hw4.sol

hw4.sol - IE 335 Operations Research - Optimization...

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IE 335 Operations Research - Optimization Solutions to Homework 4 Spring 2012 Problem 16 Let’s use the decision variables given in the hint: x i = fraction of item i allocated to Captain Hook i = 1 ,..., 4; y i = fraction of item i allocated to Captain Sparrow i = 1 ,..., 4 . If we allocate the items according to the decision variables above, then we can write the total value points of Captain Hook as 40 x 1 + 10 x 2 + 10 x 3 + 40 x 4 . Similarly, we can write the total value points of Captain Sparrow as 30 y 1 + 20 y 2 + 20 y 3 + 30 y 4 . Then, the following optimization model with a maximin objective determines how to allocate the treasure between Captain Hook and Captain Sparrow, in a way that maximizes the minimum total value points of any captain: max min { 40 x 1 + 10 x 2 + 10 x 3 + 40 x 4 , 30 y 1 + 20 y 2 + 20 y 3 + 30 y 4 } (1) s.t. x i + y i 1 for i = 1 , 2 , 3 , 4 , (2) x i 0 for i = 1 , 2 , 3 , 4 , (3) y i 0 for i = 1 , 2 , 3 , 4 . (4) Constraint (2) ensures that at most 100% of each item is allocated to the captains. Constraints (3) and (4) ensure that the fractions are nonnegative. We can convert this optimization model into a linear program, using the technique we discussed in Lecture 9. Let f be an auxiliary decision variable that represents the minimum total value points of any captain. Then the above optimization model is equivalent to the following linear program: max f s.t. f 40 x 1 + 10 x 2 + 10 x 3 + 40 x 4 , f 30 y 1 + 20 y 2 + 20 y 3 + 30 y 4 , x i + y i 1 for i = 1 , 2 , 3 , 4 , x i 0 for i = 1 , 2 , 3 , 4 , y i 0 for i = 1 , 2 , 3 , 4 . 1
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Problem 17 (a) We use the technique covered in class to convert the non-linear absolute value objective function to a linear objective function through the use of an auxillary variable t i . min m X i =1 t i s.t. t i b i - n X j =1 a i j x j for i = 1 ,...,m t i ≥ - ( b i - n X j =1 a i j x j ) for i = 1 ,...,m (b) Similarly, we use the techniques used to handle minimax objective functions with those used for absolute value objective functions to obtain the following linear program: min t s.t. t b i - n X j =1 a i j x j for i = 1 ,...,m t ≥ - ( b i - n X j =1 a i j x j ) for i = 1 ,...,m Problem 18 Using GAMS, solve the linear program you formulated in Problem 13. 2
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C : \ U s e r s \ T i m n t \ D o c u m e n t s \ g a m s d i r \ p r o j d i r \ P 1 8 . l s t 2011年2月3日 15:45:06 Page 1 1 GAMS Rev 235 WIN-VS8 23.5.2 x86/MS Windows 02/03/11 15:29:46 Page 1 2 G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m 3 C o m p i l a t i o n 4 5 6 1 variables 7 2 objval "objective function value"
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hw4.sol - IE 335 Operations Research - Optimization...

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