Wireless sensor networks frequently use multi-path routing schemes between nodes and a base station. Multi-path routing confers additional robustness against link failure, but in battery-powered networks it is desirable to choose paths which maximise the overall network lifetime - the time at which a battery is first exhausted. We introduce multi-objective evolutionary algorithms to find the routings which approximate the optimal trade-off between network lifetime and robustness. A novel measure of network robustness, the fragility, is introduced. We show that the distribution of traffic between paths in a given multi-path scheme that optimises lifetime or fragility may be found by solving the appropriate linear program. A multi-objective evolutionary algorithm is used to solve the combinatorial optimisation problem of choosing routings and traffic distributions that give the optimal trade-off between network lifetime and robustness. Efficiency is achieved by pruning the search space using k-shortest paths, braided and edge disjoint paths. The method is demonstrated on synthetic networks and a real network deployed at the Victoria & Albert Museum, London. For these networks, using only two paths per node, we locate routings with lifetimes within 3% of those obtained with unlimited paths per node. In addition, routings which halve the network fragility are located. We also show that the evolutionary multi-path routing can achieve significant improvement in performance over a braided multi-path scheme. reporsitory: https://ore.exeter.ac.uk/repository/handle/10871/23185

1. Evolutionary Multi-Path Routing for Network Lifetime and
Robustness in Wireless Sensor Networks
Ad Hoc Networks, 52 (2016), 130–145
DOI: 10.1016/j.adhoc.2016.08.005
Alma Rahat
Richard Everson
Jonathan Fieldsend
Computer Science
University of Exeter
United Kingdom
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 1 / 17

2. Wireless Sensors
Autonomous devices
Send data to a central base
station
Environmental or process
monitoring
Industrial
Heritage
Pharmaceuticals
Health-care
Battery powered
Monitor locations that are
difficult to access
Typically left unattended for
long periods of time
Radio environment can be
unreliable Sensor monitoring showcase environment
in Mary Rose Museum, UK
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 2 / 17

3. Mesh Network and Multi-Path Routing
Sensors and gateway
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 3 / 17

4. Mesh Network and Multi-Path Routing
Sensors and gateway
Network connectivity map
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 3 / 17

5. Mesh Network and Multi-Path Routing
Sensors and gateway
Network connectivity map
Mesh Topology: sensors send data
either directly (e.g. R2,i = 2, G )
or indirectly (e.g. R2,j = 2, 5, G )
to the gateway
Alternative routes
Range extension
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 3 / 17

6. Mesh Network and Multi-Path Routing
Sensors and gateway
Network connectivity map
Mesh Topology: sensors send data
either directly (e.g. R2,i = 2, G )
or indirectly (e.g. R2,j = 2, 5, G )
to the gateway
Alternative routes
Range extension
Multi-path routing: multiple paths
per node to send messages (e.g.
R2 = {R2,1, R2,2})
Time share: proportion of time each
path is active (e.g. τ2,1 and τ2,2,
where τ2,1 + τ2,2 = 1)
τ2,1
τ2,2
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 3 / 17

7. Challenges of Multi-Path Routing
D-path routing scheme for N nodes.
R = {{R1,d }D
d=1, . . . , {RN,d }D
d=1}
T = {{τ1,d }D
d=1, . . . , {τN,d }D
d=1}
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 4 / 17

8. Challenges of Multi-Path Routing
D-path routing scheme for N nodes.
R = {{R1,d }D
d=1, . . . , {RN,d }D
d=1}
T = {{τ1,d }D
d=1, . . . , {τN,d }D
d=1}
Challenges:
Some nodes are under heavy load
– the first node to exhaust its
battery dictates the network
lifetime
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 4 / 17

9. Challenges of Multi-Path Routing
D-path routing scheme for N nodes.
R = {{R1,d }D
d=1, . . . , {RN,d }D
d=1}
T = {{τ1,d }D
d=1, . . . , {τN,d }D
d=1}
Challenges:
Some nodes are under heavy load
– the first node to exhaust its
battery dictates the network
lifetime
Link failures occur with a
probability, e.g. π2,G – loss of
messages
π2,G
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 4 / 17

10. Challenges of Multi-Path Routing
D-path routing scheme for N nodes.
R = {{R1,d }D
d=1, . . . , {RN,d }D
d=1}
T = {{τ1,d }D
d=1, . . . , {τN,d }D
d=1}
Challenges:
Some nodes are under heavy load
– the first node to exhaust its
battery dictates the network
lifetime
Link failures occur with a
probability, e.g. π2,G – loss of
messages
Maximise
Time before the first node exhausts its battery: network lifetime
Messages received against any link failure: robustness
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 4 / 17

11. Node Costs
Node’s cost due to a routing
scheme R = {{Ri,1, Ri,2}},
T = {{τi,1, τi,2}}, where
vi ∈ N:
C5
2,1 = A2,5 + T5,G
N set of all nodes, {v1, . . . , vN}.
Ci
kl Cost at node vi for lth path
from vk.
Ci Total cost at node vi .
Ti,j Transmission cost at node vi .
Aj,i Reception cost at node vi .
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 5 / 17

12. Node Costs
Node’s cost due to a routing
scheme R = {{Ri,1, Ri,2}},
T = {{τi,1, τi,2}}, where
vi ∈ N:
C5
2,1 = A2,5 + T5,G
N set of all nodes, {v1, . . . , vN}.
Ci
kl Cost at node vi for lth path
from vk.
Ci Total cost at node vi .
Ti,j Transmission cost at node vi .
Aj,i Reception cost at node vi .
A2,5
T5,G
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 5 / 17

13. Node Costs
Node’s cost due to a routing
scheme R = {{Ri,1, Ri,2}},
T = {{τi,1, τi,2}}, where
vi ∈ N:
C5
2,1 = A2,5 + T5,G
C5
= τ2,1C5
2,1 + τ5,1C5
5,1 + τ5,2C5
5,2
N set of all nodes, {v1, . . . , vN}.
Ci
kl Cost at node vi for lth path
from vk.
Ci Total cost at node vi .
Ti,j Transmission cost at node vi .
Aj,i Reception cost at node vi .
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 5 / 17

14. Node Costs
Node’s cost due to a routing
scheme R = {{Ri,1, Ri,2}},
T = {{τi,1, τi,2}}, where
vi ∈ N:
C5
2,1 = A2,5 + T5,G
C5
= τ2,1C5
2,1 + τ5,1C5
5,1 + τ5,2C5
5,2
N set of all nodes, {v1, . . . , vN}.
Ci
kl Cost at node vi for lth path
from vk.
Ci Total cost at node vi .
Ti,j Transmission cost at node vi .
Aj,i Reception cost at node vi .
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 5 / 17

15. Node Costs
Node’s cost due to a routing
scheme R = {{Ri,1, Ri,2}},
T = {{τi,1, τi,2}}, where
vi ∈ N:
C5
2,1 = A2,5 + T5,G
C5
= τ2,1C5
2,1 + τ5,1C5
5,1 + τ5,2C5
5,2
N set of all nodes, {v1, . . . , vN}.
Ci
kl Cost at node vi for lth path
from vk.
Ci Total cost at node vi .
Ti,j Transmission cost at node vi .
Aj,i Reception cost at node vi .
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 5 / 17

16. Network Lifetime
Lifetime for node vi :
Li (R, T ) =
Qi
Ei + Ci
Radio communication current
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 6 / 17

17. Network Lifetime
Lifetime for node vi :
Li (R, T ) =
Qi
Ei + Ci
Radio communication currentQuiescent current
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 6 / 17

18. Network Lifetime
Lifetime for node vi :
Li (R, T ) =
Qi
Ei + Ci
Radio communication currentQuiescent current
Remaining battery charge
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 6 / 17

19. Network Lifetime
Lifetime for node vi :
Li (R, T ) =
Qi
Ei + Ci
Network lifetime: L∗(R, T ) = minvi ∈N Li (R, T )
Radio communication currentQuiescent current
Remaining battery charge
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 6 / 17

20. Network Lifetime
Lifetime for node vi :
Li (R, T ) =
Qi
Ei + Ci
Network lifetime: L∗(R, T ) = minvi ∈N Li (R, T )
Optimal time share for a fixed R .
max
T
L∗
(R , T )
Subject to constraints at any node:
Total charge ≥ total cost.
Individual time share ≥ 0.
Total time per node = 1.
Radio communication currentQuiescent current
Remaining battery charge
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 6 / 17

21. Network Lifetime
Lifetime for node vi :
Li (R, T ) =
Qi
Ei + Ci
Network lifetime: L∗(R, T ) = minvi ∈N Li (R, T )
Optimal time share for a fixed R .
max
T
L∗
(R , T )
Subject to constraints at any node:
Total charge ≥ total cost.
Individual time share ≥ 0.
Total time per node = 1.
Objective
maxR f1(R) = L∗(R, T ∗
L ),
T ∗
L is the set of optimal time
shares for a candidate set of
paths R.
Radio communication currentQuiescent current
Remaining battery charge
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 6 / 17

22. Fragility: Overall Expected Data Loss
A link may fail with a probability,
e.g. π1,2, π2,G.
π1,2
π2,G
R = {{Ri,1, Ri,2}}, T = {{τi,1, τi,2}}
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 7 / 17

23. Fragility: Overall Expected Data Loss
A link may fail with a probability,
e.g. π1,2, π2,G.
The probability of a path failure is
the sum of link failure probabilities.
p1,2 = P(R1,2 fails) ≈ π1,2 + π2,G.
π1,2
π2,G
R = {{Ri,1, Ri,2}}, T = {{τi,1, τi,2}}
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 7 / 17

24. Fragility: Overall Expected Data Loss
A link may fail with a probability,
e.g. π1,2, π2,G.
The probability of a path failure is
the sum of link failure probabilities.
p1,2 = P(R1,2 fails) ≈ π1,2 + π2,G.
Expected data loss from a node is
the proportion of messages sent
times the probability of path failure:
U1τ1,2p1,2, where U1is the total
messages sent by v1 and τ1,2 is the
proportion of time R1,2 is used.
π1,2
π2,G
R = {{Ri,1, Ri,2}}, T = {{τi,1, τi,2}}
U1τ1,2
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 7 / 17

25. Fragility: Overall Expected Data Loss
A link may fail with a probability,
e.g. π1,2, π2,G.
The probability of a path failure is
the sum of link failure probabilities.
p1,2 = P(R1,2 fails) ≈ π1,2 + π2,G.
Expected data loss from a node is
the proportion of messages sent
times the probability of path failure:
U1τ1,2p1,2, where U1is the total
messages sent by v1 and τ1,2 is the
proportion of time R1,2 is used.
π1,2
π2,G
π1,2
π2,G
R = {{Ri,1, Ri,2}}, T = {{τi,1, τi,2}}
Overall expected data loss is the loss from originating node plus the loss
from other paths due to edge sharing.
Fragility,F1,2 = data loss at R1,2 +
(data loss at other nodes | they share an edge with this route).
U1τ1,2
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 7 / 17

26. Robustness
Network Fragility: F(R, T ) = maxvi ∈N maxd Fi,d .
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 8 / 17

27. Robustness
Network Fragility: F(R, T ) = maxvi ∈N maxd Fi,d .
Optimal time share for a fixed R .
min
T
F(R , T ).
Subject to constraints at any node:
Network fragility ≥ individual path fragility.
Individual time share ≥ 0.
Total time per node = 1.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 8 / 17

28. Robustness
Network Fragility: F(R, T ) = maxvi ∈N maxd Fi,d .
Optimal time share for a fixed R .
min
T
F(R , T ).
Subject to constraints at any node:
Network fragility ≥ individual path fragility.
Individual time share ≥ 0.
Total time per node = 1.
Increasing robustness ≡ minimising network fragility
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 8 / 17

29. Robustness
Network Fragility: F(R, T ) = maxvi ∈N maxd Fi,d .
Optimal time share for a fixed R .
min
T
F(R , T ).
Subject to constraints at any node:
Network fragility ≥ individual path fragility.
Individual time share ≥ 0.
Total time per node = 1.
Increasing robustness ≡ minimising network fragility
Objective
minR f2(R) = F(R, T ∗
F ),
T ∗
F is the set of optimal time shares for a candidate set of paths R.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 8 / 17

30. Search Space Size
How big is the search space?
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

31. Search Space Size
Number of possible loopless
paths for node v3: 1
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

32. Search Space Size
Number of possible loopless
paths for node v3: 2
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

33. Search Space Size
Number of possible loopless
paths for node v3: 3
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

34. Search Space Size
Number of possible loopless
paths for node v3: 4
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

35. Search Space Size
Number of possible loopless
paths for node v3: 5
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

36. Search Space Size
Number of possible loopless
paths for node v3: 6
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

37. Search Space Size
Number of possible loopless
paths for node v3: 7
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

38. Search Space Size
Number of possible loopless
paths for node v3: 7
Number of possible D-path
routing schemes:
n
i=1
ai
D
ai : Number of available routes
from vi to vG
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

39. Search Space Size
Number of possible loopless
paths for node v3: 7
Number of possible D-path
routing schemes:
n
i=1
ai
D
ai : Number of available routes
from vi to vG
≈ 1.6 × 107 solutions (2-path)
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

40. Search Space Size
Number of possible loopless
paths for node v3: 7
Number of possible D-path
routing schemes:
n
i=1
ai
D
ai : Number of available routes
from vi to vG
≈ 5.9 × 104 solutions (2-path)
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

41. Search Space Size
Number of possible loopless
paths for node v3: 7
Number of possible D-path
routing schemes:
n
i=1
ai
D
ai : Number of available routes
from vi to vG
≈ 5.9 × 104 solutions (2-path)
Limit number of paths ≡ limit the
search space.
Shorter paths are expected to be energy
efficient. Limit search space by allowing
a k-shortest path per node [Yen, 1972].
Partially-disjoint (braided) and
edge-disjoint set of paths are known to
be robust and energy efficient [Ganesan
et al. 2001]. Add these to the path
library for robustness.
Quicker approximation of Pareto Front.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 9 / 17

42. Max-Min Lifetime Pruning
With no limits on the number of
routes per node, a linear program (LP)
can be derived to maximise network
lifetime [Chang et al., 2004]
max min
vi ∈V
Li
subject to:
Edge utilisation, uij ≥ 0
Energy usage ≤ available charge
Flow conservation
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 10 / 17

43. Max-Min Lifetime Pruning
Solving LP results in best
network lifetime and associated
edge utilisations
With no limits on the number of
routes per node, a linear program (LP)
can be derived to maximise network
lifetime [Chang et al., 2004]
max min
vi ∈V
Li
subject to:
Edge utilisation, uij ≥ 0
Energy usage ≤ available charge
Flow conservation
Remove unused edges (grey)
from graph G to extract
reduced graph G
Apply path pruning on both G
and G to construct path
library
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 10 / 17

44. Multi-Objective Evolutionary Algorithm
1: A ← InitialiseArchive(P) Initialise elite archive randomly
2: for i ← 1 : T do
3: R1, R2 ← Select(A) Select two parent solutions
4: R ← CrossOver(R1, R2)
5: R ← Mutate(R )
6: A ← NonDominated(A ∪ {R , T ∗
L } ∪ {R , T ∗
F }) Update archive
7: end for
8: PostProcess(A)
9: return A Approximation of the Pareto set
P Pruned path library constructed from both G and G
Crossover Select paths for each node from parents
Mutation Replace paths randomly from k-shortest paths for some
nodes
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 11 / 17

45. Exploiting Intermediary Time Shares
For fixed R, we showed that optimal time shares may be achieved by
solving a linear program for either network lifetime or robustness.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 12 / 17

46. Exploiting Intermediary Time Shares
For fixed R, we showed that optimal time shares may be achieved by
solving a linear program for either network lifetime or robustness.
Bi-objective linear program (R removed for brevity).
max
T
L∗
(T ) ≡ min
T
L∗
(T )
min
T
F ∗ (T )
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 12 / 17

47. Exploiting Intermediary Time Shares
For fixed R, we showed that optimal time shares may be achieved by
solving a linear program for either network lifetime or robustness.
Bi-objective linear program (R removed for brevity).
max
T
L∗
(T ) ≡ min
T
L∗
(T )
min
T
F ∗ (T )
Pareto front is a convex polyhedron and can be exactly located
[Ehrgott, 2006], but the number of solutions are not known a priori.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 12 / 17

48. Exploiting Intermediary Time Shares
For fixed R, we showed that optimal time shares may be achieved by
solving a linear program for either network lifetime or robustness.
Bi-objective linear program (R removed for brevity).
max
T
L∗
(T ) ≡ min
T
L∗
(T )
min
T
F ∗ (T )
Pareto front is a convex polyhedron and can be exactly located
[Ehrgott, 2006], but the number of solutions are not known a priori.
We apply a bi-section search as a post-processing step to locate a few
solutions on the Pareto front for varying weights λ while minimising
the weighted sum.
min
T
(λ − 1)L∗
(T ) + λF∗
(T )
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 12 / 17

49. Illustration in a Small Synthetic Network
11 nodes + gateway
10 shortest paths, 10
edge-disjoint paths and
partially-disjoint paths in
each G and G
Initial random routings:
100
Initial population size:
200 (solving for optimal
time shares)
Mutation and crossover
rate: 0.1
Number of generations:
40000
∞ high medium low
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 13 / 17

50. Illustration in a Small Synthetic Network
5 10 15 20 251
2
3
4
5
6
7
NetworkLifetime(years)
Fragility (packet loss/cycle)
11 nodes + gateway
10 shortest paths, 10
edge-disjoint paths and
partially-disjoint paths in
each G and G
Initial random routings:
100
Initial population size:
200 (solving for optimal
time shares)
Mutation and crossover
rate: 0.1
Number of generations:
40000
∞ high medium low
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 13 / 17

51. Illustration in a Small Synthetic Network
5 10 15 20 251
2
3
4
5
6
7
NetworkLifetime(years)
Fragility (packet loss/cycle)
11 nodes + gateway
10 shortest paths, 10
edge-disjoint paths and
partially-disjoint paths in
each G and G
Initial random routings:
100
Initial population size:
200 (solving for optimal
time shares)
Mutation and crossover
rate: 0.1
Number of generations:
40000
∞ high medium low
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 13 / 17

52. Illustration in a Small Synthetic Network
5 15 25 35 45
1
2
3
4
5
6
7
NetworkLifetime(years)
Fragility (packet loss/cycle)
Graph reduction and path pruning enable efficient search
11 nodes + gateway
10 shortest paths, 10
edge-disjoint paths and
partially-disjoint paths in
each G and G
Initial random routings:
100
Initial population size:
200 (solving for optimal
time shares)
Mutation and crossover
rate: 0.1
Number of generations:
40000
∞ high medium low
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 13 / 17

53. Evaluations in Synthetic and Real Networks
20π 40π 60π 80π
0.20
1.02
1.85
2.68
3.50
50π 100π 150π
0.50
1.12
1.75
2.38
3.00
5π 10π 15π 20π 25π
2.00
3.25
4.50
5.75
7.00
5π 10π 15π 20π 25π
0.6
0.8
1.0
1.2
1.4
NetworkLifetime(years)
Fragility (packet loss/cycle)
100 nodes
11 nodes
150 nodes
30 nodes (real)
2-path,
1-path
Over 97% of
optimal
network
lifetime.
Braided
multi-path
[Ganesan et
al. 2001]
(black
diamond) is
dominated.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 14 / 17

54. Importance of Post-Processing
20π 40π 60π 80π
0.20
1.02
1.85
2.68
3.50
NetworkLifetime(years)
Fragility (packet loss/cycle)
100 nodes
Post-processing
includes the interme-
diary time shares and
thus improves upon
the estimated Pareto
front of considering
only optimal time
shares.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 15 / 17

55. Summary
Multi-objective optimisation of multi-path routing scheme yield a
range of trade-off solutions between network lifetime and robustness.
Best lifetime solutions found through optimisation approximate the
optimal network lifetime within 3%.
Graph reduction and path pruning enable efficient evolutionary search.
Considering intermediary time shares improves Pareto front
approximation.
Future work
Incorporate data-driven non-linear battery models and uncertainties
therein.
Distributed multi-objective routing.
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 16 / 17

56. Extra Slide: Bisection Search
O = (F∗(TF ), L ∗(TL))
L ∗(T )
F∗(T )
λa = 1
(F∗(TL), L ∗(TL))
λb = 0
(F∗(TF ), L ∗(TF )))
θ = arctan ∆F∗
∆L ∗
λ = λa+λb
2
∆F∗
∆L ∗
if θ > θt − δθ:
λa ← λ,
else:
λb ← λ,
minT (λ − 1)L∗ (T ) + λF∗(T )
if |θ − θt| ≤ δθ:
return T ,
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 17 / 17

57. Extra Slide: Bisection Search
O = (F∗(TF ), L ∗(TL))
L ∗(T )
F∗(T )
λa = 1
(F∗(TL), L ∗(TL))
λb = 0
(F∗(TF ), L ∗(TF )))
θ = arctan ∆F∗
∆L ∗
λ = λa+λb
2
∆F∗
∆L ∗
if θ > θt − δθ:
λa ← λ,
else:
λb ← λ,
minT (λ − 1)L∗ (T ) + λF∗(T )
if |θ − θt| ≤ δθ:
return T ,
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 17 / 17

58. Extra Slide: Bisection Search
O = (F∗(TF ), L ∗(TL))
L ∗(T )
F∗(T )
λa = 1
(F∗(TL), L ∗(TL))
λb = 0
(F∗(TF ), L ∗(TF )))
θ = arctan ∆F∗
∆L ∗
λ = λa+λb
2
∆F∗
∆L ∗
if θ > θt − δθ:
λa ← λ,
else:
λb ← λ,
minT (λ − 1)L∗ (T ) + λF∗(T )
if |θ − θt| ≤ δθ:
return T ,
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 17 / 17

59. Extra Slide: Bisection Search
O = (F∗(TF ), L ∗(TL))
L ∗(T )
F∗(T )
λa = 1
(F∗(TL), L ∗(TL))
λb = 0
(F∗(TF ), L ∗(TF )))
θ = arctan ∆F∗
∆L ∗
λ = λa+λb
2
∆F∗
∆L ∗
if θ > θt − δθ:
λa ← λ,
else:
λb ← λ,
minT (λ − 1)L∗ (T ) + λF∗(T )
if |θ − θt| ≤ δθ:
return T ,
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 17 / 17

60. Extra Slide: Bisection Search
O = (F∗(TF ), L ∗(TL))
L ∗(T )
F∗(T )
λa = 1
(F∗(TL), L ∗(TL))
λb = 0
(F∗(TF ), L ∗(TF )))
θ = arctan ∆F∗
∆L ∗
λ = λa+λb
2
∆F∗
∆L ∗
if θ > θt − δθ:
λa ← λ,
else:
λb ← λ,
minT (λ − 1)L∗ (T ) + λF∗(T )
if |θ − θt| ≤ δθ:
return T ,
Rahat, Everson & Fieldsend Multi-Path for Net. Lifetime and Robustness GECCO, 18 July 2017 17 / 17