OPTIMISATION OF CAR-PARK DESIGNS ESGI91 - Arup J. Billingham, J. - - PowerPoint PPT Presentation

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

• Use algorithms to generate good routes around the polygon,

and fit parking spaces to this route.

Lanes Cars

• Fill the polygon with parking spaces, and modify to assure

exit-routes for all cars.

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.

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 1 - Continuous Filling the space with ‘car-sized’ gaps between the curves is equivalent to wanting a certain amount of road in the disc (of half road width) around each point in the domain

◮ Penalty (at a point) is a

function of the difference between the desired and actual amount of road in the disc. Function 2 - Numerical (Piecewise Linear or Bezier Curves) Penalise

◮ being outside the car park ◮ nodes coming closer than a

road width

◮ points far from road ◮ non-adjacent stretches of

road being too close

◮ (piecewise linear paths only)

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

• Ω
• f ′(z) ·
• x − f (z)
• δ(f (z) − x2 − w)
• f ′(z)

• f ′(z)

• ×

• If (y)−x2<w f ′(y)2 − kw
• dy
• +
• f ′(z) ·
• x − f (z)
• δ(f (z) − x2 − w)
• f ′(z)2 + If (z)−x2<w

• ×

• dx

• Ω
• x − f (z)
• δ(f (z) − x2 − w)f ′(z)2

• If (y)−x2<w f ′(y)2 − kw
• dy

• If (z)−x2<w f ′(z)2 − kw

• x − f (y)
• δ(f (y) − x2 − w)f ′(y)2

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

• verlapping 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

Output

Output

Output

Output

Output

Output

10 20 30 40 50 60 70 80 90 −10 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 nparked=191

20 40 60 80 100 120 −10 10 20 30 40 50 60 70 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 nparked=247

QUESTIONS