hw2sols - C O250/CM340 I NTRODUCTION TO O PTIMIZATION HW 2...

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CO250/CM340 I NTRODUCTION TO O PTIMIZATION - HW 2 - SOLUTIONS Exercise 1 (20 marks) . (a) Let x i be 1 if block i is mined and 0 otherwise ( i = 1 , 2 , 3 , 4 , 5 for A, B, C, D, E respectively). The formulation without matrices would be max 25 x 1 + 13 x 2 + 15 x 3 + 20 x 4 + 12 x 5 subject to 7 x 1 + 12 x 2 + 4 x 3 + 12 x 4 + 9 x 5 20 2 x 1 + x 2 + x 3 + 2 x 4 + x 5 4 0 x i 1 ( i = 1 , . . . , 5 ) x i integer ( i = 1 , . . . , 5 ) It was asked to formulate this in terms of matrices and vectors, which becomes max [25 13 15 20 12] x subject to Ax " 20 4 # 0 x 1 x i integer ( i = 1 , . . . , 5 ) where x = [ x 1 , x 2 , x 3 , x 4 , x 5 ] T , A = " 7 12 4 12 9 2 1 1 2 1 # , 0 = [0 0 0 0 0] T , 1 = [1 1 1 1 1] T . OR As an alternative, multiply the second inequality by - 1 and replace the constraint involving A by Ax [20 - 4] T where A is now " 7 12 4 12 9 - 2 - 1 - 1 - 2 - 1 # . (b) Introduce a variable x 6 such that x 6 = 1 represents that we exercise the option of increasing the amount of Impurity X to 5 tonnes, and x 6 = 0 stands for not exercising the option. We just need to subtract 2 x 6 1
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