Abstract:Feasible binary programs often have multiple optimal solutions, which is of interest in applications as they allow the user to choose between alternative optima without deteriorating the objective function. In this article, we present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In addition, we present the diameter binary program whose optima contains two optimal solutions of the given feasible binary program that are as diverse as possi… Show more
“…In applications, the number of optimal solutions can be too large for enumeration to be practical. Therefore, in [5], the authors present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In particular, a binary program is developed whose optima contain two optimal solutions of the given binary program that are as diverse as possible with respect to the optimal diameter.…”
Section: Lomentioning
confidence: 99%
“…Here, we develop a binary program whose optima contain two optimal solutions of a given LOP that are as far apart as possible with respect to the Kendall tau ranking distance. This program is similar to but more straightforward than the optimal diameter binary program in [5]. Also, this program is similar to those developed in [2,23], where the author in [23] is interested in two optimal solutions of separate LOPs that are as close together as possible, and the authors in [2] are interested in two optimal solutions of the same LOP that are as far apart as possible.…”
In 2019, Anderson et al. proposed the concept of rankability, which refers to a dataset's inherent ability to be meaningfully ranked. In this article, we give an expository review of the linear ordering problem (LOP) and then use it to analyze the rankability of data. Specifically, the degree of linearity is used to quantify what percentage of the data aligns with an optimal ranking. In a sports context, this is analogous to the number of games that a ranking can correctly predict in hindsight. In fact, under the appropriate objective function, we show that the optimal rankings computed via the LOP maximize the hindsight accuracy of a ranking. Moreover, we develop a binary program to compute the maximal Kendall tau ranking distance between two optimal rankings, which can be used to measure the diversity among optimal rankings without having to enumerate all optima. Finally, we provide several examples from the world of sports and college rankings to illustrate these concepts and demonstrate our results.
“…In applications, the number of optimal solutions can be too large for enumeration to be practical. Therefore, in [5], the authors present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In particular, a binary program is developed whose optima contain two optimal solutions of the given binary program that are as diverse as possible with respect to the optimal diameter.…”
Section: Lomentioning
confidence: 99%
“…Here, we develop a binary program whose optima contain two optimal solutions of a given LOP that are as far apart as possible with respect to the Kendall tau ranking distance. This program is similar to but more straightforward than the optimal diameter binary program in [5]. Also, this program is similar to those developed in [2,23], where the author in [23] is interested in two optimal solutions of separate LOPs that are as close together as possible, and the authors in [2] are interested in two optimal solutions of the same LOP that are as far apart as possible.…”
In 2019, Anderson et al. proposed the concept of rankability, which refers to a dataset's inherent ability to be meaningfully ranked. In this article, we give an expository review of the linear ordering problem (LOP) and then use it to analyze the rankability of data. Specifically, the degree of linearity is used to quantify what percentage of the data aligns with an optimal ranking. In a sports context, this is analogous to the number of games that a ranking can correctly predict in hindsight. In fact, under the appropriate objective function, we show that the optimal rankings computed via the LOP maximize the hindsight accuracy of a ranking. Moreover, we develop a binary program to compute the maximal Kendall tau ranking distance between two optimal rankings, which can be used to measure the diversity among optimal rankings without having to enumerate all optima. Finally, we provide several examples from the world of sports and college rankings to illustrate these concepts and demonstrate our results.
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