Bayesian design of stochastic inventory systems

Transcription

1 Byein deign of tochtic inventory ytem Tpn P Bgchi [email protected] NMIMS Shirpur Cmpu, Indi Abtrct Thi pper re-viit the Byein pproch to tet it efficcy in optimlly deigning the clicl (, Q) inventory model. A heuritic erch of the untructured (, Q) deciion pce find tht one cn indeed mke decent trt by the Byein pproch nd keep totl cot nerly optimlly low throughout. Keyword Stochtic Demnd, Byein Deign, (, Q) Inventory Policy, Optimiztion Inventory Control: Theory nd Prctice Inventorie re common inurnce gint uncertintie impcting mot production or ervice opertion. However, inventorie do not producing ny return. Yet inventorie help out in brekdown or crii nd lo improve cutomer ervice. So blnce i needed here to optimlly determining () when to order, nd (b) how much to order. Uncertintie re prticulrly high for new buine. One wnt here two nwer how much hould the orgniztion tock initilly, nd hould it djut deciion () nd (b) time progree? Thi pper re-viit the Byein pproch to tet it efficcy in nwering thee two quetion. Stochtic demnd condition Prominent here re the ingle period tochtic model, the (, S) model, nd the (, Q) model, decription being given in Silver et l. (1998), nd Jenen nd Brd (2003). Murry nd Silver (1966) tte tht initilly one would hve gret uncertinty concerning the le potentil of n item. Lerning, in the deciion theoretic ene, i the proce of bing one initil deciion on n informed or inpired gue to trt one buine, nd ubequently updting tht initil gue by ome rtionl logic to ure optimlity of future deciion. Byein dpttion to help better le forecting w ued firt by Murry nd Silver (1966). They dptively chnged the ditribution of le to how how to improve deciion. Other hve lo ued dpttion-imed lerning but few hve eed the potentil benefit of Byein dpttion/lerning with inventorie with the exception of Azoury nd Miller (1994), who how tht the quntity ordered under the non-byein policy would be greter thn tht under Byein policy. Byein method were ued lo to deign queuing ytem (ee Bgchi nd Cunninghm (1972) nd Morle et l. (2005). In thi tudy we inquire: Cn the Byein lerning logic (prior prior + dt poterior) of oberving nd updting tochtic informtion help reduce totl inventory operting cot?

2 A hown by Azoury nd Miller (1994), we hypotheize tht for the populr (, Q) policy, Byein lerning would let one ee how the ucceive incorportion of new dt would improve deciion (reduce the totl expect cot (EC) nd/or improve ervice level) Q* nd * re continully updted with ccumulting demnd dt. But, up to wht point uch updting would be meningful? We expect tht tht nwer my depend on the etimted unknown but ttionry demnd rte for tht might require meeting certin minimum mple ize. How long one hould mple demnd (X) might depend on the cot of uing non-optiml rther thn neroptiml EC? We do not probe thi. Here we ue well-developed tochtic inventory model from the literture the (, Q) model. It incorporte fety tock into the reorder tock level () nd ue n optimum contnt quntity Q of n order every time the current tock level touche or fll below. Optimum nd Q minimize the totl expected cot/unit time (Silver 2007). The (, Q) Inventory Model Thi tudy ue the reult of Jenen nd Brd (2003). We ume tht only ingle item i tocked nd old whoe inventory i mnged to keep the expected totl cot minimum, compriing holding, replenihment nd tock out cot. Ordering too much or too little or t the wrong time cn dirupt the optiml control of inventory, n event eily cued by uncertinty in demnd. Here the determinitic pproch clerly doe not minimize the expected totl cot. At ome intnt of time if inventory level i z, then the probbility of hortge (P ), the probbility of exce (P e ), the expected hortge (E ) nd expected exce (E e ) re, repectively, P P e P[ x P[ x z] z] 1 F( z) F( z) E z ( z x) f ( x) dx ( for continuou demnd x) E E e z z ( x z) P( x) E ( for dicrete demnd x), nd Thi tudy ued the (, Q) inventory mngement policy to erve the tet bed to probe our query. Here demnd i tochtic. (, Q) firt determine the optimum vlue ( * nd Q * ) for the reorder point () nd the order quntity (Q). It then monitor inventory continuouly through the repeted execution of order cycle. An order of ize Q * i plced when the current inventory level z touche *. The order (quntity = Q * ) i received fter led time L to replenihe tock. Optimum * nd Q * re found follow. When L i mll compred to the expected time required to exhut Q, only 1 order would be outtnding. (In prctice plnt my plce multiple order on vendor when expediting become ineffective, but we do not conider thi ce here.) An order cycle i the time between two ucceive receipt. If nd L repectively repreent the verge demnd rte nd led time, then the men demnd during led time i μ = L. The reorder point being, P i 1 F(), nd the ytem ervice level (frction of demnd during led time tht i met) i 1 P = F(). The fety tock (exce tock beyond μ) i SS = μ. The generl olution for thi itution h been given by Jenen nd Brd (2003) follow.

3 If the per SKU unit holding cot i h per unit time, then Expected holding cot/unit time = h Q The time between order i rndom with men of Q/. If the cot of replenihment/order i K then the expected replenihment cot/unit time i (K/Q). If the expected hortge cot/order cycle i C, then the expected hortge cot/unit time will be C /(Q/) = C /Q. The generl model for the expected totl cot/unit time for the (, Q) policy will thu be Q K EC (, Q) h C (1) Q Q Eqution (1) i the expreion for the expected totl cot/unit time for n inventory ytem being operted by the (, Q) policy. To optimize it one ue deciion vrible, the reorder point, nd Q, the quntity ordered in ech order cycle. Anlyticlly, (1) my be prtilly differentited with repect to nd Q nd the derivtive equted to zero. Doing thi yield two condition tht imultneouly chrcterize the two optiml vlue Q * nd *. Thee re * 2( K C Q h (2) nd C hq (3) Peteron nd Silver (1979) employ pecil ce for obtining the optiml vlue Q * nd *. The firt ce ume tht contnt cot π 1 i expended whenever tock out event occur. Thi give u quick wy to evlute C the expected hortge cot/order cycle. Thi i C 1P[ x ] 1 f ( x) dx 1[1 F( )] (4) Eqution (3) my be now utilized ince we hve C expreed in (4) function of. Thu C f ( * ) 1 hq (5) hq which give f ( * ) (6) 1 * with C (1 F( ) (7) 1 Eqution (6) help link * with Q * vi (2). Note tht eeking olution to the (, Q) policy problem by imultneouly olving (2), (6) nd (7) for rbitrry demnd ditribution F() i not trivil.

4 A vrint of the contnt cot π 1 per tock out event i cot π 2 incurred for every unit hort in tock out. Then the expected hortge cot/order cycle will be dependent on how mny unit re expected to be horted in ech order cycle (E ). Here, E ( x ) f ( x) dx nd therefore C 2 E. Thi give C 2 f ( x) dx 2 (1 F( )) (8) Combining (3) nd (8) one obtin C F )) 2 (1 ( hq which for pecified Q give the condition for the optiml reorder point * * hq F( ) 1 (9) 2 The optimum deciion ( *, Q * ) i the combintion of nd Q tht minimize EC given by (1). Owing to the non-liner nd complex nture of (1) through (9) we ued n orthogonl rry experimentl computtionl frmework to firt determine the enitivity of the two performnce (repone)-- ervice level nd totl expected cot. For thi we elected two working level for ech of the fctor monthly demnd (), holding cot (h), hortge cot (π1) nd order cot (K) hown in Tble 1, nd L 8 rry (Montgomery 2008). Tble 1 nd Figure 1 nd 2 how the reult. The following inference my be drwn: Optiml Order Quntity Q * i rective to, h nd K, but only mildly to hortge cot π1. Service level eem to be robut reltive to mot fctor conidered in the region of the cot prmeter tudied. It i cloely relted to the etting of the reorder point *. Totl Expected Cot/unit time i reltively robut with repect to hortge cot per tock out event π1, but enitive to, h nd K. Thee deduction typiclly unvilble to the inventory mnger ugget tht it would be wie to pend effort in optimlly etting the reorder point * nd order quntity Q * before one declre the opertionl policie of tochtic inventory ytem. In one ene, thi informtion i nlogou to the reltive robutne of the totl operting cot/unit time to EOQ for determinitic inventory ytem. Limited generliztion of uch deduction my be ttempted in given cot-demnd cenrio to e how ccurtely the prmeter, h, K nd π1 need to be etimted, to ure minimum cot opertion of the ytem. In thi tudy we focu on the fctor with perhp the highet uncertinty the tochtic nture of X, the demnd per unit time.

5 = h = 5 10 π1 = K = EC $/unit time Quntity or $ Tble 1: An L 8 Computtionl Experiment to uncover the Senitivity of Service Level nd Totl Expected Cot of n (, Q) inventory Sytem Orthogonl Experiment # Unit /month Totl Expected Cot/unit time h π1 K * Q * Service Level Senitivity of (, Q) Policy Performnce to Demnd Rte nd Cot Fctor Reorder Point "" Service Level Order Quntity "Q" Figure 1 Senitivity of Optiml Reorder Point ( * ), Order Quntity (Q * ) to Monthly Demnd Rte nd Cot Senitivity of Expected Cot/unit time to Demnd Rte nd Cot Fctor Figure 2 Senitivity of Expected Totl Cot/unit time to Monthly Demnd Rte nd Cot

6 Byein lerning nd Inferencing The umption of ttionrity i generlly mde to keep the nlyi trightforwrd. In the preent ce lo we ume ttionrity; pecificlly, the prmeter tht control the ditribution of demnd re umed to be unknown, but ttionry. Since the Byein lerning logic (prior prior + dt poterior) follow the pth of pre-uppoing informtion, oberving the phenomen nd then repetedly updting tochtic informtion, one i dvied to ue n pproprite ubjective probbility function for the prior. In theory thi i done by deriving the poterior denity from the likelihood function nd the prior denity, nd deriving the ditribution of the reduced-form prmeter from the initil informtion on the unknown prmeter controlling the tochtic proce (here the rndom demnd). For the preent ce, demnd i umed to be rndom, Poion ditributed with n unknown prmeter (verge rte) λ/unit time. Thi extend two dvntge: Firt, the Poion ditribution i often quite relitic when demnd i rndom nd independent of erlier nd future demnd. Secondly, from nlyticl point of view, the Byein prior-poterior conjugte fmily (Riff nd Schlifer 1961) of the ditribution of the poible vlue of λ i Gmm, twoprmeter ditribution convenient to updte. However, thi i minor retriction Byein inference my be performed uing tochtic imultion of the proce lo (Morle et l. 2005). The Byein Lerning Frmework The conjugte prior ditribution for the Poion rte prmeter i the Gmm ditribution with two prmeter α nd β, the denity function being Gmm ( x) ( x ) 1 e x, x 0, 0, 0 where 1 t ( ) t e dt 0. Here the Gmm ditribution i ued conjugte prior (Riff nd Schlifer 1961) for the rte prmeter λ Gmm h men of α/β, vrince α/β 2 nd it i known to be flexible in hpe. The interprettion i follow: There re α totl occurrence in β time intervl. Updting the Gmm prior i trightforwrd. For intnce, if r quntitie re demnded from the inventory in time period of length t, the poterior denity of λ will be Gmm ditribution with α = α + r nd β = β + t. The mximum likelihood etimted of λ i obtined from the poterior men E(λ) = (α + r)/(β + t). Thi poterior men E(λ) pproche MLE in the limit α 0 nd β 0. In the preent ppliction our intention will be to trt with ome reonbly umed prior vlue of Poion demnd rte λ (= α/β) nd then continully updte it uing r (the demnd oberved in the time pn t) time t dvnce. Bgchi nd Cunninghm (1972) provide the deciion theoretic rtionle for dopting thi procedure.

7 Tble 2 Smple Poion Rndom Demnd Vlue imulted with λ = 100. Auming prior Gmm(α = 5, β = 1) ditribution, we compute Men of the Gmm Poterior ditribution monthly demnd i ucceively oberved. Month # Oberved Monthly Demnd For Poterior Gmm (α, β): = Men of Gmm Poterior SD = SQRT(α/β^2) Upper/Lower etimte of λ i ki Sum α = α + um ki β = β + n λ etimte = α/β SD (λ et) ki Month etc Simulted Poterior Poion Men Demnd, True λ = 100/month Prior umed to be Gmm(5,1) Men etimted 95 upper limit 95 lower limit Figure 3 Trce of Continully Updted Poterior Men Etimte produced by Succeion of Smple drwn from Simulted Poion Ditribution with true λ = 100/month The Byein pproch give u wy round pecil deciion problem. Sometime one mke the bet deciion on the bi of given et of dt vilble from pt hitory, or produced by conducting ome pecil ttiticl experiment. Wht if there i no uch hitory vilble, or there i no opportunity to conduct experiment? A id bove, the Byein pproch begin with n umed prior bout the deciion environment, nd then by uing lerning logic (prior prior + dt poterior) follow the pth of pre-uppoing informtion, oberve the phenomen nd then repetedly updte informtion. Tble 2 nd Figure 3 illutrte the effect of updting the mximum likelihood etimte of the men demnd of Poion ditribution. Here Poion demnd proce with true men demnd (λ) of 100 unit/month w oberved (imulted) in ucceion of month 1, 2, 3, etc. nd the etimte of the poterior long with the ±95 limit of thi etimte w clculted. The prior of λ w umed to be Gmm(α = 5, β = 1) ditribution with n verge of 5 unit/month. The imulted monthly demnd re hown in the econd column (ki) of Tble 2.

8 Continul updting of the etimted vlue of λ produced 95 confidence bnd for λ (96.81, ) fter 100 updte. Clerly, nother umed vlue of the prior would produce nother trce of etimte. But it cn be ured bed on the tndrd devition of the etimted λ tht they would ll eventully converge towrd the true demnd time t incree. Regulr Byein updte of Demnd Rte help keep Totl Expected Cot ner Minimum One wy to operte the (, Q) inventory policy i to be myopic (Levy et l. 2007) in etting the operting vlue of nd Q t ome initil (prior) gue for the demnd rte λ, or ue only limited mount of dt to etimte λ. Such trut on n initil gue for λ nd not chnging it lter my even be fvored, for it ve the effort needed to incorporte ny emergent evidence bout true demnd the buine move forwrd. Mny prctitioner indeed do not chnge the initil umption bout or λ or even cot, though Azoury nd Miller (1994) deplore thi. Plinly, the nd Q derived uing the initil gue for λ would lmot urely be uboptiml, except by ccident (Figure 2 indicte the trong dependence of EC on (hence on λ)). To tet ny poible merit of the myopic pproch we tudied it computtionlly. Tble 3 diply the effect of etting the unknown demnd rte λ t ome unubtntited vlue (uming mitkenly thi to be the true demnd), deriving the flwed * nd Q * from it, nd incurring the conequent expected totl cot EC when the (, Q) policy i ued. It i trightforwrd to compre thee higher vlue of EC with the ner optiml EC chievble by getting cloe to the true demnd rte by uing Byein updte. Tble 3 ued 100 unit/month the true demnd rte where the ( wrongly ) preumed vlue of λ were et repectively t 25, 50, 75, 150, 200, nd 300 unit/month. The cot ued were h = $10/unit-month, π1 = $500 per bckorder event nd K = $800/order plced. Figure 3 nd 4 how the effect of flwed (differing from the true) demnd conjecture ( prior ) on EC, nd the ervice level projected. Tble 3 Cot nd Service Level Experienced when * nd Q * re et bed on prior etimted demnd, but true demnd (T) i different from the prior etimte () = Etimted demnd/month bed on flwed prior gue T = True demnd/month * = Optimum Reorder Point bed on prior Q* = Optimum Order Quntity bed on prior Holding Cot bed on nd (*, Q*) Holding Cot bed on (*, Q*) but fcing True demnd Replenihment Cot by nd (*, Q*) Replenihment Cot bed on (*, Q*) but fcing True demnd Shortge Cot by nd (*, Q*) Shortge Cot bed on (*, Q*) but fcing True demnd Expected Totl Cot by nd (*, Q*) Expected Totl Cot bed on (*, Q*) but fcing True demnd Service level experienced t True Demnd but operting t (*, Q*)

9 Service Level EC $/unit time Expected Totl Cot fcing True Demnd (= 100/month) but uing (*, Q*) bed on Prior λ Figure 4 The Expected Totl Cot for (, Q) Sytem tht ue flwed prior etimte tht i fr from the true demnd vlue Service Level fcing True Demnd (= 100/month) but uing (*, Q*) bed on Prior λ Figure 5 Service Level provided by (, Q) Sytem tht ue flwed prior etimte tht i fr from the true demnd vlue A review of Tble 3 nd Figure 4 nd 5 would ugget tht the reult of uing Byein lerning to keep continully updting the demnd etimte provide mixed mege. But cloer look revel tht the minimum totl cot (, Q) policy hould indeed be bed on demnd etimte cloe to the true demnd i poible. But Figure 4 pper to ugget tht lower gue for demnd ctully improve cutomer ervice! Prim fcie, therefore, gueing low vlue of demnd pper to be doing omething good. But uch inference i hortighted nd mot mileding. More eriouly, thi i not defect in the model or it nlyi. Recll tht our objective of etting up the (, Q) model to help find rtionl wy to mnge inventorie when demnd i tochtic included pelling out the objective firt tht of minimizing (1), the expected totl cot/unit time. Thi totl cot included three component the holding cot, the replenihment cot, nd the hortge cot. At let for thi model, therefore, mximizing cutomer ervice per e w not the objective. Cutomer ervice (F()) enter into (1) vi C (= π 1 (1 F())), the cot of hort hipment. If one require the finl (, Q) olution to ure high level of cutomer ervice, one would need to ue lrge vlue for π 1. Thu, like in ny optimiztion, one mut be cler bout the objective where doe he wnt to put priority?

10 Concluion Thi tudy h invetigted the vlue of incorporting Byein lerning into the populr (, Q) model for mnging inventorie when demnd i tochtic. The tudy find tht one cn indeed mke decent trt by the Byein pproch nd ty the coure nerly optimlly while upholding trget ervice level by uitbly electing cot nd lo keeping the expected operting cot/unit time minimum. Specificlly, thi tudy uncover the high vlue in continully updting the deciion () when to order, nd (b) how much to order, rther thn ticking to the initil gue for the demnd verge, i frequently prcticed. Thi work utilized n orthogonl rry experimentl frmework to determine the enitivity of the two performnce (repone)-- ervice level nd totl expected cot. For thi two working level for ech of the fctor monthly demnd (), holding cot (h), hortge cot (π 1 ) nd order cot (K) were elected nd L 8 rry w dopted to guide the computtion. Furthermore, importntly, myopic etimte of true demnd would urely yield uboptiml EC. Thee deduction typiclly unvilble to the inventory mnger ugget tht it would be wie to pend effort in optimlly etting the reorder point * nd order quntity Q * before one et out to declre the opertionl policie to cope with tochtic demnd. Reference Azoury Kty S nd Bruce L Miller (1994). A Comprion of the Optiml Ordering Level of Byein nd Non-Byein Inventory Model, Mngement Science, Bgchi, Tpn P nd A A Cunnighm (1972). Byein Approch to the Deign of Queuing Sytem, INFOR Journl, Vol 10(1). Jenen, P A nd J F Brd (2003). Opertion reerch: Model nd Method, John Wiley & Son Inc. Levi, T, R O Roundy nd D B Shmoy (2007). Provbly ner-optiml mpling-bed policie for tochtic inventory control model, Opertion Reerch, Vol 32, Montgomery, D C (2008). Deign nd Anlyi of Experiment, 5 th ed., John Wiley. Morle, J, M E Ctellno, A M Myorl nd C Armero (2005). Byein Deign in Queue: An Appliction to Aeronutic Mintennce, Centro de Invetigcion Opertive, Elche Murry, George R nd Edwrd A Silver (1966). A Byein Anlyi of the Style Good Inventory Problem, Mngement Science, 1966, vol. 12 (11), pge Peteron R nd E A Silver (1979). Deciion Sytem for Inventory Mngement nd Production Plnning, John Wiley. Riff, Howrd nd Robert Schlifer (1961). Applied Sttiticl Deciion Theory. Diviion of Reerch, Grdute School of Buine Adminitrtion, Hrvrd Univerity. Silver, E A (1981). Opertion Reerch in Inventory Mngement: A Review nd Critique, Opertion Reerch, vol. 29 no Silver, E A, D F Pyke, R Peteron (1998). Inventory Mngement nd Production Plnning nd Scheduling, Wiley Silver, E A (2007). Inventory Mngement: A Tutoril , Univerity of Clgry.