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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 ?