Tutorial10solution

# Tutorial10solution - Tutorial 10 Outline of Solutions Q1...

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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 one-way. 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 A-B-E while Eve’s optimal route is A-C-B-D-E.optimal route is A-C-B-D-E....
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Tutorial10solution - Tutorial 10 Outline of Solutions Q1...

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