Inventory Management Subject to Uncertain Demand
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1 Inventr Management Sbjet t Unertain emand Esma el Pınar Keskinak 7 ISYE 34 - all Inventr Cntrl Sbjet t Unertain emand In te presene nertain demand te bjetive is t minimie te epeted st r t maimie te epeted prit Tw tpes inventr ntrl mdels ied time perid - Peridi review One perid Newsvendr mdel Mltiple perids ied rder qantit - Cntins review R mdels
2 Tpes Inventr Cntrl Pliies ied rder qantit pliies Te rder qantit is alwas te same bt te time between te rders will var depending n demand and te rrent inventr levels Inventr levels are ntinsl mnitred and an rder is plaed wenever te inventr level drps belw a prespeiied rerder pint. Cntins review pli Tpes Inventr Cntrl Pliies ied time perid pliies Te time between rders is nstant bt te qantit rdered ea time varies wit demand and te rrent level inventr Inventr is reviewed and replenised in given time intervals s as a week r mnt i.e. review perid epending n te rrent inventr level an rder sie is determined t pssibl inrease te inventr level p-t a prespeiied level i.e. rderp-t level Peridi review pli
3 Inventr Cntrl - emand Variabilit Cnstant/Statinar Variable/Nn-Statinar Unertaint eterministi Stasti Enmi Order antit EO Trade between ied st and lding st Lt sie/rerder pint R r ss mdels Trade between ied st lding st and srtage st Aggregate Planning Planning r apait levels given a reast Materials Reqirements Planning MRP Ver diilt prblem! Newsvendr single perid REA THE APPENIX ON PROBABILITY REVIEW 3
4 Newsvendr Mdels Esma el Pınar Keskinak 7 ISYE 34 all Eample - INORMS INORMS Te Institte r Operatins Resear and Management Siene will ld its annal meeting in Wasingtn.C. in 8. Si mnts bere te meeting begins INORMS mst deide w man rms sld be reserved at te nerene tel. At tis time rms an be reserved at a st $5 per rm. It is estimated tat te demand r rms is nrmall distribted wit mean 5 and standard deviatin. I te nmber rms reqired eeeds te nmber rms reserved etra rms will ave t be nd at neigbring tels at a st $8 per rm. Te innveniene staing at anter tel is estimated at $. Hw man rms sld be reserved t minimie te epeted st? 4
5 Eample asin Bags Te ber r Wat-a-Markp asin Bags mst deide n te qantit a ig-pried wman s andbag t prre in Ital r te llwing Cristmas Seasn. Te nit st te andbag t te stre is $8.5 and te andbag will sell r $5. A disnt irm prases an andbags nt sld b te end te seasn r $. In additin te stre antants estimate tat tere is st $.4 r ea dllar tied p in inventr at te end te seasn ater all sales ave been made. Newsvendr mdel - Prperties One-time deisin Crrent deisins nl impat te net perid bt nt tre perids Retail: asin/seasnal items One-time events 5
6 Trade in inventr deisins Sppl Sppl < emand emand Srtage Lst sales / Lst prit Sppl > emand Eess inventr Inventr st Sppl emand Newsvendr mdel - Prperties One-time deisin Crrent deisins nl impat te net perid bt nt tre perids Retail: asin/seasnal items One-time events Relevant sts C : Unit st eess inventr st verage C : Unit st srtage st nderage emand is a randm variable wit Prbabilit densit ntin pd mlative distribtin ntin d Objetive: Cse te rdering qantit bere knw te demand t minimie Ttal epeted verage and nderage sts 6
7 7 Newsvendr mdel inding te ptimal rder qantit irst write te st ntin: d d d ] E[ rdered is i nderage st verage epeted i i : is rdered and demand is i nderage st verage :ttal Critial Rati erivatin te Critial Rati -.. Rle se Leibni's T take te derivative is rdered i nderage st verage epeted ' ' d d d d d d : In
8 8 erivatin te Critial Rati - d d d d d d d d d d d Similarl ] [.. Rle se Leibni's T take te derivative ' ' ' ' : In erivatin te Critial Rati - 3 Reall: P d P d d d d d d Als need t ek i is nve. Send derivative is +
9 Eample - INORMS = nmber rms atall reqired = nmber rms reserved Wat are C and C? Eample - INORMS = nmber rms atall reqired = nmber rms reserved : st = 5 : st = =9-4 Cst reserving Cst Cst reserving innveniene etra rms in ter tels 9
10 Eample - INORMS = nmber rms atall reqired = nmber rms reserved : st = 5 C C : st = =9-4 Cst reserving Cst Cst reserving innveniene etra rms in ter tels 4 4 Wat is? Eample - INORMS Reall: Nrmal5 We ave lkp tables r Standard Nrmal distribtin = = Cnvert t Standard Nrmal! 4 P P P P P Z Standard nrmal ind rm te lkp table = + Z
11 Eample - INORMS Reall: Nrmal5 We ave lkp tables r Standard Nrmal distribtin = = Cnvert t Standard Nrmal! 4 P P P P P Z Saet Standard nrmal Z stk Epeted demand ind rm te lkp table = + Standard Nrmal istribtin - e ensit ntin Smmetri arnd te mean Area nder te entire rve PZ
12 Standard Nrmal istribtin - e ensit ntin Smmetri arnd te mean Area nder te entire rve PZ We will se ɸ and interangeabl =PZ Standard Nrmal istribtin - e ensit ntin Smmetri arnd te mean Area nder te entire rve PZ We will se ɸ and interangeabl =PZ -=PZ -=- -
13 Eample - INORMS P Z Area nder te rve *.4 5 * 47 = -.4 Eample - INORMS W is *<5 i.e. less tan te epeted demand? Epeted st nderage Epeted ttal st Epeted st verage *=47 3
14 Eample - INORMS Wat i =5? * Epeted st nderage Epeted ttal st Epeted st verage *=339 Eample - INORMS Wat i =5? * C * C * Epeted st nderage Epeted ttal st Epeted st verage *=339 4
15 5 Wen d we ave *=Epeted demand?!!! i.e..5 need we * r.5 Area nder te rve!!! ten * I * Z P = Optimal qantit verss epeted demand rder epeted demand srtages and eess inventr st te same I *.5 tan epeted demand mre stl rder less eess inventr is I *.5 srtages are mre stl rder mre tan epeted demand I *.5 Nte: Assming te demand distribtin is smmetri arnd its mean
16 Te impat standard deviatin Wat appens t te ptimal qantit as te standard deviatin inreases? Te impat standard deviatin Wat appens t te ptimal qantit as te standard deviatin inreases? Assming te demand distribtin is smmetri arnd its mean * inreases in I srtages are mre stl * inreases in std. dev. * dereases in I eess inventr is mre stl * dereases in std. dev. * des nt ange in I srtages and eess inventr st te same rder epeted demand regardless standard deviatin 6
17 Eample - INORMS =4 < =5 =-.4 =7 > =5 =. * * Standard deviatin Standard deviatin Eample asin Bags Inpt Unit st = $8.5 Selling prie p = $5 Salvage vale s = $ Cst inventr = $.4 r ea dllar tied p in inventr at te end te seasn 7
18 Eample asin Bags Inpt Unit st = $8.5 Selling prie p = $5 Salvage vale s = $ Cst inventr = $.4 r ea dllar tied p in inventr at te end te seasn Cmpted inpt: Hlding st per bag = = = Eample asin Bags Inpt Unit st = $8.5 Selling prie p = $5 Salvage vale s = $ Cst inventr = $.4 r ea dllar tied p in inventr at te end te seasn Cmpted inpt: Hlding st per bag =.48.5 = $.4 = p = $.5 = s + = $
19 Eample asin Bags Nrmall distribted demand emand ~ Nrmal Area nder te rve= Critial rati =.86 *.8 5 * 7 =.8 Eample asin Bags Unirml distribted demand emand Unirm between 5 and 5 =5 Same epeted demand as in Nrmal distribtin.86 * 5 * Unirm densit / Area nder te rve = Critial rati = *= 9
20 Eample asin Bags Even tg bt te Nrmal and te Unirm distribtins ave te same mean =5 w did we get dierent qantities? Nrmal distribtin *=7 Unirm distribtin *= Eample asin Bags Even tg bt te Nrmal and te Unirm distribtins ave te same mean =5 w did we get dierent qantities? Nrmal distribtin *=7 Unirm distribtin *= Bease te variane eqivalentl standard deviatin and te sape te distribtin!!! Nrmal = Unirm =57.7 Nrmal Unirm
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