OPTIMISATION OF CAR-PARK DESIGNS ESGI91 - Arup J. Billingham, J. Bradshaw, M. Bulkowski, P . Dawson, P . Garbacz, M. Gilberg, L. Gosce, P . Hjort, M. Homer, M. Jeffrey, D. Papavassiliou, R. Porter, D. Wind
PROBLEM STATEMENT At present, the layout of car parks is designed by an architect with CAD tools, and the results may be extremely complicated. There is no guarantee that the architect has achieved the maximum possible number of spaces . We seek automatic algorithms which will either replace the role of the architect (or more realistically, act as a decision support tool to him/her).
APPROACHES We assume that the perimiter of the parking lot area is a polygon. Parking spaces are rectangles. All cars should be able to get out. Lanes • Use algorithms to generate good routes around the polygon, and fit parking spaces to this route. Cars • Fill the polygon with parking spaces, and modify to assure exit-routes for all cars.
HILBERT’S CAR PARK
TAPESTRY + TRIM Overlay car park polygon with Hilbert’s optimal tapestry. Then trim excess parking spaces. If necessary ensure connectivity of lanes by removing some spaces.
HILBERT’S CAR STRIP a For finite height, non perpendicular tilings may be optimal.
NON LINEAR OPTIMISATION To find the optimal packing in the finite Hilbert strip Maximise: Car density Subject to: Constraint 1: Constraint 2:
MATLAB
‘Roads’ Approach Strategy - To design a road network around which parking spaces can be efficiently packed.
Cost Functions We want to design a road network that covers the car park Ω with sufficient road separation to allow parking Function 2 - Numerical Function 1 - Continuous (Piecewise Linear or Bezier Filling the space with ‘car-sized’ Curves) gaps between the curves is Penalise equivalent to wanting a certain ◮ being outside the car park amount of road in the disc (of ◮ nodes coming closer than a half road width) around each point in the domain road width ◮ points far from road ◮ Penalty (at a point) is a ◮ non-adjacent stretches of function of the difference between the desired and road being too close actual amount of road in the ◮ (piecewise linear paths only) disc. angles less than π/ 2
Variational Approach We minimise the spatial average of Cost Function 1 over the paths f : [0 , 1] → Ω in Ω. The Euler-Lagrange Equation for the variational problem is f ′ ( z ) f ′ ( z ) f ′′ ( z ) · f ′ ( z ) ���� � �� � � f ′ ( z ) · f ′ ( z ) � � x − f ( z ) δ ( � f ( z ) − x � 2 − w ) + I � f ( z ) − x � 2 < w − � f ′ ( z ) � 2 � f ′ ( z ) � 2 � f ′ ( z ) � 3 Ω 2 � 1 � � I � f ( y ) − x � 2 < w � f ′ ( y ) � 2 − kw � × d y 0 ��� f ′′ ( z ) · f ′ ( z ) � � f ′ ( z ) · � f ′ ( z ) � 2 + I � f ( z ) − x � 2 < w + � x − f ( z ) � δ ( � f ( z ) − x � 2 − w ) � f ′ ( z ) � 2 � 1 f ′ ( y ) �� × I � f ( y ) − x � 2 < w d x d y � f ′ ( y ) � 2 0 � � 1 � �� δ ( � f ( z ) − x � 2 − w ) � f ′ ( z ) � 2 � I � f ( y ) − x � 2 < w � f ′ ( y ) � 2 − kw � � = x − f ( z ) d y Ω 0 � � 1 � I � f ( z ) − x � 2 < w � f ′ ( z ) � 2 − kw δ ( � f ( y ) − x � 2 − w ) � f ′ ( y ) � 2 + � x − f ( y ) � 0 We hope to discretize this and solve numerically.
Numerical Algorithm We loop over two optimization algorithms sequentially to minimize Cost Function 2 . We discretize the problem by ◮ partitioning the curve into N linear sections ◮ using an M × M grid of points to minimize the cost function We loop over ◮ Genetic Algorithm ◮ Simplex Algorithm
Plots
How good is a given network? In a given car park, a small number of plausible road networks . . . easy to make ‘intuitive guesses’ Which is best and how can it be optimized? Define a road network with roads and connections Develop a coarse algorithm to assess how good a network is assume ‘nose in’ parking is best distribute ‘nose in’ parking stalls along each segment on the network penalise for overlapping stalls (proportional to area) stalls that leave the domain voids in coverage Combine penalties into one ‘cost function’ For a selection of initial guesses, optimise cost function w.r.t. nodes ARUP: Optimizing Car Parking
Output n parked =191 60 50 8 7 6 5 4 3 40 30 2 1 20 9 10 11 12 13 14 10 0 −10 0 10 20 30 40 50 60 70 80 90 ARUP: Optimizing Car Parking
Output n parked =205 60 50 4 3 40 30 5 2 1 20 6 7 10 0 −10 0 10 20 30 40 50 60 70 80 90 ARUP: Optimizing Car Parking
Output n parked =247 70 60 6 5 4 3 2 1 50 7 18 19 40 8 16 17 30 20 9 10 11 12 13 14 15 10 0 −10 0 20 40 60 80 100 120 ARUP: Optimizing Car Parking
Output n parked =266 70 2 1 60 50 8 9 10 40 3 30 6 7 11 20 4 5 10 0 −10 0 20 40 60 80 100 120 ARUP: Optimizing Car Parking
Output n parked =273 70 60 6 5 4 3 2 1 50 7 17 18 40 8 30 16 20 10 9 10 11 12 13 14 15 0 −10 0 20 40 60 80 100 120 ARUP: Optimizing Car Parking
Output n parked =286 80 70 2 1 60 50 4 3 40 30 6 5 20 7 8 10 0 −10 0 20 40 60 80 100 120 ARUP: Optimizing Car Parking
n parked =191 60 50 8 7 6 5 4 3 40 30 2 1 20 9 10 11 12 13 14 10 0 −10 0 10 20 30 40 50 60 70 80 90
n parked =247 70 60 6 5 4 3 2 1 50 7 18 19 40 8 16 17 30 20 9 10 11 12 13 14 15 10 0 −10 0 20 40 60 80 100 120
QUESTIONS
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