deterministicinventorymodelsrawmaterialsfinishedproducts-091227160230-phpapp02

8/8/2019 deterministicinventorymodelsrawmaterialsfinishedproducts-091227160230-phpapp02

1/43

Inventoryofrawmaterialsand

finished

products:

deterministic

Topicobjectives

Understandtheneedforinventory

Selectand

a l

the

a ro riate

inventor

model

DrMuhammadAlSalamah,IndustrialEngineering,KFUPM


2/43

Classificationofinventory

Manufacturingclassifiesinventoryinto:

to

products Componentsinventory:incompleteproducts

Workinprocessinventory:materialsorcomponentsinthe

manufacturing

process



3/43

Whyholdinginventory

1. Economyofscale:productioncostperunitdeclineswiththesizeoftheproduction,henceproducemoreandstore

2. Uncertainty:demandismostlyuncertain,hencebuffer

againstdemand

fluctuation

. ,buymoretoday

4. Transportation:transporting

units

takes

time,

hence

store

to

5. Smoothing:frequentchangeofproductioncapacityis

expensive,

hence

fix

capacity

and

store

for

peak

production

demands

6. Logistics:incaseofminimumordersize


. ,hence

reduce

the

average

cost

by

increasing

the

order


4/43

Characteristicsofinventorysystems

1. Demand,whichismeasuredindemandrate:

Knownvs unknown

2. Leadtime,whichisthetimebetweenthetimetheorderis

placedtothetimetheorderisreceived.

3. Reviewpolicy,

which

can

be

either

continuous

or

periodic

4. Excess eman ,w c can ee t er ac or ere or ost



5/43

Costsrelatedtoinventory

Todeterminethebestinventoryplan,inmostcases,cost

minimizationisthepreferredchoice.

Aninventorycostcanbeeitherholdingcost,ordercost,or

penaltycost.



6/43

Inventoryholdingcost

Itisthetotalofallcoststhatareproportionaltotheamount

ofinventoryphysicallyonhandatanypointintime.

Theholdingcostmustaccountforthefollowingcosts:

Costofthephysicalspace

Insurance

Breakage,

spoilage,

deterioration,

etc

pportun tycost

Inventoryholdsupcapital,andwhenthiscapitalistaken

calledtheopportunitycosts.



7/43

Illustration

Acompanyhasinventorythathasavalueof1millionriyals.

.

Thecompany

can

invest

its

money

either

in

the

local

stock

market,GCCmarket,landpurchase,etc,withareturnof

about4%peryear.

What

is

the

opportunity

cost

associated

with

holding

this

Thereturnfromtheinvestmentafteroneyearis

1(1+0.04)=1,040,000riyals Assumingtheinventorydoesnotloseitsvalueduringthe

year,theopportunitycostis


1,040,000 1,000,000

=40,000

=1,000,0000.04


8/43

Chargesexpressedinpercentage

Thecomponentsoftheholdingcostareusuallyexpressedin

percentchargeperperiod.

Forexample,takethefollowingpercentchargesperyear:

Opportunitycost=22%

Insurance=1.4%

Costof

physical

storage

space

=7%

Deter orat onan spo age=

Then,theinventorypercentchargeperyearisI=32.4%.



9/43

Conversionofinterestrate

Toconverttheinterestratefromyearlytomonthlyfor

example,applytheformula

1112 yearlymonthly += IINotethatthereare12monthsinayear.

The

conversion

from

monthly

to

early

interest

rate

is

1)1( 12monthlyyearly += II



10/43

Perunitholdingcost

Theinventoryholdingcostisexpressedintermsofriyalsper

unitperperiod(year,month,week,etc).

Ifcisthemonetaryvalueofoneinventoryunitand

isthe

interestrateperyear(ormonth,etc),thentheholdingcost

perun peryear s

h= c

,

inventoryholdingcostbecomes

h

I

Iftheinventoryisheldfor3years,thetotalinventoryholding

costis



11/43

Theinventorycanbepresentedby

t

Theinventoryholdingcostisbasicallytheareaunderthe

curveoftheinventorylevelmultipliedbytheinventory

h I t



12/43

Generally,ifthelevelofinventoryisrepresentedbythe

functionI(t),thetotalinventorycostintheperiodbetweent

andt2 is

2t

1

)(

t

dttIh

Theaverageinventorycostis

2t

1

)( dttI

ht


12 tt


13/43

Ordercost

Theordercostincludesallcostsassociatedwiththe

acquisitionofinventoryitems,whether rawmaterialsor

finishedproducts.

Theordercosthastwocomponents,fixedandvariable.

LetKbethefixedorsetupcostandcbetheperunitcost,

C(x)betheorderorproductioncostofxunits:

>+

==

0if

)(

xcxK

xxC



14/43

Penaltycost

Thepenaltycost,shortagecost,orstockoutcostisthecostof

nothavingsufficientstockonhandtosatisfyademandwhen

itisreceived.

Theshortageistreatedintwoways,backorderedorlost.

Backordereditemsarecarriedovertotheperiodwhenanew

orderarrives.

calledlostsales.

Thepenaltycostpischargedperunit,irrespectiveofthe

durationoftimetofulfillthedemandedunits.



15/43

Economicorderquantitymodel

TheEOQisabasicmodelthatbalancesthefixedorderingcost

andtheholdingcost.

TheassumptionofEOQare

Thedemandrate isknownandconstant

Shortagesarenotallowed

The

lead

time

is

zero

T ecosts nvo ve aret e xe or ercostK,var a e

ordercostperunitc,andholdingcosthperunitpertime.

=

levelisassumedtobezeroandanorderofsizeQismadeand

received.


The

next

time

an

order

is

placed

is

when

the

inventory

level

dropstozeroagain.


16/43

TheobjectiveoftheEOQmodelistofindthelotsizeQthat

willminimizetheaveragecostperunittime.

Ineachcycle,thefixedandvariableordercostis

C(Q)=K+cQ

Acycleoftheinventorymodelisservedbyonelot.

The

duration

Tof

the

cycle

is

the

time

that

is

needed

to

,

T=Q/



17/43

Aplotofseveralinventorycyclesis

Q

TTime



18/43

Thetotalinventoryheldinacycleisgivenbytheareaunder

thecurve:

QT21

Then,

the

total

inventory

holding

cost

in

acycle

is

hQT21

,

hQTcQK21cost ++=



19/43

Theaveragecostperperiodiscalculatedbydividingthetotal

costbythecyclelength:

2)( hQ

TcQKQG ++=

ReplaceTbyitsequivalence:

2/)(

QQG +=

2c

Q++=



20/43

TheaveragecostfunctionG(Q)isconvex:

2

)(' 2 +=h

Q

KQG

0

2)('' 3 >=

K

QG

Theminimizer ofG(Q)isfoundby

02

)('2

=+=Q

QG



21/43

Theoptimalorderquantityis

h

K

Q

2

= Thisquantityiscalledtheeconomicorderquantity.



22/43

Illustration

AdvancedElectronicsCoproduceslaptopsatarateof60per

week.

AECneedstoplanthepurchaseofDVDdrivewhichisoneof

therawmaterialcomponentofalaptop.

Thefixedordercostis1,200riyalsandthepurchaseofa

singleDVDdriveis200riyals.

.

Whatistheeconomicorderquantity?



23/43

Firstthing,thetimeperiodhastobeexpressedinacommon

unit.

wecanchoosetotheweekasthebasisoftime.

Theweeklyinterestrateis

%43.0125.0152weekly =+=I

T e nventory o ngcost s

h=2000.43%=0.86riyalsperDVDdriveperweek



24/43

Theeconomicorderquantityis

drivesDVD2.409

86.0

,==Q

Anorderhastoplacedevery

T=409.2/60

=6.82

weeks



25/43

Reorderpoint

Whenthereisaleadtime,theorderhastobeplacedsoitwill

arrivetheinstanceinventoryrunsout.

Thereorderpointisthelevelofinventoryatwhichtheorder

hastobeplaced.

Let betheleadtime,then,thereorderpointis

R

=

,

neworderhastobeplacewhentheinventorylevelreaches

260

=

120

DVD

drives



26/43

Economicproductionquantity

TheassumptionisthattheproductionratePisgreaterthan

thedemandrate.

UnlikethecaseofEOQ,theinventoryforproducedparts

buildsupgradually.

Inthisinventorymodel,thelotsizeQisproducedduringthe

productionuptimeT1 ,andtheinventorycontinuestobe

consumeddurin

the

roduction

shutdown

T

where

the

cyclelengthisT=T1 +T2.

Thetotalunitsthatneedtobeproducedandputinthe

nventory

nacyc e

s

= .

SincepartsareproducedonlyduringT1,


1


27/43

TheinventorybuildsupatarateP anddeclinesatarate.

,

H

=

(P

)T1 =

Q(1

/P) ThetotalinventoryinacycleisHT/2.

IftheproductioncostisK+cQ,thenthetotalcostinacycleis

hHTcQK 1++ DividingbyT,theaveragecostis

KcK+ 22 cQT

++=+



28/43

Theminimizer oftheaveragecostis

)/1( Ph

KQ

=

ThisQiscalledtheeconomicproductionquantityEPQ.



29/43

Illustration

Khalijy FactoryforShoesandLeatherProductsproduces

leatherbelts.

Theyearlydemandrateis2,500belts.

Theproductionis10,000beltsperyear.

Thefixedmanufacturingcostis50riyalsandtheunit

manufacturingcostis2riyals.

.

Whatistheeconomicproductionquantity?



30/43

Theholdingcostis

= = . .

The

number

of

belts

that

have

to

be

manufactured

in

every

cycleis

belts3.745

)000,10/500,21(6.0

500,2502=

=Q

Theproductionshouldbestoppedwhentheinventorylevel

H=745.3(11,200/10,000)=655.9belts



31/43

Allunitsdiscountmodel

Thepurchaseofrawmaterialscanbeofferedinadiscount

plan.

Allunitsdiscountplanoffersdiscountsonallunitsofraw

materials.

Forexample,manufacturingcanbuyrawmaterialsfor30

riyalsperunitfororderslessthan500units,andtheprice

dro sto

29

ri als

for

orders

between

500

and

less

than

1 000

units,and28riyalsfororders1,000unitsormore:


32/43

SincetheEOQdependsontheunitcost,multiplequantities

havetobecalculatedandthequantitythatcorrespondsto

thesmallestaveragecostisselected.

TheoptimalquantitywillbeeitherthelargestEOQcalculatedus ngun cos soroneo e rea po n quan es.



33/43

Illustration

Thesetupcostfororderingrawmaterialsis800riyalsper

order.

Thediscountplanis


34/43

First,calculatetheEOQforeachunitprice:

4003020.0

1 =

=Q

4062920.0

2 =

=Q

4142820.0

3 =

=Q

o ce 1 sw n epr cerangeo r ya s,w e 2 an

Q3 are not.

Therefore and donot ualif aso timalsolutions.



35/43

Thecandidatesolutionsare400,500,and1,000.

totalcost:

hQK=

Theaverage

costs

are

2Q

Q c h G(Q)

400 30 6 20,400

500 29 5.8 19,810

Theo timalordersizeis500units.

1000 28 5.6 20,080



36/43

Inventoryformultipleitemsandlimitedresources

Acommonwarehousehasalimitonhowmanyitemsitcan

hold.

TheEOQandEPQsolutionsdonotconsiderphysicallimitation

ofthe

warehouse.

IfEOQorEPQsolutionformultipleitemsexceedsthe

inventorycapacity,thesolutionhastobemodified.

nonlinearmodel:

)(min

n

ii QG

s.t.1

1

=

=

n

i

ii

i

WQw


0 iQ


37/43

Illustration

SaudiFiltersIndustrieshasthreetypesofrawmaterials.

Raw

material K h w.

2 150 1150 87.5 12

3 50 800 21.25 18

Therawmaterialsareorderedfromasinglesuppler;hence,

whenitemsarrive,theyarriveatthesametime.

2

,

. WhatistheEOQforeachitem?



38/43

Thenonlinearmodelforthisproblemis

200018129s.t.

25.215.875.12min 321

3

221

2

121

1

++

+

++

++

Q

Q

Q

Q

Q

Q

UsingExcel

solver,

the

optimal

solution

is

EOQ1 =92

EOQ2 =51

3 =



39/43

Inventorywithchangingdemand

Thecasewherethedemandischangingisreferredtoas

dynamiclotsizing.

Define:

Di :demandinperiodi

Ii :inventorycarriedduringperiodi

Qi :order

size

in

period

i

Ki :or ersetupcost

Indynamiclotsizing,thedecisionismadethebeginningof

Ifthefixedordercostisgreaterthantheinventoryholding

cost,thenitismoreeconomicaltoorderforthecurrent


periodand

periods

to

come,

because

the

fixed

order

cost

is

incurredonlyonce.


40/43

Inperiodi,theamountofinventorycarriedis

=

The

dynamic

lot

sizing

is

modeled

by

the

mixed

integer

model:n

,...,1fors.t. 1

1

==

=

iiii

i

iiii

niIDIQ

0 0

=

=ii

I

y

,...3,2,1,0

1,0

=

=

i

i

n

Q

y



41/43

Illustration

TheSaudiModernCo.forTelephoneCablesrequiresaspecific

rawmaterial,andtherequirementsforthecoming10months

are:

Month 1 2 3 4 5 6 7 8 9 10

Theordersetupcostis1,000riyalsandtheinventoryholding

.

Whatistheordersizeforeverymonth?



42/43

Thequantitiesthatneedorderingarefoundbysolvingthe

MIPmodel

s.t.

.......m n

111

2121

==

+++++

DIQ

yy

2333

1222

= IDIQ

M

500

500

22

11

yQ

yQ

0 10

=

=I


,...3,2,1,0

,

=i

i

Q


43/43

Mischosentobethelargestdemandmultipliedby10.

Q1 =

80Q4 =130

Q8 =90

I1 =60

I2 =10

I4 =80

5 =

I6 =20

8

I9 =30

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