Ordonnancement robuste de réseaux de capteurs sans fil pour le suivi d une cible mobile sous incertitudes

Transcription

1 Ordonnancement robuste de réseaux de capteurs sans fil pour le suivi d une cible mobile sous incertitudes Charly Lersteau Marc Sevaux André Rossi ROADEF 2016, Compiègne February 11, /17

2 Context 2/17

3 Context m static sensors s 1 s 3 2/17

4 Context m static sensors Battery capacity: E i s1 s 3 2/17

5 Context m static sensors Battery capacity: E i Sensing range: R S R S s 1 s 3 2/17

6 Context m static sensors Battery capacity: E i Sensing range: R S s 1 s 3 One moving target Known trajectory 2/17

7 Context m static sensors Battery capacity: E i Sensing range: R S s 1 s 3 One moving target Known trajectory Mission Continuously monitoring the target 2/17

8 Problem s 1 s 3 Sensing activity Action of monitoring the target using a sensor Decision 0 H Timeline 3/17

9 Problem s 1 s 3 Sensing activity Action of monitoring the target using a sensor Decision Design a static schedule of sensing activities Activities must be scheduled offline s 1 s 3 0 H Timeline 3/17

10 Problem s 1 s 3 Constraints Enforce full coverage Battery lifetime One active sensor s 1 s 3 0 H Timeline 3/17

11 Problem s 1 s 3 Uncertainty The target can be early or late from the prediction Objective Maximize the earliness and tardiness that the schedule can face s 1 s 3 0 H Timeline 3/17

12 Overview Initial instance Discretization Scheduling problem Solve Schedule Two-step algorithm Discretization: transform the initial instance into a scheduling problem Solve the scheduling problem: activate sensors so as to cover the target despite its lateless or earliness 4/17

13 Discretization s 1 s 3 Area split into disjoint faces Face: surface covered by the same set of sensors f 1 f 2 f 4 f 5 f 3 0 H Timeline 5/17

14 Discretization s 1 s 3 Area split into disjoint faces Face: surface covered by the same set of sensors f 1 f 2 f 4 f 5 Trajectory can be seen: as a sequence of the traversed faces f 3 f 1 f 2 f 3 f 4 f 5 0 H Timeline 5/17

15 Discretization s 1 s 3 Area split into disjoint faces Face: surface covered by the same set of sensors f 1 f 2 f 4 f 5 Trajectory can be seen: as a sequence of the traversed faces as a sequence of sets of candidate sensors f 1 {s 1} f 2 {s 1, } f 3 f 3 {} 0 H Timeline f 4 {, s 3} f 5 {s 3} 5/17

16 Discretization s 1 s 3 Ticks: dates of transition from a face to another Time window: interval between two consecutive ticks {s 1} {s 1, } {} {, s 3} {s 3} 0 H Timeline 5/17

17 Discretization output I Set of sensors {s 1,...,s m } E i Battery lifetime of the sensor i K Set of time windows {1,..., K } t k Starting time of time window k k Duration of time window k S(k) Set of candidate sensors covering the target during time window k {s 1} {s 1, } {} {, s 3} {s 3} t 3 t 4 t 5 t 6 Timeline 6/17

18 Discretization output {s 1} {s 1, } {} {, s 3} {s 3} t 3 t 4 t 5 t 6 Timeline Representation as availability intervals s 3 s 1 t 3 t 4 t 5 t 6 Timeline 7/17

19 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates {s 1} {s 1, } {} Schedule (1) s 1 8/17

20 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates (t 1 late) {s 1} {s 1, } {} t 1 t 1 t 2 target loss! Schedule (1) s 1 8/17

21 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates {s 1} {s 1, } {} Schedule (1) s 1 Schedule (2) s 1 8/17

22 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates (t 2 early) {s 1} {s 1, } {} t 2 Schedule (1) s 1 target loss! Schedule (2) s 1 8/17

23 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates {s 1} {s 1, } {} Schedule (1) s 1 Schedule (2) s 1 Schedule (3) s 1 8/17

24 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates (t 1 late) {s 1} {s 1, } {} t 1 t 1 t 2 Schedule (1) s 1 Schedule (2) s 1 Schedule (3) s 1 8/17

25 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates (t 2 early) {s 1} {s 1, } {} t 2 Schedule (1) s 1 Schedule (2) s 1 Schedule (3) s 1 8/17

26 Robustness study Uncertainty Earliness and tardiness can occur at every tick t k Candidates {s 1} {s 1, } {} Schedule (1) s 1 Schedule (2) s 1 Schedule (3) s 1 Stability radius ρ: Maximal tick time variation allowed (in units of time) Schedule (3) has a greater stability radius than schedules (1) and (2) 8/17

27 Solution approach Definition: Optimization problem Find a feasible schedule such that its stability radius is maximized Definition: Decision problem Let ρ be an arbitrary positive value Does there exist a feasible schedule such that its stability radius is at least ρ? Answer: yes/no Suggested approach Dichotomy 9/17

28 Solution approach Algorithm Compute an upper bound UB on ρ = ρ [0, UB] Compute UB 10/17

29 Solution approach Algorithm Compute an upper bound UB on ρ = ρ [0, UB] Collect all the positive distances t k t k < UB in a list D, sorted by increasing order Compute UB D {t k t k < UB} 10/17

30 Solution approach Algorithm Compute an upper bound UB on ρ = ρ [0, UB] Collect all the positive distances t k t k < UB in a list D, sorted by increasing order Run a dichotomy on ρ by solving a decision problem at each iteration = returns a value D l from the list Compute UB D {t k t k < UB} Dichotomy 10/17

31 Solution approach Algorithm Compute an upper bound UB on ρ = ρ [0, UB] Collect all the positive distances t k t k < UB in a list D, sorted by increasing order Run a dichotomy on ρ by solving a decision problem at each iteration = returns a value D l from the list Final step: solve a linear program to maximize ρ [D l,d l+1 ) Compute UB D {t k t k < UB} Dichotomy Solve LP Schedule activities 10/17

32 Decision problem s 1 Solve the decision problem Generate an instance I ρ such that the stability radius is guaranteed = move the ticks to restrict the availability intervals s 3 t 3 t 4 t 5 t H time 1 ρ = 0 11/17

33 Decision problem s 1 Solve the decision problem Generate an instance I ρ such that the stability radius is guaranteed = move the ticks to restrict the availability intervals 1 ρ = 0 2 ρ = 1 s 1 s 3 s 3 t 3 t 4 t 5 t H t t 2 t 3 t 5 H t 6 t 4 time time 11/17

34 Decision problem s 1 Solve the decision problem Generate an instance I ρ such that the stability radius is guaranteed = move the ticks to restrict the availability intervals 1 ρ = 0 2 ρ = 1 3 ρ = 1.5 s 1 s 3 s 1 s 3 s 3 t 3 t 4 t 5 t H t t 2 t 3 t 5 H t 6 t 4 time time time t t 2 H t 4 t 3 11/17

35 Decision problem s 1 Solve the decision problem Generate an instance I ρ such that the stability radius is guaranteed = move the ticks to restrict the availability intervals 1 ρ = 0 2 ρ = 1 3 ρ = 1.5 s 1 s 3 s 1 s 3 s 3 t 3 t 4 t 5 t H t t 2 t 3 t 5 H t 6 t 4 time time time t t 2 H t 4 t 3 11/17

36 Decision problem For a given ρ Solve the decision problem as a transportation problem Suppliers: sensors of capacity E i Customers: time windows of duration k Decision variables: x ik Costs: c ik = 0 if sensor i is candidate for time window k, 1 otherwise Optimal objective value is 0 instance is feasible E 3 s 3 k 5 5 k 4 4 Sensors E 2 k 3 3 Time windows k 2 2 E 1 s 1 k /17

37 Final step Final step: Solve a linear program Dichotomy returns a value D l from the list D = ρ [D l,d l+1 ) Objective of LP: maximize ρ = D l +δ For each x ik > 0, schedule an activity of duration x ik involving the sensor i in the time window k max δ (1) x ik E i i I (2) k K i S(k) i S(k) x ik = k +(σ k+1 σ k )δ k K (3) δ 0 (4) x ik 0 k K, i S(k) (5) 13/17

38 Solution approach Complexity Scheduling problem solvable in O(m K (m+ K ) log K ), where m is the number of sensors, K the number of time windows Global problem: pseudo-polynomial Compute UB D {t k t k < UB} Dichotomy Solve LP Schedule activities 14/17

39 Experiments Experiments on random instances Language: C++ Machine: Xeon processor W3520 (2.67 GHz 8), 8GB RAM, Linux Instances up to 1000 sensors are solved in less than 20 s in average Discretization Dichotomy Average time (s) Number of sensors 15/17

40 Extensions and further research Flexible scheduling (i.e. with recourse) s2 s3 t3 t4 s1 s2 t1 t2 s2 s3 t3 t4 16/17

41 Extensions and further research Flexible scheduling (i.e. with recourse) Spatial uncertainty s2 s3 t3 t4 s1 s2 t1 t2 s2 s3 t3 t4 16/17

42 Extensions and further research Flexible scheduling (i.e. with recourse) Spatial uncertainty s2 s3 Communication costs t3 t4 s1 s2 t1 t2 s2 s3 t3 t4 16/17

43 Conclusion Lersteau C, Rossi A, and Sevaux M, Robust scheduling of wireless sensor networks for target tracking under uncertainty, European Journal of Operational Research, 2016, DOI: /j.ejor Thank you for your attention. Questions? 17/17