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MIT Center for Transportation & Logistics CTL.SC1x - Supply Chain & Logistics Fundamentals Economic Order Quantity (EOQ)

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Page 1: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

MIT Center for Transportation & Logistics

CTL.SC1x - Supply Chain & Logistics Fundamentals

Economic Order Quantity

(EOQ)

Page 2: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Replenishment Model Assumptions •  Demand

n  Constant vs Variable n  Known vs Random n  Continuous vs Discrete

•  Lead Time n  Instantaneous n  Constant vs Variable n  Deterministic vs Stochastic n  Internally Replenished

•  Dependence of Items n  Independent n  Correlated n  Indentured

•  Review Time n  Continuous vs Periodic

•  Number of Locations n  One vs Multi vs Multi-Echelon

•  Capacity / Resources n  Unlimited n  Limited / Constrained

•  Discounts n  None n  All Units vs Incremental vs One Time

•  Excess Demand n  None n  All orders are backordered n  Lost orders n  Substitution

•  Perishability n  None n  Uniform with time n  Non-linear with time

•  Planning Horizon n  Single Period n  Finite Period n  Infinite

•  Number of Items n  One vs Many

•  Form of Product n  Single Stage n  Multi-Stage

2

Page 3: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Model Assumptions: EOQ •  Demand

n  Constant vs Variable n  Known vs Random n  Continuous vs Discrete

•  Lead Time n  Instantaneous n  Constant vs Variable n  Deterministic vs Stochastic n  Internally Replenished

•  Dependence of Items n  Independent n  Correlated n  Indentured

•  Review Time n  Continuous vs Periodic

•  Number of Locations n  One vs Multi vs Multi-Echelon

•  Capacity / Resources n  Unlimited n  Limited / Constrained

•  Discounts n  None n  All Units vs Incremental vs One Time

•  Excess Demand n  None n  All orders are backordered n  Lost orders n  Substitution

•  Perishability n  None n  Uniform with time n  Non-linear with time

•  Planning Horizon n  Single Period n  Finite Period n  Infinite

•  Number of Items n  One vs Many

•  Form of Product n  Single Stage n  Multi-Stage

3

Page 4: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Notation

D = Average Demand (units/time)

c = Variable (Purchase) Cost ($/unit) ct = Fixed Ordering Cost ($/order)

h = Carrying or Holding Charge ($/inventory $/time)

ce = ch = Excess Holding Cost ($/unit/time) Q = Replenishment Order Quantity (units/order)

T = Order Cycle Time (time/order)

N = 1/T = Orders per Time (order/time)

TRC(Q) = Total Relevant Cost ($/time)

TC(Q) = Total Cost ($/time)

4

Page 5: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 5

Inventory Charts

Inventory On

Hand

Demand

Model Assumptions (EOQ) •  Demand is uniform and deterministic •  Lead time is instantaneous (0) •  Total amount ordered is received

Order

Receive

Page 6: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 6

Time

IOH

(t)

Lot Sizing: Finding Optimal Policy In

vent

ory

On

Han

d

T

Q

Inventory Policy: •  Order Q* units every T* time periods •  Order Q* units when IOH=0

Page 7: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 7

What is the total cost? TC = Purchase + Order + Holding + Shortage

TRC(Q) = ct

DQ!

"#

$

%&+ ce

Q2

!

"#

$

%&

TC(Q) = cD+ ctDQ!

"#

$

%&+ ce

Q2

!

"#

$

%&+ csE Units Short'( )*

Which costs are relevant to the order quantity decision?

Page 8: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Building Intuition Example:

What is the optimal order quantity for a product with: w  Demand = 2,000 units/year w  Cost of placing an order = 500 $/order w  Cost of product = 50 $/unit w  Holding cost = 25% of unit cost per year w  Selling price of product = 75 $/unit

TRC(Q) = ctDQ!

"#

$

%&+ ce

Q2

!

"#

$

%&

$-

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

$10,000

$11,000

$12,000

100 150 200 250 300 350 400 450 500 550 600 650 700

To

tal R

elev

ant

Co

st (

$/y

ear)

Order Quantity (Q)

Total Relevant Costs vs Order Quantity

Ordering Costs Holding Costs TRC

8

Page 9: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Solving for EOQ Analytically

9

Page 10: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 10

Finding Q* - Optimal order quantity

dTRC[Q]dQ

=−ctDQ2

+ce2

d 2TRC[Q]d 2Q

=2ctDQ3

First Order Conditions Second Order Conditions

This value is positive for any positive values for ct, D, or Q. Thus, Q* is a global minimum

Q*= 2ctDce

TRC[Q]= ctDQ

+ceQ2

dTRC[Q]dQ

=−ctDQ2

+ce2= 0

ce2=ctDQ2

Q2 =2ctDce

Q*= 2ctDce

Page 11: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 11

Q*= 2ctDce

Finding T* - Time between replenishments

Recall that: Q= units / order D= units / year T= years / order N= 1/T = orders / year

T*= 2ctDce

T*= Q*D

T*= 1D!

"#

$

%&2ctDce

=2ctDD2ce

Be sure to put T* into units that make sense (days,

weeks, months, etc.) don’t leave it in years!

Page 12: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 12

TRC[Q]= ctDQ

+ceQ2

Q*= 2ctDce

Finding TRC* - Optimal Total Relevant costs

Remember: Total relevant cost ≠ Total cost.

Need to add purchase cost back in.

TRC Q*( ) = ctD2ctDce

+

ce2ctDce2

=ctD ce2ctD

+ce 2ctD

2 ce

=ctceD

2+ctceD

2=2 ctceD

2

TRC(Q*) = 2ctceD

TC(Q*) = cD+ 2ctceD

Page 13: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Step by Step Example: EOQ Find the optimal order quantity, inventory policy, and total relevant

costs for a product with the following characteristics: w  Demand = 2,000 units/year w  Cost of placing an order = 500 $/order w  Cost of product = 50 $/unit w  Holding cost = 25% of unit cost per year w  Selling price of product = 75 $/unit

13

Page 14: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 14

Page 15: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 15

Page 16: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ Sensitivity to Changes

16

Page 17: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 17

Finding Sensitivity of EOQ Suppose I use a “wrong” order quantity, Q. How much worse will my total relevant cost be using this “wrong” Q versus Q*?

TRC Q( )TRC Q*( )

=

ctDQ

+ceQ2

2ctceD

=ctD

Q 2ctceD+

ceQ2 2ctceD

=ctD

Q 2ce+

ceQ

2 2ctD

=22

!

"##

$

%&&ctD

Q 2ce+

ceQ

2 2ctD=

2ctD

2Q ce+

ceQ

2 2ctD

=12!

"#$

%&

2ctD

Q ce+

ceQ

2ctD

!

"

##

$

%

&&

TRC Q( )TRC Q*( )

=12!

"#$

%&Q*Q

+QQ*

!

"#

$

%&

Page 18: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ Sensitivity wrt Order Size

Q Ordering

Costs (ctD/Q)

Holding Costs

(ceQ/2) Total

Relevant Costs Q/Q* TRC/TRC*

800 $1,250 $5,000 $6,250 200% 125%

600 $1,667 $3,750 $5,417 150% 108.3%

Q*=400 $2,500 $2,500 $5,000 -- --

200 $5,000 $1,250 $6,250 50% 125%

20 $50,000 $125 $50,125 5% 1002.5%

Would you rather order Q>Q* or Q<Q*?

Previous example where: D= 2000 c=50 ce = 12.5 ct = 500

Page 19: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ Sensitivity wrt Order Size

100%

110%

120%

130%

140%

150%

160%

170%

180%

190%

200%

20% 30% 40% 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% 160% 170%

Rat

io o

f T

RC

(Q)

to

TR

C(Q

*)

Ratio of Order Quantity (Q) to optimal Order Quantity (Q*)

19

Page 20: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 20

EOQ Sensitivity wrt Demand •  How sensitive is the TRC to changes in actual demand? •  Notation:

n  DF = Forecasted Demand n  QF* = EOQ using Forecasted Demand n  DA = Actual Demand (what really occurred) n  QA* = EOQ calculated with Actual Demand

QA*

QF* =

2DAct

ce

2DFct

ce

= DA

DF

QF* = 2DFct

ce

TRC QF*( )

TRC QA*( )

= 12

DADF

+DFDA

!

"##

$

%&&

QA* = 2DAct

ceTRC Q( )TRC Q*( )

=12!

"#$

%&Q*Q

+QQ*

!

"#

$

%&TRC QF

*( )TRC QA

*( )=12!

"#$

%&QA*

QF*+QF*

QA*

!

"##

$

%&&

Page 21: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ Sensitivity wrt Demand

D E Q*/Q' TC'/TC*200 0.10 0.32 1.74

1,000 0.50 0.71 1.06 1,500 0.75 0.87 1.01 1,800 0.90 0.95 1.00 2,000 1.00 1.00 1.00 3,000 1.50 1.22 1.02 4,000 2.00 1.41 1.06

20,000 10.00 3.16 1.74

D/D’

How much will TRC change if I change ct, c, or ce?

DACTUAL DA/DF Q*A/Q*F TRC*A/TRC*F

Previous example where: D= 2000 c=50 ce = 12.5 ct = 500

21

Page 22: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ – Very Robust Solution

$-

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

$8,000

$9,000

$10,000

$11,000

$12,000

100 150 200 250 300 350 400 450 500 550 600 650 700

To

tal R

elev

ant

Co

st (

$/y

ear)

Order Quantity (Q)

Total Relevant Costs vs Order Quantity

Ordering Costs

Holding Costs

TRC

22

Page 23: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ Sensitivity to Order Cycle Time

23

Page 24: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

EOQ Sensitivity wrt Order Cycle Time •  How sensitive is TRC to T*?

n  Why do we care and why is “time” different? n  How do I find the “best” T that is also practical?

•  Previous Example: n  Q* = 400 units for TRC* = $5,000 per year n  T* = 0.2 years or 73 days or 10.4 weeks n  What if I order weekly? monthly? Other?

w  TRC(1)/TRC(10.4) = .5(1/10.4 + 10.4/1) = 5.24 or 524% higher costs w  TRC(4)/TRC(10.4) = 1.49 or 50% higher costs

( ) 1 *( *) 2 *

TRC T T TTRC T T T

! "= +# $% &

24

Page 25: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Power of Two Policies n  Order in intervals of powers of two n  Select a realistic base period, TBase (day, week, month) n  Guarantees that TRC will be within 6% of optimal!

n  We want to find the smallest k where 2k ≤ T* ≤ 2k+1 w  20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, . . . w  . . . So for T*=10.4 weeks . . . k=3 so we order every 8 weeks

20

21

22

23

24

T*=10.4 weeks T*=10.4 weeks T*=10.4 weeks T*=10.4 weeks

EOQ Sensitivity wrt Order Cycle Time

25

Page 26: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 26

TRC(2k )TRC(T*)

≤1212+21

"

#$$

%

&'' ≈1.06

A Power of Two time interval is guaranteed to be within 6% of the TRC* with the optimal time

interval.

T *2≤ 2k ≤ 2 T *

To find a practical T: 1. Calculate T* 2. Pick a base time period (day, week, etc.) 3. Find the lowest value of k that satisfies:

2k-1/2 2k 2k+1/2 2k+1

T 2T

EOQ Sensitivity wrt Order Cycle Time

26

TRC(T )TRC(T*)

=12TT *

+T *T

!

"#

$

%&=122TT *

+T *2T

!

"#

$

%&

2T *−T *2T

=2T −TT *

TT *

=12

and T *T

= 2

T *2T

=TT *

Page 27: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Step by Step Example: Power of Two Find the optimal order quantity, optimal order cycle time, and TRC for

a product with the following characteristics: w  Demand = 25,000 units/year w  Cost of placing an order = 750 $/order w  Cost of product = 25.50 $/unit w  Holding cost = 15% of unit cost per year

27

Q* =3,131 units/order T*=0.125 years/order TRC*=$11,976 $/year

Assume that we want to order on a power of two weekly schedule, what is my new T and Q and what is my percent error from TRC*?

Page 28: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 28

Step by Step Example: Power of Two

Page 29: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Key Points from Lesson

29

Page 30: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)

Key Points – The EOQ model . . . •  has restrictive assumptions, but is still widely used •  trades fixed (order) and variable (holding) costs •  is relatively insensitive to changes in . . .

n  rounding of order quantities (Q) n  rounding of order cycle time (T) n  errors in forecasting (D) n  errors in cost parameters (c, ct, h, ce)

•  helps focus management on process improvements n  How do I lower ct? n  How do I lower c? n  How do I lower h?

•  is a good starting point, but not always a good end point.

30

Page 31: Economic Order Quantity (EOQ) - edX · CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) Replenishment Model Assumptions • Demand Constant

MIT Center for Transportation & Logistics

CTL.SC1x - Supply Chain & Logistics Fundamentals

Questions, Comments, Suggestions? Use the Discussion!