Moo-rings
rings
Marina Mooring Optimization
Optimization
Group 8 – Route 64
Brian Siefering
Amber Mazooji
Kevin McKenney
Paul Mingardi
Vikram Sahney
Kaz Maruyama
Presentation Overview
Overview
• Introduction
• Problem Description
Description
• Assumptions
• Model Formulation
• Analysis
• Conclusions
• Lessons Learned
• Questions
Introduction
• What is a mooring?
• Why an optimization
problem?
Mooring Buoy
Mooring Line
Line Secured to Bedrock
Problem Motivation
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 Description
Objective: Maximize Revenue!
(also increase number of moorings in marina)
Decision Variables: Boat Locations
Dock
Channel
Problem Description
Description
Marina
Top View
Cross Section
8’
Exit
Channel
Depth, z
y
(xi,yi)
x
(0,0)
Dock
4’
Assumptions
Assumptions
•• Marina:
–
– The bottom of the marina is linear, sloping down in the +y direction.
–
– Tide change is 2 feet or less.
• Moorings:
–
– Mooring lines are weightless.
–
– Moorings can and will be moved every year.
–
– At high tide, the mooring line angle is 30.
• Boats
–
– Boats are between 15’ and 40’ in length.
–
– Boats are classified into two categories based on their hull depth: boats
with hull depths less than 4’ and boats with hull depths between 4’ and
8’.
• Placement
–
–
–
–
–
–
–
–
Moorings can be precisely placed.
The minimum separation needed between boat sweeps is five feet.
Bow line length is negligible.
Boats will be able to leave moorings without specified lanes designated
in a marina.
Model Formulation
Marina
Top View
Exit
Harbor Depth
Cross Section
8’
Channel
Depth, z
y
(xi,yi)
x
(0,0)
Dock
4’
⎡ (D max −
D min )⎤
yi
z
i =
D
min +
⎢
⎥
y max
⎦
⎣
for 0 <
yi <
y max
Model Formulation
Boat Sweep Radius
Swing Radius, r2
Buoy
r1
Boat
Length, L
Buffer, B
Sea level with high tide
Tide, T
Sea level with low tide
Depth, z
θ1
h
θ2
zi + T
hi =
cos θ1
h
⎡
⎛ z ⎞⎤
r2,i = zi tan ⎢arccos⎜ ⎟⎥ + Li + B
⎝ h ⎠⎦
⎣
Model Formulation
Formulation
Mooring Location Boundary Constraints
Prevent boat location and swing circle from exceeding the marina boundaries
xi + r2,i < X max
∀i
xi − r2,i < X min
∀i
yi − r2,i < Ymin
∀i
yi + r2,i < Ymax
∀i
Model Formulation
Formulation
Mooring Location Boundary Constraint
Marina
Top View
Entrance
Cross Section
8’
Channel
Depth, z
r2,i
(xi,yi)
rk
y
(xk,yk)
x
Dock
(0,0)
4’
(xi − xk ) + ( yi − yk )
2
2
> r2,i + rk
∀i, k < i
Model Formulation
Formulation
Mooring Depth Boundary Constraint
Marina
Top View
Entrance
Cross Section
8’
Channel
r2,i
(xi,yi)
Depth, z
y
x
(0,0)
Dock
4’
⎛ Dmax −
Dmin
⎞
⎟⎟
(
yi −
r2,i )
−
Di
>
0
Dmin
+
⎜⎜
⎠
⎝
Ymax
Model Formulation
Formulation
Dock and Harbor Channel Proximity Price
Combined Proximity Pricing
Entrance
Channel
(Xmax,Ymax)
Iso price
lines
PD ,max − PD ,min
PD =
DT
Optimal Price
Region
DT
y
x
(Xmin,Ymin)
PC ,max − PC ,min
PC =
DT
Dock
DT =
( X max − X min )
2
+ (Ymax − Ymin )
2
Model Formulation
Objective Function
LF = Length Fee = ∑∑ Pj Li , j
j
i
DF = Depth Fee = ∑ PH 1 Di + PH 2 (1 − Di )
i
DPF = Dock Pr oximity Fee = ∑ PD ,max − PD
( X max − xi )2 + (Ymin − yi )2
i
CPF = Channel Pr oximity Fee = ∑ PC ,max − PC
(Ymax − y i )2 + ( X min − xi )2
i
Mooring Fee = LF + DF + DPF + CPF
Analysis
Run 1 – Constant Harbor Depth and No Channel Exit Fee
Exit
Test 1 - Fixed water depth, no harbor exit fee
500
Y Coordinate
424, 417
303, 405
400
209, 336
489, 320
373, 312
300
281, 257
188, 231
358, 217
200
277, 149
383, 134
162, 128
509, 225
434, 210
489, 130
100
327, 55
231, 46
0
0
100
200
300
X Coordinate
$
$
Doc
$
$ $ $ $
423, 46
400
509, 36
500
Analysis
Run 2 – Constant Harbor Depth and No Dock Fee
$ $ $
Exit
Test 2 - Fixed water depth, no dock fee
$
$
41, 509
500
135, 489
252, 489
369, 488
41, 433
400
108, 396
310, 399
204, 383
408, 357
Y Coordinate
43, 356
125, 311
300
308, 303
215, 277
51, 249
200
396, 241
287, 198
144, 197
216, 119
100
0
0
100
200
300
400
500
X Coordinate
Doc
Analysis
Run 3 – Fixed Harbor Depth, Equal Proximity Pricing
$ $ $
Exit
Test 3 - Fixed water depth, equal dock and
harbor channel exit fee
$
$
500
51, 499
144, 474
259, 439
76, 406
400
Y Coordinate
169, 381
98, 301
300
351, 367
194, 298
269, 316
428, 280
332, 272
263, 239
173, 203
200
366, 193
471, 171
272, 143
100
392, 99
489, 56
0
0
100
200
300
X Coordinate
400
$
Doc
$
$ $ $
500
Analysis
Run 4 – Variable water depth
Exit
Boats with
deep hulls
> 8’
Test 4 - Variable water depth, no exit proximity
600
307, 565
196, 542
410, 552
500
125, 454
329, 454
227, 444
441, 454
8 feet deep
Y Coordinate
400
300
392, 274
296, 217
200
290, 139
100
292, 51
0
375, 207
376, 118
378, 31
0, 0
0
100
200
300
X Coordinate
400
462, 274
453, 196
464, 118
454, 41
$
$
Doc
$ $ $
500
Boats with
shallow hulls
< 8’
Results
Marina Mooring Optimization with 18 Boats
Typical Marina
500
500
343, 427
470, 417
400
174, 329
287, 329
399, 329
501, 319
300
469, 244
330, 237
255, 206
200
512, 188
410, 176
VS.
Y Coordinate
Y Coordinate
400
300
139, 411
256, 411
373, 411
489, 411
139, 294
256, 294
373, 294
489, 294
139, 177
256, 177
373, 177
489, 177
139, 61
256, 61
373, 61
489, 61
200
311, 149
513, 118
100
371, 84
449, 96
279, 51
504, 41
428, 31
0
0
0, 0
0
100
100
200
300
400
500
0, 0
0
100
200
300
400
X Coordinate
X Coordinate
Revenue = $6,706.93
Revenue = $5,523.56
21.4% IMPROVEMENT !!!
500
Conclusions
• Optimization can significantly increase
marina profits.
• Optimization can significantly increase
number of moorings in marina
• Model is flexible to accommodate
constraints of any marina
Lessons Learned
• Need more powerful solver to increase number
•
•
•
of boats and constraints in optimization.
Need separate proximity pricing scheme for each
boat length category.
Would be convenient to include a boat adding
algorithm.
There are ways to make solver behave better.
Questions ?