EOQ
Answers the Question “How Much to
Order?”
Assumptions:
Instantaneous production
Immediate delivery
Deterministic demand
Constant demand: D units/year
Constant setup cost: A $/setup
Independent products
Inventory
EOQ view of Inventory
Order Quantity Q
Time
Costs
• Setup Costs
– A $/setup
– How many setups if we make Q each time?
• Why not just make D units in one setup?
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
A Model
Lot Size or Order Quantity: Q units
Average Inventory Level: Q/2units
Annual Demand: D units/year
Order Frequency: every D/Q times per year
Average Variable Cost/Year:
TVC = h*Q/2 +A*D/Q
The EOQ
Use Calculus to find the value of Q that
minimizes TVC(Q)
Or...
Total Variable Cost
Holding Costs
Transaction Costs
TVC
TVC
Order Quantity
Total Variable Cost
Holding Costs
Transaction Costs
TVC
TVC
Order Quantity
Total Variable Cost
Holding Costs
Transaction Costs
TVC
TVC
Order Quantity
Total Variable Cost
Holding Costs
Transaction Costs
TVC
TVC
Order Quantity
The Economic Order
Quantity
h Q/2 = A D/Q
Q2 = 2 A D/h
Q = SQRT(2 A D/h)
CAVEAT: Make sure you use
commensurate units!
An Example
Raw Material X
Quarterly demand: 6,000 units
Cost per unit: ~ $25/unit
Holding Cost: say 10% per year
Transaction Cost: $100/order
EOQ = SQRT(2 CT D/ CI)
= SQRT(2 * 100 *
6,000/(0.025*25))
~ 1,385 units per shipment
Robustness
100%
Robustness of the EOQ
% error in A*D
80%
% error in Q
% error in TVC
60%
40%
20%
0%
-20%
-40%
-60%
Robustness
100%
Robustness
80%
% error in h
% error in Q
% error in TVC
60%
40%
20%
0%
-20%
-40%
-60%
EPQ Answers the Question “How Much to
Produce?”
Assumptions:
Instantaneous production
Constant production rate: P > D
units/year
Immediate delivery
Deterministic demand
Constant demand: D units/year
Constant setup cost: A$/setup
Independent products
Inventory
EPQ view of Inventory
Production Quantity Q
Max Inv. Level
Length of Prod. Run
Time
A Model
Lot Size or Production Quantity: Q units
Average Inventory Level
Production run lasts: Q/P
Inventory grows at rate: (P-Q)
So, max inventory is: (P-D)Q/P = (1-D/P)Q
Average inventory is: (1-D/P)Q/2
Order Frequency: every D/Q times per year
Average Variable Cost/Year:
TVC = h*(1-D/P)Q/2 +A*D/Q
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/Q
So, Q = SQRT(2 A D/h’)
= SQRT(2AD/h(1-D/P))
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
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
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.