MthSc810 – Mathematical Programming
Fall 2011, Homework #4. Due Thursday, September 22, 2011, 6PM EDT.
Exercise 1
Implement, in AMPL, the solution you provided for Exercise 3 of Homework 2.
Solution.
Model and data are here shown in the same chunk.
param n;
set P := 1..n; # projects
param r {P};
# return of each project
param s {P};
# initial investment per project
param q {P};
# number of employees per project
param B;
# budget
param W;
# maximum number of employees
var x {P} binary; # x[i]=1 means we choose project i, 0 means we don’t
maximize total_return: sum {i in P} r [i] * x [i];
budget_cap:
sum {i in P} s [i] * x [i] <= B;
employee_cap: sum {i in P} q [i] * x [i] <= W;
data;
param n := 5;
param B := 28;
param W := 50;
param r :=
1 19 2 41
3 15 4 22
5 30;
param s :=
1 12
2 29
3 9
4 14
5 18;
param q :=
1 25
2 22
3 31
4 25
5 30;
Exercise 2
Write an AMPL model for the following problem:
A large company is mixing a set of
k
ingredients
I
=
{
1
,
2
. . . , k
}
to
produce a new type of paint. Each such ingredient has a unitary cost
c
i
,
i
∈
I
. A subset
H
⊂
I
of the ingredients are toxic chemicals. Find the
percentage
x
i
of each ingredient in the paint such that
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–
the percentage of each component does not exceed
α
;
–
the
total
percentage of toxic components does not exceed
β
;
–
there are at most
γ
ingredients in total (toxic and nontoxic);
–
the total cost is minimum.
Then solve it for the following input:
–
k
= 10;
– c
= (12
,
6
,
9
,
10
,
7
,
4
,
7
,
8
,
11
,
4);
–
H
=
{
2
,
6
,
7
,
10
}
;
–
α
= 0
.
3 (i.e., 30%);
–
β
= 0
.
1;
–
γ
= 5.
Write two separate files—one for the model and another one for the data.
Solve this
instance
and report on the results (i.e., the values of
x
). Hint: integer
variables are needed.
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 Fall '08
 Staff
 Math, Optimization, Halle Berry, Famke Janssen, param param param

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