Description
Detailed PPT explains newsvendor model in operations management.
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1
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? Determine the optimal level of product
availability
? Demand forecasting
? Profit maximization
? Other measures such as a fill rate
2
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3
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? The “too much/too little problem”:
? Order too much and inventory is left over at the end of the
season
? Order too little and sales are lost.
? Marketing?s forecast for sales is 3200 units. 4
Nov Dec Jan Feb Mar Apr May Jun Jul Aug
Generate forecast
of demand and
submit an order
to TEC
Receive order
from TEC at the
end of the
month
Spring selling season
Left over
units are
discounted
Economics:
• Each suit sells for p = $180
• TEC charges c = $110/suit
• Discounted suits sell for v = $90
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? Gather economic inputs:
? selling price,
? production/procurement cost,
? salvage value of inventory
? Generate a demand model to represent
demand
? Use empirical demand distribution
? Choose a standard distribution function
? the normal distribution,
? the Poisson distribution.
? Choose an objective:
? maximize expected profit
? satisfy a fill rate constraint.
? Choose a quantity to order.
5
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6
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0
1000
2000
3000
4000
5000
6000
7000
0 1000 2000 3000 4000 5000 6000 7000
Forecast
A
c
t
u
a
l
d
e
m
a
n
d
.
Forecasts and actual demand for surf wet-suits from the previous season
utdallas.edu/~metin
? If we underestimate the demand, we stock less than
necessary.
? The stock is less than the demand, the stockout
occurs.
? Are the number of stockout units (= unmet demand)
observable, i.e., known to the store manager?
? Yes, if the store manager issues rain checks to customers.
? No, if the stockout demand disappears silently.
? No implies demand filtering. That is, demand is
known exactly only when it is below the stock.
? Shall we order more than optimal to learn about
demand when the demand is filtered?
8
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9
Empirical distribution function for the historical A/F ratios.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
A/F ratio
P
r
o
b
a
b
i
l
i
t
y
Product description Forecast Actual demand Error* A/F Ratio**
JR ZEN FL 3/2 90 140 -50 1.56
EPIC 5/3 W/HD 120 83 37 0.69
JR ZEN 3/2 140 143 -3 1.02
WMS ZEN-ZIP 4/3 170 163 7 0.96
HEATWAVE 3/2 170 212 -42 1.25
JR EPIC 3/2 180 175 5 0.97
WMS ZEN 3/2 180 195 -15 1.08
ZEN-ZIP 5/4/3 W/HOOD 270 317 -47 1.17
WMS EPIC 5/3 W/HD 320 369 -49 1.15
EVO 3/2 380 587 -207 1.54
JR EPIC 4/3 380 571 -191 1.50
WMS EPIC 2MM FULL 390 311 79 0.80
HEATWAVE 4/3 430 274 156 0.64
ZEN 4/3 430 239 191 0.56
EVO 4/3 440 623 -183 1.42
ZEN FL 3/2 450 365 85 0.81
HEAT 4/3 460 450 10 0.98
ZEN-ZIP 2MM FULL 470 116 354 0.25
HEAT 3/2 500 635 -135 1.27
WMS EPIC 3/2 610 830 -220 1.36
WMS ELITE 3/2 650 364 286 0.56
ZEN-ZIP 3/2 660 788 -128 1.19
ZEN 2MM S/S FULL 680 453 227 0.67
EPIC 2MM S/S FULL 740 607 133 0.82
EPIC 4/3 1020 732 288 0.72
WMS EPIC 4/3 1060 1552 -492 1.46
JR HAMMER 3/2 1220 721 499 0.59
HAMMER 3/2 1300 1696 -396 1.30
HAMMER S/S FULL 1490 1832 -342 1.23
EPIC 3/2 2190 3504 -1314 1.60
ZEN 3/2 3190 1195 1995 0.37
ZEN-ZIP 4/3 3810 3289 521 0.86
WMS HAMMER 3/2 FULL 6490 3673 2817 0.57
* Error = Forecast - Actual demand
** A/F Ratio = Actual demand divided by Forecast
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? All normal distributions are characterized by two
parameters, mean = µ and standard deviation = o
? All normal distributions are related to the standard normal
that has mean = 0 and standard deviation = 1.
? For example:
? Let Q be the order quantity, and (µ, o) the parameters of the
normal demand forecast.
? Prob{demand is Q or lower} = Prob{the outcome of a standard
normal is z or lower}, where
? (The above are two ways to write the same equation, the first
allows you to calculate z from Q and the second lets you calculate
Q from z.)
? Look up Prob{the outcome of a standard normal is z or lower} in
the
Standard Normal Distribution Function
Table.
10
or
Q
z Q z
µ
µ o
o
÷
= = + ×
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? Start with an initial forecast generated from
hunches, guesses, etc.
? O?Neill?s initial forecast for the Hammer 3/2 = 3200
units.
? Evaluate the A/F ratios of the historical data:
? Set the mean of the normal distribution to
? Set the standard deviation of the normal
distribution to
11
Forecast
demand Actual
ratio A/F =
Forecast ratio A/F Expected demand actual Expected × =
Forecast ratios A/F of deviation Standard
demand actual of deviation Standard
×
=
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? O?Neill should choose a normal distribution with mean
3192 and standard deviation 1181 to represent demand for
the Hammer 3/2 during the Spring season.
? Why not a mean of 3200?
12
3192 3200 9975 0 = × = . demand actual Expected
1181 3200 369 0 = × = . demand actual of deviation Standard
Product description Forecast Actual demand Error A/F Ratio
JR ZEN FL 3/2 90 140 -50 1.5556
EPIC 5/3 W/HD 120 83 37 0.6917
JR ZEN 3/2 140 143 -3 1.0214
WMS ZEN-ZIP 4/3 170 156 14 0.9176
… … … … …
ZEN 3/2 3190 1195 1995 0.3746
ZEN-ZIP 4/3 3810 3289 521 0.8633
WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659
Average 0.9975
Standard deviation 0.3690
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0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 1000 2000 3000 4000 5000 6000
Quantity
P
r
o
b
a
b
i
l
i
t
y
.
Empirical distribution function (diamonds) and normal distribution function with
mean 3192 and standard deviation 1181 (solid line)
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? C
o
= overage cost
? The cost of ordering one more unit than what you would have
ordered had you known demand.
? In other words, suppose you had left over inventory (i.e., you over
ordered). C
o
is the increase in profit you would have enjoyed had
you ordered one fewer unit.
? For the Hammer 3/2 C
o
= Cost – Salvage value = c – v = 110 –
90 = 20
? C
u
= underage cost
? The cost of ordering one fewer unit than what you would have
ordered had you known demand.
? In other words, suppose you had lost sales (i.e., you under
ordered). C
u
is the increase in profit you would have enjoyed had
you ordered one more unit.
? For the Hammer 3/2 C
u
= Price – Cost = p – c = 180 – 110 = 70
15
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? Ordering one more unit increases the chance of overage
? Expected loss on the Q
th
unit = C
o
x F(Q), where F(Q) =
Prob{Demand <= Q)
? The benefit of ordering one more unit is the reduction
in the chance of underage:
? Expected benefit on the Q
th
unit = C
u
x (1-F(Q))
16
As more units are ordered,
? the expected benefit from
ordering one unit
decreases
? while the expected loss of
ordering one more unit
increases.
0
10
20
30
40
50
60
70
80
0 800 1600 2400 3200 4000 4800 5600 6400
E
x
p
e
c
t
e
d
g
a
i
n
o
r
l
o
s
s
.
Expected marginal benefit
of understocking
Expected marginal loss
of overstocking
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? To minimize the expected total cost of underage and
overage, order Q units so that the expected marginal
cost with the Q
th
unit equals the expected marginal
benefit with the Q
th
unit:
? Rearrange terms in the above equation ->
? The ratio C
u
/ (C
o
+ C
u
) is called the critical ratio.
? Hence, to minimize the expected total cost of underage
and overage, choose Q such that we don?t have lost
sales (i.e., demand is Q or lower) with a probability that
equals the critical ratio
17
( ) ( ) Q F C Q F C
u o
÷ × = × 1 ) (
u o
u
C C
C
Q F
+
= ) (
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? Inputs: Empirical distribution function table; p = 180; c
= 110; v = 90; C
u
= 180-110 = 70; C
o
= 110-90 =20
? Evaluate the critical ratio:
? Look up 0.7778 in the empirical distribution function
graph
? Or, look up 0.7778 among the ratios:
? If the critical ratio falls between two values in the table, choose
the one that leads to the greater order quantity
? Convert A/F ratio into the order quantity
18
7778 . 0
70 20
70
=
+
=
+
u o
u
C C
C
* / 3200*1.3 4160. Q Forecast A F = = =
Product description Forecast Actual demand A/F Ratio Rank Percentile
…
…
…
…
…
…
HEATWAVE 3/2 170 212 1.25 24 72.7%
HEAT 3/2 500 635 1.27 25 75.8%
HAMMER 3/2 1300 1696 1.30 26 78.8%
…
…
…
…
…
…
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? Inputs: p = 180; c = 110; v = 90; C
u
= 180-110 = 70; C
o
= 110-
90 =20; critical ratio = 0.7778; mean = µ = 3192; standard
deviation = o = 1181
? Look up critical ratio in the Standard Normal Distribution Function
Table:
? If the critical ratio falls between two values in the table, choose the
greater z-statistic
? Choose z = 0.77
? Convert the z-statistic into an order quantity:
? Equivalently, Q = norminv(0.778,3192,1181) = 4096.003
19
4101 1181 77 . 0 3192 = × + =
× + = o µ z Q
z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
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Demand
D_i
Probabability
p_i
Cumulative Probability of demand
being this size or less, F()
Probability of demand
greater than this size, 1-F()
4 .01 .01 .99
5 .02 .03 .97
6 .04 .07 .93
7 .08 .15 .85
8 .09 .24 .76
9 .11 .35 .65
10 .16 .51 .49
11 .20 .71 .29
12 .11 .82 .18
13 .10 .92 .08
14 .04 .96 .04
15 .02 .98 .02
16 .01 .99 .01
17 .01 1.00 .00
20
Expected demand is 1,026 parkas.
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Cost per parka = c = $45
Sale price per parka = p = $100
Discount price per parka = $50
Holding and transportation cost = $10
Salvage value per parka = v = 50-10=$40
Profit from selling parka = p-c = 100-45 = $55
Cost of understocking = $55/unit
Cost of overstocking = c-v = 45-40 = $5/unit
21
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p = sale price; v = outlet or salvage price; c = purchase
price
CSL = Probability that demand will be at or below order
quantity
CSL later called in-stock probability
Raising the order size if the order size is already optimal
Expected Marginal Benefit of increasing Q= Expected Marginal Cost
of Underage
=P(Demand is above stock)*(Profit from sales)=(1-
CSL)(p - c)
Expected Marginal Cost of increasing Q = Expected marginal cost of
overage
=P(Demand is below stock)*(Loss from
discounting)=CSL(c - v)
Define C
o
= c-v; C
u
=p-c
(1-CSL)C
u
= CSL C
o
CSL= C
u
/ (C
u
+ C
o
)
22
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C
o
= Cost of overstocking = $5
C
u
= Cost of understocking = $55
Q
*
= Optimal order size
917 . 0
5 55
55
) (
*
=
+
=
+
> s =
o u
u
C C
C
Q Demand P CSL
23
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0
0.2
0.4
0.6
0.8
1
1.2
4 5 6 7 8 9 10 11 12 13 14 15 16 87
Cumulative
Probability
24
Optimal Order Quantity = 13(‘00)
0.917
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? Expected demand = 10 („00) parkas
? Expected profit from ordering 10 („00) parkas =
$499
? Approximate Expected profit from ordering 1(„00)
extra parkas if 10(?00) are already ordered
= 100.55.P(D>=1100) - 100.5.P(D<1100)
25
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Additional
100s
Expected
Marginal Benefit
Expected
Marginal Cost
Expected Marginal
Contribution
11
th
5500×.49 = 2695 500×.51 = 255
2695-255 = 2440
12
th
5500×.29 = 1595 500×.71 = 355
1595-355 = 1240
13
th
5500×.18 = 990 500×.82 = 410
990-410 = 580
14
th
5500×.08 = 440 500×.92 = 460
440-460 = -20
15
th
5500×.04 = 220 500×.96 = 480
220-480 = -260
16
th
5500×.02 = 110 500×.98 = 490
110-490 = -380
17
th
5500×.01 = 55 500×.99 = 495
55-495 = -440
26
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? Total cost by ordering Q units:
? C(Q) = overstocking cost + understocking
cost
27
} }
·
÷ + ÷ =
Q
u
Q
o
dx x f Q x C dx x f x Q C Q C ) ( ) ( ) ( ) ( ) (
0
0 )) ( 1 ( ) (
) (
= ÷ ÷ = Q F C Q F C
dQ
Q dC
u o
Marginal cost of overage at Q* - Marginal cost of underage at Q* = 0
u o
u
C C
C
Q D P Q F
+
= s = ) ( ) (
* *
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Inventory held in addition to the
expected demand is called the
safety stock
The expected demand is 1026 parkas but
we order 1300 parkas.
So the safety stock is 1300-1026=274
parka.
28
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29
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? For any order quantity we would like to evaluate the
following performance measures:
? Expected lost sales
? The average number of units demand exceeds the order quantity
? Expected sales
? The average number of units sold.
? Expected left over inventory
? The average number of units left over at the end of the season.
? Expected profit
? Expected fill rate
? The fraction of demand that is satisfied immediately
? In-stock probability
? Probability all demand is satisfied
? Stockout probability
? Probability some demand is lost
30
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? ESC is the expected shortage in a season
? ESC is not a percentage, it is the number of units, also see
next page
¹
´
¦
s
>
=
Q Demand
Q Demand
if
if
0
Q - Demand
Shortage
}
·
=
=
÷ =
Q x
Q)f(x)dx - (x ESC
Q,0}) season a in nd E(max{Dema ESC
31
utdallas.edu/~metin
32
0
Q
Demand
During a
Season
Season
Inventory
D, Demand
During
A Season
0
0
Q
Inventory=Q-D
Upside
down
utdallas.edu/~metin
33
0
Q
Demand
Season
Shortage
D
:
D
e
m
a
n
d
D
u
r
i
n
g
s
S
e
a
s
o
n
0
0
Q
Shortage
=D-Q
Upside
down
utdallas.edu/~metin
? First let us study shortage during the lead time
4
1
4
1
10)} - (11 max{0,
4
2
10)} - (10 max{0,
4
1
10)} - (9 max{0,
) ( )} (d )} ( max{0, shortage Expected
Shortage? Expected ,
1/4 prob with 11
2/4 prob with 10
1/4 prob with 9
, 10
11
10 d
3
1 i
3 3
2 2
1 1
= + + =
= ÷ = ÷ =
¦
)
¦
`
¹
¦
¹
¦
´
¦
= =
= =
= =
= =
¿ ¿
= =
d D P Q p Q d
p d
p d
p d
D Q
i i
34
D. of pdf is f where ) ( ) (
}) 0 , (max{ shortage Expected
D
}
·
=
÷ =
÷ =
Q D
D
dD D f Q D
Q D E
? Ex:
utdallas.edu/~metin
2/6
) 10 ( 10
2
10
6
1
) 12 ( 10
2
12
6
1
10
2 6
1
6
1
) 10 ( shortage Expected
Shortage? Expected ), 12 , 6 ( , 10
2 2
12
10
2
12
10
=
|
|
.
|
\
|
÷ ÷
|
|
.
|
\
|
÷ = ÷
\
|
= ÷ =
= =
=
=
=
}
D
D
D
D
D
dD D
Unif orm D Q
35
? Ex:
utdallas.edu/~metin
? Step 1: normalize the order quantity to find its z-statistic.
? Step 2: Look up in the Standard Normal Loss Function Table the
expected lost sales for a standard normal distribution with that z-
statistic: L(0.26)=0.2824 see Appendix B table on p.380 of the
textbook
? or, in Excel L(z)=normdist(z,0,1,0)-z*(1-normdist(z,0,1,1)) see Appendix
D on p.389
? Step 3: Evaluate lost sales for the actual normal distribution:
36
26 . 0
1181
3192 3500
=
÷
=
÷
=
o
µ Q
z
( ) 1181 0.2824 334 Expected lost sales L z o = × = × =
Keep 334 units in mind, we shall repeatedly use it
utdallas.edu/~metin
? Demand=Sales+Lost Sales
D=min(D,Q)+max{D-Q,0} or min(D,Q)=D- max{D-
Q,0}
Expected sales = µ - Expected lost sales
= 3192 – 334 = 2858
? Inventory=Sales+Left Over Inventory
Q=min(D,Q)+max{Q-D,0} or max{Q-D,0}=Q-
min(D,Q)
Expected Left Over Inventory = Q - Expected Sales
= 3500 – 2858 = 642
37
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Expected total underage and overage cost with
(Q=3500)
=70*334 + 20*642
38
( )
( )
( ) ( )
$70 2858 $20 642 $187, 221
Expected profit Price-Cost Expected sales
Cost-Salvage value Expected left over inventory
= × (
¸ ¸
÷ × (
¸ ¸
= × ÷ × =
What is the relevant objective? Minimize the cost or maximize the profit?
Hint: What is profit + cost? It is 70*3192=C
u
*?, which is a constant.
utdallas.edu/~metin
39
) Q P(Demand bability InstockPro s =
Instock probability: percentage of seasons without a stock out
Q] season a during [Demand inventory] t [Sufficien
inventory sufficient has season single a y that Probabilit 0.7 y Probabilit Instock
7 . 0 y Probabilit Instock
otherwise 1 stockout, has season a if 0 Write
10
1 0 1 0 1 1 1 0 1 1
y Probabilit Instock
: seasons 10 consider example For
s =
= =
=
+ + + + + + + + +
=
utdallas.edu/~metin
40
( )
( )
|
.
|
\
|
÷
=
|
.
|
\
|
÷
s =
|
.
|
\
|
÷
s
÷
=
÷ s ÷ =
=
s
1 , 1 , 0 , Normdist
on distributi normal standard Obtaining ) 1 , 0 (
StDev by the Dividing
) , (
mean out the Taking ) , (
,1) , , Normdist(Q
) , (
o
µ
o
µ
o
µ
o
µ o µ
µ µ o µ
o µ
o µ
Q
Q
N P
Q N
P
Q N P
Q N P
N(?,?) denotes a normal demand with mean ? and standard deviation ?
utdallas.edu/~metin
? = 2,500 /week; o= 500; Q = 3,000;
Instock probability if demand is Normal?
Instock probability = Normdist((3,000-
2,500)/500,0,1,1)
41
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? = 2,500/week; o= 500; To achieve Instock
Probability=0.95, what should be Q?
Q = Norminv(0.95, 2500, 500)
42
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Recall:
Expected sales = µ - Expected lost sales = 3192 –
334 = 2858
43
2858
1
3192
89.6%
Expected sales Expected sales
Expected fill rate
Expected demand
Expected lost sales
µ
µ
= =
= ÷ =
=
Is this fill rate too low?
Well, lost sales of 334 is with Q=3500, which is less than optimal.
utdallas.edu/~metin
44
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 1000 2000 3000 4000 5000 6000 7000
Order quantity
In-stock probability
CSL
Expected fill
rate
utdallas.edu/~metin
45
CSL is 0%, fill rate is almost 100%
CSL is 0%, fill rate is almost 0%
inventory
inventory
time
time
0
0
utdallas.edu/~metin
? Determine the optimal level of product availability
? Demand forecasting
? Profit maximization / Cost minimization
? Other measures
? Expected shortages = lost sales
? Expected left over inventory
? Expected sales
? Expected cost
? Expected profit
? Type I service measure: Instock probability = CSL
? Type II service measure: Fill rate
46
utdallas.edu/~metin
? Q1: Suppose that you own a simple call option
for a stock with a strike price of Q. Suppose
that the price of the underlying stock is D at
the expiration time of the option. If D <Q, the
call option has no value. Otherwise its value is
D-Q. What is the expected value of a call
option, whose strike price is $50, written for a
stock whose price at the expiration is normally
distributed with mean $51 and standard
deviation $10?
47
doc_526646333.pptx
Detailed PPT explains newsvendor model in operations management.
utdallas.edu/~metin
1
utdallas.edu/~metin
? Determine the optimal level of product
availability
? Demand forecasting
? Profit maximization
? Other measures such as a fill rate
2
utdallas.edu/~metin
3
utdallas.edu/~metin
? The “too much/too little problem”:
? Order too much and inventory is left over at the end of the
season
? Order too little and sales are lost.
? Marketing?s forecast for sales is 3200 units. 4
Nov Dec Jan Feb Mar Apr May Jun Jul Aug
Generate forecast
of demand and
submit an order
to TEC
Receive order
from TEC at the
end of the
month
Spring selling season
Left over
units are
discounted
Economics:
• Each suit sells for p = $180
• TEC charges c = $110/suit
• Discounted suits sell for v = $90
utdallas.edu/~metin
? Gather economic inputs:
? selling price,
? production/procurement cost,
? salvage value of inventory
? Generate a demand model to represent
demand
? Use empirical demand distribution
? Choose a standard distribution function
? the normal distribution,
? the Poisson distribution.
? Choose an objective:
? maximize expected profit
? satisfy a fill rate constraint.
? Choose a quantity to order.
5
utdallas.edu/~metin
6
utdallas.edu/~metin
7
0
1000
2000
3000
4000
5000
6000
7000
0 1000 2000 3000 4000 5000 6000 7000
Forecast
A
c
t
u
a
l
d
e
m
a
n
d
.
Forecasts and actual demand for surf wet-suits from the previous season
utdallas.edu/~metin
? If we underestimate the demand, we stock less than
necessary.
? The stock is less than the demand, the stockout
occurs.
? Are the number of stockout units (= unmet demand)
observable, i.e., known to the store manager?
? Yes, if the store manager issues rain checks to customers.
? No, if the stockout demand disappears silently.
? No implies demand filtering. That is, demand is
known exactly only when it is below the stock.
? Shall we order more than optimal to learn about
demand when the demand is filtered?
8
utdallas.edu/~metin
9
Empirical distribution function for the historical A/F ratios.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
A/F ratio
P
r
o
b
a
b
i
l
i
t
y
Product description Forecast Actual demand Error* A/F Ratio**
JR ZEN FL 3/2 90 140 -50 1.56
EPIC 5/3 W/HD 120 83 37 0.69
JR ZEN 3/2 140 143 -3 1.02
WMS ZEN-ZIP 4/3 170 163 7 0.96
HEATWAVE 3/2 170 212 -42 1.25
JR EPIC 3/2 180 175 5 0.97
WMS ZEN 3/2 180 195 -15 1.08
ZEN-ZIP 5/4/3 W/HOOD 270 317 -47 1.17
WMS EPIC 5/3 W/HD 320 369 -49 1.15
EVO 3/2 380 587 -207 1.54
JR EPIC 4/3 380 571 -191 1.50
WMS EPIC 2MM FULL 390 311 79 0.80
HEATWAVE 4/3 430 274 156 0.64
ZEN 4/3 430 239 191 0.56
EVO 4/3 440 623 -183 1.42
ZEN FL 3/2 450 365 85 0.81
HEAT 4/3 460 450 10 0.98
ZEN-ZIP 2MM FULL 470 116 354 0.25
HEAT 3/2 500 635 -135 1.27
WMS EPIC 3/2 610 830 -220 1.36
WMS ELITE 3/2 650 364 286 0.56
ZEN-ZIP 3/2 660 788 -128 1.19
ZEN 2MM S/S FULL 680 453 227 0.67
EPIC 2MM S/S FULL 740 607 133 0.82
EPIC 4/3 1020 732 288 0.72
WMS EPIC 4/3 1060 1552 -492 1.46
JR HAMMER 3/2 1220 721 499 0.59
HAMMER 3/2 1300 1696 -396 1.30
HAMMER S/S FULL 1490 1832 -342 1.23
EPIC 3/2 2190 3504 -1314 1.60
ZEN 3/2 3190 1195 1995 0.37
ZEN-ZIP 4/3 3810 3289 521 0.86
WMS HAMMER 3/2 FULL 6490 3673 2817 0.57
* Error = Forecast - Actual demand
** A/F Ratio = Actual demand divided by Forecast
utdallas.edu/~metin
? All normal distributions are characterized by two
parameters, mean = µ and standard deviation = o
? All normal distributions are related to the standard normal
that has mean = 0 and standard deviation = 1.
? For example:
? Let Q be the order quantity, and (µ, o) the parameters of the
normal demand forecast.
? Prob{demand is Q or lower} = Prob{the outcome of a standard
normal is z or lower}, where
? (The above are two ways to write the same equation, the first
allows you to calculate z from Q and the second lets you calculate
Q from z.)
? Look up Prob{the outcome of a standard normal is z or lower} in
the
Standard Normal Distribution Function
Table.
10
or
Q
z Q z
µ
µ o
o
÷
= = + ×
utdallas.edu/~metin
? Start with an initial forecast generated from
hunches, guesses, etc.
? O?Neill?s initial forecast for the Hammer 3/2 = 3200
units.
? Evaluate the A/F ratios of the historical data:
? Set the mean of the normal distribution to
? Set the standard deviation of the normal
distribution to
11
Forecast
demand Actual
ratio A/F =
Forecast ratio A/F Expected demand actual Expected × =
Forecast ratios A/F of deviation Standard
demand actual of deviation Standard
×
=
utdallas.edu/~metin
? O?Neill should choose a normal distribution with mean
3192 and standard deviation 1181 to represent demand for
the Hammer 3/2 during the Spring season.
? Why not a mean of 3200?
12
3192 3200 9975 0 = × = . demand actual Expected
1181 3200 369 0 = × = . demand actual of deviation Standard
Product description Forecast Actual demand Error A/F Ratio
JR ZEN FL 3/2 90 140 -50 1.5556
EPIC 5/3 W/HD 120 83 37 0.6917
JR ZEN 3/2 140 143 -3 1.0214
WMS ZEN-ZIP 4/3 170 156 14 0.9176
… … … … …
ZEN 3/2 3190 1195 1995 0.3746
ZEN-ZIP 4/3 3810 3289 521 0.8633
WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659
Average 0.9975
Standard deviation 0.3690
utdallas.edu/~metin
13
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 1000 2000 3000 4000 5000 6000
Quantity
P
r
o
b
a
b
i
l
i
t
y
.
Empirical distribution function (diamonds) and normal distribution function with
mean 3192 and standard deviation 1181 (solid line)
utdallas.edu/~metin
14
utdallas.edu/~metin
? C
o
= overage cost
? The cost of ordering one more unit than what you would have
ordered had you known demand.
? In other words, suppose you had left over inventory (i.e., you over
ordered). C
o
is the increase in profit you would have enjoyed had
you ordered one fewer unit.
? For the Hammer 3/2 C
o
= Cost – Salvage value = c – v = 110 –
90 = 20
? C
u
= underage cost
? The cost of ordering one fewer unit than what you would have
ordered had you known demand.
? In other words, suppose you had lost sales (i.e., you under
ordered). C
u
is the increase in profit you would have enjoyed had
you ordered one more unit.
? For the Hammer 3/2 C
u
= Price – Cost = p – c = 180 – 110 = 70
15
utdallas.edu/~metin
? Ordering one more unit increases the chance of overage
? Expected loss on the Q
th
unit = C
o
x F(Q), where F(Q) =
Prob{Demand <= Q)
? The benefit of ordering one more unit is the reduction
in the chance of underage:
? Expected benefit on the Q
th
unit = C
u
x (1-F(Q))
16
As more units are ordered,
? the expected benefit from
ordering one unit
decreases
? while the expected loss of
ordering one more unit
increases.
0
10
20
30
40
50
60
70
80
0 800 1600 2400 3200 4000 4800 5600 6400
E
x
p
e
c
t
e
d
g
a
i
n
o
r
l
o
s
s
.
Expected marginal benefit
of understocking
Expected marginal loss
of overstocking
utdallas.edu/~metin
? To minimize the expected total cost of underage and
overage, order Q units so that the expected marginal
cost with the Q
th
unit equals the expected marginal
benefit with the Q
th
unit:
? Rearrange terms in the above equation ->
? The ratio C
u
/ (C
o
+ C
u
) is called the critical ratio.
? Hence, to minimize the expected total cost of underage
and overage, choose Q such that we don?t have lost
sales (i.e., demand is Q or lower) with a probability that
equals the critical ratio
17
( ) ( ) Q F C Q F C
u o
÷ × = × 1 ) (
u o
u
C C
C
Q F
+
= ) (
utdallas.edu/~metin
? Inputs: Empirical distribution function table; p = 180; c
= 110; v = 90; C
u
= 180-110 = 70; C
o
= 110-90 =20
? Evaluate the critical ratio:
? Look up 0.7778 in the empirical distribution function
graph
? Or, look up 0.7778 among the ratios:
? If the critical ratio falls between two values in the table, choose
the one that leads to the greater order quantity
? Convert A/F ratio into the order quantity
18
7778 . 0
70 20
70
=
+
=
+
u o
u
C C
C
* / 3200*1.3 4160. Q Forecast A F = = =
Product description Forecast Actual demand A/F Ratio Rank Percentile
…
…
…
…
…
…
HEATWAVE 3/2 170 212 1.25 24 72.7%
HEAT 3/2 500 635 1.27 25 75.8%
HAMMER 3/2 1300 1696 1.30 26 78.8%
…
…
…
…
…
…
utdallas.edu/~metin
? Inputs: p = 180; c = 110; v = 90; C
u
= 180-110 = 70; C
o
= 110-
90 =20; critical ratio = 0.7778; mean = µ = 3192; standard
deviation = o = 1181
? Look up critical ratio in the Standard Normal Distribution Function
Table:
? If the critical ratio falls between two values in the table, choose the
greater z-statistic
? Choose z = 0.77
? Convert the z-statistic into an order quantity:
? Equivalently, Q = norminv(0.778,3192,1181) = 4096.003
19
4101 1181 77 . 0 3192 = × + =
× + = o µ z Q
z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
utdallas.edu/~metin
Demand
D_i
Probabability
p_i
Cumulative Probability of demand
being this size or less, F()
Probability of demand
greater than this size, 1-F()
4 .01 .01 .99
5 .02 .03 .97
6 .04 .07 .93
7 .08 .15 .85
8 .09 .24 .76
9 .11 .35 .65
10 .16 .51 .49
11 .20 .71 .29
12 .11 .82 .18
13 .10 .92 .08
14 .04 .96 .04
15 .02 .98 .02
16 .01 .99 .01
17 .01 1.00 .00
20
Expected demand is 1,026 parkas.
utdallas.edu/~metin
Cost per parka = c = $45
Sale price per parka = p = $100
Discount price per parka = $50
Holding and transportation cost = $10
Salvage value per parka = v = 50-10=$40
Profit from selling parka = p-c = 100-45 = $55
Cost of understocking = $55/unit
Cost of overstocking = c-v = 45-40 = $5/unit
21
utdallas.edu/~metin
p = sale price; v = outlet or salvage price; c = purchase
price
CSL = Probability that demand will be at or below order
quantity
CSL later called in-stock probability
Raising the order size if the order size is already optimal
Expected Marginal Benefit of increasing Q= Expected Marginal Cost
of Underage
=P(Demand is above stock)*(Profit from sales)=(1-
CSL)(p - c)
Expected Marginal Cost of increasing Q = Expected marginal cost of
overage
=P(Demand is below stock)*(Loss from
discounting)=CSL(c - v)
Define C
o
= c-v; C
u
=p-c
(1-CSL)C
u
= CSL C
o
CSL= C
u
/ (C
u
+ C
o
)
22
utdallas.edu/~metin
C
o
= Cost of overstocking = $5
C
u
= Cost of understocking = $55
Q
*
= Optimal order size
917 . 0
5 55
55
) (
*
=
+
=
+
> s =
o u
u
C C
C
Q Demand P CSL
23
utdallas.edu/~metin
0
0.2
0.4
0.6
0.8
1
1.2
4 5 6 7 8 9 10 11 12 13 14 15 16 87
Cumulative
Probability
24
Optimal Order Quantity = 13(‘00)
0.917
utdallas.edu/~metin
? Expected demand = 10 („00) parkas
? Expected profit from ordering 10 („00) parkas =
$499
? Approximate Expected profit from ordering 1(„00)
extra parkas if 10(?00) are already ordered
= 100.55.P(D>=1100) - 100.5.P(D<1100)
25
utdallas.edu/~metin
Additional
100s
Expected
Marginal Benefit
Expected
Marginal Cost
Expected Marginal
Contribution
11
th
5500×.49 = 2695 500×.51 = 255
2695-255 = 2440
12
th
5500×.29 = 1595 500×.71 = 355
1595-355 = 1240
13
th
5500×.18 = 990 500×.82 = 410
990-410 = 580
14
th
5500×.08 = 440 500×.92 = 460
440-460 = -20
15
th
5500×.04 = 220 500×.96 = 480
220-480 = -260
16
th
5500×.02 = 110 500×.98 = 490
110-490 = -380
17
th
5500×.01 = 55 500×.99 = 495
55-495 = -440
26
utdallas.edu/~metin
? Total cost by ordering Q units:
? C(Q) = overstocking cost + understocking
cost
27
} }
·
÷ + ÷ =
Q
u
Q
o
dx x f Q x C dx x f x Q C Q C ) ( ) ( ) ( ) ( ) (
0
0 )) ( 1 ( ) (
) (
= ÷ ÷ = Q F C Q F C
dQ
Q dC
u o
Marginal cost of overage at Q* - Marginal cost of underage at Q* = 0
u o
u
C C
C
Q D P Q F
+
= s = ) ( ) (
* *
utdallas.edu/~metin
Inventory held in addition to the
expected demand is called the
safety stock
The expected demand is 1026 parkas but
we order 1300 parkas.
So the safety stock is 1300-1026=274
parka.
28
utdallas.edu/~metin
29
utdallas.edu/~metin
? For any order quantity we would like to evaluate the
following performance measures:
? Expected lost sales
? The average number of units demand exceeds the order quantity
? Expected sales
? The average number of units sold.
? Expected left over inventory
? The average number of units left over at the end of the season.
? Expected profit
? Expected fill rate
? The fraction of demand that is satisfied immediately
? In-stock probability
? Probability all demand is satisfied
? Stockout probability
? Probability some demand is lost
30
utdallas.edu/~metin
? ESC is the expected shortage in a season
? ESC is not a percentage, it is the number of units, also see
next page
¹
´
¦
s
>
=
Q Demand
Q Demand
if
if
0
Q - Demand
Shortage
}
·
=
=
÷ =
Q x
Q)f(x)dx - (x ESC
Q,0}) season a in nd E(max{Dema ESC
31
utdallas.edu/~metin
32
0
Q
Demand
During a
Season
Season
Inventory
D, Demand
During
A Season
0
0
Q
Inventory=Q-D
Upside
down
utdallas.edu/~metin
33
0
Q
Demand
Season
Shortage
D
:
D
e
m
a
n
d
D
u
r
i
n
g
s
S
e
a
s
o
n
0
0
Q
Shortage
=D-Q
Upside
down
utdallas.edu/~metin
? First let us study shortage during the lead time
4
1
4
1
10)} - (11 max{0,
4
2
10)} - (10 max{0,
4
1
10)} - (9 max{0,
) ( )} (d )} ( max{0, shortage Expected
Shortage? Expected ,
1/4 prob with 11
2/4 prob with 10
1/4 prob with 9
, 10
11
10 d
3
1 i
3 3
2 2
1 1
= + + =
= ÷ = ÷ =
¦
)
¦
`
¹
¦
¹
¦
´
¦
= =
= =
= =
= =
¿ ¿
= =
d D P Q p Q d
p d
p d
p d
D Q
i i
34
D. of pdf is f where ) ( ) (
}) 0 , (max{ shortage Expected
D
}
·
=
÷ =
÷ =
Q D
D
dD D f Q D
Q D E
? Ex:
utdallas.edu/~metin
2/6
) 10 ( 10
2
10
6
1
) 12 ( 10
2
12
6
1
10
2 6
1
6
1
) 10 ( shortage Expected
Shortage? Expected ), 12 , 6 ( , 10
2 2
12
10
2
12
10
=
|
|
.
|
\
|
÷ ÷
|
|
.
|
\
|
÷ = ÷
\
|
= ÷ =
= =
=
=
=
}
D
D
D
D
D
dD D
Unif orm D Q
35
? Ex:
utdallas.edu/~metin
? Step 1: normalize the order quantity to find its z-statistic.
? Step 2: Look up in the Standard Normal Loss Function Table the
expected lost sales for a standard normal distribution with that z-
statistic: L(0.26)=0.2824 see Appendix B table on p.380 of the
textbook
? or, in Excel L(z)=normdist(z,0,1,0)-z*(1-normdist(z,0,1,1)) see Appendix
D on p.389
? Step 3: Evaluate lost sales for the actual normal distribution:
36
26 . 0
1181
3192 3500
=
÷
=
÷
=
o
µ Q
z
( ) 1181 0.2824 334 Expected lost sales L z o = × = × =
Keep 334 units in mind, we shall repeatedly use it
utdallas.edu/~metin
? Demand=Sales+Lost Sales
D=min(D,Q)+max{D-Q,0} or min(D,Q)=D- max{D-
Q,0}
Expected sales = µ - Expected lost sales
= 3192 – 334 = 2858
? Inventory=Sales+Left Over Inventory
Q=min(D,Q)+max{Q-D,0} or max{Q-D,0}=Q-
min(D,Q)
Expected Left Over Inventory = Q - Expected Sales
= 3500 – 2858 = 642
37
utdallas.edu/~metin
Expected total underage and overage cost with
(Q=3500)
=70*334 + 20*642
38
( )
( )
( ) ( )
$70 2858 $20 642 $187, 221
Expected profit Price-Cost Expected sales
Cost-Salvage value Expected left over inventory
= × (
¸ ¸
÷ × (
¸ ¸
= × ÷ × =
What is the relevant objective? Minimize the cost or maximize the profit?
Hint: What is profit + cost? It is 70*3192=C
u
*?, which is a constant.
utdallas.edu/~metin
39
) Q P(Demand bability InstockPro s =
Instock probability: percentage of seasons without a stock out
Q] season a during [Demand inventory] t [Sufficien
inventory sufficient has season single a y that Probabilit 0.7 y Probabilit Instock
7 . 0 y Probabilit Instock
otherwise 1 stockout, has season a if 0 Write
10
1 0 1 0 1 1 1 0 1 1
y Probabilit Instock
: seasons 10 consider example For
s =
= =
=
+ + + + + + + + +
=
utdallas.edu/~metin
40
( )
( )
|
.
|
\
|
÷
=
|
.
|
\
|
÷
s =
|
.
|
\
|
÷
s
÷
=
÷ s ÷ =
=
s
1 , 1 , 0 , Normdist
on distributi normal standard Obtaining ) 1 , 0 (
StDev by the Dividing
) , (
mean out the Taking ) , (
,1) , , Normdist(Q
) , (
o
µ
o
µ
o
µ
o
µ o µ
µ µ o µ
o µ
o µ
Q
Q
N P
Q N
P
Q N P
Q N P
N(?,?) denotes a normal demand with mean ? and standard deviation ?
utdallas.edu/~metin
? = 2,500 /week; o= 500; Q = 3,000;
Instock probability if demand is Normal?
Instock probability = Normdist((3,000-
2,500)/500,0,1,1)
41
utdallas.edu/~metin
? = 2,500/week; o= 500; To achieve Instock
Probability=0.95, what should be Q?
Q = Norminv(0.95, 2500, 500)
42
utdallas.edu/~metin
Recall:
Expected sales = µ - Expected lost sales = 3192 –
334 = 2858
43
2858
1
3192
89.6%
Expected sales Expected sales
Expected fill rate
Expected demand
Expected lost sales
µ
µ
= =
= ÷ =
=
Is this fill rate too low?
Well, lost sales of 334 is with Q=3500, which is less than optimal.
utdallas.edu/~metin
44
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 1000 2000 3000 4000 5000 6000 7000
Order quantity
In-stock probability
CSL
Expected fill
rate
utdallas.edu/~metin
45
CSL is 0%, fill rate is almost 100%
CSL is 0%, fill rate is almost 0%
inventory
inventory
time
time
0
0
utdallas.edu/~metin
? Determine the optimal level of product availability
? Demand forecasting
? Profit maximization / Cost minimization
? Other measures
? Expected shortages = lost sales
? Expected left over inventory
? Expected sales
? Expected cost
? Expected profit
? Type I service measure: Instock probability = CSL
? Type II service measure: Fill rate
46
utdallas.edu/~metin
? Q1: Suppose that you own a simple call option
for a stock with a strike price of Q. Suppose
that the price of the underlying stock is D at
the expiration time of the option. If D <Q, the
call option has no value. Otherwise its value is
D-Q. What is the expected value of a call
option, whose strike price is $50, written for a
stock whose price at the expiration is normally
distributed with mean $51 and standard
deviation $10?
47
doc_526646333.pptx