This document describes Delta Air Lines' expendable parts inventory management system. It discusses key considerations like when to order parts and how much to order. It also outlines the various costs involved like ordering costs, carrying costs, and stockout costs. The goal is to minimize total quantifiable costs while meeting a target customer service level. The system uses a reorder point-reorder quantity model and safety stocks to prevent stockouts. It provides an example of how the economic order quantity is calculated and how safety stocks are determined. The system was implemented in 1996 and provided tangible cost savings and intangible benefits in optimizing inventory levels.

1. 1Expendable Parts IM
Expendable Parts
Inventory Management
at
Delta Air Lines
by
John D. Quillinan
September 18, 1995

2. 2Expendable Parts IM
Inventory Decisions
• When to order
• How much to order

3. 3Expendable Parts IM
Considerations
• The time when the order is released
– a shortage before the material can arrive
– average investment in inventories
• The quantity order at one time
– number of replenishment orders required
annually, and cost of processing them
– average investment in inventories

4. 4Expendable Parts IM
Inventory Costs
• Cost of Ordering
• Cost of Carrying Working Stock
• Acquisition Cost of Safety Stock
• Cost of Carrying Safety Stock
• Cost of Not Having Parts On Hand

5. 5Expendable Parts IM
Total Cost
Total cost
=
annual cycle-inventory cost
+
cost of carrying safety stock
+
cost of a stockout

6. 6Expendable Parts IM
The Goal
Minimize total quantifiable cost
• Minimize annual cycle-inventory cost and cost
of carrying safety stock
• Meet or exceed a customer service level

7. 7Expendable Parts IM
Typical Inventory History
Back orders
Stock on hand
Stock on hand
Order
quantity
Stock on order
Total available stock
Order
point
Time
Pieces

8. 8Expendable Parts IM
Terminology
• Demand - historical consumption
• EOQ - Economic Order (Reorder) Quantity
• Buffer - safety stock
• Forecast - estimate of consumption in a given
period t+n produced at time t
• Reorder Point - inventory level at which an
order is placed
(usage forecasted over lead time plus safety stock)

9. 9Expendable Parts IM
Demand Defined
Historical Consumption (constrained demand) is
defined by Delta Part Number and Station as
normal issue
+ station issue
- shop credit quantity
- scrap quantity (removed 12/29/95)

10. 10Expendable Parts IM
(r,Q) Model
• Reorder Point-Reorder Quantity Model
• Reorder Quantity, Q, is determined using the
EOQ (economic order quantity) model.
• The reorder point, r, is chosen to protect us
or provide a specified level of service during
the lead time. If demand is known with
certainty, then r is set equal to the demand
during the lead time.

11. 11Expendable Parts IM
Service Level
Service Level may be defined in terms of
• Time
– a level provided by simply carrying x periods of
supply on-hand
• Stockouts
– fraction of time that no stockout will occur, or the
probability of no stockout
• Back Orders
– fraction of demand expected to be filled from
stock, or the probability of no back order

12. 12Expendable Parts IM
A Little History
On December 6, 1995, a prototype program was
presented to Material Services.
The DT Prototype predicts future consumption and
safety stock levels for a user-specified Delta part
number, station and service level for the last
twelve months plus one month into the future.
Over the next couple months, an inventory decision
model capable of handling price breaks or
quantity discounts, and packaging and lot sizes,
was incorporated in the prototype.

13. 13Expendable Parts IM
Cost Savings (75 parts)
Target
Service
Level
Average
Service
Level
Cost
Savings
Percent
Cost
Savings
92.00% 94.25% $52,985 28.08%
92.66% 94.54% $49,959 26.48%
95.00% 96.03% $31,769 16.84%
96.00% 96.59% $25,150 13.33%
97.00% 97.33% $15,468 8.20%

14. 14Expendable Parts IM
µ = 10.5 and
σ = 39.6 −
Believe It! (or)
Not!
SAS
The Art of Forecasting

15. 15Expendable Parts IM
Forecast Models
8 Forecast Models Coded in Natural 2.
Simple (equally-weighted) 6-month moving average
Unequally-weighted 6-month moving average
Simple 12-month moving average
Exponential-smoothing (utilizing Trigg-Leach method)
Double-exponential smoothing
Holt-Winter’s multiplicative (seasonal) model with trend
Holt-Winter’s multiplicative (seasonal) model w/o trend
Time series -- simple linear regression against time

16. 16Expendable Parts IM
Basic IM Terms
A = ordering (or setup) costs, in dollars per
order lot
S = expected annual usage, pieces per year
r = carrying cost
v = actual cost, dollars per piece
Q = order quantity
k = safety factor
r = reorder point

17. 17Expendable Parts IM
Cycle-Inventory Costs
• Annual cost of ordering
(S / Q) × A
• Annual cost of carrying inventory
(Q / 2) × r × v
• Total annual cycle-inventory cost
((Q / 2) × r × v) + ((S / Q) × A)

18. 18Expendable Parts IM
A Live Example
DPN: 012202134
Station: ATL
Description: BULB, 28V, 600W, QUARTZ SEAL
Variable Description Value
A ordering cost 50.00$
S expected annual usage 9,527
r carrying cost 18.33%
v actual cost per unit $0.70
package ratio 100
supplier unit of issue EA

19. 19Expendable Parts IM
Graphical Solution
Inventory Costs
$0.00
$10.00
$20.00
$30.00
$40.00
$50.00
$60.00
$70.00
$80.00
$90.00
$100.00
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
55000
Q, Order quantity (pieces)
Annual
costs
Total Cost
Cost to Carry
Cost to Order
Minimum

20. 20Expendable Parts IM
EOQ Model
• Annual cycle-inventory cost =
((Q / 2) × r × v) + ((S / Q) × A)
• Take derivative of cost with respect to Q, set
equal to 0, and solve for Q.
d{((Q / 2) × r × v) + ((S / Q) × A)}/dQ = 0
((1 / 2) × r × v) + ((−S / Q2) × A) = 0
1/ Q2 = (r × v) / (2 × A × S)
Q = √ (2 × A × S) / (r × v)

21. 21Expendable Parts IM
Mathematical Solution
DPN: 012202134
Station: ATL
Description: BULB, 28V, 600W, QUARTZ SEAL
Variable Description Value
A ordering cost 50.00$
S expected annual usage 9,527
r carrying cost 18.33%
v actual cost per unit $0.70
package ratio 100
supplier unit of issue EA
Q economic order quantity 27,249
optimal package quantity 272
Q | packaging optimal order quantity given package ratio 27,200
Legend: Yellow area contains field descriptions
Green area for user entries
Red area is restricted and for displaying calculated values

22. 22Expendable Parts IM
Lower Ordering Cost
DPN: 012202134
Station: ATL
Description: BULB, 28V, 600W, QUARTZ SEAL
Variable Description Value
A ordering cost 25.00$
S expected annual usage 9,527
r carrying cost 18.33%
v actual cost per unit $0.70
package ratio 100
supplier unit of issue EA
Q economic order quantity 19,268
optimal package quantity 193
Q | packaging optimal order quantity given package ratio 19,300
Halfing our ordering cost reduces our order quantity
from 272 to 193 packages.

23. 23Expendable Parts IM
Some Observations
• As the cost of ordering increases, the order
quantity also increases, while the number of
replenishment orders decreases.
• As the cost of carrying inventory increases,
the order quantity decreases, while the
number of replenishment orders increases.
• Both cost drivers affect the average stock on
hand.

24. 24Expendable Parts IM
Inventory versus Order Quantity
Average
Average
Stock
on
hand
Time
Larger order quantities result in reduced annual
ordering costs, but at the cost of carrying larger
inventories

25. 25Expendable Parts IM
Additional IM Terms
σ = standard deviation of lead-time forecast
errors
E[k] = partial expectation
F[k] = cumulative probability function
P = fraction of demand expected to be filled
from stock
p0 = target service level

26. 26Expendable Parts IM
Reorder Point Logic
R = µLT + k × σ,
where
µLT = lead-time forecast.
µLT = Σ
LT
µt, t ≠ LT

27. 27Expendable Parts IM
Setting Safety Stocks
Probability of no back orders is denoted by:
P = ( S − σ × E[k] × S / Q) / S
= 1 − E[k] × σ / Q
Expected quantity short (or partial expectation):
E[k] = p{k} − k × F(k)
p{k} − k × F(k) = Q / σ × (1 − P)

28. 28Expendable Parts IM
Normally Distributed Demand
A non-linear line search was used to find the
value of k whose E[k] was closest to {Q / σ ×
(1 − p0)}
The method for setting safety stocks was used
for expendable parts having an average lead-
time forecast ≥ 10, and assumes the demand
and forecast errors can be represented by a
normal (Gaussian) distribution.

29. 29Expendable Parts IM
Slow Moving Items
In the case of slow-moving items (µLT < 10, on
average), Laplace-distributed (exponential)
forecast errors are assumed.
( )
k
Q P
=
−








1
2 2 2 1
ln
σ

30. 30Expendable Parts IM
Another Example
Bulb, 28V, 250 W, Screw Terminal
Forecasting methodology selected:
simple exponential smoothing
Standard deviation of forecast errors: 184.93
Annual forecast: 7,894
Unit price: $5.61
Price per piece: $5.61
Ordering cost: $50.00
Carrying cost: 18.33%

31. 31Expendable Parts IM
An Example continued ...
Q = √ (2 × A × S) / (r × v)
= 877
Target E[k]= Q / σ × (1 − p0)
= 877 / 184.93 × (1− 0.97) = 0.14277
Tables in the literature* tell us that k must fall
between 0.70 and 0.71, because
E[k] = 0.142879 for k = 0.70 and
E[k] = 0.140475 for k = 0.71
* first appeared in Decision Rules for Inventory Management by R.G. Brown, 1967.

32. 32Expendable Parts IM
An Example continued ...
Golden Section Search gives us
k = 0.7025
B = k × σ
≈ 130
R = µLT + k × σ
= 788
P = 1 − (0.311 − 0.7025 × 0.7588) × 184.93 / 877
= 0.969999 ≈ 0.97

33. 33Expendable Parts IM
EFS
September 1995 - TransQuest began documentation
of old system.
December 1995 - All of the prototype modules
converted into structured-format subprograms and
parameter and local data areas replaced the
specification of parameters lists within calling
programs.
January 1996 - TransQuest worked on development
of utility programs.
February 1996 - EFS loaded in test; user acceptance
testing begins.
March 30, 1996 - EFS loaded in production.

34. 34Expendable Parts IM
Tangible Benefits
• 113,000 expendable items stocked in ATL
• $7,496,368 reduction of inventory held for
safety stock

35. 35Expendable Parts IM
Intangible Benefits
• Economic Order Quantity model
• Multiple forecasting models
• Minimum sales and lot sizes quantities
• Price breaks or quantity discounts

36. 36Expendable Parts IM
Future Enhancements
Automation of outlier editing of historical data
Intermittent (or lumpy) demand models