Risk Aversion in Inventory Management

Transcription

1 OPERATIONS RESEARCH Vol. 55, No. 5, Sepember Ocober 2007, pp issn X eissn inorms doi /opre INFORMS Risk Aversion in Invenory Managemen Xin Chen Deparmen o Indusrial and Enerprise Sysems Engineering, Universiy o Illinois a Urbana Champaign, Urbana, Illinois 61801, xinchen@uiuc.edu Melvyn Sim Singapore-MIT Alliance and NUS Business School, Naional Universiy o Singapore, Singapore, dscsimm@nus.edu.sg David Simchi-Levi Deparmen o Civil and Environmenal Engineering and Engineering Sysem Division, Massachuses Insiue o Technology, Cambridge, Massachuses 02139, dslevi@mi.edu Peng Sun Fuqua School o Business, Duke Universiy, Durham, Norh Carolina 27708, psun@duke.edu Tradiional invenory models ocus on risk-neural decision makers, i.e., characerizing replenishmen sraegies ha maximize expeced oal proi, or equivalenly, minimize expeced oal cos over a planning horizon. In his paper, we propose a ramework or incorporaing risk aversion in muliperiod invenory models as well as muliperiod models ha coordinae invenory and pricing sraegies. We show ha he srucure o he opimal policy or a decision maker wih exponenial uiliy uncions is almos idenical o he srucure o he opimal risk-neural invenory (and pricing policies. These srucural resuls are exended o models in which he decision maker has access o a (parially complee inancial marke and can hedge is operaional risk hrough rading inancial securiies. Compuaional resuls demonsrae ha he opimal policy is relaively insensiive o small changes in he decision-maker s level o risk aversion. Subjec classiicaions: invenory/producion: policies, uncerainy; decision analysis: risk. Area o review: Manuacuring, Service, and Supply Chain Operaions. Hisory: Received March 2004; revision received Augus 2006; acceped Augus Published online in Aricles in Advance Sepember 14, Inroducion Tradiional invenory models ocus on characerizing replenishmen policies so as o maximize he expeced oal proi, or equivalenly, o minimize he expeced oal cos over a planning horizon. O course, his ocus on opimizing expeced proi or cos is appropriae or risk-neural decision makers, i.e., invenory planners ha are insensiive o proi variaions. Evidenly, no all invenory planners are risk neural; many are willing o rade o lower expeced proi or downside proecion agains possible losses. Indeed, experimenal evidence suggess ha or some producs, he so-called high-proi producs, decision makers exhibi riskaverse behavior; see Schweizer and Cachon (2000 or more deails. Unorunaely, radiional invenory conrol models all shor o meeing he needs o risk-averse planners. For insance, radiional invenory models do no sugges mechanisms o reduce he chance o unavorable proi levels. Thus, i is imporan o incorporae he noions o risk aversion in a broad class o invenory models. The lieraure on risk-averse invenory models is quie limied and mainly ocuses on single-period problems. Lau (1980 analyzes he classical newsvendor model under wo dieren objecive uncions. In he irs objecive uncion, he ocus is on maximizing he decision-maker s expeced uiliy o oal proi. The second objecive uncion is he maximizaion o he probabiliy o achieving a cerain level o proi. Eeckhoud e al. (1995 ocus on he eecs o risk and risk aversion in he newsvendor model when risk is measured by expeced uiliy uncions. In paricular, hey deermine comparaive-saic eecs o changes in he various price and cos parameers in he risk-aversion seing. Chen and Federgruen (2000 analyze he mean-variance rade-os in newsvendor models as well as some sandard ininie-horizon invenory models. Speciically, in he ininie-horizon models, Chen and Federgruen ocus on he mean-variance rade-o o cusomer waiing ime as well as he mean-variance rade-os o invenory levels. Marínez-de-Albéniz and Simchi-Levi (2003 sudy he mean-variance rade-os aced by a manuacurer signing a porolio o opion conracs wih is suppliers and having access o a spo marke. The paper by Bouakiz and Sobel (1992 is closely relaed o ours. In his paper, he auhors characerize he invenory replenishmen sraegy so as o minimize he expeced uiliy o he ne presen value o coss over a inie planning horizon or an ininie horizon. Assuming linear ordering cos, hey prove ha a base-sock policy is opimal. 828

2 Operaions Research 55(5, pp , 2007 INFORMS 829 Table 1. Summary o previous resuls and new conribuions. Price is no a decision Price is a decision (Capaciy k = 0 k>0 (Capaciy k = 0 k>0 Risk-neural model (Modiied base sock s S (Modiied base-sock lis price ssap Exponenial uiliy (Modiied base sock s S (Modiied base sock ssap Increasing and concave uiliy Wealh dependen (modiied? Wealh dependen (modiied? base sock base sock World-driven model parameer Sae dependen (modiied Sae dependen Sae dependen (modiied Sae dependen exponenial uiliy base sock s S base sock ssap Parially complee inancial Sae dependen (modiied Sae dependen Sae dependen (modiied Sae dependen marke exponenial uiliy base sock s S base sock ssap Indicaes exising resuls in he lieraure. Indicaes a similar exising resul based on a special case o our model. So ar all he papers reerenced above assume ha demand is exogenous. A rare excepion is Agrawal and Seshadri (2000, who consider a risk-averse reailer which has o decide on is ordering quaniy and selling price or a single period. They demonsrae ha dieren assumpions on he demand-price uncion may lead o dieren properies o he selling price. Recenly, we have seen a growing ineres in hedging operaional risk using inancial insrumens. As ar as we know, all o his lieraure ocuses on single-period (newsvendor models wih demand disribuion ha is correlaed wih he reurn o he inancial marke. This can be raced back o Anvari (1987, which uses he capial asse pricing model (CAPM o sudy a newsvendor acing normal demand disribuion. Chung (1990 provides an alernaive derivaion or he resul. More recenly, Gaur and Seshadri (2005 invesigae he impac o inancial hedging on he operaions decision, and Caldeney and Haugh (2006 show ha dieren inormaion assumpions lead o dieren ypes o soluion echniques. In his paper, we propose a general ramework o incorporae risk aversion ino muliperiod invenory (and pricing models. Speciically, we consider wo closely relaed problems. In he irs one, demand is exogenous, i.e., price is no a decision variable, while in he second one, demand depends on price and price is a decision variable. In boh cases, we disinguish beween models wih ixed-ordering coss and models wih no ixed-ordering cos. We assume ha he irm we model is a privae irm, hereore here is no conlic o ineress beween share holders and managers. Following Smih (1998, we ake he sandard economics perspecive in which he decision maker maximizes he oal expeced uiliy rom consumpion in each ime period. In 2, we discuss in more deail he heory o expeced uiliy employed in a muliperiod decision-making ramework. We exend our ramework in 4 by incorporaing a parially complee inancial marke so ha he decision maker can hedge operaional risk hrough rading inancial securiies. Observe ha i he uiliy uncions are linear and increasing, he decision maker is risk neural and hese problems are reduced o he classical inie-horizon sochasic invenory problem and he inie-horizon invenory and pricing problem. We summarize known and new resuls in Table 1. The row risk-neural model presens a summary o known resuls. For example, when price is no a decision variable, and here exiss a ixed ordering cos, k>0, Scar (1960 proved ha an s S invenory policy is opimal. In such a policy, he invenory sraegy a period is characerized by wo parameers s S. When he invenory level x a he beginning o period is less han s, an order o size S x is made. Oherwise, no order is placed. A special case o his policy is he base-sock policy, in which s = S is he base-sock level. This policy is opimal when k = 0 In addiion, i here is a capaciy consrain on he ordering quaniy (expressed as (Capaciy in he able, hen he modiied base-sock policy is opimal (expressed as (Modiied in he able. Tha is, when he invenory level is below he base-sock level, order enough o raise he invenory level o he base-sock level i possible or order an amoun equal o he capaciy; oherwise, no order is placed. I price is a decision variable and here exiss a ixed-ordering cos, he opimal policy o he risk-neural model is an ssap policy; see Chen and Simchi-Levi (2004a. In such a policy, he invenory sraegy a period is characerized by wo parameers s S and a se A s s + S /2, possibly empy depending on he problem insance. When he invenory level x a he beginning o period is less han s or x A, an order o size S x is made. Oherwise, no order is placed. Price depends on he iniial invenory level a he beginning o he period. When A is empy or all, we reer o such a policy as he ssp policy. A special case o his model is when k = 0, or which a base-sock lis price policy is opimal. In his policy, invenory is managed based on a base-sock policy and price is a nonincreasing uncion o invenory a he beginning o each period. Again, when here is an ordering capaciy consrain, a modiied base-sock invenory policy is opimal (see Federgruen and Heching 1999, Chen and Simchi-Levi 2004a.

3 830 Operaions Research 55(5, pp , 2007 INFORMS Table 1 suggess ha when risk is measured using addiive exponenial uiliy uncions, he srucures o opimal policies are almos he same as he one under he risk-neural case. For example, when price is no a decision variable and k>0, he opimal replenishmen sraegy ollows he radiional invenory policy, namely, an s S policy. A corollary o his resul is ha a base-sock policy is opimal when k = 0. Noe ha he opimal policy characerized by Bouakiz and Sobel (1992 has he same srucure as he opimal policy in our model. Finally, when k = 0 and here is an ordering capaciy consrain, a (modiied base-sock policy is opimal. The las row o Table 1 provides inormaion on he opimal policy or a decision maker wih an exponenial uiliy uncion having access o a parially complee inancial marke. Such a marke allows he risk-averse invenory planner o hedge is operaional coss and par o he demand risks. I he inancial marke is complee, insead o parially complee, our model reduces o he risk-neural case and hence we have he same srucural resuls as he risk-neural model wih respec o he marke risk-neural probabiliy. We will explain he meaning o sae dependen when we presen he model in 4. We complemen he heoreical resuls wih a numerical sudy illusraing he eec o risk aversion on he invenory policies. This paper is organized as ollows. In 2, we review classical expeced uiliy approaches in risk-averse valuaion. In 3, we propose a model o incorporae risk aversion in a muliperiod invenory (and pricing seing, and ocus on characerizing he opimal invenory policy or a risk-averse decision maker. We hen generalize he resuls in 4 by considering he inancial hedging opion. Secion 5 presens he compuaional resuls illusraing he eecs o dieren risk-averse muliperiod invenory models on invenory conrol policies. Finally, 6 provides some concluding remarks. We complee his secion wih a brie saemen on noaions. Speciically, a variable wih a ilde over i, such as d, denoes a random variable. 2. Uiliy Theory or Risk-Averse Valuaions Modelling risk-sensiive decision making is one o he undamenal problems in economics. A basic heoreical ramework or risk-sensiive decision making is he so-called expeced uiliy heory (see, e.g., Mas-Collel e al. 1995, Chaper 6. Assume ha a decision maker has o make a decision in a single-period problem beore uncerainy is resolved. According o expeced uiliy heory, he decision-maker s objecive is o maximize he expecaion o some appropriaely chosen uiliy uncion o he decision-maker s payo. Such a modeling ramework or risk-sensiive decision making is esablished mahemaically based on an axiomaic argumen. Tha is, based on a cerain se o axioms regarding he decision-maker s preerence over loeries, one can show he exisence o such a uiliy uncion and ha he decision-maker s choice crierion is he expeced uiliy (see, or example, Heyman and Sobel 1982, Chapers 2 4; Fishburn For muliperiod problems, one approach o modeling risk aversion ha seems naural is o maximize he expeced uiliy o he ne presen value o he income cash low. In calculaing he ne presen value, one may ake he ineres rae or risk-ree borrowing and lending as he discoun acor, relecing he ac ha he decision maker could borrow and lend over ime and conver any deerminisic cash low ino is ne presen value. Models based on his approach are reerred o as he ne presen value models. Sobel (2005 reers o he uiliy uncion used in such a ramework as he inerperiod uiliy uncion. Noe ha he ne presen value models have been employed by Bouakiz and Sobel (1992 and Chen e al. (2004 o analyze he muliperiod invenory replenishmen problems o a risk-averse invenory manager. However, in he economics lieraure i has long been known ha applying expeced uiliy mehods direcly o income cash lows causes he so-called emporal risk problem i does no capure he decision-maker s sensiiviy o he ime a which uncerainies are resolved (see, e.g., a summary descripion o his problem in Smih One way o overcome he emporal risk problem is o explicily model he uiliy over a low o consumpion, allowing he decision maker o borrow and lend o smooh he income low as he uncerainies unold over ime. More generally a decision maker can rade on inancial markes o adjus her consumpions over ime. Thereore, an alernaive modeling approach or he muliperiod invenory conrol problem is o direcly model consumpion, saving and borrowing decisions as well as invenory replenishmen and pricing decisions. Speciically, assume ha he decision maker has access o a inancial marke or borrowing and lending wih a riskree saving and borrowing ineres rae r. A he beginning o period, assume ha he decision maker has iniial wealh w and chooses an operaions policy (invenory/ pricing ha aecs her income cash low. A he end o period, ha is, aer he uncerainy o his period has been resolved, he decision maker observes her curren wealh level w + P and decides her consumpion level or his period, where P is he income generaed a period. The remaining wealh, w + P, is hen saved (or borrowed, i negaive or he nex period. Thus, he nex period s iniial wealh is w +1 = 1 + r w + P Equivalenly, we can model w +1 as a decision variable and calculae he consumpion, = w w r + P

4 Operaions Research 55(5, pp , 2007 INFORMS 831 The decision-maker s objecive is o maximize her expeced uiliy o he consumpion low, EU 1 T over he planning horizon 1T. We call such an approach he consumpion model. Smih (1998 provides an excellen comparison beween he consumpion model and he ne presen value model. Similar o single-period problems, axiomaic approaches were also employed o derive cerain ypes o uiliy uncions or muliperiod problems (see, e.g., Sobel 2005; Keeney and Raia 1993, Chaper 9. In paricular, he so-called addiive independence axiom 1 implies addiive uiliy uncions o he ollowing orm: T U 1 2 T = u =1 Tha is, he uiliy o he consumpion low is he summaion o he uiliy rom he consumpion in each ime period, where uncion u is increasing and concave. Sobel (2005 reers o uncions u as he inraperiod uiliy uncions. As a special case o he general inraperiod uiliy uncions, he exponenial uiliy uncions are also commonly used in economics (Mas-Collel e al and decision analysis (Smih In his case, he uiliy uncion has he orm u = a e / or some parameers a > 0 and > 0. Howard (1988 indicaes ha exponenial uiliy uncions are widely applied in decision analysis pracice. Kirkwood (2004 shows ha in mos cases, an appropriaely chosen exponenial uiliy uncion is a very good (local approximaion or general uiliy uncions. In he nex secion, we characerize he srucures o he opimal invenory policies according o he consumpion model. Ineresingly, he ne presen value model is mahemaically a special case o he consumpion model, as will be illusraed in 3. This implies ha he srucures o he opimal invenory policies or he consumpion models are also valid or he corresponding ne presen value models. A his poin i is worh menioning ha Savage (1954 uniied von Neumann and Morgensern s heory o expeced uiliy and de Finei s heory o subjecive probabiliy and esablished he subjecive expeced uiliy heory. Wihou assuming probabiliy disribuions and uiliy uncions, he Savage heory sars rom a se o assumpions on he decision maker s preerences and shows he exisence o a (subjecive probabiliy disribuion depending on he decision maker s belie on he uure sae o he world as well as a uiliy uncion. The decision maker s objecive is o maximize he expeced uiliy, wih he expecaion aken according o he subjecive probabiliy disribuion. In 4, o inroduce he ramework o risk-averse invenory managemen wih inancial hedging opporuniies, we explicily consider he decision maker s subjecive probabiliy and disinguish i rom he so-called risk-neural probabiliy releced by a (parially complee inancial marke wih no arbirage opporuniy. A similar approach has been employed by Smih and Nau (1995 and Gaur and Seshadri ( Muliperiod Invenory Models Consider a risk-averse irm ha has o make replenishmen (and pricing decisions over a inie ime horizon wih T periods. Demands in dieren periods are independen o each oher. For each period, = 1 2 le d = demand in period, p = selling price in period, p, p are lower and upper bounds on p, respecively. Observe ha when p = p or each period, price is no a decision variable and he problem is reduced o an invenory conrol problem. Throughou his paper, we concenrae on demand uncions o he ollowing orms: Assumpion 1. For = 1 2 he demand uncion saisies d = D p = p (1 where =, and are wo nonnegaive random variables wih E >0 and E >0. The random perurbaions, are independen across ime. Le x be he invenory level a he beginning o period, jus beore placing an order. Similarly, y is he invenory level a he beginning o period aer placing an order. The ordering cos uncion includes boh a ixed cos and a variable cos and is calculaed or every, = 1 2 as ky x + c y x where x = { 1 i x>0 0 oherwise. Lead ime is assumed o be zero, and hence an order placed a he beginning o period arrives immediaely beore demand or he period is realized. Unsaisied demand is backlogged. Thereore, he invenory level carried over rom period o he nex period, x +1, may be posiive or negaive. A cos h x +1 is incurred a he end o period which represens invenory holding cos when x +1 > 0 and shorage cos i x +1 < 0. For echnical reasons, we assume ha he uncion h x is convex and lim x h x =. A he beginning o period, he invenory planner decides he order-up-o level y and he price p. Aer observing he demand, she hen makes consumpion decision. Thus, given he iniial invenory level x, he order-up-o level y, and he realizaion o he uncerainy, he income a period is P x y p = ky x c y x + p D p h y D p Moreover, as discussed in he previous secion, he consumpion decision a period is equivalen o deciding on

5 832 Operaions Research 55(5, pp , 2007 INFORMS he iniial wealh level o period + 1. Le w be he iniial wealh level a period. Then, = w w r + P x y p Finally, a he las period T, we assume ha he invenory planner consumes everyhing, which corresponds o w T +1 = 0. According o he consumpion model, he invenory planner s decision problem is o ind he order-up-o levels y, he selling price p, and decide he iniial wealh level w (or equivalenly, he consumpion level or he ollowing opimizaion problem: max EU 1 2 T s.. y x x +1 = y D p = w w r + P x y p w T +1 = 0 When he uiliy uncion U 1 2 T akes he ollowing linear orm: U 1 2 T = T =1 1 + r 1 he consumpion model reduces o he radiional riskneural invenory (and pricing problem analyzed by Chen and Simchi-Levi (2004a. In his case, we denoe V x o be he proi-o-go uncion a he beginning o period wih invenory level x. A naural dynamic program or he risk-neural invenory (and pricing problem is as ollows (see Chen and Simchi-Levi 2004a or more deails: V x = c x + max ky x + g y p (3 yx p p p where V T +1 x = 0 or any x and [ g y p = E pd p c y h y D p + 1 ] V 1 + r +1 y D p (4 The ollowing heorem presens known resuls or he radiional risk-neural models. Theorem 3.1. (a I price is no a decision variable (i.e., p = p or each, V x and g y p are k-concave and an s S invenory policy is opimal. (b I he demand is addiive (i.e., is a consan, V x and max p p p g y p are k-concave and an ssp policy is opimal. (2 (c For he general case, V x and g y p are symmeric k-concave and an ssap policy is opimal. Par (a is he classical resul proved by Scar (1960 using he concep o k-convexiy; par (b and par (c are proved in Chen and Simchi-Levi (2004a using he conceps o k-convexiy, or par (b, and a new concep, he symmeric k-convexiy, or par (c. These conceps are summarized in Appendix B. In ac, he resuls in Chen and Simchi-Levi (2004a hold rue under more general demand uncions han hose in Assumpion 1. In he ollowing subsecions, we analyze he consumpion model based on he addiive uiliy uncions and is special case, he addiive exponenial uiliy model Addiive Uiliy Model In his subsecion, we ocus on he addiive uiliy uncions. According o he sequence o evens as described beore, he opimizaion model (2 can be solved by he ollowing dynamic programming recursion: V x w = max E W xwyp (5 yx p p p in which W xwyp = max {u (w w + P w 1 + r xyp } + V +1 y D p w (6 wih boundary condiion W T x w = u T w + P xyp Noe ha unlike he radiional risk-neural invenory models, where he sae variable in he dynamic programming recursion is he curren invenory level, here we augmen he sae space by inroducing a new sae variable, namely, he wealh level w. Insead o working wih he dynamic program (5 (6, we ind ha i is more convenien o work wih an equivalen ormulaion. Le U x w = V x w c x and he modiied income a period be ( ( c+1 P y p = c 1 + r y + p c +1 D 1 + r p h y D p The dynamic program (5 (6 becomes U x w = max EW xwyp (7 yx p p p in which W xwyp ( = max {u w z ky x + P z 1 + r y p } + U +1 y D p z (8

6 Operaions Research 55(5, pp , 2007 INFORMS 833 Theorem 3.2. Assume ha k = 0. In his case, U x w is joinly concave in x and w or any period. Furhermore, a wealh (w dependen base-sock invenory policy is opimal. Proo. We prove by inducion. Obviously, U T +1 x w is joinly concave in x and w. Assume ha U +1 x w is joinly concave in x and w. We now prove ha a wealhdependen base-sock invenory policy is opimal and U x w is joinly concave in x and w. Firs, noe ha or any realizaion o, P is joinly concave in y p. Thus, ( W wyp = max {u w z + P z 1 + r y p } + U +1 y D p z is joinly concave in wyp, which urher implies ha EW wyp is joinly concave in wyp. We now prove ha a w-dependen base-sock invenory policy is opimal. Le y w be an opimal soluion or he problem { } max max EW wyp y p p p Because EW wyp is concave in y or any ixed w, i is opimal o order up o y w when x<y w and no o order oherwise. In oher words, a sae-dependen base-sock invenory policy is opimal. Finally, according o Proposiion 4 in Appendix B, U x w is joinly concave. Theorem 3.2 can be exended o incorporae capaciy consrains on he order quaniies. In his case, i is sraighorward o see ha he proo o Theorem 3.2 goes hrough. The only dierence is ha in his case, a w-dependen modiied base-sock policy is opimal. In such a policy, when he iniial invenory level is no more han y w, order up o y w i possible; oherwise order up o he capaciy. On he oher hand, no order is placed when he iniial level is above y w. Recall ha in he case o a risk-neural decision maker, a base-sock lis-price policy is opimal. Theorem 3.2 hus implies ha he opimal invenory policy or he expeced addiive uiliy risk-averse model is quie dieren. Indeed, in he risk-averse case, he base-sock level depends on he wealh, measured by he posiion o he risk-ree inancial securiy. Moreover, i is no clear in his case wheher a lisprice policy is opimal or he wealh/consumpion decisions have any nice srucure. Nex, we argue ha he ne presen value model is mahemaically a special case o he consumpion model. Indeed, i he decision maker s uiliy uncions in each period = 1T 1 are all in he orm o u x = exp x/r wih R 0 +, excep in period = T, he consumpion model (5 (6 mahemaically reduces o he ne presen value model wih inraperiod uiliy uncion U = u T. The inuiion is also clear. In ac, R (commonly reerred o as he risk-olerance parameer approaching zero implies ha he decision maker becomes exremely risk averse, and hus any negaive consumpion inroduces a negaive ininie uiliy, while any nonnegaive consumpion inroduces zero uiliy. Thereore, he consumpions in period = 1T 1 have o be nonnegaive and he uiliy is always zero. The decision maker is beer o by shiing all he consumpions o he las period, which is equivalen o he ne presen value model. This also implies ha he same srucural resuls in Theorem 3.2 and hose o be presened in he nex secion also hold or he ne presen value model. Sronger resuls exis or models based on he addiive exponenial uiliy risk measure, as is demonsraed in he nex subsecion Exponenial Uiliy Funcions We now ocus on exponenial uiliy uncions o he orm u = a e /, wih parameers a > 0. is he risk-olerance acor, while a relecs he decision maker s aiude oward he uiliy obained rom dieren periods. The beauy o exponenial uiliy uncions is ha we are able o separaely make he invenory decisions wihou considering he wealh/consumpion decisions. This is discovered by Smih (1998 in he decision-ree ramework. The nex heorem saes his resul in dynamic programming language. For compleeness, a proo is presened in Appendix A. To sae he heorem, we irs inroduce some noaion. For a risk-olerance parameer R, denoe he cerainy equivalen operaor wih respec o a random variable o be E R = R ln Ee /R For a decision maker wih risk-olerance R and an exponenial uiliy uncion, he above cerainy equivalen represens he amoun o money she eels indieren o a gamble wih random payo. Similarly, we denoe he condiional cerainy equivalen operaor wih respec o a random variable given o be E R = R ln Ee /R We also consider he eecive risk olerance per period, deined as R = T = (9 1 + r Theorem 3.3. Assume ha u = a e /. The invenory decisions in he risk-averse invenory conrol model

7 834 Operaions Research 55(5, pp , 2007 INFORMS Equaions (5 (6 can be calculaed hrough he ollowing dynamic programming recursion: G x= max ky x yx p p p +E R [ P yp r G +1 y D p ] (10 and G T +1 x = 0. The opimal consumpion decision a each period = 1T 1 is wxyp d = R [ w+ ( ky x + P y p + 1 ] G 1 + r +1 y d + C in which C is a consan ha does no depend on wxypd. The heorem hus implies ha when addiive exponenial uiliy uncions are used: (i he opimal invenory policy is independen o he wealh level; (ii he opimal invenory replenishmen and pricing decisions can be obained regardless o he wealh/consumpion decisions; (iii he opimal consumpion decision is a simple linear uncion o he curren wealh level; and (iv he model parameer a does no aec he invenory replenishmen and pricing decisions. Thus, incorporaing he addiive exponenial uiliy uncion signiicanly simpliies he problem. This heorem, ogeher wih Theorem 3.2, implies ha when k = 0, a base-sock invenory policy is opimal under he exponenial uiliy risk crierion independen o wheher price is a decision variable. I, in addiion, here is a capaciy consrain on ordering, one can show ha a wealh-independen modiied base-sock policy is opimal. As beore, i is no clear wheher a lis-price policy is opimal when k = 0 and price is a decision variable. Because he ne presen value model is a special case o he consumpion model, our base-sock policy direcly implies he resul based on he ne presen value model obained by Bouakiz and Sobel (1992 using a more complicaed argumen. To presen our main resul or he problem wih k>0, we need he ollowing proposiion. Proposiion 1. I a uncion x is concave, k-concave, or symmeric k-concave in x or any realizaion o, hen or any R>0, he uncion gx = E R x is also concave, k-concave, or symmeric k-concave, respecively. Proo. We only prove he case wih K-convexiy; he oher wo cases can be proven by ollowing similar seps. Deine Mx = Eexp x. I suices o prove ha or any x 0 x 1 wih x 0 x 1 and any 0 1, Mx Mx 0 1 Mx 1 expk where x = 1 x 0 + x 1. Noe ha Mx Eexp1 x 0 + x 1 +K =expkeexp1 x 0 exp x 1 expkeexp x 0 1 Eexp x 1 =Mx 0 1 Mx 1 expk where he irs inequaliy holds because is K-convex, and he second inequaliy ollows rom he Hölder inequaliy wih 1/p = 1 and 1/q =. We can now presen he opimal policy or he risk-averse muliperiod invenory (and pricing problem wih addiive exponenial uiliy uncions. Theorem 3.4. (a I price is no a decision variable (i.e., p = p or each, G x and L y p are k-concave and an s S invenory policy is opimal. (b For he general case, G x and L y p are symmeric k-concave and an ssap policy is opimal. Proo. The main idea is as ollows: i G +1 x is k-concave when price is no a decision variable (or symmeric k-concave or he general case, hen, by Proposiion 1, G y p is k-concave (or symmeric k-concave. The remaining pars ollow direcly rom Lemma 1 and Proposiion 2 or k-concaviy or Lemma 2 and Proposiion 3 or symmeric k-concaviy. See Lemma 1, Proposiion 2, Lemma 2, and Proposiion 3 in Appendix B. We observe he similariies and dierences beween he opimal policy under he exponenial uiliy measure and he one under he risk-neural case. Indeed, when demand is exogenous, i.e., price is no a decision variable, an s S invenory policy is opimal or he risk-neural case; see Theorem 3.1, par (a. Theorem 3.4 implies ha his is also rue under he exponenial uiliy measure. Similarly, or he more general invenory and pricing problem, Theorem 3.1, par (c implies ha an ssap policy is opimal or he risk-neural case. Ineresingly, his policy is also opimal or he exponenial uiliy case. O course, he resuls or he risk-neural case are a bi sronger. Indeed, i demand is addiive, Theorem 3.1, par (b suggess ha an ssp policy is opimal. Unorunaely, we are no able o prove or disprove such a resul or he exponenial uiliy measure.

8 Operaions Research 55(5, pp , 2007 INFORMS World-Driven Model Parameers Nex, we exend he resuls or he exponenial uiliy uncion o he case o world-driven model parameers. Following Song and Zipkin (1993, we assume ha a each ime period, he business environmen could be in one o a number o possible levels. Invenory model parameers and he suicien saisics o he demand disribuion depend on he hisory o he evoluion o he business environmen. Formally, le inie se represen he se o business environmens in period. We use boldace = =1 o represen he se o rajecories o levels rom period 1 o. Each rajecory,, is reerred o as he sae o he world, which is used o model relevan economic acors ha aec he producion/invenory cos and revenue. A sae o he world uniquely deermines he cos parameers and he suicien saisics o he demand disribuion o he invenory model. Tha is, a each ime period, parameers c, h, p, and p are all uncions o (we express hem as c, h, p, and p, respecively, and he disribuions o and are also -dependen. For he sae-dependen uncerain demand, we denoe. Similarly o wha we have done earlier, deine P y p +1 ( c = c y + (p c D 1 + r 1 + r p = h ( y D p (11 which can be hough o as he decision maker s modiied income a period. The ollowing heorem is a naural exension o Theorems 3.3 and 3.4. Theorem 3.5. Separaion The opimal invenory and pricing decisions or he world-driven parameer model may be solved hrough he ollowing dynamic programming recursion: G x =c x+ in which L yp max ypyx p p p ky x+l yp (12 [ [ = E R E R +1 P y p G 1 + r +1 ]] y D p +1 (13 and wih boundary condiion G T +1 x = 0. Thus, he consumpion decisions are decoupled rom he invenory ( pricing decisions. Srucural policy The ollowing srucural resuls or he opimal invenory ( pricing policies holds. (a I price is no a decision variable (i.e., p = p or each, or each given, uncions G x are k -concave in x and a -dependen s S invenory policy is opimal. (b For he general case, G x are symmeric k - concave in x or any given and a -dependen s S A p policy is opimal. Noe ha he separaion resul in Theorem 3.5 could be exended o he siuaion where he ixed cos k is world driven. I we urher have he ollowing condiion or all, 1 + r k max k we have ha he value uncions G x are k -concave (symmeric k -concave and a -dependen s S ( - dependen ssap invenory policy is opimal. In he nex secion, we urher exend he world-driven parameer model by considering he siuaion ha he invenory planner has access o a inancial marke o hedge he risks associaed wih lucuaions in he saes o he world. 4. Muliperiod Invenory Models wih Financial Hedging Opporuniies The modern inancial marke provides opporuniies o replicae many o he changes in he sae o he world. Thereore, a risk-averse invenory planner may use he inancial marke o hedge he risks rom changes in he business environmen. For example, i he producion cos is a uncion o he oil price, he invenory planner may hedge he oil price risks hrough rading inancial securiies on oil prices. Similarly, i he demand disribuion is aeced by he general economic siuaion, inancial insrumens on he marke indices provide he possibiliy o hedging he risks o general rend in demand. In his secion, we exend our previous ramework by assuming ha he decision maker has opporuniies o hedging operaional risk hrough rading inancial securiies in a so-called parially complee inancial marke. Similarly o he previous secion, we consider a riskaverse invenory planner who has o make replenishmen (and pricing decisions over a inie ime horizon wih T periods. The invenory, pricing, and rading decisions are made a ime periods = 1T. The model parameers are world driven as deined in he las secion. In his secion, we explicily assume ha he ixed cos k is world driven. Besides he risk-ree borrowing and saving opporuniy (cash, we assume ha here are anoher N inancial securiies in he inancial marke. To simpliy noaion and analysis, assume ha hese securiies do no pay dividends during he ime horizon 1T. We denoe he prices o he securiies as a marix Q such ha is componen Q i denoes he price o securiy i a ime (measured by period dollar. We ollow he usual assumpion in he realopions lieraure ha he inancial securiy could be raded a he exac desired amoun and here is no ransacion ee.

9 836 Operaions Research 55(5, pp , 2007 INFORMS Following Smih and Nau (1995, we reer o he risks associaed wih he evoluion o he sae o he world as marke risk i can be ully hedged in he inancial marke. On he oher hand, given he sae o he world, he demand uncerainy in our model is he so-called privae risk ha canno be hedged in he inancial marke. The exisence o he privae risk conribues o he incompleeness o he inancial marke, which Smih and Nau (1995 called a parially complee inancial marke. Noe ha in his secion, we explicily disinguish subjecive probabiliies rom he probabiliy disribuion ha can be inerred rom he inancial marke, known as he risk-neural probabiliy, which will be inroduced in he ollowing subsecion. The nex subsecion is devoed o he ormal descripion o a complee inancial marke. The discussion o he complee marke and no arbirage condiions are sandard in he inance lieraure. For a discree-ime reamen o such a inancial marke, we reer readers o Pliska ( Complee Financial Marke and Risk-Neural Probabiliies Formally, a inancial marke is complee i we have he ollowing condiions. Assumpion 2. (1 Q i only depends on he sae o he world. Tha is, or any rajecory T = 1 T and is subrajecories = 1, we can uniquely deine he price sequence o inancial securiy i as he (row vecor Q i T. (2 Any cash low deermined by he sae o he world can be replicaed by rading he inancial securiies. Tha is, or any given period, he vecor is a linear combinaion o he vecors 1 and Q 1 Q N. Here, is any mapping rom o a real number represening a sae o he world adaped cash low. (3 Disclosed demand inormaion in each ime period is no correlaed wih any uure evoluion o he sae o he world. Tha is, given, he decision maker believes ha d and +1 are independen. We also assume ha Assumpion 3. The inancial marke is arbirage ree. Inuiively speaking, arbirage ree means ha one canno guaranee posiive gain only hrough rading inancial securiies on he marke. Formally, o deine arbirage opporuniies, we need o inroduce he noion o a sel-inancing rading sraegy, a well-known concep in inance. A sel-inancing rading sraegy is an N + 1-dimensional vecor o adaped sochasic processes w w =1T such ha 1 + r w 1 + Q w 1 = 1 + r w +1 + Q w +1 (14 or each ime period and or any sae-o-he-world rajecories 1 and such ha 1 is a subrajecory o. To be speciic, w w represens he posiions o cash and risky inancial securiies a he beginning o period. Tha is, he number o shares in securiy i is w i. Noe ha w w is deermined hrough he rading in period 1 based on he inormaion 1 ha was available. Equaion (14 implies ha he values o he porolio beore and aer he inancial rading in period are he same. Thereore, no money is added o or subraced rom he porolio hroughou he planning horizon according o a sel-inancing rading sraegy. Wih he help o he noion o sel-inancing rading sraegies, he arbirage-ree condiion can be represened as he ollowing: here does no exis a sel-inancing rading sraegy w w =1T such ha w 1 + w 1 Q 1 = 0 w T T 1 + w T T 1 Q T T 0 T and w T T 1 + w T T 1 Q T T >0 or some T Assumpion 2, pars (1 and (2, also imply an equivalen dual characerizaion o he no-arbirage assumpion: a securiy marke is arbirage ree i and only i here exiss a sricly posiive probabiliy disribuion (commonly reerred o as he risk-neural probabiliy onhe saes o he world such ha or all = 1T, Q i 1 1 = r 1 Q i (15 in which 1 is he risk-neural probabiliy o observing he rajecory given he subrajecory up o ime period is 1. In he sequel, we use E o denoe he condiional expecaion aken wih respec o he risk-neural probabiliy disribuion, while E +1 is used o express he expecaion aken wih respec o he decision-maker s subjecive probabiliy. When we ake expecaion on he subjecive demand disribuion, we use he noaion E. Thereore, Equaion (15 can be equivalenly expressed as Q i 1 1 = E Q i /1 + r As was poined ou by one o he reerees, our model can be exended o he case when he risk-ree borrowing and saving ineres rae r is world driven as well. In a complee inancial marke, a nonsae driven ineres rae (in erms o dollars exiss anyway, which will be used o serve our model i he uiliy uncions are in erms o payo in dollars. As a maer o ac, in a complee inancial marke, we can design a porolio such ha one dollar worh o such a porolio is always worh some ixed amoun > 0 in any given period regardless o he he sae o he world in period. Thereore, / 1 1 could be considered as he risk-ree ineres rae or ime period. For simpliciy o exposiion, we assume ha r is he same across dieren ime periods.

10 Operaions Research 55(5, pp , 2007 INFORMS Parially Complee Financial Marke Following he noaions inroduced beore, we use he N -dimensional vecor w o express he invenory planner s inancial marke posiion a ime period and he scalar w o represen he amoun o cash in he bank a period. In he beginning o each ime period, he decision maker observes he curren sae o he world, he invenory level x, and he inancial marke posiion w w, and hen makes he invenory and pricing decisions y and p. Aer observing he realized demand (and hus he income cash low P x y p, she makes he decision on he nex period marke posiion w +1 w +1 by rading a he marke price Q. Wih he amoun consumed or uiliy a period, he period + 1 cash amoun becomes w +1 = 1 + r ( w + P x y p + w w +1 Q Equivalenly, we have ( w w +1 w w +1 x y p = ( w w +1 Q + P ( x y p + w w r The objecive o he invenory planner is o ind an ordering (and pricing policy as well as a rading sraegy so as o maximize her expeced uiliy over consumpions. This maximizaion problem can be expressed by he ollowing dynamic programming recursion: V x w w = max in which ypyx p p p E W x w wyp { ( = max u w z w zxyp zz [ W x w wyp ] (16 + E +1 [ V+1 y D p z z +1 ] } (17 wih boundary condiion ( W T xwwyp T T T ( = u T w + w Q T T + P T xyp T T T Noe ha all he expecaions aken in he above dynamic programming model are wih respec o he decisionmaker s subjecive probabiliies. A special case o he parially complee marke assumpion is obained when is deerminisic or any given. This corresponds o he complee marke assumpion. Following Smih and Nau (1995, we know ha a risk-averse invenory planner wih addiive concave uiliy uncion can ully hedge he risk in a complee marke, while locking in a proi equal o he expeced (wih respec o he riskneural probabiliy proi. Thus, in his case, he invenory conrol problem reduces o a risk-neural problem. On he oher hand, under he parially complee marke assumpion, he ollowing heorem holds or a decision maker wih he addiive exponenial (subjecive expeced uiliy maximizaion crierion. This heorem can be obained direcly rom 5 o Smih and Nau (1995 and he previous secion o his paper. Deine he modiied income low in period, P y p +1, as in Equaion (11. Theorem 4.1. Separaion The invenory and pricing decisions in he risk-averse invenory model wih inancial hedging Equaions (16 (17 can be calculaed hrough he ollowing dynamic programming recursion: G x = c x + in which max ypyx p p p k y x + L yp (18 L yp [ = E R [E P y p ]] G 1 + r +1 y D p +1 (19 and wih boundary condiion G T +1 x = 0. Srucural policy I, in addiion, k E k +1 +1, hen he ollowing srucural resuls or he opimal invenory ( pricing policies hold. (a I price is no a decision variable (i.e., p = p or each, or each given, uncions G x and L yp are k -concave and a -dependen s S invenory policy is opimal. (b For he general case, G x and L yp are symmeric k -concave or any given and a -dependen ssap policy is opimal. The heorem hus implies ha when addiive exponenial uiliy uncions are used: (i he opimal invenory policy is independen o he inancial marke posiion; (ii he opimal invenory replenishmen and pricing decisions can be obained regardless o he inancial hedging decisions; (iii he coeicien a in he uiliy uncion does no aec he invenory replenishmen and pricing decisions; and (iv unlike Equaion (17, he expecaion operaor E +1 does no appear in he above dynamic programming recursion, which implies ha or he purpose o calculaing he opimal invenory decisions, we do no need o know he decision-maker s subjecive probabiliy on he sae-o-he-world evoluion. However, o obain he opimal expeced uiliy, he model requires ha he decision maker also implemen an opimal sraegy on he inancial marke.

11 838 Operaions Research 55(5, pp , 2007 INFORMS We reer readers o Smih and Nau (1995 or he deailed descripion o such an opimal rading sraegy. 2 I is also ineresing o compare he dynamic program or he inancial hedging case (18 (19 wih he one wihou he inancial hedging opporuniy in (12 (13. The only dierence in he expressions is ha he cerainy equivalen operaor wih respec o he sae-o-he-world ransiions in (13 is replaced by an expecaion operaor wih respec o he risk-neural probabiliy. I is appropriae o poin ou ha he resricion on ixed coss in he heorem is similar o he assumpions made in Sehi and Cheng (1997 or a sochasic invenory model wih inpu parameers driven by a Markov chain. Finally, when he ixed coss are all zeros and here are capaciy consrains on he ordering quaniies, our analysis shows ha a sae-dependen modiied base-sock policy is opimal. 5. Compuaional Resuls In his secion, we presen he resuls o a numerical sudy. We consider an addiive exponenial uiliy model in which = or all = 1T. Assuming he risk-ree ineres rae, r = 0, he experimenal model ocuses on how he choice o parameer can aec he enire invenory replenishmen policies. We experimened wih many dieren demand disribuions and invenory scenarios and observed similar rends in proi proile and changes in he invenory policy under he inluence o risk aversion. Hence, we highligh a ypical experimenal seup in which we consider a ixed-price invenory model over a planning horizon wih T = 10 ime periods. The invenory holding and shorage cos uncion is deined as ollows: h y = h max y0 + h + maxy 0 where h + is he uni invenory holding cos and h is he uni shorage coss. The parameers o he invenory model are lised in Table 2. Demands in dieren periods are independen and idenically disribued wih he ollowing discree disribuion: d = minmax30 z where z 0 1, and y, he loor uncion, denoes he larges ineger smaller han or equal o y. Because he Table 2. Parameers o he invenory model. Discoun acor, 1 Fixed-ordering cos, k 100 Uni-ordering cos, c 1 Uni-holding cos, h + 6 Uni-shorage cos, h 3 Uni-iem price, p 8 demand disribuion is bounded and discree, we can easily evaluae expecaions wihin he dynamic programming recursion and compue he opimal policy exacly. To evaluae he invenory policies derived, we analyze he invenory policies via Mone Carlo simulaion on S independen rials. In each rial, we generae T independen demand samples (one or each period and obain he accumulaed proi a he end o he T h period. Hence, in he policy evaluaion sage, we require ST independen demands drawn rom d. We can improve he resoluions o he policy evaluaion by increasing he number o independen rials, S. Hence, he choice o S is limied by compuaion ime, and in our experimen we choose S = For each risk parameer , we consruc he opimal risk-averse invenory policy. We now sudy numerically how he replenishmen policies change as we vary he risk-aversion level. Tha is, because he opimal policy is s S, we analyze changes in he replenishmen policy parameers as we vary he decision maker risk-aversion level. Figure 1 depics he parameers s S over he irs nine periods. Generally, or any ime period, he order-up-o level, S, decreases in response o greaer risk aversion. Ineresingly, or his paricular problem insance, he reorder level, s increases as we increase he level o risk aversion. O course, his is no rue in general. As a maer o ac, i he ixed-ordering cos, k = 0, we have s = S, and unless he policies are indieren o risk aversion, we do no expec such phenomenon o hold. Indeed, i is no diicul o come up wih examples (wih dieren values o k showing ha s decreases in response o greaer risk aversion. We poin ou ha in mos o our experimens, he order-up-o level S decreases, while he reorder poins s are monoonic (boh monoone increases and decreases are possible in response o greaer risk aversion. Unorunaely, while such a monooniciy propery is much desired, we have numerical examples ha violae his propery as we change he risk-aversion level. Figure 1. Plo o s S agains. (s, S S, ρ = 10 s, ρ = S, ρ = 20 s, ρ = 20 S, ρ = 40 s, ρ = 40 S, risk neural s, risk neural Period

12 Operaions Research 55(5, pp , 2007 INFORMS 839 To es he sensiiviy o he parameers o he opimal policy o changes in he level o risk aversion, we rack he changes in he parameers o he opimal policy as we gradually increase he parameer. I is ineresing o observe ha while he risk-averse and risk-neural policies are dieren, he policy changes resuling rom small changes in he risk-olerance level are quie small. For insance, he opimal policy remains he same as we vary = Thereore, we conclude (numerically ha he opimal policy is relaively insensiive o small changes in he decision-maker s level o risk aversion. 6. Conclusions In his paper, we propose a ramework o incorporae risk aversion ino invenory (and pricing models. The ramework proposed in his paper and he resuls obained moivae a number o exensions. Risk-Averse Ininie-Horizon Models: The riskaverse ininie-horizon models are no only imporan, bu also heoreically challenging. Assuming saionary inpu parameers, i is naural o expec ha a saionary s S policy is opimal when price is no a decision variable and a saionary ssap policy is opimal when price is a decision variable. We conjecure ha similarly o he riskneural case (see Chen and Simchi-Levi 2004b, a saionary ssp policy is also opimal even when price is a decision variable. Coninuous-Time Models: Coninuous-ime models are widely used in he inance lieraure. Thus, i is ineresing o exend our periodic review ramework o models in which invenory (and pricing decisions are reviewed in coninuous ime and inancial rading akes place in coninuous ime as well. Porolio Approach or Supply Conracs: I is possible o incorporae spo marke and porolio conracs ino our risk-averse muliperiod ramework. Observe ha a dieren risk-averse model, based on he mean-variance rade-o in supply conracs, canno be easily exended o a muliperiod ramework, as poined ou by Marínez-de- Albéniz and Simchi-Levi (2006. The Sochasic Cash-Balance Problem: Recenly, Chen and Simchi-Levi (2003 applied he concep o symmeric k-convexiy and is exension o characerize he opimal policy or he classical sochasic cash-balance problem when he decision maker is risk neural. I urns ou, similarly o wha we did in 3.2, ha his ype o policy remains opimal or risk-averse cash-balance problems under exponenial uiliy measure. Random Yield Models: So ar, we have assumed ha uncerainy is only associaed wih he demand process. An imporan challenge is o incorporae supply uncerainy ino hese risk-averse invenory problems. O course, i is also ineresing o exend he ramework proposed in his paper o more general invenory models, such as he muliechelon invenory models. In addiion, i may be possible o exend his ramework o dieren environmens ha go beyond invenory models (or example, revenue managemen models. Anoher possible exension is o include posiive lead ime. Indeed, hroughou his paper, we assume zero lead ime. I is well known ha when price is no a decision variable, he srucural resuls o he opimal policy or he risk-neural invenory models wih zero lead ime can be exended o risk-neural invenory models wih posiive lead ime (see Scar The idea is o make decisions based on invenory posiions, on-hand invenories plus invenory in ransi, and reduce he model wih posiive lead ime o one wih zero lead ime by ocusing on he invenory posiion. To conduc his reducion, we need a criical propery ha he expecaion E o he summaion o random variables equals he summaion o expecaions. Unorunaely, his propery does no hold or he cerainy equivalen operaor when he random variables are correlaed. This implies ha a replenishmen decision depends no jus on invenory posiions, bu also on he on-hand invenory level and invenories in ransi. Thus, our resuls or risk-averse invenory models wih zero lead ime canno be exended o risk-averse invenory models wih posiive lead ime. When price is a decision variable, even under risk-neural assumpions, he srucural resuls or models wih zero lead ime canno be exended o models wih posiive lead ime (see Chen and Simchi-Levi 2004b. Finally, we would like o cauion he readers abou some limiaions and pracical challenges o our model. Firs, he assumpion ha he savings and borrowing raes are idenical may no hold in pracice, especially or he majoriy o manuacuring irms, where he borrowing rae is ypically higher han he savings rae. Similar o many economic and inancial models, our resuls depend on his assumpion. Second, alhough expeced uiliy heory is commonly used or modeling risk-averse decision-making problems, i does no capure all he aspecs o human beings choice behavior under uncerainy (Rabin In pracice, he se o axioms ha expeced uiliy heory is buil upon may be violaed. We reer readers o Heyman and Sobel (1982 and Fishburn (1989 or discussions on he axiomaic game o expeced uiliy heory. Our model also bears he same pracical challenges as oher models based on expeced uiliy heory or example, speciying he decision-maker uiliy uncion and deermining relaed parameers are no easy. We noe ha some approaches or assessing he decision-makers uiliy uncions were proposed in he decision analysis lieraure; see, or example, discussions in he exbook by Clemen (1996. Neverheless, our risk-averse model provides invenory planners an alernaive way o making invenory decisions. Our numerical sudy indicaes ha he risk-averse models based on he addiive exponenial uiliy uncion are no ha sensiive o he choice o.

13 840 Operaions Research 55(5, pp , 2007 INFORMS AppendixA. Proo o Theorem 3.3 Firs, consider he las period, period T. U T x w = max yxp E a T e w ky x+pyp T / T = a T e w/ T max yxp eky x/ T Ee Pyp T / T For simpliciy, we do no explicily wrie down he consrain p T p p T. We ollow his convenion hroughou his appendix. Deine G T x = max yxp ky x + E T T Py p T We have max yxp eky x/ T Ee Pxyp T / T = e G T x/ T Thus, U T x w = a T e G T x+w/r T wih R T deined in Equaion (9. Now we sar inducion. Assume ha U +1 x w = A +1 e G +1x+w/R +1 or some consan A +1 > 0. Now we consider period : U x w = max E[ { max a e w 1/1+r z ky x+p yp / yxp z A +1 e G +1y d+z/r }] +1 where or simpliciy, we use d o denoe he demand o period, which, o course, is a uncion o he selling price o his period. For any given y p, he irs-order opimaliy condiion wih respec o z is 1 a e w 1/1+r z/ e ky x P yp / = 1 + r R +1 A +1 e z/r +1 e G +1y d/r +1 or, equivalenly (because boh a and A +1 > 0, ln a w z/1 + r + ky x P y p = ln 1 + r A +1 z G +1y d R +1 R +1 R +1 (A1 Thus, or any given y p a sae x w and he realizaion o he curren period uncerainy, he opimal banking decision z is z = G R +1 y d + R +1 ky x + P y p + R +1 w + R +1 R R R ln A r a R +1 which implies ha he opimal consumpion decision a ime period is [ w + ky x + P y p = R = R R ] G 1 + r +1 y d R 1 + r ln A r a R +1 [ w + ky x + P y p i we deine consan R ] G 1 + r +1 y d + C C = R 1 + r ln A r a R +1 Equaion (A1 also implies ha U x w = 1 + r R A R +1 max E[ e z +G +1 y ] d/r +1 yxp +1 = A e w/r max E exp{ [ G +1 y d/1 + r yxp ky x+p yp /R ]} in which A = 1 + r ( R A r /R A R a R +1 ( /R r ( /R = R R A 1 /R +1 > 0 +1 a I we deine G x = max ky x yxp [ [ R ln E exp { 1R P y p d + 1 ]}] G 1 + r +1 y d we have U x w = A exp w + G x /R AppendixB. Review on k-convexiy and Symmeric k-convexiy In his secion, we review some imporan properies o k-convexiy and symmeric k-convexiy ha are used in his paper (see Chen 2003 or more deails. The concep o k-convexiy was inroduced by Scar (1960 o prove he opimaliy o an s S invenory policy or he radiional invenory conrol problem.

14 Operaions Research 55(5, pp , 2007 INFORMS 841 Deiniion B.1. A real-valued uncion is called k-convex or k 0, i or any x 0 x 1 and 0 1, 1 x 0 + x 1 1 x 0 + x 1 + k (B1 Below we summarize properies o k-convex uncions. Lemma 1. (a A real-valued convex uncion is also 0-convex and hence k-convex or all k 0. Ak 1 -convex uncion is also a k 2 -convex uncion or k 1 k 2. (b I 1 y and 2 y are k 1 -convex and k 2 -convex, respecively, hen or 0, 1 y + 2 y is k 1 + k 2 -convex. (c I yis k-convex and w is a random variable, hen Ey w is also k-convex, provided Ey w< or all y. (d Assume ha is a coninuous k-convex uncion and y as y. Le S be a minimum poin o g and s be any elemen o he se x x Sx = g S + k Then, he ollowing resuls hold: (i S+ k = s y or all y s. (ii y is a nonincreasing uncion on s. (iii y z+ k or all yz wih s y z. Proposiion 2. I x is a K-convex uncion, hen he uncion gx = min Qy x + y yx is maxk Q-convex. Recenly, a weaker concep o symmeric k-convexiy was inroduced by Chen and Simchi-Levi (2002a, b when hey analyzed he join invenory and pricing problem wih ixed-ordering cos. Deiniion B.2. A uncion is called symmeric k-convex or k 0 i or any x 0 x 1 and 0 1, 1 x 0 + x 1 1 x 0 + x 1 + max 1 k (B2 A uncion is called symmeric k-concave i is symmeric k-convex. Observe ha k-convexiy is a special case o symmeric k-convexiy. The ollowing resuls describe properies o symmeric k-convex uncions, properies ha are parallel o hose summarized in Lemma 1 and Proposiion 2. Finally, noe ha he concep o symmeric k-convexiy can be easily exended o mulidimensional space. Lemma 2. (a A real-valued convex uncion is also symmeric 0-convex and hence symmeric k-convex or all k 0. A symmeric k 1 -convex uncion is also a symmeric k 2 -convex uncion or k 1 k 2. (b I g 1 y and g 2 y are symmeric k 1 -convex and symmeric k 2 -convex, respecively, hen or 0, g 1 y + g 2 y is symmeric k 1 + k 2 -convex. (c I gy is symmeric k-convex and w is a random variable, hen Egy w is also symmeric k-convex, provided Egy w< or all y. (d Assume ha g is a coninuous symmeric k-convex uncion and gy as y. Le S be a global minimizer o g and s be any elemen rom he se X= { x x Sgx = gs+ k and gx gx or any x x } Then, we have he ollowing resuls: (i gs = gs+ k and gy gs or all y s. (ii gy gz+ k or all yz wih s + S/2 y z. Proposiion 3. I x is a symmeric K-convex uncion, hen he uncion gx = min Qx y + y yx is symmeric maxk Q-convex. Similarly, he uncion hx = min Qx y + y yx is also symmeric maxk Q-convex. Proposiion 4. Le be a uncion deined on n m. Assume ha or any x n, here is a corresponding se Cx m such ha he se C x y y Cxx n is convex in n m.i is symmeric k-convex over he se C, and he uncion gx = in xy y Cx is well deined, hen g is symmeric k-convex over n. Proo. For any x 0 x 1 n and 0 1, le y 0 y 1 m such ha gx 0 = x 0 y 0 and gx 1 = x 1 y 1. Then, g1 x 0 + x 1 1 x 0 + x 1 1 y 0 + y 1 1 x 0 y 0 + x 1 y 1 + max 1 K = 1 gx 0 + gx 1 + max 1 K Thereore, g is symmeric K-convex. Endnoes 1. Aribues X 1 X 2 X n are addiive independen i preerences over loeries on X 1 X 2 X n depend only on heir marginal probabiliy disribuions and no on heir join probabiliy disribuion. (See Keeney and Raia 1993, Chaper 6.5, p Noe ha he above deiniion is in he muliaribue preerence seing. Preerences over money a dieren poins o ime could be reaed as muliaribue preerences. 2. We cauion reader on he dierence in he noaion o his paper and Smih and Nau (1995. In his paper, we measure he prices o inancial securiies in he period dollar, while Smih and Nau measure in period 1 dollar.

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