hw5-sol - Homework V Deterministic Optimization ISYE 6669...

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Unformatted text preview: Homework V Deterministic Optimization ISYE 6669 7.5.1 - Both Editions Define I = { 1 , 2 , 3 , 4 , 5 } to be the set of workers and J = { 1 , 2 , 3 , 4 , 5 } be the set of jobs, where 5 is a dummy job. Then the formulation for the problem is given by min i I,j J c ij x ij s.t. j J x ij = 1 i I i I x ij = 1 j J x ij { , 1 } where c ij is cost of worker i performing job j and is given in Table () Job 1 2 3 4 5 1 22 18 30 18 2 18 M 27 22 Person 3 26 20 28 28 4 16 22 M 14 5 21 M 25 28 Table 1: Cost of worker i performing job j 8.2.1 - Both Editions First label node 1 with a permanent label: [0* 7 12 21 31 44] Now node 2 receives a permanent label [0* 7* 12 21 31 44] Node Temporary Label (* denotes next assigned permanent label) 3 min (12, 7+7) = 12* 4 min (21, 7+12) = 19 5 min (31, 7+21) = 28 6 min (44, 7+31) = 38 Now labels are [0* 7* 12* 19 28 38] Node Temporary Label (* denotes next assigned permanent label) 4 min (19, 12+7) = 19* 5 min (28, 12+12) = 24 6 min (38, 12+21) = 33 1 Now labels are [0* 7* 12* 19* 24 33] Node Temporary Label (* denotes next assigned permanent label) 5 min (24, 19+7) = 24* 6 min (33, 19+12) = 31 Now labels are [0* 7* 12* 19* 24* 31] Node Temporary Label (* denotes next assigned permanent label) 6 min (31, 24+7) = 31 Now labels are [0* 7* 12* 19* 24* 31*] 31 - 24 = c 56 , 24 - 12 = c 35 , 12 - 0 = c 13 . Thus 1-3-6 is the shortest path (of length 31) from node 1 to node 6. 8.2.7 Let c ij = cost incurred if a machine is purchased at beginning of year I and is kept until beginning of year j. Then we may formulate the problem of minimizing total cost incurred over 5 years as the following transshipment problem: 2 3 4 5 6 1 208...
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hw5-sol - Homework V Deterministic Optimization ISYE 6669...

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