Point On The Plane - Line Location Problem

Location problems are problems with regard to selecting optimal sites where facilities should be located. Location problems are optimal problems combined with reality and be followed with interest by operation researchers, managers, economists, strategists, urban planers and engineers since 1960s' .The research on location problems develops rapidly these years.The traditional location problems study location problems with distance from point to point. We consider location problems with distance from point to line. We mainly discuss four point -line location problems in the plane as follows: Problem A is to determine a straight - line to mini-mize the total weighted distances from n given points; Problem B is to determine a point to mini-mize the total weighted distances from n given straight - lines; Problem C is to determine a straight - line to minimize the maximum weighted distances from n given points; Problem D is to determine a point to minimize the maximum weighted distances from n given straight - lines.Problem A and problem B are dual. They have a dual property that an optimal solution of problem A could be a straight - line connecting two given points and an optimal solution of prob-lem B could be a point of intersection of two given straight - lines . Problem C and problem D are also dual.They have a dual property that there are at least three "critical points"corresponding to an optimal straight - line in problem C and there are at least three" critical straight - lines" corre-sponding to an optimal point in problem D. From these properties, these four non - linear prob-lems could be transformed into combinatorial problems and could be solved by algorithms with polynomial - time iterations.We also consider two obnoxious center problems and some problems related to problems A, B,C and D.At last,we pose two problems on selecting optimal path in networks.