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![Page 1: 1 Managing Inventory under Risks Leadtime and reorder point Uncertainty and its impact Safety stock and service level The lot-size reorder point system.](https://reader036.fdocuments.net/reader036/viewer/2022070307/551b1969550346f70d8b635f/html5/thumbnails/1.jpg)
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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
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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
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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
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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?
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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?
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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
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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
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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
![Page 9: 1 Managing Inventory under Risks Leadtime and reorder point Uncertainty and its impact Safety stock and service level The lot-size reorder point system.](https://reader036.fdocuments.net/reader036/viewer/2022070307/551b1969550346f70d8b635f/html5/thumbnails/9.jpg)
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Random Leadtime Demand
R
2 2 2R LL R
Random Variable Mean std
Demand
Leadtime
Leadtime demand (DL)
LR
L
= LR
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Safety Stock
Time t
ROP
L L
order order order
mean demand during supply lead time
safety stock
Inventory on hand
Leadtime
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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
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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
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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
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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)
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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
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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)
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Example 7.4 (MBPF)
• µ = 20,000, σ = 5,000
β = 85% 90% 95% 99%
zβ =
ROP =
NT
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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
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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 = 250• Reorder point ROP = 1,250
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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
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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
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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
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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
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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
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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)
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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
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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
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Managing Safety Stock
Levers to reduce safety stock
- Reduce demand variability
- Reduce delivery leadtime
- Reduce variability in delivery leadtime
- Risk pooling
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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)
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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
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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
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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!!
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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
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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
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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