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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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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)?
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
EOQ with Planned Back Orders
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Planned Backorders - Solution
7
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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*
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Probabilistic Demand: Single Period Models
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory 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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Solving Single Period Model: Data Table
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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$3,1)
Profit = pMIN (x,Q)− cQ
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
=SUMPRODUCT($C$7:$C$48,E7:E48)
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Solving Single Period Model: Marginal Analysis
18
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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]
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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"# $%
cs 1− P x ≤Q#$ %&( )
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
NFL Jersey Example - solved
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Example: NFL Replica Jerseys • Data:
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
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Example: NFL Replica Jerseys • Data:
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
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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)
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Example: NFL Replica Jerseys • Data:
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
Key Points from Lesson
28
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Single Period Inventory Models
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
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