Frequency & Safety Stock
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Transcript of Frequency & Safety Stock
1
Frequency & Safety Stock
April 12, 2005
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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
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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
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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
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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?
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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
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
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Order Up To Policy
Time
Sto
ck o
n ha
ndReorder Point
Order placed
Lead Time
Reorder Point
Target pipeline inventory level
Actual Lead Time Demand
Actual Lead Time Demand
Order Quantity
Actual Lead Time Demand
Actual Lead Time Demand
How much stock is available to cover demand in this period?
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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
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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
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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.
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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?
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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
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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
• Variance in Demand in T+L (T+E[L]) D
2 +D2 L
2 = (7+2)*672 + (1052)*22
= 40,401 + 44,100 = 84,501 Std Deviation = 291
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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
• Variance in Demand in T+L (T+E[L]) D
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
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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.
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Overlapping Risk
T+ L
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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%
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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%
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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%
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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)
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Safety Lead Time as a quantity
0
100
200
300
400
500
600
700
Safety Lead Time: The next X days of forecasted demand
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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.
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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
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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