MULTI-PRODUCT INVENTORY CONTROL MODEL WITH CONSTRAINTS

Transcription

1 spot d lcommucto Vol.7, No, 6 MUIRDU INVNRY NR MD WIH NSRAINS ug Kopytov, Fdo ss, od Gglz spot d lcommucto Isttut omoosov St., Rg, V9, tv h: Fx: ml: optov@ts.lv, fdotss@ts.lv Rg Ittol School of coomcs d Busss Admstto Mz St.,., Rg, V48, tv ml: gglz@cb.lv upos of th offd wo s th costucto of multpoduct vtoy cotol modl wth dom dmd fo goods, dom ldtm d dom dvto of pcs d stuctul costts o th soucs stoc cpcty, cptl d oths. It s poposd, tht dstbutos of ths dom vbls ow. osdd ts hs b solvd s th poblm of stochstc pogmmg sttmt. Kywods: vtoy cotol, multpoduct od, dmd, ldtm, optmsto, costts o soucs. INRDUIN A bg tst to vtoy cotol poblms, sg lst dcds, s xpld by sgfct fcl ffct, whch th dcsos of ths poblms hv md vous s of coomy. h modl cosdd th gv pp hs s dug studyg poblm of stopg o th lwy []. h lwy s pt of th tspot dusty hs umb spcfc ftus; mog thm w should ot th followg: lg dstcs btw ltd obcts ts, sttos tc.; hgh dymcs of tspotto pocsss; dpdc o dom fctos wth codtos, dmds fo tspotto tc.; hgh umts fo lblty d sfty of pssg d cgo tspotto; lg fcl, mtl d lbou soucs. hfo v smll os mgmt d plg th tspotto pocsss sult cosdbl dcs of th ffccy of th lwy compy pfomc d of th ulty of th clts svc. It s ut complct mthmtcl ts to fd optml soluto fo th cssy stoc, f you wog th lwys dusty. h sch of th ffctv solutos of stoc cotol tspot compy should b bsd o umb of coomc, socl d tchcl chctstcs. Som of thm dom, d som hv lg dymc. I pctc w hv to vstgt th stochstc modls fo dfft stutos chctzg vtoy cotol systms. As th sult umb of stochstc modls vlbl to solv th vtoy cotol poblm [, ]. I pvous uthos wos som stochstc vtoy cotol modls hv b ctd [3, 4]. I th gv sch w cosd multpoduct vtoy cotol modl wth dom dmd fo goods, dom ldtm d dom dvto of pcs. It s poposd, tht dstbutos of ll dom vbls ow. W ssum, tht dlvs of dfft poducts dpdt. cpl m of th xmd ts s to df th mout of th ods d odg momts of ch poduct to chv th mmum xpss fo poducts stog, odg d losss fom dfct p tm ut. h modl c b md mo lstc f w dd som stctos o th usd soucs. I th suggstd modl w suppos tht th lmts to th stoc cpcty d usd cptl. So, th poposd modl hs two stuctul costts: whous costt d fcl costt. 8

2 ocdgs of th 5 th Ittol ofc RlStt 5 t. DSRIIN F H MD W cosd multpoduct vtoy cotol modl ud th followg codtos. h dfft typs of poducts. Fo poduct wth umb dmd, ld tm d pc dom wth ow dstbutos: dmd fo poduct D hs osso dstbuto wth tsty ; ld tm tm btw plcg od d cvg t hs oml dstbuto wth pmts µ и ; pc p fo poduct hs oml dstbutos wth pmts p d p. I th momt of tm wh th stoc lvl of poduct wth umb flls tll ct lvl w od s plcd. h utty s clld od pot, th od utty. W suppos tht utts d fo poduct wth umb costt d. A od s plcd fom custom to poduc dctly. W udl tht d optg pmts of th cosdd modl. t us toduc dd ottos fo poduct wth umb : odg cost s fucto of th od utty,.. ; holdg cost s popotol to utty of goods stoc d holdg tm wth coffct of popotolty H ; losss fom dfct of o ut of th poduct popotol to utty of dfct wth coffct of popotolty SH ; cs of dfct, th lst cot b covd by xpctd od. Suppos tht custom hs stctos o th soucs: th lmts to th whous cpcty V d usd cptl sum of moy K. Ad th xsts th pobblty of smultous cvg th ods by ll ds of poducts, d v cs wh th ods fo dfft poducts plcd t vous momts of tm plcs whous lloctd ud ch poduct pmt. Bsd tht th mxml sz of moy lloctd fo puchs of poduct wth umb s ow d w hv K. 3. RAIN F H MD g ccout tht dmd D fo poduct wth umb hs osso dstbuto wth tsty w c wt th dstbuto fo dmd fo fxd pod of tm D,,,...! Dstbuto fo dmd wth tm c b clcultd by fomul D D f d. Usg fomul w obt: µ µ D d!! d 3 π π Dymcs of vtoy lvl of poduct wth umb dug o cycl tm tvl btw two ghboug od dlvs s show Fg.. 9

3 spot d lcommucto Vol.7, No, 6 ϕ t t Fgu. Dymcs of vtoy lvl of poduct wth umb dug o cycl I th gv fgu s th utty of poduct wth umb stoc t th tm momt mmdtly ft od cvg; tm btw cvg th od fo poduct wth umb d plcg w od. Usg th dftos gv bov w gt th fomul fo lgth of cycl:. 4 hus w c dtm th utty of goods stoc t th tm momt mmdtly ft od cvg s fucto of dmd D dug ldtm s Fg. : D, D < ; 5, D. At lst, usg fomuls 3 d 5 w c dtm dstbuto fo utty of goods stoc t th tm momt mmdtly ft od cvg: D! π µ d, < ; 6. Accodg to fomul 4 th vg cycl lgth s th sum of two pts:. 7 t us cosd thm dtl. Sc dmd fo poduct hs osso dstbuto wth tsty, tm tvl btw two ghboug usts s xpotlly dstbutd wth pmt. 3

4 ocdgs of th 5 th Ittol ofc RlStt 5 t 3 If, th tm s th sum of such dpdt tm tvls. Hc, fo tm hs lg dstbuto wth pmts d. h m of such dstbuto / s clcultd s follows: /. hus, th vg tm s clcultd by fomul /, 8 wh s dfd by 6. Accodg to codtos of th modl scod dddum fomul 7 s µ ; so vg cycl tm s th followg: µ. 9 Avg holdg cost H wth o cycl: > d D d D H H H. Usg th sml cosdtos w obt vg shotg cost dug ldtm SH :! π µ d SH SH. h tsfomtos cosdd [4] gv us,! b SH SH M, wh, π d M ; µ ; b µ. t of optmsto vg totl costs xpss fo ll poducts holdg, odg d losss fom dfct p tm ut. t s dot ths vg totl cost by A. hs vg cost A c b foud s vg totl cost dug o cycl dvdd by vg cycl tm [6]:

5 spot d lcommucto Vol.7, No, 6 A, 3 wh ; H SH SH d H, dfd ccodgly by fomuls, d 9. t s dot: v volum/su fo holdg of tm of th poduct; p pc of o ut of th poduct; V whous cpcty; К mxmum cptl fo o complx od. ostts o th soucs: v V ; 4 p,,,,. 5 Fo poducts dom pcs w c wt costt 5 th fom p α, 6 wh α gv pobblty. Usg fomulto of stochstc pogmmg w hv dtmstc fom of costt 6: [ p Φ ] p α. 7 Usg 3, 4 d 7 w c ct th mthmtcl modl wth optg pmts,,,,..., fo th cosdd stuto: A m wh H SH wth costts v V; [ p Φ α p ],,,...,. 4. NUSINS Futh gudls of th cut sch th followg: to t to ccout th modl th possbl stcto o th volum of dlvy; to cosd th css wth dfft dstbutos of dom chctstcs of vtoy systm dmd fo goods, ld tm d pcs of poducts. Nxt stgs should b comput lzto of th ts to sch th optml vlus of optg pmts d solvg pctcl xmpls. 3

6 ocdgs of th 5 th Ittol ofc RlStt 5 t Rfcs [] Kopytov., ss F., Gglz. Ivtoy otol Modl fo th ypcl Rlwys ompy. I: ocdgs of th Ittol ofc RIABIIY d SAISIS RANSRAIN d MMUNIAIN RlStt'3, ctob 67, 3, Rg, tv. Rg: spot d lcommucto, 4, Vol. 5, pp [] hop S., Mdl. Supply h Mgmt. odo: tc Hll,. [3] bhu N.U. Quus d Ivtos. Nw Yo: Joh Wlly & Sos, 967. [4] Kopytov., Gglz. s of ptml Ivtoy otol. I: sctos of XXIV Ittol Sm o Stblty oblms fo Stochstc Modls. Sptmb 7, 4, Juml, tv. Rg: spot d lcommucto Isttut, 4, pp [5] Kopytov., Gglz. Ivtoy otol Modl wth Rdom d m d Rdom Dmd. I: ocdgs of th Ittol ofc o Alytcl d Stochstc Modllg chus d Applctos ASMA 5. Ju 4, 5, Rg, tv. IS S 9 Affltd ofc, pp [6] Ross Sh. M. Appld obblty Modls wth ptmzto Applctos. Nw Yo: Dov ublctos, IN.,