w6l1_ShortageCosts_ANNOTATED_v20.pdf

8/20/2019 w6l1_ShortageCosts_ANNOTATED_v20.pdf

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MIT Center for

Transportation & Logistics

CTL.SC1x -Supply Chain & Logistics Fundamentals

Single Period Inventory Models: Allowing for Stockouts


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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models

Assumptions: EOQ with Planned Backorders• Demand

!

Constant vs Variable!

Known vs Random!

Continuous vs Discrete

•

Lead Time!

Instantaneous!


Deterministic vs Stochastic

! Internally Replenished

•

Dependence of Items!

Independent

!

Correlated!

Indentured

•

Review Time!

Continuous vs Periodic•

Number of Locations!

One vs Multi vs Multi-Echelon

•

Capacity / Resources!

Unlimited!

Limited / Constrained

• Discounts!

None !

All Units vs Incremental vs One Time

•

Excess Demand!

None!

All orders are backordered!

Lost orders!

Substitution

•

Perishability! None!

Uniform with time

!

Non-linear with time

•

Planning Horizon!

Single Period

!

Finite Period!

Infinite

•

Number of Items!

One vs Many

•

Form of Product!

Single Stage

!

Multi-Stage

2


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Time

EOQ with Planned Backorders

I n v e n t o r y O n H a n d

T*

Q*

What will happen to Q* and T* if we allowfor planned backorders at some cost (cs)?


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EOQ with Planned Back Orders


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Notation

D = Average Demand (units/time)

c = Variable (Purchase) Cost ($/unit)

ct = Fixed Ordering Cost ($/order)

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

ce = c*h = Excess Holding Cost ($/unit/time)cs = Shortage Cost ($/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) 5


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Time


I

n v e n t o r y O n H

a n d T

Q

b

Q-b

T1 T2

TRC (Q,b) = ct

D

Q

!

"#

$

%&+ c

e

1

2

!

"#

$

%& T

1

T

!

"#

$

%& Q !b( )+ c

s

1

2

!

"#

$

%& T

2

T

!

"#

$

%& b( )

TRC (Q,b) = ct

D

Q

!

"#

$

%&+ c

e

1

2

!

"#

$

%&

(Q !b)

Q

!

"#

$

%& Q!b( )+ c

s

1

2

!

"#

$

%& b

Q

!

"#

$

%& b( )

TRC (Q,b) = ct

D

Q

!

"#

$

%&+ c

e

Q'b( )2

2Q

!

"

#

##

$

%

&

&&+ c

s

b2

2Q

!

"#

$

%&

From similar triangles:

Q

T =Q!b( )

T 1

=b

T 2

T 1

T =

Q!b( )Q

T 2

T =

b

Q


7/30CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models

Planned Backorders - Solution

7




Q*

PBO =

2ct D

ce

c s + c

e( )c s

=Q* c

s + c

e( )c s

b*=ceQ

PBO

*

c s + c

e( ) = 1!

c s

c s+ c

e( )

"

#

$$

%

&

''Q PBO

*

TRC (Q

,b)=

ct D

Q

!

"#

$

%&+

ce

Q'b( )2

2Q

!

"

#

##

$

%

&

&&+

c sb2

2Q

!

"#

$

%&

CR =c

s

c s

+ ce

( )

Critical Ratio

Order Q*PBO when IOH = - b*

Order Q*PBO every T*PBO time periods

Inventory Policy




0%

100%

200%

300%

400%

500%

600%

700%

800%

900%

1000%

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Q * ( P B O

) a s p e r c e n t o f Q *

( w / o b a c k o r d e r s

)

Critical Ratio Cs/(Cs+Ce)

CR =c

s

c s

+ ce

( )

Critical Ratio

If cs is very big,then Q*PBO=Q*

If cs is very small,then Q*PBO>>Q*



Probabilistic Demand:Single Period Models



Assumptions: Single Period Models• Demand

!


Known vs Random!

Continuous vs Discrete

•

Lead Time!

Instantaneous!


Deterministic vs Stochastic

! Internally Replenished

•

Dependence of Items!

Independent

!

Correlated!

Indentured

•

Review Time!

Continuous vs Periodic•

Number of Locations!

One vs Multi vs Multi-Echelon

•

Capacity / Resources!

Unlimited!

Limited / Constrained

• Discounts!

None !

All Units vs Incremental vs One Time

•

Excess Demand!

None!

All orders are backordered!

Lost orders!

Substitution

•

Perishability! None!

Uniform with time

!

Non-linear with time

•

Planning Horizon!

Single Period

!

Finite Period!

Infinite

•

Number of Items!

One vs Many

•

Form of Product!

Single Stage

!

Multi-Stage

11



Example: NFL Replica Jerseys

• Situation:!

In 2002 Reebok had sole rights to sellreplica NFL football jerseys

! Jerseys have unique names & numbers

! Peak sales last about 8 weeks

!

Lead time from contract manufacturer is 12-16 weeks

• Main Issue:

!

Reebok had to commit to an order in advance while the actualdemand was uncertain

• Question:!

How many Jerseys of each player should they order?

12

Case adapted from Parsons, J. (2004) “Using A Newsvendor Model for Demand Planning of NFL Replica Jerseys,” MITSupply Chain Management Program Thesis.

Image Source: http://commons.wikimedia.org/wiki/File:Tom_Brady_%28cropped%29.jpg




• Data:!

Unit cost = c = 10.90 $/jersey

! Unit selling price = p = 24 $/jersey

! Forecast demand = 32,000 jerseys (σ = 11,000)

" History showed demand to be Normally distributed

• Select Q* that maximizes profit where X = actual demand:

• How do I determine the “best” policy?1. Data table

2. Marginal analysis

13

Profit = p MIN x,Q!" #$( )% cQ



Solving Single Period Model:Data Table



Data Table

15

Sample spreadsheets in MS Excel and LibreOffice are available in this unit.

=$D$4*MIN($B8,E$5)-$F$4*E$5

Potential order sizes (Q)

Potential demand (x)

Probability of demand P[x]

=NORMDIST(B10,$B$2,$B$3,1)

Profit = pMIN ( x,Q)! cQ


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=SUMPRODUCT($C$7:$C$48,E7:E48)


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Expected Profits

$-

$25.00

$50.00

$75.00

$100.00

$125.00

$150.00

$175.00

$200.00

$225.00

$250.00

$275.00

$300.00

$325.00

$350.00

0 5 10 15 20 25 30 35 40 45 50

E x p e c t e d T o t a l P r

o f i t ( $ k )

Order Quantity

Expected Profit


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Solving Single Period Model:Marginal Analysis

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Marginal Analysis

For single-period problems we have two costs:ce = Excess cost when DQ ($/unit) i.e. having too little product

Assuming a continuous distribution of demand , we get

ce P[X!Q] = expected excess cost of the Qth unit ordered

cs (1-P[X!Q]) = expected shortage cost of the Qth unit ordered

If E[Excess Cost] < E[Shortage Cost] then increase Q

We are at Q* when E[Shortage Cost] = E[Excess Cost]


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Marginal Analysis

20

$-

$1.00

$2.00

$3.00

$4.00

$5.00

$6.00

$7.00

$8.00

$9.00

$10.00

$11.00

$12.00

$13.00

$14.00

0 5 10 15 20 25 30 35 40 45 50 55 60

M a r g i n a l C o s t p e r J e r s e y

Order Quantity

Marginal Shortage and Excess Costs

Excess Cost Shortage Cost

ce

P x ! Q"#

$%

c s1! P x " Q#$

%&( )


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Marginal Analysis

ce P x ! Q"# $%= c s 1& P x ! Q"# $%( )c

e P x ! Q"# $%= c s & c s P x ! Q

"# $%

ce P x ! Q"# $%+ c s P x ! Q"# $%= c s

P x ! Q"# $% ce + c s( ) = c s

P x ! Q"# $%=c

s

ce + c

s( )The Critical Ratio


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NFL Jersey Example - solved


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Example: NFL Replica Jerseys• Data:

! Total cost = c = 10.90 $/jersey

! Selling price = p = 24 $/jersey

! Forecast demand ~N(32000, 11000)

• Solution:! cs = p – c = 24 -10.90 = $13.10

! ce= c = $10.90

!

CR = (13.10)/( 10.9 + 13.10) =0.546

! Select Q where P[x!Q] = 0.546" Normal Table or use spreadsheet:

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Standard Normal Table

P[x!Q] = 0.546

Find k= 0.115

Recall k=(Q-μ)/σ

So, Q = μ+k σ = 32000+(0.115)(11000)

Q = 33,267 units

24


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Example: NFL Replica Jerseys• Data:

! Total cost = c = 10.90 $/jersey

! Selling price = p = 24 $/jersey


• Solution:! cs = p – c = 24 -10.90 = $13.10

! ce= c = $10.90

!

CR = (13.10)/( 10.9 + 13.10) =0.546

! Select Q where P[x!Q] = 0.546" Normal Table or use spreadsheet:

" =NORMINV(CR, Mean, StdDev)

"

=NORMINV(0.546, 32000, 11000)

! Q* = 33,267 - the profit maximizing quantity

But what if I can sell the left overs at a discount?

25




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Considering Other Costs

• Other costs:• g = salvage value, $/unit

• B = Penalty for not satisfying demand(beyond lost profit), $/unit

• The excess and shortage costs change:

• cs = p – c + B

• ce = c - g

• Critical Ratio = cs /(cs+ce)= (p-c+B)/(p-c+B+c-g)

=(p-c+B)/(p+B-g)


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• Data:!

Total cost = c = 10.90 $/jersey! Selling price = p = 24 $/jersey


!

Salvage value = g = 7 $/jersey

• Solution:!

cs = p – c = 24 - 10.90 = $13.10

! ce= c - g = 10.90 – 7.00 = $3.90

!

CR = (13.10)/( 3.9 + 13.10) =0.771

! Select Q where P[x!Q] = 0.771

"

Normal Table or use spreadsheet:" =NORMINV(CR, Mean, StdDev)=NORMINV(.771, 32000, 11000)

!

Q* = 40,149- the profit maximizing quantity

But, how do I determine the profitability?

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Key Points from Lesson

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Key Points

• Newsvendor problems are everywhere

• Fashion items, perishable goods, fleet sizing,contracting, space missions, etc.

• Whenever you have to make a firm bet in the face ofuncertain demand in a single period

•

Classic trade off between:

• Having too much (excess cost ce)

• Having too little (shortage cost cs)

•

Critical Ratio captures this trade-off• CR = Cs / (Cs + Ce)

• CR = Pct of demand distribution to cover= P[x!Q]


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MIT Center for

Transportation & Logistics

CTL.SC1x -Supply Chain & Logistics Fundamentals

Questions, Comments, Suggestions?

Use the Discussion!

caplice@mit edu

Documents

w6l1_ShortageCosts_ANNOTATED_v20.pdf