AGIFORS1995-EPIM-QUILLINAN

1Expendable Parts IM

Expendable PartsInventory Management

atDelta Air Lines

by

John D. Quillinan

September 18, 1995

2Expendable Parts IM

Inventory Decisions

• When to order

• How much to order

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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

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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

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Total Cost

Total cost=

annual cycle-inventory cost+

cost of carrying safety stock+

cost of a stockout

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The Goal

Minimize total quantifiable cost

• Minimize annual cycle-inventory cost and cost of carrying safety stock

• Meet or exceed a customer service level

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Typical Inventory History

Back orders

Stock on hand

Stock on hand

Order quantity

Stock on order

Total available stock

Orderpoint

Time

Piec

es

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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)

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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)

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(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.

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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

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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.

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Cost Savings (75 parts)

TargetService

Level

AverageService

LevelCost

Savings

PercentCost

Savings92.00% 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%

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µ = 10.5 and σ = 39.6 −

Believe It! (or) Not!

SAS

The Art of Forecasting

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Forecast Models

8 Forecast Models Coded in Natural 2.Simple (equally-weighted) 6-month moving averageUnequally-weighted 6-month moving averageSimple 12-month moving averageExponential-smoothing (utilizing Trigg-Leach method)Double-exponential smoothingHolt-Winter’s multiplicative (seasonal) model with trendHolt-Winter’s multiplicative (seasonal) model w/o trendTime series -- simple linear regression against time

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Basic IM Terms

A = ordering (or setup) costs, in dollars per order lot

S = expected annual usage, pieces per yearr = carrying costv = actual cost, dollars per pieceQ = order quantityk = safety factorr = reorder point

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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)

18Expendable Parts IM

A Live Example

DPN: 012202134Station: ATLDescription: BULB, 28V, 600W, QUARTZ SEAL

Variable Description ValueA ordering cost 50.00$ S expected annual usage 9,527r carrying cost 18.33%v actual cost per unit $0.70

package ratio 100supplier unit of issue EA

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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

1000

0

1500

0

2000

0

2500

0

3000

0

3500

0

4000

0

4500

0

5000

0

5500

0

Q, Order quantity (pieces)

Annualcosts

Total CostCost to CarryCost to OrderMinimum

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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)

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Mathematical Solution

DPN: 012202134Station: ATLDescription: BULB, 28V, 600W, QUARTZ SEAL

Variable Description ValueA ordering cost 50.00$ S expected annual usage 9,527r carrying cost 18.33%v actual cost per unit $0.70

package ratio 100supplier unit of issue EA

Q economic order quantity 27,249optimal package quantity 272

Q | packaging optimal order quantity given package ratio 27,200

Legend: Yellow area contains f ield descriptionsGreen area for user entriesRed area is restricted and for displaying calculated values

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Lower Ordering Cost

DPN: 012202134Station: ATLDescription: BULB, 28V, 600W, QUARTZ SEAL

Variable Description ValueA ordering cost 25.00$ S expected annual usage 9,527r carrying cost 18.33%v actual cost per unit $0.70

package ratio 100supplier unit of issue EA

Q economic order quantity 19,268optimal 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.

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.

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Inventory versus Order Quantity

Average

Average

Stockon

hand

Time

Larger order quantities result in reduced annual ordering costs, but at the cost of carrying larger inventories

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Additional IM Terms

σ = standard deviation of lead-time forecasterrors

E[k] = partial expectationF[k] = cumulative probability functionP = fraction of demand expected to be filled

from stockp0 = target service level

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Reorder Point Logic

R = µLT + k × σ,

whereµLT = lead-time forecast.

µLT = ΣLTµt, t ≠ LT

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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)

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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.

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Slow Moving Items

In the case of slow-moving items (µLT < 10, on average), Laplace-distributed (exponential) forecast errors are assumed.

( )kQ P

=−

12 2 2 1

lnσ

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Another Example

Bulb, 28V, 250 W, Screw Terminal

Forecasting methodology selected: simple exponential smoothing

Standard deviation of forecast errors: 184.93Annual forecast: 7,894Unit price: $5.61Price per piece: $5.61Ordering cost: $50.00Carrying cost: 18.33%

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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, becauseE[k] = 0.142879 for k = 0.70 andE[k] = 0.140475 for k = 0.71

* first appeared in Decision Rules for Inventory Management by R.G. Brown, 1967.

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An Example continued ...

Golden Section Search gives usk = 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

33Expendable Parts IM

EFSSeptember 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.

34Expendable Parts IM

Tangible Benefits

• 113,000 expendable items stocked in ATL• $7,496,368 reduction of inventory held for

safety stock

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Intangible Benefits

• Economic Order Quantity model

• Multiple forecasting models

• Minimum sales and lot sizes quantities

• Price breaks or quantity discounts

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Future Enhancements

Automation of outlier editing of historical data

Intermittent (or lumpy) demand models