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Unformatted text preview: Tutorial 10: Outline of Solutions Q1. Adam and Eve are planning a drive from location A to E. The time to travel and the scenic rating for the roads in the network are given below. All roads are oneway. Adam wants to reach location E as fast as possible while Eve wants to take the most scenic drive to location E (a higher scenic rating implies a prettier scene). Road Time (hours) Scenic Rating (points) A to B 3.0 3 A to C 2.5 4 B to D 1.7 5 B to E 2.8 7 C to B 1.7 4 C to F 2.5 3 D to E 2.0 9 F to E 2.0 8 (a) Formulate the linear programs that Adam and Eve need to solve respectively. Do you think they can find a common optimal driving route? (b) Suppose Eve agrees to use Adams least time policy under the condition that they will take the highly scenic road D to E. How would you modify Adams linear program in this case? (c) Suppose Adam and Eve need to pick up John from location C on their drive from A to E. How would you modify Adams linear program in this case? Solution : A C E B F D (3,3) (2.5,4) (1.7,5) (2.8,7) (1.7,4) (2.5,3) (2,9) (2,8) (hours,points) (a) Define x ij ∈ { , 1 } for arc ( i,j ) ∈ E where x ij = 1 indicates arc ( i,j ) is in the shortest path and 0 otherwise. 1 Adam’s LP: min 3 x AB + 2 . 5 x AC + 1 . 7 x BD + 2 . 8 x BE + 1 . 7 x CB + 2 . 5 x CF + 2 x DE + 2 x FE s.t. x AB + x AC = 1 x AB x CB + x BD + x BE = 0 x AC + x CB + x CF = 0 x BD + x DE = 0 x BE x DE x FE = 1 x CF + x FE = 0 x ij ≥ ∀ ( i,j ) ∈ E For Eve the constraints remains the same but the objective changes to max3 x AB + 4 x AC + 5 x BD + 7 x BE + 4 x CB + 3 x CF + 9 x DE + 8 x FE Due to the difference in the objective functions, you cannot expect the optimal solution to be the same. Solving these problems it is clear that Adam’s optimal route is ABE while Eve’s optimal route is ACBDE.optimal route is ACBDE....
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 Spring '10
 TanBanPin

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