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InventoryInventoryModelsModels

Chapter 8

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8.1 Overview of Inventory

Issues Proper control of inventory is crucial to

the success of an enterprise.

Typical inventory problems include: Basic inventory Plannedshortage

Quantity discount Periodicreview

Production lot size Single period

Inventory models are often used todevelop an optimal inventory policy,consisting of: An order quantity, denoted Q.

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Inventory analyses can be thought ofas cost-control techniques.

Categories of costs in inventorymodels:

Holding (carrying costs)

Order/ Setup costs Customer satisfaction costs

Procurement/Manufacturing costs

Type of Costs in Inventory

Models

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Holding Costs (Carryingcosts):

These costs depend on the

order size Cost of capital

Storage space rental cost

Costs of utilities

Labor Insurance

Security

Theft and breakage

Deterioration or Obsolescence

Ch= Annual holding cost per u

in inventoryH = Annual holding cost rateC = Unit cost of an item

Ch = H * C


Models

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Order/Setup Costs

These costs are independent ofthe order size.

Order costs are incurred whenpurchasing a good from a supplier.

They include costs such as

Telephone

Order checking Labor

Transportation

Setup costs are incurred whenproducing goods for sale to others.

They can include costs of Cleaning machines


Models

Co = Order cost

or setup cost

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Customer SatisfactionCosts

Measure the degree towhich a customer issatisfied.

Unsatisfied customersmay:

Switch to the competition(lost sales).

Wait until an order issupplied.

When customers are


Models

Cb = Fixedadministrativecosts of an out ofstock item

($/stockout unit).Cs = Annualized cost

of a customerawaiting an outof stock item

($/stockout unit

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Procurement/Manufacturing Cost

Represents the unitpurchase cost (includingtransportation) in case of apurchase.

Unit production cost incase of in-housemanufacturing.


Models

C = Unitpurchase ormanufacturingcost.

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Demand is a key component affectingan inventory policy.

Projected demand patterns determinehow an inventory problem ismodeled.

Typical demand patterns are: Constant over time (deterministic

inventory models)

Changing but known over time (dynamicmodels)

Variable randoml over time

Demand in Inventory Models

D = Demand rate (usually per year)

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Inventory can be classified invarious ways:

By Process By Importance By Shelf Life

Raw materials Perishable

Work in progress A, B, C Nonperishable

Finished goods

By Process By Importance By Shelf Life

Raw materials Perishable

Work in progress A, B, C Nonperishable

Finished goods

Used typically byaccountants atmanufacturing firms.Enables managementto track the production

process.

Items areclassified by theirrelativeimportancein terms of the

firms capitalneeds.

Management ofitems with shortshelf life and longshelf life is verydifferent

Inventory Classifications

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Two types of review systems areused:

Continuous review systems.The system is continuously monitored.

A new order is placed when the inventory

reaches a critical point. Periodic review systems.

The inventory position is investigated on aregular basis.

An order is placed only at these times.

Review Systems

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The item has a sufficiently long shelflife.

The item is monitored using a

continuous review system. All the cost parameters remain

constant forever (over an infinite time

horizon).

8.2 Economic OrderQuantity Model -

Assumptions Demand occurs at a known and

reasonably constant rate.

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The constant environment described bythe EOQ assumptions leads to thefollowing observation:

The optimal EOQ policy consists ofsame-size orders.Q QQ

The EOQ Model

Inventory profile

s observation results in the following inventory pro

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Q QQ

Total AnnualInventory Costs

=Total AnnualHolding Costs

Total Annualordering Costs

Total Annualprocurement Costs++

TC(Q) =(Q/2)Ch +(D/Q)Co +DC

ChCh

The optimal order SizeThe optimal order Size

2DCo2DCoQ* =Q* =

Cost Equation for the EOQ

Model

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Constructing the total annual variable cost curve

Total

Holdi

ngCo

sts

Totalo

rderin

g

costs

Add the two curves to one another

Total annual holding and

ordering costs

Q

TV(Q)

Q*

The optimal order size

o* * * * *

TV(Q) and Q*

Note:

attheo

ptimal

order

size

totalho

lding

costs

andorde

ringc

osts

areeq

ual

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The curve is reasonably flat around Q*.

Q*

Deviations from the optimal order size

cause only small increase in the total cost.

Sensitivity Analysis in EOQ

models

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The cycle time, T, represents the

time that elapses between the

placement of orders.

Note, if the cycle time is greaterthan the shelf life, items will gobad, and the model must be

T = Q/D

Cycle Time

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To find the number of orders peryears take the reciprocal of the

cycle timeN = D/Q

Example: The demand for a product is 1000units per year.

The order size is 250 units under an EOQpolicy. How many orders are placed per year? N =

1000/250 = 4 orders.

Number of Orders per Year

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In reality lead time always exists,and must be accounted for when

deciding when to place an order.The reorder point, R, is the

inventory position when an order is

placed. R is calculated by

L and D must be expressed in the same

R = LD

R = LD

Lead Time and the Reorder

Point

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In

vento

rypositio

n

LPlace the ordernow

Reorder

Point

R = Inventory at hand at the beginning of lead time

Lead Time and the ReorderPoint

Graphical demonstration: ShortLead Time

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Outstanding order

Place the order

R = inventory at handat the beginning of lead time +

one outstanding order= demand during lead time= LD

nventoryat

hand

L

Lead Time and the ReorderPoint

Graphical demonstration: LongLead Time

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Safety stocks act as buffers to handle: Higher than average lead time demand.

Longer than expected lead time.

With the inclusion of safety stock (SS), Ris calculated by

The size of the safety stock is based on

having a desired service level.

R = LD + SS

Safety stock

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LPlace the ordernow

Reorder

Point

R = LD

Safety stock

Plannedsituation

Actual

situation

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L

R = LD

Safety stock

Actualsituation

+ SS

Reorder

Point

Place the ordernow

SS=Safety stock

?

The safety stockprevents excessive

shortages.

LD

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Inventory Costs

Including safety stock

otal Annualventory Costs

=Total AnnualHolding Costs

Total Annualordering Costs

Total Annualprocurement Costs++

TC(Q) =(Q/2)Ch +(D/Q)Co +DC + ChSS

Safety stockholding cost

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ALLEN APPLIANCE COMPANY(AAC)

AAC wholesales small appliances.

AAC currently orders 600 units of the

Citron brand juicer each time

inventory drops to 205 units.

Management wishes to determine anoptimal ordering policy for the Citron

brand juicer

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Sales of Juicers over thelast weeks22Week 2 2 2 2 2Sales 111 222 222 222 222Week 2 2 2 2 22

Sales 222 222 222 222 222

Data Co = $12 ($8 for placing an order) + (20 min. to

check)($12 per hr)

Ch = $1.40 [HC = (14%)($10)]

C = $10.

H = 14% (10% ann. interest rate) + (4%miscellaneous)

D = demand information of the last 10 weekswas collected:


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Data

The constant demand rate seems to be

a good assumption. Annual demand = (120/week)(52weeks)

= 6240 juicers.


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Current ordering policy calls for Q =600 juicers.

TV( 600) = (600 / 2)($1.40) +(6240 / 600)($12) = $544.80

The EOQ policy calls for orders of size

AAC Solution:EOQ and Total Variable Cost

Savings of

16%

2(6240)(12)

1.40= 327.065 327=Q*

27) = (327 / 2)($1.40) + (6240 / 327) ( $12) =

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) = 457.89 + 6240($10) + (13)($1.40) = $62,

Under the current ordering policy AAC holds 13units safety stock (how come? Observe):

AAC is open 5 day a week. The average daily demand = 120/week)/5 = 24juicers.

Lead time is 8 days. Lead time demand is (8)(24) =192 juicers.

Reorder point without Safety stock = LD = 192. Current policy: R = 205.

Safety stock = 205 192 = 13.

For safety stock of 13 juicers the total cost isTV(327) + Procurement +

Safety stock

AAC Solution:Reorder Point and Total Cost

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Changing the order size

Suppose juicers must be ordered in increments of 100

(order 300 or 400) AAC will order Q = 300 juicers in each order.

There will be a total variable cost increase of $1.71.

This is less than 0.5% increase in variable costs.

Changes in input parameters Suppose there is a 20% increase in demand. D=7500

juicers.

The new optimal order quantity is Q* = 359.

The new variable total cost = TV(359) = $502

If AAC still orders Q = 327, its total variable costsTV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13

Only 0.4%increase

AAC Solution:Sensitivity of the EOQ Results

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For an order size of 327 juicers we

have:T = (327/ 6240) = 0.0524 year.

= 0.0524(52)(5) = 14 days.

This is useful information because:

Shelf life may be a problem.

Coordinatin orders with other items mi ht

AAC Solution:Cycle Time

working days per week

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AAC Excel Spreadsheet

=SQRT(2*$B$10*$B$14/$B$

13)

=1/E11Copy to cell

H12

=E10/B10Copy to cell

H11

=$B$10*$B$11+E14+

$B$13*B16

=(E10/2)*$B$13+

($B$10/E10)*$B$14Copy to cell H14

=$B$15*$B$10+$B$16-

INT(($B$15*$B$10+$B$16)/E10)*E10Copy to cell H13

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Service Levels andSafety Stocks

Service Levels andSafety Stocks

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8.3 Determining Safety

Stock Levels Businesses incorporate safety stock

requirements when determining

reorder points.

A possible approach to determining

safety stock levels is by specifyingdesired service level .

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The unit service

level

The percentage of

demands that are

filled withoutincurring any delay.

Applied when the

percentage of

unsatisfied demand

Two Types of Service

Level

The cycle service

levelThe probability of not

incurring a stockout

during an inventory

cycle.

Applied when the

likelihood of a

stockout, and not its

Service levels can be viewed in

two ways.

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In many cases short run demand isvariable even though long run

demand is assumed constant.

Therefore, stockout events duringlead time may occur unexpectedly ineach cycle.

Stockouts occur only if demanddurin lead time is reater than the

The Cycle Service Level

Approach

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To determine the reorder point we need

to know:

The lead time demand distribution.The required service level.

In many cases lead time demand is

approximately normally distributed. Forthe normal distribution case the reorder

point is calculated by


Approach

R = L + zL1 = service level

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=192

P(DL> R) = P(Z > (R L)/L) = .SinceP(Z > Z) = , we have Z = (R

L

)/L,

whi h iv


ApproachP(DL>R) =

Service level =P(DL

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Assume that lead time demand isnormally distributed.

Estimation of the normal distributionparameters:

Estimation of the mean weekly demand=ten weeks average demand = 120 juicersper week.

AAC -

Cycle Service Level Approach

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To find LandL the parameters (perweek) and (per week) must beadjusted since the lead time is longerthan one week.

Lead time is 8 days =(8/5) weeks = 1.6weeks.

Estimates for the lead time mean

demand and variance of demand

AAC -

Cycle Service Level Approach

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Let us use the current reorder point of

205 juicers.205 = 192 + z (11.55) z = 1.13

From the normal distribution table we

have that a reorder

point of 205 juicers results in an 87%

22222.

AAC -Service Level for a given

Reorder Point

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Management wants to improve the

cycle service level to 99%.The z value corresponding to 1%

right hand tail is 2.33.

R = 192 + 2.33(11.55) = 219juicers.

Management wants to improve the

cycle service level to 99%.The z value corresponding to 1%

right hand tail is 2.33.

R = 192 + 2.33(11.55) = 219juicers.

AAC Reorder Pointfor a given

Service Level

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AAC is willing to run out of stock an

average of at most one cycle peryear with an order quantity of 327

juicers.

What is the equivalent service levelfor this strategy?

AAC is willing to run out of stock an

average of at most one cycle peryear with an order quantity of 327juicers.

What is the equivalent service levelfor this strategy?

AAC Acceptable Number of Stockouts

per Year

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AAC Acceptable Number of Stockouts

per Year

There will be an average of

6240/327 = 19.08 lead times per year.The likelihood of stockouts = 1/19 = 0.052

This translates into a service level of 94.76

There will be an average of

6240/327 = 19.08 lead times per year.The likelihood of stockouts = 1/19 = 0.052

This translates into a service level of 94.76

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When lead time demand follows anormal distribution

service level can be calculated asfollows: Determine the value of z that satisfy the

equation

L(z) = Q*/L

Solve for R using the equation

The Unit Service Level

Approach

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=NORMDIST(B8,B5,B6,TRUE)

AAC Cycle Service Level (Excel

spreadsheet)

=NORMINV(B7,B5,B6)

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Quantity Discounts are Common Practicein Business

By offering discounts buyers are encouraged toincreasetheir order sizes, thus reducing the sellersholding costs.

Quantity discounts reflect the savings inherentin largeorders.

With quantity discounts sellers can reward their

8.4 EOQ Models with

Quantity Discounts

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Quantity Discount ScheduleThis is a list of per unit discounts and their

corresponding purchase volumes.

Normally, the price per unit declines as theorder quantity increases.

The order quantity at which the unit pricechanges is called a break point.

There are two main discount plans:All unit schedules- the price paid for all the units

purchased is based on the total purchase.

Incremental schedules - The price discount is

based only on the additional units ordered beyondeach break point.

8.4 EOQ Models with

Quantity Discounts

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AAC - All Units Quantity

Discounts

Quantity DiscountSchedule-2222 $ .1111

-222222 $ .111

-222222 $ .111-22222222 $ .2222222 $ .222

Quantity DiscountSchedule-2222 $ .2222-222222 $ .222

-222222 $ .222-22222222 $ .2222222 $ .222

AAC is offering all units quantitydiscounts to its customers.

Data

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uld AAC increase its regular order ofjuicers, to take advantage of the disco

uld AAC increase its regular order ofjuicers, to take advantage of the disco

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AAC All units discount

procedure Step 1: Find the optimal order Qi

* for

each discount level i. Use the

formula Step 2: For each discount level i modify Q

i* as follows

If Qi* is lower than the smallest quantity that

qualifies for the i th discount, increase Qi

* to that

level.

If Qi* is greater than the largest quantity that

qualifies for the ith discount, eliminate this levelfrom further consideration.

Step 3: Substitute the modified Q*i value in

Q DC Co h* ( )/2

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Step 1: Find the optimal order quantityQi

* for each

discount level i based on theEOQ formulaLowest cost order size per discount level

Discount Qualifying Pricelevel order per unit Q*

2 -2222 .2222 222

2-

222222 .222 222

2 -222222 .222 111

2 -11111111 .111 222

2 2222 .222 222

AAC All units discount

procedure

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Step 2 : Modify Q i*

Modified Q* and total Cost

Qualified Price Modified Total

Urder per Unit Q* Q* Cost

-2222 .1111 222 **** ****-111111 .111 222 222 , .222222

-222222 .222 222 222 , .1111111

-11111111 .222 222 2222 , .222222

2222 .222 222 2222 , .222222




-2222 .2222 222 **** ****-222222 .222 222 222 , .222222

-222222 .222 222 222 , .222222

-22222222 .222 222 2222 , .222222

2222 .222 222 2222 , .222222

1 299

Q1*Q1*

300

$10/unit

599331

Q2*Q2*

$9.75/unit

999999600

Q3*Q3*

336

$9.50

AAC All Units Discount

Procedure

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Step 2 : Modify Q i*




-2222 .2222 222 **** ****-222222 .222 222 222 , .222222

-222222 .222 222 222 , .222222

-22222222 .222 222 2222 , .222222

2222 .222 222 2222 , .222222




-2222 .2222 222 **** ****-222222 .222 222 222 , .222222

-222222 .222 222 222 , .222222

-22222222 .222 222 2222 , .222222

2222 .222 222 2222 , .222222

1 299

Q1*Q1*

300

$10/unit

331

Q2*Q2*

999999600

Q3*Q3*

336

$9.50

AAC All Units Discount

Procedure

Q3*Q3*Q3

*Q3*Q3

*Q3* Q3

*Q3*Q3

*Q3*Q3

*Q3*

Q3*Q3*

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Step 3: Substitute Q I* in the total

cost function

Step 4



Urder per Unit Q* Q* Cost-2222 .2222 222 **** ****

-222222 .222 222 222 , .222222

-222222 .222 222 222 , .222222

-22222222 .222 222 2222 , .2222222222 .222 222 2222 , .222222



Urder per Unit Q* Q* Cost-2222 .2222 222 **** ****

-222222 .222 222 222 , .222222

-222222 .222 222 222 , .222222

-22222222

.222 222 2222

, .222222

2222 .222 222 2222 , .222222

AAC should order 5000 juicersAAC should order 5000 juicers

AAC All Units DiscountProcedure

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Calculation of Optimal Inventory Policy Under All-Units Quantity Discounts

OPTIMAL

INPUTS Values OUTPUTS Values

Annual Demand, D = .222222 Order quantity, Q* = 2222

Per Unit Cost, C = .2222 Cycle Time (in years), T = .1111111111

Annual Holding Cost Rate, H = .222 # of Cycles Per Year, N = .2222

Annual Holding Cost Per Unit, Ch = .222 Reorder Point, R = .2222222

Order Cost, Co =.

2222 Total Annual Cost, TC(Q*) =.

1111111

Lead Time (in years), L = .111111

Safety Stock, SS = .2222

DISCOUNTS

Level Breakpoint Discount Price Q* TC(Q*) Modified Q*

2 2 .2222 222 .2222222 222

2 222.

222 222.

2222222 222

2 222 .222 222 .2222222 222

2 2222 .222 222 .2222222 2222

2 2222 .222 222 .2222222 2222

2

2

2

2

AAC All Units Discount Excel

Worksheet

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Demand rate is constant.

Production rate is larger than demand

rate.

The production lot is not received

instantaneously (at an

infinite rate), because production rate is

finite.

There is only one product to bescheduled.

8.4 Production Lot Size

Model - Assumptions

P d ti L t Si M d l

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The optimal production lot sizepolicy orders the

same amount each time. This observation results in theinventory profile

below:

Production Lot Size Model

Inventory profile


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ProductionLot Size = Q = PT1

The inventory increasesat a net rate of P - D

he production increases theventory at a rate of P.

The demand decreases theinventory at a rate of D.

Productiontime

T1

Demand accumulationduring production run

Demand accumulationduring production run = DT1

Maximum inventory = (P D)

= (P D)(Q/P) = Q(1 D/P)

Maximum inventory


Understanding the inventoryprofile

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The parameters of the total variablecosts function are similar to those used

in the EOQ model.

Instead of ordering cost, we have here a

fixed setup cost per production run (Co).

In addition, we need to incorporate theannual production rate (P) in the model.


Total Variable Cost

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TV(Q) =(Q/2)(1 - D/P)Ch +(D/Q)Co

P is the annual production rate

Ch(1-D/P)

The Optimal Order Size

Q* =2DCo

The average inventory


Total Variable Cost

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Cycle time T = Q / D.

Length of a production run T1 = Q /

P.

Time when machines are not busyproducing the product T2 = T - T1= Q(1/D - 1/P).


Useful relationships

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FARAH COSMETICS COMPANY

Farah needs to determine optimalproduction lot size for its most

popular shade of lipstick. Data

The factory operates 7 days a week, 24hours a day.

Production rate is 1000 tubes per hour. It takes 30 minutes to prepare the machinery

for production.

It costs $150 to setup the line.

Demand is 980 dozen tubes per week.

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Input for the total variable costfunction

D = 613,200 per year [(980

dozen/week (12)/ 7](365)

Ch = 0.4(0.5) = $0.20 per tube per

year.

Dozens


Solution

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Current Policy

Currently, Farah produces in lots of 84,000 tubes.

T = (84,000 tubes per run)/(613,200 tubes per year)=0.137 years

(about 50days).

T1 = (84,000 tubes per lot)/(8,760,000 tubes peryear)= 0.0096 years

(about 3.5days).

T2 = 0.137 - 0.0096 = 0.1274 years (about 46.5 days).


Solution

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The Optimal Policy

Using the input data we find

TV(Q* = 31,499) = (31,499/2) [1-(613,200/8,760,000)](0.2) +

The optimal order size

(0.2)(1-613,200/8760,000)Q* =

2(613,200)(150)= 31,499


Solution


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FARAH COSMETICS COMPANY Production Lot Size Template

(Excel)

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8.5 Planned Shortage

Model When an item is out of stock,

customers may:

Go somewhere else (lost sales). Place their order and wait

(backordering).

In this model we consider thebackordering case.

All th th r E m ti n r in

Planned Shortage Model

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The parameters of the total variable costsfunction are similar to those used in the

EOQ model.

In addition, we need to incorporate theshortage costs in the model. Backorder cost per unit per year (loss of

goodwill cost) - Cs.

Reflects future reduction in profitability.

Can be estimated from market surveys and focus

groups.

Planned Shortage Model the Total Variable Cost

Equation


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Planned Shortage Model the Total Variable Cost

Equation The Annual holding cost =

Ch[T1/T](Average inventory) =

Ch[T1/T] (Q-S)/2

The Annual shortage cost =Cb(number of backorders per year) +

Cs(T2/T)(Average number of backorders).

To calculate the annual holding cost andshortage cost we need to find The proportion of time inventory is carried, (T1/T)

The proportion of time demand is backordered,

(T2/T).

T1 T2

T

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S

Q - S

Q

T1 T2

S T

Average inventory = (Q - S) / 2

Average shortage = S / 2

Proportion of timeinventory exists

= T1/T

T1

T

Q-

S

Q

Proportion of timeshortage exists

= T2/T

Finding T1/ T and T2/ T

= (Q - S) / Q

= S / Q


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Annual holding cost:Ch[T1/T](Q-S)/2 = Ch[(Q-S) /Q](Q-S)/2

= Ch(Q-S)2/2Q

Annual shortage cost:Cb(Units in short per year) +

Cs[T2/T](Average number ofbackorders) = Cb(S)(D/Q) + CsS2/2Q

Planned Shortage Model The Total Variable Cost

Equation


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The total annual variable cost equation

The optimal solution to this problem isobtained under the following conditions

Cs > 0 ;

Cb < \/ 2CoCh / D

TV(Q,S) =

(Q -S)2

2Q Ch +

D

Q(Co + SCb) +S2

2QCS

Holdingcosts

Time dependentbackorder costs

Time independentbackorder costs

Orderingcosts

Planned Shortage Model The Total Variable Cost

Equation


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The Optimal Backorder level

S*=

Q* Ch

- DCb

Ch + CsReorder Point

R = L D - S*


The Optimal Inventory

PolicyThe Optimal OrderSize

Ch

(DCb)2

ChCs

2DC

o Q*

=

Ch + Cs

Csx

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76

SCANLON PLUMBINGCORPORATION

Scanlon distributes a portable sauna fromSweden.

Data A sauna costs Scanlon $2400.

Annual holding cost per unit $525.

Fixed ordering cost $1250 (fairly high, due to

costly transportation).

Lead time is 4 weeks.

Demand is 15 saunas per week on the average.

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Scanlon estimates a $20 goodwill cost for

each week a customer who orders a sauna

has to wait for delivery.

Administrative backordrer cost is $10.

Management wishes to know:

The optimal order quantity.

The optimal number of backorders.

Backordercosts

SCANLON PLUMBINGCORPORATION

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SCANLON PLUMBING Solution

Input for the total variable costfunction

D = 780 saunas [(15)(52)]

Co = $1,250

Ch = $525

Cs = $1,040

Cb

= $10

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x (780)(10)2

(525)(1040)525

2(780)(1250)

525+1040

1040

Q*=

74

The optimal policy

R = (4 / 52)(780) 20 = 40

_

S*

=

(74)(525)(780)(10)

525 + 104020

SCANLON PLUMBING Solution

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SCANLON PLUMBING Spreadsheet Solution

Calculation of Optimal Inventory Policy for a Planned Shortage Model

OPTIMAL ASSIGNED

INPUTS Values OUTPUTS Values OUTPUTS Values

Annual Demand, D = .22222 Order Quantity, Q* = .1111 Q = .2222Per Unit Cost, C = .222222 Backorder Level, S* = .1111 S = .2222

Annual Holding Cost Rate, H .222 Cycle Time (in years), T = .22222 T = .22222

Annual Holding Cost Per Uni .22222 # of Cycles Per Year, N = .111111N = .222222

Order Cost, Co = .222222 Reorder Point, R = .222222 R = .111111

Annual Backorder Cost, Cs .222222 Total Annual Variable Cost, .2222222 TV(Q) = .2222222

Fixed Admin. Backorder Co .2222 Total Annual Cost, TC(Q*) = .222222222 TC(Q) = .222222222

Lead Time (in years), L = .222222 % of Customers Backordere .2222 % Back. = .2222

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81

8.7 Review Systems

Continuous Review (R, Q) Policies

The EOQ, production lot size, and

planned shortage models assume that

inventory levels are continuously monitored

Items are sold one at a time.

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(R, Q) Policies

The above models call for order

point (R) order quantity (Q)

inventory policies.

Such policies can be implemented by A point-of-sale computerized system.The two-bin system.

8.7 Review Systems

Continuous Review

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(R, M) policies

When items are not necessarily sold

one at a time, the reorder point might

be missed, and out of stock situations

might occur more frequently.

The order to level (R, M) policy may be

implemented in this situation.

Continuous Review Systems

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(R,M) policies

The R, M policy replenishes inventory

up to a pre-determined level M.

Continuous Review Systems

Order Q = Q* + (R I) = (M

SS) + (R I) each time the

inventory falls to the reorder point R or

below.

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It may be difficult or impossible toadopt a continuous review system,

because of:The high price of a computerized system.

Lack of space to adopt the two-binsystem.

Operations inefficiency when orderingdifferent items from the same vendorseparately.

The periodic review system may be

Periodic Review Systems

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(T,M) Policies

In a replenishment cycle policy(T, M), the inventory position isreviewed every T time units.

An order is placed to bring theinventory level back up to a maximum

inventory level M.M is determined by

Forecasting the number of units demandedduring the review period T.

Adding the desired safety stock to the


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T =Review period

L = Lead timeSS= Safety stockQ = Inventory positionD = Annual demandI = Inventory position


Calculation of the replenishmentlevel and order size

Q = M + LD I

M = TD +SS

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89

Every three weeks AAC receivesdeliveries of different productsfrom Citron.

Lead time is eight days forordering Citrons juicers.

AAC is now reviewing its juicer

inventory and finds 210 in stock. How many juicers should AAC

order for a safety stock of 30

juicers?

AAC operates a (T, M) policy

AAC (T M) li

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Data

Review period T = 3 weeks = 3/52 = .

05769 years, Lead time = L = 8 days = 8/260 = .

03077 years,

Demand D = 6240 juicers per year,

Safety stock SS = 30 juicers,

Inventory position I = 210 juicers

AAC operates a (T, M) policy Solution

AAC operates260 days a year.

(5)(52) = 260.

AAC (T M) li

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Review period demand = TD =( 3/52)(6240) = 360 juicers,

M = TD + SS = 360 + 30 = 390juicers,

Q = M + LD I = 390 + .03077(6240) - 210 = 372 juicers.


AAC t (T M) li

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Review

point

Review

point


T

SS SS SS

Inventory position

Order Order

Replenishment level

Inventory position

L

: I + Q is designed to satisfy the demand within an intervalTo obtain the replenishment level add SS to I + Q.

M = maximum inventory

L

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Th E t d P fit

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To find an optimal order quantity we need to

balance the expected cost of over-ordering

and under ordering.

Expected Profit = (Profit whenDemand=X)Prob(Demand=X)x

The expected profit is a function of the order

size, the random demand, and the various

The Expected Profit

Function

The E pected Profit

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Developing an expression for EP(Q) Notation

p = per unit selling price of the

good.c = per unit cost of the good.

s = per unit salvage value of

unsold good.K = fixed purchasing costs

Q = order quantity.

=

The Expected Profit

Function

The Expected Profit

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Scenario 1: Demand X is less than theunits

stocked, Q.

Scenario 2: Demand X is greater than

or equal to the units stocked.

Profit = pX + s(Q - X) - cQ - K

Profit = pQ - g(X - Q) - cQ - K

) = [pX+s(Q - X) - cQ - K]P(X) + [pQ - g(X - Q) - cQ -X Q

X Q

The Expected Profit

Function

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To maximize the expected profit orderQ*

For the discrete demand case take thesmallest value of Q*

that satisfies the condition

P(D Q*) (p - c + g)/(p -s + g)

The Optimal Solution

THE SENTINEL

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THE SENTINELNEWSPAPER

Management at Sentinel wishes toknow how manynewspapers to putin a new vending machine.

Data Unit selling price is $0.30

Unit production cost is $0.38.

Advertising revenue is $0.18 pernewspaper.

Unsold newspaper can be recycled andnet $0.01.

Unsatisfied demand costs $0.10 per

Demand distribution isdiscrete uniform between30 and 49 newspapers.

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SENTINEL Solution

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100

1.0

0.500.55

30 49

0.513

3940

P(D 39) = 0.50

P(D 40) = 0.55

Q*= 40

SENTINEL SolutionFinding the optimal order

quantity Q*

SENTINEL Spreadsheet

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=(B5+B8-B6)/(B5+B8-B7)

=ROUNDUP(B10+E5*(B11-10),0)

=(E6-B10+1)/(B11-

B10+1)

SENTINEL SpreadsheetSolution

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WENDELLS BAKERY

Management in Wendells wishesto determine the number of donutsto prepare for sale, on weekday

evenings Data Unit cost is $0.15.

Unit selling price is $0.35.

Unsold donuts are donated to charityfor a tax credit of $0.05 per donut.

Customer goodwill cost is $0.25.

Operating costs are $15 per evening.

Demand is normally distribute

with a mean of 120, and astandard deviation of 20 donut

WENDELLS BAKERY

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WENDELLS BAKERY-Solution

Input to the optimal order quantityformula

p = $0.35c = $0.15

s = $0.05

g = $0.25K = $15.00

The optimal service level =p+ g - cp+ g - s

0.35+ 0.25 - 0.150.35+0.25 - 0.05

= 0.8182=

WENDELLS BAKERY-Solution

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.8182

=120 Q*

From the relationship F(Q*) = 0.8182 wefind the corresponding z value.

From the standard normal table wehave z = 0.3186.

The optimal order quantity is calculatedby

Q* = + z

*

SolutionFinding the optimal order

quantity

WENDELLS BAKERY-

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Q*) = (p - s) - (c - s)Q* - (p + g - s) ()L[(Q* - ) /]*) = (p - s) - (c - s)Q* - (p + g - s) ()L[(Q* - ) /]

For the normal distribution

L [(Q* - ) /] is obtained from thepartial expected value table.

For WendellsEP(138) = (0.35 - 0.05)(120) - (0.15 - 0.05)(138) -

(0.35 + 0.25 - 0.05)x(20)L[(138 - 120) /20] - 15 = $6.10L(0.9) = 0.1004

Solution

Calculating the expected profit

WENDELLS BAKERY

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=(B5-B7)*B10-(B6-B7)*E6-(B5+B8-B7)*B11*(EXP(-(((E6-B10)/B11)^2)/2)/

((2*PI())^0.5)-((E6-B10)/B11)*(1-

NORMSDIST((E6-B10)/B11)))-B9

=NORMINV(E5,B10,B1

1)

=(B5+B8-

B6)/(B5+B8-B7)

WENDELLS BAKERY-Spreadsheet Solution

WENDELLS

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WENDELL S The commission strategy

When commission replaces fixedwages

Compare the maximum expected profitof two strategies: $0.13 commission paid per donut sold,

$15 fixed wage per evening (calculated

before).

Calculate first the optimal quantity for

the alternative policy.

WENDELLS

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The commission strategy -

SolutionThe unit selling price changes to

c = 0.35 - 0.13 = $0.22

The optimal order:F(Q*) = (0.22 + 0.25 - 0.15) / (0.22 + 0.25 -0.05)= 0.7616.

Z = .71

Q* = + z = 120 + (0.71)(20) 134 donuts.

WENDELLS

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Will the bakerys expected profitincrease?

EP(134) = (0.22 - 0.05)(20) - (0.15 - 0.05)(134) - (0.22 + 0.25 - 0.05)x(20)L[(134 -120) / 20] = $5.80 < 6.10

The bakery should not proceed withthe alternative plan.


Solution

WENDELLS

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CommentsThe operator expected compensation

will increase, but not as much as thebakerys expected loss.

An increase in the mean sales is

probable when the commissioncompensation plan is implemented.This may change the analysis results.


Solution

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Copyright 2002 John Wiley & Sons, Inc. All rightsreserved. Reproduction or translation of this work beyond

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