rendered nactve, regardless of the relablty levels of those supplers. hus, f a partcular unrelable suppler s rendered nactve by one or more of these propostons, then mprovng relablty wll be to no aval; only through mprovng costs can the suppler break through these thresholds to become actve. From a techncal standpont, Propostons 3 are useful for smplfyng the soluton procedure. In partcular, Propostons and 3 can be used to mmedately pare the orgnal set of N supplers to a potentally smaller set of elgble supplers. hen, gven the resultng set of elgble supplers (say N E, where N E N), Proposton can be exploted as follows: Defne the n-suppler subproblem as the subproblem n whch the retaler sets 0 for n,,n E ; and solves for for,,n. hen, the retaler can be guaranteed to fnd the optmal soluton by teratvely solvng n-suppler subproblems, begnnng wth n and endng wth N E. Alternatvely, f a problem nstance s approprately-well behaved so that the KK condtons are guaranteed to have a unque soluton, then the optmal soluton can be obtaned drectly from () and (). Or, as ndcated below, one can use the same teratve procedure as above, but end wth the frst subproblem that yelds a soluton that satsfes the servce-level condton ndcated by Proposton 4. 6

18 Proposton 4. Suppose that the KK condtons are suffcent for establshng optmalty. Moreover, defne the n-suppler subproblem as the constraned problem n whch s set equal n to 0 for n,,n; and let { n } denote the optmal soluton to the n-suppler n n subproblem. If s such that SL ( ) > φn, then n s the optmal number of actve supplers for the N-suppler problem. n n Proof. Let {,...,,0,...,0} n denote any optmal soluton to the n-suppler problem. hen, n by the defnton of optmalty, n satsfes the system of equatons gven by () and (). Specfcally, for,,n: M ( n n n ) 0 and M ( ) 0. Consder, then, the general N-suppler problem. Gven that the KK condtons are suffcent for establshng optmalty, () and () have a unque soluton. Moreover, that unque soluton corresponds to, the optmal soluton to the general problem. hus, to complete the proof, t suffces to show that the feasble canddate soluton n ndeed satsfes () and () for n,,n f SL ( ) > φn M. For,,n, the canddate soluton s ( n n n ) 0 and M ( ) 0 n, where, recall, n s defned such that. hus, for,,n, the canddate soluton satsfes () and () trvally. For n,,n, the canddate soluton s 0. hus, for n, N, the canddate soluton also satsfes () trvally. herefore, to complete the proof, consder () for n, N. From (8), Note, however, that each of the (for n,,n) because these [ ( ( )] n n n ( ) ( p π v) E s( 0,R ) φ F Sk ( k, R ) M. (3) k n k s n the summaton of ths expresson s ndependent of R n k s are defned as any soluton to the system of equatons gven by () and () for k,,n (whch, gven that 0 for n,,n, s a system of equatons ndependent of R for,,n). hus, gven that n,,n, (3) smplfes to M n n n ( ) ( p π v) E [ s ( 0,R )] φ F S (,R ) [ ( k k )] ( p π v) E s( 0,R ) n [ ][ φ SL( )] 0 k < n where the nequalty follows because ( ) n SL > φ by the assumpton of the proposton, and n because φ n > φ for n,,n by the ndexng of supplers. hus, f SL ( ) > φn, then the, 7

19 canddate soluton n satsfes () and () for all, whch mples that n s the optmal soluton to the general problem. he structure of the optmal polcy, whch bols down to fndng the N cheapest actve supplers, s essentally dentcal to the soluton to a newsvendor problem n whch each suppler n a gven set has lmted capacty. In other words, f a gven set of supplers each have a supply functon gven by (a), where R s determnstc, then t s well-known that the correspondng newsvendor problem s concave and that the resultng optmal soluton dctates orderng from a more expensve suppler only after exhaustng the capactes of all less expensve supplers (Porteus, 990). hus, Propostons 4 effectvely establsh that, although concavty mght not be preserved when expandng the defnton of unrelablty beyond the noton of lmted capacty to nclude the more general construct appled n ths paper, the basc crtera for selectng supplers s preserved. We conclude ths secton by offerng nsghts on how the exstence of unrelable supplers affects a retaler s optmal orderng polcy. he nsghts come from comparng ( ) SL and, the retaler s optmal total order quantty and servce level, respectvely, aganst the optmal and SL for a newsvendor problem, whch serves as the benchmark for comparson. Proposton 5. Let N N denote the most expensve actve suppler n an optmal soluton. φ SL φ < φ <... < F. hen ( ) ( ) φ N N N Proof. Gven the defnton of N, > 0 f and only f N. hus, from Lemma, φ SL( ). And, from (), M ( ) 0 N N. herefore, from Lemma, N ( ) 0 φ SL, SL φ N <... < φ. whch, gven the ndexng of supplers, mples that ( ) 8

20 Next, from (8), for any gven, M ( ) ( p π v) { E [ s (,R )] φ E [ s (, R ) F ( ( ) )]}. S N k N k Note, however, that S ( ) Sk ( k, R ) because S k (,R k ) for all k, by the defnton of unrelablty. herefore, F ( S ( )) F ( ) M ; hence, for any gven, ( ) ( p π v) E [ s (,R )][ φ F ( ( ) )]. S But, snce suppler s an actve suppler, () mples that M ( ) 0. herefore, F ( ( ) φ S. o nterpret Proposton 5, consder the classc newsvendor problem as a benchmark. In the context of our model, the newsvendor problem represents the specal case scenaro n whch all N supplers are perfectly relable. Accordngly, by Propostons and 3, only suppler wll be actve n the newsvendor s optmal soluton. Moreover, by the corollary to Propostons and 3, the newvendor s optmal quantty to order from suppler s the quantty b that satsfes φ SL( b ) F( b ), whch s the famlar result. herefore, Proposton 5 mples that SL ( ) SL( ) F( ) F( ) b b. In other words, when compared to an otherwse equvalent retaler that does not have to deal wth unrelable supplers (.e., a newsvendor), a retaler that b does have to deal wth unrelable supplers wll order more ( ) but provde a lower level of servce to ts customers. 4 COMPARAIVE SAICS: HE CASE OF WO ACIVE SUPPLIERS In Secton 3, we developed propertes characterzng the retaler s optmal soluton, generally establshng that cost takes prorty over relablty when t comes to selectng supplers; and, as a 9

21 result, only the N least expensve supplers wll be actve, where N s such that N φ N φ SL( ). In ths secton, we develop addtonal nsghts by analyzng n further detal the case n whch N. Specfcally, we develop some ntuton wth regard to the followng questons:. What s the strategc relatonshp between and? In the sense of Bulow et al. (985), are they strategc complements or strategc substtutes? Moreover, how do changes n suppler cost affect and?. How do changes n suppler cost affect and ( ) SL? 3. Smlarly, how do changes n suppler relablty affect,,, and ( ) SL? Although we are able to answer the frst of these questons for the general case n whch each suppler s supply functon s characterzed by (), we requre addtonal problem structure to develop correspondng nsght wth respect to the second two questons. Consequently, we approach these ssues by talorng our analyss to the famly of problems nvolvng an exogenous output functon so that each suppler s supply functon can be characterzed by (a). We begn addressng the questons of ths secton wth the followng proposton, whch effectvely valdates the ntuton that the retaler s order quanttes not only are strategc substtutes, but also are such that a decrease n one suppler s cost wll result n an ncrease n that suppler s order quantty whle yeldng a decrease n the other suppler s order quantty. Proposton 6. For, let ( φ ) ( φ ) ; s decreasng n. Moreover, decreasng n φ. ; denote the optmal value of as a functon of. hen, s ncreasng n φ, and ( ; φ ) s Proof. he frst part of ths result follows drectly from (8) by takng the cross-partal dervatve: M ( ) ( p π v) E [ E [ s (,R ) s (,R ) f ( S ( ) R ) ] < 0, 0

22 whch s suffcent for establshng that ( ; φ ) 0 (opks, 998). Note also, from (8), < that M ( ) φ 0, whch mples that ( ; φ ) φ 0 Next, substtute ( ) gven the parameter φ : M ;. φ nto M () to reduce M () to a functon of the sngle varable, ( ; ) M ( ; φ ) M, ( ; φ ) ({ }; φ ) ( p π v) E s (, R ) φ F S ( ) [ ( ( ))] φ. Accordngly, snce ( ; φ ) φ 0 M, ({ }; φ ) M ({, ( ; φ )}; φ ) ( ; φ ) M, ( ; φ ) ( φ ) ; φ φ ( ; φ ) φ ( p π v) E [ s (, R )] 0. > hs mples that d d 0 φ. herefore, snce ( ; φ ) 0 > d dφ, < ( ; φ ) d ( ; φ ) ( ; φ ) dφ φ d dφ < 0. Snce the retaler wll react to a change n one suppler s cost by ncreasng one order quantty whle decreasng the other, t would be nterestng to nvestgate the combned effect. However, a defntve resoluton of ths ssue s dffcult to obtan n general because of the mplct tug of war occurrng between the two order quanttes. herefore, we develop nsght by analyzng a tractable varaton of the general problem. Specfcally, f we assume that the output functon of each suppler s exogenous, then we can ascertan that the ncrease n resultng from a decrease n suppler s cost more than compensates for the correspondng decrease n. Consequently, we fnd that a decrease n suppler s cost (as reflected by an ncrease n φ ) results n a hgher overall total quantty ordered by the retaler. Moreover, ths ncrease n the retaler s total order quantty translates nto a hgher servce level. hus, the ultmate benefactor

23 of a decrease n suppler cost s the retaler s customer market. We summarze these results wth the followng proposton. (See Appendx for formal proof.) Proposton 7. If suppler and suppler each have an exogenous output functon, then and SL( ) are ncreasng functons both of φ and of φ. o perform an analogous comparatve statcs nvestgaton wth respect to suppler relablty, we contnue to study the case of exogenous output functons, but we now need to be more precse wth our noton of more relable. Consequently, let R R (ρ,ε ), where ε s an ndependent random varable representng possble uncertanty n suppler s output functon, and ρ s a fxed parameter representng suppler s relablty level. hen defne R (ρ,ε )/ ρ > 0 so that the hgher s ρ, the more relable s suppler. Fnally, to operatonalze ths construct, assume that ε ~ G (u), where G (u) s a contnuous probablty dstrbuton; and that ε s such that, for any gven value of ρ, a sngle value of ε, namely z (,ρ ), satsfes R (ρ, z (,ρ )). he sgnfcance of z (,ρ ) s that t parttons the probablty state space characterzng ε nto two regons: one regon s such that R (ρ,ε ) < for all ε n the regon, and the other regon s such that R (ρ,ε ) for all ε n the regon. Wthout loss of generalty, we construct the state space so that R (ρ,ε ) < f and only f ε < z (,ρ ). Suppose, for example, that R (ρ,ε ) a (ρ ) b (ρ )ε. hen, z (, ρ ) ( a ( ρ )) b ( ρ ) fndngs for ths problem specfcaton. (See Appendx for proof.). Proposton 8 characterzes the relevant

24 Proposton 8. If both supplers have exogenous output functons such that K (R ) R (ρ,ε ), where R (ρ,ε )/ ρ > 0, ε s a random varable, and z (,ρ ) s defned such that R (ρ, z (,ρ )) < f and only f ε < z (,ρ ), then: (a) the KK condtons are suffcent for determnng and ; (b) s ncreasng n ρ, whle s decreasng n ρ ; (c) SL( ) s ncreasng both n ρ and n ρ ; but (d) s decreasng both n ρ and n ρ. We conclude ths secton by notng two observatons about Proposton 8 that warrant addtonal hghlghtng. Frst, recall from Secton that, n general, we have no guarantee that the retaler s general proft functon s sgnfcantly well behaved so as to ensure that the KK condtons wll be suffcent for computng the optmal soluton. hus, although the KK condtons are useful for characterzng the optmal soluton n general, ther applcablty for actually producng the optmal soluton should be verfed on a case-by-case bass. Part (a) of Proposton 8 does ust that, establshng, n effect, that the retaler s proft functon s unmodal for ths specfcaton of the problem. Second, note that, for the most part, the qualtatve results of Proposton 8 drectly parallel ther counterparts from Propostons 6 and 7. hus, n general, an ncrease n an actve suppler s relablty wll yeld effects smlar to a decrease n an actve suppler s cost, as mght be expected. However, the one notable excepton to ths general rule s the effect on : a more favorable suppler relablty results n a decrease n the retaler s total quantty ordered, whereas a more favorable suppler cost results n an ncrease n the retaler s total quantty ordered. One ntutve explanaton for ths contrast s as follows: he ultmate effect resultng from an mprovement n an actve suppler s cost or relablty s an ncrease n the retaler s servce level. But, servce level depends not on what the retaler orders n the aggregate, t depends on what the 3

25 retaler receves n the aggregate, whch s an amount characterzed by the output functons of the supplers. hese output functons, n turn, depend on the retaler s order quanttes and the supplers relabltes, but they do not depend on the supplers costs. Hence, t s natural that the retaler would need to ncrease ts aggregate order quantty n order to ncrease ts servce level when the servce level ncrease s n response to an mprovement n suppler cost snce, n such a case, order quanttes are the retaler s prmary means by whch to execute the desred effect. In contrast, the retaler need not necessarly ncrease ts aggregate order quantty to acheve an ncreased servce level when the desred servce level ncrease s n response to an mprovement n suppler relablty snce, n such a case, the retaler reaps the sde beneft of an mproved yeld. 5 CONCLUSIONS We have consdered the problem of a newsvendor that s served by multple supplers, where any gven suppler may be unrelable. By unrelable we mean that the margnal amount receved from a suppler s no more than, and typcally s less than, the margnal amount ordered. Our results ndcate that, n general, although relablty may nfluence how much s ordered from an actve suppler, cost takes prorty over relablty when t comes to suppler selecton. Even perfect relablty s no guarantee for selecton snce, n an optmal soluton, a gven suppler can be actve only f all less expensve supplers are actve, regardless of that suppler s relablty level. hs appealng result s all the more attractve because our noton of relablty s very general. In partcular, our constructon hnges on the noton of an output functon that can be ether endogenous, n the sense that ts parameters depend on the order quantty, or exogenous. Moreover, the output functon can be ether determnstc or random. 4

26 Although our model s developed wth a newsvendor, procurement envronment n mnd, there s a natural connecton wth the lterature on mult-perod manufacturng lot szng wth random yeld. hs s because, n mult-perod settngs of the lot szng model, the sngle-perod settng of the newsvendor s a basc buldng block. o make the connecton, compare our sngle-perod N-suppler newsvendor model wth a sngle-suppler N-perod model n whch an order s placed at the start of each perod after observng the nventory poston. If we assocate each suppler n the multple suppler model wth each perod n the multple perod model, then the multple suppler model bols down to a varaton of the multple perod problem n whch the decson maker must choose at the start of the frst perod how much to order for each of the next N perods. he sgnfcance of ths correspondence s that t may drectly yeld unmodalty n some of our problem settngs when the correspondng multple perod random yeld problem s unmodal. Whle we have approprately assumed that the newsvendor only pays for those good unts t receves, n many manufacturng settngs t mght be more approprate to pay for what s ordered. Snce t follows from Van Meghem and Rud (00) that n such a newsvendor settng the obectve functon becomes concave, more tradtonal optmzaton technques can be exploted to produce an optmal soluton. Interestngly, however, unmodalty appears to be attaned at the expense of a loss of some structural property: bascally, n ths varaton, t no longer s the case that cost necessarly takes prorty over relablty. Consequently, there s a basc change n the nature of rsk sharng n the supply channel. In effect, when the retaler pays only for those unts receved, the retaler bears some rsk of underproducton, but the suppler bears the rsk of defectve or neffcent output. Conversely, when the retaler pays for the entre manufacturng lot, the suppler effectvely s ndemnfed of all rsk snce the retaler ends up 5

27 burdenng not only the rsk of underproducton, but also the rsk of defectve or neffcent output. APPENDIX Proof of Lemma. From (8), M ( ) ( p π v) { E [ s (,R )] φ E [ s (,R ) F ( S ( ) )]} (A) for all. Note, however, that s (,R ) s an ncreasng functon of R by the defnton of more relable. Consequently, (, R ) s ( t, R ) S dt s an ncreasng functon of R. hs, n turn, mples that, for any gven, S () and, hence, ( ( ) ) other words, both s (,R ) and ( ( ) ) Studden (996), E S 0 S F are ncreasng functons of R. In F are ncreasng functons of R. Hence, from Karln and [ s (,R ) F ( S ( ) )] E [ s (,R )] E [ F ( S ( ) )] E [ s (,R )] SL( ). Substtutng ths nto (A) yelds M ( ) ( p π v) E [ s (, R )][ φ SL( ) ]. Proof of Lemma. Assume that 0. hen (8) and () mply that ( ) ( π ) ( ) φ 0 M ( ) 0 p v E s 0,R Pr D Sk k, R k, (A) k where, also from (), the set of k s represent any N- dmensonal vector {,, -,,, N } that smultaneously solves the N- equatons M ({,...,,,..., }) 0 N for,, N and, gven that 0. hat s, gven (8), { k} s any soluton to the followng system of equatons: 6

28 E s (,R ) φ Pr D Sk ( k,r k ) 0 k for,,n;. (A3) Notce that ths system of equatons s ndependent of R. herefore, any soluton to ths system s ndependent of R. hat s, the set of { k} s ndependent of R, whch mples that (A) reduces to ( p π v) E s ( 0,R ) [ ] φ Pr D Sk ( k,r k ) E[ s( 0,R )][ φ SL( )] 0 k herefore, SL( ) φ.. Proof of Lemma 3. Assume that 0 >. hen, from (), M ( ) 0 assumpton that s (,R ) for all and realzed values of R,. hus, from (8) and the ( p π v) { E [ s (,R )] φ E [ s (, R ) F ( S ( )]} ( p π v) { E [ s (,R )] φ SL( )} 0 herefore, E [ s (, R )] SL( ) φ.. Proof of Proposton 7. If each actve suppler has an exogenous output functon so that K (R ) R for,, then () mples that s (R ) 0 for R <, and s (R ) for R. Moreover, t means that (0), from the llustraton n Secton, provdes the optmalty condtons for { D S (,R )} E [ F( S (, R )] : φ Pr, (A4) where, ;, ;. akng the total dervatve of (A4) wth respect to φ (for, ), ( ( ) ( ) [ ( ( )] d d d d 0 E < f S,R s R E f S,R, dφ dφ dφ dφ where the nequalty follows because d dφ 0 by Proposton 6 and because s (R ) s equal > to ether 0 or for any gven value of R. herefore, for,, 7

29 d d d 0 <. d d φ φ dφ Next, for the purpose of ths proof, let G (u) be the c.d.f. of R. hen, gven (A4), 0 E SL ( ) E[ F( S (,R ) S(,R )] [ F( R S(,R )] g( u) du E [ F( S(,R )] g( u) 0 [ F( R S(,R )] g( u) du φg( u) E du, where, ;, ;. Applyng (A4) to the dervatve wth respect to φ (for, ), dsl dφ ( ) d ( ( ) ( ) ( ) [ [ ( ( )] ] d E ( ) f R S,R s R g u du E F S, R φ g 0 dφ dφ d E f ( R S(,R ) s ( R ) g( u) du > 0, d 0 φ where the nequalty follows because d dφ 0 by Proposton 6. > du Proof of Proposton 8. o smplfy notaton n ths proof, we denote R (ρ,ε ) smply as R, recognzng that the approprate dstrbuton functon G (u) s the dstrbuton of ε. Lkewse, we use z and z as shorthand for z (,ρ ) and ( ) z ρ, respectvely., Part (a). For any gven value of, the KK optmalty condton for s gven by (0): [ F( S (, ))] φ. (A5) E R he LHS of ths equaton s ndependent of, whle the RHS s ncreasng n. hus, gven, (A5) has exactly one soluton, namely ( ) of. akng the total dervatve of (A5) wth respect to,, whch s the optmal value of as a functon 8

30 0 E f s R d d ( ( ) ( )) ( ) S,R ( ) ( ) E f ( ) S(,R) s E f ( ) S (,R ) [ ( ) ( R) ] [ ( )] d (A6) d Snce s (R ) 0 for all realzed values of R, (A6) mples that ( ) d 0 d. Moreover, snce s (R ) for all realzed values of R, (A6) also mples that d ( ) d 0 Substtutng ( ). nto (0) yelds a KK condton for that s a functon only of : [ F( S ( ( ), )] φ. (A7) E R Notce, drhs d ( d ( ( ) ) E f S,R s( R ) d E f 0 d, d ( ( ( ) ) ( ) S,R ( ) where the frst nequalty follows because s (R ) whle ( ) d 0 nequalty follows because d ( ) d 0 whle the RHS s ncreasng n, condtons are suffcent for determnng optmalty. Parts b and d. From (A5) and (A7), d ; and the second. Snce the LHS of (A7) s ndependent of s unquely determned by (A7). hus, the KK and are defned as the soluton to E du 0 z z [ F( S(,R )] φ F( R ) g ( u) du F( ) g ( u) 0 φ (A8) for, ;, ; and. For the purpose of ths proof, let (,,z ) N ρ denote the, RHS of ths equaton, where, ;, ; and. hen, the followng propertes follow drectly from (A8): 9

31 (b) N (,,z, ) 0 N (,,z ρ ) ρ ;, (b) (b3) (b4) (b5) N N (,, z, ρ ) (,, z, ρ ) (,, z, ρ ) z ( R ) g ( u) du > 0 f ; 0 z f ( ) g ( u) du > 0 N F z N (,, z, ρ ) ρ z ( ) g ( z ) F( ) g ( z ) 0 R f ( R ) g ( u) du > 0. ρ 0 ; ; Gven these propertes, we next establsh how,, and react to changes n ρ (for, ) by startng wth property (b), takng total dervatves wth respect to ρ, and then applyng propertes (b) (b5) to the resultng expressons. Accordngly, from propertes (b) and (b4), N d N d N dz N d N d 0 (A9) dρ dρ z dρ dρ dρ N N d N d 0, (A0) dρ dρ where (A0) follows because. Smlarly, from propertes (b), (b4), and (b5), N 0 d dρ N d dρ N z dz dρ N ρ N > d dρ N d dρ. (A) Gven propertes (b) and (b3), (A9) mples that ( d dρ ) sgn( d dρ ) sgn, (A) 30

32 (A0) mples that ( d dρ ) sgn( d dρ ) sgn, (A3) and (A) mples that ether s decreasng n ρ, s decreasng n ρ, or both and are decreasng n ρ. However, (A) and (A3) together mply that d dρ and d d ρ have the same sgn. herefore, both and are decreasng n ρ. Consequently, gven (A) and (A3), s ncreasng n ρ. Part c. By defnton, ( ) E[ F( S (, R ) S (, R )] SL z z z ( ) ( ) ( ) [ ( )] F R R g u du g u du G z F( R ) g( u) du herefore, applyng (A8), SL 0 z [ G ( z )] F( R ) g ( u ) du [ G ( z )] F( ). z z ( ) F( R R ) g( u) g( u ) dudu [ G( z )][ G ( z )] F( ) 0 0 [ G ( z )] φ [ G( z )] φ Let SL( ), z, z,, ρ. (A4) ρ denote the RHS of (A4). hen, for, ;, ; and, the followng propertes follow drectly from (A4): SL (,z,z, ρ, ρ ) (c) [ G ( z )][ G ( z )] f ( ) < 0 z SL(,z,z, ρ, ρ ) (c) g ( z ) F( R ) g ( u ) du [ G ( z )] F( ) φ 0 z ; 0 ; 3

33 SL, z, z, ρ, ρ ρ z z R ρ ( ) (c3) f ( R R ) g( u ) g ( u ) dudu > hus, for, ;, ; and, dsl dρ ( ) SL(,z,z, ρ, ρ ) d dz dz SL(,z,z, ρ, ρ ) dρ 0 dρ dρ (, z, z,, ) ρ ρ d > 0 SL >, dρ ρ < where the fnal nequalty follows because d dρ 0 from the proof of part d. REFERENCES Anupnd, A., A. Akella Dversfcaton under supply uncertanty. Management Scence 39, Bulow, J. I., J. D. Geanakoplos, P. D. Klemperer Multmarket olgopoly: Strategc substtutes and complements. he Journal of Poltcal Economy 93, Chen, J. F., D. D. Yao, S. H. Zheng. 00. Optmal replenshment and rework wth multple unrelable supply sources. Operatons Research 49, Carallo, F. W., R. Akella,. E. Morton A perodc revew, producton plannng model wth uncertan capacty and uncertan demand optmalty of extended myopc polces. Management Scence 40, Gerchak, Y A newsvendor wth multple, unrelable, all-or-nothng supplers. Workng paper, Unversty of Waterloo, Waterloo, Ontaro. Karln, S., Studden, W. J chebycheff systems, wth applcatons n Analyss and Statstcs, Interscence Publshers, New York, NY. Heng, M., Y. Gerchak he structure of perodc revew polces n the presence of random yeld. Operatons Research 38, Khoua, M he sngle-perod (news-vendor) problem: Lterature revew and suggestons for future research. Omega 7, Mnner, S. 00. Multple-suppler nventory models n supply chan management: A revew. Internatonal Journal of Producton Economcs 8,