Model for Optimal Management of the Spare Parts Stock at an Irregular Distribution of Spare Parts

Transcription

1 Joural of Evromeal cece ad Egeerg A 7 (08) 8-45 do:0.765/6-598/ D DAVID UBLIHING Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars veozar Madzhov Fores Research Isue, Bulgara Academy of ceces, ofa 756, Bulgara Absrac: The arcle preses a model for opmal maageme of he (pare ars) sock whe hey are rregularly dsrbued. The s are dsrbued rregularly ad he supply ad purchase coss deped o he umber of elemes. I s ecessary o fx he opmal umber of supples, for whch he oal purchase ad sorage are mmal. Key words:, opmum maageme, sock, foresry.. Iroduco The queso of he opmal sock maageme has bee dscussed oe way or aoher may publcaos [-7]. Mos ofe, moo omeclaure deerma models are dscussed. They are of eres, because hey eable us o sudy he deology of research mehods used more sophscaed sysems. raccally, a large umber of dffere ypes of socks are sored he warehouses, whch mposes ha dffere kds of omeclaure models for sock maageme of (pare ars) are developed. Aoher mpora feaure of he maageme sysem he pracce s he presece of resrcos, more oal umber of delveres durg he year, he maxmal sorage space, maxmal volume of each separae delvery, mmal quay of he ordered produco, ec. The compaes he real ecoomy usually do o dspose of ulmed facal resource, o be vesed sock, bu jus he oppose hey dspose of a lmed workg capal, by whch hey maa he sock. The curre model reflecs he rregulary of foresry produco,.e. he fac ha he mechazao load, respecvely he eed of s hgh, ad he Correspodg auhor: veozar Madzhov, Asssa rofessor, h.d., research feld: Fores research. purchase coss ad sock fll coss deped o he umber of elemes. Ths s a commo suao, especally oday s marke codos, whe supply depeds largely o marke demad. I s ecessary o fx he opmal umber of supples, for whch he oal coss for sock purchase ad sorage are mmal.. Maerals ad Mehods The curre model reflecs he rregulary of foresry produco,.e. he fac ha he mechazao load, respecvely he eed of s hgh, ad he purchase coss ad sock fll coss deped o he umber of elemes. Ths s a commo suao, especally oday s marke codos, whe supply depeds largely o marke demad. I s ecessary o fx he opmal umber of supples, for whch he oal coss for sock purchase ad sorage are mmal. If he sorage fuco of s m(), he he cos of elemes for he perod [, ] ca be calculaed by he Eq. (): () If he meframe 0 = 0 ll he sock s flled mes hrough equal me ervals ad he mome: m d /

2 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars ;. / ; / ; If a he mome 0 elemes are purchased, he a he mome he umber of he remag uused elemes accordg o Eq. () wll be: () Le us desgae wh he coss of sorage of oe eleme for he whole perod from 0 ll ad we receve, ha for he me / he sorage coss of oe eleme wll be. /. The accordg o Eq. (), he sorage coss of he whole volume of elemes for he me from 0 ll are: 0 [ ] () If a he mome all elemes have bee used up,.e. ( ) =, he we purchase addoal quay of elemes / - / ad smlarly he sorage coss for he perod (, ) wll be: / ; m d d d 0 0 d d [ ] (4) Tll he mome all elemes have bee used up,.e. ( ) = ec.. Therefore he sorage coss С for he whole perod (0, ) wll be: C d (5) d 0 The purchase coss of -s bach of wh a volume Р I may be preseed by k + l, where k are orgazaoal coss (busess rp coss, ravel expeses, ec.), whch are o depeda o he umber of he elemes. Therefore he coss С for elemes purchase are: The he oal coss С are: where as, k l C k l... k l C C k l C (6) As 0 = 0 ad = cos, he В s varable, whch s o depeda o ad order o fd he smalles value of he oal coss s eough o sudy he equao: (7) (8) a exreme. I hs case, we have o oe ha ( ) depeds o, because /. Therefore afer we fd ou he meag of, we ca deerme he ecessary volume of he purchased elemes by he followg formulas: We assume ha he fuco B l...; k B / m 0 k B d k ; / / ; / m a b /. looks lke: (9)

3 40 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars As he researches doe Ref. [] show ha fucos of hs kd are applcable o maches wh a operag lfe e years. The preseed model ad he relao Eq. (9) allow defg he opmal delvery umber o he bass of varable real values of demad, presumed ha he coss of purchase ad sorage of sock are mmal. Accordg o Eq. (), we receve for he cos of elemes (): I complace o Eq. (7) Р s equal o: I s kow ha The ; ; 6 4. b. b a a 6 4 6,. b k. (0) as = cos, he he frs added does o deped o ad s eough o mmze he equao, where. l R b a 4 l d, () k 6. Resuls ad Dscusso 0 b m d a. b a и. d b. I sudy of Eq. (9), we foud ou ha he opmal umber of delveres () durg he meframe ( ) for he preseed levels of sorage coss (). The resuls of k hs sudy are llusraed Fg.. I sudy of Eq. (9), we foud ou ha he opmal umber of delveres () he logsc sysem depeds o he al coss (k). The opmal values are preseed Fg.. We dffereae Eq. (9) ad ge: dr l k d ; d R l 6d. 4 We equae o zero ad ge a cubc equao k l d 0 (0), whch ca be easly solved aalycally. The posve roos of Eq. (0) correspod o he exreme of Eq. (7). If s o a eger, we roud up o a whole. If he fuco has a exreme, s d R mmal. We eed o check wheher 0 ad f so, he he fuco has a soluo. To be able o solve he cubc Eq. (0) we se l / k p; d / k q ad he he equao obas he oulook: () Eq. () s solved by he Carda equaos for a cubc equao. I our case, s mos covee o work wh he rgoomerc soluo. Three cases ca be cosdered depedg o p q : () A < 0 ad p < 0 Eq. () s solved uder he formulas: y cos p cos, p q 0 y q. p p cos,

4 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars k=, s=0 k=, s=0 R, BGN 0 5 Opmal sorage umber k=, s=0 k=, s=40 k=, s=50 0 Opmal Оптимален umber брой 5 4 5, umber Fg. Opmal umber of delveres () he logsc sysem depedg o he sorage coss (). Opmal umber of delveres R, BGN Opmal umber of delveres, umber Fg. Opmal umber of delveres () he logsc sysem depedg o he al coss (к).

5 4 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars y () A 0 ad p > 0 Eq. () s solved lke hs: p cg, y, p cg cos ec ; I s kow ha (8) g g, 4 () A 0 ad p < 0 Eq. () s solved lke hs: p y cosec, p y, cosec cg ; g g, 4 p s. q Eq. () s solved ad s posve roo s he requesed value of. Le us ow sudy he case, whch he fuco m looks lke: a Ae. () Accordg o Eq. () for elemes coss () we ge: I complace wh Eq. (6) Р looks lke: Cosderg ha Gve he fac ha m g q m d A a 0 e a A e a a p Ae a a k, he:. cos a s a, he: cos a s a /. A a a cos s a k k The s a sa s a k, a Gve he fac ha = cos, he he secod added does o deped o ad s eough o mmze he equao: q R k () where. A a a cos s. A q cos a a We dffereae Eq. () ad we ge: dr q k ; s a. d R q. Equaed o zero ad we ge a squared equao k (4) whch s solved aa lycally. The posve roos of Eq. (4) correspod o he exreme of Eq. (7). If s assumed ha s o a whole umber, he s rouded o he whole. If he fuco has a exreme, s mmal. I s ecessary d R o check wheher 0, ad f so, he he fuco s resolved. The equao roo s go afer q rework ad, where k s. A q cosa a s a. Afer subsug q formula for, we ge: sacos a sa ak s a q 0

6 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars 4 I order ha he equao has real posve roos, we d R d R q have o prove ha 0,.e. 0. I order ha he precedg equao s > 0, s ecessary q > 0 ad k > 0, bu k s posve by codo. Furher below we wll sudy he flucuao of he oal coss (R ), depedg o he model parameer, amely he delvery umber () durg he perod ( ). The objec of he sudy s he flucuao of he model parameer umber of delveres () depedg o he maageme facors ad coeffce of he sorage fuco m a b. The maageme facors are he sorage facors (s), he al coss (k) ad he sorage fuco ( m ). The coeffces of he sorage fucos ( m ) are а, b ad γ. Afer he oal coss fuco rework we equae o zero he frs dervave of he fuco ad ge a cubc equao, whch posve roos cocde o he exreme of he fuco. The hree roos of he cubc equao are locaed a he cross po of he graphc wh he ordae R = 0. The cubc equao has oe posve ad wo egave roos, oe of whch has egave values close o zero. The posve roo s he soluo of he equao ad gves he value of he delveres umber () durg he perod depedg o he maageme facors, ad he egave values of delveres umber () have o physcal meag. Fg. preses he flucuao of supply umber () durg he perod depedg o he coeffce values (a) by he sock fuco ( m ). The resul aalyss shows ha wh he crease of he coeffce (a) he umber of delveres durg he year () grows quckly,.e. he value of he coeffce (a) flueces srogly he delveres umber durg he year (). The daa for obag he coeffces of he regresso equao (а, b, γ) are from drec observaos. The sudy, led wh dffere values of he coeffce (b) (Fg. 4), over he delveres umber (), shows ha he coeffce value (b) of he sock fuco ( m ) flueces sgfcaly over he delvery umber flucuao durg he year (). The sudy, coduced by dffere values of delvery umber () a dffere values of he coeffce (γ) of he sock fuco ( m ) (Fg. 5), proves ha he coeffce value (γ) praccally does o fluece he delveres umber (). R BGN, umber Fg. The delveres umber () flucuao durg he perod depedg o he coeffce flucuao (a) by he sock fuco ( m ).

7 44 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars R, BGN R, BGN, umber Fg. 4 The delveres umber () flucuao durg he perod depedg o he coeffce flucuao (b) by he sock fuco ( m ). N, umber Fg. 5 The delveres umber () flucuao durg he perod depedg o he coeffce flucuao (γ) by he sock fuco ( m ). 4. Ma Coclusos A mahemacal model for opmzao of he ecessary umber for maeace of he mache workload a a rregular cosumpo has bee elaboraed. The fluece of he ma parameers of he model over he flucuao of he delveres volume umber has bee suded. Leraure [] Vbe, V., ad usheko, B. 98. O he Mehod of Raog Cosumpo of Tracor pare ars. Trudy ChHI 66: ( Russa) [] Dmrov, B cefc Iveory Maageme-ofa. ( Bulgara) [] Kosa, G., Aleksadrov, K., e al. 00. urvey of Operaos-ofa. ( Bulgara) [4] prdoov, G., ad Tasev, G Jusfcao of he arameers of he ysem for Dsrbuo of pare ars

8 Model for Opmal Maageme of he pare ars ock a a Irregular Dsrbuo of pare ars 45 bewee he Us of ROB. Rousse, Repor o he subjec No. 850/ (o publshed). ( Bulgara) [5] pyrdoov, G., ad Tasev, G. 98. ome Theorecal ad Appled Aspecs of he Repar ad Maeace of Agrculural Machery. Rousse. ( Bulgara) [6] Hadley, G A comparso of Order uaes Compued Usg he Average Aual Cos ad he Dscoued Cos. Maageme cece 0 (): [7] Harao, H Opmal order quay NORM. Operao Research (): 9-4.