1
15.053
February 26, 2002
z
Sensitivity Analysis
z
presented as FAQs
–
Points illustrated on a running example of
glass manufacturing.
–
If time permits, we will also consider the
financial example from Lecture 2.
2
Glass Example
z
x
1
= # of cases of 6oz juice glasses (in 100s)
z
x
2
= # of cases of 10oz cocktail glasses (in 100s)
z
x
3
= # of cases of champagne glasses (in 100s)
max
5 x
1
+
4.5 x
2
+
6 x
3
($100s)
s.t
6 x
1
+
5 x
2
+
8 x
3
≤
60
(prod. cap. in hrs)
10 x
1
+
20 x
2
+ 10 x
3
≤
150
(wareh. cap. in ft
2
)
x
1
≤
8
(60z. glass dem.)
x
1
≥
0, x
2
≥
0, x
3
≥
0
3
FAQ. Could you please remind me what a
shadow price is?
z
Let us assume that we are maximizing.
A shadow price is the increase in the
optimum objective value per unit increase
in a RHS coefficient, all other data
remaining equal.
z
The shadow price is valid in an interval.
4
FAQ. Of course, I knew that.
But can you
please provide an example.
z
Certainly.
Let us recall the glass example given in the
book.
Let’s look at the objective function if we change
the production time from 60 and keep all other values
the same.
11/14
53 11/14
63
11/14
53
62
11/14
52
3/14
61
51
3/7
60
difference
Optimal
obj. value
Production
hours
The
shadow
Price is
11/14.
5
More changes in the RHS
15/22
56 17/22
67
*
56 1/11
66
11/14
55 5/14
65
11/14
54
4/7
64
difference
Optimal
obj. value
Production
hours
The
shadow
Price is
11/14 until
production
= 65.5
6
FAQ. What is the intuition for the shadow
price staying constant, and then changing?
z
Recall from the simplex method that the
simplex method produces a “basic
feasible solution.”
The basis can often be
described easily in terms of a brief verbal
description.
Glass Example
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The verbal description for the
optimum basis for the glass problem:
1.
Produce Juice Glasses
and cocktail glasses only
2.
Fully utilize production
and warehouse capacity
z =
5 x
1
+ 4.5 x
2
6 x
1
+ 5
x
2
=
60
10 x
1
+ 20 x
2
=
150
x
1
= 6 3/7
x
2
= 4 2/7
z =
51 3/7
8
The verbal description for the
optimum basis for the glass problem:
1.
Produce Juice Glasses
and cocktail glasses only
2.
Fully utilize production
and warehouse capacity
z =
5 x
1
+ 4.5x
2
6 x
1
+ 5
x
2
=
60 +
∆
10 x
1
+ 20 x
2
=
150
x
1
= 6 3/7 + 2
∆
/7
x
2
= 4 2/7 –
∆
/7
z =
51 3/7 + 11/14
∆
For
∆
= 5.5,
x
1
= 8, and the
constraint x
1
<= 8
is binding.
9
FAQ. How can shadow prices be used
for managerial interpretations?
z
Let me illustrate with the previous
example.
z
How much should you be willing to pay for
an extra hour of production?
Glass Example
10
FAQ. Does the shadow price always
have an economic interpretation?
z
The answer is no, unless one wants
to really stretch what is meant by an
economic interpretation.
z
Consider ratio constraints
11
Apartment Development
z
x
1
= number of 1bedroom apartments built
z
x
2
= number of 2bedroom apartments built
z
x
3
= number of 3bedroom apartments build
z
x
1
/(x
1
+ x
2
+ x
3
)
≤
.5
Î
x
1
≤
.5x
1
+ .5x
2
+ .5x
3
z
Î
.5x
1
–5
.x
2
.5x
3
≤
0
z
The shadow price is the impact of increasing
the 0 to a 1.
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 Spring '05
 Prof.JamesOrlin
 Operations Research, Linear Programming, Optimization, Shadow price, Reduced cost

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