A significant amount of traffic flows from point S to point T in the one-way street network shown below. Points A, B, C, and D are junctions in the network, and the arrows mark the direction of traffic flow. The fuel cost in rupees for travelling along a street is indicated by the number adjacent to the arrow representing the street. Motorists travelling from point S to point T would obviously take the route for which the total cost of travelling is the minimum. If two or more routes have the same least travel cost, then motorists are indifferent between them. Hence, the traffic gets evenly distributed among all the least cost routes. The government can control the flow of traffic only by levying appropriate toll at each junction. For example, if a motorist takes the route S-A-T (using junction A alone), then the total cost of travel would be Rs 14 (i.e., Rs 9 + Rs 5) plus the toll charged at junction A.
Question: 1
If the government wants to ensure that no traffic flows on the street from D to T, while equal amount of traffic flows through junctions A and C, then a feasible set of toll charged (in rupees) at junctions A, B, C, and D respectively to achieve this goal is:
Show Hint
When blocking a route, raise tolls at junctions that appear exclusively in that route’s shortest path.
We first compute the base travel cost for each route from S to T without tolls:
S-A-T: \(9 + 5 = 14\)
S-B-C-T: \(2 + 3 + 2 = 7\)
S-B-D-T: \(2 + 1 + 6 = 9\)
S-D-T: \(7 + 6 = 13\)
To block D-T traffic, we set toll(D) high enough to make D-T route cost higher than alternatives.
We also adjust tolls at A and C so that S-A-T and S-B-C-T routes have equal cost, achieving equal flow through A and C.
The toll combination \(A=1, B=5, C=3, D=3\) satisfies both conditions.
\[
\boxed{1,5,3,3}
\]
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Question: 2
If the government wants to ensure that all motorists travelling from S to T pay the same amount (fuel costs and toll combined) regardless of the route they choose and the street from B to C is under repairs (and hence unusable), then a feasible set of toll charged (in rupees) at junctions A, B, C, and D respectively to achieve this goal is:
Show Hint
Equalizing total costs across routes forces an even traffic distribution.
With B-C blocked, remaining feasible routes are:
S-A-T, S-B-D-T, S-D-T.
We assign tolls so that total costs of all remaining routes are equal:
\(A=0, B=5, C=3, D=1\) balances costs perfectly.
\[
\boxed{0,5,3,1}
\]
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Question: 3
If the government wants to ensure that the traffic at S gets evenly distributed along streets from S to A, from S to B, and from S to D, then a feasible set of toll charged (in rupees) at junctions A, B, C, and D respectively to achieve this goal is:
Show Hint
For equal distribution at the origin, match effective costs from origin through each branch.
We set tolls so that initial segment cost from S to A, S to B, and S to D are equal, factoring in downstream costs to T.
\(A=0, B=5, C=2, D=2\) ensures the three entry paths have the same effective cost.
\[
\boxed{0,5,2,2}
\]
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Question: 4
If the government wants to ensure that all routes from S to T get the same amount of traffic, then a feasible set of toll charged (in rupees) at junctions A, B, C, and D respectively to achieve this goal is:
Show Hint
Equalizing costs across all complete routes balances route selection probabilities.
We assign tolls so that S-A-T, S-B-C-T, S-B-D-T, and S-D-T have identical total costs.
The combination \(A=1, B=5, C=4, D=2\) achieves this.
\[
\boxed{1,5,4,2}
\]
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Question: 5
The government wants to devise a toll policy such that the total cost to the commuters per trip is minimized. The policy should also ensure that not more than 70% of the total traffic passes through junction B. The cost incurred by the commuter travelling from point S to point T under this policy will be:
Show Hint
Traffic distribution constraints can force slightly higher costs to reduce congestion at a key junction.
To minimize cost while limiting B usage to $\leq 70%$, we set a small toll at B to divert some flow via A or D, keeping total trip cost minimal.
The optimal design yields a commuter cost of Rs 10.
\[
\boxed{\text{Rs 10}}
\]
