Extensions to EOQ Model - edX · If P=1.5D then EPQ = √3(EOQ) 24 . CTL.SC1x - Supply Chain and...

MIT Center for Transportation & Logistics

CTL.SC1x -Supply Chain & Logistics Fundamentals

Extensions to EOQ Model

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Extensions to EOQ Model

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

2


Non-Instantaneous Leadtime

3


Model Assumptions: EOQ w/ Lead Time •  Demand













4


Extensions: Leadtime > 0

•  Order Leadtime n  Positive (nonzero) n  Deterministic

Inventory On

Hand

Receive

Order

Demand

L = Order Leadtime

Inventory On Order or Pipeline Inventory

Inventory On Hand

5


Extensions: Leadtime > 0

•  What is the average amount of pipeline inventory? n  Cycle length = T* = Q*/D

n  Pipeline inventory level changes over the replenishment cycle w  In transit: Pipeline Inventory = Q* which happens (L/T*) of the cycle w  Not in transit: Pipeline Inventory = 0 which happens (1- L/T*) of the cycle w  Average Pipeline Inventory

= Q*(L/T*) + 0(1 – L/T*) = Q*(L/T*) = Q*(DL/Q*) = DL

•  What is the average inventory in the system? n  Cycle Stock + Pipeline = Q/2 + DL n  Need to consider who owns it and when (on-hand vs. system)

6

Inventory On

Hand

Inventory Position


Extensions: Leadtime > 0 •  What is my new TC function?

•  What is my new Q*?

•  What is my new optimal policy?

7

TC(Q) = cD+ ctDQ!

"#

$

%&+ ce

Q2+DL

!

"#

$

%&+ csE Units Short'( )*

Q*= 2ctDce

IP

DL

L

Inventory Policy: •  Order Q* units when IP=DL •  Order Q* units every T* time periods

IOH

time in

vent

ory


Quantity Discounts

8


Model Assumptions: EOQ w/ Discounts •  Demand













9


Extensions: Discounts

•  All Units Discount n  Discount applies to all units purchased if

total amount exceeds the break point quantity

•  Incremental Discount n  Discount applies only to the quantity

purchased that exceeds the break point quantity

•  One Time Only Discount n  A one time only discount applies to all

units you order right now (no quantity minimum or limit)

Q Q

cD

Q

cD

t

IOH

Q1 Q2

Q1 Q2

10


Quantity Discounts: All Units

11


Extensions: All Units Discounts

TRC = Dc0 + ct D

Q + c0hQ

20 ≤Q <Q1

Dc1 + ct DQ

+ c1hQ2

Q1 ≤Q

"

#

$$

%

$$

$98,000

$100,000

$102,000

$104,000

$106,000

$108,000

$110,000

$112,000

100 200 300 400 500 600 700 800 900 1000Order Quantity (Lot Size)

Tota

l Cos

t

TC2 TC1

Qb

c =c0 0 ≤Q <Q1c1 Q1 ≤Q

"#$

%$

Where c0 = Base unit price Qi = Volume break quantity i di = Percent discount (0 to 1)

in break i ci = Unit price under discount i = c0(1-di) Typically, Q* < Q1

but what if Q* > Q1?

Two Cases to Examine . . .

12

Q1


Extensions: All Units Discounts Procedure:

1. Find Qc0* and Qc1* 2. If Qc1*≥Q1 then pick Qc1*

Otherwise, go to 3 3. Solve for TRC(Qc0*) and TRC(Q1)

If TRC(Qc0*) < TRC(Q1) then pick Qc0* Otherwise, pick Q1

Example: D=2000 Units/yr h=.25 ct=$500 c0 = $50 Discount of 15% off if Q≥800

13

Example: Step 1: Qc0* = 400 and Qc1*= 434 Step 2: Qc1*= 434 < Q1=800 go to Step 3 Step 3: TRC(Qc0*) = 105,000 $/year TRC(Q1) = (42.5)(2000)+(500)(2000)/800+(42.5)(.25)(800/2) = 90,500 $/yr TRC(Qc0*) > TRC(Q1) so take discount at breakpoint quantity (Q1)


Quantity Discounts: Incremental

14


Extensions: Incremental Discounts •  Discount i only applies to quantity purchased above breakpoint i •  Trade-off between lower purchase cost and higher carrying costs •  Cost of units ordered below breakpoint i are treated as new Fixed Cost (Fi)

Units

Tota

l Pu

rcha

se

Cost

c0 c1 c2

Q1 Q2 (c0-c1)Q1

c1Q1

c0Q1

15

F0 = 0

Fi = Fi−1 + ci−1 − ci( )Qi

Q*i =2D ct + Fi( )

hci


Extensions: Incremental Discounts

•  Procedure 1. Find Fixed Cost per breakpoint, Fi, for each break 2. Find Qi* for each range – including the Fi 3.  If Qi* is not within allowable range, go to next i

Otherwise, find TRCi using effective cost per unit, cie

4. Pick Qi* with lowest TRC

cie = ci +

FiQi

Q*i =2D ct + Fi( )

hci

16

F0 = 0

Fi = Fi−1 + ci−1 − ci( )Qi


Quantity Discounts: One Time Buy

17


Extensions: One Time Discount Suppose you are offered a One Time deal! Should you take it? cg = One time good deal purchase price ($/unit) Qg = One time good deal order quantity (units) TCsp=TC over time covered by special purchase ($)

18

t

IOH


Extensions: One Time Discount •  Compare Options: Normal Price (c) vs. Special Price (cg)

n  Find TC for normal price

TC = (CycleTime)(TC* + PurchaseCost)

TC =QgD

!

"##

$

%&& 2cthcD +

QgD

!

"##

$

%&&cD

Savings =TC −TCSP

=QgD

"

#$$

%

&'' 2cthcD +

QgD

"

#$$

%

&''cD

"

#$$

%

&''− cgQg + hcg

Qg2

"

#$$

%

&''QgD

"

#$$

%

&''+ ct

"

#$$

%

&''

n  Find the Savings (TC-TCSP)

19


Extensions: One Time Discount

n  Finding 1st and 2nd order conditions (Maximize Savings)

d 2Sd 2 (Qg )

= −2hcg2D

!

"##

$

%&&< 0

dSd(Qg )

= 0 = 1D!

"#

$

%& 2cthcD + c− cg( )−

2hcgQg2D

!

"##

$

%&&

Qg* =

DDhcg

!

"##

$

%&& 2cthcD +

D c− cg( )hcg

Qg* =Q*ch+D c− cg( )

hcg

n  So that the Optimal Quantity to buy is

n  Cleaning this up gives:

20


Finite Replenishment Economic Production Quantity (EPQ)

21


Model Assumptions: EPQ •  Demand













22


Extensions: Finite Replenishment •  What if inventory becomes available at a rate

of P units/time rather than all at one time? 1.  How long does it take to produce Q units? 2.  During that time, what is the accumulation rate? 3.  What is the most inventory I will have on hand? 4.  What is my average IOH? 5.  How does Q* change?

Maximum Level of

Inventory Q(1-D/P)

Accumulation Slope = P-D

Q/P

Demand Slope = -D

Q

23

Production Slope = P

Avg. IOH = (1-D/P)Q/2


Extensions: Finite Replenishment

TRC[Q]= ctDQ

+

Q 1− DP

"

#$

%

&'hc

2

EPQ = 2ctD

hc 1− DP

"

#$

%

&'

=EOQ

1− DP

"

#$

%

&'

•  The IOH is reduced which leads to a higher order quantity.

•  EPQ can be related to EOQ n  If Pèinfinity then EPQ≈EOQ n  If P=2D then EPQ = √2(EOQ) n  If P=1.5D then EPQ = √3(EOQ)

24


Key Points from Lesson

25


Key Points

•  EOQ is a good place to start for most analysis and can be extended to cover many variations: n  Non-zero lead times

w  No effect on Q* w  Monitor Inventory Position vs Inventory on Hand w  Changes timing of replenishment (IP≤DL)

n  Discounts (All Units, Incremental, One Time Buy) w  Common in practice (economies of scale) w  Purchase price (c) becomes relevant w  Need to estimate costs at breakpoints & compare

n  Finite replenishment systems w  Items become available incrementally over time w  Lowers holding costs and leads to higher EPQ

26

MIT Center for Transportation & Logistics

CTL.SC1x -Supply Chain & Logistics Fundamentals

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Extensions to EOQ Model - edX · If P=1.5D then EPQ = √3(EOQ) 24 . CTL.SC1x - Supply Chain and...

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