Complete Solution of a Constrained Tropical Optimization Problem with Application to Location Analysis

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists of minimizing a nonlinear objective function defined on vectors over an idempotent semifield by means of a conjugate transposition operator, subject to constraints in the form of linear vector inequalities. A complete direct solution to the problem under fairly general assumptions is given in a compact vector form suitable for both further analysis and practical implementation. We apply the result to solve a multidimensional minimax single facility location problem with Chebyshev distance and with inequality constraints imposed on the feasible location area.

[1]  U. Zimmermann Linear and combinatorial optimization in ordered algebraic structures , 1981 .

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

[3]  Dileep R. Sule,et al.  Logistics of Facility Location and Allocation , 2001 .

[4]  N. Krivulin An algebraic approach to multidimensional minimax location problems with Chebyshev distance , 2011, 1211.2425.

[5]  Nikolai Krivulin,et al.  Direct solutions to tropical optimization problems with nonlinear objective functions and boundary constraints , 2013, ArXiv.

[6]  Karel Zimmermann,et al.  Biobjective center – balance graph location model * , 1999 .

[7]  R. Weiner Lecture Notes in Economics and Mathematical Systems , 1985 .

[8]  A. Zilinskas Review of Logistics of facility location and allocation by Dileep R. Sule, Marcel Dekker 2001 , 2003 .

[9]  K. Zimmermann Optimization problems with unimodal functions in max-separabal constraints , 1992 .

[10]  S. N. N. Pandit,et al.  A New Matrix Calculus , 1961 .

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

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

[13]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

[14]  D. Peeters,et al.  LOCATION OF PUBLIC SERVICES: A SELECTIVE METHOD-ORIENTED SURVEY , 1980 .

[15]  Geert Jan Olsder,et al.  Synchronization and Linearity: An Algebra for Discrete Event Systems , 1994 .

[16]  Nikolai Krivulin,et al.  A multidimensional tropical optimization problem with a non-linear objective function and linear constraints , 2013, ArXiv.

[17]  B. Carré An Algebra for Network Routing Problems , 1971 .

[18]  Michel Minoux,et al.  Graphs, dioids and semirings : new models and algorithms , 2008 .

[19]  R. Cuninghame-Green Minimax algebra and applications , 1991 .

[20]  John R. Beaumont,et al.  Studies on Graphs and Discrete Programming , 1982 .

[21]  K. Zimmermann,et al.  A service points location problem with Min-Max distance optimality criterion , 1993 .

[22]  H. A. Eiselt,et al.  Foundations of Location Analysis , 2011 .

[23]  Zvi Drezner,et al.  Continuous Center Problems , 2011 .

[24]  B. Ciffler Scheduling general production systems using schedule algebra , 1963 .

[25]  I. Anderson,et al.  Graphs and Networks , 1981, The Mathematical Gazette.

[26]  R. A. Cuninghame-Green,et al.  Describing Industrial Processes with Interference and Approximating Their Steady-State Behaviour , 1962 .

[27]  L. Hogben Handbook of Linear Algebra , 2006 .

[28]  K. Edee,et al.  ADVANCES IN IMAGING AND ELECTRON PHYSICS , 2016 .

[29]  Esmaeel Moradi,et al.  Single Facility Location Problem , 2009 .

[30]  S. Yau Mathematics and its applications , 2002 .

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

[32]  Nikolai Krivulin,et al.  A constrained tropical optimization problem: complete solution and application example , 2013, ArXiv.

[33]  N. Krivulin A new algebraic solution to multidimensional minimax location problems with Chebyshev distance , 2012, 1210.4770.

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

[35]  Jacques-François Thisse,et al.  Constrained Location and the Weber-Rawls Problem , 1981 .

[36]  Jean-Paul Chilès,et al.  Wiley Series in Probability and Statistics , 2012 .

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

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

[39]  K. G. Farlow,et al.  Max-Plus Algebra , 2009 .