from normal distribution table It is interesting to notice in the above computation that the starting stock level each

Transcription

1 Homeork Solon

2 Par A. Ch a b from normal dsrbon able Ths, order qany s 39-7 b o b5 from normal dsrbon able Ths, order qany s 9-7 I s neresng o noce n he above compaon ha he sarng sock level each eek. R *, s less han he average eekly demand. Based on he defnons provded n class, hs essenally ranslaes o a negave safey sock! As yo can see from he above, hs s he resl of he very lo vale of he backorder cos compared o he vale of he holdng cos.

3 Ch 3 a Posson Process, snce hs demand process accmlaes he random demand generaed by he 5 parallel saons n he assembly sage, and hese saons are no synchronzed n any ay; n fac, each of hem has a qe random / hghly varable behavor see also he dscsson on he Posson dsrbon provded n pgs n yor exbook, and revs he maeral from he I665 class on he Posson process and s properes. b Based on he provded nformaon, each saon n he assembly sage ll prodce 3 ns per hor, on average, and herefore, reqres 3 chasss ns. Snce here are 5 sch saons, he average horly demand s 45 chasss ns, and herefore, he average demand over he lead me perod of 5 mnes s 45 / 4.5 ns. Snce he demand dsrbon s Posson, he above resl also ranslaes o a varance of.5 ns and a sandard devaon of c When veed from he sandpon of he assembly sage, he consdered sysem s essenally a basesock nvenory model h s lead me demand dsrbon characerzed n pars a and b above, and s basesock level R deermned by he nmber of paper cards essenally KABAS ha conrol he maeral flo from he chasss sage o he assembly sage. Hence, or man problem here s he deermnaon of he mnmal basesock level ha ll garanee he reqred servce level fll rae. We kno ha n he case of dscree dsrbons G, he fll rae reslng from any gven basesock level R s eqal o GR- Gr, here r s he mpled reorder pon. eng r m, e are lookng for he mnmal m sch ha Gm Posson m;.5.99

4 m and he KABA level s Rm. Ch 7 7. Formlae for some of he qanes: θ l F Holdng cos per year *hi σ θ becase demand s POISSO I,r*c / r - θ B,r*c Order cos per year *FA a,b The fll raes able s a he end of hs problem s solon. c l θ σ r F Type S S B I $/n ns/mo mos ns ns ns ns order freq fll rae backorder nvenory approx exac As e observed n class, Type servce specfcaons are mos srngen han he correspondng Type ones. Therefore, hen sch a specfcaon s sed as an approxmaon for fll rae, hch s anoher erm for he Type servce level, ll nderesmae s re vale, leadng o a mch larger r and hgher nvenory. c c l θ σ r F Type S S B I $/n ns/mo mos ns ns ns ns order freq fll rae backorder nvenory approx exac Ths approxmaon s very accrae becase s based on he acal formla ha characerzes fll rae, and hen s hs large, he dropped erm Br s neglgble. d c l θ σ r F S B I $/n ns/mo mos ns ns ns ns order freq fll rae backordenvenory exac oe ha hen s redced, e ge slghly hgher servce a a mch smaller nvenory nvesmen. B of corse, e order ce as ofen. If e neglec he cos or capacy consderaons of placng orders, e can alays mnmze nvenory coss y choosng. B f e consder eher order freqency capacy or fxed order cos, hen O may gve a perfecly reasonable. Formlae sed n he fll raes able: pr θ r e -θ /r! r cdf of Posson random varable Gr p k by def on pg. 69 of he exbook k Br θpr {θr -Gr} eqn.63 on pg.. Ths s he backorder level formla

5 for he base sock model. The vales of Br are comped becase hey are sed n he follong B,r formla, hch s a,r model formla. r B, r B x eqn.38 on pg. 78 x r Type servce G r eqn.36 on pg. 78 B r Type servce eqn.37 on pg. 79 xac S,r B r B r eqn.35 on pg. 78 Fll raes able for Problem.7:

6 θ 7.5 r pr Gr Br 5 3 Type S B,r Type S xac S B,r xac S

7 Par B. Problem. Solon Mehodology In hs ml-prodc nesboy problem, he objecve s o fnd order qanes for nespaper,,,, sch ha he daly prof s maxmzed hle he oal egh of nespapers s less han or eqal o W. Maxmzng he prof s eqvalen o mnmzng he oal cos, hch consss of he nderage and overage coss. e p - c be he n nderage cos and o c - s be he n overage cos for nespaper. Then he cos conrbed by prodc s C, o max{, - } max{, - } Takng he expeced vale of he cos h respec o he demand and smmng p for he prodcs, he opmzaon problem P can be formlaed as Mnmze o x g x dx x g x dx sbjec o W here g x s he probably densy fncon of he demand. oe ha n he basc nesboy model, here here s no egh consran, gves o separae opmal orderng qany for nespaper. So f he combnaon of hese qanes does no volae he egh consran,.e. W, hen s a feasble and opmal solon o o P. If > W, hen he opmal solon ll sasfy he egh consran as o eqaly. The reason s ha he expeced cos of each nespaper, C, o x g x dx x g x dx, s a convex fncon n. For <, o C, s decreasng n. For any solon ha gves a oal egh srcly less han W,

8 he objecve fncon can be mproved by ncreasng some of he s nl he egh consran s sasfed a eqaly. Assmng W o >, e can replace he neqaly sgn h eqaly n he consran n P and oban he same opmal solons. In hs case e may nrodce a agrange mlpler θ and fnd he opmal solon o P by solvng he nconsraned problem: Mnmze W dx x g x dx x g x o C,,..., θ θ The opmaly condons are: G G o C,...,, θ and W C θ From, e have o G θ, so e can re n erms of θ as o G θ θ *. Then he problem becomes fndng a vale of θ sch ha * θ sasfes. θ can be solved sng bsecon search over he nerval beeen a loer bond and pper bond for θ. oe ha θ can be nerpreed as he penaly cos of volang he egh consran by one n, so a loer bond for θ s. Also, snce G s a cmlave dsrbon fncon, θ has o sasfy o θ,.e. o θ. Therefore, mn can be aken as he nal pper bond for θ n he bsecon search. rng bsecon search, se θ o be he mdpon beeen he pper and loer bonds. Sop f W mn < θ. θ s opmal. If W mn, replace he pper bond h he crren vale of θ. If < W, replace he loer bond h he crren vale of θ. Repea nl an opmm s fond.

9 . Applcaon Remark: oce ha n he follong calclaons, he noaon a,b employed n he problem daa, has been nerpreed as a normal dsrbon h mean eqal o a and s. devaon eqal o b. o crcal rao / o Ieraon oer Upper θ 3 Toal Wegh A Ieraon 6, he dfference beeen he oal egh and he alloable egh s.9 l b, less han he egh of he lghes paper paper,.5 lb, so e sop here. Rondn g don s o negers, e ge 99, 7 and Tha frees p.5 lbs and allos he nesboy o carry an addonal copy of Paper and each. The fnal anser s, 7 and 3 46.

10 Problem Thnkng of each of he 7 remanng csomers as a Bernoll random varable h a sccess probably of.7, s easy o see ha he dsrbon characerzng he remanng aendees s a Bnomal dsrbon Bn,p h n7 and p.7. For compaonal prposes, e can approxmae hs dsrbon by a normal, hrogh an applcaon of he Cenral m Theorem provded n he homeork, here he r.v s. appearng n he heorem, correspond o he Bernoll random varables menoned above. Hence, n he follong e shall ake: Then, he problem becomes a esvendor problem on he above demand, h: nderage cos 7-5, and overage cos 5 Therefore, from normal able Ths, clb manager shold order 3already regsered members47expeced members 77 cps a he prce of $5.

11 xra Cred Problem

12 Problem. Sho ha Proof: are ndependen are ndependen and n n n n n n n

13 Also, Therefore, e ge.

14 Problem 3 e gx be he dsrbon fncon of he random demand n he nevendor problem. Then c smax{,} p cmax{,} c xgxdx smax{,} p c x gxdx c xgxdx smax{,} p x gxdx c gxdx c xgxdx smax{,} p xgxdx gxdx c c smax{,} p xgxdx xgxdx xgxdx gxdx c c smax{,} p pmn{, } {pmn{, } smax{,} c} p c qaon Therefore, mn c s max{,} p c max{,} mn { pmn{, } smax{,} c} p c max pmn{,} smax{,} c p c oe ha he las erm p c can be dropped from he objecve fncon becase s ndependen of. Therefore, he o objecves are eqvalen. Remark: Maybe a beer ay o ndersand he resl of q., s by re-rng as: pmn{,} smax{,} c p c p cmax{,} c smax{,}