Algebraic solution to a constrained rectilinear minimax location problem on the plane

We consider a constrained minimax single facility location problem on the plane with rectilinear distance. The feasible set of location points is restricted to rectangles with sides oriented at a 45 degrees angle to the axes of Cartesian coordinates. To solve the problem, an algebraic approach based on an extremal property of eigenvalues of irreducible matrices in idempotent algebra is applied. A new algebraic solution is given that reduces the problem to finding eigenvalues and eigenvectors of appropriately defined matrices.

[1]  J. Golan.  Semirings and Affine Equations over Them: Theory and Applications , 2003 .

[2]  K. Zimmermann,et al.  One class of separable optimization problems: solution method, application , 2010 .

[3]  D. Hearn,et al.  Geometrical Solutions for Some Minimax Location Problems , 1972 .

[4]  R. L. Francis.  A Geometrical Solution Procedure for a Rectilinear Distance Minimax Location Problem1 , 1972 .

[5]  Eigenvalues and Eigenvectors of Matrices in Idempotent Algebra , 2006 .

[6]  R. L. Francis.  A geometrical solution procedure for a rectilinear minimax location problem , 1972 .

[7]  N. Krivulin.  An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem , 2011 .

[8]  R. Cuninghame-Green.  Minimax Algebra and Applications , 1994 .

[9]  Karel Zimmermann,et al.  Disjunctive optimization, max-separable problems and extremal algebras , 2003, Theor. Comput. Sci..

[10]  P. Butkovic.  Max-linear Systems: Theory and Algorithms , 2010 .

[11]  Solution of generalized linear vector equations in idempotent algebra , 2006 .

[12]  Nikolai K. Krivulin,et al.  Algebraic solutions to multidimensional minimax location problems with Chebyshev distance , 2011, ArXiv.

[13]  V. Kolokoltsov,et al.  Idempotent Analysis and Its Applications , 1997 .

[14]  Geert Jan Olsder,et al.  Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications , 2005 .

[15]  Geert Jan Olsder,et al.  Max Plus at Work-Modelling and Analysis of Synchronized Systems , 2006 .