Hw1_Sol - x ijg = students from neighborhood i assigned to...

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Math464 - HW 1 Solutions Due on Thursday, Jan 21 1 Linear Optimization (Spring 2010) Brief solutions to Homework 1 1. The standard form LP is the following. min 3 x 1 + ( x + 3 - x - 3 ) s.t. - x 0 2 - 4( x + 3 - x - 3 ) - e 1 = 4 x 1 - 5 x 0 2 - ( x + 3 - x - 3 ) + s 2 = 2 x 1 , x 0 2 , x + 3 , x - 3 , e 1 , s 2 0 2. Let x j = # units of product j made and sold, for j = 1 , 2. Here is the LP. max (6 - 3) x 1 + (5 . 4 - 2) x 2 (net income) s.t. 3 x 1 + 4 x 2 20000 (total machine hrs) 3 x 1 + 2 x 2 4000 + 0 . 45 · 6 x 1 + 0 . 30 · 5 . 4 x 2 (cash availability) x 1 ,x 2 0 (non-negativity) 3. Let x j = # units of product j made, for j = 1 , 2. Here is the LP. max (9 - 1 . 2) x 1 + (8 - 0 . 9) x 2 (daily profit) s.t. (1 / 4) x 1 + (1 / 3) x 2 90 (assembly hours) (1 / 8) x 1 + (1 / 3) x 2 80 (testing hours) x 1 ,x 2 0 (non-negativity) 4. Let x j = number of times process j is used, for j = 1 , 2 , 3. The LP is given below. max 38 (4 x 1 + x 2 + 3 x 3 ) + 33 (3 x 1 + x 2 + 4 x 3 ) - (51 x 1 + 11 x 2 + 40 x 3 ) (net revenue) s.t. 3 x 1 + x 2 + 5 x 3 8 , 000 , 000 (crude A supply) 5 x 1 + x 2 + 3 x 3 5 , 000 , 000 (crude B supply) x 1 , x 2 , x 3 0 (non-negativity) 5. We choose the decision variables as follows:
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Unformatted text preview: x ijg = # students from neighborhood i assigned to school j in grade g, for i ∈ I,j ∈ J,g ∈ G. The LP is given below: min ∑ i ∈ I ∑ j ∈ J d ij ∑ g ∈ G x ijg ! (total distance traveled) s.t. ∑ i ∈ I x ijg ≤ C jg , ∀ j ∈ J,g ∈ G (max capacity of a school for a grade) ∑ j ∈ J x ijg = S ig ∀ i ∈ I,g ∈ G (assign all students) all x ijg ≥ (non-negativity) One could also solve a separate problem for each grade g ∈ G , as the description of the problem essentially separates the problem for each grade into an individual assignment task. Of course, the above single LP will solve the entire problem in one step....
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This note was uploaded on 02/15/2011 for the course EECS 6.231 taught by Professor Bertsekas during the Spring '10 term at MIT.

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