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EOQ
Answers the Question “How Much to Order?”Assumptions:Instantaneous productionImmediate deliveryDeterministic demandConstant demand: D units/yearConstant setup cost: A $/setupIndependent products
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EOQ view of Inventory
Time
Inve
ntor
y Order Quantity Q
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Costs
• Setup Costs– A $/setup– How many setups if we make Q each time?
• Why not just make D units in one setup?
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Inventory Cost
Usually billed as a “holding cost” Essentially interest on the money tied
up in inventory h $/unit/year
Example: Holding 100 units for 6 months costs: ?
Inventory holding Cost h*Average Inventory
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A Model
Lot Size or Order Quantity: Q unitsAverage Inventory Level: Q/2unitsAnnual Demand: D units/yearOrder Frequency: every D/Q times per
yearAverage Variable Cost/Year: TVC = h*Q/2 +A*D/Q
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The EOQ
Use Calculus to find the value of Q that minimizes TVC(Q)
Or...
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Total Variable Cost
Order Quantity
TVC
Holding Costs
Transaction Costs
TVC
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Total Variable Cost
Order Quantity
TVC
Holding Costs
Transaction Costs
TVC
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Total Variable Cost
Order Quantity
TVC
Holding Costs
Transaction Costs
TVC
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Total Variable Cost
Order Quantity
TVC
Holding Costs
Transaction Costs
TVC
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The Economic Order Quantity
h Q/2 = A D/QQ2 = 2 A D/hQ = SQRT(2 A D/h)
CAVEAT: Make sure you use commensurate units!
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An Example
Raw Material X Quarterly demand: 6,000 units Cost per unit: ~ $25/unit Holding Cost: say 10% per year Transaction Cost: $100/order
EOQ = SQRT(2 CT D/ CI)
= SQRT(2 * 100 * 6,000/(0.025*25)) ~ 1,385 units per shipment
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Robustness of the EOQRobustness
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
% error in A*D
% error in Q
% error in TVC
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RobustnessRobustness
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
%error in h
% error in Q
% error in TVC
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EPQ Answers the Question “How Much to Produce?”Assumptions:Instantaneous production Constant production rate: P > D units/yearImmediate deliveryDeterministic demandConstant demand: D units/yearConstant setup cost: A$/setupIndependent products
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EPQ view of Inventory
Time
Inve
ntor
y
Production Quantity Q
Max Inv. Level
Length of Prod. Run
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A Model
Lot Size or Production Quantity: Q unitsAverage Inventory LevelProduction run lasts: Q/P Inventory grows at rate: (P-Q)So, max inventory is: (P-D)Q/P = (1-D/P)QAverage inventory is: (1-D/P)Q/2Order Frequency: every D/Q times per yearAverage Variable Cost/Year: TVC = h*(1-D/P)Q/2 +A*D/Q
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The EPQ
Use Calculus to find the value of Q that minimizes TVC(Q)
Or use the previous answer... TVC = h*(1-D/P)Q/2 +A*D/Q = h’Q/2 +A*D/QSo, Q = SQRT(2 A D/h’) = SQRT(2AD/h(1-D/P))
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An Example
Raw Material X Quarterly demand: 6,000 units Cost per unit: ~ $25/unit Holding Cost: say 10% per year Transaction Cost: $100/order Quarterly Production Rate: 8,000 units
EOQ = SQRT(2 CT D/ CI)
= SQRT(2*100*6,000/(0.025*25*(1-6/8)))
~ 2,771 units per run
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A Model
Divide the planning horizon into time buckets t = 1, 2, ..., T
Dt = units of demand in period t
ct = unit production cost in period t
At = setup cost in period t
ht = inventory holding cost in period t
Qt = the lot size in period t
It = units in inventory at the end of period t
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Heuristics
Lot-for-lot: Make what is required each period.
Fixed Order Quantity: Order the EOQ Period Order Quantity: Calculate the
EOQ, Q. Convert to order frequency: T = Q/D. Orders sized to last for time T.