Transportation Procurement in the Digital Age...Discounting All Units ... Transportation Procurement in the Digital Age Author: Yossi Created Date: 6/29/2004 11:10:57 AM

Inventory Management IILot Sizing

Caplice

Lecture 5ESD.260 Fall 2003

© Chris Caplice, MIT2MIT Center for Transportation & Logistics – ESD.260

Outline

EOQ ModelSensitivity AnalysisDiscounting

All UnitsIncrementalOne Time Special


Assumptions: Basic EOQ ModelDemandConstant vs VariableKnown vs RandomContinuous vs Discrete

Lead timeInstantaneousConstant or Variable (deterministic/stochastic)

Dependence of itemsIndependentCorrelatedIndentured

Review TimeContinuous vs Periodic

Number of EchelonsOne vs Many

Capacity / ResourcesUnlimited vs Limited

DiscountsNoneAll Units or Incremental

Excess DemandNoneAll orders are backorderedLost ordersSubstitution

PerishabilityNoneUniform with time

Planning HorizonSingle PeriodFinite PeriodInfinite

Number of ItemsOneMany


Notation

TC = Total Cost (dollar/time)

D = Average Demand (units/time)

Co = Ordering Cost (dollar/order)

Ch = Holding Cost (dollars/dollars held/time)

Cp = Purchase Cost (dollars/unit)

Q = Order Quantity (units/order)

T = Order Cycle Time (time/order)


Lot Sizing: Many Potential Policies

I(t)

TimeT

Q

Objective: Pick the policy with the lowest total cost

Inve

ntor

y O

n H

and


What is the total cost?

TC = Purchase + Order + Holding + Shortage Cost

Which costs are relevant? Purchase Costs Ordering CostsHolding CostsShortage Costs

( )2o h p

D QTC Q C C CQ

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠


Economic Order Quantity (EOQ)

2][

QCCQ

DCQTC pho +=

2* o

h p

DCQC C

= * 2 o h pTC DC C C=


Simple EOQ Example

Annual demand of widgets is 2,000. The cost of ordering is $500. Widgets are procured for $50 each and are sold for $95 each. Holding cost is estimated to be 25%.

What is the EOQ, Total Cost, and Cycle Time?

Q* = 400 unitsTC* = $5,000 / yearT* = 0.20 years or 73 days

Annual Cost vs. Order Quantity

$-

$5,000

$10,000

$15,000

$20,000

$25,000

50 150

250

350

450

550

650

750

850

950

1050

1150

1250

1350

1450

1550

1650

1750

1850

1950

Lot Size

Ann

ual C

ost

Total Cost versus Lot (Order) Size


The Effect of Non-Optimal Q

Q DCo/Q ChCpQ/2 TC2000 $500 $12,500 $13,000500 $2,000 $3,125 $5,125400 $2,500 $2,500 $5,000200 $5,000 $1,250 $6,25020 $50,000 $125 $50,125

So, how sensitive is TC to Q?


Sensitivity of EOQ

How does TC change with Q?

*

*

* *

( )2

( ) 2

( ) 1( ) 2

o p h

o p h

D QTC Q C C CQ

TC Q C C C D

TC Q Q QTC Q Q Q

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

=

⎡ ⎤= +⎢ ⎥

⎣ ⎦


Sensitivity of EOQ

Sensitivity TC wrt Q

90%110%130%150%170%190%

50%

63%

75%

88%

100%

113%

125%

138%

150%

163%

175%

188%

200%

213%

225%

238%

250%

263%

Q/Q*

TC/T

C*


Sensitivity of EOQ

What if our estimate of demand was off?E = Actual Demand / Estimated DemandQ’ = Estimated EOQQ* = Actual EOQ, given actual demand that occurred

D ActualE = = D Estimate′ CC

CD2 = Qph

o′′

E 2E + 1 =

TCTC

*

'E =

QQ

'*


Sensitivity of EOQ to Parameter Error

So, for D’ = 2000 units/year, if the actual demand was …

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


Insights from EOQ

There is a direct trade off between lot size and total inventoryTotal cost is relatively insensitive to changes

Very robust with respect to changes in:Q – rounding of order quantitiesD – errors in forecastingCh, Co, Cp – errors in cost parameters

Thus, EOQ is widely used despite its highly restrictive assumptions


Introduce Discounts to Lot Sizing

Types of discountsAll units discountIncremental discountOne time only discount

How will different discounting strategies impact your lot sizing decision?


All Units Discount

Unit Price [Cpi ]

Price Break Quantity [PBQI]

$50.00 0

$45.00 500

$40.00 1000


All Units Discount

Need to introduce purchase cost into TC function

2QCC +

QDC + DC =] CTC[Q, piho

pipi


All Units Discount

Annual Costs

$80,000

$85,000

$90,000

$95,000

$100,000

$105,000

$110,000

$115,000

100

300

500

700

900

1100

1300

1500

1700

1900

Lot Size

Dol

lars


All Units Discount

Annual Costs

$80,000

$85,000

$90,000

$95,000

$100,000

$105,000

$110,000

$115,000

100

300

500

700

900

1100

1300

1500

1700

1900

Lot Size

Dol

lars


All Units Discount: Method

1 Cpi $40.00 $45.00 $50.00

2 PBQ 1000 500 0

3 EOQ[Cpi] 447 421 400

4 Qpi 1000 500 400

5 DCpi $80,000 $90,000 $100,000

6 CoD/Qpi $1,000 $2,000 $2,500

7 ChCpiQpi/2 $5,000 $2,812 $2,500

8 TC[Qpi] $86,000 $94,812 $105,000

Same Example: D=2000 Units/yrCh=.25Co=$500

Cpi Price Breaks: $50 for 0 to <500 units$45 for 500 to <1000 units$40 for 1000+ units


Incremental Discount



Annual Cost

$80,000$90,000

$100,000$110,000$120,000$130,000$140,000$150,000

100

300

500

700

900

1100

1300

1500

1700

1900

2100

2300

2500

2700

2900

Lot Size

$/ye

ar



Index i=3 i=2 i=1

1 Cpi $40.00 $45.00 $50.00

2 PBQi 1000 500 0

3 Fi $7500 $2500 $0

4 EOQ[Cpi] 1789 1033 400

5 Qpi 1789 400

6 Cpe $44.19 $50.00

7 DCpe $88,384.57 $100,000

8 CoD/Qpi $558.97 $2,500

9 (Ch C pe Qpi)/2 $9,882.50 $2,500

10 TC[Qpi] $98,826.04 $105,000


One Time Discount


Let, Cpg = One time deal purchase price ($/unit)Qg = One time special order quantity (units)TCsp=TC over time covered by special purchase ($)

Then, g g

pg h pg osp g g

Q QTC [ /D] = + + Q QC C C C

2 D

One Time Discount


One Time Discount

C + DQ

2Q

CC + QC =/D] QTCsp[ ogg

pghgpgg

QQ

C + D

)Q-Q(2

QCC +

DQ

2Q

CC + )Q-Q(C + QC =/D] Q[TCw

go

wgwph

wwpghwgpwpggnsp


One Time Discount

CQC +

CC

)DC-C( = Q

pg

wp

hpg

pgpg

*

2

1 - QQ

CCC =] QSAVINGS[

w

g*

p

pgog

*

⎦

⎤

⎣

⎡


One Time Discount

A quick numerical example...

D = 2000 units / yearCo = $500.00 / orderCh = 25%Cp = $50.00 / unitQ* = 400 unitsTC* = $5000.00 / year

Cpg = $45.00 / unitQ*g = 1333Savings* = $2448.25

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Transportation Procurement in the Digital Age...Discounting All Units ... Transportation Procurement in the Digital Age Author: Yossi Created Date: 6/29/2004 11:10:57 AM