Airplane maintenance; Boeing needs to build 5 maintenance centers for the Euro-A

Question

Airplane maintenance;

Boeing needs to build 5 maintenance centers for the Euro-Asian area. The construction cost for each center is 300 million euros in the European area (between 20W and 40E) and 150 million euros in the Asian area (between 40E and 160E), Each center can service up to 60 airplanes each year. The centers should service the airports with the highest number of Boeing customers, as detailed in the table below (airport name, geographical coordinates, expected number of airplanes/year needing maintenance). Aeroporto Coordinate N. aviogetti London Heathrow 51N 0 W 30 Frankfurt 51N 8 E 35 Lisboa 38N 9 W 12 Z¨urich 47N 8 E 18 Roma Fiumicino 41N 12E 13 Abu Dhabi 24N 54E 8 Moskva Sheremetyevo 55N 37E 15 Vladivostok 43N 132E 7 Sydney 33S 151E 32 Tokyo 35N 139E 40 Johannesburg 26S 28E 11 New Dehli 28N 77E 20 The total cost is given by the construction cost plus the expected servicing cost, which depends linearly on the distance an airplane needs to travel to reach the maintenance center (weighted by 50 euro/km). We assume earth is a perfect sphere, so that the shortest distance between two points with geographical coordinates (1, 1) and (2, 2) is given by: d(1, 1, 2, 2) = 2r asins sin2 1 2 2 + cos 1 cos 2 sin2 1 2 2 , where r, the earth radius, is 6371km. Formulate a (nonlinear) mathematical program to minimize the operation costs.

Airplane maintenance: Solution

1. Indices: • i n = number of maintenance centers; • j m = number of airports. 2. Parameters: • j : latitude of airport j; • j : longitude of airport j; • Aj : expected number of airplanes/year leaving from airport j; • r: earth radius; • P: capacity of centers (number of airplanes); • C1: construction cost between 20W e 40E

• C2: construction cost between 40E e 160E • : proportionality between distance and cost; 3. Variables: • xi : latitude of center i (90S xi 90N) • yi : longitude of center i (20W yi 160E) • dij : geodesic distance between center i and airport j (dij 0) • wij : number of airplanes going to center i and coming from airport j (0 wij Aj ) • zi = 1 if center i is built between 20W e 40E, 0 otherwise 4. Objective function: minX in C1zi + C2(1 zi) + X jm wijdij 5. Constraints: • Distances: dij = 2r asins sin2 xi j 2 + cos xi cos j sin2 yi j 2 i n, j m;

• (maintenance) X in wij = Aj j m; • (capacity) X jm wij P i n; • (z definition) yi 40 360 zi i n yi 40 360 (1 zi) i n • (variables domains) xi , yi , dij R+ i n, j m wij Z+ i n, j m zi {0, 1} i n.

Explanation / Answer

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