location problem in the planein kent the council

Question

Location problem in the plane:

In Kent, the council to respond to the people and government needs, it decided to establish 3 community care homes. The towns are recorded with their corresponding x and y coordinates as well as their population, see Table 1.

(i)Use Excel to solve this location problem in the plane. (Hint: As the problem is not convex and not linear, you cannot use the Simplex but the other 2 approaches shown in Solver (Evolutionary approach and GRG Non Linear)). Also you may need to use several starting points as poor local minima may be found instead, Try both approaches before you finalise your decision and comment on the way you attempted it.

(ii) What do you expect the solution quality will be if you had to solve the discrete version of the problem using all the town locations as potential sites? Which weaknesses you may be faced with when implementing the optimal solution of (i) if it exists.

(iii) Now consider that any facility built cannot serve more than 350000 in total. Use the locations found in (i) and determine the new allocation (This is now a TP which can be solved using Simplex LP in solver). Express in % the extra cost incurred due to such a restriction.

TOWN/CITY	X	Y	POPULATION
Ashford	600985	142805	58,178
Broadstairs	639320	167760	24,370
Canterbury	614880	157830	42,249
Chatham	575785	167920	70,540
Dartford	554200	174325	50,000
Deal	637510	152745	29,248
Dover	631650	141835	39,078
Faversham	601530	161425	18,000
Folkestone	622765	135915	53,411
Gillingham	577350	168385	99,773
Gravesend	564730	174170	51,150
Herne Bay	617900	167945	31,000
Maidstone	576150	155705	75,000
Margate	635460	170580	58,465
Northfleet	562235	174310	13,590
Ramsgate	638365	165180	37,967
Rochester	574375	168475	25,000
Royal Tunbridge Wells	558360	139265	45,000
Sevenoaks	552375	155295	18,588
Sheerness	591955	174725	20,000
Sittingbourne	590740	163660	55,000
Tonbridge	559080	146600	31,600

Table 1: Coordinates and population of Kent Towns