Paulo Lafeta Ferreira
Introduction to Artificial Intelligence – CS 581
Professor: Pushpak Karnick

“Traveling Salesman” must visit each city
exactly once and then return back to the
starting point.

Complexity of the problem (NP-hard) grows
exponentially.

Approximate solutions can be made using
Genetic Algorithms.
Map:
89 cities:
http://www.tsp.gatech.edu
Optimal solution found.
Path size: 351 mi
Map:
9,152 Cities:
Path size found: 44,232 mi
Solution nonoptimal.
Path found is at maximum,
0.012% greater than optimum
solution.
Map:
71,009 cities:
Best path found in a recent test:
Path size: 241,188 mi
0.024% near optimal
solution

This problem has 1,904,711 cities from around the
world.
Best path found:
396,967,677 mi
0.04996% greater than
optimal path.
Sweden:

24,978 cities

Path size: 45,049 mi

Used 96 clusters dual Intel Xeon 2.8 Ghz

Time running: approximately 3 years

In an usual computer: ~84 years

Path
 Crossovers:
▪ Partially Mapped (PMX)
▪ Order (OX)
▪ Cycle (CX)

Ordinal
 Crossover:
▪ Cut-and-splice

Adjacency
 Crossovers:
▪ Alternating Edges
▪ Subtour-chunks
▪ Heuristic

Partially Mapped crossover (PMX):
p1 = [1 2 3 | 4 5 6 7 | 8 9] and
p2 = [4 5 2 | 1 8 7 6 | 9 3]
o1 = [x x x | 1 8 7 6 | x x] and
o2 = [x x x | 4 5 6 7 | x x]
o1 = [x 2 3 | 1 8 7 6 | x 9] and
o2 = [x x 2 | 4 5 6 7 | 9 3]

Order crossover (OX):
p1 = [1 2 3 | 4 5 6 7 | 8 9] and
p2 = [4 5 2 | 1 8 7 6 | 9 3]


o1 = [x x x | 4 5 6 7 | x x] and
o2 = [x x x | 1 8 7 6 | x x]


o1 = [2 1 8 | 4 5 6 7 | 9 3]
o2 = [3 4 5 | 1 8 7 6 | 9 2]

Cycle crossover (CX):
p1 = [1 2 3 4 5 6 7 8 9] and
p2 = [4 1 2 8 7 6 9 3 5]
o1 = [1 x x 4 x x x x x]
…
01 = [1 x x 4 x x x 8 x]
…
o1 = [1 2 3 4 x x x 8 x]

Alternating edges crossover:
p1 = [2 3 8 7 9 1 4 5 6] and
p2 = [7 5 1 6 9 2 8 4 3]
o1 = [2 5 8 7 9 1 6 4 3]

Randomly choose an edge from the first
parent, then select an appropriate edge from
the second parent, and so.

Subtour-chunks crossover

Similar to “Alternating Edges”, but instead of
switching only one element, switch a
random-length subtour from one of the
parents.

Heuristic crossover

Choose a random city, then compares two
edges that emanate from this city from both
parents.
 Choose the one with lower cost.
 The city on the other end of this edge, is the next
starting point.

Ordinal representation:

Reference C: C = [1 2 3 4 5 6 7 8 9]

List l of references,
l = [1 1 2 1 4 1 3 1 1]

In path encoding, equal to:
[1 2 4 3 8 5 9 6 7]

Cut-and-splice crossover

Take a part from one parent, and splice it with
the remainder from the second parent.

Maintain the best individual found so far in
the population.

Applied only if the population is equal or
greater than 4.

Randomize one or two children when both
are equal after reproduction.

This increased the performance of the GA in
general, due to the high frequency of GA
being stuck in a local optima.

Other option: randomize entire population if
best solution does not change after Y
iterations.

After a while, single crossover methods start
lacking variety in their reproductions.

Possible solutions:
 Randomize part of the population.
and/or:
 Change crossover method in real time, during
execution.

During execution, switch crossover method when
“best solution” doesn’t change during n iterations.

Conversion of all individuals is necessary if switch is
between a Path and an Adjacency crossover.

Crossovers used:




Heuristic (Adjacency)
Partially Mapped (Path)
Order (Path)
Subtour-chunks (Adjacency)
Optimal tour: 6656
Best path cost
Pop = 16, Iter = 10000
15290
14620
12277
9470
8285
7561
6765
PMX
OX
CX
Cut and
splice
Alternating
Edges
Subtour
chunks
Heuristic
6659
Heuristic +
PMX + OX +
Subtour
6758
All
Optimal tour: 27603
Best path cost
Pop = 8, Max Iter = 20000
36228
33448
32748
27748
Heuristic +
PMX + OX +
Subtour
27872
PMX
28877
27748
OX
CX
Cut and splice
Alternating
Edges
Subtour
chunks
27601
Heuristic
Xqf - Heuristic
900
800
Optimal
tour: 564
700
Best cost
600
500
400
Heuristic
300
200
100
0
Heuristic
30000
60000
90000
120000
150000
180000
210000
240000
270000
825
688
646
638
614
609
607
605
605
720
700
Heur+PMX+OX+
Subtour
680
Best cost
660
640
620
600
580
560
540
520
Series1
30000
60000
90000
120000
150000
180000
210000
240000
270000
697
633
619
608
598
597.7
592
590
586
Optimal tour: 9352
Qatar - Number of Iterations x Best cost found
80000
Num Iterations: 40000
Heuristic
Partially Mapped
Order
Cycle
Alternating Edges
Subtour-chunks
Cut-and-splice
Multiple
17638
29266
16799
43006
66098
27744
58970
15797
70000
60000
50000
40000
30000
20000
10000
0
1000
5000
10000
20000
30000
Heuristic
Partially Mapped
Order
Cycle
Alternating Edges
Subtour-chunks
Cut-and-splice
Multiple
40000
Optimal tour: 9352
Qatar - Population x Best Cost found
90000
Num Iterations: 10000
80000
Heuristic
Order
Subtourchunks
Cut-andsplice
Multiple
2
42200
76978
4
17715
22026
8
19813
25802
16
14227
30584
81890
36417
36230
43530
83278
44357
71579
15157
65950
19605
65650
15827
70000
60000
Heuristic
Partially Mapped
50000
Order
Cycle
40000
Alternating Edges
Subtour-chunks
30000
Cut-and-splice
20000
Multiple
10000
0
2
4
8
Population
16
Num. Iterations: 200000; Population = 4;
Multiple: 9910
Heuristic: 9830
Optimal tour: 9352
Num. Iterations: 400000; Population: 8
Multiple: 236667
Heuristic: 423593
734 cities
Optimal tour: 79114

Heuristic showed best performance between
the 7 implemented.
 Converges quickly to a “good” solution
 But after an amount of iterations it gets “stuck”

Followed by the Order.

Ordinal with its Cut-and-splice is easy to
implement and works well in some GA
problems, but not in TSP.

Multiple crossovers shows its advantages in
long term executions.

Heuristics seems to be the best choice for
low number of iterations and small number of
cities.

Multiple technique showed to be a safer bet
in long executions and bigger inputs, due to
its continuous discovery of new solutions.

What might improve the Multiple technique:
 Store information about crossovers performance
during execution and then select the current
best techniques more frequently.
 Vary number of iterations necessary to switch
crossover, based on their current performance.
Thank you!