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Bicriteria and restricted 2-facility Weber problems
Locatron Science, Vol. 5, No. I, pp. 59-74, 1997 Published by Elsevier Science Ltd. Printed m Great Britain.
a9
Pergamon
ABSTRACTS IN LOCATION ANALYSIS
Bicriteria and restricted 2-facility Weber problems Stefan Nickel Mathematical Methods of Operations Research Vol. 45, 1997, pp. 167-195 In this paper we look at two interesting extensions to the classical 2-facility Weber problem in R*: at first problems are investigated where we do not allow the optimal locations to be in a specific region. Efficient algorithms for this global optimization problem are presented as well as new structural results. Secondly we consider 2-facility Weber problems with two decision makers where each decision maker can choose his own preferences for the location problem. We give an efficient algorithm for determining all pareto locations for this multicriteria problem as well as a polynomial description of the set of all pareto locations (in W”). All the results presented in this paper are based on a discretization of the original continuous problem using geometrical and combinatorial arguments. The time complexity of all the presented algorithms is 0 (dM IogM), where M is the number of existing facilities.
Duality theorem for a generalized Fermat-Weber Wilfred Kaplan and Wei H. Yang Mathematical Programming Vol. 76, 1997, 285-297
problem
The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane space and to k given points in the plane. This problem was generalized by Witzgall to n-dimensional to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multifacility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.
Specially structured uncapacitated facility location problems Philip C. Jones, Timothy J. Lowe, Georg Muller, Ning Xu, Yinyu Ye and James L. Zydiak Operations Research Vol. 43, No.4, 1995, p. 661 ff. This paper considers a specially structured uncapacitated facility location problem. We show that several problems, including certain tool selection problems, substitutable inventory problems, supplier sourcing problems, discrete lot sizing problems, and capacity expansion problems, can be formulated as instances of the problem. We also show that the problem with m facilities and n customers can be solved in 0 (mn), as a shortest path problem on a directed graph. 59
a9
Pergamon
ABSTRACTS IN LOCATION ANALYSIS
Bicriteria and restricted 2-facility Weber problems Stefan Nickel Mathematical Methods of Operations Research Vol. 45, 1997, pp. 167-195 In this paper we look at two interesting extensions to the classical 2-facility Weber problem in R*: at first problems are investigated where we do not allow the optimal locations to be in a specific region. Efficient algorithms for this global optimization problem are presented as well as new structural results. Secondly we consider 2-facility Weber problems with two decision makers where each decision maker can choose his own preferences for the location problem. We give an efficient algorithm for determining all pareto locations for this multicriteria problem as well as a polynomial description of the set of all pareto locations (in W”). All the results presented in this paper are based on a discretization of the original continuous problem using geometrical and combinatorial arguments. The time complexity of all the presented algorithms is 0 (dM IogM), where M is the number of existing facilities.
Duality theorem for a generalized Fermat-Weber Wilfred Kaplan and Wei H. Yang Mathematical Programming Vol. 76, 1997, 285-297
problem
The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane space and to k given points in the plane. This problem was generalized by Witzgall to n-dimensional to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multifacility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual.
Specially structured uncapacitated facility location problems Philip C. Jones, Timothy J. Lowe, Georg Muller, Ning Xu, Yinyu Ye and James L. Zydiak Operations Research Vol. 43, No.4, 1995, p. 661 ff. This paper considers a specially structured uncapacitated facility location problem. We show that several problems, including certain tool selection problems, substitutable inventory problems, supplier sourcing problems, discrete lot sizing problems, and capacity expansion problems, can be formulated as instances of the problem. We also show that the problem with m facilities and n customers can be solved in 0 (mn), as a shortest path problem on a directed graph. 59