Management Science 461 Lecture 4 a P center

Management Science 461 Lecture 4 a – P center September 30, 2008

Summary Source: Daskin, Network and Discrete Location 2

Graphical Example A 14 B 10 13 E 17 23 C Locate at A: E is __km away Locate at B: C is __km away 16 12 D Locate at D: A is __km away 3

P-center (Minimax) Problem n n Given P facilities, find the minimum coverage distance such that all demands are covered In effect, we are minimizing the maximum demand from any customer to the nearest facility No matter how insignificant, the most “remote” demand node drives the solution Can be solved as a linear problem – but how to incorporate a max function? 4

P-center Formulation Ensure coverage, limit number of facilities Can’t claim credit for non-existent facilities Force W equal to max distance Binary constraints, nonnegativity 5

Set Covering Solutions 10 A D 7 8 12 C 9 F 13 13 9 7 B 17 E 6

P-center Solutions 10 A D 7 8 12 C 9 F 13 13 9 7 B 17 E 7

P-center and Set Covering Set covering n P-center Solvertable can help … 8

Algorithm for integer distances n n Let DL = 0; DH = big number Set DC = (DL+DH)/2 (rounded down) Solve set cover problem with DC. How many facilities does it take? ¨ If # facilities <= P, Set DH = ¨ Otherwise, set DL = DC+1 n DC DH = D L ¨ Yes; Stop ¨ No; go to step 2 9