Robust Stochastic Lot-Sizing by Means of Histograms

Transcription

1 Robust Stochastc Lot-Szng by Means of Hstograms Abstract Tradtonal approaches n nventory control frst estmate the demand dstrbuton among a predefned famly of dstrbutons based on data fttng of hstorcal demand observatons, and then optmze the nventory control usng the estmated dstrbutons. These approaches often lead to fragle solutons whenever the preselected famly of dstrbutons was nadequate. In ths work we propose a mnmax robust model that ntegrates data fttng and nventory optmzaton for the sngle-tem mult-perod perodc revew stochastc lot-szng problem. In contrast wth the standard assumpton of gven dstrbutons, we assume that hstograms are part of the nput. The robust model generalzes the Bayesan model, and t can be nterpreted as mnmzng hstory dependent rsk measures. We prove that the optmal nventory control polces of the robust model share the same structure as the tradtonal stochastc dynamc programmng counterpart. In partcular, we analyze the robust model based on the ch-square goodness-of-ft test. If demand samples are obtaned from a known dstrbuton, the robust model converges to the stochastc model wth true dstrbuton under generous condtons. Its effectveness s also valdated by numercal experments. 1. Introducton The stochastc lot-szng model has been extensvely studed n the nventory lterature. Most of the research has focused on models wth complete nformaton about the dstrbuton of customer demand. However, n most real-world stuatons, the demand dstrbuton s not known; only hstorcal data s avalable. A common approach s to hypothesze a famly of demand dstrbutons and then to estmate the parameters specfyng the dstrbuton usng the hstorcal data. Once the probablty dstrbuton has been dentfed, the nventory problem s solved followng ths estmated dstrbuton. Ths mples that the nventory polcy s determned under the assumpton that the ftted dstrbuton adequately characterzes the demand to be realzed n the future. The estmated demand dstrbuton may not be accurate and hence the approach of fttng the dstrbuton and optmzng the nventory decsons sequentally may not work as expected. As shown n Lyanage and Shanthkumar (2005 for the newsvendor model, such an approach may generate suboptmal solutons. Besdes, n dstrbuton fttng, one needs to assume a parametrc famly of a demand dstrbuton n the frst place, and ths hypothess may also go awry. For nstance, we may ft the hstorcal data to a lognormal dstrbuton whle t actually follows a unform dstrbuton. The robust nventory models, wthout assumng a parametrc famly of dstrbutons, provde an approach to address ambguty n the demand dstrbuton. A bref revew of these robust models s provded n Secton 1.1. These models adopt a mnmax approach targetng to mnmze the worst case expected cost maxmzed over the set of dstrbutons. Wthout excepton, the exstng lterature ether consders a pre-specfed set for demand dstrbutons wthout detaled dscussons 1

2 about how to generate the set, e.g., Notzon (1970, or derves the set of dstrbutons based on certan statstcs of the hstorcal data such as the sample mean and varance, e.g., Bertsmas and Thele (2006 and See and Sm (2010. Comparng wth the classcal approach wth separate fttng and optmzaton, the robust models based on hstorcal statstcs may mss mportant nformaton about the demand dstrbuton conveyed n the hstorcal data set, e.g., the shape of the dstrbuton, whch, n the separate, two-phase approach, s usually used to determne the parametrc famly of the dstrbutons. In ths paper, we merge the merts of both approaches, namely, ( to fully utlze hstorcal data as n the classcal approach and ( to concurrently optmze the demand dstrbuton and the nventory decson wthout assumng a dstrbuton famly as n the robust models. We analyze the sngle-tem stochastc fnte-horzon perodc revew lot-szng model, under the assumpton that the demand s subject to an unknown dstrbuton and only hstorcal demand observatons (gven by hstograms are avalable. As all practtoners n nventory control start wth hstograms and then ft an underlyng demand dstrbuton, ths assumpton reflects the practcal value of ths research. By adoptng the mnmax robust optmzaton approach, rather than frst estmatng the demand dstrbuton and then optmzng nventory decsons, we combne these two steps to mnmze the worst case expected cost over a set of demand dstrbutons, whch s defned as all possble dstrbutons satsfyng the ch-square goodness-of-ft test. The advantage of ths approach s twofold. Frst, the hstorcal data s used n the same manner as n the goodness-of-ft test, thus we use all the nformaton conveyed by the hstorcal data that can be utlzed by the goodness-of-ft test n dstrbuton fttng. Second, we avod the assumpton about the parametrc famly of dstrbutons, whch s a must n dstrbuton fttng. We show that the (s, S polcy remans optmal, dscuss the behavor of the model as the number of samples ncreases, and demonstrate through a numercal study that ths model outperforms ( the classcal approach where dstrbuton estmaton and nventory optmzaton are separate and ( a robust model where the set of dstrbutons s defned by sample mean and varance. Our two man contrbutons are as follows. Frst, we develop a robust mnmax model that only requres hstorcal data, and allows correlated demand. Note that most mnmax models (see, e.g., Notzon 1970 and Ahmed et al as well as Bayesan nventory models (e.g., updatng demand dstrbutons as suggested n Iglehart (1964 n the lterature can be nterpreted as specal cases of our framework. Unlke the classcal nventory model, whch solves a sngle-varate optmzaton problem n each perod, the robust model needs to dentfy the orderng quantty and probablty dstrbuton represented by a vector of decson varables smultaneously. Despte ths complexty, the optmal polcy of the robust model stll shares the same structure as the correspondng polcy n the classcal stochastc lot-szng model. In partcular, the optmal polcy s a state-dependent base-stock polcy for the mult-perod nventory problem wthout fxed procurement costs, and a state-dependent (s, S polcy f the fxed procurement cost s consdered. Whle the frst contrbuton manly serves as an extenson to exstng models, the second major contrbuton regards combnng the statstcal test n dstrbuton fttng wthn a sngle nventory control model. We consder a specal case of the general robust framework when the set of demand 2

3 dstrbutons s drectly related to the ch-square goodness-of-ft test. Such a dstrbuton set can be defned by a set of second-order cone constrants and hence t s tractable to compute the (s, S levels for each perod. To the best of our knowledge, ths s the frst endeavor to ntegrate the goodness-of-ft statstcal test wth nventory optmzaton and to explctly consder the shape of the dstrbuton n a robust framework. We also prove that the robust model based on the ch-square test converges to the stochastc model wth true demand dstrbuton under generous condtons f samples are drawn from ths dstrbuton and they grow ndefntely. In partcular, f the demand dstrbutons are dscrete, the robust model converges to the stochastc model wth the true demand dstrbuton as the number of ndependent samples drawn from the true dstrbuton for each perod tends to nfnty. Moreover, the rate of convergence s n the order of 1/ k, where k s the number of samples. Slghtly weaker results are obtaned for contnuous dstrbutons. These convergence results ensure the effectveness of the robust approach when the sample sze s suffcently large. When the sample sze s relatvely small, the performance of the robust model s llustrated by means of computatonal experments. We argue that the robust model based on the ch-square test outperforms the tradtonal approach, whch optmzes the nventory decsons by usng ftted dstrbutons, as well as the mnmax robust model where the set of dstrbutons s based on the set proposed by Delage and Ye (2010. We also provde nsghts on the performance of the robust model wth dfferent parameters and sample szes. In Secton 2 we descrbe our robust model, whch ncorporates hstorcal data, and present the optmalty equaton n a compact form. The structure of the optmal polces s characterzed n Secton 3. Secton 4 consders a specal case wth robustness defned by the ch-square goodnessof-ft test. We also dscuss selected convergence results for the ch-square test based models n the same secton. The computatonal results are presented n Secton 5. Fnally, addtonal extensons are presented n Secton 6. We conclude the ntroducton wth the lterature revew. 1.1 Lterature Revew Ths work s bult upon two streams of lterature: stochastc nventory control and robust optmzaton. The dscrete-tme stochastc nventory model has been studes snce 1950s. Scarf (1960 proposes the concept of K-convexty and proves that the (s, S polcy s optmal n the presence of a fxed orderng cost. Snce then, the research n ths area has flourshed. We refer the reader to Zpkn (2000 for a detaled revew. The concept of K-convexty has been generalzed to attack varous problems related to nventory control, e.g., Chen and Smch-Lev (2004. Effcent algorthms, e.g., Guan and Mller (2008 and Halman et al. (2009, have also been proposed to solve other more general stochastc nventory problems. Robust optmzaton was poneered by Soyster (1973, whch proposes robust lnear programmng formulatons for lnear programs wth coeffcent uncertanty. Ths lne of research has enjoyed popularty n recent years. Some of the mportant works nclude but are not lmted to Ben-Tal and Nemrovsk (2000 and Bertsmas and Sm (2004 for robust lnear programmng, Ben-Tal and Nemrovsk (1998 for robust convex optmzaton, and Kouvels and Yu (1997 and Bertsmas and Sm (2003 for robust dscrete optmzaton. More relevant to ths research, Iyengar (2005 and 3

4 Nlm and El Ghaou (2005 develop a robust optmzaton framework for dynamc programmng models, extend the Bellman recurson to the robust counterpart, and nvestgate ts computatonal complexty. Delage and Ye (2010 propose a data-drven robust framework for any sngle-stage optmzaton problem, whch mnmzes the maxmum expectaton over a set of dstrbutons defned by the sample mean and varance. They dentfy suffcent condtons under whch the correspondng robust problem s polynomally solvable and provde probablstc arguments for usng ths model by consderng a confdence regon for the mean and varance as a random vector. In ths paper, we apply the dea of robust optmzaton to nventory control models. Ths noton of robust nventory control s not new n the lterature. The earlest work n mnmax nventory control s attrbuted to Scarf (1958, where mnmzaton of the maxmum expected cost of the newsvendor model over all dstrbutons wth a gven mean and varance s consdered. Gallego and Moon (1993 present another proof of Scarf s result and consder varous extensons of the model. The recent work by Natarajan et al. (2008 extends the result of Scarf (1958 by consderng the set of dstrbutons wth a gven mean, varance and semvarance nformaton. Peraks and Roels (2008 mnmze the maxmum regret of the newsvendor model over a convex set of dstrbutons wth certan moments and shape. Notzon (1970 s among the earlest works that consders a mnmax multple-perod nventory model. The demand n each perod s assumed to be ndependent and ts dstrbuton functon s ambguous but wthn a specfed class of dstrbuton functons. The mnmax control polcy mnmzes the maxmum expected cost. The optmalty of the (s, S polcy s proved. Bertsmas and Thele (2006 analyze dstrbuton-free nventory problems, n whch demand n each perod s assumed to be a random varable that takes values n a gven range. The demand s assumed to be a random varable controlled only by two values: the lower and upper estmators. To capture the trade-off between robustness and optmalty, a parameter s defned to control the budgets of uncertanty at every tme perod. They show that for a varety of problems, the structures of the optmal polcy reman the same as n the assocated model wth complete nformaton about the dstrbuton of customer demand. A related model from the base-stock perspectve s analyzed n Benstock and Özbay (2008. See and Sm (2010 consder a factor-based demand model wth gven mean, support, and devaton measures. To obtan tractable replenshment polces, the worst case expected cost among all dstrbutons satsfyng the demand model s mnmzed by solvng a second order cone optmzaton problem. Ahmed et al. (2007 propose an nventory control model whch mnmzes a coherent rsk measure nstead of the overall cost functon. They show that rsk averson treated n the form of coherence rsk measures s equvalent to the mnmax formulatons, and t s proved that the optmal polces conserve the propertes of the stochastc dynamc programmng counterparts. They do not consder demand dependent evolutons. Lyanage and Shanthkumar (2005 frst provde concrete examples n a sngle perod (newsvendor settng, whch llustrate that separatng dstrbuton estmaton and nventory optmzaton, as done n the classcal approach, may lead to suboptmal solutons. They propose the use of operatonal statstcs where t s assumed that the demand dstrbuton functon belongs to a spe- 4

5 cfc (predetermned famly and estmate the (sngle parameter of the famly wthn an nventory optmzaton model. In addton, selected recent papers also consder lost-sale nventory problems wth censored demand data,.e., the observed hstorcal demand data excludes the lost-sale nformaton as the lost sales are not observable. Huh and Rusmevchentong (2009 propose nonparametrc adaptve polces to solve ths problem and provde a bound for the asymptotc performance, whch nterestngly s the same as the converenge rate of our model under dscrete dstrbutons. The models by Notzon (1970 and Ahmed et al. (2007 do not take the hstorcal data nto account, and they predefne the class of dstrbuton functons. The robust optmzaton approaches from Bertsmas and Thele (2006 as well as See and Sm (2010 do not use any hstorcal data except to determne the support, expectaton and devaton measures. On the other hand, Lyanage and Shanthkumar (2005 use hstorcal data but predetermne the famly of dstrbutons. In fact, they consder only dstrbutons characterzed by a sngle unknown parameter. Ths s the only work besdes the one proposed n ths paper that concurrently optmzes the orderng quantty and apples technques n dstrbuton fttng to determne the demand dstrbuton. Our research combnes both strateges by ntegratng dstrbuton fttng wth robust optmzaton. Specfcally, we consder the set of demand dstrbutons that satsfy a certan data fttng crteron wth respect to hstorcal data and characterze an optmal polcy that mnmzes the maxmum expected cost. 2. Formulaton of Robust Stochastc Lot-Szng The classcal mult-perod nventory problem consders a fnte plannng horzon of T perods. We assume that all shortages are backlogged. For each perod t = 1,..., T, let D t be a random varable representng demand n that perod. The sequence of events s as follows. At the begnnng of each perod t, the decson maker revews the net nventory level x t, and places an order for q t (possbly zero unts. The procurement cost n each perod t = 1,..., T 1 ncludes two components: a fxed procurement cost K f q t > 0, and a unt procurement cost c t for each unt ordered. Assumng zero lead tme, ths order arrves mmedately and ncreases the nventory level up to y t, where y t = x t + q t. After observng demand D t, nventory holdng cost s charged at a rate of h t for any unt of excess nventory after satsfyng demand D t, and a unt backorder cost b t s ncurred for any unt of unsatsfed demand. The net nventory at the begnnng of perod t + 1 s reduced to x t+1 = y t D t. Thus, the total cost for perod t gven the net nventory levels before and after orderng (x t and y t respectvely as well as demand D t n that perod s C t ( x t, y t, D t = KI(y t x t + c t (y t x t + h t ( y t D t + + bt (y t D t t = 1,..., T, (1 where x + = max(x, 0, x = max( x, 0, I(x = 1 f x > 0 and I(x = 0 otherwse. In the standard dynamc programmng formulaton, we consder Ṽt(x t, t = 1,..., T, whch denotes the optmal expected cost over horzon [t, T ], gven that the net nventory level at the 5

6 begnnng of perod t s x t and an optmal polcy s adopted over horzon [t, T ]. We assume Ṽ T +1 (x T +1 = 0. Let θ [0, 1] be the dscount rate. The optmalty equaton reads { Ṽ t (x t = mn E [C t (x t, y t, D ] t + θe [Ṽt+1 (y t D ]} t t = 1,..., T. (2 y t x t Note that the dstrbuton of D t, t = 1,..., T s requred to solve ths dynamc programmng formulaton. In practce, the demand dstrbuton s not known. Rather, an nventory manager has at her dsposal only hstorcal data. Dependng on the realzed past demand n the plannng horzon, the manager may choose dfferent aggregatons of hstorcal data to forecast the demand dstrbuton. For example, the demand data of the last n observatons are consdered, whch s analogous to the movng average forecast, or the realzed demand n perods 1 to t 1 s accounted for when forecastng the demand n perod t. Hstorcal observatons are often aggregated to a hstogram wth respect to unknown dstrbuton D t. The bns are [D t,, D t,+1, whch denotes the th possble range of the demand n perod t (all observatons wthn a gven range are ndstngushable. Let the vector d t = [d 1,..., d t 1 ] denote the realzed demand n perods 1 to t 1, where d τ, τ = 1,..., t 1 corresponds to the realzed demand n perod τ. The number of observatons fallng wthn the th bn s a functon of the realzed demand d t and s denoted by N t, (d t. Fnally, we defne n t (d t = N t,(d t, whch corresponds to the total number of avalable observatons under realzed demand d t. In practce, the decson maker observes only these hstograms,.e., the hstorcal samples. We assume that D t,1 = 0 and D t,mt+1 = +, ( where M t corresponds to the number of bns n the hstogram for tme perod t. Let P t, = P Dt [D t,, D t,+1 be the probablty that demand n perod t falls n the nterval [D t,, D t,+1 under the ftted dstrbuton. Clearly, n t (d t P t, s the expected number of observatons that fall n ths nterval accordng to the ftted dstrbuton. The classcal approach to dentfy the best dstrbuton representng the observed data s to use a goodness-of-ft test. The objectve s to ft a dstrbuton that closely follows the observed data. Under ths crteron, there should be a set of dstrbutons dependng on d t, whch satsfy the gven goodness-of-ft test. We denote ths set by P t (d t. Throughout ths paper, we assume that P t (d t s compact for any t and d t. As defned n the dynamc programmng feld, a decson rule µ t at tme t s a functon of net nventory x t, whch decdes the orderng quantty at tme t gven x t,.e., y t = µ t (x t. We formally state our problem n the context of a two-player game, whch s also presented n Iyengar (2005. The frst player chooses the decson rule µ t at tme t and pays the cost. The second player chooses a dstrbuton of D t n P t (d t after observng the order quantty, and receves a reward equal to the cost pad by the frst player. Therefore, the second player may select a dfferent dstrbuton for dfferent x t and µ t. Let P t (x t, µ t (x t denote the dstrbuton chosen by player 2 at tme t gven 6

7 net nventory x t and decson rule µ t. The the set of all dstrbutons avalable to player two s Q µt = {P(x t, µ t (x t P t (d t over all x t, d t }. In Q µt we merely express that for each x t, µ t, d t, we mght have a dfferent dstrbuton. Moreover, a polcy π s defned as the decson rule to be used at every perod,.e., π = (µ 1,..., µ T. A polcy π also yelds a set of dstrbutons Q π whch can be used by the second player or adversary, where Q π = Q µ 1 Q µ 2 Q µ T. (3 As the second player wll maxmze her reward, gven polcy π, net nventory x t, and realzed demand d t, the cost pad by player one from perod t to T s [ T ] ( Vt π (x t, d t = max Q Q EQ, D θ τ t C τ x τ, µ τ (x τ, D τ + θ T +1 t V T +1 (x T +1, d T +1, π τ=t where C τ ( x τ, µ τ (x τ, D τ denotes the cost ncurred n perod τ n (1, and V T +1 (x T +1, d T +1 s the termnal cost. Also note that Q defnes the dstrbutons D τ, τ = t,..., T. Unless stated otherwse, we assume that V T +1 ( = 0. We also have x τ+1 = µ τ (x τ D τ and d τ+1 = [d τ, D τ ]. Snce the frst player wll choose a polcy that mnmzes the cost, the optmal cost from perod t to T gven net nventory x t at tme t, and the realzed demand d t from perod 1 to t 1, s [ T ] ( V t (x t, d t = mn max π Q Q EQ, D θ τ t C τ x τ, µ τ (x τ, D τ + θ T +1 t V T +1 (x T +1, d T +1, (4 π for t = 1,..., T. τ=t Note that the model mnmzes the maxmum expected cost arsng from any dstrbuton n the set P t (d t for any t, whch s known as the mnmax robust approach. We next state an optmalty equaton, whch s essental to establsh the optmal control polces. Proposton 2.1. The optmalty equaton of the robust model s { V t (x t, d t = mn max P t, (C t (x t, y t, D t, + θv t+1 (y t D t,, [d t, D t, ] } (5 y t x t P t P t(d t for t = 1,..., T, where P t (d t s the set of dstrbutons satsfyng the goodness-of-ft condton at perod t, and C t (x t, y t, D t, s defned by (1. Proof. It follows from Theorem 2.1 n Iyengar (2005 when P t (d t s arbtrary. If P t (d t s convex, the proposton can also be proved by the Von Neumann s mnmax theorem (see, e.g., Von Neumann An mmedate observaton from Proposton 2.1 s that we mnmze the worst case expected cost over a set of dstrbutons. Therefore, our robust stochastc model may not be as conservatve 7

8 as the classcal mnmax models, where the worst case s defned by the realzed demand nstead of dstrbuton, e.g., the mnmax model dscussed n Secton 2.4 of Notzon (1970. Note that the Bayesan nventory models assume a pror demand dstrbuton, and the posteror dstrbuton at tme t s obtaned by updatng the pror dstrbuton usng d t, e.g., Iglehart (1964 updates the demand dstrbuton belongng to the exponental and range famles after observng realzed demand nformaton. Our model only requres the set of dstrbutons P t (d t to be a functon of the realzed demand d t. Therefore, we can defne t as a sngleton updated by a Bayesan rule. In ths case, the robust mnmax model s reduced to a Bayesan nventory model, whch ndcates that the Bayesan models are specal cases of our mnmax model. Proposton 2.1 also gves us an nterpretaton of the robust model from a rsk measure perspectve when set P t (d t s convex. Ahmed et al. (2007 establsh the correspondence between coherent rsk measures and mnmax models over convex sets of dstrbutons. From ths perspectve, our mnmax robust model essentally mnmzes a coherent rsk measure wth respect to the total cost. If we consder P t (d t P t for any d t and t,.e., the set of dstrbutons s ndependent of any realzed demand n prevous perods, then the mnmax robust model (5 mnmzes a coherent rsk measure n any perod t and t reduces to the model consdered n Ahmed et al. (2007. When the set of dstrbutons P t (d t depends on demand realzaton d t, model (5 also mnmzes a coherent rsk measure n every perod t. However, ths model s dfferent from that n Ahmed et al. (2007 n the sense that the rsk measure n perod t s updated by the realzed demand n prevous perods. Intutvely, f the decson maker lost a sgnfcant amount n the prevous perod, he or she would tend to be more rsk-averse n subsequent perods. Therefore, t s reasonable to adjust the rsk measure based on the realzed demand nformaton d t. In addton, let constant p t denote the sellng prce of the product n perod t. We can maxmze the expected total proft from perods 1 to T by subtractng term p t D t,p t, n the objectve functon of (5. All of the results, such as the optmal polcy and the convergence propertes, stll hold for such an objectve functon. In addton, f we suppose that all the dstrbutons n set P t (d t could have the same expectaton ˆD t (d t,.e., constrant D t,p t, = ˆD t (d t s ncluded n the defnton of P t (d t, then the models that mnmze cost and that maxmze proft are equvalent to each other. However, as long as the demands follow certan dstrbutons, whch are not necessarly known to the decson maker, the expected total revenue s ndependent of any nventory decson,.e., the order quantty n any perod t. Therefore, t s suffcent to consder the cost mnmzaton model presented n (5. 3. Propertes of Optmal Polces In ths secton we study optmal polces of the general robust stochastc model (5. Notzon (1970 and Ahmed et al. (2007 show the optmalty of (s, S polcy when the set of dstrbutons n the mnmax model s ndependent of the realzed demand d t (Ahmed et al also assume the set of dstrbutons s convex. Here we extend the optmalty of (s, S polcy to the more general model n (5. 8

9 We assume that the reader s famlar wth standard concepts n nventory theory such as K-convexty and (s, S polces (see, e.g., Zpkn 2000 and Porteus Let us defne U (y, d = h t (y t D t, + + b t (y t D t, + θv t+1 (y t D t,, [d t, D t, ], (6 whch corresponds to the expected cost ncurred from perod t to T f the nventory level after recevng the order n perod t s y t and the demand n perod t s D t,. Consder the functon f(y, d = max P P(d U (y, dp. 1 Snce optmalty of the (s, S polcy follows drectly from K-convexty, frst we are gong to establsh that the functon f(y, d s K-convex n y. Lemma 3.1. If U (y, d s K-convex n y for any gven d, then f(y, d s a K-convex functon n y for any gven d. Proof. Please refer to the Onlne Supplement. Lemma 3.1 shows that K-convexty s preserved under maxmzaton over a set of dstrbutons. Base on ths property, we show the K-convexty of the cost-to-go functons. Proposton 3.1. If V t+1 (x t+1, d t+1 s a K-convex functon n x t+1 for any fxed d t+1, the costto-go functon V t (x t, d t s a K-convex functon n x t for any fxed d t, and for any t = 1,..., T. Proof. The proposton s trvally true for t = T +1. Suppose that the proposton holds for perod t + 1, and consder perod t. To smplfy the notaton, let us defne f t (y t, d t = c t y t + max P t P t(d t Therefore, the optmalty equaton n (5 s equal to [ P t, ht (y t D t, + + b t (y t D t, + θv t+1 (y t D t,, [d t, D t, ] ]. V t (x t, d t = c t x t + mn y t x t {KI(y t x t + f t (y t, d t }. Accordng to Lemma 3.1, f V t+1 (x t+1, d t+1 s K-convex n x t, f t (y t, d t s K-convex n y t. Let S t (d t be a global mnmzer of f t (y t, d t for any gven d t. Moreover, let s t (d t be the smallest element of the set {s t (d t s t (d t S t (d t, f t (s t, d t = f t (S t, d t +K}. Accordng to the propertes of K-convex functons (see, e.g., Zpkn 2000 and Porteus 2002, we have { K ct x V t (x t, d t = t + f t (S t (d t, d t f x t s t (d t, c t x t + f t (x t, d t otherwse. K-convexty of V t (x t, d t follows from K-convexty of f t (y t, d t. 1 Note that here we drop subscrpt t n order to smplfy the notaton. (7 9

10 From the structure of V t+1 (, we can derve an optmal polcy. Theorem 3.1. A state dependent (s, S polcy s optmal for the robust stochastc model. More precsely, for any t and d t, there exsts S t (d t and s t (d t such that S t (d t x t unts are ordered n perod t f x t s t (d t and no order s placed otherwse. Proof. The structure of the polcy follows drectly from the proof of Proposton 3.1 and general theory of K-convexty (see, e.g., Zpkn 2000 and Porteus If there s no fxed cost, then V t (x t, d t s convex n x t for any t. Therefore, a state dependent base-stock polcy s optmal, and the base-stock level gven the realzed demand d t s S t (d t. A drawback from the practcal pont of vew s the fact that s t and S t depend on d t. We next characterze a specal case where dfferent values of d t correspond to the same (s, S levels. Suppose that d t and d t denote two dfferent demand realzatons from perod 1 to t 1. Let us assume that f demand realzatons n perods 1 to t 1 are d t or d t, then the same demand realzaton n perod t to T generates the same hstogram n any perod t,..., T. Then vectors d t and d t correspond to the same (s, S levels. To formalze ths property, let s t (d t and S t (d t (respectvely s t (d t and S t (d t denote the (s, S levels correspondng to hstory d t (respectvely d t. For any τ t, let the vector [d t, d t, d t+1,..., d τ 1 ] denote the realzed demand up to perod τ 1 where the demands from perods 1 to t 1 are aggregated n vector d t, and the realzed demand n perods t to τ 1 s labeled by d t, d t+1,..., d τ 1, respectvely. Proposton 3.2. Let V T +1 (x T +1, d T +1 = V T +1 (x T +1, d T +1 for any x T +1, d T +1, d T +1, and consder any τ = t,..., T. Suppose that realzatons d t and d t gve the same number of samples n nterval [D τ,, D τ,+1 for any as long as the realzed demand n perods t to τ 1 s the same,.e., N τ, ([d t, d t, d t+1,..., d τ 1 ] = N τ, ([d t, d t, d t+1,..., d τ 1 ] for any and any realzaton [ d t, d t+1,..., d τ 1 ] of [ D t, D t+1,..., D τ 1 ]. Then we have s t (d t = s t (d t, S t (d t = S t (d t, and V t (x t, d t = V t (x t, d t for any x t. Proof. Consder perod T. Accordng to the assumpton stated, N T, (d T = N T, (d T for any, and hence we have n T (d T = n T (d T and P T (d T = P T (d T. By assumpton on V T +1(, we obtan s T (d T = s T (d T and S T (d T = S T (d T from Theorem 4.1. Moreover, the result V T (x T, d T = V T (x T, d T follows from (5. Suppose that the proposton s true for any perod τ > t. Hence, V t+1 (x t+1, [d t, D t, ] = V t+1 (x t+1, [d t, D t, ] for any x t+1 and. Moreover, we have N t, (d t = N t, (d t for any, whch mples n t (d t = n t (d t and P t (d t = P t (d t. Accordng to Theorem 4.1 and (5, the results hold for perod t. Suppose that we use the same bn ntervals [D t,, D t,+1 for any perod t n the plannng horzon. Furthermore, let us assume that we update the hstogram n tme perod t only based on the realzed demand n perods 1 to t 1, or, for example, gven a fxed n, we update the hstogram n tme perod t only based on realzed demand n tme perods t n through t 1. Observe that these 10

11 two scenaros do not allow any forecastng based on the just realzed demand. From Proposton 3.2, t now follows that the number of dfferent (s, S levels at tme t cannot exceed the number of bns to the power of t. Ths observaton substantally reduces the computatonal burden. 4. Robust Models Based on the Ch-Square Test The most wdely used goodness-of-ft test s the ch-square test (see, e.g., Chernoff and Lehmann 1954 wth the statstcal test (N t, (d t n t (d t P t, 2 χ 2 t t = 1,..., T, n t (d t P t, where parameter χ 2 t controls how close the observed sample data s to the estmated expected number of observatons accordng to the ftted dstrbuton (P t, =1,...,Mt. More specfcally, suppose that k s the number of bns, c s the number of estmated parameters for the ftted dstrbuton (e.g., c = 2 for normal dstrbutons due to the mean and varance, and consder the null hypothess H 0 that the observatons are ndependent random samples drawn from the ftted dstrbuton. Chernoff and Lehmann (1954 show that f H 0 s true, the test statstc converges to a dstrbuton functon that les between the dstrbuton functons of chsquare dstrbutons wth k 1 and k c 1 degrees of freedom. Let α denote the sgnfcance level, and consder χ 2 k 1,1 α such that F (χ2 k 1,1 α = 1 α, where F (x s the dstrbuton functon of the ch-square dstrbuton wth k 1 degrees of freedom. It s often recommended that we reject the null hypothess at the sgnfcance level α f the test statstc s greater than χ 2 k 1,1 α (see, e.g., Law and Kelton In our context, k = M t and α, whose nterpretaton s as above, s gven by the decson maker. Snce P t, should defne a probablty dstrbuton, we have P t, = 1 and P t, 0. Let P t denote the vector of (P t,. The set of dstrbutons that satsfy the ch-square test s { P t (d t = P t A t P t = b t, } (N t, (d t n t (d t P t, 2 χ 2 t, P t 0 t = 1,..., T. (8 n t (d t P t, The lnear constrants A t P t = b t capture at least the fact that P t, = 1. They can also be used to model more complcated propertes of the dstrbuton set, such as constrants on the expected value, any moment or desred percentles of the dstrbutons. It s straghtforward to establsh the compactness of P t (d t. We next gve an alternatve optmalty equaton that explots the structure of (8. We frst provde an alternatve characterzaton of P t (d t. We assume that every norm s the Eucldean norm. Lemma 4.1. The set of demand dstrbutons P t (d t defned n (8 s equvalent to the projecton of the set { (P t, Q t A tp t = b t, on the space of P t. [ ] } N t, (d t 2 Q t, n t (d t 2 n t (d t χ 2 t, Pt, Q t, P 2 t, + Q t, 11

12 Proof. Snce P t, = 1 and N t,(d t = n t (d t, we have (N t, (d t n t (d t P t, 2 = n t (d t P t, = N t, (d t 2 2N t, (d t + n t (d t P t, N t, (d t 2 n t (d t. n t (d t P t, n t (d t P t, As χ 2 t and n t (d t are fnte, we have P t, > 0 for any. Therefore, s equvalent to N t, (d t 2 n t (d t P t, n t (d t χ 2 t N t, (d t 2 Q t, n t (d t 2 n t (d t χ 2 t, 1 P t, Q t,, P t,, Q t, > 0. Obvously, we have [ ] 1 Pt, 1 Q t,, P t,, Q t, > 0 P t, Q t, 1, P t,, Q t, 0 0. P t, 1 Q t, [ ] Pt, 1 Note that the egenvalues of the matrx are P t, + Q t, ± (P t, Q t, 2 + 4, therefore the postve semdefnte constrant s equvalent to 1 Q t, 2 P t, + Q t, (P t, Q t, 2 [ ] Pt, Q t, 2 P 2 t, + Q t,, whch proves the proposton. Lemma 4.1 shows that the set P t (d t can be defned by a set of lnear and second order cone constrants (see, e.g., Lobo et al Note that the second order cone constrants are a specal class of postve semdefnte constrants and they have better computatonal propertes than general postve semdefnte constrants. Ths alternatve defnton of the set P t (d t also suggests a compact optmalty equaton. Proposton 4.1. The optmalty equaton of the robust stochastc model (5 s equvalent to V t (x t, d t = for any t = 1,..., T. mn y t,u t,p t,u t,λ t s.t. KI(y t x t + c t (y t x t + p T t b t 2 u t, N t, (d t ( +λ t nt (d t 2 + n t (d t χ 2 t [ ] p T t U t, A t, λ t p T t U t, A t, + λ t for every 2u t, Proof. Please refer to the Onlne Supplement. U t, h t (y t D t, + θv t+1 (y t D t,, [d t, D t, ] U t, b t (y t D t, + θv t+1 (y t D t,, [d t, D t, ] y t x t, for every for every (9 Note that ths s not the standard optmalty equaton snce V t+1 ( s present n constrants and not the objectve functon. We use t later to obtan computatonally tractable control polces. 12

13 4.1 Computaton of (s, S Levels Next we gve a computatonal approach to compute s t (d t and S t (d t. Theorem 4.1. Let S t (d t be an optmal soluton to the mnmzaton problem mn c t y t + p T t b t 2 ( u t, N t, (d t + λ t nt (d t 2 + n t (d t χ 2 t y t,u t,p t,u t,λ t [ ] s.t. p T t A t, U t, λ t p T t A t, U t, + λ t for every 2u t, U t, h t (y t D t, + θv t+1 (y t D t,, [d t, D t, ] for every U t, b t (D t, y t + θv t+1 (y t D t,, [d t, D t, ] for every, and let s t (d t be the smallest element of the set where f t (y t, d t s defned by (7. {s t (d t s t (d t S t (d t, f t (s t, d t = f t (S t, d t + K}, A state dependent (s, S polcy s optmal for the robust stochastc model (5 wth P t (d t defned by (8, and the (s, S levels are gven by s t (d t and S t (d t respectvely. If there s no fxed cost, a state dependent base-stock polcy s optmal, and the base-stock level gven the realzed demand d t s S t (d t. Proof. The mnmzaton problem to calculate S t (d t follows from the alternatve optmalty equaton (9. Consder the models where the hstorcal data used for perod t s ndependent of the realzed demand from perods 1 to t 1,.e., the number of observatons N t, n the th bn and the total number of avalable observatons n t are constant for any realzed demand d t. Therefore, the set of dstrbutons that satsfy the ch-square test s defned by { P t = P t A t P t = b t, } (N t, n t P t, 2 χ 2 t, P t 0 n t P t, t = 1,..., T. (10 In ths case, the optmalty equaton of the robust model s reduced to { V t (x t = mn max P t, (C t (x t, y t, D t, + θv t+1 (y t D t, } t = 1,..., T, (11 y t x t P t P t where P t and C t (x t, y t, D t, are defned by (10 and (1 respectvely. Alternatvely, t can be wrtten as V t (x t = for any t = 1,..., T. mn y t,u t,p t,u t,λ t s.t. KI(y t x t + c t (y t x t + p T t b t 2 [ p T t U t, A t, λ t 2u t, ] p T t U t, A t, + λ t U t, h t (y t D t, + θv t+1 (y t D t, U t, b t (D t, y t + θv t+1 (y t D t, y t x t, ( u t, N t, + λ t n 2 t + n t χ 2 t for every for every for every The correspondng optmal (s, S polcy levels are also ndependent of the realzed demand d t. 13 (12

14 Theorem 4.2. The (s, S polcy s optmal for the robust stochastc model (11. In partcular, let S t be the optmal soluton to the mnmzaton problem mn y t,u t,p t,u t,λ t s.t. c t y t + p T t b t 2 [ p T t A t, U t, λ t 2u t, and let s t be the smallest element of the set where f t (y t = c t y t + max ( u t, N t, + λ t n 2 t + n t χ 2 t ] p T t A t, U t, + λ t for every U t, h t (y t D t, + θv t+1 (y t D t, for every U t, b t (D t, y t + θv t+1 (y t D t, for every, P t P t {s t s t S t, f t (s t = f t (S t + K}, P t, [ ht (y t D t, + + b t (y t D t, + θv t+1 (y t D t, ]. The polcy s to order S t x t unts n perod t f x t s t, and no order s placed otherwse. Wthout fxed procurement cost, a base-stock polcy s optmal, that s, S t x t unts are ordered n perod t f x t S t, and no order s placed otherwse. 4.2 Summary of Convergence Results In ths subsecton, we explore the case of the bns n the hstogram beng defned by dstnctve values n the sample data, and we study the performance of the robust model when the number of samples ncrease. The mportant results are summarzed n the remanng part of ths subsecton, and the detals of the analyss are presented n Appendx A. The majorty of the results are bult on the followng convergence property: as χ 2 t approaches to 0, the cost-to-go functon V t (x t, d t of the robust model converges to the correspondng cost-to-go functon of the stochastc model where the demand dstrbutons follow the emprcal dstrbuton defned by the hstogram (c.f. Proposton A.1. If there exsts no fxed procurement, then based on Proposton A.1, f ( χ 2 t converges to 0 and ( the emprcal dstrbuton functons converge pontwse to the true dstrbuton functon, then the cost-to-go functon of the robust model converges to that of the stochastc model wth the true demand dstrbuton (c.f. Proposton A.2. In partcular, f the demand dstrbutons for each tme perod follow ndependent contnuous dstrbutons, the convergence of the robust model holds as long as the sample sze approaches nfnty and χ 2 t converges to 0 (c.f. Corollary A.1. For the models wth both fxed and varable procurement cost, we consder the case where the demands follow a dscrete dstrbuton over a fnte set. Smlar to Proposton A.2, the cost-togo functon of the robust model converges to that of the stochastc model wth the true demand dstrbuton under the condtons that ( χ 2 t converges to 0 and ( the emprcal dstrbuton converges to the true dstrbuton (c.f. Proposton A.4. We have also dentfed a condton under whch the convergence holds wthout χ 2 t approachng 0 (c.f. Proposton A.5. Moreover, f the demand dstrbutons are ndependent across dfferent perods, as long as the sample sze goes to 14

15 nfnty, the cost-to-go functon of the robust model converges to that of the stochastc model wth the true demand dstrbuton, and the rate of convergence s O(1/ k, where k denotes the sample sze (c.f. Corollary A.3. The convergence study not only provdes the asymptotc performance of the robust model when the sample sze approaches nfntely, but also guarantees that the robust models wth small bn szes and small χ 2 values perform well n the presence of a sgnfcant number of samples. 5. Computatonal Results In ths secton, we descrbe computatonal experments and present numercal results to support the effectveness of the mnmax robust model based on the ch-square test. In partcular, the robust model proposed n Secton 4 s compared wth ( the approach whch frst fts the hstorcal data and then solves the nventory optmzaton model usng the ftted dstrbuton and ( the robust model based on Delage and Ye (2010. These two comparsons are presented n the followng two subsectons, respectvely. 5.1 Comparson wth Separated Data-Fttng and Inventory Optmzaton As we have mentoned n the prevous sectons, the tradtonal approach s to ft the hstorcal data wth a dstrbuton and then apply stochastc nventory optmzaton usng the ftted dstrbuton. The man objectve of our experments s to compare performances of ths separated approach and the studed mnmax robust model wth respect to optmalty and robustness. At the same tme, we would lke to assess senstvty of the robust model to the choces of the bn szes and χ 2 parameters, and provde an emprcal approach to choose these values. We consder nventory control problems wthout fxed orderng costs. Followng the notaton n the prevous sectons, we let T denote the plannng horzon and c t, h t, b t denote the varable order cost, unt nventory holdng cost, and backorder cost for any perod t, t = 1,..., T, respectvely. The demand dstrbutons for any perod t are assumed to be..d. In the robust model, we restrct ourselves to the case of equal bn szes and these, together wth χ 2, are the same for every perod n the plannng horzon. To smplfy the notaton, par ɛ, χ 2 denotes the choce of the bn sze ɛ and χ 2 n the robust model. The procedure of the computatonal experments s as follows. Step 1. Suppose that the underlyng demand dstrbuton has support {0, 1,..., D}. We randomly generate a dstrbuton among all dstrbutons whose support s a subset of {0, 1,..., D}. In partcular, we pck dstrbuton p = P ( D t = = U D =0 U, for any = 0, 1,..., D, where U for all are..d. random varables unformly dstrbuted n the nterval [0, 1]. We refer to the dstrbuton p = {p } as the true dstrbuton. Step 2. Generate n random samples accordng to the true dstrbuton selected n Step 1. 15

16 Step 3. Ft the samples obtaned n Step 2 usng Crystal Ball and then choose the l best-ftted dstrbutons accordng to the χ 2 goodness-of-ft statstc. Step 4. Solve the standard stochastc nventory control problem wth dstrbutons generated n Steps 1 and 3. Step 5. Solve the robust nventory control model usng a set of bn-sze and χ 2 combnatons. Step 6. Evaluate the total expected cost wth respect to the true dstrbuton p correspondng to the polces of the stochastc models and robust models computed n Steps 4 and 5. We use ths step to nvestgate the optmalty of the robust models. Step 7. The n samples generated n Step 2 defne the emprcal dstrbuton ˆp such that ˆp = the number of tmes value appears n the n samples n for any = 0, 1,..., D. Let δ = p ˆp. We generate m random permutatons of vector δ and denote the jth permutaton of the coordnates by δ j. Vector ˆp j = ˆp + δ j also defnes a dstrbuton. 2 Note that ˆp j s equal to p f δ j = δ,.e., when δ j s not permuted. For each dstrbuton defned by vector ˆp j, we can evaluate the correspondng cost for each polcy computed n Steps 4 and 5. Therefore, we obtan m costs for each polcy and we report the condtonal value-at-rsk 3 (CVaR at the 5% level of the m costs for each polcy. The purpose of ths step s to understand the robustness of dfferent approaches. Let us consder a 10-perod problem. The support for the demand dstrbuton s assumed to be the set {0, 1,..., 29},.e., D = 29. The cost parameters c t, h t and b t are generated ndependently accordng to unform dstrbutons wthn the ntervals [12, 15], [2, 5] and [22, 25], respectvely. Followng the computatonal procedure, we frst draw n = 20 samples from the selected true dstrbuton. Fttng the samples usng Crystal Ball, the three best-ftted dstrbutons accordng to the ch-square values are negatve bnomal, Posson, and beta. The true dstrbuton p, sample frequency ˆp and the three dstrbutons are dsplayed n Fgure 1. In Steps 4 and 5 of our procedure, we compute the base-stock levels correspondng to dfferent models: the stochastc model usng the true dstrbuton, the stochastc model usng the three best-ftted dstrbutons, and robust models wth dfferent bn-sze and χ 2 value combnatons. In partcular, the followng set of bn-sze and χ 2 value combnatons are consdered: 3, 1, 3, 3, 3, 5, 5, 1, 5, 3, 5, 5. As stated n our analyss, the robust model pcks the demand dstrbuton based on the onhand nventory after the order s receved,.e., the order-up-to level y t. Although we use the same hstogram n each perod, the demand dstrbuton returned by the robust model depends also on t. We use the robust model wth the bn-sze/χ 2 value 3, 3 to llustrate these propertes. 2 If ˆp j contans any negatve component, we set ˆp j to be the postve part of ˆp j plus a random permutaton of ts negatve part, and we repeat ths process untl ˆp j 0. 3 Gven random varable X, the condtonal value-at-rsk at a quantle-level q s defned as E[X X < µ] where µ s defned by P (X < µ = q. 16

17 True Dstrbuton Frequency Best Ft 2nd Best Ft 3rd Best Ft Demand Fgure 1: True Dstrbuton, Frequency and Ftted Dstrbutons wth 20 Samples Demand Fgure 2: Demand Dstrbutons Returned by the Robust Model wth Bn Sze = 3 and χ 2 = 3 In Fgures 2 and 3, and Table 1, we use a smple representatve sample of cost parameters. Fgure 2 shows the robust dstrbutons for the last perod t = 10 and the frst perod t = 1 when the nventory levels after recevng the order y t are 0 and 20 respectvely. For both perods, the dstrbutons returned by the robust model for y t = 20 have lower probabltes n the regon 15 to 26 than those for y t = 0. The ntuton behnd ths observaton s that the robust model pcks a demand dstrbuton maxmzng the expected cost. For any possble value of the demand, we ncur a certan cost correspondng to U t, (y t, d t defned n (6. Therefore, the robust model chooses a lower probablty for demand values wth lower costs. Value y t = 20 s very close to the demand when the demand falls n the regon 15 to 26. The amount we over- or under-order s low and hence the correspondng over- or under-order cost s also low. 4 Therefore, the correspondng costs assocated wth the demand values are lower than the costs correspondng to other demand values. As a result, the robust model assgns lower probabltes n these regons compared wth the case when y t = 0. If we compare the robust dstrbutons when y t = 20 for perod 10 and perod 1, we observe that the probablty for perod 10 s hgher for small demand values. Ths can also be explaned by the tradeoff between the over- and under-order costs. In the last perod, the over-order cost s c 10 + h 10 and the under-order cost s b 10 snce we set V T +1 ( = 0. For any earler perod t < 10, 4 In ths secton, the over-order (under-order, respectvely cost ncludes not only the nventory holdng cost h t (backorder cost b t, respectvely ncurred n perod t, but also the mpact of over-order (under-order, respectvely n perod t based on the cost-to-go functon V t+1(. 17

18 Fgure 3: Base-Stock Levels Computed Usng Dfferent Models the over-order costs are sgnfcantly lower as we can carry the nventory to the next perod and save the order cost c t, but the under-order cost s b t + c t+1 snce we not only pay the backorder cost but also procure the product n perod t + 1 to satsfy the unmet demand n perod t. When y t = 20, we pay the over-order costs when the demands are low (e.g., n the regon 0 to 11, and the under-order costs are ncurred when the demand are hgh (e.g., n the regon 21 to 26. As the over-order costs are hgher and the under-order costs are lower n the last perod, t mples that the rato between the costs for low demands and the costs for hgh demands s greater n perod 10 than perod 1. Ths s the reason why the robust model assgns hgher probabltes for low demands n perod 10. On the other hand, the robust dstrbutons when y t = 0 are almost the same for the two perods wth t = 10 and t = 1. In ths case, we only have the under-order cost no matter f the demand s hgh or low. Although the under-order cost s hgher n perod 1 than perod 10, the ratos between the costs for low and hgh demands are almost the same for perods 1 and 10. Therefore, the worst case dstrbutons are smlar for these two perods. The base-stock levels computed n Steps 4 and 5 are dsplayed n Fgure 3. For any of the stochastc or robust models, the base-stock level for perod 10 s sgnfcantly lower than the remanng perods. As explaned before, ths s caused by the fact that the overorder cost s much hgher whle the underorder cost s lower n perod 10 because of V T =1 ( = 0, and thus we should order less n that perod. In addton, the base-stock level for perod 4 s slghtly lower for most of the models snce perod 4 has the hghest order and nventory holdng cost whle ts backorder cost s relatvely low. For the three robust models wth the bn-sze 3, the base-stock levels are nondecreasng wth respect to the χ 2 value, snce the sets of dstrbutons are ncluson-wse ncreasng n the χ 2 value. In our nstances, the backorder cost s much hgher than the nventory holdng cost. Intutvely, the worst case dstrbuton should assgn hgher probabltes for hgh demand values. Therefore, the larger the χ 2 value s, the hgher the probabltes for hgh demand values n the worst case dstrbuton, and hence we should order more to mnmze the worst case expected cost. As a result, 18

19 the base-stock levels are hgher for the robust models wth greater χ 2 values. However, f we set the bn-sze to 5 for the robust models, the base-stock levels are the same when the χ 2 values are equal to 1, 3 and 5. Ths observaton ndcates that the base-stock levels are less senstve to the χ 2 values when we have larger bns. We use Steps 6 and 7 to understand the performance of dfferent models. The results are summarzed n Table 1. The frst four columns correspond to the results for the stochastc models usng true dstrbuton p and the three best-ftted dstrbutons, respectvely. The next four columns show the results for the robust models. Note that the last column corresponds to the robust models wth bn-sze 5 and χ 2 values 1, 3 and 5. These three robust models have the same performance for ths example as they have the same base-stock levels. We show the expected cost for dfferent models wth respect to the true dstrbuton n the frst lne, whch corresponds to the output of Step 6. In the second lne, we report the output of Step 7,.e., the CVaR at 5% level for the costs of m = 1000 dstrbutons generated by ˆp plus random permutatons of p ˆp. For the purpose of ( comparson, the numbers n Table 1 are calculated by subtractng the cycle stock order cost,.e., T ( t=1 c D t =1 p, from the orgnal cost or CVaR, and normalzng wth respect to that of the stochastc model usng true dstrbuton. Stochastc Models Robust Models True Best 2nd Best 3rd Best 5, 1 or Dst Ft Ft Ft 3, 1 3, 3 3, 5 3 or 5 Cost CVaR Table 1: Performance of Dfferent Models for the Instance n Fgure 1 Obvously, the stochastc model usng the true dstrbuton gves the lowest expected cost. The output of Step 7, CVaR, also ndcates that ths model s robust wth respect to perturbatons n the nput dstrbuton as t has the thrd lowest CVaR, whch s only 2.61% hgher than the lowest CVaR. For the three stochastc models usng ftted dstrbutons, the models usng the 1st and 3rd bestftted dstrbutons have a very smlar performance. The best-ft case has the best performance among the ftted stochastc models as ts CVaR s 0.25% better than the 3rd best-ft stochastc model and the cost s only 0.13% hgher than that. The performance of the model usng the 2nd best dstrbuton s much worse compared wth the other two. Its cost and CVaR values are at least 12% hgher than those of the remanng two models. The three robust models wth bn-sze 3 outperform all of the stochastc models usng ftted dstrbutons n terms of both optmalty (cost and robustness (CVaR. The robust models wth bn-sze 5 also have better values of the cost and CVaR than the stochastc model usng the 2nd bestftted dstrbuton. In partcular, the robust models wth bn-sze/χ 2 value combnatons of 3, 3 and 3, 5 are sgnfcantly better than the stochastc models usng ftted dstrbutons. They reduce the cost by more than 3% and CVaR by more than 7% when comparng wth the ftted stochastc models. Among the robust models we prefer the model wth bn-sze/χ 2 value combnaton 3, 3, snce t mproves the cost by 0.38% at the prce of a 0.35% ncrease n CVaR. 19