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A Knapsack Approach to Sensor-Mission Assignment with
Uncertain Demands
Diego Pizzocaro a Matthew P. Johnson b Hosam Rowaihy c
Stuart Chalmers d Alun Preece a Amotz Bar-Noy b Thomas La Porta c
a Cardiff University b The City University of New York c Pennsylvania State Universityd University of Aberdeen
SPIE Europe Security+Defence 2008
1. Motivation
2. Formal model
3. Pre-existent algorithms
4. Novel algorithm
5. Performance evaluation
6. Conclusion
Outline
• An homogeneous sensor network which is already deployed.
• It is usually required to support multiple missions to be accomplished
simultaneously.
• Missions compete for the exclusive usage of a sensor:
we need a scheme to assign individual sensors to missions.
• Missions have an uncertain demand for sensing resource capabilities
(due to weather, unexpected events, ...)
Motivation
XTarget 1
XTarget 2
XTarget X
Target
XTarget
XTarget X
Target
XTarget
XTarget
XTarget
Formal Model
• We model this assignment problem by introducing the Sensor Utility Maximization (SUM) model.
• SUM can be represented as a bipartite graph:
‣ Vertex sets:Sensors {Si}, Missions {Mj}
‣ For each mission (Mj): dj = utility demand pj = priority
‣ For each sensor-mission pair: eij = utility that Si could contribute to Mj
pij = eij / dj * pj profit of Si for mission Mj
• Goal: a sensor assignment that maximizes the total profit.
S1
S2
S3
S4
M1
M2
Sensors
Missions(e11, p11)
(d1, p1)
(d2, p2)
(e12, p
12)
Formal Model (contd)
• Integer Linear Programming (ILP) formulation:
• Most important constraint (knapsack-style): Total utility cumulated by each mission Mj ≤ its demand dj
• This assumption is based:
‣ on the uncertainty of the demands,
‣ on the principle that sensing resources are in high demand and should not be wasted.(from “JP 2-01 joint and national intelligence support to military operations”, see ref, in the paper)
Maximize:!m
j=1
!ni=1 pijxij , where pij = eij/dj ! pj.
Such that:!n
i=1 xijeij " dj, for each mission Mj # M ,!m
j=1 xij " 1, for each sensor Si, and
xij # {0, 1}, for each variable xij
Pre-existent Algorithms
• Problem hardness: it is NP-Complete
- Proof by showing that SUM is a generalization of the knapsack problem
• We adapted three pre-existent algortihms to solve SUM:
1. Mission-side greedy
- Sort missions by decreasing priority ( pj )
- For each mission: Assign sensors with highest utilities ( eij ) to the mission,
until the cumulated utility does not exceed its demand ( dij ).
2. Sensor-side greedy
- Unsorted sensors
- For each sensor: Assign the sensor to the mission where that sensor is of most use
(the one which maximizes the profit of each sensor pij)
3. State-of-the-art approximation algorithm for GAP
- Proved that SUM is a special case of Generalized Assignment Problem (GAP)
- Provide a very good guaranteed approximation of the optimal solution
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
XTarget 1
XTarget 2
XTarget 3
Sensor 3
Sensor 2
Sensor 3
S1 M1
M2
M3
S2
S3
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
S1 M1
M2
M3
S2
S3
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
S1
M1
M2
p11 = 0.1
p12 = 0.3
M3
p13 = 0.4
S1 M1
M2
M3
S2
S3
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
S1
M1
M2
p11 = 0.1
p12 = 0.3
M3
p13 = 0.4
S2
M1
M2
p21 = 0.3
p22 = 0.2
S1 M1
M2
M3
S2
S3
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
S1
M1
M2
p11 = 0.1
p12 = 0.3
M3
p13 = 0.4
S2
M1
M2
p21 = 0.3
p22 = 0.2
S3
M2
M3
p32 = 0.6
p33 = 0.2
S1 M1
M2
M3
S2
S3
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
S1
M1
M2
p11 = 0.1
p12 = 0.3
M3
p13 = 0.4
S2
M1
M2
p21 = 0.3
p22 = 0.2
S3
M2
M3
p32 = 0.6
p33 = 0.2 S1 p13 = 0.4
S2 p21 = 0.3
S3 p32 = 0.6+
-
S1 M1
M2
M3
S2
S3
Novel algorithm• It is an improved version of Sensor-side greedy:
‣ Ordered Sensor-side greedy
- IDEA: Sorts sensors by decreasing max profit offer between all the missions to which the sensor can contribute ( max{pij} ).
- Like basic Sensor-Side Greedy: Assign sensor to the mission where that sensor is of most use (the one which maximizes the profit pij).
- Further improvement: If a sensor cannot be assigned to the mission which maximize its profit, we consider the second most profitable mission, and so on ...
XTarget 1
Sensor 2
Sensor 3
Sensor 1
XTarget 3
XTarget 2
• Simulation environment implemented in Java.
• Missions and Sensors are deployed in random positions in the field.
• Utility of sensor Si to mission Mj is a function
of the distance Dij between sensor and mission.
• For each experiment:
- Constant number of sensors (200, 500, 1000)
- Number of simultaneous missions was varied from 10 to 150.
• Experiments divided into two main groups to test algs’ efficiency:
eij =
!"
#
11+D2
ij/cif Dij ! SR
0 otherwise
Performance evaluation
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Number of Missions
66
68
70
72
74
76
78
80
82
84
% O
ptim
al F
ractio
na
l S
olu
tio
n
Optimality10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Number of Missions
0
2000
4000
6000
8000
10000
12000
14000
16000
18000R
un
nin
g T
ime
(m
s)
1000 sensors
Time
Performance evaluation (Optimality)
• The graphs show the sum of the profits pij of each sensor-mission assignments(i.e. the value of the objective function in the ILP formulation)
• We show only the best two algs: Sensor-Side Greedy and GAP approx algorithm
‣ Optimality results differ only by 1% in average.
‣ The best solution quality: GAP approx algorithm
500 sensors
10 30 50 70 90 110 130 150
Number of Missions
84
88
92
% O
ptim
al S
olu
tio
n
Ordered Sensor-side greedy
GAP approx alg
1000 sensors
10 30 50 70 90 110 130 150
Number of Missions
88
92
96
% O
ptim
al S
olu
tio
n
Ordered Sensor-side greedy
GAP approx alg
Performance evaluation (Time)
• Best effective running time: Ordered Sensor-side greedy
Effective running time (1000 sensors)
10 30 50 70 90 110 130 150
Number of Missions
0
2000
4000
6000
8000
Ru
nn
ing
Tim
e (
ms)
Ordered Sensor-side greedy
GAP approx alg
‣ Best trade off optimality/running time: Ordered Sensor-side greedy
Conclusion
• We considered an homogeneous sensor network which was already deployed, to support competing simultaneous missions with uncertain demands.
• We developed a formal model and a new greedy algorithm.
• Simulation results show that: our novel Ordered Sensor-side greedy algorithm offers the best trade-off between optimality of the solution and effective running time.
• Future work:
‣ Heterogeneous sensor types
‣ Sharing of a sensing resource
Thanks for listening