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The complete vertex $$p$$-center problem. (English) Zbl 1452.90224
Summary: The vertex $$p$$-center problem consists of locating $$p$$ facilities among a set of $$M$$ potential sites such that the maximum distance from any demand to its closest located facility is minimized. The complete vertex $$p$$-center problem solves the $$p$$-center problem for all $$p$$ from 1 to the total number of sites, resulting in a multi-objective trade-off curve between the number of facilities and the service distance required to achieve full coverage. This trade-off provides a reference to planners and decision makers, enabling them to easily visualize the consequences of choosing different coverage design criteria for the given spatial configuration of the problem. We present two fast algorithms for solving the complete $$p$$-center problem: one using the classical formulation but trimming variables while still maintaining optimality and the other converting the problem to a location set covering problem and solving for all distances in the distance matrix. We also discuss scenarios where it makes sense to solve the problem via brute-force enumeration. All methods result in significant speedups, with the set covering method reducing computation times by many orders of magnitude.
##### MSC:
 90C10 Integer programming 90C11 Mixed integer programming 90C27 Combinatorial optimization 90C90 Applications of mathematical programming
##### Software:
BEAMR; GitHub; MENU-OKF; MOD-DIST; SITATION; SSCFLPsolver; TSPLIB
Full Text:
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