Nonparametric Data-Driven Algorithms for Multi-Product Inventory Systems with Censored Demand

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1 Submtted to Operatons Research manuscrpt OPRE R3 Nonparametrc Data-Drven Algorthms for Mult-Product Inventory Systems wth Censored Demand Cong Sh, Wedong Chen Industral and Operatons Engneerng, Unversty of Mchgan, Ann Arbor, MI 48109, {shcong, Izak Duenyas Technology and Operatons, Ross School of Busness, Unversty of Mchgan, Ann Arbor, MI 48109, We propose a nonparametrc data-drven algorthm called DDM for the management of stochastc perodcrevew mult-product nventory systems wth a warehouse-capacty constrant. The demand dstrbuton s not known a pror and the frm only has access to past sales data often referred to as censored demand data). We measure performance of DDM through regret, the dfference between the total expected cost of DDM and that of an oracle wth access to the true demand dstrbuton actng optmally. We characterze the rate of convergence guarantee of DDM. More specfcally, we show that the average expected T -perod cost ncurred under DDM converges to the optmal cost at the rate of O1/ T ). Our asymptotc analyss sgnfcantly generalzes approaches used n Huh and Rusmevchentong 2009) for the uncapactated sngle-product nventory systems. We also dscuss several extensons and conduct numercal experments to demonstrate the effectveness of our proposed algorthm. Key words : nventory, mult-product, censored demand, nonparametrc algorthms, asymptotc analyss Hstory : Receved July 2014; revsons receved February 2015, June 2015, September 2015; accepted December Introducton The study of stochastc mult-product nventory systems dates back to Venott 1965). Most, f not all, of the papers on stochastc mult-product nventory systems assume that the stochastc future demand s gven by a specfc exogeneous random varable, and the nventory decsons are made wth full knowledge of the future demand dstrbuton. However, n practce, the demand dstrbuton s usually not known a pror. Even wth past demand data often censored) collected, the selecton of the most approprate dstrbuton and ts parameters remans dffcult see Huh and Rusmevchentong 2009), Huh et al. 2011), Besbes and Muharremoglu 2013) for more dscussons on censored demand n nventory systems). 1

2 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 2 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 Model overvew and research ssue. In our perodc-revew mult-product lost-sales nventory system over a fnte horzon of T perods, the demands across perods t = 1,... T are..d.) random vectors D t wth each component representng a dfferent product), respectvely. There s a jont warehouse-capacty constrant M mposed on the total number of products that can be held n nventory. The frm has no access to the true underlyng demand dstrbuton a pror, and can only observe sales data.e., censored demand) over tme. We develop a nonparametrc data-drven adaptve nventory control polcy π = y t t 1) where the decson y t represents the order-up-to level n perod t. We measure performance of our proposed polcy π through regret denoted by R T Cπ) Cπ ), where Cπ) s the total expected cost of π and Cπ ) s the total expected cost of a clarvoyant optmal polcy π wth access to the true underlyng demand dstrbuton a pror. The research queston s to devse an effectve nonparametrc data-drven polcy π that drves the average regret R T /T to zero wth a fast convergence rate. Man results and contrbutons. We propose a nonparametrc data-drven algorthm called DDM for stochastc mult-product nventory systems wth a warehouse-capacty constrant. We characterze the rate of convergence guarantee of DDM. More specfcally, we show that the average regret R T converges to zero at the rate of O1/ T ). Our algorthm DDM s a stochastc gradent descent type of algorthm, smlar n sprt to Burnetas and Smth 2000), Kunnumkal and Topaloglu 2008) and Huh and Rusmevchentong 2009). The work closest to ours s Huh and Rusmevchentong 2009) who studed an uncapactated nventory system wth a sngle product. The novelty of our work les n both algorthmc desgn and performance analyss of DDM. Frst, unlke the uncapactated sngle-product case, the gradent estmator n DDM could be sometmes ndetermnable n the presence of a warehouse-capacty constrant on multple products. Second, the projecton step n DDM has to factor n both postve nventory carry-over of all products and the warehouse-capacty constrant. To mantan feasblty of the soluton n each step, we solve two addtonal optmzaton problems. The optmzaton problems can be effcently solved by greedy algorthms, but the soluton structure makes the asymptotc performance analyss nvarably harder than that n the uncapactated sngle-product case where no optmzaton procedures are needed). The key techncal challenge n our analyss s to derve an upper bound of the dstance between the target order-up-to level and the actual mplemented order-up-to level due to the warehouse-capacty constrant and postve nventory carry-over from prevous perods). Note that the upper bound on ths dstance functon s almost mmedate n the uncapactated sngle-product case whle the development of an upper bound s sgnfcantly more complex n our mult-product settng. Thrd, we relate the nventory process to a GI/G/1 queue. We then develop a stochastc domnance argument and nvoke a classcal result on the expected busy perod n GI/G/1 queue due to Loulou 1978).

3 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 3 We compare the computatonal performance of DDM wth several exstng parametrc and nonparametrc approaches n the lterature. Our results show that DDM outperforms these benchmark algorthms n terms of both consstency and convergence rate. We also consder two nterestng extensons, one wth a more general warehouse-capacty constrant where dfferent products may have dfferent dmenson or szes, and the other one wth dscrete demand and order quanttes. Relevant lterature. Our work s relevant to the followng research streams. Mult-product stochastc nventory systems. There s a large body of lterature devoted to varous classes of such problems. In ths paper, we focus our attenton on the classcal stochastc multproduct nventory systems under a warehouse-capacty constrant, frst studed by Venott 1965). He provded condtons that ensure that the base-stock orderng polcy s optmal n a perodcrevew nventory system wth a fnte horzon. Subsequently, Ignall and Venott 1969) showed that n the statonary demand case, a myopc orderng polcy s optmal under certan mld condtons. Beyer et al. 2001, 2002) establshed the optmalty of myopc polces n backlogged systems wth separable costs by appealng to the suffcent condton provded by Ignall and Venott 1969), whch was further extended by Cho et al. 2005) under a relaxed demand assumpton. Our work focuses on a nonparametrc varant n whch the demand dstrbuton s not known a pror. Nonparametrc nventory systems. Burnetas and Smth 2000) developed a gradent descent type algorthm for orderng and prcng when nventory s pershable; they showed that the average proft converges to the optmal but dd not establsh the rate of convergence. Huh and Rusmevchentong 2009) proposed gradent descent based algorthms for lost-sales systems wth censored demand. Subsequently, Huh et al. 2009) proposed algorthms for fndng the optmal base-stock polcy n lost-sales nventory systems wth postve lead tme. Huh et al. 2011) appled the concept of Kaplan-Meer estmator to devse another data-drven algorthm for censored demand. Other nonparametrc approaches n the nventory lterature nclude sample average approxmaton SAA) e.g., Kleywegt et al. 2002), Lev et al. 2007, 2015)) whch uses the emprcal dstrbuton formed by uncensored samples drawn from the true dstrbuton. Concave adaptve value estmaton e.g., Godfrey and Powell 2001), Powell et al. 2004)) successvely approxmates the objectve cost functon wth a sequence of pecewse lnear functons. The bootstrap method e.g., Bookbnder and Lordahl 1989)) estmates the newsvendor quantle of the demand dstrbuton. The nfntesmal perturbaton approach IPA) s a samplng-based stochastc gradent estmaton technque that has been used to solve stochastc supply chan models see, e.g., Glasserman 1991)). Maglaras and Eren 2015) employed maxmum entropy dstrbutons to solve a stochastc capacty control problem. For parametrc approaches, such as Bayesan learnng see, e.g., Larvere and Porteus 1999), Chen and Plambeck 2008)) or operatonal statstcs see, e.g., Lyanage and Shanthkumar 2005),

4 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 4 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 Chu et al. 2008)) n stochastc nventory systems, we refer readers to Huh and Rusmevchentong 2009) for an excellent dscusson of the key dfferences between nonparametrc and parametrc approaches. Our paper contrbutes to the lterature by studyng mult-product nventory systems under a warehouse-capacty constrant, whch s sgnfcantly more complex to analyze. Onlne convex optmzaton. The am of onlne convex optmzaton s to mnmze the cumulatve loss functon defned over a convex compact set wth onlne learnng process snce the optmzer does not know the convex) objectve functon a pror see Hazan 2015), Shalev-Shwartz 2012) for an overvew). Znkevch 2003) has shown that the average T -perod cost usng a gradent descent based algorthm converges to the optmal cost at the rate of O1/ T ). Ths result was further extended by Flaxman et al. 2005) n a bandt settng. Under addtonal techncal assumptons, a modfed algorthm by Hazan et al. 2006) acheves a faster convergence rate Olog T/T ). Our problem dffers from the conventonal onlne convex optmzaton problems n that the target levels or the terates) may not be acheved due to polcy-dependent dynamc nventory constrants. Stochastc approxmaton. The proposed gradent descent type of algorthm also resembles the ones used n the Stochastc Approxmaton SA) lterature see Nemrovsk et al. 2009) and references theren), whch should be carefully contrasted wth ours. Frst, SA algorthms am to solve a sngle-stage stochastc optmzaton problem by makng successve experments whle the cost of experments s gnored. On the other hand, our algorthm ams to mnmze the cumulatve loss suffered along the learnng progress for a mult-stage closed-loop stochastc optmzaton problem. Puttng nto context, SA focuses on measurng the termnal regret E[Πy T ) Πy ), whereas our [ T algorthm focuses on measurng the cumulatve loss over tme E Πy t) Πy )). Second, n the analyss of robust SA algorthms wth general convex costs, the step sze s chosen to be O1/ t) to obtan a convergence rate of O1/ t) n the termnal regret crteron by approprately averagng the terate solutons. The standard robust SA approaches cannot be adapted to our settng where the terates cannot move freely due to polcy-drven dynamc nventory constrants. General notaton. For any real vectors x, y R n, y x means component-wse greater or equal to; x + = max{x, 0}) n =1; x = x ) n =1; the jon operator x y = max{x, y }) n =1; the meet operator x y = mn{x, y }) n =1; for any ntegers t and s wth t s, x [t,s = s j=t x j and x [t,s) = s 1 j=t x j; or 2 means 2-norm; 1 means 1-norm. The notaton means s defned as. 2. Mult-Product Stochastc Inventory Systems We consder a stochastc T -perod n-product nventory system under a warehouse-capacty constrant M e.g., Ignall and Venott 1969), Beyer et al. 2001)). The frm has no knowledge of

5 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 5 the true underlyng demand dstrbuton a pror, but can observe past sales data.e., censored demand data), and make adaptve nventory decsons based on the avalable nformaton. Random demand and regularty assumptons. For each perod t = 1,..., T and each product = 1,... n, we denote the demand of product n perod t by a random varable Dt. For notatonal convenence, we use D t = Dt 1,..., Dt n ) to denote the random demand vector n perod t, and d t = d 1 t,..., d n t ) to denote ther realzatons. Assumpton 1. We make the followng assumptons and regularty condtons on demand. a) For each product, Dt s..d. across tme perod t. b) For each product and for each perod t, Dt s ndependent but not necessarly dentcally dstrbuted) of Ds j for all j and s = 1,..., T. c) For each product and for each perod t, Dt s a contnuous random varable defned on a fnte support [0, M, whose CDF F D ) s dfferentable and densty F x) > 0 for all x [0, M. D d) For each product and for each perod t, E[Dt l for some real number l > 0. Assumptons 1a) and 1b) assume some form of statonarty of demand, whch s predomnant n the nonparametrc learnng lterature see, e.g., Lev et al. 2007), Huh et al. 2009, 2011), Huh and Rusmevchentong 2009), Besbes and Muharremoglu 2013)). Assumpton 1c) ensures the per-perod cost functon defned n 3) s dfferentable, fnte-valued and strctly jontly) convex, whch guarantees a unque mnmzer. Assumpton 1d) rules out degenerate demands. System dynamcs and objectves. Let f t denote the nformaton collected up to the begnnng of perod t, whch ncludes all the realzed demands and past decsons. A feasble closed-loop polcy π s a sequence of functons y t = π t x t, f t ), t = 1,..., T, mappng begnnng nventory x t and f t state) nto endng nventory y t decson) whle satsfyng y t x t and the warehouse-capacty constrant see Bertsekas 2000) for dscussons on closed-loop optmzaton problems). Note that when the demand dstrbuton s known a pror, t suffces to consder polces of the form y t = π t x t ), due to the assumed across-tme ndependence of demands see Bertsekas and Shreve 2007)). Gven a feasble polcy π, we descrbe the sequence of events below. Note that x π t, yt π and q π t s are functons of π; for ease of presentaton, we make ther dependence on π mplct.) a) At the begnnng of perod t, the frm observes the startng nventory x t = x 1 t,..., x n t ). b) The frm decdes to order q t = qt 1,..., qt n ) 0, and the endng nventory y t = x t + q t, where y t = yt 1,..., yt n ). We assume nstantaneous replenshment. The total nventory level s restrcted by a warehouse-capacty constrant see Ignall and Venott 1969)),.e., { } y t Γ y t R n + : y t M. 1) =1

6 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 6 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 c) The demand D t s realzed, denoted by d t, whch s satsfed to the maxmum extent usng onhand nventory. Unsatsfed demand unts are lost, and the frm only observes the sales quantty or censored demand),.e., mnd t, yt) for each product n perod t. The state transton can be wrtten as x t+1 = x t + q t d t ) + = y t d t ) +. d) The producton, overage and underage costs at the end of perod t s then c q t + h y t d t ) + + p d t y t ) +, where c = c 1,..., c n ), h = h 1,..., h n ) and p = p 1,..., p n ) are the perunt purchasng, holdng and lost-sales penalty cost vectors, respectvely. We note that the cost mnmzaton model wth lost-sales assumes that p c see Zpkn 2000)) snce the frm loses revenue and goodwll from the sale and the revenue has to be greater than the producton cost. Our approach also works for tme-nvarant random purchasng cost vector.) Assumng the salvage value of any left-over product at the end of plannng horzon equals ts producton cost, the total expected cost ncurred by π can be wrtten as [ T Cπ) = E c y t x t ) + h y t D t ) + + p D t y t ) + E[c x T +1, 2) = c x 1 + T E [ c y t + h c) y t D t ) + + p D t y t ) +, where the second equalty follows from x t+1 = y t d t ) + and some smple algebra. If the underlyng dstrbuton D t s gven a pror, the stochastc nventory control problem specfed above can be formulated usng dynamc programmng see Beyer et al. 2001)) wth state varables x t, control varables y t wth x t y t Γ), random dsturbances D t, and state transton x t+1 = y t d t ) +. It turns out that ths problem s n fact myopcally solvable, whch s dscussed next. Clarvoyant optmal polcy. We frst characterze the clarvoyant optmal polcy where the dstrbuton of D t s known a pror. We defne Π ) to be the per-perod expected cost functon, Πa) = Π t a) E [ c a + h c) a D t ) + + p D t a) +. 3) Let y be a unque crtcal determnstc) vector defned by y arg mn a Γ:a 0 Πa). 4) Theorem 1. Under Assumpton 1, when the demand dstrbuton s known a pror, orderng up to y defned n 4) n each perod s optmal, wth expected per-perod cost Πy ). Although not central to our man focus, the proof of Theorem 1 s qute nvolved, whch reles on verfyng a suffcent condton called substtute property provded by Ignall and Venott 1969)

7 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 7 to establsh the optmalty of myopc polcy. We relegate the full detaled proof to the Electronc Companon. 3. Nonparametrc Data-Drven Inventory Control Polces When the frm has no knowledge of the true underlyng dstrbuton of D t a pror, we am to fnd a provably good adaptve data-drven nventory control polcy that makes the total expected system costs close to the optmal strategy. The proposed data-drven algorthm DDM mantans a vector trplet of sequences z t, ŷ t, y t ) t 0. The frst sequence z t ) t 0 represents the constrant-free target nventory levels where the warehouse storage constrant s waved. The second sequence ŷ t ) t 0 represents the target nventory levels when the warehouse storage constrant s taken nto account. However, the target nventory levels ŷ t ) t 0 may not be always feasble due to warehouse capacty constrant and postve nventory carry-over. Thus, we use the thrd sequence y t ) t 0 to represent the actual mplemented nventory levels after orderng. We frst present a compact descrpton of our data-drven mult-product algorthm DDM). Data-Drven Mult-product Algorthm DDM). Step 0. Intalzaton.) Set the ntal nventory levels y 0 = ŷ 0 = z 0 to be any values wthn Γ and then set the ntal values t = 0, τ 0 = 0 and k = 0. For each perod t = 0,..., T 1, repeat the followng steps: Step 1. Settng the constrant-free and constraned target nventory levels.) Case 1: If y t ŷ t.e., yt ŷt for all = 1,..., n), the algorthm updates the constrant-free target nventory levels z t+1 by γm 1 t z t+1 = ŷ t η t G t ŷ t ), where η t = for some γ > 0 5) n max {p c, h }) for each product = 1,..., n, and the th component of G t s defned as h, f ŷ G t > d t, tŷ t ) = p c ), f ŷt d t. 6) Note that γ = 1 for achevng the tghtest theoretcal bound. Then the algorthm sets the constraned target nventory levels ŷ t+1 by solvng ŷ t+1 = arg mn w Γ w z t )

8 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 8 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 Record the break pont τ k := t and ncrease the value k by 1. Case 2: Else f y t ŷ t.e., there exsts an such that yt < ŷt), the algorthm keeps both the constrant-free and constraned target nventory levels unchanged,.e., z t+1 = z t and ŷ t+1 = ŷ t. Step 2. Solvng for the actual mplemented target nventory levels.) Defne the set J and ts complement as J { } { : x t+1 > ŷ t+1, J : x t+1 ŷt+1}. 8) For each product J, we set the actual mplemented levels y t+1 = x t+1, f x t+1 > ŷ t+1. 9) If J, then we set the actual mplemented levels y t+1 by solvng mn ŷ t+1 y t+1) 2 s.t. y t+1 M x j t+1, y t+1 x t+1, J. 10) Ths concludes the descrpton of the algorthm Algorthm Overvew of DDM and Propertes Step 1: Stochastc Gradent Descent). Let T = {τ 0, τ 1,..., τ m } wth τ m T, whch s the set of break ponts of DDM. In each perod τ k + 1 k = 1,..., m), we update the constrant-free target levels z t+1 by a stochastc gradent descent step. Conceptually, we update the mnmzer along the negatve drecton of the true gradent of Π ). However, snce the true cost functon Π ) s not avalable to us wthout knowng the underlyng demand dstrbuton), we can only rely on the observed sales data d t to provde us an estmator of the true gradent of Πŷ t ) at the ponts ŷ t. The estmator G tŷ t ) defned n 5) can be computed usng the sales censored demand) data observed by the frm n perod t T. When t T, we have yt ŷt for all = 1,..., n. Hence, the event {ŷt d t} s equvalent to the case where the endng nventory n perod t s at most yt ŷt, whch s an observable event; the event {ŷt > d t} s equvalent to the case where the endng nventory n perod t s strctly greater than yt ŷt, whch s also observable. In ths case, G t defned n 6) s an unbased estmator of the true gradent Πŷ t ) at ŷ t,.e., E[G t ŷ t ) = Πŷ t ), where the expectaton s taken over the demand n perod t. On the other hand, when t / T, G tŷ t ) may be ndetermnable because the actual mplemented nventory levels could fall below the target order-up-to levels. To be more specfc, when yt < ŷt and yt d t, the frm only observes the stockout

9 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 9 but not the lost-sales quantty. Therefore, the frm cannot dstngush between yt d t < ŷt and yt < ŷt d t, and hence cannot determne the value of G tŷ t ). In perods when t / T, we keep the target order-up-to levels unchanged. We then carry out a greedy projecton of the constrant-free target nventory levels z t+1 onto the warehouse storage constrant set Γ va 7), more specfcally, mn ŷ t+1 zt+1) 2 s.t. =1 ŷ t+1 M, ŷ t+1 0,. 11) =1 We also make two smple observatons that wll be useful n Secton 4. a) A smple observaton leads to the lower and upper bounds of zt+1 for each product,.e., ŷt η t h zt+1 ŷt + η t p c ). In fact, zt+1 has to ht one of the two boundares. b) Another mportant observaton s that when the product n the frst step updates ts constrant-free target level zt+1 through a postve drecton,.e., zt+1 = ŷt + η t p c ) ŷt 0, we must have ŷt+1 zt+1. To see ths, suppose otherwse ŷt+1 > zt+1, we can decrease ŷt+1 to zt+1, thereby strctly mprovng the objectve value of 11) whle mantanng feasblty. On the other hand, when the product n the frst step updates ts constrant-free target level zt+1 through a negatve drecton, we have zt+1 = ŷt η t h ŷt, Thus, ths leads to the followng property that wll be useful n the performance analyss, ŷ t+1 ŷ t + η t p c ), = 1,..., n. 12) Step 2: Mantanng Feasblty). The target nventory levels ŷ t+1 derved n the second step may not be achevable or mplementable, due to the physcal nventory carry-over and the warehouse capacty constrant. We then need to carry out an addtonal optmzaton procedure as follows. Ths step tres to order as many products as possble to reach the target level, and t s easy to solve quanttatvely but hard to analyze. Frst we dvde all the products nto two groups, namely, the set J and ts complement as defned n 8). We then have the followng two cases. Case 1. For each product J,.e., the begnnng nventory level of product s already greater than ts target level. It s natural to not order any more product and hence we follow 9). Case 2. Now we focus on the set J. Snce the remanng nventory space now becomes M xj t+1, we solve the optmzaton problem 10) to determne the actual mplemented levels y t+1. Note that the optmzaton problem s well-defned snce M x j t+1 = M y j t d j t) + M y j t 0, where the nequalty follows from the fact that the algorthm keeps y t Γ.

10 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 10 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 The optmzaton 10) attempts to rase our nventory level as close as possble to the target nventory level ŷt+1 for each product J; however, t s possble that some of the products n J cannot ht the target level due to nventory constrants. Snce we mnmze the 2-norm type of objectve functon, t can be readly verfed that the optmzaton 10) makes the shortfalls defned as ŷt+1 yt+1 as even as possble across the products n the set J. Note that f the optmal objectve value of 10) s equal to 0, then the algorthm goes to Case 1 n the next perod and updates the target nventory levels. Otherwse t goes to Case 2 and mantans the target nventory levels; whle mantanng these target levels, the nventory levels wthn J are decreasng and more nventory space s freed over tme, and the shortfalls wll decrease to zero. 4. Performance Analyss of the Nonparametrc Data-Drven Algorthm The regret of our data-drven algorthm, denoted by R T, s defned as the dfference between the optmal clarvoyant cost gven the demand dstrbuton a pror) and the cost ncurred by our data-drven algorthm whch learns the demand dstrbuton over tme). That s, for any T 1, [ T R T E Πy t ) T Πy ), where y t are the actual mplemented order-up-to levels of our nonparametrc closed-loop) algorthm DDM, and y s the clarvoyant optmal soluton n 4). Theorem 2 below states the man result n ths paper. Theorem 2. Under Assumpton 1, the average regret R T /T of our data-drven algorthm DDM approaches 0 at the rate of 1/ T. That s, there exsts some constant K, such that for any T 1, [ T 1 T R T 1 T E Πy t ) Πy ) K, T where y t are actual mplemented order-up-to levels of our nonparametrc closed-loop) algorthm DDM, and y s the clarvoyant optmal soluton n 4). It s known that n the general convex case wthout assumng smoothness and strong convexty), ths rate of O1/ T ) s unmprovable see, e.g., Theorem 3.2. of Hazan 2015)). Our key contrbuton here s to establsh ths best possble rate even wth nventory and capacty constrants.e., the terates cannot move freely due to polcy-drven dynamc nventory constrants). Then the proof of Theorem 2 s the drect consequence of the followng two key lemmas.

11 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 11 Lemma 1. For any T 1, there exsts a constant K 1 R such that [ T T 1 T ) = E Πŷ t ) Πy ) K 1 T, where ŷ t are target order-up-to levels of DDM, and y s the clarvoyant optmal soluton n 4). Lemma 2. For any T 1, there exsts some constant K 2 R such that [ T T 2 T ) = E Πy t ) Πŷ t ) K 2 T, where y t and ŷ t are actual mplemented and target order-up-to levels of DDM, respectvely Bound on 1 - Onlne Convex Optmzaton Proof of Lemma 1) The proof of Lemma 1 bulds upon the deas and technques used n onlne convex optmzaton see, e.g., Znkevch 2003) and Flaxman et al. 2005)). It s shown the cost functon Π ) s jontly convex, and G ) s an unbased estmator of the true expected gradent of Π ) under censored demand wthn the set of breakponts. In addton, ths gradent estmator s bounded,.e., G ) 2 2 nmax {p c, h }) 2. We relegate the proof of Lemma 1 to the Electronc Companon Bound on 2 - Stochastc Domnance and a GI/G/1 Queue Proof of Lemma 2) The man focus of ths paper s to establsh the result n Lemma 2. Frst we derve a bound of the gap between the cost functons assocated wth the actual mplemented level y t and the desred target level ŷ t, usng the dstance functon y t ŷ t. Lemma 3. The dfference n cost functons E[Πy t ) Πŷ t ) E [h p c)) y t ŷ t. Gven Lemma 3, we need to develop an upper bound on the dstance functon y t ŷ t, whch s the crux of our performance analyss. Lemmas 4 and 5 below play a major role n the development of such an upper bound. Ther proof strategy reles heavly on the constructon of DDM and also the structural propertes of optmzaton problems 10) and 11), whch s qute nvolved. Lemma 4 below provdes an upper bound on the dstance functon for products n the set J n whch the begnnng nventory level already exceeds the target order-up-to level.

12 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 12 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 Lemma 4. In each perod t + 1, we bound the dstance functon for all J { : x t+1 > ŷt+1}. y t+1 ŷt+1 y t ŷ t + η t h + p j c j) d t. J J J j J J Lemma 5 below provdes an upper bound on the dstance functon for products n the complement set J n whch the begnnng nventory level s below the target order-up-to level. If ths s the case for all products,.e., all products belong to J, then the target levels can always be acheved. If not, we solve 10) to re-dstrbute our target levels such that the dfference between the target level and the actual mplemented level s as even as possble across dfferent products. Lemma 5. In each perod t + 1, we bound the dstance functon for all J { : x t+1 ŷt+1} as follows. If J =, we have ŷ t+1 yt+1 = 0. Otherwse, f J, we have ŷ t+1 yt+1 ŷ t y t + η t p c ) d j t. Wth the upper bounds on the dstance functon n two mutually exclusve sets J and J obtaned from Lemmas 4 and 5, we provde an overarchng upper bound n Lemma 6. Lemma 6. In each perod t + 1, we bound the sum of dstance functons as follows. y t+1 ŷt+1 y t ŷ t + η t =1 =1 =1 h + 2 p c ))) mn j=1,...,n dj t Next, we wsh to fnd a stochastc process that can be used to bound the sum of dstance functons. It s now convenent to ntroduce the noton of stochastc order and convex order see Shaked and Shanthkumar 2007)). Consder two random varables X and Y. X s sad to be stochastcally smaller than Y denoted by X st Y ) f PX > x) PY > x), x R. Also, X s sad to be smaller than Y n the convex order denoted as X cx Y ) f E[φX) E[φY ) for all convex functons φ : R R, provded the expectatons exst. Note that convex order s weaker,.e., X st Y X cx Y. Next, correspondng to the sum of dstance functons, we consder a stochastc process Z t t 0) [ Z t+1 = Z t + S + t D t, Z 0 = 0, t ) +. where S t n =1 h + 2p c )), and D t s a random varable satsfyng D t st D j t, j.

13 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 13 Lemma 7. The total expected dstance functon [ T [ T E y t ŷt E Z t, =1 where Zt+1 s a stochastc process defned above. We observe that the stochastc process Z t s very smlar to a GI/G/1 queue, except that the servce tme s scaled by 1/ t n each perod t. Now consder a GI/G/1 queue W n n 0) defned by the followng Lndley s equaton: W 0 = 0, and W t+1 = [W t + S t D t +, 13) where the sequences S t and D t consst of ndependent and dentcally dstrbuted random varables. Let τ 0 = 0, τ 1 = nf{t 1 : W t = 0} and for k 1, τ k+1 = nf{t > τ k : W t = 0}. Let B k = τ k τ k 1. The random varable W t s the watng tme of the t th customer n the GI/G/1 queue, where the nter-arrval tme between the t th and t + 1 th customers s dstrbuted as D t, and the servce tme s dstrbuted as S t. Then, B k s the length of the k th busy perod. Let ρ = E[S 1 /E[ D 1 represent the system utlzaton. It s well-known that n a GI/G/1 queue, f ρ 1, then the queue s stable and the random varable B k s ndependent and dentcally dstrbuted. Note that ths stablty condton ρ 1 can always be satsfed by approprately scalng the unts of cost parameters. We nvoke the followng result from Loulou 1978) to bound E[B, the expected busy perod of a GI/G/1 queue wth nter-arrval dstrbuton D n and servce dstrbuton S n. Theorem 3 Loulou 1978). Let X n = S n D n, and α = E[X 1. Let σ 2 be the varance of X 1. If E[X 3 1 = β <, and ρ < 1, E[B σ 6β α exp σ + α ). 3 σ We can now obtan an upper bound on our expected busy perod E[B for the stochastc process W t defned n 13), by settng X 1 = n =1 h + 2p c )) D 1 whose expectaton s negatve snce ρ 1). Wth the explct form of E[B, Lemma 8 gves the upper bound of our dstance functon below. The dea s to connect the upper-boundng stochastc process whch evolves as a GI/G/1 queue) wth the expected busy perod of ths queue where there exsts an explct upper bound that does not depend on the tme horzon T ). Lemma 8. The total expected dstance functon E [ T =1 y t ŷt 2E[BS T,

14 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 14 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 where E[B σ α exp 6β σ 3 + α σ ), and S = n =1 h + 2 p c )). The proof of Lemma 2 then follows from Lemma 3 and Lemma 8, wth ther proofs provded n the Electronc Companon. 5. Extensons Improvng the convergence rate. If we change Assumpton 1c) slghtly to enforce a unform lower bound δ > 0 on the densty of demand,.e., F x) δ > 0 for all x [0, M and all = 1,..., n, D and also change the step sze η t = O1/t) n the algorthm, one can readly show that the cost functon s δ-strongly convex, and the rate of convergence of DDM can be mproved to Olog T/T ). Dfferent product dmensons or szes. Our basc model defned n Secton 2) assumes that all products have exactly the same dmenson or szes. However, n general, dfferent products may have dfferent dmenson or szes. Let v 1, v 2,..., v n denote the szes of the dfferent products, and { } y t Γ y t R n + : v y t M, 14) By a smple cost tranformaton, we show n the Electronc Companon that our algorthm DDM now defned n terms of transformed varables) and ts performance analyss reman the same. Dscrete demand and order quanttes. In practce, the demand and orderng quanttes are often ntegers. We provde a modfed algorthm denoted by DDM-Dscrete) n the Electronc Companon to handle such dscrete cases, whch acheves the same convergence rate O1/ T ) wth the ad of lost-sales ndcators.e., the frm knows whether lost-sales has occurred n each perod). =1 6. Numercal Experments We compare the performance of DDM wth several exstng parametrc and nonparametrc approaches n the lterature brefly descrbed below). Our results show that DDM outperforms these benchmark algorthms n terms of both consstency and convergence rate. We relegate the detaled expermental setup and numercal results fgures) to the Electronc Companon. 1. Algorthm a1 Known Dstrbuton): Clarvoyant Optmal Polcy. 2. Algorthm a2 Uncensored): Uncensored SAA. Ths s a sample average approxmaton SAA) algorthm wth uncensored demand a hypothetcal stuaton). The target nventory level s the quantle of the emprcal demand dstrbuton usng uncensored demand data. 3. Algorthm b1 Parametrc): MLE Censored. Assumng the correct parametrc form has been pre-specfed, ths parametrc polcy uses censored demand data to construct maxmum lkelhood estmators MLE) for the parameters n the demand dstrbuton.

15 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R Algorthm c1 Nonparametrc): Burnetas-Smth B-S) Polcy. The B-S polcy s a nonparametrc polcy whch was developed by Burnetas and Smth 2000). 5. Algorthm c2 Nonparametrc): CAVE Polcy. The CAVE polcy, developed by Godfrey and Powell 2001), s a nonparametrc approach by approxmatng the underlyng objectve functon usng a seres of pecewse lnear functons. Comparson wth parametrc MLE algorthms. The numercal results are presented n Fgure EC.1. Our results ndcate that DDM performs very well, and s consstent.e., t converges to the optmal soluton). In contrast, MLE Censored s sgnfcantly slower than DDM, and also suffers from nconsstency,.e., t often fals to converge to the optmal soluton. Ths s due to a spraldown effect. More specfcally, f the ntal nventory level s lower than the optmal target level, the censored demand s lkely to gve an even lower estmate n the next perod because the frm cannot observe the lost-sales quantty). Then the target nventory level wll be set lower and lower, resultng n dvergent cost. The consstency of MLE Censored hnges on the almost) perfect ntal estmaton of target levels, whch s often mpractcal. In fact, n three of the examples n Fgure EC.1, the MLE Censored algorthm dd not converge; n the only one where t dd converge, we actually pcked startng target levels close enough to the optmal levels so that t would converge, whch of course would not be possble n practce. Comparson wth nonparametrc algorthms. The numercal results are presented n Fgure EC.2. Our results show that DDM consstently outperforms both the B-S polcy and the CAVE polcy. We also fnd out that the B-S polcy has an extremely slow convergence rate whle the CAVE polcy s much faster but stll slower than DDM. Fgure EC.2 also dsplays the performance of the Uncensored SAA polcy assumng the uncensored demand nformaton). It s nterestng to note that the DDM polcy performs very close to the Uncensored SAA polcy n all of our examples. Extreme cases wth uneven lost-sales penalty costs. DDM performs consstently very well for extreme cases wth some pathologcal parameters see Fgure EC.3). Supplemental Materal An electronc companon to ths paper s avalable at Acknowledgments The authors are grateful to the area edtor Professor Chung-Paw Teo, the anonymous assocate edtor, and the three anonymous referees for ther detaled comments and suggestons, whch have helped to sgnfcantly mprove both the content and the exposton of ths paper. We thank Huseyn Topaloglu for sharng the CAVE mplementaton. Ths research s partally supported by NSF grants CMMI and CMMI

16 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems 16 Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 References Bertsekas, D. P Dynamc Programmng and Optmal Control. 2nd ed. Athena Scentfc. Bertsekas, D. P., S. E. Shreve Stochastc Optmal Control: The Dscrete-Tme Case. Athena Scentfc. Besbes, O., A. Muharremoglu On mplcatons of demand censorng n the newsvendor problem. Management Scence 596) Beyer, D., S. P. Seth, R. Srdhar Stochastc mult-product nventory models wth lmted storage. Journal of Optmzaton Theory and Applcatons Beyer, D., S. P. Seth, R. Srdhar Average-cost optmalty of a base-stock polcy for a mult-product nventory model wth lmted storage. G. Zaccour, ed., Decson & Control n Management Scence, Advances n Computatonal Management Scence, vol. 4. Sprnger US, Bookbnder, J. H., A. E. Lordahl Estmaton of nventory re-order levels usng the bootstrap statstcal procedure. IIE Transactons 214) Boyd, S., L. Vandenberghe Convex Optmzaton. Cambrdge Unversty Press, New York, NY, USA. Burnetas, A. N., C. E. Smth Adaptve orderng and prcng for pershable products. Operatons Research 483) Chen, L., E. L. Plambeck Dynamc nventory management wth learnng about the demand dstrbuton and substtuton probablty. Manufacturng & Servce Operatons Management 102) Cho, J., J. J. Cao, H. E. Romejn, J. Geunes, S. X. Ba A stochastc mult-tem nventory model wth unequal replenshment ntervals and lmted warehouse capacty. IIE Transactons 3712) Chu, L. Y., J. G. Shanthkumar, Z.-J. M. Shen Solvng operatonal statstcs va a bayesan analyss. Operatons Research Letters 361) Flaxman, A. D., A. T. Kala, H. B. McMahan Onlne convex optmzaton n the bandt settng: Gradent descent wthout a gradent. Proceedngs of the Sxteenth Annual ACM-SIAM Symposum on Dscrete Algorthms. SODA 05, Glasserman, P Gradent Estmaton Va Perturbaton Analyss. Kluwer nternatonal seres n engneerng and computer scence: Dscrete event dynamc systems, Sprnger. Godfrey, G. A., W. B. Powell An adaptve, dstrbuton-free algorthm for the newsvendor problem wth censored demands, wth applcatons to nventory and dstrbuton. Management Scence 478) Hazan, E Introducton to onlne convex optmzaton. Book Draft. Computer Scence, Prnceton Unversty. Avalable at Hazan, E., A. Kala, S. Kale, A. Agarwal Logarthmc regret algorthms for onlne convex optmzaton. In 19th COLT Huh, W. H., P. Rusmevchentong A non-parametrc asymptotc analyss of nventory plannng wth censored demand. Mathematcs of Operatons Research 341) Huh, W. H., P. Rusmevchentong, R. Lev, J. Orln Adaptve data-drven nventory control wth censored demand based on kaplan-meer estmator. Operatons Research 594) Huh, W. T., G. Janakraman, J. A. Muckstadt, P. Rusmevchentong An adaptve algorthm for fndng the optmal base-stock polcy n lost sales nventory systems wth censored demand. Mathematcs of Operatons Research 342) Ignall, E., A. F. Venott Optmalty of myopc nventory polces for several substtute products. Management Scence 155) Kleywegt, A. J., A. Shapro, T. Homem-de Mello The sample average approxmaton method for stochastc dscrete optmzaton. SIAM J. on Optmzaton 122) Kunnumkal, S., H. Topaloglu Usng stochastc approxmaton methods to compute optmal base-stock levels n nventory control problems. Operatons Research 563)

17 Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Artcle submtted to Operatons Research; manuscrpt no. OPRE R3 17 Larvere, M. A., E. L. Porteus Stalkng nformaton: Bayesan nventory management wth unobserved lost sales. Management Scence 453) Lev, R., G. Peraks, J. Uchanco The data-drven newsvendor problem: New bounds and nsghts. Operatons Research 636) Lev, R., R. O. Roundy, D. B. Shmoys Provably near-optmal samplng-based polces for stochastc nventory control models. Mathematcs of Operatons Research 324) Lyanage, L. H., J. G. Shanthkumar A practcal nventory control polcy usng operatonal statstcs. Operatons Research Letters 334) Loulou, R An explct upper bound for the mean busy perod n a GI/G/1 queue. Journal of Appled Probablty 152) Maglaras, C., S. Eren A maxmum entropy jont demand estmaton and capacty control polcy. Producton and Operatons Management 243) Nemrovsk, A., A. Judtsky, G. Lan, A. Shapro Robust stochastc approxmaton approach to stochastc programmng. SIAM J. on Optmzaton 194). Powell, W., A. Ruszczyńsk, H. Topaloglu Learnng algorthms for separable approxmatons of dscrete stochastc optmzaton problems. Mathematcs of Operatons Research 294) Shaked, M., J. G. Shanthkumar Stochastc Orders. Sprnger Seres n Statstcs, Physca-Verlag. Shalev-Shwartz, S Onlne learnng and onlne convex optmzaton. Found. Trends Mach. Learn. 42) Venott, Jr., A. F Optmal polcy for a mult-product, dynamc, nonstatonary nventory problem. Management Scence 123) pp Znkevch, M Onlne convex programmng and generalzed nfntesmal gradent ascent. Tom Fawcett, Nna Mshra, eds., Proceedngs of the 20th Internatonal Conference on Machne Learnng ICML). AAAI Press, Cambrdge, MA, USA, Zpkn, P Foundatons of Inventory Management. McGraw-Hll, New York. Bref Bo: Cong Sh s an assstant professor n the Department of Industral and Operatons Engneerng at the Unversty of Mchgan. Hs research les n stochastc optmzaton and onlne learnng theory wth applcatons to nventory and supply chan management, and revenue management. He won frst prze n the 2009 George Ncholson Student Paper Competton. Wedong Chen s a Ph.D. canddate n the Department of Industral and Operatons Engneerng at the Unversty of Mchgan. Hs research les n stochastc optmzaton and onlne learnng theory wth applcatons to nventory and supply chan management, and revenue management. Izak Duenyas s the Donald C. Cook Professor of Busness Admnstraton and Professor of Technology and Operatons at the Ross School of Busness. Hs recent research nterests are n prcng and revenue management, sourcng and procurement, and producton, nventory and capacty control.

18 e-companon to Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems ec1 Ths page s ntentonally blank. Proper e-companon ttle page, wth INFORMS brandng and exact metadata of the man paper, wll be produced by the INFORMS offce when the ssue s beng assembled.

19 ec2 e-companon to Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Electronc Companon to Nonparametrc Data-Drven Algorthms for Mult-Product Inventory Systems wth Censored Demand Cong Sh, Wedong Chen, Izak Duenyas Industral and Operatons Engneerng, Unversty of Mchgan, {shcong, Technology and Operatons, Ross School of Busness, Unversty of Mchgan, EC.1. Clarvoyant Optmal Polcy Proof of Theorem 1 Based on 3), we defne a myopc feasble closed-loop) polcy π as a sequence of functons ȳ t = π t x t ), t = 1,..., T, mappng begnnng nventory state) x t nto endng nventory decson) ȳ t, whch also myopcally mnmzes per-perod cost Π t ) wth begnnng nventory x t,.e., ȳ t x t ) arg mn Π t a). a Γ:a x t EC.1) The above feasble polcy π s myopc, because t only optmzes per-perod cost n each perod the mmedate reward). Ths s n contrast wth standard dynamc programmng or approxmate dynamc programmng approaches. To ease the presentaton of establshng optmalty of π, followng Ignall and Venott 1969), we keep x t, ȳ t, Π t tme-generc,.e., ȳx) arg mn a Γ:a x Πa). EC.2) It s mportant to see that ȳx) s the unque mnmzer of EC.2), due to Assumpton 1 ensurng strct jont) convexty of Πy) over the feasble regon, and the fact that the constrant set s affne see Boyd and Vandenberghe 2004)). Lemma EC.1. The optmzaton problem defned n EC.2) has a unque mnmzer ȳx). Proof of Lemma EC.1. Due to Assumpton 1, the cost functon Π ) s dfferentable and fntevalued. The dervatves nsde expectaton are bounded, and also the expectaton s a multple ntegraton over fnte ranges. Hence ths guarantees the valdty of nterchange between dfferentaton and expectaton. Next we argue that Π ) are strctly jontly) convex over the feasble regon. For all and j, 2 Πa) a ) 2 = h + p c )F D a ) > 0; 2 Πa) a a j = 0,

20 e-companon to Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems ec3 where Assumpton 1c) ensures F a ) > 0 for all a [0, M. Hence, the Hessan matrx s postve D defnte wth all strctly postve egenvalues) over the entre feasble regon, ensurng Π to be strctly jontly) convex. Now consder the optmzaton problem wth a gven startng nventory x) defned n EC.2). Snce Πy) s strctly jontly) convex and the constrant set s affne, ȳx) s the unque mnmzer. See Boyd and Vandenberghe 2004) for dscussons of unque mnmzer n convex optmzaton problems and also Example 5.4.). Q.E.D. Next we shall show that the myopc polcy π defned above s optmal. Ignall and Venott 1969) provded a suffcent condton called substtute property together wth two mld regularty assumptons) under whch the myopc polcy s optmal. Defnton EC.1 Substtute property). For any nventory levels x, x Γ, f x x, then ȳx) x ȳ x) x. Defnton EC.2 Regularty condtons n Ignall and Venott 1969)). The two regularty condtons n Ignall and Venott 1969) are: a) x x ȳx) mples ȳx) = ȳx ) for x, x Γ; b) The state transton permts ether pure, partal, or no backloggng lost-sales). The regularty condton a) s satsfed by ȳx) beng the unque mnmzer of EC.2) by Lemma EC.1, and the regularty condton b) s mmedate snce we consder a standard lost-sales model. We can now proceed to establsh the optmalty of myopc polces for the mult-product lostsales system by showng that the suffcent condton substtute property) gven above holds for our system. Proposton EC.1. Under Assumpton 1, when the demand dstrbuton s known a pror, the myopc orderng polcy defned n EC.1) s optmal for the mult-product lost-sales nventory systems. To prove Proposton EC.1, we need to derve several mportant propertes of the myopc polcy. Now consder the two possble startng nventory levels x and x, wth x x. For notatonal superscrpt) convenence, we use θ nstead of y to be the global mnmzer of Π ) over Γ. Recall that θ = y arg mn a Γ Πa), and also the myopc order-to-up level ȳx) arg mn a Γ:a x Πa). For smplcty, we defne the boundary of our warehouse storage constrant, { } Γ y R n + : y = M. =1

21 ec4 e-companon to Sh, Chen, and Duenyas: Nonparametrc Algorthms for Mult-Product Inventory Systems Note that y Γ means that the total order-up-to levels have reached the total storage lmt M. If y / Γ, then the warehouse storage constrant s not tght. Now denote the j th partal dervatve of Π ) by Π j ). We then develop some useful propertes of the myopc order-up-to levels ȳ ). Lemma EC.2. Let x Γ and θ be the global mnmzer of Π ) over Γ, ) x j θ j ȳ j x) = x j. ) x j θ j ȳ j x) θ j. Proof of Lemma EC.2. The proof s straghtforward. Statement ) holds because x j θ j the startng nventory s hgher than the global mnmzer) for product j, t s sub-optmal to order any more product j. Statement ) holds because f x j θ j the startng nventory s lower than the global mnmzer), t s sub-optmal to rase the nventory above the global mnmzer. Q.E.D. Lemma EC.3. Let x Γ and θ be the global mnmzer of Π ) over Γ, ) θ Γ ȳx) Γ; ) ȳx) / Γ, x j θ j ȳ j x) = θ j. In Lemma EC.3, statement ) states that f the global mnmzer occupes the entre storage space, then the myopc order-up-to levels wll also occupy the entre storage space. Ths s because our myopc polcy wll always order as much as possble to approach the global mnmzer. Statement ) states that f the total myopc order-up-to level has not reached the storage lmt M, then f x j θ j, the myopc polcy wll rase nventory level for product j to the global mnmzer θ j. Proof of Lemma EC.3. We prove ) by contradcton. Suppose that θ Γ and ȳx) / Γ, then θ = M =1 and ȳ x) < M. =1 It s obvous that there exsts at least one j such that ȳ j x) < θ j. Snce θ mnmzes Π ) over Γ, t s clear that θ ether reaches the global mnmzer of Π ) over the entre real lne R or s smaller than t due to the storage constrant, so the dervatve Π jθ) 0. Therefore, snce Π ) s strctly convex, Π jȳx)) < Π jθ) 0. On the other hand, snce ȳx) / Γ and ȳx) s a mnmzer of Π ) over set {y y x, y Γ}, t s clear that ȳx) ether reaches θ or s greater than t because of the ntal on-hand nventory, so Π jȳx)) 0, whch results n a contradcton, thereby provng ). To prove ), we observe from the contraposton of ),.e., ȳx) / Γ θ / Γ. Then for any product j, ȳ j x) s not restrcted by the storage constrant, and thus f θ j x j, then θ j can always be reached, mplyng that ȳ j x) = θ j. Ths completes the proof. Q.E.D.