COM3524 Exam – practice “takeaway” problem Jan 2021
COM3524 1
1. A delivery company with a single driver needs to optimise their route to deliver to
all cities which are shown on the following map. Distances are in km and the
company’s depot is located in city A. Dotted lines indicate the existence of roads.


Figure 1 Map showing location of depot and cities. Units are in km

Location Xi (km) Yi (km)
A (Depot) 50 15
B 80 10
C 90 50
D 40 50
E 30 30
Table 1 Co-ordinates of dept and cities (in km)
The delivery company commissions you to develop a bio-inspired algorithm to find the
best delivery route for them (meaning that every delivery site is visited only once and the
route length is minimised). In the first instance, you decide to use an Ant Colony
Optimisation Algorithm (ACO).

0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
A
B
C D
E
a) Explain briefly in a maximum of 100 words what is meant by an ACO and the key
biological behaviour that it is inspired by. [10%]

[Answer] An Ant Colony Optimisation algorithm (ACO) is a heuristic approach that can be applied to
solving a number of real-world problems, particularly those relating to routing or networking. It is based
on the behaviour of ants which communicate stigmergically by the deposition of a pheromone trail in
order to aid the process of the location and collection of food sources. Ants returning to the next lay a
pheromone trail that decays over time. Shorter, more desirable routes will accumulate pheromone and
hence be followed by more ants, causing further deposition. This process is the inspiration behind the ACO
approach [99 words]
b) The probability of ant k located at node r, choosing to transition to an allowed node,
s, is given by the following formula:


Where τ is the pheromone strength on edge rs, and η is the inverse of distance
between nodes r and s.
a, b are weighting parameters
At the start of the simulation, assume that pheromone is equally distributed on all
model edges (=1) and a, b are both set to one.
In the first update step, what is the probability that an ant located at the depot will
transition to i) node B, ii) node D and iii) node E? [20%]
[Answer] In this simplified case τ, a, b all =1, so the problem reduces to:

= ∑ ! ∈$%%&'()
Where ηrs is the inverse of distance between nodes r and s.
Use Pythagoras to calculate magnitude of distance vectors for each ALLOWED transition :
Depot to node B= sqrt( xb - xdepot)^2 + sqrt( yb - ydepot)^2) = sqrt((80 – 50)^2 + (10 – 15)^2
= sqrt (30^2 + (-5)^2) = 30.4 so ηAB »1/30.4
So ηAB » 0.033
åÎ
=
ALLOWEDt rtrt
rsrsk
rsp ba
ba
ht
ht
COM3524 Exam – practice “takeaway” problem Jan 2021
COM3524 3
Repeating for other nodes:
ηAD » 1/36.4 » 0.0275
ηAE = 1/25 = 0.04
so i) pAB » 0.033/(0.033 + 0.0275 + 0.04) » 0.33
ii) pAD = 0.0275/(0.033 + 0.0275 + 0.04) » 0.27
iii) pAE = 0.04/(0.033 + 0.0275 + 0.04) » 0.4
Note: you can retain precision (and save time!) by not bothering to take the individual inverses and
simply using the relative ratio of relative distances between nodes, which is equivalent in this case!
c) You fully implement your ACO algorithm and find it works reasonably well.
However, you are keen to explore other bio-inspired approaches to solve the
problem and decide to supervise a student project to develop an Evolutionary
algorithm to apply to the same problem, so you can compare them.
The student reads a textbook on Evolutionary Algorithm approaches and presents
you with the following plan for the algorithm that they propose to implement
1. Generate random initial population of any route permutations of ABCDE
2. For every chromosome
3. Calculate fitness = 1/total route length
4. Select Parents for Mating Pool based on relative fitness
5. Implement probabilistic One Point Crossover to generate new offspring
6. Implement probabilistic Random Swap Mutation on offspring
Give two reasons why this proposed algorithm might give rise to some invalid
solutions to the specific delivery problem described on page 1.
[20%]
[Answer] – in this case, only a subset of transitions are permitted due to the presence of roads
connecting only some cities/nodes. Step 1 will therefore create a population of solutions where some
transitions are invalid as roads do not exist between neighbouring points in the chromosomes e.g.
ACBED contains the invalid transition A – C.
Step 5 combines the two parent genomes randomly and this may result in either repeated nodes (e.g.,
missing nodes (e.g ABCCD), or invalid node transitions.
Step 6 randomly swaps individual entries of a chromosome and may also result in invalid node
transitions.
{Any two of the above are correct answers}
d) Your student takes note of your suggestions and implements a revised version of
the Evolutionary Algorithm, that appears to work.
The delivery company asks you to implement your preferred algorithm for them,
with their primary requirement that delivery routes for up to 10 possible delivery
cities are identified as quickly and efficiently as possible. They want the software
to be delivered to them, fully functional and optimised as much as possible, by the
end of the week.
i) From your knowledge of the characteristics of the two approaches, explain
which algorithm you would choose to package for the client, and why.
[20%]
ii) A month later, they phone to let you know that they have been taken over by
a larger company, and will now be delivering to up to 100 cities nationwide.
They ask you whether you think your previously delivered algorithm will still
work efficiently in this scenario. What is your response? [10%]
[ANSWER] i) In terms of efficiency requirement: The ACO works on a smaller population of
solutions each iteration (no. ants/solutions == no. nodes =5), whereas an EA typically works on
populations of O(100) chromosomes that need to be evaluated and operated on at multiple steps
within any given iteration. Therefore for a small problem size of <10 it is likely that the ACO is
more efficient (note that this was an outcome discussed in a lab session follow up).
You can also comment on the quick turnaround requested by the client – EAs are usually more
complex to program and test due to the multiple steps with probabilistic calculations involved, so
ACO would probably be the best option here too!
ii) For a larger problem size, the EA would probably scale better as the chromosome size tends to
stay fixed. Again, this was an outcome of a lab session.
You will obtain marks for any justified argument here. You would be expected to cite sources to
support your arguments where possible.
e) The new parent delivery company imposes a new business model. Under this new
model, its staff receive commission for every parcel that they deliver. The expanded
fleet of vehicles at other depots means that not every driver needs to visit every
city on every trip, but they must still start and finish at their depot and visit at least
3 other locations. The aim of your original client is to keep their inter-city travel
distance as low as possible, whilst maximising their commission.
The number of parcels to be delivered in city i, can be denoted as Ni
Your student is still working on their Evolutionary Algorithm (irrespective of whether
it is actually being used by the client!). How would the student need to modify steps
1 and 2 of their algorithm (from part c) in order to take this into account? [20%]
[Answer] Step 1 needs to be modified to permit chromosomes of differing length >=3 (as well as
removing any that are invalid e.g. containing repeated entries or transitions between locations not
connected directly by roads.
COM3524 Exam – practice “takeaway” problem Jan 2021
COM3524 5
The fitness function would need to be adapted to take into account the gain/profit of visiting each
location rather than just the cost (distance travelled to get to/from there).
An example could be fitness = total profit – total cost (i.e. distance travelled) (or some other valid
form that maximises profit and penalises distance)
So for each location the consideration is something like:
cNi – distance to reach and return to that node from the rest of the route
(where c is the commission rate)
So fitness of chromosome = cN - sum of segment lengths,
Or cN/sum of segment lengths
Where N =total number of cities visited.
Note: there is no exact mathematical formalism that is “correct” here, but your answer will be
expected to be logical, justified and consistent.



END OF QUESTION PAPER