Mechanical Design (48650) – Spring Semester 2012
Optimisation Assignment – BUILDING A ROAD
Teams of THREE to FIVE [3 to 5] – Worth 20% of the Semester Grade
Due: Thursday 23rd March 2012 by 23:59 pm
Where? My Assignment Box [No 15] situated in Building 2 between Floors 5 & 6 below the big computer lab on level 6. This is the only way to submit – it eliminates confusion.
Late submissions (without prior permission) will be penalised at -10% per day late. Submissions more than one week late will not be accepted.
Students organise themselves into teams; share out the work; plan a timetable with delivery dates; hold regular face to face meetings and issue minutes, and; show fellow team members how to do things. Acknowledge outside assistance from other students in the report.
1. Background
The cost of building a road depends upon many things. For the purposes of this assignment it will depend on total distance covered and quantity of material cut from a hill, the quantity of material to fill hollows and the net amount of excess earth trucked to/from the area to make up for that used internally. The area being considered covers the coordinates from x = -5 to x = +5 kilometres and from y = +5 to y = -5 kilometres. The height of a point (z in metres) within the area is found from the equation:
z = sin [(x-y2)?/14] + sin [(2x+y)?/9)] - 0.37 {illustrated in the contour map below}
Five pairs of roads can start from the West at x=-5 and y =+4, +2, 0, -2 and -4 always travelling eastward. The pair starting at y = +4 finish at y = 1 and y = -1: the pair starting at y = +2 finish at y = +5 and y = -2: the pair starting at y = 0 finish at y = -3 and y = +3: the pair starting at y = -2 finish at y = -5 and y = +1, and: the pair starting at y = -4 finish at y = +1 and -1. The roads run horizontally at z = zero metres.
The costs (in units of $100,000) are:
Road construction cost is 7/km Cost to fill a hollow is 3/km.metre
Cost to cut into a hill is 2/km.metre. Cost to truck net excess earth to/from the area is 3.5/km.metre
The cut or fill for each step is zav . ?s, where zav = (zleft + 2zmid + zright)/4 and ?s is the step road length.
2. Teams
Teams of three start at x = -5 and y = +4, 0 and -4. Teams of 4 start at x = -5 and y = +4, +2, 0 and -2. Teams of five will start at x = -5 and y =+4, +2, 0, -2 and -4. Each team member will do 2 roads starting at one point and a team will cover all roads allocated to them.
3. The Task
3.1 Maps: Produce a contour map better than that shown below and also a three-dimensional surface graph of the ground. Divide the x and y ranges into steps of 0.25 km.
[Hint; Look up the “range” naming function in “Help” – it allows you to easily write a general formula that can be written to any cell to calculate z and it will automatically step through the x- and y-ranges.]
3.2 Optimisation: Using an optimisation package such as “Solver” in Excel, determine the road path which gives the minimum cost for each starting and finishing point examined by the team. Plot the road route and z-elevation profile for each road. A small bonus will be awarded for the team with the lowest cost, viable solution.
[Hints: Move from x = -5 to x = +5 in steps of 0.5 km [or 0.25 km, preferably]. Determine the (i) the road distance for the step and it’s cost; (ii) the z at the half x-step co-ordinates; (iii) the “fill” or “cut” for that step; (iv) the cost of the fill or cut for that step; (v) the net fill or cut for the road and it’s cost; (vi) the total road cost, and; (vii) optimise the y-path to minimise total cost. Check for correct operation by putting all costs except road construction to zero – you should get a straight line road path. Warning! Try several initial path guesses to get the “best” path. “Play’ with the option settings in Solver.]