Obtaining the Optimal Order Quantities Through Asymptotic Distributions of the Stockout Duration and Demand

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1 he Seond Inernonl Symposum on Sohs Models n Relbly Engneerng Lfe Sene nd Operons Mngemen Obnng he Opml Order unes hrough Asympo Dsrbuons of he Sokou Duron nd Demnd Ann V Kev Nonl Reserh omsk Se Unversy omsk Russ k57@ynderu Nl V Sepnov Al Eonoms nd Lw Insue Brnul Russ Absr he esmors of demnd s men nd sndrd devon bsed on he observed sokou durons nd he demnds n he frmework of he newsvendor problem wh ompound Posson demnd re onsdered We propose lso n dpve lgorhm deermnng he order quny for he ne perod usng only he resul of urren one Keywords Newsvendor; sokou duron; prmeers esmon; sympo dsrbuon; ompound Posson demnd; dpve lo szng I PROBLEM SAEMEN Consder sngle-produ supply hn onssng of suppler buyer nd usomers A he begnnng of me perod he buyer purhses quny nd replenshmen durng he perod s mpossble whle he demnd s rndom hs model s known s he newsvendor problem or sngle perod nvenory model see for emple [ ] Le he rndomness of he demnd be modelled by ompound Posson proess e we ssume h usomers rrve ordng Posson proess wh unknown nensy λ nd requre mouns of vryng sze ndependen of he rrvl proess; he mouns requred eh rrvl (bh szes) re d onnuous rndom vrbles wh unknown dsrbuon defned by probbly densy funon (pdf) p ξ ( ) Denoe nd he frs nd seond momens respevely of he bh sze dsrbuon Lo sze s purhsed fed pre per un d nd we do no onsder he os of lefovers ulzon nd los sles Le X() be rndom usomer demnd [ ] p( ) be probbly densy funon of X() = X Epeed prof for he buyer he end of he perod S = d + p( ) d + p( ) d where > d s sellng (rel) pre per un of he produ he buyer s neresed n deermnng n opml vlue of by mmzng he epeed prof Obvously he sk hs unque soluon deermned by he equon nd orrespondng prof d p( ) d = () S = p( ) d () he resul s well known; see for emple [] So o fnd S we need o know dsrbuon of X In generl he dsrbuon s ompled nd dfferen ppromon mehods for represenon of hs dsrbuon hve been proposed see [3 4] nd [5] n he frmework of sngle perod nvenory model I seems h he norml ppromon s he frs one hs been onsdered whh go bk o F Lundberg Aordng X λ enrl lm heorem onverges n dsrbuon o λ sndrd norml rndom vrble N ( ) s λ for ny prlly used dsrbuon p ξ ( ) Noe h only frs wo momens of he bh sze dsrbuon re needed here For norml ppromon we reeve from () /6 $3 6 IEEE DOI 9/SMRLO

2 where ( ) Ψ = Φ ( ) = λ + λ Ψ d Φ ( ) = ep π d So o fnd pprome vlue of n pre we need o esme wo prmeers of he demnd: he men E X Vr X = λ { } = λ nd he vrne { } Mos of he ppers hve foused on he se when he demnd s fully observed Erly ppers where he ensored demnd s onsdered re [6 9] Here we propose o use he hrerss of he duron of he sokous he proedures re bsed on wo ensored smples: he sellng durons e he durons whh he buyer hs nonzero nvenory level durng me perod : f here re los sles; nd he demnds durng me perod : f here re lefovers he end of he perod If here re los sles hen sellng duron equls mnus sokou duron I s more ommon o use he erm sokou duron nd we hve used he erm sellng duron n our prevous ppers h s why n hs pper he wo erms re used II ASYMPOIC DISRIBUION OF HE SELLING DURAION Le us fnd sympo ( s lrge enough) dsrbuon u ( ) of he lengh of me kes o sell he lo sze for he model under onsderon o solve he problem for ny bh sze s dsrbuon le us onsder dffuson ppromon of demnd proess X() dx ( ) = λ d + λ dw( ) where w( ) s he Wener proess Dffuson mehods hve been ppled o nvenory models n vrey of domns o begn wh he ppers [ ] In [] heorel jusfon of dffuson ppromon wh some numerl resuls s gven Denoe τ( ) he frs ourrene me of he rossng of s ( ) level by ( ) g( s ) = E e τ = X gven X() = ; { } (3) sy = e u( y ) dy s he Lple rnsform of ondonl densy u( ) Denoe X ( ) X ( ) Δ = + Δ nd {} he epeon wh respe o rndom vlue Consder ( ( )) { } Δ E Δ s Δ +τ +Δ sδ g( s ) = E e = e E { g( s + Δ )} = g( s ) + g( s ) λ g( s ) + sg ( s ) + λ + Δ + o ( Δ ) I follows s Δ Δ g( s ) g( s ) s + g s = g s = λ ( ) ; ( ) Denoe he vlue of neres τ () = τ nd g( s) = gτ ( s) From (4) we ge s g ( s) = ep τ + λ he nverse Lple rnsform of gτ ( s) Le u( ) ep λ = 3 π λ λ E{ τ} = q >> Vr { τ} = q 3 3 λ λ (4) (5) hen sup u( ) ep s π λ q nd we n onsder ( ) norml rndom vrble λ τ q/ λ s sndrd q Le us onsder he se of eponenl bh sze dsrbuon Here = = f ξ ( ) = ep If he lo s bough by he frs usomer he ondonl λ probbly densy funon u( ) = λe nd he probbly of hs even s e

3 If he lo s over when he n-h usomer (n > ) mke purhse he ondonl pdf λ u( ) = λe I λ n n λ u( n) = e ( n )! nd he probbly of hs even λ ξ = ( ) n n p ( ) d f ( y) dy ep e p ( ) d = where I ( ) nd zeroh order s he modfed Bessel funon of he frs knd Le us ompre he e resul wh he pprome norml one For eponenl bh sze dsrbuon he dmensonless vrble λ hs pprome norml dsrbuon wh he men q = nd he vrne q ( ) n n ( n )! = ep e e d = n ( ) ep = n ( n )! where δ ( ) f n = p ( ) n n = e n ( n )! f n s ondonl pdf of he demnd gven h number of usomers [ ] equls n So we reeve Fg he e (sold lne) nd pprome norml (dshed lne) pdf of λ for q = n n n λ λ λ n n= ( n )! ( n )! u( ) = λ e + e e = λ λ = λ e + ( s!) s= s kng no oun followng equon from [3] we fnlly ge λ λ + = I ( s!) s= s Fg he e (sold lne) nd pprome norml (dshed lne) pdf of λ for q = 3 In Fg nd he grphs of he e (sold lne) nd pprome (dshed lne) probbly densy funons of λ re shown for dfferen vlues of prmeer q As q nreses he ppromon beomes more ure

4 III PARAMEERS ESIMAION We ssume h he usomer ssfes he demnd s fr s possble when he moun requred s more hn he remnder; nd he lo szes re he sme for eh perod Suppose we hve observed n sle perods nd n m of hem we hve hd lefovers nd n n m ses we hve hd los sles Dfferen esmors n be onsrued bsed on he dsrbuons obned bove In [4] we esme prmeers μ = λ nd σ = λ seprely by lefovers perods nd by los sles perods We esme he probbles μ μ P{ X < } = Φ nd P{ τ < } = Φ by σ σ orrespondng ros: m h n = nd n m = h ; nd esme n he ondonl mens by orrespondng ondonl smple mens: m n m = nd = n m Here m = = 3 3 ( ) / / μ = λ = μ nd σ = λ λ = σ μ nd So we hve reeved wo prs of esmors ( ) σ ˆ = Ψ ( h) + F( h) Ψ ( h) + F ( h) μ ˆ = ; Ψ ( h) + F( h) ( ) Ψ( h) + F( h) μ ˆ = s Ψ( h ) + F ( h ) ( ) [ Ψ( h) + F( h) ] ( Ψ( ) + ( )) / σ ˆ = s 3/ s h F h ep Ψ ( u) where F( u) = πu nd s = In [4] he resuls of smulon re gven for unform eponenl nd be dsrbuons of bh sze nd weghed m n m ( ) esmors ˆ ˆ ˆ m n m ( ) μ = μ + μ nd σ ˆ = σ ˆ + σ ˆ n n n n re lso onsdered We reommend usng weghed esmors ˆμ nd ˆσ I llows us o nrese he ury of demnd s sndrd devon s esmon whh s omprvely low nd he ury of ˆμ does no dffer sgnfnly from he bes one proposed he oher mehod of he prmeer s esmon s o use he equons for he ondonl mens only he mehod does no llow us o wre he esmors n losed form he numer soluon of rnsendenl equon s needed More del dsusson n be found n [5] And now we presen he smple lgorhm bsed on he sme observons h llows us o fnd he lo szes sep by sep n mul perods model usng only he resul of urren perod IV ADAPIVE LO SIZING ALGORIHM FOR MULI PERIOD MODEL A he begnnng of he -h perod he buyer purhses some quny A he end of he perod wo senros re possble: he sze of produ sold durng he perod ws equl o < e here were lefovers ; he buyer rn shor me < e here were los sles Le us onsder he followng deson rule onernng he order s quny he ne perod A he begnnng of he (+)-h perod he buyer buys quny = + Δ + where Δ = κ ( ) f he frs senro ws relzed oherwse Δ = κ ( ); κ > κ > re some oeffens hus κ ( ) f < + = + κ ( ) f < Durng he mul perod sesson he vlues of orders hve o fluue round opml vlue nd equon hs o hold κe{ X X < } = κ E{ τ τ < } (6) Usng norml ppromons for he demnd nd sellng duron we ge

5 λ E X X λ Φ { < } = ( λ ) d ( )ep λ = = π λ = λ f λ λ / λ E Φ { < } = σ ( / λ) = d = ( ) ep πσ σ se of prmeers s = s smple men of = lo szes derved by he lgorhm ABLE I CHARACERISICS OF ADAPIVE ALGORIHM S SIMULAION # λ d/ κ κ o where / λ = σ f σ y f ( y) = y + ep Φ π ( y) kng no oun h for >> see (3) follows d = + ψ λ λ / λ λ λ = σ / λ Fg3 Resul of smulon Run #6 3 σ = λ λ λ So we n rewre (6) s follows λ λ κ f κ f λ λ hus reommended ro of he oeffens κ / κ = d d f Ψ / f Ψ In ble he resuls of smulon re presened for dfferen ses of prmeers for he se of eponenl bh sze dsrbuon wh he men he number of runs for eh Fg4 Resul of smulon Run #7 We see h vlues re lose o he orrespondng pprome opml vlues In Fg 3 nd 4 he onseuve vlues = re presened for he runs #6 nd

6 ACKNOWLEDGMEN hs work ws suppored by he Compeve Growh Progrm of he omsk Se Unversy for 3- yers REFERENCES [] K J Arrow h E Hrrs nd J Mrshk Opml nvenory poly Eonomer vol 9 (3) pp [] E A Slver D F Pyke nd R Peerson Invenory mngemen nd produon plnnng nd shedulng New York: Wley 998 [3] R Ser nd C Chor Comprson of ppromons for ompound Posson proesses ASIN Bullen vol 45 (3) pp [4] W Hürlmnn A Gussn eponenl ppromon o some ompound Posson dsrbuons ASIN Bullen vol 33 pp [5] M J G Domney nd R M Hll Performne of ppromons for ompound Posson dsrbued demnd n he newsboy problem Inernonl Journl of Produon Eonoms vol 9() pp [6] R M Hll Prmeer esmon nd performne mesuremen n los sles nvenory sysems Inernonl Journl of Produon Eonoms vol 8 pp 5 99 [7] S Nhms Demnd esmon n los sles nvenory sysems Nvl Reserh Logss vol 4 pp [8] H-S Lu nd A H-L Lu Esmng he demnd dsrbuons of sngle perod ems hvng frequen sokous Europen Journl of Operonl Reserh vol 9 pp [9] N Agrwl nd SA Smh Esmng negve bnoml demnd for rel nvenory mngemen wh unobservble los sles Nvl Reserh Logss vol 43 pp [] J A Bher A onnuous me nvenory model Journl of Appled Probbly vol 3 pp [] M Puermn A dffuson proess model for sorge sysem n Sudes n he Mngemen Senes Logss vol I M A Gesler Eds Amserdm: Norh-Hollnd Press 975 pp [] A Kev V Subbon nd O Zmeev Dffuson ppomon n nvenory mngemen wh emples of pplon n Leure Noes n Compuer Sene A Dudn e l Eds Swzerlnd: Sprnger 4 pp [3] Hndbook of mheml funons wh formuls grphs nd mheml bles M Abrmowz nd I Segun Eds New York: Dover Publons 97 [4] A Kev V Subbon nd N Sepnov Esmng he ompound Posson demnd's prmeers for sngle perod problem for lrge lo sze Pro 5h IFAC Symp on Informon Conrol Problems n Mnufurng (INCOM 5) Elsever (IFAC-PpersOnLne) My 5 vol 48 pp [5] Ann V Kev Nl V Sepnov nd Alendr O Zhukovsky Demnd esmon for fs movng ems nd unobservble los sles unpublshed 56 56