A Novel MIP-based Airspace Sectorization for TMAs · ATM seminar, June 28, 2017 3 • International...

A Novel MIP-based Airspace Sectorization for TMAs Tobias Andersson Granberg, Tatiana Polishchuk, Valentin Polishchuk, Christiane Schmidt

ATM seminar, June 28, 2017 2

Introduction: Air transportation, Workload/Taskload, Sectorization

Grid-based IP formulation

Experimental Study: Arlanda Airport

Conclusion/Outlook


• International Air Transport Association (IATA) projected that the number of passengers will double to reach 7 billion/year by 2034



• Terminal Maneuvering Area (TMA) is particularly affected by congestion



• Terminal Maneuvering Area (TMA) is particularly affected by congestion• Improve design of arrival and departure procedures ➜ higher throughput



• Terminal Maneuvering Area (TMA) is particularly affected by congestion• Improve design of arrival and departure procedures ➜ higher throughput• In Air Traffic Management (ATM): humans-in-the-loop!




๏ Planes constantly monitored/guided by air traffic controllers (ATCOs)




๏ Planes constantly monitored/guided by air traffic controllers (ATCOs)๏ ATCOs assigned to sectors of the airspace (TMA)




๏ Planes constantly monitored/guided by air traffic controllers (ATCOs)๏ ATCOs assigned to sectors of the airspace (TMA)๏ Workload of the ATCOs should be balanced





• Plus: geometric constraints on the sector shape






Workload?






Workload?Various tasks contribute to airspace’s complexity and drive ATCO’s mental workload






Workload?Various tasks contribute to airspace’s complexity and drive ATCO’s mental workload• Taskload: objective demands of the ATCO’s monitoring task






Workload?Various tasks contribute to airspace’s complexity and drive ATCO’s mental workload• Taskload: objective demands of the ATCO’s monitoring task • We use heat maps of the density of weighted clicks as an input [Zohrevandi et al., 2016].






Workload?Various tasks contribute to airspace’s complexity and drive ATCO’s mental workload• Taskload: objective demands of the ATCO’s monitoring task • We use heat maps of the density of weighted clicks as an input [Zohrevandi et al., 2016].• BUT: we do not depend on specific maps.


A sectorization of a simple polygon P is a partition of P into k disjoint subpolygons S1,…,Sk (Si ∩ Sj = ∅ ∀i≠j), such that .



Si: sectors



Si: sectors

Sectorization Problem:



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load(c) Balanced taskload



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load(c) Balanced taskload(d) Connected sectors



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load(c) Balanced taskload(d) Connected sectors(e) Nice shape (smooth boundary and an easily memorable shape)



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load(c) Balanced taskload(d) Connected sectors(e) Nice shape (smooth boundary and an easily memorable shape)(f) Interior conflict points ( Points that require increased attention from ATCOs

should lie in the sector’s interior.)



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors.Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load(c) Balanced taskload(d) Connected sectors(e) Nice shape (smooth boundary and an easily memorable shape)(f) Interior conflict points ( Points that require increased attention from ATCOs

should lie in the sector’s interior.)(g) Convex sectors ((straight-line) flight cannot enter and leave a convex sector

multiple times)

ATM seminar, June 28, 2017


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Mixed Integer Program (MIP):



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Mixed Integer Program (MIP):• Not new for sectorization



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Mixed Integer Program (MIP):• Not new for sectorization• But so far: synthesis methods



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- Variable per elementary airspace piece



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- Variable per elementary airspace piece- Glue together pieces to sectors



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- Variable per elementary airspace piece- Glue together pieces to sectors

• We: variable per potential edge of sector boundary



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๏ Square grid in the TMA



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๏ Square grid in the TMA๏ G = (V,E):

๏ Every grid node connected to its 8 neighbors



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๏ Every grid node connected to its 8 neighbors๏ N(i) = set of neighbors of i (including i)



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๏ Every grid node connected to its 8 neighbors๏ N(i) = set of neighbors of i (including i)๏ length of an edge (i, j)`i,j



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๏ Every grid node connected to its 8 neighbors๏ N(i) = set of neighbors of i (including i)๏ length of an edge (i, j)

Main idea: use an artificial sector, S0, that encompasses the complete boundary of P, using all counterclockwise (ccw) edges.

`i,j



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๏ Every grid node connected to its 8 neighbors๏ N(i) = set of neighbors of i (including i)๏ length of an edge (i, j)

Main idea: use an artificial sector, S0, that encompasses the complete boundary of P, using all counterclockwise (ccw) edges. We use sectors in S* = S ∪ S0 with S = {S1,…,Sk} .

`i,j



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: edge (i,j) used for sector s



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All ccw boundary edges in S0




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If (i,j) used for some sector, (j, i) has to be used as well.




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If (i,j) used for some sector, (j, i) has to be used as well.Sector cannot contain (i,j) and (j,i).




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(i,j) in Sl, (j,i)

has to be in

another

sector




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No edge in two sectors.

(i,j) in Sl, (j,i)

has to be in

another

sector




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Minimum size

(i,j) in Sl, (j,i)

has to be in

another

sector




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Minimum size

Indegree=outdegree for all vertices

(i,j) in Sl, (j,i)

has to be in

another

sector




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Minimum size


A node has at most one ingoing edge per sector

(i,j) in Sl, (j,i)

has to be in

another

sector




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Minimum size


A node has at most one ingoing edge per sector

(i,j) in Sl, (j,i)

has to be in

another

sector


⟹ Union of the |S| sectors completely covers the TMA.



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(a) Balanced Size



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(a) Balanced SizeWe need to assign area to sector selected by boundary edges!



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(a) Balanced SizeWe need to assign area to sector selected by boundary edges!Area of polygon P with rational vertices and can be computed efficiently [Fekete et al., 2015]:



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(a) Balanced SizeWe need to assign area to sector selected by boundary edges!Area of polygon P with rational vertices and can be computed efficiently [Fekete et al., 2015]:• We introduce reference point r.



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(a) Balanced SizeWe need to assign area to sector selected by boundary edges!Area of polygon P with rational vertices and can be computed efficiently [Fekete et al., 2015]:• We introduce reference point r.• We compute the area of the triangle of each directed edge e of P.



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(a) Balanced SizeWe need to assign area to sector selected by boundary edges!Area of polygon P with rational vertices and can be computed efficiently [Fekete et al., 2015]:• We introduce reference point r.• We compute the area of the triangle of each directed edge e of P.• We sum up the triangle area for all edges of P:



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- cw triangles contribute positive



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- cw triangles contribute positive- ccw triangles contribute negative



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• fi,j: signed area of the triangle (i,j) and r



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Assigns area of sector s to as



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Sum of areas = area of S0



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Sum of areas = area of S0

as � aLB 8s 2 S, aLB = c1 · a0/|S|



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(b) Bounded Taskload / (c) Balanced Taskload



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(b) Bounded Taskload / (c) Balanced TaskloadWe need to associate task load with a sector.



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(b) Bounded Taskload / (c) Balanced TaskloadWe need to associate task load with a sector.• Overlay heat map with a grid.



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(b) Bounded Taskload / (c) Balanced TaskloadWe need to associate task load with a sector.• Overlay heat map with a grid.• Extract values at the grid points.



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(b) Bounded Taskload / (c) Balanced TaskloadWe need to associate task load with a sector.• Overlay heat map with a grid.• Extract values at the grid points.• Use discretized heat map.



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(b) Bounded Taskload / (c) Balanced TaskloadWe need to associate task load with a sector.• Overlay heat map with a grid.• Extract values at the grid points.• Use discretized heat map.• Each discrete heat map point q: “heat value” hq



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(b) Bounded Taskload / (c) Balanced TaskloadWe need to associate task load with a sector.• Overlay heat map with a grid.• Extract values at the grid points.• Use discretized heat map.• Each discrete heat map point q: “heat value” hq• Let the sign of fi,j be pi,j



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Objective Function



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Objective Function• Choice not obvious.



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Objective Function• Choice not obvious. • Used in literature:



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• Taskload imbalance(constraint c)



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• Taskload imbalance(constraint c) • Number of sectors (input)



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• If we want to balance the area of the sectors, but are not interested in the sector taskload, this objective function ensures that sectors are connected (constraint d)



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• With taskload: only connected sectors if c2 allows it:



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• With taskload: only connected sectors if c2 allows it:Given the current complexity map: user must allow larger imbalances between controller’s taskload, if having connected sectors is a necessary condition.



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• With taskload: only connected sectors if c2 allows it:Given the current complexity map: user must allow larger imbalances between controller’s taskload, if having connected sectors is a necessary condition.• With constraint (f), interior conflict points:



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• Points with higher complexity should be in sector’s interior



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• Points with higher complexity should be in sector’s interior• In relation to complexity of other points ➜ no absolute threshold



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• Points with higher complexity should be in sector’s interior• In relation to complexity of other points ➜ no absolute threshold• Force points of relatively high complexity to be in interior



Si: sectors

Sectorization Problem:Given: The coordinates of the TMA, defining a polygon P , the number of sectors |S|, and a set C of constraints on the resulting sectors. Find: A sectorization of P with k = |S| , fulfilling C.(a) Balanced size(b) Bounded task load(c) Balanced taskload(d) Connected sectors(e) Nice shape (smooth boundary and an easily memorable shape) (f) Interior conflict points ( Points that require increased attention from ATCOs


multiple times)



Si: sectors



multiple times)

✔



Si: sectors



multiple times)

✔✔



Si: sectors



multiple times)

✔✔✔



Si: sectors



multiple times)

✔✔✔✔



Si: sectors



multiple times)

✔✔✔✔ Preprocessing



Si: sectors



multiple times)


✔



Si: sectors



multiple times)


✔

in a publication based on this work [ICNS 2017]



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AMPL and CPLEX 12.6 on a single server with 24GB RAM and four kernels running on Linux. Each instance was run until a solution with less than 1% gap had not been found, or for a maximum of one CPU-hour. No instance finished with an optimality gap of more than 6%.

(c)Balanced taskload, (d) Connected sectors, (e) Nice shape



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AMPL and CPLEX 12.6 on a single server with 24GB RAM and four kernels running on Linux. Each instance was run until a solution with less than 1% gap had not been found, or for a maximum of one CPU-hour. No instance finished with an optimality gap of more than 6%.

(c)Balanced taskload, (d) Connected sectors, (e) Nice shape

(c)Balanced taskload, (d) Connected sectors, (e) Nice shape, (g) Interior Conflict Points

wi,j = hi +hj. ←𝛾 = 0.9 𝛾= 0.5➞.



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5 sectors: (c)Balanced taskload, (d) Connected sectors, (e) Nice shape, (f) Interior conflict points



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wi,j = hi +hj, 𝛾= 0.5 c2=0.9



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wi,j = hi +hj, 𝛾= 0.5 c2=0.9

No “c2-balanced” solution



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wi,j = hi +hj, 𝛾= 0.5 c2=0.9

5 sectors: (a)Balanced size, (d) Connected sectors, (e) Nice shape, (f) interior conflict points




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wi,j = hi +hj, 𝛾= 0.5 c2=0.9

5 sectors: (a)Balanced size, (d) Connected sectors, (e) Nice shape, (f) interior conflict points

c1=0.7




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Influence of choosing wi,j :



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Influence of choosing wi,j :|S|=2 ➜ One cut through rectangle



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(c)Balanced task load, (d) Connected sectors, (e) Nice shape, (g) Interior Conflict Points



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(c)Balanced task load, (d) Connected sectors, (e) Nice shape, (g) Interior Conflict Points

𝛾=1 𝛾=0.5 and 𝛾=0.5 and


Conclusion/Outlook


Conclusion/Outlook

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Conclusion

Outlook


Conclusion/Outlook

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Conclusion• Sectorization method that balances sector task load

Outlook


Conclusion/Outlook

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Conclusion• Sectorization method that balances sector task load• Includes various geometrical constraints

Outlook


Conclusion/Outlook

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Conclusion• Sectorization method that balances sector task load• Includes various geometrical constraints• Results for Stockholm TMA

Outlook


Conclusion/Outlook

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Conclusion• Sectorization method that balances sector task load• Includes various geometrical constraints• Results for Stockholm TMA• Highly flexible approach

Outlook


Conclusion/Outlook

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Conclusion• Sectorization method that balances sector task load• Includes various geometrical constraints• Results for Stockholm TMA• Highly flexible approach• Fine-grained view on the TMA

Outlook


Conclusion/Outlook

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Conclusion• Sectorization method that balances sector task load• Includes various geometrical constraints• Results for Stockholm TMA• Highly flexible approach• Fine-grained view on the TMA• Easily integrates SUA etc.

Outlook


Conclusion/Outlook

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• We extended with convex sectors (ICNS 2017)Outlook


Conclusion/Outlook

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• We extended with convex sectors (ICNS 2017)• Allow usage of a few reflex vertices

Outlook


Conclusion/Outlook

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• We extended with convex sectors (ICNS 2017)• Allow usage of a few reflex vertices➡limit the total deviation from a maximum interior degree of 180 of reflex

vertices per sector

Outlook


Conclusion/Outlook

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vertices per sector• Detailed evaluation by ATCOs

Outlook


Conclusion/Outlook

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vertices per sector• Detailed evaluation by ATCOs• Combine with MIP for SIDs and STARs to an integrated design

Outlook


Conclusion/Outlook

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• Sectorization method that balances sector task load • Includes various geometrical constraints • Results for Stockholm TMA • Highly flexible approach • Fine-grained view on the TMA • Easily integrates SUA etc.

• We extended with convex sectors (ICNS 2017) • Allow usage of a few reflex vertices ➡limit the total deviation from a maximum interior degree of 180 of reflex

vertices per sector • Detailed evaluation by ATCOs • Combine with MIP for SIDs and STARs to an integrated design

Outlook

THANK YOU.Conclusion

A Novel MIP-based Airspace Sectorization for TMAs · ATM seminar, June 28, 2017 3 • International...

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