ARTICLE IN PRESS. Int. J. Production Economics

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1 Int. J. Production Economics 3 () Contents lists available at ScienceDirect Int. J. Production Economics journal omeage: Backorder enalty cost coefficient b : Wat could it be? George Liberooulos a,, Isidoros Tsikis a, Stefanos Delikouras b a Deartment of Mecanical Engineering, University of Tessaly, Volos, Greece b Steen M. Ross Scool of Business, University of Micigan, Ann Arbor, Micigan, USA article info Article istory: Received 3 Aril 7 Acceted 3 July 9 Available online Setember 9 Keywords: Economic order quantity Stockout Backorders Perturbed demand abstract Te classical economic order quantity (EOQ) model wit lanned enalized backorders (PB) relies on ostulating a value for te backorder enalty cost coefficient, b, wic is suosed to reflect te intangible adverse effect of te future loss of customer goodwill following a stockout. Recognizing tat te effect of te future loss of customer goodwill sould be not a direct enalty cost but a cange in future demand, Scwartz [966. A new aroac to stockout enalties. Management Science (), B538 B544] modified te classical EOQ-PB model by eliminating te backorder enalty cost term from te objective function and assuming tat te long-run demand rate is a decreasing, strictly convex function of te customer s disaointment factor (defined as te comlement of te demand fill rate) following a stockout, wic in turn is an increasing, strictly convex function of te demand fill rate. He called te new model a erturbed demand (PD) model. Scwartz rovided convincing justification for is PD model and resented several variations of it in a follow-u aer, but e did not solve any of tese models. In tis aer, we solve Scwartz s original PD model and its variations, and we discuss te imlications of teir solutions, tus filling a ga in te literature left by Scwartz. Moreover, aving been convinced tat Scwartz s aroac is more valid tan te classical aroac for reresenting te effect of te loss of customer goodwill following a stockout, but also recognizing tat te classical aroac is far more oular tan te PD aroac, because of its simlicity and because of tradition, we use te solution of te PD model to infer te value of b in te classical model, tus roviding one ossible answer to te question, wat could b be? A noteworty imlication of te solution of Scwartz s original PD model is tat te otimal fill rate is always or, rendering te inferred value of b in te classical model or N, resectively. Susecting tat te roerty of te PD function wic is most likely resonsible for roducing tis bangbang tye of result is strict convexity, we sow tat for te case were te PD function is roortional to an integer ower, say n, of te fill rate, te otimal fill rate is always or, if and only if n4, in wic case te PD function is strictly convex in te fill rate. & 9 Elsevier B.V. All rigts reserved.. Introduction Anyone wo as taken or taugt a course in inventory management is likely to ave ondered at ow to quantify Corresonding autor. Tel.: addresses: glib@mie.ut.gr (G. Liberooulos), itsikis@gmail. com (I. Tsikis), sdeli@umic.edu (S. Delikouras). te cost incurred by a stockout. A stockout may incur an immediate, direct cost to te firm, as well as a future, indirect cost. Te direct cost deends on weter te unfilled demand is backordered and eventually fulfilled wit a delay, or is cancelled. In te first case, te direct cost is related to te delayed delivery and may include extra administration costs, material andling and transortation costs for exediting te backordered items, fixed or variable contractual enalties, te loss of rofit from /$ - see front matter & 9 Elsevier B.V. All rigts reserved. doi:.6/j.ije.9.7.5

2 G. Liberooulos et al. / Int. J. Production Economics 3 () selling te backordered items at a discounted rice, te interest on te rofit tied u in te backorder, etc. In te second case, te direct cost is te lost rofit of te cancelled demand. In many ractical situations, art of an unfilled demand is backordered and art of it is cancelled. In most cases, te direct cost may be calculated wit some effort. Te indirect cost is muc arder to evaluate. It is related to te loss of customer goodwill due to te stockout, wic may lead to a temorary or ermanent decline in future demand and market sare, esecially in a cometitive market environment. Te quantification of te indirect cost of stockouts as long been an unsatisfactorily resolved issue in te literature. Te difficulty in determining an aroriate enalty rate for te indirect cost of stockouts as romted many researcers to relace tis rate by a constraint on te customer service level. For examle, C- entikaya and Parlar (998) take tis aroac for te economic order quantity (EOQ) model wit lanned backorders wic is at te center of our study in tis aer too. Tis aroac may seem more aealing to ractitioners, but it only transoses te roblem of estimating an aroriate enalty rate for stockouts to one of determining an aroriate customer service level. Te EOQ model wit lanned backorders is one of te earliest models in inventory teory tat deals wit stockouts. It relies on ostulating a value for te backorder enalty cost coefficient, denoted by b, wic is suosed to reflect te intangible adverse effect of te loss of customer goodwill following a stockout. We refer to tis model as te enalized backorders (encefort, PB) model. Te PB model is based on te following assumtions. A firm buys a single tye of items from a sulier, olds tem in inventory, and sells tem to its customers uon demand. Te demand for items, denoted by D, is continuous and constant over time, rocurement and delivery of te items are instantaneous, and unfilled demand is backordered. Finally, te gross rofit (selling rice minus urcase rice) er item sold, denoted by, te fixed order cost, denoted by k, te inventory olding cost er item er unit time, denoted by, and te backorder enalty cost er item er unit time, denoted by b, are known and constant over time. Te decision variables are te order quantity, denoted by Q, and te fraction of demand tat is met from stock, known as fill rate, denoted by F. All te arameters of te PB model, excet b, may be more or less secified. Scwartz (966) was one of te first to note tat te effect of te loss of goodwill sould not be a direct enalty cost of te tye considered in te PB model, because te effect of goodwill loss is incurred not at te time of te stockout incident, but at a later time, due to te customer s disaointment caused by te stockout and is subsequent decision to lower is future demand. Wit tis in mind, Scwartz (966) modified te PB model by eliminating te exlicit backorder enalty cost term from te objective function and assuming tat te long-run demand rate and ence te long-run average reward of te firm is a function of te customer s disaointment factor, wic e defined as te fraction of demand not met from stock. Scwartz called te resulting model a erturbed demand (encefort, PD) model. To derive an analytical form of te erturbed demand as a function of te disaointment factor, Scwartz (966) assumed te following customer resonse to stockouts. Wen a customer laces an order and finds out tat it cannot be delivered, e canges is a riori ordering attern in te future by reducing te amount e would oterwise ave bougt in eac of a number of future eriods. Te total amount tat te customer does not buy because of te disaointment, denoted by B, and te maximum otential demand rate in a cycle wit no disaointments, denoted by A, are finite. Te above assumed customer resonse led to te following strictly convex long-run PD function: D ðf A þð F ÞB ; ðþ were F is te long-run average fill rate, and ence F is te fraction of demand not met from stock, i.e., Scwartz s disaointment factor. We note tat trougout tis aer, we sall be using te notation X for variables and functions in te PD model wose equivalent variable/function in te PB model is denoted by X, to distinguis between te two models. Te PD model roosed by Scwartz (966) relaces te indeterminable task of subjectively coosing b in te PB model wit te better defined task of estimating arameters A and B of te PD function, D (F ). Scwartz (966) roosed a rocedure for measuring arameters A and B from observed demand data. Tis rocedure is based on te assumtion tat wen a customer faces a stockout, e reduces te size of is next order by some amount, te following one by a smaller amount, te next by a still smaller, and so on, so tat as time asses, e tends to forget about te disaointment; terefore, is subsequent orders will aroac teir original level, A. Scwartz (966) rovided convincing justification for is PD model, but e did not solve it. In a follow-u aer, Scwartz (97) continued is investigation of te PD model by formulating tree different variations of it in wic e relaced te exlicit fixed order cost wit a constraint on te order quantity, te interorder time, and te starting inventory in eac cycle, resectively. For eac variation e considered bot cases wit backlogging and lost sales. In all variations, e merely stated in a few lines te first-order condition for te otimal quantity of unfilled demand, but in none of tese variations did e solve tis condition or rovide any furter analysis, discussion, or insigt. In tis aer, we solve exactly te original PD model introduced by Scwartz (966) and its tree variations considered in Scwartz (97), in te case of backlogging, tus filling a ga in te literature left by Scwartz. Moreover, we discuss te imlications of te solutions. In te last sentence of is conclusions, Scwartz (97) wrote, Te Perturbed Demand aroac to goodwill stockout enalties is bot substantially more valid and more ractical tan any reviously considered in te literature of inventory teory. We agree wit te osition tat te PD aroac to goodwill stockout enalties is in

3 68 G. Liberooulos et al. / Int. J. Production Economics 3 () general more valid tan te classical PB aroac, altoug we must oint out tat te PD function () roosed by Scwartz (966) is based on a secific consumer resonse assumtion and is terefore one of many ossible alternative functions. Te main reason we agree wit Scwartz s aroac is tat tis aroac byasses te difficulty of defining te roblem of ow to coose a good let alone te best value for te backorder enalty cost coefficient (or te equivalent customer service level) in te classical aroac. Anoter reason is tat te classical aroac as te following aradox embedded in it. It assumes tat tere is a backorder enalty cost wic is suosed to reflect te future loss of demand due to te loss of customer goodwill following stockouts, yet at te same time it assumes tat te demand is constant. Wile te PD aroac introduced by Scwartz (966, 97) sawned several follow-u aers, to date, te classical PB aroac is still redominant in te inventory management researc literature and in ractice. One ossible exlanation for tis redominance is tradition and te fact tat many ERP systems and oter decision suort systems used in ractice rely on te inut of userdefined safety stocks or equivalent customer service levels wic imly secific backorder/stockout cost coefficients. Anoter ossible exlanation is tat, wile te PD aroac is more valid tan te PB aroac, it is more comlicated and ence less aealing to ractitioners tan te classical aroac. Having been convinced tat te PD model is more valid tan te PB model but also recognizing tat te PB model is more aealing and widely used tan te PD model because of its simlicity and because of tradition, we use te solution of te PD model to infer te value of b in te equivalent PB model, tus roviding one ossible answer to te question, wat could b be in te PB model? Te way we infer b is by setting te otimal decision variables, Q and F, in te PD model, equal to te resective variables, Q* and F* (wic are functions of b), in te PB model, and solving for b. Once tis is done, te resulting demand rate D ðf Þ in te PD model is in general different tan te constant demand rate D in te PB model. Tis difference, owever, can be justified if one tinks of D as a sortrun constant demand rate and D ðf Þ as a long-run constant demand rate. Te idea ere is tat as time asses, if one uses te correct decision variables based on te correct value of b in te PB model, te sort-run demand rate D will drift towards D ðf Þ, assuming tat te fill rate F is ket constant, so tat in te long run, it will settle to D ðf Þ. A noteworty imlication of te solution of Scwartz s original PD model is tat te otimal fill rate is always or, making te inferred value of b in te classical model or N. Susecting tat te roerty of te PD function wic is most likely resonsible for roducing tis bang-bang tye of result is strict convexity, we sow tat for te case were te PD function is roortional to an integer ower, say n, of te fill rate, te otimal fill rate is always or, if and only if n4, in wic case te PD function is strictly convex in te fill rate. If n=, in wic case te PD function is not strictly convex in te fill rate, te bang-bang result does not old. Tis reinforces our susicion tat te roerty of D (F ) wic is most likely resonsible for roducing tis bang-bang tye of result is strict convexity. We recognize tat te PB model is a bit tactical relative to current inventory researc. It is only an aroximation to te stocastic (Q, r) inventory model wit backorders. As C- entikaya and Parlar (998) oint out, te relationsi between te two models is analogous to te relationsi between two classical inventory/roduction models, namely te deterministic multi-eriod model wit backorders (Zangwill, 969) and te stocastic multieriod model wit backorders for wic an (s, S) olicy is otimal (Scarf, 96). Te simle PD model in tis aer and its comutable results in terms of inferred backorder costs and decision variables may rovide insigt into te analysis of te (Q, r) model wit stocastic demands. Te rest of tis aer is organized as follows. In Section we review some of te relevant literature on te effect of stockouts. In Section 3, we summarize some more or less known results on te otimal decision variables of te classical PB model and tree variations of it wic corresond to te tree variations described by Scwartz (97), for te case of backlogging. In Section 4, we derive analytical exressions for te otimal decision variables of te resective PD models and te inferred value of b in te PB models. In Section 5, we exlore te role of te convexity of te PD function on te otimal fill rate of te PD model. Finally, we draw our conclusions in Section 6.. Literature review Te effect of stockouts on current sales and future demand as been studied by te Oerations Management (OM) community as well as by te Logistics Researc (LR) community. Some of te related work reorted in te OM literature as been based on develoing decision trees to model te consequences of stockouts (e.g., Cang and Niland, 967) and using surveys to estimate te arameters of te trees (e.g., Oral et al., 97; Oral, 98). Most of te researc on te effects of stockouts on current and future sales in te OM literature, owever, as focused on develoing matematical inventory control models in wic demand is resumed to be a function of a certain direct or indirect quantitative measure of stockouts, suc as fill rate, average delivery delay, etc. Examles of suc work are Hanssmann (959), Scwartz (966, 97), Hill (976), Caine and Plaut (976), Robinson (988), and Argon et al. (). Te work in tis aer follows tis stream of researc and in articular Scwartz (966, 97). Tere as also been a closely related stream of researc in wic demand is resumed to be a function of inventories. Examles of suc work are Urban (995) and Balakrisnan et al. (4). Scwartz s work and te works tat followed it ave remained witin te framework of a single decision maker formulation and ence ave not looked into te underlying cometition interactions between suliers. Given tat te future defection of a customer deends on wat oter otions e as, several researcers ave addressed

4 G. Liberooulos et al. / Int. J. Production Economics 3 () service-related issues witin a game teoretic framework. Tere is a large body of OM literature tat as looked at roduct and/or sulier substitution or switcing wen stockouts occur. Examles of suc work are Li (99), Ernst and Coen (99), Ernst and Powell (995, 998), Liman and McCardle (997), Netessine et al. (6), Bernstein and Federgruen (4a, b), and Dana and Petruzzi (). In all of te above works, te two factors cometition in roduct availability and its future effect ave been studied more or less searately. To te best of our knowledge, te only excetions tat ave assumed tat customer demand is a function of revious service encounters, are Gans (), Hall and Porteus (), Gaur and Park (7), Liberooulos and Tsikis (7), Liu et al. (7), and Olsen and Parker (8). Most of te work on te effects of stockouts reorted in te LR literature as focused on identifying and exlaining consumer reaction to stockouts in retail settings. Suc reaction may include item (brand or variety) or urcase quantity switcing, cancellation or deferral of urcase, store switcing, etc. A number of studies ave relied on ostulating some decision model wit alternative ossible outcomes and courses of action of consumers and retailers following a stockout and estimating te arameters (robabilities, costs, etc.) of tat model via interviews and/or mail surveys. Examles of survey-based studies include Nielsen (968a, b), Walter and Grabner (975), Sycon and Srague (975), Scary and Becker (978), Scary and Cristoer (979), Zinszer and Lesser (98), Emmelainz et al. (99), Zinn and Liu (), Camo et al. (, 4), and van Woensel et al. (7). Two excetions tat focus on BB rater tan BC markets are Dion et al. (99) and Dion and Banting (995). Anoter grou of studies ave been based on laboratory exeriments. Examles are Carlton and Erenberg (976), Motes and Castleberry (985), and Fitzsimons (). Te above works ave focused mainly on te immediate imact of stockouts on urcase incidence and coice decisions and not on te cumulative effects of stockouts over time. Tere are some studies tat ave looked at ow stockouts affect future long-term demand of retailers, based on istorical beavioral data analysis. Examles of suc studies are Straugn (99), Camo et al. (3), Liberooulos and Tsikis (8), and Anderson et al. (6). Finally, tere exist some relatively recent surveyand exeriment-based studies on consumers ercetions of and reactions to waiting and service. Some examles are Taylor (994), Carmon et al. (995), Hui and Tse (996), Kumar et al. (997), Zou and Soman (3), and Municor and Rafaeli (7). It is wort noting tat some of te emirical studies mentioned above rovide suorting evidence tat seems to validate Scwartz s assumtion tat wen a customer laces an order and finds out tat it cannot be delivered, e canges is a riori ordering attern in te future by reducing te amount e would oterwise ave bougt, but tat as time asses, e tends to forget about te disaointment so tat is subsequent orders would aroac teir original level. More secifically, Carlton and Erenberg (976) conducted an exeriment in wic a anel of consumers was reeatedly offered te oortunity to buy certain artificial brands of a detergent, and sowed tat wen a stockout condition was introduced and subsequently witdrawn, market sares and category sales returned to teir re-stockout levels wit no aarent long-term effects. Motes and Castleberry (985) reeated te same tye of exeriment using a real otato ci brand and also found tat category sales returned to teir re-stockout levels. In anoter study, Scary and Becker (978) reorted te effects of a regional beer strike in wic stockouts occurred in selected brands. Using brand sare as te deendent variable, stockout effects were judged to be more sort- tan long-run. Dion and Banting (995) reorted te results of a study on te erceived consequences for business-to-business market buyers of being stocked out by teir sulier and teir reurcase loyalty on te next urcase occasion. Te results sowed tat buyers often sougt an alternate sulier in te face of a stockout, but te majority returned to te original sulier on te next urcase occasion. Zinn and Liu () reorted results of an interview-based study of consumer sort-term resonse to stockouts. By comaring te ercetions of consumers wo recently exerienced a stockout wit tose wo did not, tey sowed tat consumers aear able to isolate a recent stockout exerience from teir ercetion of oter dimensions of te store s image. In anoter study, Camo et al. (3) exlored te imact of retail stockouts on weter, ow muc, and wat to buy, by adjusting traditional urcase incidence, quantity and coice models, so as to account for stockout effects. Teir study, wic was based on scanner anel data of a large Euroean suermarket cain, sowed tat stockouts may reduce te robability of urcase incidence and lead to te urcase of smaller quantities. To te best of our knowledge, to date, tere as been no emirical work aimed at estimating te size of te backorder (or any oter stockout-related) cost coefficient. One excetion is Badinelli (986), wo reeatedly asked decision makers to secify teir marginal excange rate between on-and inventory and backorders, and ten used te relatively more exact inventory olding cost to estimate a disvalue cost function of te stockout erformance measure troug regression. In a somewat related earlier work, Gardner and Dannenbring (979) roosed tat inventory decisions be seen as olicy tradeoffs on a 3D resonse surface sowing te otimal relationsis among aggregate customer service (defined as te comlement of te fill rate), workload (defined as te relenisment frequency) and investment, (defined as te sum of cycle and safety stock), regardless of te articular cost structure of te firm; owever, tey did not rovide any information on ow to obtain objective cost information. 3. Te PB model In tis section, we discuss te otimal decision variables in te classical PB model, namely, te order quantity, Q, and te fill rate, F. Te only constraint on Q is

5 7 G. Liberooulos et al. / Int. J. Production Economics 3 () tat it must be nonnegative. Te fill rate must satisfy rfr. Note tat if F=, te firm oerates in a ure maketo-order mode, backordering all te demand and not keeing any inventory. If F=, on te oter and, te firm oerates in a ure make-to-stock mode, keeing all items in inventory and not allowing any backorders. Finally, if ofo, te firm uses a mixed make-to-order and maketo-stock olicy. We also derive and discuss te otimal decision variables for tree variations of te PB model wic are equivalent to te variations tat Scwartz (97) considered for te PD model. In tese variations, te exlicit fixed order cost is relaced wit a constraint on te order quantity, te interorder time, and te starting inventory in eac cycle, resectively. Te classical PB model and its tree variations make u a total of four cases. For eac case, it is straigtforward to derive an exression of te average rofit of te firm as a function of te decision variables Q and F. Table sows te average rofit function, denoted by te P(Q, F), and te constraints for te four cases. Te quantities Q, Q/D, and QF in te last column of Table are te order quantity, te interorder time, and te starting inventory in eac cycle, resectively, and Q min, T min, and I min are ositive, finite numbers denoting te minimum values of tese quantities, resectively. Parameters Q min and T min may be set eiter externally by te sulier, or internally by te firm to incur an imlicit fixed order exense, if te exlicit fixed order cost k is not known or is difficult to Table Objective function and constraints for te classical PB model and its tree variations. Case P(Q, F) Constraints 3 4 D k D Q D QF D QF D QF QF bqð FÞ bqð FÞ bqð FÞ bqð FÞ rfr, QZ rfr, QZQ min rfr, Q/DZT min rfr, QFZI min obtain. Similarly, arameter I min may be set internally by te firm to incur an imlicit fixed order exense, or as a safety stock against fluctuations in demand, because in reality demand may vary. Proosition gives te otimal order quantity and fill rate tat maximize te average rofit subject to te constraints, for all four cases of te PB model, sown in Table. Proosition. Te otimal order quantity and average rofit as a function of F, Q*(F) and P(F), resectively, and te overall otimal order quantity, fill rate, and average rofit, F*, Q*, and P, were P=P(Q*, F*), for te classical PB model and its tree variations sown in Table, are given in Table. Te roof of Proosition is trivial and is terefore omitted. It suffices to mention tat te metodology to solve te otimization roblems sown in Table consists of te following four stes: () exress te otimal order quantity as a function of F, say Q*(F), () write an exression for te average rofit as a function of F only, say P*(F), after aving relaced Q by Q*(F), i.e., P*(F)=P(Q*(F), F), (3) maximize P*(F), subject to rfr, to determine te otimal fill rate F*, and (4) evaluate Q*(F*) to determine te otimal order quantity Q*. Te imlementation of tese stes can be found in many textbooks on inventory management (e.g. Zikin, ), at least for te classical PB model (case ). For cases 4, it can be easily carried out in a similar manner. Discussion of Proosition. From column 4 of Table, we can observe tat in all four cases, te otimal fill rate, F*, is a function of te backorder enalty cost coefficient, b. More secifically, in cases 3, F* is given by te familiar newsvendor fraction, b/(+b), wereas in case 4, it is given by te square route of tis fraction. From Tables and, it is easy to see tat if we set Q min =DT min, cases and 3 are identical to eac oter. Tis means tat tere are really only tree cases of te PB model to consider; owever, we urosely leave te results for bot cases and 3 in Table, even toug tere are identical, because in Section 4 we will relate tem to te results of te resective cases of te PD model, wic are not identical. From Table, it is also easy to see tat in cases 4, te Table Otimal decision variables and objective function for te classical PB model and its tree variations. Case Q*(F) P*(F) F* Q* P* sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kd F þ bð FÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D kd½f þ bð FÞ Š b þ b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kdð þ bþ b sffiffiffiffiffiffiffiffiffiffiffiffiffiffi kdb D ð þ bþ Q min D Q minf b Q minð FÞ b þ b Q min D bq min ð þ bþ 3 DT min D DT minf b DT minð FÞ b þ b DT min D bdt min ð þ bþ 4 I min F D I minf b I minð FÞ F ffiffiffi b þ b ffiffiffi þ b I min b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ð bð þ bþ bþi min

6 G. Liberooulos et al. / Int. J. Production Economics 3 () average rofit, P(Q, F), is strictly decreasing in te order quantity Q and tat Q is only restricted by a lower limit. For tis reason, te otimal order quantity Q* is simly set at tis lower limit, as can be seen in column 5 of Table. In cases and 3, tis limit is indeendent of F and is equal to Q min and DT min, resectively. In case 4, it is given by I min /F, wic becomes I min /F* once te otimal fill rate F* is secified. Case is te only case were P(Q, F) isnot strictly decreasing in Q, because of te extra fixed order Table 3 Objective function and constraints for te original PD model and its tree variations. Case P (Q, F ) Constraints 3 4 D ðf Þ k D ðf Þ Q F Q D ðf Þ Q F D ðf Þ Q F D ðf Þ Q F rf r, Q Z rf r, Q ZQ min rf r, Q /D (F )ZT min rf r, Q F ZI min cost term, kd/q, wic is increasing in Q. In tis case, te otimal order quantity is given by te familiar square root formula in Table. To summarize, in all cases, F* is a function of b. Moreover, in cases and 4, Q* is a function of F*, and ence also a function of b. In cases and 3, on te oter and, Q* does not deend on b. 4. Te PD model For eac of te four variations of te PB model discussed in Section 3, we can construct an equivalent PD model. Table 3 sows te average rofit function, denoted by te P (Q, F ), and te constraints for te four equivalent PD models, were Q and F are te decision variables. Case is te original PD model introduced by Scwartz (966) and cases 4 are te variations of te PD model considered in Scwartz (97). Proosition gives te otimal order quantity and fill rate tat maximize te average rofit subject to te constraints, for all four cases of te PD model, sown in Table 3. It also gives te inferred backorder enalty cost coefficient, b, and te resulting otimal order quantity for te resective cases of te PB model, sown in Table. Proosition. Suose tat D (F ) is given by (). Ten te otimal decision variables for te original PD model and its Table 4 Otimal decision variables and conditions under wic tey old for te original PD model and its tree variations, and inferred backorder enalty cost coefficient and resulting otimal order quantity for te resective PB models. Case PD model PB model F Q Condition b Q* N ffiffiffi A B=ð þ BÞo, ka ka N, ffiffiffiffiffiffiffiffi k ffiffiffi ffiffiffiffiffiffiffiffi A B=ð þ BÞ4 k ffiffiffi ffiffiffiffiffiffiffiffi A B=ð þ BÞ ¼ k N N, N ffi kd ffi kd N, F Q min ABoQ min or F F Q min AB ¼ Q min ; B4:5or AB4Q min ; B4:5; P ðf Þ4A Q min = ABZQ min ; Br:5 or N AB4Q min ; B4:5; P ðf ÞoA Q min = F, AB4Q min ; B4:5; P ðf A Q min = F F N 3 F 3 AT min þð F 3 ÞB B=ð þ BÞoT min F 3 F 3 DT min AT min B=ð þ BÞZT min N 4 N AB=ð þ BÞoI min N I min AB=ð þ BÞ4I min N I min, N, I min AB=ð þ BÞ ¼I min, N N, I min

7 7 G. Liberooulos et al. / Int. J. Production Economics 3 () tree variations sown in Table 3, along wit te conditions under wic tey old, as well as te inferred backorder enalty cost coefficient and resulting otimal order quantity for te resective PB models sown in Table, are given by Table 4, were AB F ¼ smallest real root of ½ þð F ÞBŠ Q minf ; ðþ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 3 ¼ þ B ð þ BÞ : ð3þ B Tmin Te roof of Proosition is given in te Aendix for case and in Liberooulos et al. (9) for cases 4. Altoug te metodology to solve te otimization roblems sown in Table 3 is standard and consists of te same four stes for solving te four cases of te PB model, outlined in Section 3, imlementing tis metodology on te PD model is not trivial, as is te case in te PB model, because te PD function D (F ) given by () significantly comlicates te average rofit function P (Q, F ) sown in column of Table 3. Discussion of Proosition. From columns 4 of Table 4, we can see tat te results for cases and 4 of te PD model are similar to eac oter, and so are te results for cases and 3. Tis was also true for te resective cases of te PB model, discussed in Section 3. Te most striking similarity between cases and 4 of te PD model is tat in bot cases te otimal fill rate F is always or. Tis means tat in tese two cases, it is always otimal to only old inventory and not allow any backorders ðf ¼ Þ, or to only allow backorders and not old any inventory ðf ¼ Þ. Holding inventory and allowing backorders is never otimal. On te oter and, in bot cases and 3 te otimal fill rate F is always or equal to a quantity between and, even if te inventory olding cost rate,, is extremely large, as long as it is not infinite. Tis quantity is denoted by F and F 3 for cases and 3, resectively, and deends on te model arameters; terefore, it can assume a continuum of values. Tis means tat in tese two cases, it is always otimal eiter to only old inventory and not allow any backorders ðf ¼ Þ, or to allow backorders for some time and old inventory for te rest of te time ðof oþ. Te only interretation tat we can give about wy in cases and 4 F can be, wereas in cases and 3 it assumes a continuum of finite nonzero values, is te following. In cases and 4, F can be, because Q can go to N, andwenq goes to N, F must be, as we will exlain sortly. In cases and 3, on te oter and, Q cannot go to N, and terefore F does not ave to be (in fact, it cannot be, because te first derivative of te average rofit P ðf Þ is always ositive at F =). Note, owever, tat if we imose a finite uer limit, say Q max, on te order quantity, ten it can be sown tat in cases and 4, F cannot be but instead assumes a continuum of ositive values, just like in cases and 3. At te same time, if te minimum order quantity Q min in case, or te minimum interorder time T min in case 3, goes to N,tenQ will also go to N, and terefore F will go to, just like in cases and 4. In cases and 4, te decisive condition of weter to only old inventory ðf ¼ Þ or only allow backorders ðf ¼ Þ is ffiffiffi ffiffiffiffiffiffiffiffi A B=ð þ BÞo k and AB/(+B)oI min, resectively. From tese conditions, we can see tat in bot cases, increasing te reward and demand related arameters,, A, orb, tends to favor te solution F ¼, i.e., only old inventory. On te oter and, increasing te cost related arameters, or k, in case, and similarly increasing or I min, in case 4, tends to favor te solution F ¼, i.e., only allow backorders. In case, te arameter tat affects mostly te decisive condition is te rice margin, because it aears linearly in tis condition, wereas arameters, k and A aear in a square root, and arameter B aears in a term tat ranges between and. In case 4, on te oter and, arameters, A,, and I min affect equally strongly te decisive condition, because tey aear linearly in tis condition. In contrast, te effect of arameter B is weaker, because B aears in a term tat ranges between and. From columns and 3 of Table 4, we can see tat in all cases, if F ¼, ten Q is finite. If F ¼, owever, wic is true only in cases and 4, ten Q ¼,aswe mentioned earlier. Te reason for tis is sligtly different in eac case. More secifically, in bot cases, te aroriate decisive condition determines weter F ¼ or. Te tradeoff at stake, favoring one or te oter solution, is between incurring ig inventory costs (and, in case, ig ordering costs as well) on one and, and losing long-term demand and terefore revenue, on te oter and. If te model arameters in te decisive condition are suc tat F ¼, ten it is otimal for te firm to oerate strictly wit lanned backorders and no inventory. Since backorders incur no direct cost, te firm can ave as many of tem as it leases for free. Tis muc is true for bot cases and 4. Te difference in wy Q ¼, between te two cases, is te following. In case, given tat te firm ays an order cost k every time it orders a quantity Q, ten wy not ave Q be infinite to Table 5 Otimal decision variables and conditions under wic tey old for te original PD model in wic te erturbed demand function is given by (4). n PD model A 9k F Q Condition, 3 k ka ka ka ka A o 9 k A Z 9 k A ok A 4k A ¼ k 4 A ok ka A 4k, ka A ¼ k,

8 G. Liberooulos et al. / Int. J. Production Economics 3 () avoid aying te order cost? Hence, Q ¼. In case 4, on te oter and, if F ¼, ten Q must be infinite, not to avoid aying te order cost, since tere is no suc cost, but because oterwise, te minimum-inventory constraint, Q F ZI min, will be violated. Of course, in reality, te order quantity cannot be infinite. Tis can be andled in te model by assuming tat te order quantity as an uer limit, say Q max, wic is large enoug so tat Q max Z ffiffiffiffiffiffiffiffiffiffiffiffiffi ka=, in case, and Q max ZI min, in case 4, and ten resolving te otimization roblem wit te additional constraint Q rq max to obtain F. As was mentioned earlier, it can be sown tat if we imose suc a limit, F cannot be but instead assumes a continuum of ositive values, just like in cases and 3. A similar discussion interreting te results of Table 5 for cases and 3 can be found in Liberooulos et al. (9). From column 5 of Table 4, it can be seen tat in cases and 4, te inferred value of b is or N, because in tese cases F is always equal to or, as was discussed earlier. In cases and 3, on te oter and, te inferred value of b is eiter N or equal to a finite number, because in tese cases F is always or equal to a number between and. From column 3 of Table 4, it can be seen tat in te subcase of case of te PD model, were ffiffiffi ffiffiffiffiffiffiffiffi A B=ð þ BÞZ k, as well as in case 3, te otimal order quantity Q is a function of D (F*). From te last column of te same table, it can also be seen tat in te resective cases of te PB model, if we use te inferred value of b, sown in te second to last column of Table 4, ten te resulting otimal order quantity Q* is given by te same function, but wit D ðf Þ in te lace of D. Ata first glance, tis seems to suggest tat in tese cases, te inferred value of b, wic by definition guarantees tat F ¼ F, does not guarantee tat Q ¼ Q. Tis furter suggests tat in tese cases, tere exist no two models a PB and a resective PD model wit te same otimal decision arameters. Tis is true in te sort run. As was mentioned in Section, owever, if te firm uses te otimal arameters F* and Q* in te PB model, ten as time asses, no matter wat te initial value of D is, te average demand rate will drift towards D (F*) wic is equal to D ðf Þ, assuming tat te fill rate F* is ket constant and equal to F, so tat in te long run, its average value will be equal to D ðf Þ. Terefore, in te long run, te otimal order quantity Q* in te PB model will be equal to te otimal order quantity in te resective PD model. 5. On te role of te convexity of D (F )onf From Proosition, we saw tat in Scwartz s original PD model (case ), in wic D (F ) is given by (), te otimal fill rate, F, is or, rendering te inferred value of b in te PB model equal to or N, resectively. It is natural to susect tat te roerty of D (F ) wic is most likely resonsible for roducing tis bang-bang tye of result is strict convexity; terefore, a question tat arises logically is weter tis bang-bang result olds for all strictly convex PD functions. Unfortunately, it is ractically imossible to rovide a clear answer to tis question by analytical means. Instead, we can only rovide a clue by means of te following roositions. Proosition 3. Suose tat te PD function, D (F ), is ositive, increasing, continuous and twice differentiable in [, ]. Ten, te following olds concerning te otimal fill rate, F, in te original PD model sown in Table 3 (case ):. If D ðf Þ=F rk=, for all F A[,], ten F ð¼ Þ.. If D ðf Þ=F 4k=, for some F A[,], and d D ðf Þ=dF 4 and dd ðf Þ=dF Z4ðD ðf Þ=F Þ, for all F A[,], ten F is eiter or. Te roof of Proosition 3 is in te Aendix. Discussion of Proosition 3. Te first art of Proosition 3 states a sufficient condition tat D (F ) must satisfy in order for te otimal fill rate to be. If tis condition is not met, ten te second art states two oter sufficient conditions tat D (F ) must satisfy in order for te otimal fill rate to be eiter or. Te first of tese two conditions is tat D (F ) must be strictly convex. Te second condition is tat D (F ) must satisfy te inequality dd ðf Þ=dF Z4D ðf Þ=F. It is interesting to note tat Scwartz s PD function given by () is strictly convex, and terefore satisfies te first condition, but it does not satisfy te second condition, wic can be exressed as B(5F 4)Z4. Yet, by Proosition (case ), F is still or. To el understand wat te condition dd ðf Þ=dF Z4D ðf Þ=F means and ow it is related to convexity, suose tat D (F ) is simly roortional to a ositive ower of F, namely, D ðf AF n; ð4þ were A is a ositive real number and n is a nonnegative real integer. Ten, te first condition (strict convexity) imlies tat n4, wereas te second condition ðdd ðf Þ=dF 44D ðf Þ=F Þ imlies tat n44. Clearly, if te second condition is satisfied, i.e., if n44, ten te first condition is also satisfied, i.e., n4; ence, by Proosition 3, F is or. In oter words, te second condition is more restrictive tan strict convexity. Wat aens if onr4, owever? Ten, te second condition no longer olds, but strict convexity olds. Does te bang-bang tye of result still old? Te following roosition gives an answer to tis question. Proosition 4. Suose tat D (F ) is given by (4). Ten, te otimal decision variables for te original PD model sown in Table 3 (case ), along wit te conditions under wic tey old, are given in Table 5, for different values of n. Te roof of Proosition 4 is in te Aendix. Discussion of Proosition 4. Proosition 4 essentially states tat if D (F ) is given by (4), ten te bang-bang tye of results olds if D (F ) is strictly convex and does not old if D (F ) is not strictly convex. Tis result togeter wit te result of Proosition, case, according to wic te bang-bang tye result folds if D (F ) is given by (), wic is strictly convex, reinforces our susicion tat te roerty of D (F ) wic is most likely resonsible

9 74 G. Liberooulos et al. / Int. J. Production Economics 3 () for roducing tis bang-bang tye of result is strict convexity. Of course, a more fundamental question is weter it is reasonable to assume tat te PD function is strictly convex. Badinelli (986) argues tat in realistic cases, an inventory manager could reason tat starting from a situation of excellent service (i.e., one wit little or no backorders and low stockout risk) an incremental increase in stockouts would look bad. As te situation grows worse, subsequent increases migt not look as serious as te first. Hence te manager s stockout cost as a function of te average backorders or stockout risk would exibit a diminising marginal cost wic would yield a concave disvalue function. Te analogy of tis beavior, in te context of our model, is tat te demand rate as a function of te disaointment factor ( F ) would exibit a diminising marginal decrease wic would yield a PD function tat is convex in F. 6. Conclusions Te work in tis aer was motivated by our desire to find a lausible answer to te question, wat could te backorder enalty cost coefficient b be? To tis end, we roosed to infer te value of b for te PB model by connecting b to te loss in te long-run average demand rate wic is affected by backorders according to Scwartz s PD model (). We alied tis rocedure to te original PD model and tree variations of it in wic we relaced te exlicit fixed order cost wit a constraint on te order quantity, te interorder time, and te starting inventory in eac cycle, resectively. Our first main finding is tat for te original PD model and te variation of te PD model wit te minimum starting inventory in eac cycle, te otimal fill rate is always or, wic imlies tat te inferred backorder enalty cost b in te resective PB models is or N, resectively. In te former case, te otimal order quantity is infinite, wereas in te latter case it is finite. Based on te results in Section 5, our second main finding is tat we ave strong reasons to susect tat te roerty of D (F ) wic is most likely resonsible for roducing tis bang-bang tye of result is strict convexity. Future researc following tis work could be directed toward reeating tis rocedure for oter PD models, for examle models tat assume tat te long-run average demand rate is eiter a different function of te long-run average fill rate tan te one given by Eqs. () and (4), or a function of some oter customer service related erformance measure, suc as te long-run average backorder waiting time or number of backorders. Some suc functions ave been roosed in te literature. For examle, Ernst and Coen (99) roosed a PD rate wic is a linear function of te fill rate. Using our notation, teir function can be written as D ðf A½ Bð F ÞŠ; were A is te maximum otential demand rate corresonding to a fill rate equal to and B is a ercentage. Zikin () (roblem 3.,. 69) roosed te PD function D ðw a ½f ðw ÞŠ b ; were W is te average waiting time, a and b are ositive constants wit b4, and f( ) is an increasing function wit f()=. Given tat te average waiting time can be exressed as a function of Q and F as well as te demand rate itself, namely W ¼ Q ð F Þ D ðw ; Þ if we substitute W from te equation above into D (W ), we can see tat te PD rate is a rater comlicated function of Q and F satisfying D ðq ; F a ½f ðq ð F Þ =D ðq ; F ÞÞŠ b : A less comlicated alternative would be to relace te average waiting time W wit te average number of backorders, say R, in Zikin s PD function, i.e., assume tat D ðr a ½f ðr ÞŠ b : Given tat te average number of backorders R can be exressed as a function of Q and F as follows: R ¼ Q ð F Þ ; ten D (R ) can be rewritten as a function of Q and F as follows: D ðq ; F a ½f ðq ð F Þ =ÞŠ b : In all te models above, te arameters of te PD function ave to be estimated. As was mentioned in Section 4, Scwartz (966) roosed a rocedure for measuring arameters A and B in is model from observed demand data. In general, tis is not a trivial task; owever, it is a better defined task tat icking a value for b. Finally, two oter wortwile directions for future researc following tis work would be to include direct backorder costs besides te indirect loss-of-customer-goodwill costs, to examine models wit lost sales instead of order backlogging, and to extend tis analysis to stocastic inventory models. Acknowledgments Te work in tis aer was suorted by Action Heraclitus: Researc Scolarsis wit Priority in Basic Researc of te Oerational Program for Education and Initial Vocational Training II, wic is managed by Greece s Ministry of National Education and Religious Affairs and is co-financed by te Euroean Social Fund, te Euroean Regional Develoment Fund, and Greece s Public Sector.

10 G. Liberooulos et al. / Int. J. Production Economics 3 () Aendix Proof of Proosition. To solve te otimality conditions of te four otimization (maximization) roblems corresonding to te four cases of te PD model sown in Table 3 of Section 4, we use Descartes s rule of signs, wic was first ublised by Renee Descartes in 637. Tis rule states tat if te terms of a olynomial f(x) are written in a customary fasion tat is wit te terms given in decreasing order of te exonent of x ten te number of ositive real roots of te olynomial is eiter equal to te number of sign canges in te coefficients of successive terms of f(x) or is less tan tat number by an even number (until or is reaced). If any coefficients are, tey are simly ignored. Similarly, te number of negative real roots of te olynomial is eiter equal to te number of sign canges in te coefficients of successive terms of f( x) or is less tan tat number by an even number (until or is reaced) (e.g., see Young and Gregory, 97). In wat follows, we develo te solution of te otimization roblem for case. Te solution for cases 4 can be found in Liberooulos et al. (9). Solution of te original PD model (case of Table 3). In order to find te otimal order quantity as a function of F, Q ðf Þ, we set te first artial derivative of P (Q, F ) wit resect to Q equal to and solve te resulting equation. Tis equation is quadratic in Q and as two solutions, one of wic is negative. Te only ositive and terefore accetable solution is ffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðf kd ðf Þ ka F ¼ F ½ þð F : ð5þ ÞBŠ Let P ðf Þ be te average rofit as a function of F wen te otimal order quantity is used, i.e., ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðf P ðq ðf Þ; F D ðf Þ F kd ðf Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ka ¼ þð F F ÞB þð F : ð6þ ÞB To find te otimal fill rate, F, we set te first derivative of P ðf Þ equal to, solve te resulting equation, and examine te values of te average rofit and its derivative at te end oints of te interval [, ]. Te first derivative of te average rofit P ðf Þ is dp ðf Þ df ffiffiffiffiffiffiffiffiffiffiffi AB ¼ ½ þð F ÞBŠ ka½ þð F ÞBŠ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð7þ þð F ÞB Setting te first derivative of P ðf Þ equal to, erforming a cange of variables from F to Y, were Y=+( F )B, and rearranging terms, yields te following 5t degree olynomial equation in Y: Y 5 þ ð þ BÞY 4 þðþbþ Y 3 A B ¼ : ð8þ k According to Descartes s rule of signs, te olynomial on te ls of Eq. (8) as exactly one ositive real root and exactly two or zero negative real roots. For eac real root, Y n, tere corresonds a real root, F n, of te rs of exression (7), wic is given by F n ¼ ðy n Þ=B. Since F reresents te long-run, average fill rate, it must take values in te interval [, ]. Note tat if Y n o, ten F n 4, wereas if Y n 4+B, ten F n o. Tis imlies tat for eac negative real root, Y n, if tere are any, te corresonding root F n is 4. It also imlies tat te root F n corresonding to te only ositive real root, Y n, lies in te interval [, ] if and only if Y n ½; þ BŠ. Tis means tat at most one real root of te rs of Eq. (7) may lie in te interval [, ]. Wit te above result in mind, to find te otimal fill rate, F, we roceed by examining te average rofit and its derivative at te end oints, and. From (7), it is easy to see tat te first derivative of te average rofit at te two end oints, and, is given resectively by dp ðf Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AB ðþbþ þ B ka df ¼ ; ð9þ ð þ BÞ dp ðf Þ df F ¼ F ¼ ¼ AB ffiffiffiffiffiffiffiffiffiffiffi ka þ B : ðþ From (6) it is also easy to see tat te average rofit at and is given resectively by P ðþ ¼ A þ B ; ðþ P ðþ ¼A ffiffiffiffiffiffiffiffiffiffiffi ka: ðþ Now, suose tat P ðþ4p ðþ, wic from () and ffiffiffiffiffiffiffiffiffiffiffi () is true if and only if ABoðB þ Þ ka. Te latter condition, wic can be rewritten as ffiffiffi ffiffiffiffiffiffiffiffi A B=ð þ BÞo k, imlies tat te first derivative of te average rofit at F =, wic is given by (9), is always negative. Tis means tat as F increases starting from, te average rofit, wic starts at P ðþ, eiter continuously decreases in te interval [, ], or continuously decreases until it reaces a minimum at te only real root of te rs of exression (7) wic may ossibly lie in te interval [, ], and ten continuously increases since tere is at most one real root in te interval [, ] until it reaces P ðþ at F =. Given our initial assumtion tat P ðþ4p ðþ, tis furter imlies tat te maximum average rofit in te interval [, ] is attained at F =. Now, suose tat P ðþop ðþ, wic from () and () is true if and only if ffiffiffi ffiffiffiffiffiffiffiffi A B=ð þ BÞ4 k. Ten, te first derivative of te average rofit at F =, wic is given by (), is always ositive. Tis means tat as F decreases starting from, te average rofit, wic starts at P ðþ, eiter continuously decreases in te interval [, ], or continuously decreases until it reaces a minimum at te only real root of exression (7) wic may ossibly lie in te interval [, ], and ten continuously increases since tere is at most one real root in te interval [, ] until it