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# A Knapsack Approach to Sensor-Mission Assignment with Uncertain Demands

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### A Knapsack Approach to Sensor-Mission Assignment with Uncertain Demands

1. 1. SPIE Europe Security+Defence 2008 A Knapsack Approach to Sensor-Mission Assignment with Uncertain Demands Diego Pizzocaro a Matthew P. Johnson b Hosam Rowaihy cStuart Chalmers d Alun Preece a Amotz Bar-Noy b Thomas La Porta ca Cardiff University b The City University of New York c Pennsylvania State University d University of Aberdeen
2. 2. Outline1. Motivation2. Formal model3. Pre-existent algorithms4. Novel algorithm5. Performance evaluation6. Conclusion
3. 3. Motivation• 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. X X Target Target X Target 2 X X Target X Target 1 X Target Target X X Target X Target Target• Missions have an uncertain demand for sensing resource capabilities (due to weather, unexpected events, ...)
4. 4. Formal Model• We model this assignment problem by introducing the Sensor Utility Maximization (SUM) model. Sensors• SUM can be represented as a bipartite graph: S1 (e11, p1 1 ) Missions (e 12 ,p ‣ (d1, p1) M1 Vertex sets: 12 ) Sensors {Si}, Missions {Mj} S2 ‣ For each mission (Mj): S3 M2 (d2, p2) dj = utility demand pj = priority S4 ‣ 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.
5. 5. Formal Model (contd)• Integer Linear Programming (ILP) formulation: m n Maximize: j=1 i=1 pij xij , where pij = eij /dj × pj . n Such that: i=1 xij eij ≤ dj , for each mission Mj ∈ M , m j=1 xij ≤ 1, for each sensor Si , and xij ∈ {0, 1}, for each variable xij• 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)
6. 6. 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
7. 7. 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} ). X S1 M1 Sensor 2 Target 2 X Sensor 3 S2 M2 Target 1 Sensor 3 X S3 M3 Target 3 - 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 ...
8. 8. 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} ). S1 M1 S2 M2 S3 M3 - 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 ...
9. 9. 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} ). S1 M1 M1 .1 =0 S2 M2 p11 S1 p12 = 0.3 M2 p13 =0 S3 M3 .4 M3 - 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 ...
10. 10. 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} ). S1 M1 .3 M1 M1 p21 = 0 = 0.1 S2 M2 S2 p11 p22 = p12 = 0.3 S1 0.2 M2 M2 p13 =0 S3 M3 .4 M3 - 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 ...
11. 11. 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} ). S1 M1 .6 M2 p32 = 0 .3 M1 M1 S3 p21 = 0 = 0.1 S2 M2 p S233 = 0 p11 .2 M3 p22 = p12 = 0.3 S1 0.2 M2 M2 p13 =0 S3 M3 .4 M3 - 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 ...
12. 12. 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} ). + S1 M1 .6 M2 S3 p32 = 0.6 p32 = 0 .3 M1 M1 S3 p21 = 0 = 0.1 S2 M2 p S233 = 0 p11 S1 p13 = 0.4 .2 M3 p22 = p12 = 0.3 S1 0.2 M2 M2 p13 =0 S3 M3 .4 S2 p21 = 0.3 - M3 - 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 ...
13. 13. 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} ). X Sensor 2 Target 2 X Sensor 1 Target 1 Sensor 3 X Target 3 - 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 ...
14. 14. Performance evaluation• 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   1 2 if Dij ≤ SR 1+Dij /c of the distance Dij between sensor and mission. eij =  0 otherwise• 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: 18000 84 % Optimal Fractional Solution 16000 82 14000 80 Running Time (ms) Optimality Time 12000 78 76 10000 74 8000 72 6000 70 4000 68 2000 66 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Number of Missions Number of Missions 1000 sensors
15. 15. 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) 500 sensors 1000 sensors 96 92 GAP approx alg GAP approx alg % Optimal Solution % Optimal Solution 88 92 84 Ordered Sensor-side greedy 88 Ordered Sensor-side greedy 10 30 50 70 90 110 130 150 10 30 50 70 90 110 130 150 Number of Missions Number of Missions• 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
16. 16. Performance evaluation (Time) Effective running time (1000 sensors) Running Time (ms) 8000 GAP approx alg 6000 4000 2000 Ordered Sensor-side greedy 0 10 30 50 70 90 110 130 150 Number of Missions • Best effective running time: Ordered Sensor-side greedy‣ Best trade off optimality/running time: Ordered Sensor-side greedy
17. 17. 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
18. 18. Thanks for listening