1. Managing Inventory under Risks
Leadtime and reorder point
Uncertainty and its impact
Safety stock and service level
The lot-size reorder point system
Managing system inventory

2. Leadtime and Reorder PointQ
Usage
rate R
Inventory level
Average inventory = Q/2
Reorder
point
Time
Receive
order
Place
order
Receive
order
Place
order
Receive
order
Leadtime

3. When to Order?
ROP (reorder point): inventory level that triggers a new order
ROP = LR (1)
Example:
R = 20 units/day
Q*= 200 units
L = leadtime with certainty
μ = LR = leadtime demand
L (days) 
ROP 2 40 7
140 14
280 22
440

4. Motorola Hong Kong Revisited
It takes the supplier 3 full working days to deliver the material to Motorola
Consumption rate is 90 kg/day
At what inventory level should Mr. Chan place an order?

5. Uncertainty and Its Impact
Sandy is in charge of inventory control and ordering at Broadway Electronics. The average demand for their best-selling battery is on average 1,000 units per week with a standard deviation of 250 units
With a one-week delivery leadtime from the supplier, Sandy needs to decide when to order, i.e., with how many boxes of batteries left on-hand, she should place an order for another batch of new stock
What is the difference between Mr. Chan’s task at Motorola and this one?

6. Forecast and Leadtime Demand
Often we forecast demands and make stocking decisions accordingly trying to satisfy arriving customers from on-hand stock
Often, forecasting for a whole year is easier than for a week
Leadtime demands usually can not be treated as deterministic

7. Inventory Decision Under Risk
Inventory on hand
When you place an order, you expect the remaining stock to cover all the leadtime demands
Any order now or later can only satisfy demands after the leadtime L
When to order? ROP1?
order
ROP1
ROP2 L L

8. ROP under Uncertainty
When DL is uncertain, it often makes sense to order a little earlier, i.e., at an inventory level higher than the mean
ROP =  + IS (2)
IS = safety stock or extra inventory
IS = zβ ×s (3)
zβ = safety factor

9. Random Leadtime Demand
Random Variable
Mean
std
Demand
Leadtime
Leadtime demand (DL) R L
 = LR

10. Safety Stock Leadtime Inventory on hand L Time t L order order order
ROP
mean demand during supply lead time
safety stock
Time t L
Leadtime

11. Some Relations ROP safety stock safety stock safety factor
safety factor service level
Given demand distribution, there is a one-to-one relationship, so we also have
ROP Is zβ β

12. Safety Stock and Service Level
Service level is a measure of the degree of stockout protection provided by a given amount of safety inventory
Cycle service level:
the probability that all demands in the leadtime are satisfied immediately
SL = Prob.( LT Demand ≤ ROP) =β

13. Service Level under Normal Demands
Service Level: SL = ? (The area of the shaded part under the curve)
Is= ROP – µ = 200
Mean: µ = 1,000
ROP = 1,200
SL = Pr (LD  ROP) = probability of meeting all demand
(no stocking out in a cycle)

14. Compute Cycle Service Level
Given Is and σ
Use normal table, we find β from zβ
Use excel:
SL= NORMDIST(ROP, ,σ,True) (5)
(4)

15. Example 7.3 (MBPF) zβ = or SL = NORMDIST(24,000,20,000, 5,000, True)
ROP = 24,000, µ = 20,000, σ = 5,000
zβ =
β =
or SL = NORMDIST(24,000,20,000, 5,000, True) NT
9-EX1

16. Compute Safety Stock Given β, we obtain zβ from the normal table
Use (3), we obtain the safety stock
Use (2), we obtain ROP
Given β, we can also have
zβ = NORMSINV (β) (6)
ROP = NORMINV(β, µ, σ ) (7)

17. Example 7.4 (MBPF) µ = 20,000, σ = 5,000 β = 85% 90% 95% 99% zβ =
β = 85% 90% 95% 99%
zβ =
ROP = NT

18. Price of High Service Level
NORMSINV ( 0.999)·200
NORMSINV ( 0.99)·200
Safety Stock
NORMSINV ( 0.97)·200
NORMSINV ( 0.95)·200
NORMSINV ( 0.90)·200
NORMSINV ( 0.85)·200
Service Level
Suppose I said I want 100% service level and it turns out it costs me $8.1b and the company only has $2m to invest. What should I do? Look at the curve.
More towards perfection, more costly.
Go to the computer example!
9-EX2

19. Example, Broadway
Sandy orders a 2-week supply whenever the inventory level drops to 1,250 units.
What is the service level provided with this ROP ?
If Sandy wants to provide an 95% service level to the store, what should be the reorder point and safety stock ?
Average weekly demand µ = 1,000
Demand SD  = 250
Reorder point ROP = 1,250

20. The Service Level Safety stock Is = Safety factor zβ = Service level
By normal table
β =
By excel
SL= NORMDIST (1250, 1000, 250, True) NT
9-EX1

21. Safety Stock for Target SL
For 95% service rate
By the normal table
z0.95 =
ROP =
Is =
By excel
ROP =NORMINV (0.95, 1000, 250) NT
9-EX1

22. Lot Size-Reorder Point System
Having determined the reorder point, we also need to determine the order quantity
Note that we can forecast the annual demand more accurately and hence treat it as deterministic
Then, the order quantity can be obtained using the standard EOQ

23. The Average Inventory Let the order quantity be Q
The average inventory level = (Q+Is+ Is)/2
= Q/2 +Is
The holding cost
= HQ/2+HIs
The ordering cost
= S(R/Q)
The optimal inventory cost = HQ* + HIs
Inventory on hand
Q +Is
order
ROP
mean demand during supply lead time
safety stock
Time t
Leadtime

24. Example, Broadway R=52000/year (52 weeks) H=$1/unit/year S=$200/order
Lot-size Reorder point
Order quantity
Q* =
For 95% service rate
Is = 250zβ =
Inventory cost =
Sandy’s current policy
µ= 1000, Q = 2000
ROP = 1,250, SL =84%
Holding cost =
Ordering cost
Inventory cost
9-EX1

25. Managing System Inventory
There are different stocking points with inventories and at each stocking point, there are inventories for different functions
Total average inventory includes three parts:
Cycle + Safety + Pipeline inventories
Total Average Inventory = Q/2 + Is + RL (8)

26. Pipeline Inventory
If you own the goods in transit from the supplier to you (FOB or pay when order), you have a pipeline inventory
Average pipeline inventory equals the demand rate times the transit time or leadtime by Little’s Law
Pipeline inventory = RL

27. Sandy’s Current System Inventory
Q=2,000, L =1 week, R = 1,000/week
ROP = 1250, Safety stock = Is = 250
Total system average inventory:
not own pipeline
I = 2000/2+250 = 1250
owns pipeline
I = 2000/ = 2250

28. Managing Safety Stock Levers to reduce safety stock
- Reduce demand variability
- Reduce delivery leadtime
- Reduce variability in delivery leadtime
- Risk pooling

29. Demand Aggregation By probability theory
Var(D1 + …+ Dn) = Var(D1) + …+ Var(Dn)
= nσ2
As a result, the standard deviation of the aggregated demand is
(9)

30. The Square Root Rule Again
We call (9) the square root rule:
For BMW Guangdong
Monthly demand at each outlet is normal with mean 25 and standard deviation 5
Replenishment leadtime is 2 months. The service level used at each outlet is 0.90
The SD of the leadtime demand at each outlet of our dealer problem
The leadtime demand uncertainty level of the aggregated inventory system

31. Cost of Safety Stock at Each Outlet
The safety stock level at each outlet
Is = z0.9σ
= 1.285×7.07
= 9.08
The monthly holding cost of the safety stock
TC(Is) = H×Is
= 4,000x9.08
= 36,340RMB/month

32. Saving in Safety Stock from Pooling
System-wide safety stock holding cost without pooling
4×C(Is) = 4 ×36,340=145,360 RMB/month
System-wide safety stock holding cost with pooling
C(Isa ) = 2 ×36,340=72,680 RMB/month
Annual saving of 12x(145,360-72,680)
= 872,160 RMB!!

33. BMW’s System Inventory
With SL = 0.9: L = 2, Q = 36 (using EOQ), R=100/month
z0.9 =1.285, Is =(1.285)(14.4)= 18.5
ROP = 2x =218.5
Total system average inventory:
not own pipeline
I = 36/ = 36.5
owns pipeline
I = 36/ = 236.5

34. Takeaways (1) ROP =  + IS = RL + zβσ
Leadtime demand usually must be treated as random, and hence creates risks for inventory decision
We use safety stock to hedge the risk and satisfy a desired service level
Together with the EOQ ordering quantity, the lot-size reorder point system provide an effective way to manage inventory under risk
Reorder point under normal leadtime demand
ROP =  + IS = RL + zβσ

35. Takeaways (2) For given target SL For given ROP
ROP =  + zβσ
= NORMINV(SL, ,σ)
For given ROP
SL = Pr(DL  ROP)
= NORMDIST(ROP, , σ, True)
Safety stock pooling (of n identical locations)
Total system average inventory
= Q/2 + Is not own pipeline
= Q/2 + Is+RL owns pipeline