Deterministic Inventory Models

Transcription

1 Tpic 7 eterministic Inventry Mdels LEARNING OUTOMES By te end f tis tpic, yu suld be able t: 1. Explain te imprtance f inventry cntrl;. mpute te EO and EP t determine w muc t rder r prduce; 3. mpute te ROP in determining wen t rder; and 4. Slve inventry prblems tat allw quantity discunts. INTROUTION Inventry means any stred resurce tat is used t satisfy a current r a future need; fr example, te raw materials and finised gds. Terefre te inventry cntrl is crucial fr every cmpany. Reducing n-and inventry level means reducing csts and tus increases te cas flw. On te ter and frequent stckuts may dissatisfy te custmers. Tus a cmpany must make a balance between lw and ig inventry levels. Te majr factr t be cnsidered in acieving tis balance is te cst minimisatin.

2 TOPI 7 ETERMINISTI INVENTORY MOELS INVENTORY EISIONS Tere are tw fundamental decisins tat we ave t make wen cntrlling inventry: 1. Hw muc t rder r prduce?. Wen t rder? Te majr bjective in cntrlling inventry is t minimize te ttal inventry csts wic include: 1. st f te items (purcase cst r material cst),. st f rdering, 3. st f carrying r lding inventry and 4. st f stck uts. 7. EONOMI ORER UANTITY (EO) Tis is te basic mdel in acieving an ptimal rdering quantity wic minimises te ttal inventry csts. Hwever tere are sme assumptins t be upeld: (a) (b) (c) (d) (e) (f) emand is knwn and cnstant. Te lead time, te time between te placement f rder and te receipt f te rder, is knwn and cnstant. Te receipt f inventry is instantaneus. uantity discunts are nt pssible. Te nly variable csts are te rdering cst and te lding r carrying cst. Orders are placed s tat stck uts r srtages are avided cmpletely.

3 116 TOPI 7 ETERMINISTI INVENTORY MOELS Tese assumptins lead t an inventry usage f a saw tt sape, as swn in Figure 7.1. Figure 7.1: Inventry usage ver time in a simple mdel 7..1 Inventry sts in te EO Situatin Let be te amunt rdered. Tus an inventry levels increases frm 0 t units wen an rder arrives. Since te inventry level canges daily, we use te average inventry level t determine te annual lding r carrying cst. Terefre, Maximum inventry level = Average inventry level = Te fllwing variables are imprtant in develping matematical expressins fr te annual rdering and carrying csts. Ordering quantity = EO = * Annual emand = Ordering cst per rder = Hlding r carrying cst per unit per year =

4 TOPI 7 ETERMINISTI INVENTORY MOELS 117 Annual Ordering st = (number f rders per year) (rdering cst per rder) = = Annual Hlding st = (average inventry) (carrying cst per unit per rder) = = Figure 7. sws te grap f te rdering cst, lding cst and te ttal inventry cst. Te lwest pint n te ttal cst ccurs were te rdering cst is equal t te lding cst. Figure 7.: Ttal cst as a functin f rder quantity

5 118 TOPI 7 ETERMINISTI INVENTORY MOELS 7.. Finding te EO Te abve grap sws tat te ttal csts are minimised wen Annual Ordering st = Annual Hlding st = = = In sme cntext, te lding r carrying cst is expressed as a percentage f te unit cst, I. Terefre te lding cst will be calculated as: = I were = cst per unit Finding te ROP (Re-Order Pint) Te secnd inventry questin is wen t rder. Fr tis, we suld knw te lead time first. Lead time, L, (r delivery time) means te time between te placing and receipt f an rder. Tis culd be few days r few weeks and inventry must meet te demand during tese days. Terefre, ROP = (demand per day) (lead time) = d L Tis means, an rder is placed wen te inventry level reaces te ROP, and te new inventry arrives at te same instant te inventry is reacing 0.

6 TOPI 7 ETERMINISTI INVENTORY MOELS Ecnmic Order uantity (EO) Mdel Annual Ordering st = Annual Hlding st = EO = * = Ttal Annual Inventry st (including purcasing cst) = Re Order Pint, ROP = d L + + Example 7.1 A Hardware Sp in Subang Jaya sells,500 brackets in a year and te sales is relatively cnstant trugut te year. Tese brackets are purcased frm a supplier frm Ip fr RM15.00 eac, and te lead time is tw days. Te lding cst per bracket per year is RM1.50 (r 10% f te unit cst) and te rdering cst per rder is RM Tere are 50 wrking days per year. 1. Wat is te EO?. Given te EO, wat is te average inventry? Wat is te annual inventry cst? 3. In minimising te cst, w many rders wuld be made eac year? Wat wuld be te annual rdering cst. 4. Given te EO, wat is te ttal annual inventry cst (including purcase cst)? 5. Wat is te time between rders? 6. Wat is te ROP?

7 10 TOPI 7 ETERMINISTI INVENTORY MOELS Slutin: Annual emand, = 500 Ordering cst per rder, = RM18.75 Hlding r carrying cst per unit per year, Unit cst, Lead time, L Number f wrking days, Wd = RM150 and I = 0.10 = RM15.00 = days = 50 days 1. EO = * = = = 50 brackets 50. Average inventry = 15 Annual inventry cst = # f rders per year = (1.50) RM Annual rdering cst = 500 (18.75) RM Ttal Annual Inventry st (including purcasing cst) = + + = (500 15) = RM Time between rders = Wd per year 50 5 days # rders per year ROP daily demand lead time brackets 50

8 TOPI 7 ETERMINISTI INVENTORY MOELS EONOMI PROUTION UANTITY (EP) In te prductin envirnment, te inventry cntinuusly flws r builds up ver a perid f time after an rder as been placed r wen units are prduced and sld simultaneusly. Hence, te daily demand rate must be taken int accunt in tis mdel. Figure 7.3, sws te inventry levels as a functin f time. Figure 7.3: Inventry usage ver te time in a prductin run mdel In tis mdel, we ave setup cst, s i.e. cst f setting up te prductin facility t manufacture te desired prduct. Te lding cst per unit per year, remains uncanged. Hwever te annual lding cst canges due t te cange in average inventry level. Te fllwing variables are used in develping matematical expressins fr te annual setup and carrying csts fr a prductin run mdel: Prductin uantity = EP = * Annual emand = Setup st per prductin run = s Hlding r carrying cst per unit per year = aily prductin rate aily demand rate Lengt f prductin run in days = p = d = t

9 1 TOPI 7 ETERMINISTI INVENTORY MOELS Inventry sts in te EP Situatin Maximum inventry level = (ttal prductin in t days) (ttal demand in t days) = pt dt = tp d As te prductin quantity is were = pt wic gives t =. p Maximum inventry level = pt dt = tp d p p = d = 1 d p Average Inventry = 1 d p Annual Hlding st = (average inventry) (carrying cst per unit per rder) d = 1 p d = 1 p Annual Setup st = (number f prductin per year) (setup cst per rder) = = s s

10 TOPI 7 ETERMINISTI INVENTORY MOELS Finding te EP Te ttal inventry csts are minimised wen Annual Setup st = Annual Hlding st s = d = 1 p s d 1 p = s d 1 p Ecnmic Prductin uantity (EP) Mdel Annual Setup st = Annual Hlding st = EP = * = s d (1 ) p s d (1 ) p Ttal Annual Inventry st (including purcasing cst) = Re Order Pint, ROP = d L s d + p + (1 )

11 14 TOPI 7 ETERMINISTI INVENTORY MOELS Example 7. Te Hardware Sp in Subang Jaya (refer te Example 7.1), wants t recnsider its decisin f buying te brackets and is cnsidering in-use, prductin. Te setup cst wuld be RM5.00 and 50 brackets culd be prduced in a day nce te macine as been setup. Te cst f prducing ne bracket is estimated as RM Te lding cst wuld be 10% f tis cst. 1. Wat is te daily demand rate?. Wat is te EP? 3. Hw lng will it take t prduce te ptimal prductin quantity? Hw muc inventry is being sld during tis time? 4. Given te EP, wat is te maximum inventry level? Wat is te average inventry? Wat is te annual inventry cst? 5. In minimising te cst, w many prductin runs wuld tere be eac year? Wat wuld be te annual setup cst? 6. Given te EP, wat is te ttal annual inventry cst (including prductin cst)? 7. If te lead time is ne and a alf-day, wat is te ROP? Slutin: Annual emand, = 500 aily prductin rate, p = 50 Setup cst per rder, s = RM5.00 Hlding r carrying cst per unit per year, Unit cst, Lead time, L Number f wrking days, Wd = RM1.48 and I = 0.10 = RM14.80 = 0.5 day = 50 days

12 TOPI 7 ETERMINISTI INVENTORY MOELS emand rate, d = annual demand # f wrking days 50. EP = * = s d (1 ) p = = 35 brackets 3. Prductin time, t = Prductin ty days Prductin rate 50 Inventry sld during t days = t d = = 65 brackets 4. Maximum inventry = prductin inventry sld = = 60 Average inventry = Max inventry r Average inventry = d p 50 d RM19.40 p 50 Annual inventry cst = 5. # f prductin run per year = runs s 5 RM Annual setup cst = 6. Ttal Annual Inventry st (including purcasing cst) = s d + p + (1 ) = ( ) = RM

13 16 TOPI 7 ETERMINISTI INVENTORY MOELS ROP daily demand lead time brackets 50 mparing bt te example 7.1 and 7.7, te in-use prductin is ceaper. 7.4 UANTITY ISOUNT MOEL Te deterministic inventry cntrl mdels, previusly cnsidered ave been develped under te assumptin tat te unit cst f an item was independent f te quantity prduced r rdered. Hwever, in many business and industries, quantity discunts are prvided as incentives fr te purcase f larger quantities f prducts. Wen quantity discunts are available and all te EO assumptins are met, it is pssible fr us t find te quantity tat minimises te ttal annual inventry cst (r simply said, te ttal cst). As te discunts are usually lwering te material cst r te unit cst f a prduct, based n te rder quantity, we ave t calculate te ttal cst based upn te rder quantity in determining te ptimal rdering quantity. Ttal st = material cst + rdering cst + carrying cst = + + r = s d + 1 p + Te prcess fr determining te minimum ttal cst as fllws: 1. Fr eac discunt price (), cmpute te EO, * = r s d 1 p. If * < minimum fr discunt, adjust te quantity t = minimum fr discunt. 3. Fr eac * r adjusted, cmpute te ttal cst.

14 TOPI 7 ETERMINISTI INVENTORY MOELS eck fr te lwest ttal cst and select tat ptin fr ptimal rdering quantity. Example 7.3 A cmpany tat prduces aluminium fil, ffers te fllwing discunt scedule fr its 4 4 seets: iscunt ategry Order Size iscunt (5%) Unit st (RM) A 0 t B 1500 t Mre tan Sar Mini Market rders aluminium fils frm te cmpany abve. Te mini market s rdering cst is RM100. Te carrying cst is 0% f te unit cst and te annual demand is 000 seets. Wat d yu recmmend? Slutin: iscunt ategry Order Size = 000, Unit st (RM) = RM100, I = 0.. EB = * = I Adjusted Ordering uantity, Ttal st I+ A 0 t B 1500 t 499 Mre tan Exceeded mparing te discunt ptins f A, B and, we find tat ptin A, witut discunt, reduces te annual inventry cst. Terefre its better t keep te rdering quantity at 447 seets.

15 18 TOPI 7 ETERMINISTI INVENTORY MOELS 7.5 AB ANALYSIS A cmpany s inventry is divided int tree grups i.e. Grup A, Grup B and Grup based n te verall inventry value. In anter wrd, AB categrises inventry by its igest value and greatest vlume. Managing te greatest value f inventry is crucial as it saves a lt fr te cmpany. But tis may cnsist nly abut 5 15% f te inventry. Te table belw mentins a brief descriptin f eac grup wit general guidelines as w t categrise tem. Inventry Grup Inventry Value (%) Inventry (%) Imprtance f Inventry Management A Yes B Mderate N Segmentatin may nt always ccur s neatly. Te bjective, tug, is t try t separate te imprtant frm te unimprtant. Were te line actually breaks depends n te particular inventry under questin and n w muc persnnel time is available. Figure 7.4: Percentage f ttal list f different stck items

16 TOPI 7 ETERMINISTI INVENTORY MOELS Steps t AB Analysis (a) (b) (c) Srt prducts frm largest t smallest annual $ vlume. ivide int A, B and classes. Fcus n A prducts. (i) evelp class A suppliers mre. (ii) Give tigter pysical cntrl f A items. (iii) Frecast A items mre carefully. (iv) nsider B prducts nly after A prducts. Example 7.4 Te fllwing table describes a cmpany s inventry items wit unit cst and annual usage. Illustrate using an AB plan. Part Unit st Annual Usage $ Part Ttal Value % f Ttal Value % f Ttal uantity % ummulative 9 $30, A , , , B , , , , , , $85,400

17 130 TOPI 7 ETERMINISTI INVENTORY MOELS ATIVITY In a supermarket, an electrical prduct csts RM1,000, te annual lding cst is 5% and te annual demand is 10,000 units. Te rdering cst per rder is RM150. Wat is te rdering size t minimise te inventry csts? (Answer: 70). In a basic EP mdel, te daily prductin rate is 5 units. Given tat te annual demand is 6000 units, setup cst, s = RM100 and annual lding cst, = RM5, find te ptimal prductin quantity.(assume te annual wrking days is 300). (Answer: 173) 3. In an electrical sp, te cst-minimising rder quantity btained wit te basic EO mdel fr an electrical item was 00 units and te ttal annual inventry cst was RM600. Wat is te inventry carrying cst per unit per year fr tis item? (Answer: RM3.00) 4. emand fr a certain prduct rdered by a mini market tends t be cnstant at a mntly rate f 1000 units. Annual lding cst fr tis item is RM5.00 per unit and te cst f placing an rder is RM0.00. (a) Find te ptimal rder size and w ften tis rder suld be placed? (b) Wat is te relevant ttal inventry cst, excluding te purcasing cst? (c) If te mini market makes rders every week, wat is te percentage increase in te relevant ttal inventry cst? (d) Suppse tis prduct can be rdered in multiples f 100 units, wat is te best rder size? (Answer: (a) 310 units, every 9 days (c) 8%)

18 TOPI 7 ETERMINISTI INVENTORY MOELS (a) KH textile sells 175 pieces f and twels per mnt. Te unit cst f a twel is RM.50 and te cst f placing an rder as been estimated at RM1.00. Te stre uses an inventry carrying carge f 7% per year. Find te ptimal rder quantity, rder frequency, and te ttal annual inventry cst. (Answer: 73 units, 8 rders, RM184.44) (b) If te rder is placed nline, te rdering cst sall decrease t RM4.00. Wat wuld be te new ecnmic rder quantity, rder frequency, and te ttal annual inventry cst? Explain tese results. (158 units, 13 rders, RM106.48) 6. Te annual demand fr an item is 1000 units. Te cst t prcess an rder is RM50.00 and te annual inventry lding cst is 5% f te unit cst per item per year. Te unit cst per item is RM50.Wat is te ptimal rdering quantity, given te fllwing quantity discunt scedule? Order size iscunt(%) 0 t t 49 3 mre tan 50 5 (Answer: 50, RM49, ) 7. Mutu Plumbers as a demand fr 1000 pumps every year. A pump csts RM50.00 and te cst t place an rder is RM Annual lding cst per pump is 5% f te unit cst. If te pumps are rdered in quantities f 00, Mutu Plumbers can get a discunt f 3% n te unit cst f te pumps. Suld Mutu rder te pumps in te quantities f 00?

19 13 TOPI 7 ETERMINISTI INVENTORY MOELS 8. Te annual demand fr a certain electrnic prduct is 6750 units. An electrnic cmpany in Prt Klang manufactures tis prduct and its daily prductin capacity is 15 units. Te daily demand during te prductin prcess is 30 units. If te setup cst is RM and te carrying cst is RM1.00 per unit per year, wat is te ptimal prductin quantity tat minimises te annual inventry cst? (Answer: 163 units) 9. A refrigeratr plant in Bangi as a cnstant demand f 8000 units a year. Te plant can prduce 00 units per day and every prductin run takes a setup cst f RM Te lding cst fr refrigeratr is RM0.00 per year. urrently te plant prduces 400 units in eac prductin and te plant perates 50 days in a year. (a) Wat is te daily demand fr tis prduct? (b) Wat is te prductin time fr 400 units? (c) Under te current plicy, wat wuld be te annual setup cst? (d) Under te current plicy, wat is te annual lding cst? (e) Under te current plicy, wat is te ttal f annual setup cst and lding cst? 10. Referring t questin 8 abve, wat wuld be te ptimal prductin quantity minimising te ttal annual setup and lding cst? Hw muc wuld tis save te plant in te annual inventry cst cmpared wit te current plicy?

20 TOPI 7 ETERMINISTI INVENTORY MOELS 133 In tis tpic we ave cvered te fllwing: Te imprtance f inventry management. mpute te EO and EP t determine w muc t rder r prduce. mpute te ROP in determining wen t rder. Handle inventry prblems tat allw quantity discunts. Te AB analysis f inventry management. Render, B. et al. (001). uantitative analysis fr management (9t ed.). NJ: Prentice Hall. Taa, H. A. (003). Operatins researc An intrductin. New Jersey: Prentice-Hall. Hillier, F. S., & Lieberman, G. J. (1990). Intrductin t peratins researc. Singapre; Mc-Graw Hill.