1
15.093J/2.098J Optimization Methods
Assignment 5 Solutions
Exercise 5.1
BT, Exercise 10.2.
The decision variables are
x
i
,
i
= 1
,
20.
x
i
= 1 if the player
p
i
is selected; otherwise,
p
i
=0. We have:
The total number of players in the team is 12:
20
i
=1
x
i
=12
The team has at least 3 play makers:
5
i
=1
x
i
≥
3
The team has at least 4 shooting guards:
11
i
=4
x
i
≥
4
The team has at least 4 forwards:
16
i
=9
x
i
≥
4
The team has at least 3 centers:
20
i
=16
x
i
≥
3
The team has at least 2 NCAA players:
x
4
+
x
8
+
x
15
+
x
20
≥
2
Average rebounding statistics constraint:
20
i
=1
r
i
x
i
≥
12
r
Average assists statistics constraint:
20
i
=1
a
i
x
i
≥
12
a
Average scoring statistics constraint:
20
i
=1
s
i
x
i
≥
12
s
Average height statistics constraint:
20
h
i
x
i
≥
12
h
i
=1
Average defense ability statistics constraint:
20
d
i
x
i
≥
12
d
i
=1
Player
p
5
is not in the team if the player
p
9
is in the team:
x
5
≤
1
−
x
9
Player
p
2
and
p
19
can only be selected together:
x
2
=
x
19
At most 3 players from the same team (
p
1
,p
7
,p
12
,p
16
) are selected:
x
1
+
x
7
+
x
12
+
x
16
≤
3
With these constraints, the problem for the coach is to maximize the scoring average or total the score
20
i
=1
s
i
x
i
Exercise 5.2
BT, Exercise 10.4.
Consider
x
ij
is the number of module
i
we need to purchase in the year
j
,where
i
= 1
,
5 representing module A,
B, C, D, and complete engine respectively and
j
= 1
,
3. We have:
x
ij
∈
Z
+
for all
i
and
j
. We also need to know
how many complete engines that will be broken into modules each year, let denote this quantity as
x
6
j
,wehave
,
x
6
j
∈
Z
+
and
x
6
j
≤
x
5
j
.
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 Spring '10
 Bertsekas
 Optimization, demand constraint

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