Minimax Regret Single-Facility Ordered Median Location Problems on Networks

We consider the single-facility ordered median location problem with uncertainty in the parameters (weights) defining the objective function. We study two cases. In the first case, the uncertain weights belong to a region with a finite number of extreme points, and in the second case, they must satisfy some order constraints and belong to some box (convex case). To deal with the uncertainty, we apply the minimax regret approach, providing strongly polynomial time algorithms to solve these problems. Finally, we also extend the proposed methodology to other problems with order constraints, which are not necessarily convex.

[1]  Oded Berman,et al.  Algorithms for the robust 1-center problem on a tree , 2000, Eur. J. Oper. Res..

[2]  I. Averbakh,et al.  Minimax regret p-center location on a network with demand uncertainty , 1997 .

[3]  Sergey Bereg,et al.  Facility location problems with uncertainty on the plane , 2005, Discret. Optim..

[4]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .

[5]  Christos D. Zaroliagis,et al.  Shortest Paths in Digraphs of Small Treewdith. Part II: Optimal Parallel Algorithms , 1998, Theor. Comput. Sci..

[6]  Eitan Zemel,et al.  An O(n) Algorithm for the Linear Multiple Choice Knapsack Problem and Related Problems , 1984, Inf. Process. Lett..

[7]  F. Fernández,et al.  Robustness in the Pareto-solutions for the multi-criteria minisum location problem , 2001 .

[8]  Nimrod Megiddo,et al.  Linear-time algorithms for linear programming in R3 and related problems , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[9]  Igor Averbakh,et al.  On the complexity of a class of combinatorial optimization problems with uncertainty , 2001, Math. Program..

[10]  Justo Puerto,et al.  Robust Positioning of Service Units , 2003 .

[11]  Subhash Suri,et al.  Offline maintenance of planar configurations , 1991, SODA '91.

[12]  N. Megiddo Linear-time algorithms for linear programming in R3 and related problems , 1982, FOCS 1982.

[13]  Igor Averbakh,et al.  Complexity of Robust Single Facility Location Problems on Networks with Uncertain Edge Lengths , 2003, Discret. Appl. Math..

[14]  Justo Puerto,et al.  Location Theory - A Unified Approach , 2005 .

[15]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..

[16]  Igor Averbakh The Minmax Relative Regret Median Problem on Networks , 2005, INFORMS J. Comput..

[17]  Arie Tamir,et al.  The k-centrum multi-facility location problem , 2001, Discret. Appl. Math..

[18]  Justo Puerto,et al.  Algorithmic results for ordered median problems , 2002, Oper. Res. Lett..

[19]  J. Puerto,et al.  A unified approach to network location problems , 1999 .

[20]  Richard Cole,et al.  Slowing down sorting networks to obtain faster sorting algorithms , 2015, JACM.

[21]  Edith Cohen Efficient Parallel Shortest-Paths in Digraphs with a Separator Decomposition , 1996, J. Algorithms.

[22]  Eitan Zemel On search over rationals , 1981, Oper. Res. Lett..

[23]  Bintong Chen,et al.  Minmax‐regret robust 1‐median location on a tree , 1998 .

[24]  Eduardo Conde A note on the minmax regret centdian location on trees , 2008, Oper. Res. Lett..

[25]  Nimrod Megiddo,et al.  An O(n log2 n) Algorithm for the k-th Longest Path in a Tree with Applications to Location Problems , 1981, SIAM J. Comput..

[26]  Nimrod Megiddo Combinatorial Optimization with Rational Objective Functions , 1979, Math. Oper. Res..

[27]  Oded Berman,et al.  An improved algorithm for the minmax regret median problem on a tree , 2003, Networks.

[28]  George L. Vairaktarakis,et al.  Incorporation dynamic aspects and uncertainty in 1‐median location problems , 1999 .

[29]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[30]  Rainer E. Burkard,et al.  A Note on the Robust 1-Center Problem on Trees , 2002, Ann. Oper. Res..

[31]  Biing-Feng Wang,et al.  Improved algorithms for the minmax-regret 1-center and 1-median problems , 2008, TALG.

[32]  F. Fernández,et al.  Multi-criteria analysis with partial information about the weighting coefficients , 1995 .

[33]  Eduardo Conde,et al.  An improved algorithm for selecting p items with uncertain returns according to the minmax-regret criterion , 2004, Math. Program..

[34]  Loukas Georgiadis,et al.  An O(nlogn) version of the Averbakh-Berman algorithm for the robust median of a tree , 2008, Oper. Res. Lett..

[35]  Leslie G. Valiant,et al.  Parallelism in Comparison Problems , 1975, SIAM J. Comput..

[36]  Christos D. Zaroliagis,et al.  Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms , 2000, Algorithmica.

[37]  Eduardo Conde,et al.  Minmax regret location-allocation problem on a network under uncertainty , 2007, Eur. J. Oper. Res..

[38]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[39]  Rainer E. Burkard,et al.  Robust location problems with pos/neg weights on a tree , 2001, Networks.

[40]  Daniel Vanderpooten,et al.  Approximation of min-max and min-max regret versions of some combinatorial optimization problems , 2007, Eur. J. Oper. Res..

[41]  Arie Tamir,et al.  On the Solution Value of the Continuous p-Center Location Problem on a Graph , 1987, Math. Oper. Res..

[42]  Justo Puerto,et al.  A flexible approach to location problems , 2000, Math. Methods Oper. Res..

[43]  Oded Berman,et al.  Minmax Regret Median Location on a Network Under Uncertainty , 2000, INFORMS J. Comput..

[44]  Igor Averbakh,et al.  On the complexity of minmax regret linear programming , 2005, Eur. J. Oper. Res..