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homework-4-solutions

# homework-4-solutions - MthSc810 Mathematical Programming...

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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 non-toxic); 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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