a$.$ In a TSP problem with $5$ cities, cities are connected as shown in the figure:

the following table shows the distances between different cities

C$0$

C$1$

C$2$

C$3$

C$4$

C$0$

$0\mathrm{.}00$

$40\mathrm{.}00$

$34\mathrm{.}00$

$66\mathrm{.}00$

$62\mathrm{.}00$

C$1$

$40\mathrm{.}00$

$0\mathrm{.}00$

$69\mathrm{.}00$

$27\mathrm{.}00$

$40\mathrm{.}00$

C$2$

$34\mathrm{.}00$

$69\mathrm{.}00$

$0\mathrm{.}00$

$92\mathrm{.}00$

$74\mathrm{.}00$

C$3$

$66\mathrm{.}00$

$27\mathrm{.}00$

$92\mathrm{.}00$

$0\mathrm{.}00$

$37\mathrm{.}00$

C$4$

$62\mathrm{.}00$

$40\mathrm{.}00$

$74\mathrm{.}00$

$37\mathrm{.}00$

$0\mathrm{.}00$

$[$Distance of $0$ means distance is not applicable.

$]$The following table shows the pheromone in units between different cities

C$0$

C$1$

C$2$

C$3$

C$4$

C$0$

$0\mathrm{.}00$

$0\mathrm{.}00$

$35\mathrm{.}38$

$20\mathrm{.}03$

$22\mathrm{.}01$

C$1$

$0\mathrm{.}00$

$0\mathrm{.}00$

$0\mathrm{.}00$

$18\mathrm{.}65$

$30\mathrm{.}33$

C$2$

$35\mathrm{.}38$

$0\mathrm{.}00$

$0\mathrm{.}00$

$22\mathrm{.}16$

$15\mathrm{.}24$

C$3$

$20\mathrm{.}03$

$18\mathrm{.}65$

$22\mathrm{.}16$

$0\mathrm{.}00$

$20\mathrm{.}33$

C$4$

$22\mathrm{.}01$

$30\mathrm{.}33$

$15\mathrm{.}24$

$20\mathrm{.}33$

$0\mathrm{.}00$

Assume that an ant has followed the following route:$[2,0,4,1,3,]$ and back to source

Calculate the total cost of the above route.

Calculate the ant's switching probabilities for the first $2$ steps in the above route assuming pheromone exponent parameter $=0\mathrm{.}02$ and heuristic exponent parameter $=0\mathrm{.}88.$

Calculate the updated pheromone amounts after applying the ACO evaporation step assuming $=0\mathrm{.}21.$

Calculate the updated pheromone amounts after applying the ACO depositing step assuming Q$=27\mathrm{.}35.$

In the light of the change in the pheromone matrix, comment on the convergence time for this algorithm.