DMOR OEM 2007 Homework 2 Solution Key
Solution problem 1:
(questions, comments, suggestions please to Zeki Caner Taskin,
[email protected]
)
a) The idea is to define a
pair
of decision variables
for each
absolute value
term. Define decision variables
e
+
i
and
e

i
i
= 1
, . . . ,
8 that represent the amount
of positive and negative discrepancy, respectively, between the amount of fuel
consumption predicted by the model for data point
i
, and the realized fuel
consumption.
Using these decision variables, we can linearize the objective
function using these decision variables as follows.
min
e
+
1
+
e

1
+
e
+
2
+
e

2
+
· · ·
+
e
+
8
+
e

8
(
A
)
s.t.
e

1

e
+
1
= 1100

17
p
1

22
p
2
e

2

e
+
2
= 1420

25
p
1

22
p
2
. . .
e

8

e
+
8
= 2850

60
p
1

41
p
2
e

i
, e
+
i
≥
0
The structure of the LP model guarantees that at most one of
e
+
i
and
e

i
can
be positive in an optimal solution, and hence
e
+
1
+
e

1
=

1100

17
p
1

22
p
2

,
and so on. The optimal solution is
p
1
= 23
.
98
, p
2
= 34
.
41, with an objective
function value of 412
.
26.
Another way of thinking about this is the following. Ideally, we want to find
p
1
, p
2
so that our model can
perfectly
explain the relationship between altitude,
distance and fuel usage. We can define the following LP model for finding
p
1
, p
2
.
min
0
(
B
)
s.t.
0 = 1100

17
p
1

22
p
2
0 = 1420

25
p
1

22
p
2
. . .
0 = 2850

60
p
1

41
p
2
Since we just want to find a feasible pair of multipliers
p
1
, p
2
to be used in our
linear regression, we don’t need an objective function. Any feasible solution of
this model is a good set of multipliers that can be used in the linear regression
model to explain the relationship between altitude, distance and fuel usage.
However, it is highly unlikely that a
perfectly linear
relationship exists between
those factors.
In other words, the LP presented above is highly likely to be
infeasible. Therefore we convert the constraints to “soft constraint” presented
in class, and associate
e
+
i
and
e

i
i
= 1
, . . . ,
8 variables with each constraint to
measure the amount of violation. We set the objective function to minimize the
total amount of violation. The resulting model is exactly the same as Model A.
b) Even though this part looks nonlinear, it is not. The trick is that all nonlinear
terms are just functions of the data, not the decision variables. Therefore, we
1
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can just calculate the values of
D/A
and
D
2
for each data point, and then use
the same modeling idea as in part a).
min
e
+
1
+
e

1
+
e
+
2
+
e

2
+
· · ·
+
e
+
8
+
e

8
s.t.
e

1

e
+
1
= 1100

(17
/
22)
p
1

17
2
p
2
e

2

e
+
2
= 1420

(25
/
22)
p
1

25
2
p
2
. . .
e

8

e
+
8
= 2850

(60
/
41)
p
1

60
2
p
2
e

i
, e
+
i
≥
0
The optimal solution is
p
1
= 1329
.
59
, p
2
= 0
.
25, with an objective function
value of 1357
.
78. Since the sum of errors in the first model is less, it is a better
model in explaining the relationship between the factors given.
Most common mistakes:
•
p
1
and
p
2
are not necessarily nonnegative in this problem. They should
be left as unrestricted.
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 Fall '09
 VLADIMIRLBOGINSKI
 Optimization, objective function, Shortest path problem, everglades, Zeki Caner Taskin

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