MooMoo--ringsringsMarina Mooring OptimizationMarina Mooring Optimization

Group 8Group 8 –– Route 64Route 64Brian Siefering Amber Mazooji

Kevin McKenney Paul Mingardi

Vikram Sahney Kaz Maruyama

Presentation OverviewPresentation Overview

•• IntroductionIntroduction•• Problem DescriptionProblem Description•• AssumptionsAssumptions•• Model FormulationModel Formulation•• AnalysisAnalysis•• ConclusionsConclusions•• Lessons LearnedLessons Learned•• QuestionsQuestions

IntroductionIntroduction

•• What is a mooring?What is a mooring?•• Why an optimizationWhy an optimization

problem?problem?

Mooring Buoy

Mooring Line

Line Secured to Bedrock

Problem MotivationProblem Motivation

“Moorings are so scarce that in towns such as Orleans or Truro, the wait to get one can be as long as 20 years. In Sandwich, which has no moorings, only slips, the waiting list is closed with 1,200 waiting for a mere 200 spaces.”

Cape Cod Times 8/10/2003

Problem DescriptionProblem DescriptionObjective: Maximize Revenue!Objective: Maximize Revenue!

(also increase number of moorings in marina)(also increase number of moorings in marina)

Decision Variables: Boat LocationsDecision Variables: Boat Locations

Channel

Dock

Problem DescriptionProblem DescriptionMarina Cross Section Top View

8’

Exit

y

x (0,0) Dock

Depth, z

Channel

(xi,yi)

4’

AssumptionsAssumptions•• Marina:Marina:

–– The bottom of the marina is linear, sloping down in the +y direcThe bottom of the marina is linear, sloping down in the +y direction.tion.–– Tide change is 2 feet or less.Tide change is 2 feet or less.

•• Moorings:Moorings:–– Mooring lines are weightleMooring lines are weightless.ss. –– Moorings can and will beMoorings can and will be movemovedd eveeverry yey year.ar.–– At high tide, the mooring line angle is 30.At high tide, the mooring line angle is 30.

•• BoatsBoats–– Boats are between 15’ and 40’ in length.Boats are between 15’ and 40’ in length.–– Boats are classified into two caBoats are classified into two categories basetegories based on their hull ded on their hull deppth:th: boatsboats

with hull depths less than 4’ and boats with hull depths betweenwith hull depths less than 4’ and boats with hull depths between 4’ and4’ and 8’.8’.

•• PlacementPlacement–– Moorings can be precisely placeMoorings can be precisely placed.d.–– The minimum separation needed beThe minimum separation needed between boat sweeps is five feet.tween boat sweeps is five feet.–– Bow line length is negligible.Bow line length is negligible.–– Boats will be able to leave moorBoats will be able to leave moorings without specified lanes desings without specified lanes designatedignated

in a marina.in a marina.

Model FormulationModel Formulation Harbor Depth

Dock

Exit

8’

Depth, z

Marina Top View

Cross Section

Channel

(xi,yi)

(0,0) x

y

4’

)(⎡ Dmax −Dmin ⎤⎥⎦

<y y0fori i <ymax =D +z ⎢⎣

i min ymax

Model FormulationModel Formulation Boat Sweep Radius

BLh z zr iii ++⎥ ⎦

⎤ ⎢ ⎣

⎡ ⎟ ⎠ ⎞

⎜ ⎝ ⎛ = arccostan ,2

1cosθ Tzh i

i +

=

Sea level with high tide

Sea level with low tide Depth, z h

h

Buoy

Buffer, B

Swing Radius, r2

θ1

θ2

Tide, T r1

Boat Length, L

Model FormulationModel Formulation

Mooring Location Boundary Constraints Prevent boat location and swing circle from exceeding the marina boundaries

rx ii + ,2 < X max i ∀

rx ii − ,2 < X min i ∀

ry ii − ,2 Y < min i ∀

ry ii + ,2 Y < max i ∀

Model FormulationModel FormulationMooring Location Boundary Constraint

Marina Cross Section Top View

8’

Entrance

y

x (0,0)

4’

Dock

Depth, z

Channel

(xk,yk)

(xi,yi)

r2,i

rk

( xx ) + ( yy ) > r ,2 i + rk ∀ i k ii − k 2

i − k 2 , <

Model FormulationModel FormulationMooring Depth Boundary Constraint

Dock

Entrance

8’

Depth, z

Marina Top View

Cross Section

Channel

(0,0) x

y

(xi,yi)

r2,i

4’

⎛⎜⎜D max −D min ⎞⎟⎟( )−Di >D min+ −ry i ,2 iY max⎝ ⎠

0

Model FormulationModel FormulationDock and Harbor Channel Proximity PriceDock and Harbor Channel Proximity Price

Combined Proximity Pricing

Dock

Entrance Channel

(Xmin,Ymin)

x

y

(Xmax,Ymax)

Optimal Price Region

Iso price lines

DT

PD PD max, − PD min,=

DT

PC PC max, − PC min,=

DT

2 2DT = (X − X min ) + (Y − Ymin )max max

Model FormulationModel Formulation

∑∑==j i

jij LPFeeLengthLF ,

( )∑ −+==i

iHiH DPDPFeeDepthDF 121

( ) ( )∑ −+−−==i

iiDD yYxXPPFeeoximityDockDPF 2min

2maxmax,Pr

( ) ( )∑ −+−−==i

iiCC xXyYPPFeeoximityChannelCPF 2min

2maxmax,Pr

CPFDPFDFLFFeeMooring +++=

Objective Function

AnalysisAnalysisRun 1 – Constant Harbor Depth and No Channel Exit Fee

Test 1 - Fixed water depth, no harbor exit fee

509, 225358, 217

509, 36

434, 210

231, 46 423, 46327, 55

188, 231

383, 134

281, 257

303, 405

489, 320

162, 128

373, 312

489, 130277, 149

424, 417

209, 336

0

100

200

300

400

500

0 100 200 300 400 500

X Coordinate

Y C

oord

inat

e

Doc

Exit

$$$

$ $ $ $

AnalysisAnalysisRun 2 – Constant Harbor Depth and No Dock Fee

Test 2 - Fixed water depth, no dock fee

43, 356

41, 509

41, 433

108, 396

215, 277

216, 119

310, 399

125, 311

51, 249

308, 303

369, 488

396, 241

144, 197

135, 489

204, 383

252, 489

287, 198

408, 357

0

100

200

300

400

500

0 100 200 300 400 500

X Coordinate

Y C

oord

inat

e

Doc

Exit

$$

$ $ $

AnalysisAnalysisRun 3 – Fixed Harbor Depth, Equal Proximity Pricing

Test 3 - Fixed water depth, equal dock and harbor channel exit fee

263, 239

269, 316194, 298

332, 272

76, 406

144, 474

366, 193

392, 99

51, 499

169, 381

428, 280

471, 171

489, 56

351, 367

259, 439

272, 143

173, 203

98, 301

0

100

200

300

400

500

0 100 200 300 400 500X Coordinate

Y C

oord

inat

e

Doc

Exit

$$

$ $ $

$$

$ $ $

AnalysisAnalysisRun 4 – Variable water depth

Test 4 - Variable water depth, no exit proximity

290, 139

392, 274

378, 31

464, 118

375, 207

462, 274

227, 444

454, 41

410, 552

453, 196296, 217

125, 454

376, 118

196, 542

292, 51

307, 565

441, 454329, 454

0, 00, 00

100

200

300

400

500

600

0 100 200 300 400 500

X Coordinate

Y C

oord

inat

e

Doc

Exit

$ $ $

$$

Boats with shallow hulls

< 8’

Boats with deep hulls

> 8’

8 feet deep

ResultsResultsTypical Marina

489, 61373, 61

373, 177 489, 177

256, 61

256, 177

489, 294373, 294256, 294

489, 411373, 411256, 411

139, 61

139, 177

139, 294

139, 411

0, 00, 00, 00, 00

100

200

300

400

500

0 100 200 300 400 500

X CoordinateY

Coo

rdin

ate

Marina Mooring Optimization with 18 Boats

513, 118

255, 206

428, 31

449, 96

512, 188

469, 244

501, 319

371, 84

330, 237

311, 149

504, 41

399, 329

279, 51

343, 427

410, 176

287, 329174, 329

470, 417

0, 00, 00

100

200

300

400

500

0 100 200 300 400 500

X Coordinate

Y Co

ordi

nate

VS.

Revenue = $6,706.93 Revenue = $5,523.56

21.4% IMPROVEMENT !!!

ConclusionsConclusions

•• Optimization can significantly increase Optimization can significantly increase marina profits.marina profits.

•• Optimization can significantly increase Optimization can significantly increase number of moorings in marinanumber of moorings in marina

•• Model is flexible to accommodate Model is flexible to accommodate constraints of any marinaconstraints of any marina

Lessons LearnedLessons Learned

•• Need more powerful solver to increase number Need more powerful solver to increase number of boats and constraints in optimization.of boats and constraints in optimization.

•• Need separate proximity pricing scheme for each Need separate proximity pricing scheme for each boat length category.boat length category.

•• Would be convenient to include a boat adding Would be convenient to include a boat adding algorithm.algorithm.

•• There are ways to make solver behave better.There are ways to make solver behave better.

Questions ?Questions ?