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 PointIn

vent

ory

leve

lQ

Receive order

Placeorder

Receive order

Placeorder

Receive order

Leadtime

Reorderpoint

Usage rate R

Time

Average inventory = Q/2

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

0

2

7

14

22

0

40

140

280

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

• 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?L

order

Inventory on hand

ROP1

ROP2

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 ×3

zβ = safety factor

9

Random Leadtime Demand

R

2 2 2R LL R

Random Variable Mean std

Demand

Leadtime

Leadtime demand (DL)

LR

L

= LR

10

Safety Stock

Time t

ROP

L L

order order order

mean demand during supply lead time

safety stock

Inventory on hand

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

Mean: µ = 1,000 ROP = 1,200

Service Level: SL = ? (The area of the shaded part under the curve)

SL = Pr (LD ROP) = probability of meeting all demand(no stocking out in a cycle)

Is= ROP – µ = 200

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)

ROPI

z S (4)

15

Example 7.3 (MBPF)

• 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 =

ROP =

NT

18

Price of High Service Level

0.5 0.6 0.7 0.8 0.9 1.0

Saf

ety

Sto

ck

Service Level

NORMSINV ( 0.85)·200

NORMSINV ( 0.90)·200

NORMSINV ( 0.95)·200

NORMSINV ( 0.97)·200

NORMSINV ( 0.99)·200

NORMSINV ( 0.999)·200

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

Time t

ROP

order

mean demand during supply lead time

safety stock

Inventory on hand

Leadtime

Q +Is

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 = 2000/2+250+1000 = 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

na (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

07.725

14.1407.724 a

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 = 2x100+ 18.5 =218.5• Total system average inventory:

not own pipeline

I = 36/2+18.5 = 36.5

owns pipeline

I = 36/2+18.5+200 = 236.5

34

Takeaways (1)

• 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

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

nzI sa