Marina Mooring Optimization 15066j System Optimization and Analysis Summer 2003 Professor Stephen C. Graves Team Members: Brian Siefering Amber Mazooji Kevin McKenney Paul Mingardi Vikram Sahney Kaz MaruyamaIntroduction Each summer, boat owners flock to marinas to sign up for a mooring spot to tie up their boats. A mooring is a buoy that is anchored to the ocean floor for the purpose of securing a boat for storage. A picture of a mooring is shown in Figure 1. Figure 2 is a diagram of a mooring buoy and anchor line. A typical marina is shown in Figure 3, where moorings are positioned in a grid like pattern. (Photo removed for publication.) Figure 1. Picture of Boat on a Mooring. Mooring Buoy Mooring Line Line Secured to Bedrock Figure 2. Diagram of Mooring Buoy and Anchor Line. (Photo removed for publication.) Figure 3. Typical Marina Mooring Layout. Each summer, many marina customers are put on waiting lists because the mooring supply cannot always meet the demand. Whether or not a customer gets a mooring and where the mooring is located is based strictly on seniority. As customers with optimal mooring locations leave the marina, their moorings are handed down to the customers who have been with the marina the longest. Mooring that are freed up are then released to customers on the waiting list. 2Additionally, the same moorings are placed in the same grid locations from year to year. Boats are then assigned on a first come first serve basis as previously described without taking into account variables which distinguish between boats. This propagates the sub-optimal usage of the marina water space. By instead taking into account variables such as boat length, hull depth, and mooring location, a marina can optimize their revenues while maximizing the number of satisfied customers. Problem Description The goal of this optimization program is to maximize revenue for the assignment of boat moorings at a marina. The model was based on the marina shown in Figure 4. The marina is modeled as a rectangular harbor as shown in Figure 5 with a manifest of boats of different lengths and hull depths. The black dots represent the boat locations while the dotted circles represent the circle that the boat swings through under varying tide and wind conditions. The depth of the water in the marina increases as y increases from the dock to the channel. The costs of mooring positions in the harbor are based on the vicinity to both the dock and to the ocean channel. In other words, customers desire their boats to be close to the dock so that it takes less time to row out to their boat. Additionally, if the boat is closer to the marina exit, it will take less time for the customer to exit the harbor. Mooring costs are dependent on the swing radius that the boat occupies as well as the depth of the water the boat requires. Channel Dock Figure 4. Green Pond Harbor. 3Marina Cross Section Top View 8’ Exit y x (0,0) DockDepth, z Channel (xi,yi) 4’ Figure 5. Marina Model. 4Notation The following is a summary of the variables and notation used in the model calculations and discussion. i = Index distinguishing between boats in model xi = x location of boat i [ft] yi = y location of boat i [ft] Dmin = Minimum depth of marina at low tide[ft] Dmax = Maximum depth of marina at low tide[ft] Xmin = Minimum X coordinate for marina [ft] Xmax = Maximum X coordinate for marina [ft] Ymin = Minimum Y coordinate for marina [ft] Ymax = Maximum Y coordinate for marina [ft] T = Tide change for marina [ft] B = Minimum buffer distance between adjacent boats [ft] j = Index distinguishing between boat length categories, (j=1: 15-20', j=2L 20-30', j=3: 30-40') Li = Length of boat i [ft] Li,j = Integer variable, 1 if boat i is in length range j, 0 otherwise Di Hull Depth of boat I [ft] Di,k = Integer variable, 1 if boat I is in hull depth range k, 0 otherwise θ1 = Angle of mooring line to vertical during high tide (fixed) θ2,i = Angle of mooring line to vertical during low tide for boat i hi = Mooring line length for boat i [ft] r1,i = Mooring sweep radius at low tide for boat i [ft] r2,i = Total boat sweep radius at low tide for boat i [ft] zi = Harbor depth at low tide for boat i location [ft] Pk = Price for boats in hull depth range k[$] Pj = Price for boats in length range j [$] PD = Variable Price associated with location in vicinity of the dock [$] PC = Variable Price associated with location in vicinity of the channel entrance [$] PD,max = Maximum Price for Dock Proximity PC,max = Maximum Price for Harbor Channel Entrance Proximity PD,min = Minimum Price for Dock Proximity PC,min = Minimum Price for Harbor Channel Entrance Proximity DT = Diagonal Distance (Greatest Distance) in Harbor 5Assumptions 1) The bottom of the marina is linear, sloping down in the +y direction. This relationship was chosen in order to impose a realistic constraint on the mooring placement problem. Harbor bottoms are generally non-uniform and, in addition, may be non-linear. The problem can quite easily be adapted to any new harbor ocean floor equation. 2) Moorings can be precisely placed. Mooring anchors are placed by divers and may not be placed exactly in the desired location. This model contains a buffer term which will be introduced later and which will accommodate for some uncertainty. For the most part, however, this model ignores such variability. 3) Mooring lines are weightless. In reality, mooring cables have a mass which cause the line to sag in the water, decreasing the overall distance from the anchor to the mooring. The model assumes that the cables can be stretched out straight, which accommodates for the longest cable length and as a result, the worst case scenario. 4) Moorings can and will be moved every year. This is a reasonable assumption for moorings in the northeast as they are removed each winter to protect them from the elements. In a tropical environment, this assumption may not be valid because the moorings are left in the water year in and year out. 5) Tide change is 2 feet or less. This is based on actual data for the harbor chosen, however this constraint can vary quite significantly between geographic locations and even between nearby harbors. 6) At high tide, the mooring line angle is 30o . When a boat is attached to a mooring and subjected to a horizontal force (either wind or tide), the mooring will be pulled to one side causing the