Managing Inventory under Risks
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Leadtime and reorder point
Uncertainty and its impact
Safety stock and service level
The lot-size reorder point system
Managing system inventory
1
Leadtime and Reorder Point
Inventory level
Q
Usage
rate R
Average inventory = Q/2
Reorde
r
point
Receive
order
Place
order
Receive
order
Time
Place
order
Receive
order
Leadtime
2
When to Order?
ROP (reorder point):
inventory level that triggers a
new order
ROP = LR
(1)
L (days)

0
0
2
40
R = 20 units/day
7
140
Q*= 200 units
14
280
L =
leadtime with certainty
22
440
μ=
LR = leadtime demand
Example:
ROP
3
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?
4
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?
5
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
6
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 ROP1
after the leadtime L
ROP2
• When to order? ROP1?
Inventory on hand
order
L
L
7
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
8
Random Leadtime Demand
Random Variable
Demand
Leadtime
Mean
Leadtime demand (DL)
 = LR
std
sR
sL
R
L
s
Ls 2R  R 2 s 2L
9
Safety Stock
Inventory on hand
order
order
order
ROP
mean demand
during supply
lead time
safety stock
Time t
L
L
Leadtime
10
Some Relations
safety stock
ROP
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
11
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
12
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)
13
Compute Cycle Service Level
• Given Is and σ
z 
IS
s

ROP  
s
• Use normal table, we find β from zβ
• Use excel:
SL= NORMDIST(ROP, ,σ,True)
(4)
(5)
14
Example 7.3 (MBPF)
• ROP = 24,000, µ = 20,000, σ = 5,000
zβ =
β=
NT
or
SL = NORMDIST(24,000,20,000, 5,000, True)
9-EX1
15
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)
16
Example 7.4 (MBPF)
• µ = 20,000, σ = 5,000
β
=
85%
90%
95%
99%
zβ =
ROP =
NT
17
Price of High Service Level
NORMSINV ( 0.999)·200
Safety Stock
NORMSINV ( 0.99)·200
0.5
NORMSINV ( 0.97)·200
NORMSINV ( 0.95)·200
NORMSINV ( 0.90)·200
NORMSINV ( 0.85)·200
0.6
0.7
0.8
0.9
1.0
Service Level
9-EX2
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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
s = 250
• Reorder point
ROP = 1,250
19
The Service Level
• Safety stock
Is =
• Safety factor
zβ =
• Service level
– By normal table NT
β=
– By excel
9-EX1
SL= NORMDIST (1250, 1000, 250, True)
20
Safety Stock for Target SL
• For 95% service rate
– By the normal table
NT
z0.95 =
ROP =
Is
=
– By excel
9-EX1
ROP =NORMINV (0.95, 1000, 250)
21
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
22
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
23
Example, Broadway
• R=52000/year (52 weeks)
H=$1/unit/year
S=$200/order
Lot-size Reorder point
• Order quantity 9-EX1
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
=
24
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)
25
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
26
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
27
Managing Safety Stock
Levers to reduce safety stock
- Reduce demand variability
- Reduce delivery leadtime
- Reduce variability in delivery leadtime
- Risk pooling
28
Demand Aggregation
• By probability theory
Var(D1 + …+ Dn) = Var(D1) + …+ Var(Dn)
= nσ2
• As a result, the standard deviation of the
aggregated demand is
sa  n s
(9)
29
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
s  5 2  7.07
• The leadtime demand uncertainty level of the
aggregated inventory system
s a  s 4  2  7.07  14.14
30
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
31
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!!
32
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
33
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
34
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)
I sa  z s n
• Total system average inventory
= Q/2 + Is
= Q/2 + Is+RL
not own pipeline
owns pipeline
35