Managing Rentals with Usage-Based Loss

Transcription

1 Corell Uiversity School of Hotel Admiistratio The Scholarly Commos Articles ad Chapters School of Hotel Admiistratio Collectio Maagig Retals with Usage-Based Loss Vicet W. Slaugh Corell Uiversity School of Hotel Admiistratio, Bahar Biller Geeral Electric, Global Research Ceter Sridhar R. Tayur Caregie Mello Uiversity Follow this ad additioal works at: Part of the Operatios ad Supply Chai Maagemet Commos Recommeded Citatio Slaugh, V. W., Biller, B., & Tayur, S. R. (2015). Maagig retals with usage-based loss [Electroic versio]. Retrieved [isert date], from Corell Uiversity, School of Hotel Admiistratio site: This Article or Chapter is brought to you for free ad ope access by the School of Hotel Admiistratio Collectio at The Scholarly Commos. It has bee accepted for iclusio i Articles ad Chapters by a authorized admiistrator of The Scholarly Commos. For more iformatio, please cotact hlmdigital@corell.edu.

2 Maagig Retals with Usage-Based Loss Abstract Motivated by ew ad iovative retal busiess models, this paper develops a ovel discrete-time model of a retal operatio with radom loss of ivetory due to customer use. The ivetory level is chose before the start of a fiite retal seaso, ad customers ot immediately served are lost. Our aalysis framework uses stochastic comparisos of sample paths to derive structural results that hold uder good geerality for demads, retal duratios, ad retal uit lifetimes. Cosiderig differet \recirculatio" rules i.e., which retal uit to choose to meet each demad we prove the cocavity of the expected profit fuctio ad idetify the optimal recirculatio rule. A umerical study clarifies whe cosiderig retal uit loss ad recirculatio rules matters most for the ivetory decisio: Accoutig for retal uit loss ca icrease the expected profit by 7% for a sigle seaso ad becomes eve more importat as the time horizo legthes. We also observe that the optimal ivetory level i respose to icreasig loss probability is o-mootoic. Fially, we show that choosig the optimal recirculatio rule over aother simple policy allows more retal uits to be profitably added, ad the profit-maximizig service level icreases by up to 6 percetage poits. Keywords service operatios, capacity plaig ad ivestmet, ivetory theory ad cotrol supply chai maagemet, stochastic methods Disciplies Operatios ad Supply Chai Maagemet Commets Required Publisher Statemet Spriger. Fial versio published as: Slaugh, V. W., Biller, B., & Tayur, S. R. (2016). Maagig retals with usage-based loss. Maufacturig & Service Operatios Maagemet, 18(3), doi: /msom Reprited with permissio. All rights reserved. This article or chapter is available at The Scholarly Commos:

3 Maagig Retals with Usage-Based Loss Vicet W. Slaugh Tepper School of Busiess, Caregie Mello Uiversity Bahar Biller Geeral Electric, Global Research Ceter Sridhar R. Tayur Tepper School of Busiess, Caregie Mello Uiversity Draft: Jauary 7, 2015 Motivated by ew ad iovative retal busiess models, this paper develops a ovel discrete-time model of a retal operatio with radom loss of ivetory due to customer use. The ivetory level is chose before the start of a fiite retal seaso, ad customers ot immediately served are lost. Our aalysis framework uses stochastic comparisos of sample paths to derive structural results that hold uder good geerality for demads, retal duratios, ad retal uit lifetimes. Cosiderig differet recirculatio rules i.e., which retal uit to choose to meet each demad we prove the cocavity of the expected profit fuctio ad idetify the optimal recirculatio rule. A umerical study clarifies whe cosiderig retal uit loss ad recirculatio rules matters most for the ivetory decisio: Accoutig for retal uit loss ca icrease the expected profit by 7% for a sigle seaso ad becomes eve more importat as the time horizo legthes. We also observe that the optimal ivetory level i respose to icreasig loss probability is o-mootoic. Fially, we show that choosig the optimal recirculatio rule over aother simple policy allows more retal uits to be profitably added, ad the profit-maximizig service level icreases by up to 6 percetage poits. Key words : Service Operatios; Capacity Plaig ad Ivestmet; Ivetory Theory ad Cotrol; Supply Chai Maagemet; Stochastic Methods 1. Itroductio Advaces i olie commercial models have produced a ew geeratio of iovative busiesses built upo retig goods. The flexibility ad affordability promised by retig a wide array of products have led to retal busiesses specializig i just about every aspect of our busiess ad 1

4 2 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss persoal lives. Besides the traditioal retal products such as movies, cars, ad hotel rooms, less commo goods available to ret rage from bicycles to jets, cribs to coffis, ad furiture to campig gear. Accordig to IBISWorld idustry aalysts, the aual reveue of fiftee differet retal idustries i the Uited States each exceeded $1 billio i 2013, while the aual reveue of each of the car, heavy equipmet, ad idustrial equipmet retal idustries surpassed $25 billio. Luxury goods have received particular attetio as fertile groud for retal busiesses that make those goods available to ew customer classes. For example, Ret the Ruway is a compay that allows customers to ret high-fashio dresses for either four or eight days at approximately 10% of the retail price of a dress (Wortham 2009). Customers ca view the selectio of dresses ad their availability through a website, ad receive style ad fit advice from Ret the Ruway cosultats ad customer reviews. Dresses are shipped to customers ad retured by mail. However, the critical decisio about the umber of dresses that will comprise Ret the Ruway s seasoal retal ivetory must be made shortly after pre-seaso fashio shows, which are several moths i advace of the retal seaso (Bikley 2011). Choosig the umber of retal uits to procure before the start of a retal seaso without the possibility of repleishmet durig the seaso is a importat problem that may retal busiesses face. Despite the seemigly fudametal ature of this problem, operatios maagemet literature offers very little aalytical support whe lost sales ad discrete time periods atural assumptios for may retal systems are cosidered. I this paper, we aalyze a sigle-product retal system usig a discrete-time framework. We focus o the usage-based loss of retal uits over a fiite retal horizo durig which o additioal retal uits may be ordered, e.g., whe log procuremet lead times prohibit i-seaso reorderig. I particular, we cosider each retal uit to have a radom lifetime, which is characterized by a geeral probability distributio o the umber of times the uit ca be reted before its retiremet from the retal ivetory. Our goal is to uderstad the role of this ucertaity arisig from the usage-based loss of retal uits o the maagemet of retal ivetory.

5 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 3 I additio to Ret the Ruway, whose dresses are susceptible to both destructive icidets ad wearig out over time, other retal systems face the challege of losig ivetory that ca be difficult to replace i the middle of the retal seaso. For example, a Paris-based bicycle sharig program that bega with 20,600 bicycles i 2007 had more tha 8,000 bikes stole ad aother 8,000 bikes severely damaged ad i eed of replacemet withi two years (Erlager ad De La Baume 2009). Ivetory loss ca also occur whe customers exercise a optio to purchase a product. Users of Redbox, a automated movie ad game retal kiosk, ret a DVD for $1.20 a day. If the DVD is ot retured i 20 days, the the customer pays $24 for the accrued daily retal charge ad ows the DVD. Aother example is Ret-A-Ceter, a compay with over $3 billio i reveue i 2012 ad which rets furiture, appliaces ad electroics to customers who ca ow the item if it is reted beyod a certai duratio. I its 2012 aual report, Ret-A-Ceter states that approximately 25% of its retal agreemets result i customer owership. Existig work supportig capacity plaig for retal busiesses relies primarily o queueig models. Although Poisso or compoud Poisso arrival processes may adequately represet demads for some retal busiesses, better choices may exist for modelig demad i retal systems characterized by discrete retal time slots. For example, busiess travelers occupy a hotel room for a discrete umber of days ad are more likely to begi retig a hotel room o Moday ight tha a Saturday ight. At Ret the Ruway, for example, whose customers primarily ret dresses for evets o Fridays ad Saturdays, a discrete-time demad model with a period of oe week more accurately represets a customer demad patter tha a Poisso arrival process. Therefore, extedig the discrete-time ivetory theory to iclude loss of retal ivetory offers a advatage for a retal system like Ret the Ruway. We develop a model that makes o distributioal assumptios ad captures (a) operatioal details such as radom retal uit lifetimes (with costat, icreasig or decreasig failure rates) ad radom retal duratios, (b) very geeral demad models with features such as seasoality or auto-correlatios ad (c) recirculatio rules that are used i practice for choosig amog available retal uits to satisfy demads. We make the followig cotributios regardig the ivetory maagemet of retal systems:

6 4 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 1. Model ad Framework. To the best of our kowledge, we are the first to cosider the loss of retal uits accordig to distributios over the umber of times that each uit ca be reted before loss. Thus, our model icludes a state variable that represets the umber of times that a retal uit has bee reted out (i.e., a cout-based model) or a state variable that represets a retal uit s coditio (i.e., a coditio-based model). It also accommodates a arbitrary demad process ad geeral distributios for lifetime ad duratio of each retal uit. 2. Structural Results: (a) We establish the cocavity of the expected profit fuctio i the iitial ivetory of retal uits for geometric lifetime distributios. Not surprisigly, this structural property holds idepedet of the retal uit recirculatio rules as the loss probability is costat over time. (b) For geeral lifetime distributios, it becomes ecessary to cosider the rules that allocate retal uits to satisfy customer demad for both cout-based or coditio-based models. (c) Cout-Based Model: We establish the cocavity of the expected profit fuctio for the static priority recirculatio rule; i.e., the uits to be reted are prioritized accordig to a list that does ot chage over the retal horizo. We show that the cocavity of the expected profit fuctio also holds for a policy that spreads the retal load evely over all uits, allocatig the retal uit that has bee reted out the fewest umber of times. Referrig to this recirculatio rule as the eve spread policy, we prove its optimality whe retal uit loss probabilities are o-decreasig i the umber of times that the uit has bee reted. (d) Coditio-Based Model. We demostrate aalogous results for the coditio-based model, showig the cocavity of the expected profit fuctio for the best-first policy i which the retal uit i the best coditio receives the highest allocatio priority. Also, we prove that the best-first policy is optimal whe the state trasitio probability matrix is totally positive of order 2, a coditio that implies that the retal uit failure rate is icreasig as its coditio worses. 3. Maagerial Isights from Numerical Study (a) Failig to accout for usage-based loss of retal ivetory leads to a sigificat reductio i the expected profit. For a 5% probability of loss each time a uit is reted, we fid that igorig

7 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 5 the loss of retal uits reduces the expected profit by 7.3% ad 33.0% for a half-year ad a full-year retal horizo, respectively. (b) The optimal respose to the icreasig loss probability is to first icrease the umber of retal uits, the decrease the umber of retal uits ad fially stock zero retal uits. (c) For a retal uit lifetime distributio with icreasig loss probability, the retal uit recirculatio rule plays a importat role accordig to the rate at which the loss probability icreases. We focus o the cout-based model, as similar results apply for the coditio-based model, ad compare the eve spread policy to the static priority recirculatio rule. Choosig the eve spread policy icreases the optimal iitial ivetory level with a correspodig icrease of up to 6 percetage poits i the profit-maximizig service level. The remaider of the paper is orgaized as follows. Sectio 2 reviews the retal ivetory maagemet literature. Sectio 3 itroduces our retal ivetory model. We establish the structural properties of this model for geometric lifetime distributios i Sectio 4 ad for geeral lifetime distributios i Sectio 5, where we further idetify the optimal retal uit recirculatio rule uder certai coditios. The umerical aalysis follows i Sectio 6. We coclude with a summary of fidigs ad future research directios i Sectio Literature Review Early research o retal ivetory maagemet exclusively uses queueig models as a foudatio for aalysis. The iitial advaces i queueig theory by Takács (1962) ad Riorda (1962) for the telephoe trukig problem fidig the statioary probabilities of a multi-server pure loss system have sparked two semial papers o the problem of sizig a fleet of retal equipmet. Taiiter (1964) formulates a optimizatio problem for M/G/c/c ad G/M/c/c retal systems based o the limitig distributios of the system states derived by Takács (1962). The decisio variable is the capacity of the retal system ad the problem is studied both asymptotically ad over a fiite horizo. Whisler (1967), o the other had, shows that the optimal policy structure for a retal system with lost sales, periodic reorderig, ad ostatioary state trasitio probabilities

8 6 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss as i Riorda (1962) has upper critical values above which ivetory should be discarded ad lower critical values below which ivetory should be ordered. Our work differs from these studies by its focus o the ivetory decisio prior to the retal seaso, the challege of hadlig radom usage-based loss of retal uits ad stochastic retal duratio, ad the use of a discrete-time model for demad represetatio. The early research o retal ivetory maagemet with lost sales is followed by a extesive study of the M/M/c queueig model with backlogged demads. Specifically, the problem is posed as fidig the optimal umber of servers to employ i a multi-server queuig system, where servers represet retal uits ad service time correspods to the retal duratio; see Huag et al. (1977), Jug ad Lee (1989), Gree et al. (2001), ad Zhag et al. (2012). Motivated by the time-specific ature of customers retals, however, we restrict our focus to lost sales models i this paper. Table 1 compares our retal ivetory model to the other retal ivetory models that also make the assumptio of lost sales. I additio to the cotiuous-time retal ivetory models of Taiiter (1964) ad Whisler (1967) tabulated here, Papier ad Thoema (2008) build o the M/M/c/c queueig model i Harel (1988), where approximatios, as well as lower ad upper bouds, are developed for the lost sales rate as a fuctio of the system capacity. Extedig this model to accout for a compoud Poisso arrival process, Papier ad Thoema (2008) coduct a statioary queueig aalysis to obtai structural results for a fleet sizig problem ad provide a approximatio suitable for implemetatio. The use of the M/M/c/c or M/G/c/c queueig model as a basis for studyig capacity maagemet for retal systems further follows i Savi et al. (2005), Gas ad Savi (2007), Adelma (2008), Hampshire et al. (2009), ad Levi ad Shi (2011). However, our work is differet from this stream of research by our cosideratio of a discrete-time retal model with a fiite retal seaso, radom usage-based ivetory loss, ad a arbitrary demad model possessig the ability to capture ay distributioal characteristic. I cotrast to the cotiuous-time queueig models, Cohe et al. (1980) use a discrete-time model to represet a retur process to a blood bak with the goal of determiig a optimal order-up-to

9 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 7 Table 1 Compariso of lost sales retal ivetory models. CONTINUOUS TIME DISCRETE TIME Taiter (1964) Whisler (1967) Papier ad Thoema (2008) Cohe (1980) Baro et al. (2011) Our Paper Ivetory Decisio Oe Time Repeated Oe Time Repeated Oe Time Oe Time Time Horizo Fiite Fiite Ifiite Fiite Fiite Fiite Demad Process IID Iterarrival Times IID Iterarrival Times Compoud Poisso Statioary; Also Nostatioary Retal Duratio Geeral IID Geeral IID Geeral IID Determiistic Geeral IID Arbitrary Arbitrary Geeral IID with a Restricted Retur Process Geeral IID Ivetory Loss N/A N/A N/A Costat Decay N/A Usage-Based Radom Loss level i every period. Reflectig hospitals tedecy to order sigificatly more uits of blood tha eeded, a costat percetage of the quatity reted by hospitals is retured to the blood bak ad the rest is cosumed after a retal duratio of a fixed umber of periods. A costat percetage of the ivetory leftover at the blood bak is, o the other had, cosidered to have decayed. The problem of fidig the optimal ivetory level uder a periodic review policy is formulated as a dyamic program ad a approximate solutio is provided. I compariso, we examie the oe-time pre-seasoal orderig problem ad cosider the loss of ivetory as radom, istead of beig a costat proportio. Furthermore, we do ot require the assumptio of a idepedet ad idetically distributed demad process, ad we allow radomess i the retal duratio. Closest to our model is Baro et al. (2011), who determie the optimal pre-seaso order quatity for a video retal store with lost sales but o ivetory loss. I particular, Baro et al. (2011) cosider a retur process that is mootoe; i.e., the percetage of the retal uits reted i period t ad retured by period is always greater tha or equal to the percetage of the uits reted i period t + 1 ad retured by the same period. The key result is the cocavity of the expected umber of retals i the umber of retal uits procured. We are, o the other had, the first to establish this result for a retal system with radom usage-based ivetory loss. We also address the issue of retal uit ivetory allocatio, which arises oly i our retal ivetory model as a result of accoutig for radom lifetimes of the retal uits.

10 8 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss To aalyze models i which retal uit lifetimes do ot follow a geometric distributio, we use sample path aalysis i a very geeral settig to prove the two mai results of our work: the cocavity of the expected profit fuctio ad the optimal retal uit recirculatio rule. This approach has bee used i various settigs to model complexities of productio ad ivetory systems. Examples iclude the umber of customers ad their utilities for a model with dyamic substitutio by Mahaja ad va Ryzi (2001) ad the processig times for multi-statio productio lies by Muth (1979) ad Tayur (1993). Also, our proofs of cocavity bear similarities to that of Shathikumar ad Yao (1987) i their study of systems with multi-server statios. 3. Retal Model: A Sample Path Approach I this sectio, we describe a sample path approach to modelig a retal ivetory system that allows us to aalyze the system icludig rules for recirculatig retal uits uder geeral assumptios about the demad process. Motivated by the problem of selectig the umber of retal uits to procure before the start of a fiite retal seaso, we begi our aalysis with the followig two-stage model of a sigle-product, discrete-time retal ivetory system with lost sales. I the first stage, the size of the retal ivetory is chose to be y. Each retal uit is procured before the start of the seaso ad has a salvage value at the ed of the retal seaso that depeds o whether the retal uit retires from the ivetory before the ed of the seaso. Hece, the uit procuremet cost accouts for ot just the purchase price, but is adjusted to also iclude the salvage value for a dress i good coditio ad the cost of holdig the item for the duratio of the retal seaso. I the secod stage, demads occur over N periods ad the uits purchased i the first stage are reted to satisfy the customer demads. Each customer is assumed to ret a sigle uit, ad for simplicity we begi by cosiderig the case i which each retal lasts for a determiistic duratio of A periods. Thus, fulfillig oe uit of demad requires that oe uit of the ivetory is withdraw for the period i which the demad is received ad for the A 1 succeedig periods, resultig i the retal havig a determiistic duratio of A periods. A critical aspect of retal ivetory plaig is to accout for the loss of retal uits. Misuse by customers, customer optios to purchase reted items or simply the deterioratio of the retal

11 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 9 uit s quality over time preset reasos for why a uit would be retired from the retal ivetory. To accout for radom usage-based loss of ivetory i our model, we assume that each retal uit m {1, 2,..., y} fails after a radom umber of retals, l m, characterized by a geeral probability mass fuctio. More precisely, upo completio of its l m th retal, uit m satisfies o further demads, although it does have a expected salvage value that is eared at the ed of the horizo. I additio, the demad d is received i period {1, 2,..., N}. Take together, the demads d 1, d 2,..., d N ad the retal uit lifetimes l 1, l 2,..., l y comprise a sample path, which we deote by ξ; i.e., ξ = {d 1, d 2,..., d N, l 1, l 2,..., l y }. Whe retal uit loss probabilities chage based o the umber of times reted, we must also specify the recirculatio rule γ to fully characterize the system s operatio. We use R γ (y, ξ) for the umber of uits reted ad L γ (y, ξ) for the umber of sales lost i period as a fuctio of the iitial ivetory of y retal uits ad the sample path ξ of demads ad retal uit lifetimes for a recirculatio rule γ. For coveiece, the total umber of retals ad lost sales over the etire horizo are defied as R γ (y, ξ) := N =1 Rγ (y, ξ) ad L γ (y, ξ) := N =1 Lγ (y, ξ), respectively. We also let W γ (y, ξ) deote the umber of uits that are successfully retured to the system i the begiig of period ad available to be reted agai i that period, ad defie Z γ (y, ξ) := R γ A (y, ξ) W γ (y, ξ) as the umber of retal uits that would have bee retured i period but were lost. A reward r is eared every time a uit is reted, ad c is the uit cost of a lost sale. The retal system operates for period of the secod stage as follows: (1) Of all the items reted i period A, W γ (y, ξ) uits are retured while Z γ (y, ξ) retire from the retal ivetory. After returs are received but before retals are made, the total ivetory available to ret out i period is I γ (y, ξ) := y 1 t=1 Rγ t (y, ξ) + t=1 W γ t (y, ξ). (2) The demad D is realized as d. If d I γ (y, ξ), the d uits are reted out. Otherwise, I γ (y, ξ) uits are reted out. More succictly, R γ t (y, ξ) := d I γ (y, ξ), where a b deotes the miimum of a ad b. The retal uit recirculatio rule determies which retal uit is allocated to satisfy each uit of demad, ad cosequetly

12 10 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss Retals Lost Sales 12 Retals (Total) Lost Sales (Total) Demad Period Number of Retal Uits (a) Illustratio with two retal uits (y = 2). (b) Effect of umber of retal uits y. Figure 1 Number of retals ad lost sales for Example 1 with a retal duratio of two periods (A = 2). determies W γ (y, ξ) ad Z(y, γ ξ). (3) Excess demad L γ (y, ξ) := [I(y, γ ξ) d ], which ca be alteratively writte as d γ R(y, γ ξ), is lost. Therefore, give the sample path ξ, the dyamics of the retal system s operatio ca be represeted recursively as follows, where I γ 0 (y, ξ) = y ad R γ t (y, ξ) = 0 for t 0: I γ +1(y, ξ) = I γ (y, ξ) R γ (y, ξ) + W γ +1(y, ξ). R γ +1(y, ξ) = d +1 I γ +1(y, ξ). (1) L γ +1(y, ξ) = d +1 R+1(y, γ ξ). Example 1. Figure 1 illustrates this retal system with a demad sequece of {d 1,..., d 8 } = {1, 0, 2, 0, 3, 1, 2, 1} for eight periods (N = 8). Each retal lasts for two periods (A = 2); i.e., a uit that is reted i period will ext be available to be reted agai i period + 2. If the system would operate with oly oe retal uit (i.e., y = 1), the that uit would be reted i periods 1, 3, 5, ad 7 for a total of four retals, while six uits of the demad would be lost. Figure 1a shows how the demad is divided ito retals ad lost sales for a system with y = 2 retal uits. Thus, the additio of the secod retal uit allows a additioal uit of demad to be satisfied i periods 3, 5, ad 7, so that there are ow 7 uits of fulfilled demad ad 3 uits of lost sales. Figure 1b shows how the umber of retals ad lost sales chage with the umber of retal uits y. We observe that the umber of retals is cocave i y ad that the umber of lost sales is covex i y o this

13 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 11 sample path. I other words, the umber of additioal retals produced by oe additioal retal uit (i.e., the slope of the retals curve) is decreasig i y. As the retur process depeds o the specific recirculatio rule γ, we will describe W γ +1(y, ξ) ad Z γ +1(y, ξ) as eeded whe referrig to specific rules. To accout for the retur of retal uits that are reted i periods N A+1, N A+2,..., N, we defie W γ N+1 (y, ξ), W γ N+2 (y, ξ),..., W γ N+A (y, ξ) as the returs ad Z γ N+1 (y, ξ), Zγ N+2 (y, ξ),..., Zγ N+A (y, ξ) as the lost uits i each of the correspodig periods. The total umber of lost retal uits is deoted by Z γ (y, ξ) := N+A =A+1 Zγ (y, ξ). Oe way to model retal uit loss is to cosider geometrically distributed retal uit lifetimes. The memorylessess of the geometric distributio leads to a costat probability of retal uit loss over time. However, if a retal uit does ideed have a higher probability of wearig out over time, the a retal uit lifetime distributio with a icreasig failure rate (i.e., a loss probability icreasig with the umber of times the uit has bee reted) would be a suitable choice. Bikes, cars ad large equipmet are examples of assets for which a icreasig loss probability as a fuctio of the umber of retals could be used to model the retal uit lifetime. Furthermore, lifetimes that are determiistic whe eforced by safety regulatios that require their disposal after a certai umber of uses ca be aalyzed as a special case of a icreasig loss probability. Next, we discuss how to icorporate the salvage value of a retal uit ito our retal ivetory model. This is a importat issue because the salvage value of a retal uit that retires from the retal ivetory durig the retal seaso may differ from the salvage value of a uit that is still fuctioal at the ed of the seaso. I that case, we separately defie the procuremet cost s g for the uit that ca be still reted at the ed of the retal seaso ad the procuremet cost s b for the uit that has already retired from ivetory. The relatio s g s b idicates that the uit retirig from the retal ivetory has bee damaged. Hece, it has lost a portio of its value. The relatio s g s b may, o the other had, represet the purchase of the retal uit by the customer who is retig it as discussed i Sectio 1 for the retal compaies Redbox ad Ret-A-Ceter. To accout for the cost of ivetory loss i the objective fuctio of our retal ivetory model, the reductio i the salvage value of a lost retal uit (s b s g ) is multiplied by the umber of lost

14 12 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss retal uits ad subtracted from the reveue as part of the profit fuctio, which we deote by Π γ (y, ξ). Cosequetly, we obtai the profit fuctio as follows: Π γ (y, ξ) = r N D (r + c)l γ (y, ξ) s g y (s b s g )Z γ (y, ξ). =1 We are ow ready to formulate the retal ivetory optimizatio problem as the maximizatio of the expected profit fuctio π γ (y) := E[Π γ (y, ξ)] subject to y 0. We ivestigate the cocavity of this expected profit fuctio i the iitial ivetory of y retal uits for geometric lifetime distributios i Sectio 4 ad for geeral lifetime distributios i Sectio 5. We ca also exted this basic model to iclude a radom duratio for each retal. We defie A m,i as the radom variable deotig the retal duratio for the ith demad served by the mth retal uit for i 1 ad m = 1, 2,..., y. Each retal lasts for ay umber of periods betwee a miimum of A mi ad a maximum of A max ; i.e., A m,i {A mi, A mi + 1,..., A max }. We cosider A m,i to be idepedet ad idetically distributed accordig to a geeral probability mass fuctio characterized by h(a) := P{A m,i = a}, a = A mi, A mi + 1,..., A max. For ay sample path, we let D a, represet the umber of uits of demad i period, = 1, 2,..., N, that have a duratio of a periods. Thus, our model of stochastic retal duratio differs from that of Cohe et al. (1980), who model uit demads as havig differet duratios with respect to costat proportios. It also differs from that of Baro et al. (2011), who require the followig coditio o the retal duratio. We let R γ a,(y, ξ) deote of the umber of retals of duratio a that begi i period, ad R γ a(y, ξ) deote the umber of retals of duratio a that occur over the etire retal horizo. Similarly, L γ a(y, ξ) represets the umber of retals of duratio a that are lost over the etire retal horizo. We allow W γ a,(y, ξ) ad Z γ a,(y, ξ) to represet the umber of retal uits retured ad lost, respectively, i period after a retal duratio of a periods with Z γ a (y, ξ) := N+A =A+1 Zγ a,(y, ξ). Furthermore, we defie a m,i to deote the realized retal duratio of the ith demad served by the retal uit m. It is importat to ote that the sample path ξ ow cosists of ot oly the demad realizatios d, = 1, 2,..., N, ad the retal uit lifetimes l m, m = 1, 2,..., y, but also the retal duratios a m,i, i 1 ad m = 1, 2,..., y. Also, we use r a for deotig the reward eared with a

15 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 13 retal that has a duratio of a periods. With this otatio, the profit o ay sample path ca be expressed as: Π γ (y, ξ) = N =1 A max a=a mi r a D a, A max a=a mi (r a + c)l γ a(y, ξ) s g y (s b s g )Z γ (y, ξ). Whe retal duratio is radom, the covexity of the umber of lost sales L(y, ξ) i y ad thus, the cocavity of the umber of retals R(y, ξ) i y, might ot hold for every sample path ξ. As a example, we cosider the additio of two retal uits to our ivetory system, where the first additioal uit fulfills oe customer demad with a very log duratio ad the secod additioal uit fulfills several customer demads with short retal duratios. I this case, the umber of additioal customer demads satisfied by oe extra retal uit is ot ecessarily o-icreasig i y. Therefore, we proceed by aalyzig the structural properties of the expected umber of lost sales ad the expected umber of retals. We are the first to cosider this modelig aspect simultaeously with radom loss of retal ivetory i the followig sectio. It is worth otig that the radom retal duratio accouts for each customer s decisio to keep the retal uit for a differet umber of periods, but it ca also iclude the radom service time eeded to repair the retal uit depedig o its coditio upo retur. 4. Retal Ivetory Loss with Geometric Lifetime Distributios This sectio cosiders a model i which each retal uit m has a loss probability of p with each retal. More specifically, the radom variable l m, which deotes the umber of times the uit m {1, 2,..., y} is reted before retirig from the retal ivetory, follows a geometric distributio with a expected value of 1/p. We assume that r + c p(s b s g ). This coditio implies that the beefit, r + c, of covertig a lost sale ito a retal is greater tha or equal to the expected cost, p(s g s b ), of the retal uit loss. Due to the costat failure rate, the recirculatio rule has o effect o E [L γ (y, ξ)], E [R γ (y, ξ)], or E [W γ (y, ξ)] for ay. Therefore, we omit the superscript i the otatio used i this sectio. We establish the cocavity of the expected profit fuctio by first presetig a coditio related to the retal retur process for which the expectatio of the umber of lost sales L(y, ξ) :=

16 14 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss N =1 L (y, ξ) is covex. Correspodigly, the expectatio of the umber of retals R(y, ξ) := N =1 R (y, ξ) is cocave i the iitial ivetory of y retal uits for this coditio. Lemma 1. If the expected umber of retal uits retured E [ t=1 W t(y, ξ)] is cocave ad odecreasig i y for = 1, 2,..., N, the the expected umber of lost sales E[L(y, ξ)] is covex ad o-icreasig while the expected umber of retals E[R(y, ξ)] is cocave ad o-decreasig i y. All proofs appear i the appedix. Propositio 1. Whe retal uit lifetimes are geometrically distributed, the expected profit π(y) is cocave i y for ay retal uit recirculatio rule. 5. Retal Ivetory Loss with Geeral Lifetime Distributios Whe the lifetimes of the retal uits follow a geeral distributio, the umber of retal uits retured i period may deped o the policy used to choose amog available retal uits to satisfy the demad i previous periods. Also, the umber of retals R(y, ξ) ad the umber of lost sales L(y, ξ) might ot ecessarily be cocave ad covex, respectively, i y due to a retal uit that has a particularly log or short lifetime. Therefore, we ivestigate whether it is possible to establish the cocavity of the expected umber of retals E[R(y, ξ)] as well as the covexity of the expected umber of lost sales E[L(y, ξ)] i the iitial ivetory of y retal uits. We modify the assumptio of r + c p(s b s g ) for determiistic retal duratio to be A max a=a mi r a h(a) + c p(s b s g ) for radom retal duratio. Because we have ot yet foud a direct algebraic proof, we compare sample paths via couplig, as described i Chapter 4 of Lidvall (1992). A couplig approach allows us to compare the value of a additioal retal uit i two systems that differ oly i the umber of retal uits. Due to the retal uit lifetime distributios ad the recirculatio rule, aalysis of the chage i the expected umber of retals would otherwise be extremely difficult. Our approach uses the followig steps: 1. Establish demad values d 1, d 2,..., d N, which do ot require ay distributioal assumptios. 2. Operate the system with y retal uits, each of which has a lifetime l m, m = 1,..., y. The ith demad, i 1, served by retal uit m has duratio a m,i.

17 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss Add a additioal retal uit the (y + 1)st uit to the system that has a lifetime l ad serves demads with duratios {a 1, a 2,...}. To be clear, the system has retal uits with lifetimes l 1, l 2,..., l y, l. 4. To the system described i Step 2 (i.e., igorig Step 3), add a (y + 1)st uit that has a lifetime l y+1 ad serves demads with duratios {a y+1,1, a y+1,2,...}. 5. To the system described i Step 4, add a additioal retal uit the (y + 2)d uit so that the system has retal uits with lifetimes l 1, l 2,..., l y, l y+1, l. This additioal retal uit has the same lifetime l ad serves demads with same duratios {a 1, a 2,...} as the additioal uit added to the system i Step 3. For otatioal coveiece, we defie ξ(y) as the sample path cosistig of the demads d 1, d 2,..., d N of all N periods, the retal uit lifetimes l 1, l 2,..., l y, ad the demad duratios {a m,1, a m,2,...} for m = 1, 2,..., y, as well as the lifetime l ad retal duratios {a 1, a 2,...} for a additioal retal uit. For example, ξ(y) ad ξ(y + 1) cotai all of the sample path iformatio ecessary to aalyze the systems described i Steps 3 ad 5, respectively. We cosider two types of decisios for retal uit allocatios. First, we examie a cout-based retal uit state i which the allocatio decisio is based o the umber of times that each uit has bee reted. The, we study a coditio-based retal uit state i which the allocatio decisio is based o the curret state of each retal uit. Each of these models may be relevat for Ret the Ruway. Specifically, the dress s physical coditio may ot be observed requirig a cout-based model if it is ot carefully ispected or if the cause of a dress failure is difficult to observe as the dress s coditio degrades. For example, a zipper may be more likely to fail over time eve if idicatios of impedig failure may ot be observed. O the other had, a dress s physical coditio may be observed if it relates to the coditio of the fabric. Sati dresses are susceptible to developig mior damage to idividual threads due to their loose weaves, ad the repeated iroig of silk taffeta dresses may cause them to lose their ideal appearace aroud pleats ad seams. A coditio-based model would the be more appropriate for this settig.

18 16 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss Retals (Retal Uit Lost) Retals (Retal Uit Lost) Retals (Retal Uit Retured) Lost Sales Retals (Retal Uit Retured) Lost Sales 4 4 Demad Demad Period Period Figure 2 (a) Static Priority Policy (b) Eve Spread Policy Retal uit recirculatio schemes for Example 2 with y = 3. The umber iside a box idetifies the retal uit that satisfies a demad Cout-Based Retal Uit State For the aalysis i this sectio, we let l i deote the probability that a retal uit has a lifetime of i retals; i.e., it is the probability that a retal uit retires from the retal ivetory after its ith retal, or P r(l m = i). Uder the assumptio of determiistic retal duratio i which each retal has a duratio of A periods, we assume that r + c (s b s g )l i for 1 i N/A; i.e., the beefit of a additioal retal to a customer (i.e., r + c) is greater tha or equal to the expected reductio i the salvage value due to retal ivetory loss (i.e., (s b s g )l i, i 1). Similar to the presetatio i Sectio 4 for geometrically distributed lifetimes, this coditio takes the form Amax a=a mi r a h(a) + c (s b s g )l i, i N/A mi, for radom retal duratio so that the expected profit of servig a customer is ot egative. I this sectio, we examie two recirculatio rules: the eve spread policy ad the static priority policy. I the eve spread policy, each demad is served by a available retal uit that has bee reted out the fewest umber of times amog all available retal uits. We ote that the priority is assiged to retal uits i the order of icreasig hazard rate uder the eve spread policy. The static priority policy, o the other had, allocates retal uits accordig to a priority list that does ot chage over the course of the retal horizo.

19 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 17 Example 2. Figure 2 shows how differet retal uit recirculatio schemes ca affect the umber of retals ad lost sales for a example sample path ξ. Usig the same demad values as i Example 1, we ow state the lifetimes of available retal uits as {l 1, l 2,..., l 5 } = {2, 4, 3, 4, 2}. For y = 1 ad y = 2 with l 1 = 2 ad l 2 = 4, both the eve spread ad static priority policies satisfy the same umber of demads; i.e., R(1, ξ) = 2 ad R(2, ξ) = 5. However, whe y = 3, the eve spread recirculatio rule eables oe more retal over the retal horizo tha the static priority rule. Uder the static priority rule, retal uit 1 is lost after servig a demad i period 3, while it is lost after servig a demad i period 5 uder the eve spread rule. This allows oe extra demad to be served i period 5 for the eve spread rule because it has oe more retal uit available tha the static priority rule. Figure 3a shows that the eve spread rule also serves oe more demad tha the static priority rule whe y = 4 ad that both policies serve all te uits of demad whe y 5. Figure 3a also demostrates that the umber of retals is ot ecessarily cocave i y; i.e., the additio of retal uit 1 with lifetime l 1 = 2 satisfies fewer additioal uits of demad tha the additio of retal uit 2 with lifetime l 2 = 4. I Figure 3b, we use the demad values from Example 1 but istead let the lifetime of each retal uit be a discrete uiform radom variable betwee 2 ad 4 (i.e., l 2 = l 3 = l 4 = 1/3), ad estimate the expected umber of retals with a simulatio executed for a sufficietly large umber of replicatios so that the stadard error of the experimet is egligible. Eve though cocavity is violated o idividual sample paths, the expected umber of retals is revealed to be a cocave fuctio of the umber of retal uits. The eve spread ad static priority policies result i the same umber of retals regardless of the sample path for y 2 ad y 5. However, the expected umber of retals for the eve spread policy exceeds that of the static priority policy by 0.33 whe y = 3 ad by 0.26 whe y = 4. Whe y = 3, the eve spread policy results i at least oe more retal tha the static priority policy o 44.1% of the sample paths, ad the static priority policy exceeds the eve spread policy o 10.9% of all sample paths.

20 18 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss Eve Spread Static Priority Eve Spread Static Priority Expected Number of Retals Number of Retal Uits Expected Number of Retals Number of Retal Uits (a) Example sample path. (b) Expected umber of retals. Figure 3 Effect of the umber of retal uits ad retal uit recirculatio rule for Example The Static Priority Recirculatio Rule The static priority recirculatio rule, deoted by the superscript SP, selects the uits to be reted accordig to a priority list, which is the same for every time period. That is, whe a retal uit is eeded to satisfy demad, the oe with the highest priority amog the set of available retal uits is chose. Therefore, whe the retal duratio is determiistic, the static priority recirculatio rule is the same as the policy selectig the retal uit that has bee reted the most. Propositio 2. I a retal system with radom retal duratio ad geeral retal uit lifetime distributios, π(y) is cocave ad o-decreasig i y for the static priority recirculatio rule The Eve Spread Recirculatio Rule We ow cosider the eve spread recirculatio policy deoted by the superscript ES which satisfies a demad with the retal uit that has bee reted the fewest umber of times. Defiig R γ,m(y, ξ) as the umber of times that retal uit m is reted i period uder some policy γ, a eve spread compliat policy selects a retal uit m to satisfy a demad based o available retal uits that miimize 1 t=1 Rγ t,m(y, ξ). Whe the loss probability for retal uits is o-decreasig i the umber of times reted, recirculatio priority correspods to a hazard rate orderig. We assume that ties are broke by some static priority list for allocatig retal uits. We first ivestigate the relatio betwee the expected umber of retals ad the iitial retal ivetory level (Propositio 3) ad discuss the cocavity of the

21 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 19 expected profit fuctio i the iitial retal ivetory (Propositio 4). We the demostrate the optimality of the eve spread policy to maximize the expected profit whe the loss probability of each retal uit icreases with the umber of times that the uit has bee reted (Propositio 5). Propositio 3. For a retal system with radom retal duratio that follows the eve spread recirculatio rule, the expected umber of retals E[R ES (y, ξ)] is cocave ad o-decreasig i the iitial ivetory of y uits. For the special case of s g = s b, the expected profit ca be expressed i terms of the liear procuremet cost ad the expected umber of retals. Therefore, Propositio 3 also implies the cocavity of the expected profit fuctio π ES (y) i y whe s g = s b. However, whe s g s b, we must show that the expected profit fuctio π ES (y) remais cocave i the iitial ivetory of y retal uits. Propositio 4. Whe the retal duratio is radom ad the loss probability of each retal uit icreases with the umber of times that the uit has bee reted, the expected profit π ES (y) is cocave i the iitial ivetory of y retal uits uder the eve spread recirculatio rule. Whe the retal uit loss probability is icreasig i the umber of times that the uit has bee reted, we idetify the eve spread policy as the optimal retal uit recirculatio rule to maximize the expected profit. Our key argumet is a pairwise iterchage argumet i which switchig a allocatio that violates the eve spread policy to coform to the eve spread policy icreases the expected umber of retals. We require additioal otatio to compare sample paths ad make this argumet, which we describe alog with a overview of the steps of the proof: 1. Fid the first allocatio decisio over the retal horizo that violates the eve spread policy. We deote this existig policy with the superscript V for violatig. Assume that this violatig decisio occurs i some period. Specifically, a retal uit j is allocated to demad whe some other retal uit i is available ad 1 t=1 RV t,j(y, ξ) > 1 t=1 RV t,i(y, ξ). The availability of retal uits i ad j implies that 1 t=1 RV t,j(y, ξ) < l j ad 1 t=1 RV t,i(y, ξ) < l i.

22 20 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 2. Cosider a switched forward allocatio path of uits i ad j i periods, + 1,..., N so that retal uit i is allocated istead of retal uit j. We refer to this allocatio with the superscript S for switched. 3. Chage values i ξ related to the lifetimes ad retal duratios after period for uits i ad j with two ew partial sample path vectors ξ (1) ad ξ (2). Specifically, we geerate two sets of radom duratios (a (1),1, a (1),2,...) ad (a (2),1, a (2),2,...) for demads after period served by two differet retal uits ad iverse probability mass fuctio values η (1) ad η (2) for the coditioal lifetime distributios of the two retal uits. The latter iformatio allows the determiatio of the lifetimes l (1) ad l (2). 4. Calculate the umber of retals over the etire horizo uder four scearios (with correspodig otatio for the total umber of retals used for coveiece): (1) R V (ξ (1), ξ (2) ) for the violatig allocatio with ξ (1) applied to retal uit i ad ξ (2) to retal uit j; (2) R V (ξ (2), ξ (1) ) for the violatig allocatio with ξ (2) applied to uit i ad ξ (1) to uit j; (3) R S (ξ (1), ξ (2) ) for the switched allocatio with ξ (1) applied to uit i ad ξ (2) to uit j; ad (4) R S (ξ (2), ξ (1) ) for the switched allocatio with ξ (2) applied to uit i ad ξ (1) to uit j. 5. Compare scearios to observe that E [R S (y)] E [R V (y)], which implies that E [Π S (y)] E [Π V (y)] uder certai assumptios o the cost parameters. 6. Go to Step 1 ad repeat util the switched allocatio is equivalet to the eve spread allocatio. Propositio 5. If the loss probability of each retal uit icreases with the umber of times that the uit has bee reted ad the retal duratio is radom, the eve spread recirculatio rule maximizes the expected profit Coditio-Based Retal Uit State We ow study a differet model of retal uits i which each retal uit has a kow state i the set {1, 2,..., S} that may chage after each time that the uit is reted. O a sample path ξ, we defie s mi as the state of retal uit m after it is reted for the ith time, m {1, 2,..., y}

23 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss 21 ad i {1, 2,..., l m }. The iitial state of each retal uit is defied as s m0 = 1 ad a retal uit s retiremet from recirculatio correspods to s m,lm = S. A trasitio probability matrix P govers the evolutio of each retal uit s state upo each istace i which the uit is reted. We defie P (i, j) as the probability that a retal uit trasitios from state i to state j after each retal with i, j {1, 2,..., S}. We also assume that A max a=a mi r a h(a) + c (s b s g )P (i, S) for i = 1, 2,..., S 1 so that the expected value of offerig a retal is ever egative. Oe simple recirculatio policy based o the observed retal uit state is to allocate the retal uits i icreasig order of their state. I other words, the retal uit that is i the best coditio is give the highest allocatio priority. We label this policy as the best-first policy. Similarly, the worst-first policy gives the highest priority to the retal uit i the worst coditio for which it ca still be reted out. For either policy, we show that the expected umber of retals is cocave i the iitial ivetory level. Propositio 6. The expected profit π(y) is cocave i the iitial ivetory of y retal uits uder either the best-first or worst-first policy with radom retal duratio. We ext cosider the optimal retal uit recirculatio policy whe retal uit selectio decisios are based o the retal uit coditio. We assume that the trasitio matrix is totally positive of order 2; i.e., that P (i, j)p (i, j ) P (i, j )P (i, j) for all i < i, j < j. Brow ad Chagaty (1983) show that this property implies that the first passage time from state 1 to some state C j = {i : i > j} has a icreasig failure rate for j = 1,..., S 1. Propositio 7. If the trasitio matrix P is totally positive of order 2, the the best-first policy maximizes the expected profit for a system with radom retal duratio. 6. A Idustrial Numerical Aalysis: Ret the Ruway Motivated by the high-fashio dress retal busiess Ret the Ruway, we itroduce the model parameters represetig a retal system with usage-based loss of ivetory i Sectio 6.1. We discuss the impact of the retal ivetory loss o the optimal procuremet decisio i Sectio 6.2

24 22 Slaugh, Biller, ad Tayur: Maagig Retals with Usage-Based Loss ad the effect of the retal uit recirculatio rule o retal ivetory maagemet i Sectio 6.3. All umerical testig is performed via sample average approximatio, as described i Kleywegt et al. (2002) ad Shapiro (2003) Retal Model Parameters The product we cosider is a middle-tier dress as described i Eisema ad Wiig (2012); i.e., a full-price retal provides a et reveue of $59, which is the differece betwee $90 i reveue ad $31 i costs of cleaig, shippig, packagig ad credit card processig. However, customers are allowed to ret a secod style for $25 ad a secod size for free; thus, a uit may ot achieve $59 i et reveue every time it is reted. We assume that these three scearios for a retal retig as the primary dress with et reveue of $59, retig as the secodary dress with et reveue of $20, ad retig as the free secod size with et cost of $5 occur with probabilities 50%, 20%, ad 30%, resultig i a expected et reveue of r = $32 per retal. Eisema ad Wiig (2012) report that Ret the Ruway purchases a middle-tier dress with a retail price of $750 for $226. We assume a aual uit holdig cost that is equal to 20% of the purchase price of the dress to accout for the cost of storage ad the cost of capital. At the ed of a fashio seaso, dresses i a variety of coditios are sold i New York City at what is kow as a sample sale. Based o websites such as Yaetta (2013) that report o these sales, we let a dress i good coditio sell for 80% 85% off of the $750 retail price ad a dress i bad coditio (i.e., a dress that retires from the retal ivetory) to sell for 95% off of the retail price. Adjustig these sample sale prices for stagig ad trasactio costs, we assume a dress that does ot retire from the retal ivetory by the ed of the seaso to have a salvage value of $100 ad a dress that retires from the retal ivetory to have a salvage value of $30. We also calculate procuremet costs separately for these two types of dresses by combiig their purchase prices, holdig costs ad salvage values. For a 26-week horizo, the cost of procurig a dress is s g := $149, which cosists of a purchase price of $226, a holdig cost of $23 ad a salvage value of $100. A dress retirig from the retal ivetory icurs a additioal pealty of $70, resultig i a procuremet cost of