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Adding feasibility constraints to a ranking rule under a monotonicity constraint

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Abstract
We propose a new point of view in the long-standing problem where several voters have expressed a linear order relation (or ranking) over a set of candidates. For a ranking a > b > c to represent a group's opinion, it would be logical that the strength with which a > c is supported should not be less than the strength with which either a > b or b > c is supported. This intuitive property can be considered a monotonicity constraint, and has been addressed before. We extend previous approaches in the following way: as the voters are expressing linear orders, we can take the number of candidates between two candidates to be a measure of the degree to which one candidate is preferred to the other. In this way, intensity of support is both counted as the number of voters who indicate a > c is true, as well as the distance between a and c in these voters' rankings. The resulting distributions serve as input for a natural ranking rule that is based on stochastic monotonicity and stochastic dominance. Adapting the previous methodology turns out to be non-trivial once we add some natural feasibility constraints.
Keywords
Linear Order, Group Decision Making, Weak Order, Monotonicity, Stochastic Dominance, Integer Linear Programming

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Citation

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Chicago
Perez Fernandez, Raul, Michaël Rademaker, Pedro Alonso, Irene Díaz, and Bernard De Baets. 2015. “Adding Feasibility Constraints to a Ranking Rule Under a Monotonicity Constraint.” In Advances in Intelligent Systems Research, ed. JM Alonso, H Bustince, and M Reformat, 89:1302–1309. Paris, France: Atlantis Press.
APA
Perez Fernandez, R., Rademaker, M., Alonso, P., Díaz, I., & De Baets, B. (2015). Adding feasibility constraints to a ranking rule under a monotonicity constraint. In JM Alonso, H. Bustince, & M. Reformat (Eds.), Advances in Intelligent Systems Research (Vol. 89, pp. 1302–1309). Presented at the 16th World congress of the International Fuzzy Systems Association (IFSA) ; 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), Paris, France: Atlantis Press.
Vancouver
1.
Perez Fernandez R, Rademaker M, Alonso P, Díaz I, De Baets B. Adding feasibility constraints to a ranking rule under a monotonicity constraint. In: Alonso J, Bustince H, Reformat M, editors. Advances in Intelligent Systems Research. Paris, France: Atlantis Press; 2015. p. 1302–9.
MLA
Perez Fernandez, Raul, Michaël Rademaker, Pedro Alonso, et al. “Adding Feasibility Constraints to a Ranking Rule Under a Monotonicity Constraint.” Advances in Intelligent Systems Research. Ed. JM Alonso, H Bustince, & M Reformat. Vol. 89. Paris, France: Atlantis Press, 2015. 1302–1309. Print.
@inproceedings{7258308,
  abstract     = {We propose a new point of view in the long-standing problem where several voters have expressed a linear order relation (or ranking) over a set of candidates. For a ranking a {\textrangle} b {\textrangle} c to represent a group's opinion, it would be logical that the strength with which a {\textrangle} c is supported should not be less than the strength with which either a {\textrangle} b or b {\textrangle} c is supported. This intuitive property can be considered a monotonicity constraint, and has been addressed before. We extend previous approaches in the following way: as the voters are expressing linear orders, we can take the number of candidates between two candidates to be a measure of the degree to which one candidate is preferred to the other. In this way, intensity of support is both counted as the number of voters who indicate a {\textrangle} c is true, as well as the distance between a and c in these voters' rankings. The resulting distributions serve as input for a natural ranking rule that is based on stochastic monotonicity and stochastic dominance. Adapting the previous methodology turns out to be non-trivial once we add some natural feasibility constraints.},
  author       = {Perez Fernandez, Raul and Rademaker, Micha{\"e}l and Alonso, Pedro and D{\'i}az, Irene and De Baets, Bernard},
  booktitle    = {Advances in Intelligent Systems Research},
  editor       = {Alonso, JM and Bustince, H and Reformat, M},
  isbn         = {9789462520776},
  issn         = {1951-6851},
  language     = {eng},
  location     = {Gijon, Spain},
  pages        = {1302--1309},
  publisher    = {Atlantis Press},
  title        = {Adding feasibility constraints to a ranking rule under a monotonicity constraint},
  volume       = {89},
  year         = {2015},
}

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