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7/24/2019 NVM Example
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xample: Mrs. Kandells Christmas Tree Shop
Order for Christmas trees must be placed in Sept
If she orders too few, the unit shortage costis cu= 55 25=
$30
If she orders too many, the unit overage costis co= 25 5= $0
Sales 22 24 26 28 30 32 34 36
Probability .05 .10 .15 .20 .20 .15 .10 .05
!ast"ata
Cost per tree# $25 !rice per tree#$55 before "ec 25
$5 after "ec 25
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7/24/2019 NVM Example
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Stockout and Markdown Risks
D total demand before Christmas
Fx% the demand distribution,
D& Q stoc'out, at a cost of# cuD Q%() cuma*+D
Q, 0
D- Q
o.erstoc', at a cost of coQD%(
= coma*+Q D, 0
/ rs/ 1andell has only one chanceto orderuntil the sales bein#
no information to re.ise theforecast
after the sales start# too late to order more/
2/ She has to decide an order 4uantity Qnow
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7/24/2019 NVM Example
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Model development
Stoc'out cost ) cuma*+D Q, 0
O.erstoc' cost = coma*+Q
D, 0
otal cost ) 67% ) cuD Q%(
( coQ D%(
8*pected cost, 8 67% % ) 8cuD Q%(( co
Q D%(%
) cu8D Q%(( co8Q D%
(
=
+
=
+
=
+++=+=
Q
x
o
Qx
u
x
ou xPxQcxPQxcxPxQcQxc00
)(])([)(])([)(])()([
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7/24/2019 NVM Example
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odel Development: enerali!ation
=
+
=
+ +=Q
x
o
Qx
u xPxQcxPQxcQGE0
)(])([)(])([))((
==
+==Qx
u
Q
x
dxxPQxcdxxPxQcQGEQg )()()()())(()(0
0
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7/24/2019 NVM Example
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Model solution
7% is a con.e* function# it has a uni4ue minimum
when 7% is at minimum .alue, 97% ) cu:cu( co%
inimi;e 7%
==+==
Qx
u
Q
x
dxxPQxcdxxPxQcQGEQg )()()()())(()(0
0
0)()()()(0
0 =
+
== Qxu
Q
x
dxxPQxcdxxPxQcdQ
d
0)(=
dQ
Qgd
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7/24/2019 NVM Example
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he Critical
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7/24/2019 NVM Example
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Mrs. Kandells "ro#lem$ solved:
= cu/co+ cu% ) 30:30 ( 0% ) 0/>5
cu= 55 25= $30co= 25 5= $0
!ast"ata
D 22 24 26 28 30 32 34 36
Probability 0.05 0.1 0.15 0.2 0.2 0.15 0.1 0.05
F(D ) 0.05 0.15 0.3 0.5 0.7 0.85 0.95 1
optimum ? 3
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7/24/2019 NVM Example
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ewsvendor model: e&ect o' critical ratio
= cu/co+ cu% ) 30:30 ( 0% ) 0/>5 optimum# 3
D 22 24 26 28 30 32 34 36
Probability 0.05 0.1 0.15 0.2 0.2 0.15 0.1 0.05
F(D ) 0.05 0.15 0.3 0.5 0.7 0.85 0.95 1
o.erstoc' cost less sini@cant order more
o.erstoc' cost dominates order less
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7/24/2019 NVM Example
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Summar(
hen demand is uncertain, we minimi;e expected costs
ews.endor model# sinle period, with o.erB and underBstoc'
cost
ritical ratio determines the optimum order point
ritical ratio aects the direction and manitude of order
4uantit
