TY - JOUR

T1 - Relaxation method for the solution of the minimax location‐allocation problem in euclidean space

AU - Chen, R.

AU - Handler, G. Y.

PY - 1987/12

Y1 - 1987/12

N2 - A method previously devised for the solution of the p‐center problem on a network has now been extended to solve the analogous minimax location‐allocation problem in continuous space. The essence of the method is that we choose a subset of the n points to be served and consider the circles based on one, two, or three points. Using a set‐covering algorithm we find a set of p such circles which cover the points in the relaxed problem (the one with m < n points). If this is possible, we check whether the n original points are covered by the solution; if so, we have a feasible solution to the problem. We now delete the largest circle with radius rp (which is currently an upper limit to the optimal solution) and try to find a better feasible solution. If we have a feasible solution to the relaxed problem which is not feasible to the original, we augment the relaxed problem by adding a point, preferably the one which is farthest from its nearest center. If we have a feasible solution to the original problem and we delete the largest circle and find that the relaxed problem cannot be covered by p circles, we conclude that the latest feasible solution to the original problem is optimal. An example of the solution of a problem with ten demand points and two and three service points is given in some detail. Computational data for problems of 30 demand points and 1–30 service points, and 100, 200, and 300 demand points and 1–3 service points are reported.

AB - A method previously devised for the solution of the p‐center problem on a network has now been extended to solve the analogous minimax location‐allocation problem in continuous space. The essence of the method is that we choose a subset of the n points to be served and consider the circles based on one, two, or three points. Using a set‐covering algorithm we find a set of p such circles which cover the points in the relaxed problem (the one with m < n points). If this is possible, we check whether the n original points are covered by the solution; if so, we have a feasible solution to the problem. We now delete the largest circle with radius rp (which is currently an upper limit to the optimal solution) and try to find a better feasible solution. If we have a feasible solution to the relaxed problem which is not feasible to the original, we augment the relaxed problem by adding a point, preferably the one which is farthest from its nearest center. If we have a feasible solution to the original problem and we delete the largest circle and find that the relaxed problem cannot be covered by p circles, we conclude that the latest feasible solution to the original problem is optimal. An example of the solution of a problem with ten demand points and two and three service points is given in some detail. Computational data for problems of 30 demand points and 1–30 service points, and 100, 200, and 300 demand points and 1–3 service points are reported.

UR - http://www.scopus.com/inward/record.url?scp=84989685250&partnerID=8YFLogxK

U2 - 10.1002/1520-6750(198712)34:6<775::AID-NAV3220340603>3.0.CO;2-N

DO - 10.1002/1520-6750(198712)34:6<775::AID-NAV3220340603>3.0.CO;2-N

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AN - SCOPUS:84989685250

SN - 0894-069X

VL - 34

SP - 775

EP - 788

JO - Naval Research Logistics

JF - Naval Research Logistics

IS - 6

ER -