inventory control models part ii applied management science for decision making, 1e © 2012 pearson...

Inventory Control ModelsInventory Control Models

Part IIPart II

Applied Management Science for Decision Making, 1e Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro , PhD© 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro , PhD

The Production Order Quantity ModelThe Production Order Quantity Model

USED WHENEVER THE VENDORCANNOT DELIVER THE ORDER

( Q* ) ALL IN ONE DAY

USED WHENEVER THE FACTORYCANNOT PRODUCE THE ORDER

( Q* ) ALL IN ONE DAY

OR

Service SectorService Sector

Variable InterpretationsVariable Interpretations

P or p

The delivery rateThe delivery rateofof

purchased itemspurchased items

P or p

The production rateThe production rateofof

manufactured itemsmanufactured items

ManufacturingManufacturing

Service SectorService Sector

Variable InterpretationsVariable Interpretations

The optimal The optimal EOQEOQ whenwhen

purchased items purchased items areare

received in partialreceived in partialshipmentsshipments

The optimal The optimal EOQEOQwhen when

manufactured itemsmanufactured itemscannot cannot

all be producedall be producedin a single dayin a single day

ManufacturingManufacturing

Qp* Qp*

Production Order Quantity ModelProduction Order Quantity Model Cycle ChartCycle Chart

Maximum Inventory Level Maximum Inventory Level ( ( IMAXIMAX ) )Replenishment Rate Replenishment Rate ( ( PP ) )Consumption Rate Consumption Rate ( ( DD ) )

TimeTime Cycle Chartsenhance

understandingof basic

inventoryconcepts

Graphically depicts the relationship betweenGraphically depicts the relationship between

SAW TOOTH VERSIONSAW TOOTH VERSION

Production Order Quantity ModelProduction Order Quantity ModelCycle ChartCycle Chart

MAXIMUM INVENTORY LEVELMAXIMUM INVENTORY LEVEL

AVERAGE INVENTORY LEVELAVERAGE INVENTORY LEVEL

PP PPDD DD

CONSUMPTIONCONSUMPTIONRATE ONLYRATE ONLY

REPLENISHMENTREPLENISHMENTRATERATE

UN

ITS

ELAPSED TIME

CONSUMPTION OCCURSEVEN AS

REPLENISHMENTIS TAKING PLACE

0

Production Order Quantity ModelProduction Order Quantity ModelCycle ChartCycle Chart

MAXIMUM INVENTORY LEVELMAXIMUM INVENTORY LEVEL

AVERAGE INVENTORY LEVELAVERAGE INVENTORY LEVEL

PP PPDD DD

CONSUMPTIONCONSUMPTIONRATE ONLYRATE ONLY

REPLENISHMENTREPLENISHMENTRATERATE

0

ELAPSED TIME

An EOQ of 100 unitsis delivered at the rate of20 units per week over

five weeks

INVENTORYLEVEL PEAKSAT THIS POINT

INVENTORYFALLS TO ZERO OR

REORDER POINT

UN

ITS

SAW TOOTH VERSIONSAW TOOTH VERSION

CYCLE CHART DISCUSSIONCYCLE CHART DISCUSSION

The replenishment rate ( PP or pp ) is diminished by the consumption rate ( DD or dd ).

Average inventory will always be less than [ Q* / 2 ] .

Average inventory is essentially [ IMAX / 2IMAX / 2 ] .

Production Order Quantity Production Order Quantity FormulaFormula

2DS

H [ 1 – d / p ]

THE CONSUMPTION OR THE CONSUMPTION OR USAGE RATE ( D )USAGE RATE ( D )

THE REPLENISHMENT OR THE REPLENISHMENT OR PRODUCTION RATE ( P )PRODUCTION RATE ( P )

THE THE FINITE CORRECTION FACTORFINITE CORRECTION FACTOR PRODUCES A LARGER PRODUCES A LARGERVALUE OF Q* IN ORDER TO COMPENSATE FOR PIECEMEALVALUE OF Q* IN ORDER TO COMPENSATE FOR PIECEMEAL

REPLENISHMENT AND CONSUMER DEMANDREPLENISHMENT AND CONSUMER DEMAND

Qp* = √

Production Order Quantity ModelProduction Order Quantity ModelEXAMPLEEXAMPLE

DA = 1000 units ( annual demand ) S = $10.00 ( cost per order ) H = $.50 ( annual unit carry cost ) P = 8 units ( daily supply rate ) D = 6 units ( daily usage rate )

How manyunits shouldbe ordered at a time?

How long to receivethe entire

order?

Production Order Quantity ModelProduction Order Quantity ModelSOLUTIONSOLUTION

2(1000)(10.00)

.50 [ 1 – 6 / 8 ]

20,000 .50 [ .25 ]

Qp* = √=√ = 160,000 = 400 units√

The Basic EOQ ModelThe Basic EOQ ModelSOLUTIONSOLUTION

2(1000)(10.00)

.50

20,000 .50

Q* = √= √ = 40,000 = 200 units√

Q* IS ONLYHALF OF WHAT

IT WAS UNDER THEPRODUCTION ORDER

QUANTITY MODEL


Qp* 400 P 8

ORDER RECEIPT TIME PERIOD ( t )

t = = = 50 days


TOTAL VARIABLE COSTS ( TVC )

TVC =Q*p

2X 1 - D

PX H +

D

Q*pX S

= 400

2X 1 - 6

8 X .50 + 1,000

400X 10.00

= [ 200 x (.25)(.50) ] + [ 2.5 x 10.00 ] = [ 25.00 ] + [ 25.00 ] = $50.00

Post Solution Comments

The Qp* must be larger than Q* since it is being drawn down even as it arrives on a piecemeal basis. The larger Qp* produces no increase in carry costs.

Annual carry costs are actually less than they are

under the basic EOQ model ( Q* ) .

““P” and “D” can be expressed asP” and “D” can be expressed asdaily, weekly, monthly, and annualdaily, weekly, monthly, and annualfigures without changing the valuefigures without changing the valueof the finite correction factor [1-d/p]of the finite correction factor [1-d/p]

ProductionOrder

QuantityModel

WE SELECT THE“INVENTORY”

MODULE

WE SELECT THE“PRODUCTION ORDER

QUANTITY MODEL”

THE DIALOGUE BOXALLOWS US TOINSERT A TITLE

FOR THIS PROBLEM

THE DATA INPUTTABLE

APPEARS

ANNUAL DEMAND = 1,000 UNITS

ORDER COST = $10.00

UNIT CARRY COST = $.50

DAILY REPLENISHMENT RATE = 8 UNITS

DAILY CONSUMPTION RATE = 6 UNITS

Optimal Order Quantity

Total Variable Costs

Total Variable Cost

Annual Carry Cost

Annual Order Cost

ProductionOrder

QuantityModel

Templateand

Sample Data

Sensitivity Analysisalso shows the

optimal solution

The The Backorder Inventory Backorder Inventory ModelModel

USED WHENEVER THEFIRM IS APPEALING TO ITS CUSTOMERS TO BUY AND THEN

WAIT FOR THEIR PURCHASES UNTILA NEW SHIPMENT

( Q* ) IS ORDERED AND RECEIVED

Backorder ModelBackorder Model Variables Variables

Backorder cost – BB

Optimal number of backorders – S*S*

Optimal order quantity under a backordering

scenario – QQbb**

Number of units going into stock after all

backorders have been filled – bb or [ QQbb* - S** - S* ]

Backorder Model Cycle ChartBackorder Model Cycle Chart

Optimal Order Quantity,Optimal Order Quantity,

Optimal Number of Backorders,Optimal Number of Backorders,

Remaining Units Going IntoRemaining Units Going Into

Stock after Backorders Stock after Backorders

Have Been FilledHave Been Filled

Graphically depicts the relationship betweenGraphically depicts the relationship between

Cycle chartsenhance

understandingof basic inventory

concepts

Backorder Model Cycle ChartBackorder Model Cycle ChartPICKET FENCE VERSIONPICKET FENCE VERSION

Q* or EOQ Q* or EOQ b = Q*- S* b = Q*- S*

S*

0

INITIALCYCLE

CYCLE2

CYCLE3

CYCLE4

PO

SIT

IVE

INV

EN

TO

RY

NEGATIVENEGATIVEINVENTORYINVENTORY

Backorder Model Cycle ChartBackorder Model Cycle ChartPICKET FENCE VERSIONPICKET FENCE VERSION

Q* or EOQ = 92 unitsQ* or EOQ = 92 units b = Q*- S* = 88 unitsb = Q*- S* = 88 units

S* = 4 UNITS

0

INITIALCYCLE

CYCLE2

CYCLE3

CYCLE4

PO

SIT

IVE

INV

EN

TO

RY


ASSUME ANEOQ OF92 UNITS

88 UNITS GO INTO INVENTORYAFTER THE FOUR

STOCKOUTS HAVE BEEN

FILLED

THE FIRMALLOWS

BACKORDERSTO REACH

4 UNITS

THIS MODELASSUMESTHAT THE

NEW EOQ ISORDERED WHEN

BACKORDERSEQUAL FOUR

UNITS

Backorder Model FormulaBackorder Model Formula

B

HBDSQb 2*

OPTIMAL ORDER QUANTITYIN BACKORDER-TOLERATED

SITUATIONS

UNIT BACKORDERCOST

HH

ANNUAL CARRY COSTPER UNIT

Backorder Model FormulaBackorder Model Formula

BH

HQbS **

OPTIMAL ORDERQUANTITY UNDERBACKORDERING

UNIT BACKORDERCOST

THE OPTIMALNUMBER

OFBACKORDERS

The Backorder ModelThe Backorder ModelEXAMPLEEXAMPLE

DA = 500 units ( annual demand ) S = $4.00 ( order cost ) H = $ .50 ( annual unit carry cost ) B = $10.00 ( unit backorder cost )

How many units shouldbe ordered?

What are the number of

backorders?

The Backorder ModelThe Backorder ModelSOLUTIONSOLUTION

2(500)(4.00) (10.00 + .50) .50 10.00

= 8000 x 1.05

= 8400

≈ 92 units

Qb* =√√

√

X

The Backorder ExampleThe Backorder ExampleSOLUTIONSOLUTION

.50

.50 + 10.00S* = 91.65 x

= ( 91.65 x .0476 )

≈ 4 units

Relationship Between ROP and S*Relationship Between ROP and S*

IF LEADTIME IS ZERO, THE REORDER POINT OCCURS WHEN THE OPTIMAL NUMBER OF BACKORDERS IS REACHED

If leadtime ( L ) = 0 , then ROP = S*

Relationship Between ROP and S*Relationship Between ROP and S*

IF LEADTIME IS NOT ZERO, THE REORDER POINT OCCURS BEFORE THE OPTIMAL NUMBER OF

BACKORDERS IS REACHED

If leadtime ( L ) > 0, then ROP > S*

Backorder Reorder PointBackorder Reorder PointEXAMPLEEXAMPLE

Given: L = 0 days Qb* = 92 units S* = 4 units

Order 92 units when the number of backorders

accumulate to 4

Backorder Reorder PointBackorder Reorder PointEXAMPLEEXAMPLE

Given: d = 2 units, L = 6 days, Qb* = 92 units, S* = 4 units

Order 92 units when there are 8 units still left in the

account balance.

THE INITIAL ROP = d x L = [ 2 units x 6 days ] = 12 units12 units

THE FINAL ROP = Initial ROP – S* = [ 12 units – 4 units ] = 8 units8 units

ROP under BackorderingROP under BackorderingEXAMPLE WHERE L IS POSITIVEEXAMPLE WHERE L IS POSITIVE

S* = 4 units ( - 4 )


REGIONREGION

POSITIVEINVENTORY

REGION

Qb* Qb*

00

FINAL ROP FINAL ROP

12 – 4 = +8 12 – 4 = +8IN

ITIAL R

OP = 12 U

NITS

INITIAL ROP = 12 UNITS

THE BACKORDER MODELMAY OR MAY NOT BE AVAILABLE ONYOUR PARTICULAR

SOFTWARE

Inventory ControlInventory Control

Part IIPart II


The The ABC ABC Classification SystemClassification System

Its purpose is to assist inventory specialistsin establishing policies that focus their limited resources on the relatively few

critical materials, components, and products...and not the many trivial ones.


ABC Classification SystemABC Classification System

Purchasing personnel are relatively few in number in the firm.

There are thousands of components, materials, and finished good inventory accounts in medium and large firms.

There needs to be a priority system for establishing and updating inventory control doctrines ( Q* / R ).

RATIONALE

Class “A” Class “A” Inventory ItemsInventory Items

LARGE APPLIANCESLARGE APPLIANCES AUTOMOBILESAUTOMOBILES FURNITUREFURNITURE DIAMONDSDIAMONDS

Comprise only 15% of thetotal items in stock yetrepresent 70% - 80% ofthe total dollar volume.

Class “B” Class “B” Inventory ItemsInventory Items

MID-SIZED APPLIANCESMID-SIZED APPLIANCES LAWN MOWERSLAWN MOWERS MOST SUITS & COATSMOST SUITS & COATS

Comprise 30% of thetotal items in stock and represent 15% - 25% ofthe total dollar volume

Class “C”Class “C” Inventory Items Inventory Items

TOASTERS / BLENDERSTOASTERS / BLENDERS NUTS, BOLTS, SCREWSNUTS, BOLTS, SCREWS STATIONERY SUPPLIESSTATIONERY SUPPLIES MOST ACCESSORIESMOST ACCESSORIES

Comprise 55% of the total items in stock yet

represent only 5% ofthe total dollar volume

Policies Based On ABCPolicies Based On ABC

“A” items would be inventoried in a more secure areaitems would be inventoried in a more secure area



“ “A” items would warrant more care in forecastingA” items would warrant more care in forecasting




“ “A” item records would be verified more frequentlyA” item records would be verified more frequently




“ “A” item records would be verified more frequentlyA” item records would be verified more frequently

“ “A” items would justify closer attention to customerA” items would justify closer attention to customer service levelsservice levels


“A” items would be inventoried in a more secure area

“A” items would warrant more care in forecasting

“A” item records would be verified more frequently

“A” items would justify closer attention to customer service levels

“A” items would qualify for real-time inventory tracking systems and more sophisticated ordering rules

Additional Criteria for ABCAdditional Criteria for ABC

Anticipated engineering changesAnticipated engineering changes

Delivery problemsDelivery problems

Quality problemsQuality problems

High unit production costsHigh unit production costs

WE FOCUS ON THE RELATIVELY FEW ITEMS WITH MAJOR PROBLEMSWE FOCUS ON THE RELATIVELY FEW ITEMS WITH MAJOR PROBLEMS

ABCAnalysis

WE FIRST SCROLLWE FIRST SCROLLTOTO

““INVENTORY”INVENTORY”

WE THEN SELECT THEWE THEN SELECT THESUB MENUSUB MENU

““ABC Analysis”ABC Analysis”

THE DIALOG BOXTHE DIALOG BOXAPPEARSAPPEARS

WE HAVE FIVE ( 5 )WE HAVE FIVE ( 5 )ITEMS THAT NEEDITEMS THAT NEEDTO BE CLASSIFIEDTO BE CLASSIFIED

THE DATA INPUT TABLETHE DATA INPUT TABLE

FIVE ITEMS, TOGETHER WITHFIVE ITEMS, TOGETHER WITHTHEIR ANNUAL DEMANDS, THEIR ANNUAL DEMANDS,

AND UNIT VALUESAND UNIT VALUES

WE DESIRE WE DESIRE 20% OF ALL20% OF ALLITEMS BE ITEMS BE

““A” CATEGORYA” CATEGORY

WE DESIRE WE DESIRE 30% OF ALL30% OF ALLITEMS BE ITEMS BE

““B” CATEGORYB” CATEGORY

One “A” ItemOne “A” ItemOne “B” ItemOne “B” Item

Three “C” ItemsThree “C” Items

ABCAnalysis

Templateand

Sample Data

ABC Classification ABC Classification SystemSystem


Reorder Point ModelsReorder Point Models

Known Stockout Cost Model

Service Level Model

Variable Demand / Constant Lead Time Model

Constant Demand / Variable Lead Time Model

Variable Demand / Variable Lead Time Model


Reorder Point Models

The original reorder point formula

ROP = d x L

computes the mean demandduring lead time, that is, theaverage demand during thewaiting period for the item.

Where d = daily demandWhere d = daily demand L = lead timeL = lead time ( in days )( in days )


However, actual demand duringlead time can be much higher

than the mean (average) demand.

For this reason, the reorder pointshould contain a built-in safety stock ( SS or B ) that will meetunexpectedly higher demands

and consequently reduce stockout costs.


The reorder point formulanow becomes

ROP = d x L + SS

But larger safety stocksinvolve higher carry costs.

We must find a safetyWe must find a safetystock that minimizesstock that minimizesboth carry costs andboth carry costs andexpected stockoutexpected stockout

costs annuallycosts annually

Reorder Point Reorder Point Cost TradeoffCost Tradeoff

ANNUALANNUALSAFETYSAFETY

( BUFFER ) ( BUFFER ) STOCKSTOCKCARRY CARRY COSTSCOSTS

ANNUALANNUALEXPECTEDEXPECTEDSTOCKOUTSTOCKOUT

COSTSCOSTS

REORDERREORDERPOINTPOINTTOTALTOTALCOSTSCOSTS

00 Safety ( Buffer ) Stock in Units

Co

stKNOWN STOCKOUT COST MODEL

∞

THESELECTED

SAFETY( BUFFER )

STOCKLEVEL

( SS or B )

Known Stockout Cost ModelKnown Stockout Cost Model

Stockout cost per unit is known.

Lead time is known and constant.

Lead time demand is variable.

IT IS ALSO ASSUMED THAT THE ANNUAL NUMBERIT IS ALSO ASSUMED THAT THE ANNUAL NUMBER OF ORDERING PERIODS IS KNOWN ( n )OF ORDERING PERIODS IS KNOWN ( n )

ASSUMPTIONS

Demand During Lead TimeDemand During Lead Time ( D ( DL L ))

15% 25% 30% 20% 10%

30 40 50 60 70

DEMAND IN UNITS

Probability

LEAD TIME DEMAND IS APPROXIMATELY NORMALLY DISTRIBUTED AND RANGES BETWEEN THIRTY AND SEVENTY UNITS

EXAMPLEEXAMPLE


15% 25% 30%30% 20% 10%

30 40 50 60 70

DEMAND IN UNITS

Probability

SINCE THERE IS A 30% CHANCE THAT LEAD TIME DEMAND WILL BE FIFTY ( 50 ) UNITS WE WOULD AT LEAST SET THE REORDER POINT AT 50 UNITS. OTHERWISE, WE ARE EXTREMELY VULNERABLE TO

LARGE AND RECURRING STOCKOUTS.

THE HIGHESTPROBABILITY


15% 25% 30% 20% 10%

30 40 50 60 70

DEMAND IN UNITS

Probability

THE SAFETY OR BUFFER STOCK LEVEL AT THE MINIMUM REORDER POINT IS ZERO BY DEFINITION

SAFETYSTOCK

IS ZEROHERE

Reorder Point Carry CostReorder Point Carry Cost

X

=

ANNUAL UNITCARRY COST

SAFETY OR BUFFERSTOCK in units

ANNUAL SAFETY STOCKCARRY COSTS

Reorder Point of 50 UnitsReorder Point of 50 UnitsANNUAL SAFETY STOCK CARRY COSTS

‘0’ Safety Stock x $5.00 per unit = $0.00 per year carry cost

( BUFFER STOCK )

Reorder Point of 50 UnitsReorder Point of 50 Units

15% 25% 30%30% 20% 10%

30 40 5050 60 70

ReorderReorderPointPoint

Probability

ActualDemand

10 Stockouts

20 Stockouts

A DEMAND OF 60 UNITS WOULDCREATE A STOCKOUT OF 10 UNITS

WITH A 20% PROBABILITY



EXPECTED STOCKOUTS PER ORDER PERIOD

EXPECTED STOCKOUTS

2 + 2

Reorder Point Stockout CostReorder Point Stockout Cost

X

X

=

Lead Time ExpectedStockouts in units

Annual Number of Lead Time Periods

( ordering periods )

Stockout Cost per unitAnnual ExpectedStockout Costs

Reorder Point of 50 UnitsReorder Point of 50 UnitsANNUAL EXPECTED STOCKOUT COSTS

NUMBEROF

STOCKOUTS

EXPECTEDSTOCKOUTS

STOCKOUT

COST ( per unit )

NUMBER OFORDER PERIODS

( per year )

10 2 $40.00 6

COST

$480.00

20 2 $40.00 6 $480.00

Σ = $960.00

Lead TimeDemand

of 60

Lead TimeDemand

of 70

Reorder Point Total CostReorder Point Total Cost

ANNUAL SAFETY STOCKCARRY COSTS

+

ANNUAL EXPECTED STOCKOUT COSTS

THE REORDER POINT THAT HAS THE LOWEST TOTAL COST IS SELECTED

Reorder Point Score BoardReorder Point Score Board

REORDERREORDER

POINTPOINT

ANNUAL CARRY ANNUAL CARRY COSTSCOSTS

ANNUAL STOCKOUT ANNUAL STOCKOUT COSTSCOSTS

TOTAL COSTSTOTAL COSTS

5050 $0.00$0.00 $960.00$960.00 $960.00$960.00

Reorder Point of 60 UnitsReorder Point of 60 UnitsANNUAL SAFETY STOCK CARRY COSTS


( BUFFER STOCK )

RAISING THE REORDER POINTTO 60 UNITS AUTOMATICALLYCREATES A SAFETY STOCK

OF TEN UNITS


15% 25% 30% 20%20% 10%

30 40 50 6060 70


Probability

ActualDemand

10 Stockouts



EXPECTED STOCKOUT

1



NUMBEROF

STOCKOUTS

EXPECTEDSTOCKOUTS

STOCKOUT

COST ( per unit )


( per year )

1010 1 $40.00 6

COST

$240.00

Σ = $240.00LEAD TIMEDEMAND

of 70


REORDERREORDER

POINTPOINT

ANNUALANNUAL

CARRY COSTSCARRY COSTS

ANNUALANNUAL

STOCKOUT COSTSSTOCKOUT COSTS TOTAL COSTSTOTAL COSTS

50 Units $0.00 $960.00 $960.00

60 Units $50.00 $240.00 $290.00

Reorder Point of 70 UnitsANNUAL CARRY COSTS


( BUFFER STOCK )

RAISING THE REORDER POINTTO 70 UNITS AUTOMATICALLY

CREATES A SAFETY STOCKOF 20 UNITS


15% 25% 30% 20% 10%

30 40 50 60 7070


Probability

0Stockouts

BASED ON HISTORICAL DEMAND DATA,NO DEMAND SHOULD OCCUR THAT

CREATES A STOCKOUT

EXPECTEDSTOCKOUTS

ZERO



NUMBEROF

STOCKOUTS

EXPECTEDSTOCKOUTS

STOCKOUT

COST ( per unit )


( per year )

0 0 $40.00 6

COST

$0.00

Σ = $0.00LEAD TIMEDEMAND

of 70


REORDERREORDER

POINTPOINT

ANNUALANNUAL

CARRY COSTSCARRY COSTS

ANNUALANNUAL

STOCKOUT COSTSSTOCKOUT COSTS TOTAL COSTSTOTAL COSTS

50 Units $0.00 $960.00 $960.00

60 Units $50.00 $240.00 $290.00

70 Units $100.00 $0.00 $100.00

The ConclusionThe Conclusion

The lowest cost option:

ROP = 70 units

SS = 20 units

KnownStockout

CostModel

WE SCROLL TOWE SCROLL TO““INVENTORY”INVENTORY”TO FIND THETO FIND THE

KNOWN STOCKOUT COSTKNOWN STOCKOUT COSTREORDER POINT MODELREORDER POINT MODEL

WE SELECT THEWE SELECT THE“Reorder Point / Safety Stock”“Reorder Point / Safety Stock”

( Discrete Distribution )( Discrete Distribution )SUB MENUSUB MENU

WE HAVE FIVE ( 5 )WE HAVE FIVE ( 5 )DISCRETE DEMANDSDISCRETE DEMANDSDURING LEAD TIMEDURING LEAD TIME

THE DATA INPUT TABLETHE DATA INPUT TABLE

THE COMPLETEDTHE COMPLETEDDATA INPUT TABLEDATA INPUT TABLE

THE KNOWN STOCKOUT COSTTHE KNOWN STOCKOUT COSTREORDER POINT MODELREORDER POINT MODEL

Reorder Point = 70 UNITSReorder Point = 70 UNITS

Safety Stock = 20 UNITSSafety Stock = 20 UNITS




KnownStockout

CostModel

Templateand

Sample DataInsert Selected

Reorder Point, etc.

This is the conditional payoff matrix for this problem. The strategiesare the safety stocks, the events are the five possible demands, and

the conditional payoffs are the cost consequences (expectedstockouts + carry costs) of selecting a particular safety stock

and a certain demand materializing.

Selected Reorder Point

5 discrete demand possibilities during any given lead time ( with probabilities )

Safety stocks associatedwith all 5 discrete demands

For ROP = 50 , total cost = $960.00 , H = $ 0.00For ROP = 60 , total cost = $290.00 , H = $ 50.00For ROP = 70 , total cost = $100.00 , H = $100.00

Conditional Payoffs

EMV Criterion Maxi-Min Criterion

Service Level ModelService Level Model

FOR EXAMPLE, THE FIRM MAY DESIRE A SERVICE LEVELTHAT MEETS 95% OF THE DEMAND, OR CONVERSELY,

RESULTS IN STOCKOUTS ONLY 5% OF THE TIME.

REORDER POINT DETERMINATION

When the stockout cost per unit ( Cs ) isdifficult or impossible to determine, thefirm may elect to establish a policy of

keeping enough safety stock on hand tosatisfy a prescribed level of customer

service.

Service Level ModelService Level ModelREORDER POINT DETERMINATION

SALES DATA ARE USUALLY ADEQUATE FOR COMPUTINGTHE MEAN AND STANDARD DEVIATION

Assuming that the demand during lead time( DL ) follows a normal distribution, only themean ( µ ) and standard deviation ( σ ) arerequired to define the reorder point as wellas the safety stock ( SS or B ) for any given

service level.

Service Level ModelService Level ModelREORDER POINT DETERMINATION EXAMPLE

A firm stocks an item that has a normally-distributed demand during the lead time period. The average or mean demand during the lead time period is 400 units and the standard deviation is 15 units. The firm wants a reordering policy that limits stockouts to only 5% of the time.

Requirement: 1. How much safety stock should the firm maintain? 2. What should be the reorder point?


.03.03 .04.04 .05 .06.06 .07.07

1.4 .92364.92364 .92507.92507 .92647.92647 .92785.92785 .92922.92922

1.5 .93699.93699 .93822.93822 .93943.93943 .94062.94062 .94179.94179

1.6 .94845.94845 .94950.94950 .95053 .95154.95154 .95254.95254

1.7 .95818.95818 .95907.95907 .95994.95994 .96080.96080 .96164.96164

AREAS UNDER THE STANDARD NORMAL CURVE

Z

95% SERVICE LEVEL IS REPRESENTED BY z = 1.65 standard normal deviates


ROPROP

95% STOCKAGE PROBABILITY5%

STOCKOUTPROBABILITY

SAFETY STOCK

Z = + 1.65

μ LEAD TIME MEAN DEMAND = 400 UNITS

0 units0 units

REORDER POINT AND SAFETY STOCK FOR 95% SERVICE LEVEL

Service Level ModelService Level ModelFORMULAE

Reorder Point ( R ) = μ + ( z )( σ )

Safety Stock ( SS ) = ( z )( σ )

μ = MEAN DEMAND DURING LEAD TIME

z = NUMBER OF STANDARD NORMAL DEVIATES

σ = STANDARD DEVIATION OF LEAD TIME DEMAND

Service Level ModelService Level ModelEXAMPLE – 95% SERVICE LEVEL

Reorder Point ( R ) = 400 + ( 1.65 )( 15 ) = 425 units

Safety Stock ( SS ) = ( 1.65 )( 15 ) = 25 units

μ = 400 units

z = 1.65 ( 95% service level )

σ = 15 units

Service Level ModelService Level ModelEXAMPLE – 99% SERVICE LEVEL

Reorder Point ( R ) = 400 + ( 2.33 )( 15 ) = 435 units

Safety Stock ( SS ) = ( 2.33 )( 15 ) = 35 units

μ = 400 units

z = 2.33 ( 99% service level )

σ = 15 units 2.3 .99010

.03Z

ServiceLevel

ReorderPointModel

TO FIND THE SERVICE LEVELTO FIND THE SERVICE LEVELREORDER POINT, SCROLL TOREORDER POINT, SCROLL TO

“ “ INVENTORY “INVENTORY “

SELECT THE SUB MENUSELECT THE SUB MENUENTITLEDENTITLED

“ “ Reorder Point / Safety Stock “Reorder Point / Safety Stock “( Normal Distribution )( Normal Distribution )

THE DATA INPUT TABLE REQUIRESTHE DATA INPUT TABLE REQUIRES

- DAILY DEMAND DURING LEAD TIMEDAILY DEMAND DURING LEAD TIME

- THE STANDARD DEVIATION OF DAILYTHE STANDARD DEVIATION OF DAILY DEMAND DURING LEAD TIMEDEMAND DURING LEAD TIME

- THE SERVICE LEVEL DESIRED ( i.e. 95% )THE SERVICE LEVEL DESIRED ( i.e. 95% )

- LEAD TIME IN DAYSLEAD TIME IN DAYS

- THE STANDARD DEVIATION OF LEAD TIMETHE STANDARD DEVIATION OF LEAD TIME

IF THE DAILY DEMAND IF THE DAILY DEMAND AND/ORAND/OR

LEADTIME ARE CONSTANT,LEADTIME ARE CONSTANT,THEN THEIR STANDARDTHEN THEIR STANDARD

DEVIATION(S) = 0DEVIATION(S) = 0

The mean demand during lead time was given as 400 units.

Since this model requires both daily demand during thelead time, and the lead time ( in days ), we will assume

here that daily demand = 50 units, and lead time = 8 days.

THE 95% “SERVICE LEVEL”THE 95% “SERVICE LEVEL”REORDER POINT = 423 UNITSREORDER POINT = 423 UNITS

THE 95% “SERVICE LEVEL” THE 95% “SERVICE LEVEL” SAFETY STOCK = 23 UNITSSAFETY STOCK = 23 UNITS

ServiceLevel

ReorderPointModel

Templates for three ( 3 ) differentreorder point models.

( we choose the first model )

The Reorder Point = 400 + 25 = 425 !

Stochastic Reorder Point ModelsStochastic Reorder Point Models

These models generally assume that any variability in either the demand rate or lead time can be adequately described by a normal distribution. This, however, is not a strict requirement.

These models will provide approximate reorder points even in cases where the actual probability distributions depart substantially from normal.

In all models shown, stockout costs are assumed to be unknown.

WHEN DEMAND AND / OR LEAD TIME ARE VARIABLE

Stochastic Reorder Point ModelsStochastic Reorder Point Models MODELS CONSIDERED IN THIS PRESENTATION

I. variable demand rate / constant lead time

II. constant demand rate / variable lead time

III. variable demand rate / variable lead time

Stochastic Reorder Point ModelsStochastic Reorder Point Models THE VARIABLES

d = constant demand rate

d = average demand rate

L = constant lead time

L = average lead time

σD = standard deviation of demand rate

σL = standard deviation of lead time

Stochastic ReorderStochastic Reorder Point Models Point Models

Jack’s Pizza Parlor uses 1,000 cans of tomatoes per month at an average rate of 40 per day for each of 25 days per month. Usage can be approximated by a normal distribution with a standard deviation of 3 cans per day. Lead time is constant at 4 days. Jack desires a service level of 99% , that is, a stockout risk of only 1%

Requirement:

1. Determine the reorder point ( ROP )

2. Determine the safety or buffer stock ( SS or B )

LEAD TIME IS CONSTANT

DEMAND RATE IS VARIABLE

Reorder Point SolutionReorder Point Solution

Given:Given:

d = 40 cans dailyd = 40 cans daily

σσDD = 3 cans daily = 3 cans daily

L = 4 daysL = 4 days

__

ROP = ( d x L ) + ( z ) ( L ) ( σD )_

= ( 40 x 4 ) + 2.33 ( 4 ) ( 3 )

= 160 + 2.33 ( 6 )

= 160 + 13.98

= 174 cans

PIZZA PARLOR

Safety Stock SolutionSafety Stock Solution

SS or B = ( z ) ( L ) ( σD )

= 2.33 ( 4 ) ( 3 )

= 2.33 ( 6 )

≈ 14 cans

PIZZA PARLOR

ReorderPoint

WhereLead Time

IsConstant

Demand RateIs

Variable

Templatefor

pizza parlorreorder point

Reorder point = 174Safety stock = 14

Stochastic ReorderStochastic Reorder Point Models Point Models

An oil-driven generator uses 2.1 gallons per day. Lead time is normally distributed with a mean of 6 days. The standard deviation of lead time is 2 days. The service level is 98%, that is, the stockout risk is 2%

Requirement:



LEAD TIME IS VARIABLE

DEMAND RATE IS CONSTANT


Given:Given:

d = 2.1 gallons dailyd = 2.1 gallons daily

σσLL = 2 days = 2 days

L = 6 daysL = 6 days

ROP = ( d x L ) + ( z ) ( d ) ( σL )_

= ( 2.1 x 6 ) + 2.055 ( 2.1 ) ( 2 )

= 12.6 + 2.055 ( 4.2 )

= 12.6 + 8.631

= 21.23 gallons

__

THE GENERATOR

Safety Stock SolutionSafety Stock Solution

SS or B = ( z ) ( d ) ( σL )

= 2.055 ( 2.1 ) ( 2 )

= 2.055 ( 4.2 )

= 8.63 gallons

THE GENERATOR

ReorderPoint

WhereLead Time

IsVariable

Demand RateIs

Constant

Stochastic Reorder Point ModelsStochastic Reorder Point Models

Beer consumption at a local tavern is known to be normally distributed with a mean of 150 bottles daily and a standard deviation of 10 bottles daily. Delivery time is also normally distributed with a mean of 6 days and a standard deviation of 1 day. The service level is 90%

Requirement:



LEAD TIME IS VARIABLE

DEMAND RATE IS VARIABLE


Given: d = 150 bottles daily

σD = 10 bottles daily

L = 6 days

σL = 1 day

_

TAVERN EXAMPLE

_


ROP = ( d x L ) + ( z ) ( L )( σD ) + ( d ) ( σL )_ _ _ _2 22

= ( 150 x 6 ) + ( 1.28 ) (6)(10) + (150) (1)2 2 2

= 900 + 1.28 600 + 22,500

= 900 + 1.28 ( 151.986 ) ≈ 1,095 bottles

TAVERN EXAMPLE

Safety Stock SoluSafety Stock Solutiontion

SS or B = ( z ) ( L )( σD ) + ( d ) ( σL )

_ _2 22

= (1.28 ) (6)(10) + (150) (1)2 2 2

= 1.28 600 + 22,500

= 1.28 ( 151.986 ) ≈ 195 bottles

TAVERN EXAMPLE

StochasticReorder

PointModel

Demandand

Lead TimeVariable

TO SOLVE A STOCHASTICTO SOLVE A STOCHASTICREORDER POINTREORDER POINT

AND AND SAFETY STOCKSAFETY STOCK

PROBLEMPROBLEM

THE DATA INPUT TABLETHE DATA INPUT TABLEREQUIRES DAILY DEMANDREQUIRES DAILY DEMAND

AND STANDARD DEVIATIONAND STANDARD DEVIATION( if any )( if any )

LEAD TIME ( DAYS ) LEAD TIME ( DAYS ) AND STANDARD DEVIATIONAND STANDARD DEVIATION

( if any )( if any )

STOCHASTIC MODELSTOCHASTIC MODEL

REORDER POINT = 1,095 UNITSREORDER POINT = 1,095 UNITS

SAFETY STOCK = 195 UNITSSAFETY STOCK = 195 UNITS

THE LOCAL TAVERNTHE LOCAL TAVERN

StochasticReorder Point

Model

DemandVariable

Lead TimeVariable

Reorder Point ModelsReorder Point Models

Inventory ControlInventory Control


Solved Problem

Quantity Discount Model

Ivonne CallenIvonne Callen

Computer-BasedComputer-Based

ManualManual

Ivonne Callen ProblemIvonne Callen Problem

Ivonne Callen sells beauty supplies. Her annual demand for a particular skin lotion is 1,000 units.The cost of placing an order is $20.00, while the holdingcost per unit per year is 10 percent of the cost.The item currently costs $10.00 if the order quantity isless than 300. For orders of 300 units or more, the costfalls to $9.80.To minimize total cost, how many units should Ivonneorder each time she places an order?What is the minimum total cost?

Ivonne Callen Problem

D = 1,000 units

S = $20.00

I = 10%

P = $10.00 if Q < 300 units

P = $ 9.80 if Q > 300 units


To minimize total costs, how many units should Ivonne order each time she places an order ?

What is the minimum total cost ?

EOQ at $10.00 Item Cost

Q*1 =

2 x D x S

I x P√2 (1,000 ) ( 20 )

( .10 ) ( 10.00 )√=

= √40,000

1.00= 200 units

Total Cost at $10.00 Item Cost

TC = [ Q* / 2 ] (I)(P) + [ D / Q* ] (S) + [ D x P ]

= [ 200 / 2 ] (.10)(10.00) + [ 1,000 / 200 ] (20.00) + [1,000 x 10.00]

= $100.00 + $100.00 + $10,000.00

= $10,200.00

Total Cost at $9.80 Item Cost

TC = [ Q* / 2 ] (I)(P) + [ D / Q* ] (S) + [ D x P ]

= [ 300 / 2 ] (.10)(9.80) + [ 1,000 / 300 ] (20.00) + [1,000 x 9.80]

= $147.00 + $66.67 + $9,800.00

= $10,013.67


Therefore, she should order 300 units.

CONCLUSIONCONCLUSION

Solved Problem

Serial Rate Production Model

The Handy Manufacturing CompanyThe Handy Manufacturing Company


ManualManual

The Handy Mfg. Company

The Handy Manufacturing Company manufactures smallair conditioner compressors. The estimated demand forthe year is 12,000 units. The setup cost for the productionprocess is $200.00 per run, and the carrying cost is $10.00per unit per year. The daily production rate is 100 units perday, and demand has been 50 units per day.

Determine the number of units to produce in each batch, the number of batches that should be run each year, andthe time interval, in days, between each batch.( Assume 240 operating days. )

The Data

D = 12,000 units

S = $200.00

H = $10.00 / unit / year

p = 100 units / day

d = 50 units / day

240 working days / yearUse

Production OrderQuantity Model

Production Run Model

Qp* = √ 2 (D)(S)

H ( 1 - d / p )

=√ 2 (12,000)(200.00)

(10.00)( 1 - 50 / 100 )

Production Run Model

Qp* = √ 4,800,000

(10.00)(.5)

=√ 4,800,000

5=√960,000 = 979.8

units

Batches Run Annually

n = D / Q*p

n = 12,000 / 980

n ≈ 12 production runs

Time Between Runs

t = 240 days / n

= 240 days / 12

= every 20 days

Solved Problem

Reorder Point Model

Mr. Beautiful Mr. Beautiful


ManualManual

Mr. Beautiful ProblemMr. Beautiful ProblemMr. Beautiful, an organization that sells weight training sets, has an ordering cost of $40.00 for the BB-1 set. The carry cost for theBB-1 set is $5.00 per set per year. To meet demand, Mr. Beautiful orders large quantities of BB-1 (7)seven times a year. The stockout cost for BB-1 is estimated to be$50.00 per set. Over the past several years, Mr. Beautiful has ob-served the following demand during lead time for BB-1:

Lead Time Demand Probability 40 .10 50 .20 60 .20 70 .20 80 .20 90 .10

Starting with a ROP of 60 units, what level of safety stockshould be maintained for BB-1?

Mr. Beautiful ProblemMr. Beautiful Problem

40 50 60 70 80 90

.10 .20 .20 .20 .20 .10

ROP

Expected Stockouts

[.20 x 10] + [.20 x 20] + [.10 x 30] = 9 sets

10 sets

20 sets

30 sets


Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7At ROP = 60 sets , SS = 0 sets

Stockout Cost:

[.20 x 10 sets + .20 x 20 sets + .10 x 30 sets] x 7 orders x $50.00 [ 2 + 4 + 3 ] x 7 x $50.00 = $3,150.00

Carrying Cost:

SS = 0 sets x $5.00 / set / year = $0.00

TOTAL COST = $3,150.00


40 50 60 70 80 90

.10 .20 .20 .20 .20 .10

ROP

Expected Stockouts

[.20 x 10] + [.10 x 20] = 4 sets

10 sets

20 sets


Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7At ROP = 70 sets , SS = 10 sets

Stockout Cost:

[.20 x 10 sets + .10 x 20 sets] x 7 orders x $50.00 [ 2 + 2 ] x 7 x $50.00 = $1,400.00

Carrying Cost:

SS = 10 sets x $5.00 / set / year = $50.00

TOTAL COST = $1,450.00


40 50 60 70 80 90

.10 .20 .20 .20 .20 .10

ROP

Expected Stockouts

[.10 x 10] = 1 set

10 sets

Mr. Beautiful ProblemMr. Beautiful Problem Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7

At ROP = 80 sets , SS = 20 sets

Stockout Cost:

[.10 x 10 sets] x 7 orders x $50.00 [ 1 ] x 7 x $50.00 = $350.00

Carrying Cost:

SS = 20 sets x $5.00 / set / year = $100.00

TOTAL COST = $450.00


40 50 60 70 80 90

.10 .20 .20 .20 .20 .10

ROP

Expected Stockouts

[ .00 x 0 ] = 0 sets

Mr. Beautiful ProblemMr. Beautiful Problem Carry Cost / Set / Year = $5.00 Stockout Cost / Set = $50.00 Order Periods / Year = 7

At ROP = 90 sets , SS = 30 sets

Stockout Cost:

[.0 x 0 sets] x 7 orders x $50.00 [ 0 ] x 7 x $50.00 = $0.00

Carrying Cost:

SS = 30 sets x $5.00 / set / year = $150.00

TOTAL COST = $150.00


Reorder

Point

Stockout

Cost

Carry

Cost

Total

Cost

60 sets $3,150.00 $0.00 $3,150.00

70 sets $1,400.00 $50.00 $1,450.00

80 sets $350.00 $100.00 $450.00

90 sets $0.00 $150.00 $150.00

Mr. Beautiful should select a ROP = 90 sets with a SS = 30 sets

We Scroll Down ToINVENTORY

WE SELECT THE SUB MENUWE SELECT THE SUB MENU““Reorder Point / Safety Stock”Reorder Point / Safety Stock”

( Discrete Distribution )( Discrete Distribution )

THERE ARE SIX ( 6 ) DEMANDSTHERE ARE SIX ( 6 ) DEMANDSTHAT CAN OCCUR THAT CAN OCCUR DURING LEADTIMEDURING LEADTIME

THE DATA INPUT TABLE REQUIRES:THE DATA INPUT TABLE REQUIRES:

- STARTING REORDER POINT WHERESTARTING REORDER POINT WHERE THE SAFETY STOCK IS ZERO ( 0 )THE SAFETY STOCK IS ZERO ( 0 )

- CARRY COST PER UNIT PER YEARCARRY COST PER UNIT PER YEAR

- STOCKOUT COST PER UNIT PER YRSTOCKOUT COST PER UNIT PER YR

- NUMBER OF ORDERS PER YEAR ( 7 )NUMBER OF ORDERS PER YEAR ( 7 )

THE DISCRETE THE DISCRETE PROBABILITYPROBABILITY

DISTRIBUTION OF DISTRIBUTION OF LEAD TIME DEMANDLEAD TIME DEMAND

THE ROP = 90 SETSTHE ROP = 90 SETSTHE SS = 30 SETSTHE SS = 30 SETS

Expected Stockout Cost = $0.00Expected Stockout Cost = $0.00

Carry ( Holding ) Costs = $150.00Carry ( Holding ) Costs = $150.00

inventory control models part ii applied management science for decision making, 1e © 2012 pearson...

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