Inventory Control Models Chapter 6
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1 Lecture 9 Intrductin Inventry Cntrl Mdels Chapter 6 T accmpany uantitative Analysis fr Management, Eleventh Editin, by Render, Stair, and Hanna n Inventry is an expensive and imprtant asset t many cmpanies. n Inventry is any stred resurce used t satisfy a current r future need. n Cmmn examples are raw materials, wrkin-prcess, and finished gds. n Mst cmpanies try t balance high and lw inventry levels with cst minimizatin as a gal. n Lwer inventry levels can reduce csts. n Lw inventry levels may result in stckuts and dissatisfied custmers. 6- Intrductin n All rganizatins have sme type f inventry cntrl system. n Inventry planning helps determine what gds and/r services need t be prduced. n Inventry planning helps determine whether the rganizatin prduces the gds r services r whether they are purchased frm anther rganizatin. n Inventry planning als invlves demand frecasting. Intrductin Inventry planning and cntrl Planning n What Inventry t Stck and Hw t Acquire It Frecasting Parts/Prduct Demand Feedback Measurements t Revise Plans and Frecasts Cntrlling Inventry Levels Figure Imprtance f Inventry Cntrl n Five uses f inventry: n The decupling functin n String resurces n Irregular supply and demand n uantity discunts n Aviding stckuts and shrtages n Decuple manufacturing prcesses. n Inventry is used as a buffer between stages in a manufacturing prcess. n This reduces delays and imprves efficiency. Imprtance f Inventry Cntrl n String resurces. n Seasnal prducts may be stred t satisfy ffseasn demand. n Materials can be stred as raw materials, wrkin-prcess, r finished gds. n Labr can be stred as a cmpnent f partially cmpleted subassemblies. n Cmpensate fr irregular supply and demand. n Demand and supply may nt be cnstant ver time. n Inventry can be used t buffer the variability
2 Imprtance f Inventry Cntrl n Take advantage f quantity discunts. n Lwer prices may be available fr larger rders. n Extra csts assciated with hlding mre inventry must be balanced against lwer purchase price. n Avid stckuts and shrtages. n Stckuts may result in lst sales. n Dissatisfied custmers may chse t buy frm anther supplier. Inventry Decisins n There are tw fundamental decisins in cntrlling inventry: n Hw much t rder. n When t rder. n The majr bjective is t minimize ttal inventry csts. n Cmmn inventry csts are: n Cst f the items (purchase r material cst). n Cst f rdering. n Cst f carrying, r hlding, inventry. n Cst f stckuts Inventry Cst Factrs Inventry Cst Factrs ORDERING COST FACTORS Develping and sending purchase rders Prcessing and inspecting incming inventry Bill paying Inventry inquiries Utilities, phne bills, and s n, fr the purchasing department Salaries and wages fr the purchasing department emplyees Supplies, such as frms and paper, fr the purchasing department Table 6.1 CARRYING COST FACTORS Cst f capital Taxes Insurance Spilage Theft Obslescence Salaries and wages fr warehuse emplyees Utilities and building csts fr the warehuse Supplies, such as frms and paper, fr the warehuse n Ordering csts are generally independent f rder quantity. n Many invlve persnnel time. n The amunt f wrk is the same n matter the size f the rder. n Carrying csts generally varies with the amunt f inventry, r the rder size. n The labr, space, and ther csts increase as the rder size increases. n The actual cst f items purchased can vary if there are quantity discunts available Ecnmic Order uantity n The ecnmic rder quantity (EO) mdel is ne f the ldest and mst cmmnly knwn inventry cntrl techniques. n It is easy t use but has a number f imprtant assumptins. n Objective is t minimize ttal cst f inventry. Ecnmic Order uantity Assumptins: 1. Demand is knwn and cnstant.. Lead time is knwn and cnstant. 3. Receipt f inventry is instantaneus. 4. Purchase cst per unit is cnstant thrughut the year. 5. The nly variable csts are the cst f placing an rder, rdering cst, and the cst f hlding r string inventry ver time, hlding r carrying cst, and these are cnstant thrughut the year. 6. Orders are placed s that stckuts r shrtages are avided cmpletely
3 Inventry Usage Over Time Inventry Csts in the EO Situatin Inventry Level Order uantity Maximum Inventry Level Cmputing Average Inventry Average inventrylevel Minimum Inventry Figure 6. 0 Time INVENTORY LEVEL DAY BEGINNING ENDING AVERAGE April 1 (rder received) April April April April Maximum level April 1 10 units Ttal f daily averages Number f days 5 Average inventry level 5/5 5 units Table Inventry Csts in the EO Situatin Mathematical equatins can be develped using: number f pieces t rder EO * ptimal number f pieces t rder D annual demand in units fr the inventry item C rdering cst f each rder hlding r carrying cst per unit per year Inventry Csts in the EO Situatin Mathematical equatins can be develped using: number f pieces t rder EO * ptimal number f pieces t rder D annual demand in units fr the inventry item C rdering cst f each rder hlding r carrying cst per unit per year Number f Annual rdering cst rders placed per year D C Ordering cst per rder Average Annual hlding cst inventry Carrying cst per unit per year Inventry Csts in the EO Situatin Finding the EO Ttal Cst as a Functin f Order uantity Cst Minimum Ttal Cst Curve f Ttal Cst f Carrying and Ordering Accrding t the graph, when the EO assumptins are met, ttal cst is minimized when annual rdering cst equals annual hlding cst. D C Carrying Cst Curve Ordering Cst Curve Slving fr DC DC C h Figure 6.3 Optimal Order uantity Order uantity 6-17 DC EO *
4 Ecnmic Order uantity (EO) Mdel Sumc Pump Cmpany Summary f equatins: Annual rdering cst D Annual hlding cst C n Sumc Pump Cmpany sells pump husings t ther cmpanies. n The firm wuld like t reduce inventry csts by finding ptimal rder quantity. n Annual demand 1,000 units n Ordering cst $10 per rder n Average carrying cst per unit per year $0.50 EO * DC C h * DC ( 1000, )( 10) 40, units C h Sumc Pump Cmpany Ttal annual cst Order cst + Hlding cst D TC C , 00 ( 10) + ( 0. 5) 00 $ 50 + $ 50 $ 100 Purchase Cst f Inventry Items n Ttal inventry cst can be written t include the cst f purchased items. n Given the EO assumptins, the annual purchase cst is cnstant at D C n matter the rder plicy, where n C is the purchase cst per unit. n D is the annual demand in units. n At times it may be useful t knw the average dllar level f inventry: (C) Average dllar level Purchase Cst f Inventry Items n Inventry carrying cst is ften expressed as an annual percentage f the unit cst r price f the inventry. n This requires a new variable. I Annual inventry hlding charge as a percentage f unit price r cst n The cst f string inventry fr ne year is then thus, IC * DC IC Sensitivity Analysis with the EO Mdel n The EO mdel assumes all values are knw and fixed ver time. n Generally, hwever, sme values are estimated r may change. n Determining the effects f these changes is called sensitivity analysis. n Because f the square rt in the frmula, changes in the inputs result in relatively small changes in the rder quantity. DC EO C h
5 Sensitivity Analysis with the EO Mdel n In the Sumc Pump example: EO ( 1000, )( 10) 00 units n If the rdering cst were increased fur times frm $10 t $40, the rder quantity wuld nly duble EO ( 1000, )( 40) 400 units n In general, the EO changes by the square rt f the change t any f the inputs. Rerder Pint: Determining When T Order n Once the rder quantity is determined, the next decisin is when t rder. n The time between placing an rder and its receipt is called the lead time (L) r delivery time. n When t rder is generally expressed as a rerder pint (ROP). Demand ROP per day d L Lead time fr a new rder in days Prcmp s Cmputer Chips Rerder Pint Graphs n Demand fr the cmputer chip is 8,000 per year. n Daily demand is 40 units. n Delivery takes three wrking days. ROP d L 40 units per day 3 days 10 units n An rder based n the EO calculatin is placed when the inventry reaches 10 units. n The rder arrives 3 days later just as the inventry is depleted. Figure EO Withut The Instantaneus Receipt Assumptin n When inventry accumulates ver time, the instantaneus receipt assumptin des nt apply. n Daily demand rate must be taken int accunt. n The revised mdel is ften called the prductin run mdel. Annual Carrying Cst fr Prductin Run Mdel n In prductin runs, setup cst replaces rdering cst. n The mdel uses the fllwing variables: number f pieces per rder, r prductin run C s setup cst hlding r carrying cst per unit per year p daily prductin rate d daily demand rate t length f prductin run in days Figure
6 Annual Carrying Cst fr Prductin Run Mdel Maximum inventry level (Ttal prduced during the prductin run) (Ttal used during the prductin run) since we knw (Daily prductin rate)(number f days prductin) (Daily demand)(number f days prductin) (pt) (dt) Maximum inventry level Ttal prduced pt t p d pt dt p d 1 p p p Annual Carrying Cst fr Prductin Run Mdel Since the average inventry is ne-half the maximum: and d Average inventry 1 p d Annualhlding cst 1 p Annual Setup Cst fr Prductin Run Mdel Setup cst replaces rdering cst when a prduct is prduced ver time. Determining the Optimal Prductin uantity By setting setup csts equal t hlding csts, we can slve fr the ptimal rder quantity replaces Annual setup cst Annual rdering cst D C s D C Annual hlding cst Annual setup cst Slving fr, we get d D 1 C p * h C s DC s d Ch 1 p Prductin Run Mdel Brwn Manufacturing Summary f equatins d Annualhlding cst 1 p Annual setup cst Optimal prductin quantity D * C s DCs d Ch 1 p Brwn Manufacturing prduces cmmercial refrigeratin units in batches. Annual demand D 10,000 units Setup cst C s $100 Carrying cst $0.50 per unit per year Daily prductin rate p 80 units daily Daily demand rate d 60 units daily 1. Hw many units shuld Brwn prduce in each batch?. Hw lng shuld the prductin part f the cycle last?
7 Brwn Manufacturing Example * DC s 1.. d Ch 1 p * 10, , 000, ( 1 ) 4 4,000units 16, 000, 000 Prductin cycle p 4 000, 50 days 80 uantity Discunt Mdels n uantity discunts are cmmnly available. n The basic EO mdel is adjusted by adding in the purchase r materials cst. Ttal cst Material cst + Ordering cst + Hlding cst where D Ttal cst DC + C + D annual demand in units C rdering cst f each rder C cst per unit hlding r carrying cst per unit per year uantity Discunt Mdels n uantity Because discunts unit are cst cmmnly is nw variable, available. n The basic EO mdel is adjusted by adding in the Hlding cst C purchase r materials cst. h IC I hlding cst as a percentage f the unit cst (C) Ttal cst Material cst + Ordering cst + Hlding cst D Ttal cst DC + C + where D annual demand in units C rdering cst f each rder C cst per unit hlding r carrying cst per unit per year uantity Discunt Mdels n A typical quantity discunt schedule can lk like the table belw. n Hwever, buying at the lwest unit cst is nt always the best chice. DISCOUNT NUMBER Table 6.3 DISCOUNT UANTITY DISCOUNT (%) DISCOUNT COST ($) 1 0 t ,000 t 1, ,000 and ver uantity Discunt Mdels Ttal cst curve fr the quantity discunt mdel Figure 6.6 Ttal Cst $ TC Curve fr Discunt 1 0 1,000,000 Order uantity TC Curve fr Discunt 3 TC Curve fr Discunt EO fr Discunt 6-41 Brass Department Stre n Brass Department Stre stcks ty race cars. n Their supplier has given them the quantity discunt schedule shwn in Table 6.3. n Annual demand is 5,000 cars, rdering cst is $49, and hlding cst is 0% f the cst f the car n The first step is t cmpute EO values fr each discunt. ( )( 5, 000)( 49) EO1 700 cars per rder ( 0. )( 5. 00) ( )( 5, 000)( 49) EO 714 cars per rder ( 0. )( 4. 80) ( )( 5, 000)( 49) EO3 718 cars per rder ( 0. )( 4. 75) 6-4 7
8 Brass Department Stre Example n The secnd step is adjust quantities belw the allwable discunt range. n The EO fr discunt 1 is allwable. n The EOs fr discunts and 3 are utside the allwable range and have t be adjusted t the smallest quantity pssible t purchase and receive the discunt: ,000 3,000 Brass Department Stre The third step is t cmpute the ttal cst fr each quantity. DISCOUNT NUMBER UNIT PRICE (C) ORDER UANTITY () ANNUAL MATERIAL COST ($) DC ANNUAL ORDERING COST ($) (D/)C ANNUAL CARRYING COST ($) (/) TOTAL ($) 1 $ , , ,000 4, , ,000 3, ,8.50 The final step is t chse the alternative with the lwest ttal cst. Table Use f Safety Stck n If demand r the lead time are uncertain, the exact ROP will nt be knwn with certainty. n T prevent stckuts, it is necessary t carry extra inventry called safety stck. n Safety stck can prevent stckuts when demand is unusually high. n Safety stck can be implemented by adjusting the ROP. Use f Safety Stck n The basic ROP equatin is ROP d L d daily demand (r average daily demand) L rder lead time r the number f wrking days it takes t deliver an rder (r average lead time) n A safety stck variable is added t the equatin t accmmdate uncertain demand during lead time where ROP d L + SS SS safety stck Use f Safety Stck ROP with Knwn Stckut Csts n With a fixed EO and an ROP fr placing rders, stckuts can nly ccur during lead time. n Our bjective is t find the safety stck quantity that will minimize the ttal f stckut cst and hlding cst. n We need t knw the stckut cst per unit and the prbability distributin f demand during lead time. n Estimating stckut csts can be difficult as there may be direct and indirect csts. Figure
9 Safety Stck with Unknwn Stckut Csts n There are many situatins when stckut csts are unknwn. n An alternative apprach t determining safety stck levels is t use a service level. n A service level is the percent f time yu will nt be ut f stck f a particular item. Service level 1 Prbability f a stckut r Prbability f a stckut 1 Service level Single-Perid Inventry Mdels n Sme prducts have n future value beynd the current perid. n These situatins are called news vendr prblems r single-perid inventry mdels. n Analysis uses marginal prfit (MP) and marginal lss (ML) and is called marginal analysis. n With a manageable number f states f nature and alternatives, discrete distributins can be used. n When there are a large number f alternatives r states f nature, the nrmal distributin may be used Marginal Analysis with Discrete Distributins We stck an additinal unit nly if the expected marginal prfit fr that unit exceeds the expected marginal lss. P prbability that demand will be greater than r equal t a given supply (r the prbability f selling at least ne additinal unit). Marginal Analysis with Discrete Distributins n The expected marginal prfit is P(MP). n The expected marginal lss is (1 P)(ML). n The ptimal decisin rule is t stck the additinal unit if: P(MP) (1 P)ML n With sme basic manipulatin: 1 P prbability that demand will be less than supply (r the prbability that ne additinal unit will nt sell). P(MP) ML P(ML) P(MP) + P(ML) ML P(MP + ML) ML r ML P ML + MP Steps f Marginal Analysis with Discrete Distributins ML 1. Determine the value f fr the prblem. ML + MP. Cnstruct a prbability table and add a cumulative prbability clumn. 3. Keep rdering inventry as lng as the prbability (P) f selling at least ne additinal ML unit is greater than ML + MP Café du Dnut n The café buys dnuts each day fr $4 per cartn f dzen dnuts. n Any cartns nt sld are thrwn away at the end f the day. n If a cartn is sld, the ttal revenue is $6. n The marginal prfit per cartn is. MP Marginal prfit $6 $4 $ n The marginal lss is $4 per cartn since cartns can nt be returned r salvaged
10 Café du Dnut s Prbability Distributin DAILY SALES (CARTONS OF DOUGHNUTS) Table 6.6 PROBABILITY (P) THAT DEMAND WILL BE AT THIS LEVEL Ttal 1.00 Café du Dnut Example ML Step 1. Determine the value f fr the decisin rule. ML + MP ML $ 4 4 P ML + MP $ 4 + $ 6 P 0.67 Step. Add a new clumn t the table t reflect the prbability that dughnut sales will be at each level r greater Marginal Analysis fr Café du Dnut Café du Dnut Example DAILY SALES (CARTONS OF DOUGHNUTS) PROBABILITY (P) THAT DEMAND WILL BE AT THIS LEVEL PROBABILITY (P) THAT DEMAND WILL BE AT THIS LEVEL OR GREATER Ttal 1.00 Step 3. Keep rdering additinal cartns as lng as the prbability f selling at least ne additinal cartn is greater than P, which is the indifference prbability. P at 6 cartns 0.80 > 0.67 Table ABC Analysis n The purpse f ABC analysis is t divide the inventry int three grups based n the verall inventry value f the items. n Grup A items accunt fr the majr prtin f inventry csts. n Typically abut 70% f the dllar value but nly 10% f the quantity f items. n Frecasting and inventry management must be dne carefully. n Grup B items are mre mderately priced. n May represent 0% f the cst and 0% f the quantity. n Grup C items are very lw cst but high vlume. n It is nt cst effective t spend a lt f time managing these items. INVENTORY GROUP Table 6.8 Summary f ABC Analysis DOLLAR USAGE (%) INVENTORY ITEMS (%) ARE UANTITATIVE CONTROL TECHNIUES USED? A Yes B 0 0 In sme cases C N
11 Cpyright All rights reserved. N part f this publicatin may be reprduced, stred in a retrieval system, r transmitted, in any frm r by any means, electrnic, mechanical, phtcpying, recrding, r therwise, withut the prir written permissin f the publisher. Printed in the United States f America
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