EOQ

Answers the Question “How Much to Order?”Assumptions:Instantaneous productionImmediate deliveryDeterministic demandConstant demand: D units/yearConstant setup cost: A $/setupIndependent products

EOQ view of Inventory

Time

Inve

ntor

y Order Quantity Q

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 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

The EOQ

Use Calculus to find the value of Q that minimizes TVC(Q)

Or...

Total Variable Cost

Order Quantity

TVC

Holding Costs

Transaction Costs

TVC

Total Variable Cost

Order Quantity

TVC

Holding Costs

Transaction Costs

TVC

Total Variable Cost

Order Quantity

TVC

Holding Costs

Transaction Costs

TVC

Total Variable Cost

Order Quantity

TVC

Holding Costs

Transaction Costs

TVC

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!

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 of the EOQRobustness

-60%

-40%

-20%

0%

20%

40%

60%

80%

100%

% error in A*D

% error in Q

% error in TVC

RobustnessRobustness

-60%

-40%

-20%

0%

20%

40%

60%

80%

100%

%error in h

% error in Q

% error in TVC

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

EPQ view of Inventory

Time

Inve

ntor

y

Production Quantity Q

Max Inv. Level

Length of Prod. Run

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

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))

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.