The General Model for Least Convex Disparity RIM Quantifier Problems
Round 1
Reviewer 1 Report
A good contribution.
In the recent paper Dug Hun Hong generalizes the results
published in
D. H. Hong, S. Han, The general least square deviation OWA operator
problem,
Mathematics, 2019 7, 326. https://doi.org/10.3390/math7040326
It can be seen on page 5, where the earlier problem is displayed
by equation 2 and the generalized problem is give by equation 3.
The earlier problem is a special case of the new problem, namely when
the
regular increasing monotone quantifier is defined by the square function
(see line 90).
Then using a transformation trick Hong provides an explicit solution to
problem 3 for
a very large class of regular increasing monotone quantifiers.
If I were Hong, I would not use the same notation F for the OWA operator
(lines 38-39) and for a strictly convex function on [0, 1] in equation
2.
Also it would be better to use different symbols in the left-hand side
and
the right-hand side for the function to be minimized in problem 2.
Author Response
Thank you for your good comments on my paper. Based on your refort, I have made as many modifications as possible. Thank you.
Reviewer 2 Report
The topic addressed in the paper is interesting. It is on looking for optimal weighing functions for RIM quanti fiers in case of a fixed orness value. The author suggests a new optimality criterion that generalizes an already known one (or even two models). Particularly those known optimality criteria are described in the paper as moedels (1) and (2), and the newly proposed as model (3).
Recommendations to the author for improving the manuscript:
The model (3) is in general not a least squares model, but is much more general. I
propose to change the name of this model - could be least convex disparity ?, and
also to change its denotation to VF (f) = ... (or something similar to stress that
the criterion depends substantially on F).
The numerical example provided at the end of the paper is just on the least squares
model. I would expect another convex function F to stress the generality of the new
model.
In example 1 a reference is missing.
Throughout the whole paper (very often) value of a function at x , r, and so on
(i.e., f(x), f'(x), Q(r)), is used in the meaning of the corresponding functions f, f',
Q, e.t.c. This is a mathematical mistake (though being sometimes neglected).
Page 2, line 36: absolutely continuous function -> absolutely continuous functions (absolutely continuous is also sufficient)
Page 2, formulae below lines 49 and 50: on the left-side the variable is r, on the right-side x Quite often in the text (starting with Def. 2) the function is f, f ', Q, e.t.c., and f(x), Q(x), e.t.c. is the corresponding value at x. This must be check and corrected in the whole paper
Page 3, line67: a Lebesgue measure of zero -> Lebesgue measure zero
Page 3, line 2 of Def. 3: should be {[a_i,b_i]}_{i\in I} where I is an index set.
Author Response
Thank you for your good comments on my paper.
Based on your referee's refort, I have made as many changes as possible.
Thank you.
