single period inventory models - … · transportation & logistics ... (c s)? ctl.sc1x - supply...

MIT Center for Transportation & Logistics

CTL.SC1x -Supply Chain & Logistics Fundamentals

Single Period Inventory Models: Allowing for Stockouts

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

Assumptions: EOQ with Planned Backorders •  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


Time

EOQ with Planned Backorders In

vent

ory

On

Han

d

T*

Q*

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


EOQ with Planned Back Orders


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


Time

EOQ with Planned Backorders In

vent

ory

On

Han

d T

Q

b

Q-b

T1 T2

TRC(Q,b) = ctDQ!

"#

$

%&+ ce

12!

"#$

%&T1T

!

"#

$

%& Q −b( )+ cs

12!

"#$

%&T2T

!

"#

$

%& b( )

TRC(Q,b) = ctDQ!

"#

$

%&+ ce

12!

"#$

%&(Q −b)Q

!

"#

$

%& Q −b( )+ cs

12!

"#$

%&bQ!

"#

$

%& b( )

TRC(Q,b) = ctDQ!

"#

$

%&+ ce

Q −b( )2

2Q

!

"

###

$

%

&&&+ cs

b2

2Q

!

"#

$

%&

From similar triangles:

QT=Q −b( )T1

=bT2

T1T=Q −b( )Q

T2T=bQ


Planned Backorders - Solution

7


EOQ with Planned Backorders

Q*PBO =2ctDce

cs + ce( )cs

=Q*cs + ce( )cs

b*=ceQPBO

*

cs + ce( )= 1−

cscs + ce( )

"

#

$$

%

&

''QPBO*

TRC(Q,b) = ctDQ!

"#

$

%&+ ce

Q −b( )2

2Q

!

"

###

$

%

&&&+ cs

b2

2Q

!

"#

$

%&

CR = cscs + ce( )

Critical Ratio Order Q*PBO when IOH = - b* Order Q*PBO every T*PBO time periods

Inventory Policy


EOQ with Planned Backorders

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*

(PB

O)

as p

erce

nt

of

Q*

(w

/o b

acko

rder

s)

Critical Ratio Cs/(Cs+Ce)

CR = cscs + 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

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

11


Example: NFL Replica Jerseys •  Situation:

n  In 2002 Reebok had sole rights to sell replica NFL football jerseys

n  Jerseys have unique names & numbers n  Peak sales last about 8 weeks n  Lead time from contract manufacturer is 12-16 weeks

•  Main Issue: n  Reebok had to commit to an order in advance while the actual

demand was uncertain

•  Question: n  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,” MIT Supply Chain Management Program Thesis.

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


Example: NFL Replica Jerseys •  Data:

n  Unit cost = c = 10.90 $/jersey n  Unit selling price = p = 24 $/jersey n  Forecast demand = 32,000 jerseys (σ = 11,000)

w  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


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


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

Ex

pec

ted

To

tal P

rofi

t ($

k)

Order Quantity

Expected Profit


Solving Single Period Model: Marginal Analysis

18


Marginal Analysis

For single-period problems we have two costs: ce = Excess cost when D<Q ($/unit) i.e. having too much product cs = Shortage cost when D>Q ($/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]


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

Mar

gin

al C

ost

per

Jer

sey

Order Quantity

Marginal Shortage and Excess Costs

Excess Cost Shortage Cost

ceP x ≤Q"# $%

cs 1− P x ≤Q#$ %&( )


Marginal Analysis

ceP x ≤Q"# $%= cs 1− P x ≤Q"# $%( )ceP x ≤Q"# $%= cs − csP x ≤Q"# $%

ceP x ≤Q"# $%+ csP x ≤Q"# $%= cs

P x ≤Q"# $% ce + cs( ) = cs

P x ≤Q"# $%=cs

ce + cs( )The Critical Ratio


NFL Jersey Example - solved



n  Total cost = c = 10.90 $/jersey n  Selling price = p = 24 $/jersey n  Forecast demand ~N(32000, 11000)

•  Solution: n  cs = p – c = 24 -10.90 = $13.10 n  ce= c = $10.90 n  CR = (13.10)/( 10.9 + 13.10) =0.546 n  Select Q where P[x≤Q] = 0.546

w  Normal Table or use spreadsheet:

23




Standard Normal Table

P[x≤Q] = 0.546 Find k= 0.115 Recall k=(Q-μ)/σ So, Q = μ+kσ = 32000+(0.115)(11000) Q = 33,267 units

24



n  Total cost = c = 10.90 $/jersey n  Selling price = p = 24 $/jersey n  Forecast demand ~N(32000, 11000)

•  Solution: n  cs = p – c = 24 -10.90 = $13.10 n  ce= c = $10.90 n  CR = (13.10)/( 10.9 + 13.10) =0.546 n  Select Q where P[x≤Q] = 0.546

w  Normal Table or use spreadsheet: w  =NORMINV(CR, Mean, StdDev) w  =NORMINV(0.546, 32000, 11000)

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

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

25




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)



n  Total cost = c = 10.90 $/jersey n  Selling price = p = 24 $/jersey n  Forecast demand ~N(32000, 11000) n  Salvage value = g = 7 $/jersey

•  Solution: n  cs = p – c = 24 - 10.90 = $13.10 n  ce= c - g = 10.90 – 7.00 = $3.90 n  CR = (13.10)/( 3.9 + 13.10) =0.771 n  Select Q where P[x≤Q] = 0.771

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

n  Q* = 40,149- the profit maximizing quantity

But, how do I determine the profitability?

27


Key Points from Lesson

28


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 of uncertain 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]

MIT Center for Transportation & Logistics

CTL.SC1x -Supply Chain & Logistics Fundamentals

Questions, Comments, Suggestions? Use the Discussion!

[email protected]

single period inventory models - … · transportation & logistics ... (c s)? ctl.sc1x - supply...

Documents