A Knapsack Approach to Sensor-Mission Assignment with Uncertain Demands

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






S1 M1

M2

M3

S2

S3






S1

M1

M2

p11 = 0.1

p12 = 0.3

M3

p13 = 0.4

S1 M1

M2

M3

S2

S3






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






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






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






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


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)


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

Spiritual

A Knapsack Approach to Sensor-Mission Assignment with Uncertain Demands