Some Properties of Location Problems with Block and Round Norms

Jacques Thisse, J. E. Ward and R. E. Wendell
Additional contact information
J. E. Ward: Purdue University, West Lafayette, Indiana
R. E. Wendell: University of Pittsburgh, Pittsburgh, Pennsylvania

Operations Research, 1984, vol. 32, issue 6, 1309-1327

Abstract: The point-objective problem and the Weber problem are two well-known formulations for locating a new facility with respect to a set of fixed facilities. When locations are represented as points on a plane, the point-objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points. Similarly, the Weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points, where transportation cost is a function of distance. Generalizing solution properties for these problems from distance measures given by the Euclidean, rectilinear, I p , and one-infinity norms; this paper develops solution properties under the broad classes of distance measures given by block and round norms. For the point-objective problem, we show that (i) the efficient set for all round norms is the convex hull of the set of fixed points and (ii) the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm. For the Weber problem, we prove that (i) an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and (ii) this set tends to the convex hull for a sequence of block norms approaching a round norm. Finally, we use these results to propose a synthesis of some of the main properties in continuous and network location theory.

Keywords: 185 properties of location problems with block and round norms; 641 point objective location problems (search for similar items in EconPapers)
Date: 1984
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Citations: View citations in EconPapers (8)

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Working Paper: Some properties of location problems with block and round norms (1984)
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