economic order quantity (eoq) - edx · ctl.sc1x - supply chain and logistics fundamentals lesson:...
TRANSCRIPT
MIT Center for Transportation & Logistics
CTL.SC1x - Supply Chain & Logistics Fundamentals
Economic Order Quantity
(EOQ)
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
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
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
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
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
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?
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)
Solving for EOQ Analytically
9
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
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!
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
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 14
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 15
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)
EOQ Sensitivity to Changes
16
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*
!
"#
$
%&
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
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
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*
!
"##
$
%&&
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
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)
EOQ Sensitivity to Order Cycle Time
23
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
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
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 *
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*?
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ) 28
Step by Step Example: Power of Two
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Economic Order Quantity (EOQ)
Key Points from Lesson
29
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
MIT Center for Transportation & Logistics
CTL.SC1x - Supply Chain & Logistics Fundamentals
Questions, Comments, Suggestions? Use the Discussion!