Question

A dancing instructor wants to pair his students for the annual paired dance competition. For better harmony, pairs must have close heights. If the dancing instructor has n female and m male students, formulate the model of choosing k pairs with maximum harmony (minimum total height difference). (You can take height of ith female as fi and ith male as mi)

Answer 1

A mathematical model to pair dance students for maximum harmony could involve sorting the students by height, then selecting pairs with the least height difference to minimize total height disparity across selected pairs.

Step-by-step explanation:

To formulate a model for a dancing instructor to pair students with the goal of maximum harmony based on height, one could use a mathematical approach that minimizes the total height difference within the pairs. Assuming there are n female students with heights fi and m male students with heights mi, the objective is to choose k pairs such that the sum of the absolute differences in height for each pair is minimized. This can be represented as a minimization problem.

The process of forming these pairs can be broken down into a few steps:

1. Organize the female students' heights (fi) and male students' heights (mi) into two lists.
2. Sort both lists to efficiently find the closest matches.
3. Iteratively select pairs starting with the closest matches in height and eliminate those pairs from the lists.
4. Continue the process until k pairs have been formed or no more pairs can be created.

To implement this model as an algorithm, one might consider greedy methods, dynamic programming, or optimization techniques such as linear programming. The specifics of the model would depend on additional constraints such as whether all students must be paired or if there are preferences that need to be taken into account beyond height.