Frequency & Safety Stock

1

Frequency & Safety Stock

April 12, 2005

2

Agenda

• Frequency and Safety Stock– Continuous Review – EOQ– Periodic Review – Order-up-to– With a forecast – Safety lead time

Continuous Review Model• Assumptions

– Fixed ordering cost– Orders can be placed at any time– Relatively constant, but uncertain demand– Variable lead time

• Often called (Q, r) policy– Q is order quantity– r is the re-order point

4

Continuous Review Basics

• Basic tool to manage risk

Time

Sto

ck o

n ha

nd

Safety Stock

Reorder Point

Order placed

Lead Time

Actual Lead Time Demand

Avg LT Demand

Order when inventory hits the Average Lead time demand + safety

stock

Safety stock = number of standard deviations in lead

time demand

Order the EOQ

5

Safety Stock Basics

• Lead time demand has std deviation • Safety stock levels

– Choose z to get correct probability that lead time demand exceeds safety stock z

6

Question

• What is the impact of increasing the frequency of orders in this context?

• Inventory costs?

• Ordering costs?

• Service level if we hold safety stock constant?

7

Continuous Review Basics

• Basic tool to manage risk

Time

Sto

ck o

n ha

nd

Safety Stock

Reorder Point

Order placed

Lead Time


Avg LT Demand

Experience riskMORE often!!!

Should you increase your safety stock?

8

Periodic Review Models

• Standing contract with carrier or supplier– E.g., weekly shipment– Daily replenishment– Every 4 hours replenishment

• Order-up-to Policies– Bring the inventory up to a target level with

each shipment

9

Order Up To Policy

Time

Sto

ck o

n ha

ndReorder Point

Order placed

Lead Time

Reorder Point

Target pipeline inventory level



Order Quantity



How much stock is available to cover demand in this period?

10

Order Up To Policy: Inventory

Time

Sto

ck o

n ha

ndReorder Point Reorder Point

On Average this is the Expected

demand between orders

Order Quantity

So average on-hand inventory is DT/2+ss

On Average this is the safety stock

11

Order Up To Policy: Inventory

Time

Sto

ck o

n ha

ndReorder Point Reorder Point

After an order is placed, it is the

Order up to level

Order Quantity

So average Pipeline inventory is OUL – DT/2

Before an order is placed it is smaller by the demand in

the period

12

Safety Stock in Periodic Review

• Probability of stock out is the probability demand in T+L exceed the order up to level, S

• Set a time unit, e.g., days• T = Time between orders (fixed)• L = Lead time, mean E[L], std dev L

• Demand per time unit has mean D, std dev D

• Assume demands in different periods are independent• Let Ddenote the standard deviation in demand per unit

time• Let Ldenote the standard deviation in the lead time.

13

Safety Stock in Periodic Review

• Probability of stock out is the probability demand in T+L exceed the order up to level, S

• Expected Demand in T + L D(T+E[L])

• Variance in Demand in T+L (T+E[L]) D

2 +D2 L

2

• Order Up to Level: S= D(T+E[L]) + safety stock• Question: What happens to service level if we

hold safety stock constant, but increase frequency?

14

Impact of Frequency

• What if we double frequency, but hold safety stock constant?

• Expected Demand in T/2 + L D(T/2+E[L])

• Variance in Demand in T/2+L (T/2+E[L]) D

2 +D2 L

2

• Order Up to Level: S = D(T/2+E[L]) + safety stock

But now we face the risk of failure twice as often

This is reduced by

TD2/2

15

Example• Time period is a day• Frequency is once per week

T = 7

• Daily demand Average 105 Std Dev 67

• Lead time Average 2 days Std Dev 1 days

• Expected Demand in T+L D (T + E[L]) = 105 (7 + 2) = 945


2 +D2 L

2 = (7+2)*672 + (1052)*22

= 40,401 + 44,100 = 84,501 Std Deviation = 291

16

Example Cont’dExpected Demand in T+L

D (T + E[L]) = 105 (7 + 2) = 945 If we ship twice a week this drops to 578 If we ship thrice a week this drops to 456


2 +D2 L

2 = (7+28)*672 + (1052)*22

= 40,401 + 44,100 = 84,501

Std Deviation = 291 If we ship twice a week this drops to 262 If we ship thrice a week this drops to 252

17

Example Cont’d

• With weekly shipments: To have a 98% chance of no stockouts in a year, we need .9996 chance of no stockouts in a week .999652 ~ .98

• With twice a week shipments, we need .9998 chance of no stockouts between two shipments .9998104 ~ .98

• With thrice a week shipments, we need .9999 chance of no stockouts between two shipments .9999156 ~ .98

Note that if Lead time is greater than T this is very conservative.

18

Overlapping Risk

T+ L

19

Example Cont’d• Assume Demand in L+T is Normal

• Hold risk constant 98% chance of no shortages all year

Once a week Twice a week Thrice a weekD(T+E[L]) 945 578 455 Std dev in Demand 291 262 252 Order up to Level 1,920 1,506 1,392 Safety Stock 975 929 938 On Hand Inventory 1,342 1,112 1,060 % Reduction 0% 17.1% 21.0%Echelon Inventory 1,552 1,322 1,270 % Reduction - 14.8% 18.2%

20

Lead time = 28• When lead time is long relative to T

• Safety stock is less clear (Intervals of L+T overlap)

• Very Conservative Estimate Once a week Twice a week Thrice a week

D(T+E[L]) 3,675 3,308 3,185 Std dev in Demand 449 431 425 Order up to Level 5,179 4,832 4,764 Safety Stock 1,504 1,525 1,579 On Hand Inventory 1,871 1,709 1,702 % Reduction 0% 8.7% 9.1%Echelon Inventory 4,811 4,649 4,642 % Reduction 0.0% 3.4% 3.5%

21

Lead time = 28• When lead time is long relative to T

• Safety stock is less clear (Intervals of L+T overlap)

• Aggressive Estimate: Hold safety stock constant

Once a week Twice a week Thrice a weekD(T+E[L]) 3,675 3,308 3,185 Std dev in Demand 449 431 425 Order up to Level 5,179 4,811 4,689 Safety Stock 1,504 1,504 1,504 On Hand Inventory 1,871 1,688 1,626 % Reduction 0.0% 9.8% 13.1%Echelon Inventory 4,811 4,628 4,566 % Reduction 0.0% 3.8% 5.1%

22

Periodic Review against a Forecast

• A forecast of day-to-day or week-to-week requirements

• Two sources of error– Forecast error (from demand variability)– Lead time variability

• Safety Lead Time replaces/augments Safety Stock• Example 6 days Safety Lead Time• Safety Lead Time translates into a quantity through

the forecast, e.g., the next 6 days of forecasted requirements (remember the forecast changes)

23

Safety Lead Time as a quantity

0

100

200

300

400

500

600

700

Safety Lead Time: The next X days of forecasted demand

24

The Ship-to-Forecast Policy

• Periodic shipments every T days

• Safety lead time of S days

• Each shipment is planned so that after it arrives we should have S + T days of coverage.

25

If all goes as planned

0

100

200

300

400

500

600

700

Safety Lead Time: The next X days of forecasted demand

Planned Inventory

Ship to this level

26

New Strategy• Ship-to-Average

– Average usage or longer term forecast

• Motivation: flex inventory, keep mfg and shipments constant

• Effects:– Relatively constant demands on production

and transport capacity– Reduced reliance on forecasts– Reduced complexity– Have to manage inventories

Frequency & Safety Stock

Documents

Transcript of Frequency & Safety Stock