Lecture 26 Order Quantities (Revisited) Books Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming College,

Lecture 26

Order Quantities (Revisited)

Books• Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming

College, Emeritus, Stephen N. Chapman, Ph.D., CFPIM, North Carolina State University, Lloyd M. Clive, P.E., CFPIM, Fleming College

• Operations Management for Competitive Advantage, 11th Edition, by Chase, Jacobs, and Aquilano, 2005, N.Y.: McGraw-Hill/Irwin.

• Operations Management, 11/E, Jay Heizer, Texas Lutheran University, Barry Render, Graduate School of Business, Rollins College, Prentice Hall

Objectives• Objectives of inventory management• Lot size decision• Inventory models • EOQ• Robust model• Reorder point• Production order quantity model• Quantity discount model• Probabilistic Models and Safety Stock• Probabilistic Demand• Other probabilistic models• Fixed period system• EOQ consequences• Period order quantity model

Objectives of Inventory Management

Determine:• How much should be ordered at one time?• When should an order be placed?

Lot-Size Decision Rules

Lot-for-lot. Order exactly what is needed.

Fixed-order quantity. Arbitrary

Order “n” periods supply. Satisfy demand for a given period of demand.

Inventory Models for Independent Demand

Basic economic order quantity Production order quantity Quantity discount model

Need to determine when and how much to order

Basic EOQ Model

1. Demand is known, constant, and independent2. Lead time is known and constant3. Receipt of inventory is instantaneous and

complete4. Quantity discounts are not possible5. Only variable costs are setup and holding6. Stockouts can be completely avoided

Important assumptions

Inventory Usage Over Time

Order quantity = Q (maximum

inventory level)

Usage rate Average inventory on

handQ2

Minimum inventory

Inve

ntor

y le

vel

Time0

Minimizing Costs

Objective is to minimize total costs

Ann

ual c

ost

Order quantity

Curve for total cost of holding

and setup

Holding cost curve

Setup (or order) cost curve

Minimum total cost

Optimal order quantity (Q*)

The EOQ Model

Q = Number of pieces per orderQ* = Optimal number of pieces per order (EOQ)D = Annual demand in units for the inventory itemS = Setup or ordering cost for each orderH = Holding or carrying cost per unit per year

Annual setup cost = (Number of orders placed per year) x (Setup or order cost per order)

Annual demand

Number of units in each orderSetup or order cost per order

=

Annual setup cost = SDQ

= (S)DQ

The EOQ Model


Annual holding cost = (Average inventory level) x (Holding cost per unit per year)

Order quantity

2= (Holding cost per unit per year)

= (H)Q2


Annual holding cost = HQ2

The EOQ Model


Optimal order quantity is found when annual setup cost equals annual holding cost


Annual holding cost = HQ2

DQ

S = HQ2

Solving for Q*2DS = Q2HQ2 = 2DS/H

Q* = 2DS/H

An EOQ Example

Determine optimal number of needles to orderD = 1,000 unitsS = $10 per orderH = $.50 per unit per year

Q* =2DS

H

Q* =2(1,000)(10)

0.50= 40,000 = 200 units

An EOQ Example

Determine optimal number of needles to orderD = 1,000 units Q* = 200 unitsS = $10 per orderH = $.50 per unit per year

= N = =Expected number of

orders

DemandOrder quantity

DQ*

N = = 5 orders per year 1,000200

An EOQ Example

Determine optimal number of needles to orderD = 1,000 units Q* = 200 unitsS = $10 per order N = 5 orders per yearH = $.50 per unit per year

= T =Expected time between orders

Number of working days per year

N

T = = 50 days between orders250

5

An EOQ Example

Determine optimal number of needles to orderD = 1,000 units Q* = 200 unitsS = $10 per order N = 5 orders per yearH = $.50 per unit per year T = 50 days

Total annual cost = Setup cost + Holding cost

TC = S + HDQ

Q2

TC = ($10) + ($.50)1,000200

2002

TC = (5)($10) + (100)($.50) = $50 + $50 = $100

Robust Model

The EOQ model is robust It works even if all parameters and

assumptions are not met The total cost curve is relatively flat

in the area of the EOQ

An EOQ Example

Management underestimated demand by 50%D = 1,000 units Q* = 200 unitsS = $10 per order N = 5 orders per yearH = $.50 per unit per year T = 50 days

TC = S + HDQ

Q2

TC = ($10) + ($.50) = $75 + $50 = $1251,500200

2002

1,500 units

Total annual cost increases by only 25%

An EOQ Example

Actual EOQ for new demand is 244.9 unitsD = 1,000 units Q* = 244.9 unitsS = $10 per order N = 5 orders per yearH = $.50 per unit per year T = 50 days

TC = S + HDQ

Q2

TC = ($10) + ($.50)1,500244.9

244.92

1,500 units

TC = $61.24 + $61.24 = $122.48

Only 2% less than the total cost of $125 when the

order quantity was 200

Reorder Points

EOQ answers the “how much” question The reorder point (ROP) tells when to

order

ROP =Lead time for a new

order in daysDemand per day

= d x L

d = D

Number of working days in a year

Reorder Point Curve

Q*

ROP (units)

Inve

ntor

y le

vel (

unit

s)

Time (days)Lead time = L

Slope = units/day = d

Reorder Point Example

Demand = 8,000 iPods per year250 working day yearLead time for orders is 3 working days

ROP = d x L

d = D

Number of working days in a year

= 8,000/250 = 32 units

= 32 units per day x 3 days = 96 units

Production Order Quantity Model

Used when inventory builds up over a period of time after an order is placed

Used when units are produced and sold simultaneously

Production Order Quantity ModelIn

vent

ory

leve

l

Time

Demand part of cycle with no production

Part of inventory cycle during which production (and usage) is taking place

t

Maximum inventory


Q = Number of pieces per order p = Daily production rateH = Holding cost per unit per year d = Daily demand/usage ratet = Length of the production run in days

= (Average inventory level) xAnnual inventory holding cost

Holding cost per unit per year

= (Maximum inventory level)/2Annual inventory level

= –Maximum inventory level

Total produced during the production run

Total used during the production run

= pt – dt


Q = Number of pieces per order p = Daily production rateH = Holding cost per unit per year d = Daily demand/usage

ratet = Length of the production run in days

= –Maximum inventory level

Total produced during the production run

Total used during the production run

= pt – dtHowever, Q = total produced = pt ; thus t = Q/p

Maximum inventory level = p – d = Q 1 –Q

pQp

dp

Holding cost = (H) = 1 – H dp

Q2

Maximum inventory level2


Q = Number of pieces per order p = Daily production rateH = Holding cost per unit per year d = Daily demand/usage rateD = Annual demand

Q2 =2DS

H[1 - (d/p)]

Q* =2DS

H[1 - (d/p)]p

Setup cost = (D/Q)S

Holding cost = HQ[1 - (d/p)]12

(D/Q)S = HQ[1 - (d/p)]12

Production Order Quantity Example

D = 1,000 units p = 8 units per dayS = $10 d = 4 units per dayH = $0.50 per unit per year

Q* =2DS

H[1 - (d/p)]

= 282.8 or 283 hubcaps

Q* = = 80,0002(1,000)(10)

0.50[1 - (4/8)]


When annual data are used the equation becomes

Q* =2DS

annual demand rateannual production rate

H 1 –

Note:

d = 4 = =D

Number of days the plant is in operation

1,000

250

EPQ Problem: HP Ltd. Produces premium plant food in 50# bags. Demand is

100,000 lbs/week. They operate 50 wks/year; HP produces 250,000 lbs/week.

Setup cost is $200 and the annual holding cost rate is $.55/bag. Calculate the

EPQ. Determine the maximum inventory level. Calculate the total cost of using the EPQ policy.

H

2

IS

Q

DTC MAX

EPQ

p

d1H

2DSEPQ

p

d1QIMAX

EPQ Problem Solution

p

d1H

2DSEPQ BagsEPQ 850,77

250000

000,100155.

)200)(000,100)(50(2

p

d1QIMAX

H

2

IS

Q

DTC MAX

EPQ

bagsMAXI 710,46000,250

000,1001850,77

690,25$55.2

710,46200

850,77

000,000,5

TC

Quantity Discount Models

Reduced prices are often available when larger quantities are purchased

Trade-off is between reduced product cost and increased holding cost

Total cost = Setup cost + Holding cost + Product cost

TC = S + H + PDDQ

Q2


Discount Number Discount Quantity Discount (%)

Discount Price (P)

1 0 to 999 no discount $5.00

2 1,000 to 1,999 4 $4.80

3 2,000 and over 5 $4.75

A typical quantity discount schedule


1. For each discount, calculate Q*

2. If Q* for a discount doesn’t qualify, choose the smallest possible order size to get the discount

3. Compute the total cost for each Q* or adjusted value from Step 2

4. Select the Q* that gives the lowest total cost

Steps in analyzing a quantity discount


1,000 2,000

Tota

l cos

t $

0

Order quantity

Q* for discount 2 is below the allowable range at point a and must be adjusted upward to 1,000 units at point b

ab

1st price break

2nd price break

Total cost curve for

discount 1

Total cost curve for discount 2

Total cost curve for discount 3

Quantity Discount Example

Calculate Q* for every discount Q* =2DSIP

Q1* = = 700 cars/order2(5,000)(49)

(.2)(5.00)

Q2* = = 714 cars/order2(5,000)(49)

(.2)(4.80)

Q3* = = 718 cars/order2(5,000)(49)

(.2)(4.75)


Calculate Q* for every discount Q* =2DSIP

Q1* = = 700 cars/order2(5,000)(49)

(.2)(5.00)

Q2* = = 714 cars/order2(5,000)(49)

(.2)(4.80)

Q3* = = 718 cars/order2(5,000)(49)

(.2)(4.75)

1,000 — adjusted

2,000 — adjusted


Discount Number

Unit Price

Order Quantity

Annual Product

Cost

Annual Ordering

Cost

Annual Holding

Cost Total

1 $5.00 700 $25,000 $350 $350 $25,700

2 $4.80 1,000 $24,000 $245 $480 $24,725

3 $4.75 2,000 $23.750 $122.50 $950 $24,822.50

Choose the price and quantity that gives the lowest total cost

Buy 1,000 units at $4.80 per unit

Quantity Discount Example: Collin’s Sport store is considering going to a different hat supplier. The present supplier charges $10/hat and requires minimum

quantities of 490 hats. The annual demand is 12,000 hats, the ordering cost is $20, and the inventory carrying cost is 20% of the hat cost, a new supplier is offering

hats at $9 in lots of 4000. Who should he buy from?

• EOQ at lowest price $9. Is it feasible?

• Since the EOQ of 516 is not feasible, calculate the total cost (C) for each price to make the decision

• 4000 hats at $9 each saves $19,320 annually. Space?

hats 516$1.80

20)2(12,000)(EOQ$9

$101,66012,000$9$1.802

4000$20

4000

12,000C

$120,98012,000$10$22

490$20

490

12,000C

$9

$10

Probabilistic Models and Safety Stock

Used when demand is not constant or certain

Use safety stock to achieve a desired service level and avoid stockouts

ROP = d x L + ss

Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit

x the number of orders per year

Safety Stock Example

Number of Units Probability

30 .2

40 .2

ROP 50 .3

60 .2

70 .1

1.0

ROP = 50 units Stockout cost = $40 per frameOrders per year = 6 Carrying cost = $5 per frame per year

Safety Stock Example

ROP = 50 units Stockout cost = $40 per frameOrders per year = 6 Carrying cost = $5 per frame per year

Safety Stock

Additional Holding Cost Stockout Cost

Total Cost

20 (20)($5) = $100 $0 $100

10 (10)($5) = $ 50 (10)(.1)($40)(6) = $240 $290

0 $ 0 (10)(.2)($40)(6) + (20)(.1)($40)(6) =$960 $960

A safety stock of 20 frames gives the lowest total cost

ROP = 50 + 20 = 70 frames

Safety stock 16.5 units

ROP

Place order

Probabilistic DemandIn

vent

ory

leve

l

Time0

Minimum demand during lead time

Maximum demand during lead time

Mean demand during lead time

Normal distribution probability of demand during lead time

Expected demand during lead time (350 kits)

ROP = 350 + safety stock of 16.5 = 366.5

Receive order

Lead time

Probabilistic Demand

Safety stock

Probability ofno stockout

95% of the time

Mean demand

350

ROP = ? kits Quantity

Number of standard deviations

0 z

Risk of a stockout (5% of area of normal curve)

Probabilistic Demand

Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined

ROP = demand during lead time + ZsdLT

where Z = number of standard deviations

sdLT = standard deviation of demand during lead time

Probabilistic Example

Average demand = m = 350 kitsStandard deviation of demand during lead time = sdLT = 10 kits5% stockout policy (service level = 95%)

Using Appendix I, for an area under the curve of 95%, the Z = 1.65

Safety stock = ZsdLT = 1.65(10) = 16.5 kits

Reorder point = expected demand during lead time + safety stock= 350 kits + 16.5 kits of safety stock= 366.5 or 367 kits

Other Probabilistic Models

1. When demand is variable and lead time is constant

2. When lead time is variable and demand is constant

3. When both demand and lead time are variable

When data on demand during lead time is not available, there are other models available


Demand is variable and lead time is constant

ROP = (average daily demand x lead time in days) + ZsdLT

where sd = standard deviation of demand per day

sdLT = sd lead time


Average daily demand (normally distributed) = 15Standard deviation = 5Lead time is constant at 2 days90% service level desired

Z for 90% = 1.28From Appendix I

ROP = (15 units x 2 days) + Zsdlt

= 30 + 1.28(5)( 2)

= 30 + 9.02 = 39.02 ≈ 39

Safety stock is about 9 iPods


Lead time is variable and demand is constant

ROP = (daily demand x average lead time in days)

= Z x (daily demand) x sLT

where sLT = standard deviation of lead time in days


Daily demand (constant) = 10Average lead time = 6 daysStandard deviation of lead time = sLT = 398% service level desired

Z for 98% = 2.055From Appendix I

ROP = (10 units x 6 days) + 2.055(10 units)(3)

= 60 + 61.65 = 121.65

Reorder point is about 122 cameras


Both demand and lead time are variable

ROP = (average daily demand x average lead time) + ZsdLT

where sd = standard deviation of demand per day

sLT = standard deviation of lead time in days

sdLT = (average lead time x sd2)

+ (average daily demand)2 x sLT2


Average daily demand (normally distributed) = 150Standard deviation = sd = 16Average lead time 5 days (normally distributed)Standard deviation = sLT = 1 day95% service level desired Z for 95% = 1.65

From Appendix I

ROP = (150 packs x 5 days) + 1.65sdLT

= (150 x 5) + 1.65 (5 days x 162) + (1502 x 12)

= 750 + 1.65(154) = 1,004 packs

Fixed-Period (P) Systems

Orders placed at the end of a fixed period Inventory counted only at end of period Order brings inventory up to target level

Only relevant costs are ordering and holding Lead times are known and constant Items are independent from one another

Fixed-Period (P) SystemsO

n-ha

nd in

vent

ory

Time

Q1

Q2

Target quantity (T)

P

Q3

Q4

P

P

Fixed-Period (P) Example

Order amount (Q) = Target (T) - On-hand inventory - Earlier orders not yet received

+ Back orders

Q = 50 - 0 - 0 + 3 = 53 jackets

3 jackets are back ordered No jackets are in stockIt is time to place an order Target value = 50

Fixed-Period Systems

Inventory is only counted at each review period

May be scheduled at convenient times Appropriate in routine situations May result in stockouts between periods May require increased safety stock

EOQ Assumptions

• Demand is relatively constant and is known.• The item is produced or purchased in lots or batches

and not continuously.• Order prep costs & inventory-carrying costs are

constant and known.• Replacement occurs all at once.

EOQ Consequences

Average inventory = EOQ lot size / 2

# of orders per year

= Annual demand / lot size

Basic EOQ Model

• Demand is constant over time• Inventory drops at a uniform rate over time• When the inventory reaches 0, the new order is placed and

received, and the inventory level again jumps to Q units• The optimal order quantity will occur at a point where the

total setup cost is equal to the total holding cost

ic

2AS =Q*Basic EOQ:

i

S2A =

D

Basic EOQ Model (cont.)

• Benefit of EOQ model:– It is a robust model, meaning that it gives

satisfactory answers even with substantial variation in the parameters.

• Reorder Points:– Lead Time - the time between the placement and

receipt of an order.– The when-to-order decision is expressed in terms of

a reorder point, the inventory level at which an order should be placed.

Time

Inve

nto

ry L

evel

Maximum Inventory Level

AverageInventory Level

Inventory Level Over Time (Basic EOQ Model)


• Production Order Quantity Model is useful when:– Inventory continuously flows or builds up over a period of

time after an order has been placed or when units are produced and sold simultaneously.

– Takes into account the daily production (or inventory flow) rate and the daily demand rate.

– All other EOQ assumptions are valid.

)]PA(-i[1

S2A = *Q

D

D

Production OrderQuantity

Time

Inve

nto

ry L

evel

Production Portionof Cycle

Maximum Inventory Level

DemandPortionof Cycle

DemandPortionof Cycle

Inventory Level Over Time (Production Model)

Period Order Quantities

• Calculate or determine EOQ• Determine avg. weekly usage• Divide EOQ by avg. weekly usage to determine

period• Order the amount needed during the next period to

satisfy demand during that period

Practice Question 1. Sarah’s Silk Screening

• Sarah’s Silk Screening sells souvenir shirts. Sarah is trying to decide how many to produce for the upcoming naming of the College of Management. The University will allow her to sell the shirts only on one day, the day that the school naming is announced. Sarah will sell the T-shirts for $20 each. When the event day is over, she will be allowed to sell the remaining stock to the Bookstore for $4 each. It costs Sarah $8 to make the specialty shirt. She estimates mean demand to be 545, with a standard deviation is 115. How many shirts should she make?

Practice Question 2. The Great Southern Automotive Co.

• The Great Southern Automotive Co. buys steering wheels from a supplier. One particular steering wheel has a known and constant demand rate of 2,000 units per year. The fixed cost of ordering is $100 and the inventory holding cost is $2 per unit per year. It takes 2 weeks for an order to arrive. Compute

• • The optimal EOQ• The reorder point• The average inventory level• The time between successive orders• The total annual cost• • If demand was variable with a standard deviation of 4 units per week, and the

firm aims for 98% customer satisfaction, what would the reorder point be?

End of Lecture 26

Documents

Lecture 26 Order Quantities (Revisited) Books Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming College,