# Routes and Networks Dear students,
Routes and Networks is another important topic from which questions are asked in MBA entrance exams. CAT also asks questions from this topic.This set is from CAT 2006 Paper and will be very useful in understanding the application of the concepts of this topic in CAT DI-LR section.

Answer questions 1 to 5 on the basis of the information given below.

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.

1.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:

a. 2, 5, 3, 2

b. 0, 5, 3, 1

c. 1, 5, 3, 2

d. 2, 3, 5, 1

e. 1, 3, 5, 1

2.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:

a. 1, 5, 3, 3

b. 1, 4, 4, 3

c. 1, 5, 4, 2

d. 0, 5, 2, 3

e. 0, 5, 2, 2

1. 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:

a. 0, 5, 2, 2

b. 0, 5, 4, 1

c. 1, 5, 3, 3

d. 1, 5, 3, 2

e. 1, 5, 4, 2

4.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:

a. 0, 5, 4, 1

b. 0, 5, 2, 2

c. 1, 5, 3, 3

d. 1, 5, 3, 2

e. 0, 4, 3, 2

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 per cent 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:

a. Rs 7

b. Rs 9

c. Rs 10

d. Rs 13

e. Rs 14

Solution –

1. Let the toll charged at junctions A, B, C, and D be a, b, c and d respectively. Then the so that equal amount is collected through all route we have,
9+a+5=2+b+2+a+5=10+d+c=13+d.
Option C is satisfy above equations so, answer is option C.

2. Since the cost of travel including toll on routes S-A-T, S-B-C-T, S-B-A-T and S-D-CT is the same.
∴ 14 + a = 7 + b + c = 9 + a + b = 10 + c + d
∴ b = 5, d = 2, c – a = 2
Only option e satisfies these criteria.
Hence, option e.

3. Since the cost of travel including toll on all routes is the same.
∴ 14 + a = 7 + b + c = 13 + d = 9 + a + b = 10 + c + d
∴ b = 5, d = 2, c = 3 and a = 1 Hence, option d.

4. But we want traffic along S-A, S-B and S-D to be the same. As routes lead to C from both B and D, we can increase the toll at C so that the cost of travelling along S-B-C-T and S-D-C-T is more than that along the other three routes.
Now, 14 + a = 9 + b = 13 + d
∴ a = 0, b = 5 and d =1
Also, 7 + b + c > 14 and 10 + d + c > 14
∴ c > 3
Hence, option a.

5. If toll charges at all junctions are made 0, 100% traffic will pass through S-B-C-T. This is not possible.
If toll charges at A and B are made 0, then 100% traffic will pass through S-B-A-T.
This is also not possible.
If toll charges at C and D are made 0, that at B are made Rs.3, then the traffic will get equally divided between S-D-C-T and S-B-C-T.
Thus, the cost incurred will be Rs 10.

Set analysis:-

Level of difficulty:- Easy to moderate
Expected time to solve:- 5 to 7 minutes.

Hope you learnt!

See you with the next one tomorrow!

Happy learning!
Gajendra sir
CAT Coach