MooMoo--rings ringsMarina Mooring Optimization Marina Mooring OptimizationGroup 8Group 8 ––Route 64Route 64Brian Siefering Amber Mazooji Kevin McKenney Paul Mingardi Vikram Sahney Kaz MaruyamaPresentation Overview Presentation Overview••IntroductionIntroduction••Problem Description Problem Description••AssumptionsAssumptions••Model FormulationModel Formulation••AnalysisAnalysis••ConclusionsConclusions••Lessons LearnedLessons Learned••QuestionsQuestionsIntroductionIntroduction••What is a mooring?What is a mooring?••Why an optimizationWhy an optimization problem?problem?Mooring Buoy Mooring Line Line Secured to BedrockProblem Motivation Problem 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 8/10/2003Problem 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 LocationsChannel DockProblem Description Problem DescriptionMarina Cross Section Top View 8’ Exit y x (0,0) Dock Depth, z Channel (xi,yi) 4’Assumptions Assumptions• •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 between4’ 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 FormulationHarbor 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 ymaxModel FormulationModel FormulationBoat 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, LModel Formulation Model FormulationMooring 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 Formulation Model 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 Formulation Model 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⎝ ⎠ 0Model Formulation Model FormulationDock and Harbor Channel Proximity PriceDock and Harbor Channel Proximity PriceCombined 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 − Xmin ) +(Y − Ymin )max maxModel FormulationModel Formulation∑∑==jijijLPFeeLengthLF,()∑−+==iiHiHDPDPFeeDepthDF 121()()∑−+−−==iiiDDyYxXPPFeeoximityDockDPF2min2maxmax,Pr()( )∑−+−−==iiiCCxXyYPPFeeoximityChannelCPF2min2maxmax,PrCPFDPFDFLFFeeMooring+++=Objective FunctionAnalysisAnalysisRun 1 – Constant Harbor Depth and No Channel Exit FeeTest 1 - Fixed water depth, no harbor exit fee509, 225358, 217509, 36434, 210231, 46 423, 46327, 55188, 231383, 134281, 257303, 405489, 320162, 128373, 312489, 130277, 149424, 417209, 33601002003004005000 100 200 300 400 500X CoordinateY CoordinateDocExit$$$$ $ $ $AnalysisAnalysisRun 2 – Constant Harbor Depth and No Dock Fee Test 2 - Fixed water depth, no dock fee43, 35641, 50941, 433108, 396215, 277216, 119310, 399125, 31151, 249308, 303369, 488396, 241144, 197135, 489204, 383252, 489287, 198408, 35701002003004005000 100 200 300 400 500X CoordinateY CoordinateDocExit$$$ $ $AnalysisAnalysisRun 3 – Fixed Harbor Depth, Equal Proximity PricingTest 3 - Fixed water depth, equal dock and harbor channel exit fee263, 239269, 316194, 298332, 27276, 406144, 474366, 193392, 9951, 499169, 381428, 280471, 171489, 56351, 367259, 439272, 143173, 20398, 30101002003004005000 100 200 300 400 500X CoordinateY CoordinateDocExit$$$ $ $ $$$ $ $AnalysisAnalysisRun 4 – Variable water depthTest 4 - Variable water depth, no exit proximity290, 139392, 274378, 31464, 118375, 207462, 274227, 444454, 41410, 552453, 196296, 217125, 454376, 118196, 542292, 51307, 565441, 454329, 4540, 00, 001002003004005006000 100 200 300 400 500X CoordinateY CoordinateDocExit$ $ $ $$Boats with shallow hulls< 8’Boats with deep hulls > 8’8 feet deepResultsResultsTypical Marina489, 61373, 61373, 177 489, 177256, 61256, 177489, 294373, 294256, 294489, 411373, 411256, 411139, 61139, 177139, 294139, 4110, 00, 00, 00, 001002003004005000 100 200 300 400 500X CoordinateY CoordinateMarina Mooring Optimization with 18 Boats513, 118255, 206428, 31449, 96512, 188469, 244501, 319371, 84330, 237311, 149504, 41399, 329279, 51343, 427410, 176287, 329174, 329470, 4170, 00, 001002003004005000 100 200 300 400 500X CoordinateY CoordinateVS.Revenue =