$$ In a TSP problem with $4$ cities, cities are connected as shown in the figure:

the following table shows the distances between different cities

$\backslash$table$[[,$C$0,$C$1,$C$2,$C$3],[$C$0,0\mathrm{.}00,40\mathrm{.}00,16\mathrm{.}00,80\mathrm{.}00],[$C$1,40\mathrm{.}00,0\mathrm{.}00,54\mathrm{.}00,65\mathrm{.}00],[$C$2,16\mathrm{.}00,54\mathrm{.}00,0\mathrm{.}00,83\mathrm{.}00],[$C$3,80\mathrm{.}00,65\mathrm{.}00,83\mathrm{.}00,0\mathrm{.}00]]$

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

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

$\backslash$table$[[,$C$0,$C$1,$C$2,$C$3],[$C$0,0\mathrm{.}00,46\mathrm{.}55,47\mathrm{.}34,0\mathrm{.}00],[$C$1,46\mathrm{.}55,0\mathrm{.}00,2\mathrm{.}06,35\mathrm{.}35],[$C$2,47\mathrm{.}34,2\mathrm{.}06,0\mathrm{.}00,44\mathrm{.}43],[$C$3,0\mathrm{.}00,35\mathrm{.}35,44\mathrm{.}43,0\mathrm{.}00]]$

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

a$.$ Calculate the total cost of the above route.

b$.$ Calculate the ant's switching probabilities for the first $3$ steps in the above route assuming pheromone exponent parameter $\backslash$alpha $=0\mathrm{.}56$ and heuristic exponent parameter $\backslash$beta $=0\mathrm{.}09.$

c$.$ Calculate the updated pheromone amounts after applying the ACO evaporation step assuming $\backslash$rho $=0\mathrm{.}25.$

d$.$ Calculate the updated pheromone amounts after applying the ACO depositing step assuming Q$=13\mathrm{.}91.$

e$.$ Comment on the update with regard to the optimization task convergence