Inventory control for a non-stationary demand perishable product: comparing policies and solution methods

Transcription

1 Invenory conrol for a non-saionary demand perishable produc: comparing policies and soluion mehods Karin G.J. Pauls-Worm 1, Eligius M.. Hendrix 2 1 Operaions Research and Logisics, Wageningen Universiy, orcid , karin.pauls@wur.nl 2 Compuer Archiecure, Universidad de Málaga, orcid , eligius@uma.es Absrac. his paper summarizes our findings wih respec o order policies for an invenory conrol problem for a perishable produc wih a maximum fixed shelf life in a periodic review sysem, where chance consrains play a role. A Sochasic Programming (SP) problem is presened which models a pracical producion planning problem over a finie horizon. Perishabiliy, non-saionary demand, fixed ordering cos and a service level (chance) consrain make his problem complex. Invenory conrol handles his ype of models wih so-called order policies. We compare hree differen policies: a) producion iming is fixed in advance combined wih an orderup-o level, b) producion iming is fixed in advance and he producion quaniy akes he agedisribuion ino accoun and c) he decision of he order quaniy depends on he age-disribuion of he iems in sock. Several heoreical properies for he opimal soluions of he policies are presened. In his paper, four differen soluion approaches from earlier sudies are used o derive parameer values for he order policies. For policy a), we use MILP approximaions and alernaively he so-called Smoohed Mone Carlo mehod wih sampled demand o opimize values. For policy b), we ouline a sample based approach o deermine he order quaniies. he flexible policy c) is derived by SDP. All policies are compared on feasibiliy regarding he α-service level, compuaion ime and ease of implemenaion o suppor managemen in he choice for an order policy. Keywords: Invenory Perishable produc MILP Chance consrain Mone Carlo 1 Inroducion Alhough in Invenory Conrol lieraure, focus is ofen on infinie horizons wih saionary demand, in he realiy of reail, demand of perishable food producs is ypically non-saionary, i.e. uncerain and also flucuaing, parly due o promoions. In his case, we can consider he planning problem as a finie horizon problem wih non-saionary demand. he conribuion of his paper is o ouline a pracical invenory conrol problem of perishable producs as a Sochasic Programming (SP) model wih a finie horizon and evaluae several soluion approaches from earlier sudies o handle i. We focus on perishable producs ha are processed and ge a bes-before-dae on heir package, such as packed cheese, cu and packed leuce, yoghur, ec. hose producs have a fixed maximum shelf life. Producers and reail organisaions have arrangemens regarding delivery performance, including he necessary available shelf life for he consumer and he service level. Afer he maximum shelf life M, he produc canno be used anymore for he inended purpose and is considered wase. An α-service level requiremen refers o he probabiliy o be ou of sock, i.e. i should be smaller han 1 α. his implies we are dealing wih a chance consrain for each period. For every period (day, week,...) he producer has o decide wheher or no o order and how much, considering a fixed ordering cos, holding cos and disposal cos. his resuls in replenishmen cycle lenghs of a varying number of periods. he producer has conrol over he issuing of producs, so in order o minimise wase, a Firs-in-Firs-Ou (FIFO) policy is used. Excess demand is backlogged. he quesion is wha are he mos appropriae invenory policies o handle he invenory conrol problem in pracice. 1

2 For producs wih non-saionary demand, order decisions and producion planning will flucuae, especially for perishable producs where smoohing of ordering or producion is impossible, because he older iems in sock can perish. he flucuaions in demand ask for a specific sraegy. Bookbinder and an (1988) disinguish hree sraegies o deal wih ordering of non-perishable producs wih nonsaionary demand in a periodic review. he firs sraegy is called he saic uncerainy sraegy; he iming and size of he orders are se a he beginning of he ime horizon. he replenishmen schedule defines when o order beforehand, denoed by Y = 1 when an order is placed and Y = 0 if no. his resuls in replenishmen cycles R of differen lengh. In case of long lead ime, adapaion of he order quaniy jus before demand is no possible, so also he producion quaniy is deermined a he beginning of he planning horizon. We call his a YQ policy. We use he variable Y in he policy name insead of R o make clear ha for every period a decision is made. his policy is appropriae when here is considerable lead ime and is invesigaed in (Pauls-Worm e al., 2016). he second sraegy o deal wih non-saionary demand is he saic-dynamic uncerainy sraegy, where iming of he orders is se a he beginning of he ime horizon, bu he order quaniy may be adaped in response of he invenory levels observed during he ime horizon. We call his a YS policy. In a heurisic approach, Bookbinder and an (1988) spli he problem in wo sages. he firs sage deermines he iming and he second he quaniy. arim and Kingsman (2004) considered his approach as a basis for an MILP model formulaion for non-saionary sochasic demand for he simulaneous deerminaion of he iming and size of he replenishmen orders. he hird sraegy o deal wih non-saionary demand is he dynamic uncerainy sraegy where he order quaniy is decided a he beginning of every period. We call his a Q(X) policy, where X is he invenory level a he beginning of he period. A policy according o he dynamic uncerainy sraegy is discussed already in he 1960s. Karlin (1960a) shows ha a criical number policy is opimal, were he criical numbers are a reorder level s, and an order-up-o level S, resuling in an (R,s,S ) policy. Karlin (1960b) and Veino Jr (1963, 1965) developed opimal myopic policies for cerain cases. Moron (1978) shows ha near myopic bounds are close o opimal, under he assumpion of disposal of excess sock. Moron and Penico (1995) derive nearmyopic bounds for he more general case. Zipkin (1989) developed opimal criical number policies for a cyclic demand paern. He shows ha he criical numbers in he opimal policy smooh he flucuaion in he demand daa. his wai-and-see approach in he criical number policies following he dynamic uncerainy sraegy could require an order wih seup cos in almos every period. his migh be undesirable for he producion planning of a company, bu in case of large seup cos relaive o he holding cos, his is neiher opimal (Bookbinder and an, 1988). Bookbinder and an (1988) formulaed heir sraegies for non-perishable producs. Perishable producs require special aenion wih respec o order policies aking non-saionary demand ino accoun. Order policies for perishable producs wih a fixed lifeime are reviewed by ahmias (1982), Goyal and Giri (2001), and Karaesmen e al. (2011). Almos all papers assume saionary demand. Fries (1975) shows ha wih a maximum shelf life of M 2, neiher an (R,S) nor an (R,s,S) policy is opimal. ahmias (1975) and Fries (1975) observe ha in general an opimal order policy for perishable producs wih a fixed life ime should ake he age-disribuion of he producs in sock ino accoun. Even when all perishable iems are of he same age, base sock (order-up-o level) policies are no opimal, as argued by ekin e al. (2001) and Haijema e al. (2007). Some papers, e.g. (Haijema e al., 2007), (Broekmeulen and van Donselaar, 2009), (Minner and ranschel, 2010) assume a cyclic demand paern, wih a weekly demand paern per day, bu saionary expeced demand per week. hey assume negligible seup cos and follow a dynamic uncerainy sraegy, which migh no be opimal in case of fixed seup cos. 2

3 Food producers ofen have conracs wih heir cusomers, regarding service level. From he above menioned papers, only Bookbinder and an (1988), arim and Kingsman (2004) and Minner and ranschel (2010) consider service level consrains. he oher papers use penaly cos for each backlogged iem. Unforunaely, he size of a penaly o guaranee a desired service level is no sraighforward o deermine as in general when dealing wih chance consrains. Pauls-Worm e al. (2014) presened an SP invenory model ha minimises he expeced oal coss, including seup cos, uni procuremen cos, holding cos and cos of wase, for a perishable produc wih non-saionary sochasic demand wih an α-service level consrain under a FIFO issuing policy. A YS policy is convenien in pracical planning for a food producer; one knows beforehand in which periods o produce. Using an order-up-o level S defines a pracical rule o deermine he order quaniy aking uncerainy ino accoun. In Pauls-Worm e al. (2014), we used an MILP approximaion o derive values for a YS sysem, based on he cumulaive disribuion funcion (cdf) for he demand during he replenishmen cycle and he expeced age-disribuion of he invenory during he replenishmen cycle. However, we know ha his approach may fail in meeing service level requiremens because he approximaed amoun of wase is underesimaed and he amoun of fresh iems is overesimaed, due o Jensen Inequaliy. herefore, we presened in (Hendrix e al., 2015) a compuaional mehod based on he so-called Smoohed Mone Carlo mehod wih sampled demand o opimise values for a YS sysem. he resuling MILP approach uses enumeraion, bounding and ieraive nonlinear opimisaion. he order quaniy is deermined by order-up-o level S minus he oal invenory in sock. However, i could be more cos-efficien o consider he age-disribuion of all iems in sock in deermining he order quaniy. Le X denoe he invenory age-disribuion a he beginning of a period. Using a sample based YQ(X) policy, one can ake he age-disribuion ino accoun. Finally, we can consider a more flexible Q(X) policy according o a dynamic uncerainy sraegy, derived by SDP. A Q(X) policy deermines he order quaniy a he sar of every period based on he age-disribuion of he iems in sock. Besides variaion in uncerainy sraegy, he presened policies vary in ease of implemenaion in pracice. An order-up-o level policy requires only informaion abou he oal available invenory and a se of orderup-o levels S. When he age-disribuion is considered in he policy, informaion is required abou he age-disribuion of iems in sock, and an order quaniy has o be deermined applying a able or compuaion process. Wih he exra informaion required, also he calculaion ime o find a soluion may increase. o summarise, we evaluae hree differen policies: YS policies (producion iming is fixed, order-up-o level), a YQ(X) policy (producion iming is fixed, producion quaniy akes age-disribuion ino accoun) and an Q(X) policy (decides every period on order quaniy depending on age-disribuion). An overview of invesigaed sraegies and policies is presened in able 1. We compare he policies for 81 insances and invesigae in which siuaions in pracice which policy is mos suiable. Secion 2 gives he SP model of he pracical problem. Secion 3 exposes several heoreical properies YS and YQ(X) policy soluions. Secion 4 oulines he Smoohed Mone Carlo MILP approach o deermine parameers for a YS policy. he YQ(X) policy is depiced in Secion 5 and Secion 6 presens he flexible Q(X) policy. Secion 7 compares he policies and Secion 8 concludes. able 1 Overview of invesigaed sraegies and policies Uncerainy sraegy Policy Soluion Mehod Paper Insances repored Saic-dynamic YS MILP (Pauls-Worm e al., 2014) 81+5 insances Saic-dynamic YS SMC-MILP (Hendrix e al., 2015) 1 insance Saic-dynamic YQ(X) Sample Based Mehod (Hendrix e al., 2015) 1 insance Dynamic Q(X) SDP (Hendrix e al., 2012) 1 insance 3

4 2 Sochasic Programming model he pracical invenory conrol problem formulaed as an SP model in (Pauls-Worm e al., 2014) is described below. Due o sochasic demand d, he invenory levels I b (excep he saring invenory I b0 ) and he order quaniies Q are random variables. We assume a lead ime of zero. he probabiliy in he chance consrains is noaed as P(.), and E(.) denoes he expeced value operaor. We use (x) + o express max {x, 0}. able 2 provides a lis of symbols. able 2 Lis of symbols Indices period index, = 1,.., wih he ime horizon b age index, b = 1,..,M, wih M he fixed maximum (inernal) shelf life Daa d ormally disribued demand wih expecaion µ > 0 and variance (CV x µ ) 2 where CV is a given coefficien of variaion k fixed ordering or seup cos, k > 0 c uni procuremen cos, c > 0 h uni holding cos, for iems ha are carried over from one period o he nex, h > 0 w uni disposal cos (w > 0) or salvage value (w < 0) for iems becoming wase α required service level, 0 < α < 1 Variables Q ordered and delivered quaniy a he beginning of period I b invenory level of iems wih age b a he end of period, iniial invenory fixed I b0 = 0, I 1 R, I b 0 for b = 2,.., M. Invenory of age M a he end of period is considered wase he expeced oal coss over he ime horizon are minimized. M 1 M 1 Min E( C) E g( Q ) h I b wi M E g( Q ) hi b wi M (1) 1 b1 1 b1 where he procuremen coss are given by he funcion g(x) = k + cx if x > 0 and g(0) = 0. (2) he chance consrain requiring he α-service level is expressed by P(I 1 0) α 1,.., (3) meaning ha he probabiliy of no being ou-of-sock a he end of period should be greaer han or equal o α. he probabiliy of a sock-ou is 1 α. Because of a FIFO issuing policy, he invenory levels of he older iems are zero in case of a shorage, so only he invenory of he freshes iems can be negaive. he invenory dynamics for he FIFO issuing is described by M 1 I b Ib 1, 1 d I j, 1 j b 1,.., ; b 2,.., M (4) and for he freshes iems M 1 I 1 Q d Ib, 1 1,..,. (5) b1 Iems of age M become wase a he end of he period and canno be used in he nex period. he invenory of he freshes iems can be negaive. he unme demand will be backlogged. he invenory a he beginning of a period is defined by X = (I 1,1,..., I M1,1 ). (6) 4

5 As described in Secion 1, several order policies can be defined as soluions of he SP model. One way is o define order-up-o levels S. he decision maker replenishes in period he invenory up-o he level S, where he order quaniy Q is defined by M 1 Q ( X ) S X b (7) b1 In Secion 3 we discuss properies of policy soluions according o he saic-dynamic uncerainy sraegy for his SP model. In Secion 4 we develop a YS policy and in Secion 5 a YQ(X) policy, boh following a saic-dynamic uncerainy sraegy. In Secion 6 we presen a more flexible Q(X) policy according o he dynamic uncerainy sraegy o obain parameers for he SP model. 3 Properies of a soluion of he saic-dynamic uncerainy sraegy heoreical properies abou he opimal soluion of specific cases help o limi heir soluion space. herefore, we firs focus on he properies of feasible soluions. In Secion 3.1, he concep of replenishmen cycles and iming are discussed. Secion 3.2 shows in which cases he so-called basic order-up-o level is he opimal quaniy. For he oher order momens we sudy he mahemaical implicaions of esimaing he service level by a Mone Carlo sampling approach in Secion Replenishmen cycles and limis on order iming vecor Y A replenishmen cycle is he number of periods R he order quaniy Q aims o fulfil. For non-saionary demand, replenishmen cycle lengh R depends on order momen. When Y = 1, hen Y +R+1 = 1 and no orders ake place in beween. In case of perishable iems wih maximum shelf life M, he replenishmen cycle canno be longer han he shelf life M, so R M. So, Propery 1. Le Y be an order iming vecor of he SP model, i.e. Y = 0 Q = 0. Y provides an infeasible soluion of he SP model, if i conains more han M 1 consecuive zeros. Le F be he se of all feasible order iming vecors Y of lengh. he number of elemens F of horizon and shelf life of M < 1 follows recursive rule F +1 = 2 F F M wih he iniial erms F = 2 1 for < M + 1 and F M+1 = 2 M 1; see (Alcoba e al., 2015). F is exponenial in he horizon. However, in pracice (Pauls-Worm e al., 2014), i is sufficien o plan ahead for = 12 periods. 3.2 Basic order-up-o level and opimal order quaniies For a cerain replenishmen cycle lengh R = 1,.., M we can define a basic order-up-o level invenory ha should be available a he beginning of period o cover demand of R periods. Ŝ R as he Definiion 1. Le d d +R 1 be he sochasic demand during a replenishmen cycle of lengh R wih cumulaive disribuion funcion (cdf) G R. he basic order-up-o level Ŝ R wih probabiliy α o fulfil demand is defined by G Sˆ ) such ha ˆ S G 1 ( ). R ( R For some replenishmen cycles, R R M 1 Ŝ R may be no enough, so X b Q Sˆ R b1, because producs in sock can become wase during he replenishmen cycle. everheless, for some replenishmen cycles, he basic order-up-o level is sufficien and specifies he opimal order quaniy. he following bounds can be derived. For he proofs, see (Hendrix e al., 2015). 5

6 Propery 2. Le Y be an order iming vecor of he SP model wih corresponding cycle lengh R and X defined by (6). For order momen having Y M = 1, R = M, he opimal order quaniy is Q = Ŝ R. Second, a replenishmen cycle may be of jus one period; during he cycle no wase occurs. Propery 3. Le Y be an order iming vecor of he SP model and X defined by (6). For an order momen M 1 having Y +1 = Y = 1 he opimal order quaniy is Q Sˆ X. he bes order quaniy a a negaive sock level is of an order-up-o ype. Propery 4. Le Y be an order iming vecor of he SP model and X defined by (6), Y = 1 wih replenishmen cycle lengh R. If X 1 0, he opimal order quaniy is S ˆ X. 1 b1 b Q R Sample based Mone Carlo esimaion of he service level If heoreical properies of Secions 3.1 and 3.2 do no apply, one can ry o find he order quaniies fulfilling he chance consrains using samples of he demand series. Le d be he sochasic demand vecor (d,..., d +R 1 ) from, during replenishmen cycle lengh R, X he saring invenory and Q he order quaniy. Le f (Q, X, d) = I 1,+R 1 define he end invenory of iems wih an age of one period given a realisaion d of d following he invenory dynamics wih possible perishing according o (4) and (5). Consider indicaor funcion δ : R R R {0, 1} 1 if f ( Q, X, d ) 0 ( Q, d ) (8) 0 oherwise ha ranslaes he service level in consrain (3) for period + R 1 o a( Q) P( I1, R1 0) Ed ( Q, d ) (9) he idea is ha given sample pahs d 1,.., d of d, he probabiliy (service level) (9) is esimaed by 1 aˆ ( Q) ( Q, d r ). (10) r1 As is know from handbooks on saisics (e.g. (Lyman O and Longnecker, 2001)), given a se of independen random samples d r, (10) is an unbiased esimaor of a(q) wih sandard deviaion 1 2 ( aˆ( Q)) ( a( Q) a( Q) ), (11) which is used in Mone Carlo approaches o deermine he number of samples o reach a desired probabilisic accuracy. A rule of humb is o have an accuracy of 2σ. Aiming a α = 90%, 95%, 98%, a sample size of = 5000 gives a rule of humb accuracy of abou for he esimaor a ˆ( Q). 4 YS policy: sample based SMC-MILP approach Consider a YS policy, i.e. he decision maker is provided an order iming vecor Y, i.e. Y = 0 Q = 0. he quesion is o generae values for S such ha he α-service level consrains are fulfilled for all insances and expeced coss are minimized. Properies 2 and 3 can be used o order-up-o level S = Ŝ R for specific momens. Sample-based esimaion can be used for oher service levels. Following he radiion of using scenarios (sample pahs), one can wrie he problem of finding discree iming Y and coninuous order-up-o levels S as a Mone Carlo based Mixed Ineger Linear Programming (MC-MILP) model as specified in Secion 4.1. Such sample-based model for mos insances canno be solved in reasonable ime, e.g. (Rijpkema e al., 2016). herefore, in Secion 4.2 we skech he use of an equivalen 6

7 MILP model based on he Smoohed Mone Carlo mehod, see (Hendrix and Olieman, 2008). A specific algorihm is designed ha uses enumeraion and bounding for he ineger par Y of he problem leaving us wih ieraively solving an LP problem in he coninuous variables S. his algorihm has been described earlier in (Hendrix e al., 2015). 4.1 radiional scenario based MC-MILP opimisaion of he YS policy he sample-based approach for he YS policy can be handled by adding o he SP model a sample (scenario) index r = 1,.., o he variables I br and Q r. he objecive (1) is exended owards M 1 1 Min ky cqr h Ibr wi Mr (12) 1 r 1 b1 wih order quaniy M 1 Q r S Ib, 1, r b1 r 1,.., ; 1,.., (13) and he convenional order relaion S MY 1,.., (14) wih a big-m value. he consrains (4) and (5) are exended o each sample M 1 I br Ib 1, 1, r dr I j, 1, r r 1,.., ; 1,.., ; b 2,.., M (15) j b and for he freshes iems M 1 I 1r Qr dr Ib, 1, r r 1,.., ; 1,.., (16) b1 oice ha due o he nonnegaiviy of he variables, he funcion (x) + = max{0, x} can be rewrien by addiional variables and inequaliies in order for (13), (15) and (16) o be linear, as described in (Pauls- Worm e al., 2014). Furhermore, for he chance consrains one adds a binary variable δ r {0, 1} represening he indicaor value ha specifies wheher demand is fulfilled in period in sample r I1 r m (1 r ) r 1,.., ; 1,.., (17) wih m an upper bound on he ou of sock I 1. his defines aˆ : R 1 2 { 0,,,.., 1} giving he reached service level under he se of samples. he corresponding chance consrains are 1 aˆ : r 1,.., (18) r1 he variables of he model are Y, δ r {0, 1}, S, I br, Q r 0. oice ha Y and S do no depend on sample r and he oher variables Q r, I br and δ r ha describe he simulaion or evaluaion par, do so. Solving MC-MILP is in mos cases pracically impossible due o he large number of binary variables δ and many soluions δ ha represen he same obained service levels a(s). he number of samples = 5000 menioned in Secion 3.3, implies defining for each period = 5000 binary variables δ r. 4.2 Smoohed Mone Carlo MILP approach o he YS policy Considering MC-MILP from he poin of view of an LP problem in he coninuous variables S given Y, he funcion aˆ : 1 2 { 0,,,.., 1} in (18) is piecewise consan, i.e. changing he values of S a bi may no change he evaluaed value of ˆ. According o Hendrix and Olieman (2008), he a 7

8 reached service level can be made pracically a coninuous funcion by following he MC smoohing approach. Le M b 1 br z r I be he oal amoun of produc lef over a he end of period in sample r. Measuring how close aˆ is o change value using he leas nonnegaive oal invenory [ in] [ ou] p min { z z 0} and he leas negaive invenory p min { z z 0}. he r r r suggesed smoohing funcion o (S) is [ in] ( 1 2p o ) 2 ( ) ( ) 1 S [ in] [ ou. (19) p S p ] S r r r Fig. 1. Illusraion of Smoohed MC from (Hendrix and Olieman, 2008), where he esimaed probabiliy on he y- axis depends on varying one parameer on he x-axis, in our case he order-up-o level S Hendrix and Olieman (2008) show ha aˆ o is coninuous in ineresing values of S, as illusraed in Figure 1 and he funcion aˆ o deviaes a mos 1 2 from he reached service level aˆ which is much smaller han he possible esimaion error. Problem LPS(Y) is defined as MC- MILP replacing consrain (18) by aˆ o 1,.., (20) as a smooh opimizaion problem ha in principle can be solved by a nonlinear opimisaion rouine. his can be used in enumeraion rouine Algorihm 1. Algorihm 1. YSsmooh in: samples d r, cos daa, α, Se he bes funcion value C U := Generae he se of feasible order iming Y for all Y if for he lower bound on cos LBc(Y) < C U solve LPS(Y) using Ŝ R values S and cos C if C < C U save he bes found values C U := C, S * := S, Y * = Y Ŝ R, ou: Y *, S * 8

9 5 YQ(X ) policy: sample based algorihm In policy YQ(X), he decision maker is provided a replenishmen schedule Y, i.e. Y = 0 Q = 0. he order quaniy depends on he age disribuion X of he iems in sock. Pracically, his is more complex han an order-up-o level as he decision maker requires an informaion sysem advising he order quaniy given he acual age composiion of invenory. he resuls of Secion 3 also hold for he YQ(X) policy. Quaniies of order momens fulfilling Properies 2, 3 or 4, can be se o basic order-up-o level Ŝ R. Oherwise, for a posiive invenory X, he sample-based esimaion of Secion 3.3 can be used. A an order momen, i.e. Y = 1 where he invenory posiion is posiive and R > 1, he order quaniy may be larger han he basic order-up-o level M 1 Q ˆ ( X ) SR X b (21) b1 due o he occurrence of expeced wase during he replenishmen cycle. o compue he opimal order quaniy for his case, we focus on he oal invenory a he end of he replenishmen cycle as funcion of he saring invenory X, he order quaniy Q and he demand d,.., d +R 1. Definiion 2. he funcion Z : M R is defined as he ransformaion z = Z(Q, X, d) giving M he oal invenory z I b 1 b, R1 following he dynamics (4), (5) wih saring invenory X, order quaniy Q and demand vecor (d 1,.., d +R 1 ). Acually, we are looking for he minimum value Q for which he chance consrain holds; his is he value of Q for which P(Z(Q, X, d,.., d +R 1 ) 0) = α. We can use he following propery. Propery 5. Le funcion Z be defined by Definiion 2, R M, saring invenory X and z = Z(0, X, d,.., d +R 1 ) wih cdf Γ. he opimal order quaniy in period is Q = ( Γ 1 (1 α)) +. he cases where M 1 b1 Q Sˆ X is he opimal soluion are given by Properies 2 and 3. he R b value may deviae in oher cases due o he wase during he replenishmen cycle. aking his value as benchmark provides a corollary which follows direcly from he former properies. Corollary 1. Le Z be given by Definiion 2, R M, M 1,,,.., ) R X b1 b X d dr 1 M 1 1 Sˆ R X (1 ) b1 b Γ z Z( Sˆ wih cdf Γ and saring invenory X. he opimal order quaniy is Q. his means ha for cases following Properies 2 and 3 and if X 1 0 (no wase occurs during he cycle), M 1 he opimal choice is Q Sˆ X as Γ 1 (1 α) = 0. In oher cases, wase can be generaed and R b1 b Γ 1 (1 α) < 0. o esimae he quanile Γ 1 (1 α), Mone Carlo simulaion can be used as discussed in Secion 3.3. Le D be an marix wih samples d r. For a saring invenory X, giving he order M 1 quaniy Q Sˆ X, one can evaluae z r = Z(Q, X, d r,.., d +R 1,r ) being he oal invenory of R b1 b sample r a he end of he cycle. he adjusing amoun Γ 1 (1 α) is esimaed by A ( X ) ( quanile ({ z, r 1,.., },1 )) r, (22) where quanile ({}, α) is he α sample quanile of se {}. So he order quaniy for any saring invenory according o Corollary 1, can be approximaed by 9

10 Q ( X ) Sˆ R M 1 b1 X b A ( X ). (23) I should be noed ha he order quaniy is in fac based on a condiional chance consrain and may lead o unnecessarily high service levels, i.e. given he age disribuion, (23) can only focus on fulfilling he chance consrain in he fuure, no maer how likely he curren age disribuion X is, see (Rossi e al., 2008). o provide some inuiion, consider a discree demand and invenory age-disribuion X a he beginning of he period and I he invenory age-disribuion a he end of he period, or alernaively replenishmen cycle. hen we have P(I 1 0) = Σ X P(X = X ) P(I 1 0 X ). Having a policy ha assures for all possible siuaions X he condiional probabiliy P(I 1 0 X ) α is sufficien o imply P(I 1 0) α, bu no necessary. he order quaniies for he YQ(X) policy are now defined eiher by he heoreical resuls, or by he sample-based esimaion in (22) and (23). he nex quesion is o generae he bes advice for he order iming Y. Algorihm (2) enumeraes he possible replenishmen schedules replenishmen schedules Y. Propery 1 can be used o leave ou hose wih oo large periods beween wo orders. For each vecor Y, he average cos is evaluaed for a large simulaion run ha uses differen random numbers han he ones in marix D used o deermine he order quaniies by (22) and (23). Algorihm 2. YQ in: samples d r, cos daa, α, Se he bes funcion value C U := Generae he se of feasible order iming Y for all Y if for he lower bound on cos LBc(Y) < C U Deermine C by simulaing sample pahs During he simulaion if Y = 1 Ŝ R, ou: Y * if saring invenory X no posiive or R = 1 ake Q Sˆ else simulae he replenishmen cycle wih pahs from X Deermine he order quaniy Q from (23) if C < C U save he bes found values C U := C, Y * = Y R M 1 b1 X b he YQ(X) policy akes he age-disribuion ino accoun. For he decision maker he required use of sampling and possibly inerpolaion is more inconvenien han using a simple order-up-o sraegy wih a lis of order-up-o levels of he YS policy. he quesion arises in which cases he policy YQ(X) performs significanly beer han he YS policy. 6 Flexible Q(X) policy according o he dynamic uncerainy sraegy In his policy, a he beginning of every period, he order quaniy is deermined, based on he agedisribuion of he available invenory. We obain his policy by Sochasic Dynamic Programming (SDP), an appropriae echnique o solve he SP model, as i is clearly separable in. he SDP approach o solve he SP model has been described in (Hendrix e al., 2012). he sae values are given by X, he ransiion is provided by equaions (4) and (5), so we have ransiion funcion Φ: I Φ X, Q, d ) 1,.., (24) ( 10

11 he chance consrains can be wrien as M 1 1 Q Γ ( ) X b 1,.., (25) b1 he wase I M a he end of period is a funcion of he invenory a he beginning of he period and he demand: I M = f (X,d ). We can wrie he expeced conribuion o he objecive funcion in period as funcion of sae X and decision Q : EC ( X, Q ) g( Q ) E{ wf ( X, d ) h1 Φ ( X, Q, d )} (26) where 1 is he all-ones vecor. he SDP objecive funcion can be wrien down via he Bellmann equaion using a value funcion V: V X ) min EC ( Q, X ) E[ V ( Φ ( X, Q, d ))] (27) ( 1 Q subjec o fulling (25). he argmin of (27) represens he opimal policy Q(X ). o implemen his policy, a he beginning of every period, he decision maker needs an informaion sysem wih he opimal sraegy in ables, advising on he order quaniy given he acual age composiion of he invenory. 7 Comparison of policies o compare he differen policies, we use he erraic demand paern and he design of experimens used in (Pauls-Worm e al., 2014). he demand paern is depiced in able 3. In 81 experimens, we sysemaically vary fixed seup cos k = 1500, 500, 2000, disposal cos w = -0.5, 0, 0.5, α-service level = 90%, 95%, 98%, CV = 0.1, 0.25, he oher values are kep consan: he produc has a shelf life of M = 3, uni producion cos c = 2 and uni holding cos h = 0.5. egaive disposal cos means he produc has a salvage value, which is usually much less han he uni producion cos c, zero disposal cos means ha only he uni producion cos is los in case of wase, and posiive disposal cos means ha here is a cos o discard he wased iems. able 3 Erraic demand paern Daa E(d ) We compare he policies of able 1, i.e. he YS MILP policy wih parameers generaed by MILP (Pauls- Worm e al., 2014), a YS SMC-MILP policy wih parameers generaed by he smoohed Mone Carlo MILP approach described in Secion 4, a YQ(X) policy according o he sample based algorihm presened in Secion 5, and he flexible Q(X) policy generaed by SDP as discussed in Secion 6. he invenory sysem is simulaed for all policies using he same (pseudo) random number series of 10,000 runs and compared on expeced oal coss wih he reference value for he YS SMC-MILP policy se o 100, as repored in able 4. he calculaion ime o generae parameer values for he policies differs significanly. Values for he YS MILP policy are calculaed wihin a second, while values for he YS SMC-MILP policy ake abou 100 seconds. Calculaions for he YQ(X ) policy use abou 274 seconds, and for he Q(X ) policy abou 150 seconds in Malab on an Inel 7 processor. he calculaion ime is deermined based on insances wih a seup cos k = 1500, where he replenishmen cycles vary in lengh, resuling in he longes compuaion ime. able 4 shows ha in mos insances he expeced oal coss are very close. he YS MILP policy provides ofen he lowes coss, bu does no always mee he service level requiremens. However, his is mosly in he las period = 12 due o end-of-horizon effecs. Simulaion shows ha in 96.4% of he periods he service level requiremens are me, wih an error olerance of 1%. he oher policies are designed 11

12 such ha hey always mee he service level requiremens. he YS MC-MILP policy is in general he policy ha mees he service level bes a lowes coss. he shaded cells show he insances where a more complex policy is preferred regarding he coss. able 4. Simulaed cos resuls of four order policies relaive o YS SMC-MILP coss = 100, shaded cells highligh he insances where a more complex policy is preferred regarding he coss CV = 0.10 CV = 0.25 CV = 0.33 k = 1500 Insance YS MILP YQ (X ) Q (X ) SDP Insance YS MILP YQ (X ) Q (X ) SDP Insance YS MILP YQ (X ) Q (X ) SDP Service level 90% w = w = w = Service level 95% w = w = w = Service level 98% w = w = w = CV = 0.10 CV = 0.25 CV = 0.33 k = 500 Insance YS MILP YQ (X ) Q (X ) SDP Insance YS MILP YQ (X ) Q (X ) SDP Insance YS MILP YQ (X ) Q (X ) SDP Service level 90% w = w = w = Service level 95% w = w = w = Service level 98% w = w = w = CV = 0.10 CV = 0.25 CV = 0.33 k = 2000 Insance YS MILP YQ (X ) Q (X ) SDP Insance YS MILP YQ (X ) Q (X ) SDP Insance YS MILP YQ (X ) Q (X ) SDP Service level 90% w = w = w = Service level 95% w = w = w = Service level 98% w = w = w = In case of low seup cos (k = 500), one orders almos every period. he expeced oal coss of he differen order policies are almos equal. Only wih increasing uncerainy and higher service levels, he Q(X) SDP migh significanly save coss. However, he dynamic uncerainy sraegy Q(X) SDP policy may raise he expense on planning, which is no par of his model. In case of high seup cos (k = 2000), he saic-dynamic uncerainy YS and YQ(X) policies have mosly he same producion plans based on he 12

13 derived basic order-up-o levels and herefore, he coss show no significan differences. he Q(X) SDP policy has in mos cases significanly higher coss. his is due o overachievemen of he service level, as an SDP policy mees a condiional service level requiremen as is illusraed in (Rossi, 2013) and (Pauls-Worm and Hendrix, 2015). Moreover, he Q(X) SDP policy is allowed o order every period, which is very cosly in case of high seup cos. he YQ(X) policy mees a condiional service level requiremen, bu here he producion momens are fixed, which resuls in a less nervous sysem (unc e al., 2013). he so-called seup-oriened sysem nervousness can be prevened agains a minor cos increase in case of a non-perishable produc (unc e al., 2013). he resuls in his paper give a similar indicaion for a perishable produc. Due o his behaviour, in insances 80 and 81 he YQ(X) policy has lower coss, comparable o he coss of Q(X ) SDP. his means he YQ(X) policy migh be preferred in pracice as i fixes he order iming. As also shown in (Pauls-Worm e al., 2014), an inermediae level of seup cos is he mos ineresing siuaion. he replenishmen cycles are varying in lengh, resuling in wase during he replenishmen cycles. Finding he opimal order iming is more difficul as discussed in Secion 3, and considering he age-disribuion of he iems in sock becomes more imporan. his is confirmed by he coss of he differen order policies, in he insances wih higher uncerainy and higher service levels (16 18, 22 27). he YQ(X ) policy, aking he age-disribuion of he iems in sock ino accoun, gives clearly lower coss, as expeced. 8 Conclusion We sudied a pracical invenory conrol problem wih non-saionary demand for a perishable produc, invesigaing he quesion which order policies migh be appropriae for his problem. We evaluaed wo differen sraegies o deal wih he uncerainy in he pracical problem, resuling in hree differen policies. he invenory conrol can be handled according o a saic-dynamic uncerainy sraegy, where we disinguish a YS policy and a YQ(X ) policy. In invenory lieraure of perishable producs his sraegy is lile sudied. We firs looked a possible bounds of he soluion space discussing properies of he policies. ex we skeched several soluion mehods for he conrol policies. Firs a compuaional mehod based on he Smoohed Mone Carlo mehod wih sampled demand, called he YS SMC-MILP policy has been oulined and a sample based mehod o calculae values for he YQ(X ) policy. hese policies were compared o an YS MILP policy and a more flexible Q(X ) policy generaed by SDP, according o a dynamic uncerainy sraegy. he experimenal evaluaion comprises 81 insances wih he same erraic demand paern, bu wih varying seup cos, service level, cos of wase and uncerainy measured in he Coefficien of Variaion. able 5 presens an overview of he compared policies and soluion mehods. For mos insances, he expeced oal coss of he policies are very close, and a YS policy gives a cos efficien and easy o implemen soluion. From a producion planning perspecive, he saic-dynamic uncerainy sraegy is he mos convenien sraegy o follow. In siuaions of relaively low seup cos, (producion akes place every period), or high seup cos (each M periods a producion akes place), MILP generaes appropriae parameer values. In siuaions of inermediae seup cos, were he replenishmen cycles are highly varying, he opimized SMC-MILP parameers migh be more suiable, alhough more calculaion ime is needed; abou 100 seconds compared o less han 1 second. 13

14 able 5. Comparing policies and soluion mehods for 81 insances wih horizon = 12 Uncerainy Policy Soluion Minimal α-service level Comp A Implemenaion Sraegy Mehod ime ype Feasibiliy Pre-de. Info needed ordering Saicdynamic YS MILP Expeced 96.4% <1s Yes Order-up-o level α-sl oal Invenory Saicdynamic YS SMC- Expeced 100% 100s* Yes Order-up-o level MILP α-sl oal Invenory Saicdynamic Condiional α-sl YQ(X) Sample Based Mehod Dynamic Q(X) SDP Condiional α-sl * In Malab on an Inel 7 processor 100% 274s* Yes Invenory disr. Compuer program o deermine Q based on sampling 100% 150s* o Invenory disr. Find Q in able based on invenory disribuion Average o. coss (indexed) When also he required service level is high or he CV is 0.25 or more, a YQ(X ) policy, where he agedisribuion of he invenory is aken ino accoun, migh have less invenory coss. However, for he decision maker he required use of samples and possibly inerpolaion is more inconvenien han using a simple order-up-o sraegy wih a lis of order-up-o levels of he YS policy. Also he compuaion ime of abou 274 seconds migh be a disadvanage. he dynamic uncerainy sraegy Q(X ) policy is only appropriae in siuaions wih relaively low seup cos, a CV of 0.33 and a high service level of 98%. he same implemenaion drawback as for he YQ(X ) policy applies, wih a compuaion ime of around 150 seconds. Based on he findings of his paper, i is up o managemen o decide which uncerainy sraegy and which order policy is mos appropriae. Acknowledgmens. his paper has been suppored by he Spanish Minisry (I C2-2- R), in par financed by he European Regional Developmen Fund (ERDF). References Alcoba, A.G., Hendrix, E.M.., García, I., Orega, G., Pauls-Worm, K.G.J., Haijema, R., On Compuing Order Quaniies for Perishable Invenory Conrol wih on-saionary Demand, in: Gervasi, O. e al. (Eds.), Compuaional Science and Is Applicaions -- ICCSA Springer Inernaional Publishing, pp Bookbinder, J.H., an, J.Y., Sraegies for he probabilisic lo-sizing problem wih service-level consrains. Manage. Sci. 34, Broekmeulen, R., van Donselaar, K.H., A heurisic o manage perishable invenory wih bach ordering, posiive lead-imes, and ime-varying demand. Compu. Oper. Res. 36, Fries, B.E., Opimal Ordering Policy for a Perishable Commodiy wih Fixed Lifeime. Operaions Research 23, Goyal, S.K., Giri, B.C., Recen rends in modeling of deerioraing invenory. Eur. J. Oper. Res. 134, Haijema, R., van der Wal, J., van Dijk,.M., Blood plaele producion: Opimizaion by dynamic programming and simulaion. Compu. Oper. Res. 34, Hendrix, E.M.., Haijema, R., Rossi, R., Pauls-Worm, K.G.J., On Solving a Sochasic Programming Model for Perishable Invenory Conrol, in: Murgane, B. e al. (Eds.), Compuaional Science and Is Applicaions ICCSA Springer Berlin Heidelberg, pp

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