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

2 Summary Source: Daskin, Network and Discrete Location

3 Graphical Example 14 A E D C B Locate at A: E is __km away Locate at B: C is __km away Locate at D: A is __km away

4 P-center (Minimax) Problem 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?

5 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, non- negativity

6 Set Covering Solutions 10 A E D C B F

7 P-center Solutions 10 A E D C B F

8 P-center and Set Covering Set coveringP-center Solvertable can help …

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