Upload
beryl-webster
View
228
Download
8
Tags:
Embed Size (px)
Citation preview
11
Managing Uncertainty with Inventory I
John H. Vande Vate
Spring, 2007
22
Topics
• Integrate Obermeyer (wholesaler) with the Retail Game (retail pricing)
• Continuous Review Inventory Management
• Periodic Review Inventory Management
• Safety Lead Time
33
Item Season Sales1 10342 19423 10974 10685 15786 20007 14298 11459 1571
10 124811 200012 170813 177014 153715 161116 2000
Average 1546
The Retail Game Revisited
• How much inventory to bring to the market? 2000?
• What will demand be?
• How to estimate it?
That’s not demand! It’s supply
• How to estimate demand for this item?
44
Estimating Demand
• How fast was it selling?
Week Inventory Price Weeks Sales1 2000 60 942 1906 60 853 1821 60 1704 1651 60 1555 1496 60 1266 1370 60 647 1306 60 1058 1201 48 2299 972 48 253
10 719 48 17911 540 48 16312 377 48 22313 154 48 15414 0 48 015 0 48 016 0
Average 209/week
• So an estimate of season demand for this item is
• 2473 = 2000 – 154 + 3*209
55
New Estimate
• Should we order 1664?
• What are the issues?
• If salvage value exceeds our cost?
• If salvage value is less than our cost?
Item Season Sales1 10342 19423 10974 10685 15786 24737 14298 11459 1571
10 124811 249012 170813 177014 153715 161116 2927
Average 1664
66
Risk & Return
• Will Demand be 1664?• How to measure our uncertainty about
demand?– Method 1: Standard deviation of diverse
forecasts– Method 2: Historical A/F ratios + Point
forecast
• Trade off Risks (out of stock and overstock) vs Return (sales)
77
Measuring Risk and Return • Profit from the last item
$profit if demand is greater, $0 otherwise
• Expected Profit$profit*Probability demand is greater than our choice
• Risk posed by last item$risk if demand is smaller, $0 otherwise
• Expected Risk$risk*Probability demand is smaller than our choice
Example: risk = Salvage Value - CostWhat if Salvage Value > Cost?
88
Balancing Risk and Return• Expected Profit
$profit*Probability demand is greater than our choice
• Expected Risk$risk*Probability demand is smaller than
our choice
• How are probabilities related?
99
Risk & RewardDistribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 2 4 6 8 10 12
Prob. Outcome is smaller
Prob. Outcome is larger
Our choice
How are they related?
1010
Balance
• Expected Revenue$profit*(1- Probability demand is smaller than our
choice)
• Expected Risk$risk*Probability demand is smaller than our choice
• Set these equalprofit*(1-P) = risk*Pprofit = (profit+risk)*Pprofit/(profit + risk) = P = Probability demand is smaller
than our choice
1111
Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 2 4 6 8 10 12
Making the Choice
Prob. Demand is smaller
Our choice
profit/(profit - risk)
Cumulative Probability
1212
Swimsuit Case p 49
• Fixed Production Cost $100K• Variable Production Cost $80• Selling Price $125• Salvage Value $20• Profit is $125 - $80 = $45• Risk is $80 - $20 = $60• Profit + Risk is $125 - $20 = $105• Order to an expected stock out probability
57% = 1-$45/$105 = 1-43% • Several Sales Forecasts
1313
Forecasts
0%
5%
10%
15%
20%
25%
30%
8,000 10,000 12,000 14,000 16,000 18,000
1414
Inferred Cum. Probability
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
8,000 10,000 12,000 14,000 16,000 18,000
57% stockout:11,490 units
11490=10000+2000*
[43%-Pr(10000)]/[Pr(12000)-Pr(10000)]
43%
Net Profit as a function of Quantity
$(400)
$(300)
$(200)
$(100)
$-
$100
$200
$300
$400
$500
$600
8,000 9,000 10,000 11,000 12,000 13,000 14,000 15,000 16,000 17,000 18,000
Th
ou
san
ds
Gross Profits from sales
Costs of liquidations
Net Profits= Gr. Profits from sales – Cost of liquidation-fixed
cost
1616
What to order?
• So, we want P to be (Selling Price – Cost)
(Selling Price– Salvage)
• Assume Cost = $30,
• But what’s the selling price?
• In a wholesale environment this is easier. In a retail environment, it is messierSome protection from vendor some times
Retail Game
1717
The Value of P as a function of Average Selling Price
• If Cost is $30
74%
76%
78%
80%
82%
84%
86%
88%
$48 $50 $52 $54 $56 $58 $60
Selling Price
P
(Selling Price – Cost)(Selling Price– Salvage)
1818
The Quantity as a function of Average Selling Price
• If Cost is $30
2,000
2,050
2,100
2,150
2,200
2,250
2,300
$48 $50 $52 $54 $56 $58 $60
P=Pr(D<=Q)N-1(P)=QMean:1664, stdev=555
1919
Not Overly Sensitive
$-
$5,000
$10,000
$15,000
$20,000
$25,000
$30,000
$35,000
$40,000
$45,000
$50,000
$48 $50 $52 $54 $56 $58 $60
2100 2150
2200 2250
Differences are small
2020
Extend Idea
• Ship too little, you have to EXPEDITE the rest
• Ship Q
• If demand < Q– We sell demand and salvage (Q – demand)
• If demand > Q– We sell demand and expedite (demand – Q)
• What’s the strategy?
2121
Same idea
• Ignore profit from sales – that’s independent of Q
• Focus on salvage and expedite costs• Look at last item
– Chance we salvage it is P– Chance we expedite it is (1-P)
• Balance these costs– Unit salvage cost * P = Unit expedite cost (1-P)– P = expedite/(expedite + salvage)
2222
Safety Stock
• Protection against variability– Variability in demand and
– Variability in lead time
– Typically described as days of supply
– Should be described as standard deviations in lead time demand
– Example: BMW safety stock • For axles only protects against lead time variability
• For option parts protects against usage variability too
2323
Inventory
• Inventory On-hand
• Inventory Position: On-hand and on-order
2424
Continuous Review Basics
Time
Inve
ntor
y
Safety Stock
Reorder Point
Order placed
Lead Time
Actual Lead Time Demand
Avg LT Demand
On Hand
Position
Order Up to Level
EOQIf lead time is long, …
2525
Assumptions
• Fixed Order Cost
• Constant average demand
• Typically assume Normally distributed lead time demand
2626
Safety Stock Basics
• Lead time demand N(, )
• Typically Normal with – Average lead time demand – Standard Deviation in lead time demand
• Setting Safety Stock– Choose z from N(0,1) to get correct
probability that lead time demand exceeds z,– Safety stock is z
2727
Only Variability in Demand
• If Lead Times are reliable– Average Lead Time Demand
L * D
– Standard Deviation in lead time demand
L = LD
– Sqrt of Lead time * Standard Deviation in demand
– Units (Example)• L is the Lead Time in days, • D is the standard deviation in daily demand
Sq. Root because we are adding up L independent (daily)
demands.
2828
Implementation
• Inventory On-hand• Inventory Position: On-hand and on-order• When Inventory Position reaches a re-order point
(ROP), order the EOQ• This takes the Inventory Position to the Order-
Up-To Level: EOQ + ROP• That’s because review is continuous – we always
re-order at the ROP• Often called a (Q,r) policy (when inventory
reaches r, order Q)
2929
Example 3-7 page 61
Order cost $4,500 (e.g., transport cost)Cost of TV $250Holding cost 18%Lead time 2 weeks
Month September October November December January February March April May June July AugustSales 200 152 100 221 287 176 151 198 246 309 98 156
Avg Monthly Demand 191.17 UnitsStd Deviation in Monthly Demand 66.53 Units
Avg Weekly Demand 44.12 UnitsStd Deviation in weekly Demand 32.09 Units
Service Level 97% This is the fraction of time we expect to run out of stock before the next order arrives
z value 1.88 Standard DeviationsSafety Stock 85.34 UnitsEconomic Order Quantity 677 UnitsReorder Point 174 UnitsOrder-Up-To Level 851 UnitsAverage Inventory Position 512.2 The inventory position rises and falls between the Reorder Point and the Order-Up-To LevelAverage Inventory On Hand 424.0 The inventory on hand rises and falls between the Safety Stock and the Economic Order Quantity plus the Safety StockAverage Pipeline Inventory 88.23 The difference is the average pipeline inventory
Model assumes constant average monthly sales with
variability around that average: no seasonality or
growth in our sales
NormInv(0.97)√(L) D
ss+ L* AvgDROP+EOQ
[ROP+(EOQ+ROP)]/2
ss+EOQ/2
3030
Lead Time Variability
If Lead Times are variable• D = Average (daily) demand• D = Std. Dev. in (daily) demand• L = Average lead time (days)• sL = Std. Dev. in lead time (days)• Average lead time demand
– DL
• Std. Dev. in lead time demand– L = L2
D + D2 s2L
• Remember: Std. Dev. in lead time demand drives safety stock
3131
Levers to Pull
• Std. Dev. in lead time demand– L = L2
D + D2 s2L
Reduce Lead Time
Reduce Variability
in Lead Time
Reduce Variability in Demand
3232
Periodic Review
• Orders can only be triggered at certain times
• Examples– Batched transmissions (e.g., every night,
week, …)– Imposed by transportation (e.g., weekly
vessel)
• Examples of Continuous Review?
3333
No Ordering Cost
• Example?
• Cost typically viewed as – Inventory cost
• Service Level seen as a constraint– Probability of stock out in an order cycle
• Key Assumption: NO COST TO CHANGE ORDER SIZE
• Is this typically the case?
3434
Order-Up-To Policy
• Order-up-to Policy: At each period place an order to bring inventory position up to a level S
• What problem might we encounter?
3535
(S,s) Policy
• To avoid small orders
• In each period, if the inventory position is below s, place an order to bring it up to S.
3636
Order Up To Policy
Time
Sto
ck o
n ha
ndReorder Point
Order placed
Lead Time
Reorder Point
Target Inventory Position
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?
3737
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
3838
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 S – DT/2
Before an order is placed it is smaller by the demand in
the period
3939
Safety Stock in Periodic Review
• Probability of stock out is the probability demand in T+L exceeds 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.
4040
Safety Stock in Periodic Review
• Probability of stock out is the probability demand in T+L exceeds 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?
4141
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
4242
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 2 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
4343
Example Cont’d
Expected Demand in T+LD (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
4444
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
4545
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 975 928 937 On Hand Inventory 1,342 1,112 1,060 % Reduction 0% 17.1% 21.0%Average Inventory 1,552 1,322 1,270 % Reduction - 14.8% 18.2%
NormInv(0.9996) S-D(T+E[L])
DT/2+ssOHI+DE[L]
4646
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 1,708 1,701 % Reduction 0% 8.7% 9.1%Average Inventory 4,811 4,648 4,641 % Reduction 0.0% 3.4% 3.5%
Assume independence
4747
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%Average Inventory 4,811 4,628 4,566 % Reduction 0.0% 3.8% 5.1%
4848
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)
4949
Safety Lead Time as a quantity
0
100
200
300
400
500
600
700
Safety Lead Time: The next X days of forecasted demand
5050
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.
• Coverage: Inventory on hand should meet S+T days of forecasted demand
5151
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
5252
Safety Stock Basics
• n customers
• Each with lead time demand N(, )
• Individual safety stock levels– Choose z from N(0,1) to get correct
probability that lead time demand exceeds z,– Safety stock for each customer is z– Total safety stock is nz
5353
Safety Stock Basics
• Collective Lead time demand N(n, n)• This is true if their demands and lead times are
independent!• Collective safety stock is nz• Typically demands are negatively or positively
correlated• What happens to the collective safety stock if
demands are – positively correlated?– Negatively correlated?
5454
Risk Pooling Case 3.3 p 64Historical Data for Product A
1 2 3 4 5 6 7 8Massachusetts 33 45 37 38 55 30 18 58New Jersey 46 35 41 40 26 48 18 55Pooled 79 80 78 78 81 78 36 113
Average Std DevCoeff of
Var
Avg. Lead time
DemandSafety Stock
Reorder Point EOQ
Order Up To Level
Avg. Inventory
Massachusetts 39.25 13.18 0.34 39.25 24.78 64 132 196 91New Jersey 38.63 12.05 0.31 38.63 22.66 61 131 192 88Pooled 77.88 20.71 0.27 77.88 38.95 117 186 303 132
Historical Data for Product B1 2 3 4 5 6 7 8
Massachusetts 0 2 3 0 0 1 3 0New Jersey 2 4 0 0 3 1 0 0Pooled 2 6 3 0 3 2 3 0
Average Std DevCoeff of
Var
Avg. Lead time
DemandSafety Stock
Reorder Point EOQ
Order Up To Level
Avg. Inventory
Massachusetts 1.13 1.36 1.21 1.13 2.55 4 22 26 14New Jersey 1.25 1.58 1.26 1.25 2.97 4 24 28 15Pooled 2.38 1.92 0.81 2.38 3.62 6 32 38 20
Inventory ComparisonMassachusetts New Jersey Total Pooled Reduction
Product A 91 88 179 132 26%Product B 14 15 28 20 30%Total 105 103 207 152 27%
Week
Week
97%
ss+EOQ/2
5555
Risk Pooling Case 3.3 p 64Historical Data for Product A
1 2 3 4 5 6 7 8Massachusetts 33 45 37 38 55 30 18 58New Jersey 46 35 41 40 26 48 18 55Pooled 79 80 78 78 81 78 36 113
Average Std DevCoeff of
Var
Avg. Lead time
DemandSafety Stock
Reorder Point EOQ
Order Up To Level
Avg. Inventory
Massachusetts 39.25 13.18 0.34 39.25 24.78 64 132 196 91New Jersey 38.63 12.05 0.31 38.63 22.66 61 131 192 88Pooled 77.88 20.71 0.27 77.88 38.95 117 186 303 132
Historical Data for Product B1 2 3 4 5 6 7 8
Massachusetts 0 2 3 0 0 1 3 0New Jersey 2 4 0 0 3 1 0 0Pooled 2 6 3 0 3 2 3 0
Average Std DevCoeff of
Var
Avg. Lead time
DemandSafety Stock
Reorder Point EOQ
Order Up To Level
Avg. Inventory
Massachusetts 1.13 1.36 1.21 1.13 2.55 4 22 26 14New Jersey 1.25 1.58 1.26 1.25 2.97 4 24 28 15Pooled 2.38 1.92 0.81 2.38 3.62 6 32 38 20
Inventory ComparisonMassachusetts New Jersey Total Pooled Reduction
Product A 91 88 179 132 26%Product B 14 15 28 20 30%Total 105 103 207 152 27%
Week
Week
Pooling Inventory can reduce safety stock
The impact is less than the sqrt of 2 law
It predicts that if 2 DCs need 47 units then a single DC will need
33
The impact is greater than the sqrt of 2 law
It predicts that if 2 DCs need 5.5 units
then a single DC will need 4
5656
Inventory (Risk) Pooling
• Centralizing inventory can reduce safety stock
• Best results with high variability and uncorrelated or negatively correlated demands
• Postponement ~ risk pooling across products
5757
Next Time
• Read Mass Customization Article
• Read To Pull or Not To Pull by Spearman