HW11_sol

HW11_sol - IE 335 Operations Research Optimization...

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IE 335 Operations Research - Optimization Solutions to Homework 11 Spring 2011 Problem 51 Let us deﬁne the decision variable x i for i = 1 ,..., 6 as: x i = ± 1 if a monitoring device is installed at intersection i 0 otherwise Minimize 42 x 1 + 53 x 2 + 61 x 3 + 44 x 4 + 78 x 5 + 39 x 6 minimize the total cost x 1 + x 2 1 monitoring road a x 1 + x 4 1 monitoring road b x 2 + x 3 1 monitoring road c x 2 + x 5 1 monitoring road d x 3 + x 5 1 monitoring road e x 3 + x 6 1 monitoring road f x 4 + x 5 1 monitoring road g x 5 + x 6 1 monitoring road h x 1 x 4 if installed at intersection 1, so is intersection 4 x 2 1 - x 1 if installed at 2, then cannot be at 1 x 2 1 - x 3 if installed at 2, then cannot be at 3 x i ∈ { 0 , 1 } for i = 1 ,..., 6 Problem 52 Let us deﬁne the decision variable y i for i = 1 ,..., 12 as: y i = ± 1 if block i is removed 0 otherwise 1

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maximize 12000 y 1 + 25000 y 8 + 100000 y 12 - 10000( y 1 + y 2 + y 3 + y 4 + y 5 + y 6 ) - 20000( y 8 + y 9 + y 10 ) - 30000 y 12 - 100000( y 7 + y 11 ) subject to y 7 y 1 ,y 7 y 2 ,y 7 y 3 if block 7 is extracted, blocks 1,2,3 must be extracted y 8 y 2 ,y 8 y 3 ,y 8 y 4 if block 8 is extracted, blocks 2,3,4 must be extracted y 9 y 3 ,y 9 y 4 ,y 9 y 5 if block 9 is extracted, blocks 3,4,5 must be extracted y 10 y 4 ,y 10 y 5 ,y 10 y 6 if block 10 is extracted, blocks 4,5,6 must be extracted y 11 y 7 ,y 11 y 8 ,y 11 y 9 if block 11 is extracted, blocks 7,8,9 must be extracted y 12 y 8 ,y 12 y 9 ,y 12 y 10 if block 12 is extracted, blocks 8,9,10 must be extracted Problem 53 11-37 Let us deﬁne the decision variable c j for j = 1 ,..., 22 as: x j = ± 1 if proposal j is selected 0 otherwise maximize 22 X j =1 r j x j maximize merit score subject to 5 X j =1 x j 3 , 11 X j =6 x j 3 , 17 X j =12 x j 3 , 22 X j =18 x j 3 At least 3 from each region 22 X j =1 b j x j 15 Budget constraint of 15 million 22 X j =1 S j x j 11 # of smokers at least 11 million x 2 + x 7 + x 11 + x 19 2 At least 2 states must be selected x 1 + x 4 + x 13 + x 14 + x 21 2 from each of these outlier groups x i ∈ { 0 , 1 } for i = 1 ,..., 22 Problem 54 11-38 Let us deﬁne the decision variable x ij for i = 1 ,..., 60 and for j = 1 ,..., 20 as: x ij = ± 1 if sector i is assigned to disposal j 0 otherwise 2
60 X i =1 20 X j =1 d ij f i x ij minimize total distance times volume moved subject to 20 X j =1 x ij = 1 for i = 1 ,..., 60 Each sector is assigned site j as its primary disposal point 60 X i =1 f i x ij u j for j = 1 ,..., 20 Total volume assigned to one disposal point within capacity 60 X i =1 r i x ij b j for j = 1 ,..., 20 Removal rate assigned to each disposal point within receiving range x ij ∈ { 0 , 1 } for i = 1 ,..., 60 and for j = 1 ,..., 20 Problem 59 Refer to next pages for GAMS. 3

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HW11_sol - IE 335 Operations Research Optimization...

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