Question

16. The downtown core of Gotham City consists of one-way streets, and the traffic flow has been measured at each intersection. For the city block shown in Figure 2.19, the numbers represent the average numbers of vehicles per minute entering and leaving intersections A, B, C, and D during business hours. (a) Set up and solve a system of linear equations to find the possible flows fi,.. f (b) If traffic is regulated cles per minute, what will the average flows on the other streets be? on CD so that fa= 10 vehi- (c) What are the minimum and maximum possible flows on each street? C (d) How would the solution change if all of the direc- tions were reversed? 10 fi 10 A f4 15 15 C D 15 20 1 -1 0 2 1 1 11 1 2 2 1

Answer 1

Solution (a) :

We will use junction law for this problem.

The incoming traffic at every junction should be equal to outgoing traffic. Means every vehicle arriving at a junction must leave the junction.

For junction A:

f210+ 10

220.

For junction B:

+ f3 = 20 +5

if3=25. II

For junction C:

3f4=15+ 15)

34=30. - -1II

For junction D:

$f_2+f_4=15+10$

24 25 V

Now, let's assume f₄= t. Then from equation IV, we get;

f2t 25

f2=25- t -V

From equation III:

$f_3+t=30$

$\Rightarrow f_3=30-t_{---------------------------------------VI}$

From equation II:

$f_1+f_3=25$

30-t=25

t-5 -VII

From equation I:

$t-5+f_2=20$

$\Rightarrow f_2=25-t_{--------------------------------VIII}$

Solution (b):

If f₄ = 10, then:

$f_1=t-5=10-5=5$

$\\f_2=25-t=25-10=15\\f_3=30-t=30-10=20\\f_4=t=10$

Solution (c):

The constraint on t is that it can take values from 0 to 30 max. To decide maximum and minimum traffic, we will consider only non-negative values. Because negative value just mean reverse direction here and minimum possible value for traffic is 0.

	$f_1$	$f_2$	$f_3$	$f_4$
Expression
Minimum traffic	0	0	0	0
Maximum possible traffic	25	25	30	30

Solution (d):

If all the directions are reversed there would be no change in the final answer. Because we will get the same expressions as above.