Modified Kleene Star Algorithm Using Max-Plus Algebra and Its Application in the Railroad Scheduling Graphical User Interface

REVIEWER 1

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1

The paper discusses a modified Kleene Star algorithm to address the drawback of traditional Kleene Star algorithm and effectively reduce the complexity of the algorithm. Meanwhile, the algorithm is applied in a railroad network scheduling problem and construct a graphical user interface to implement scheduling efficiently.

Overall, the problem is well formalized and motivated, and the background is clear. Also, the design is detailed and the evaluations are comprehensive.

Thank you for the comments

REVIEWER 2

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This paper describes three algorithms of determining eigenvectors of irreducible matrices in max-plus algebra, discusses their time complexity, and demonstrates an application of them: scheduling in railroad network system.

It is interesting to see how max-plus algebra can be applied to solve a very practical problem, train scheduling. However, the three algorithms are all published before, including the proposed best one (Modified Kleene Star Algorithm). Could you make this point clear in Introduction? Also, it looks like the complexity analysis is one of the key contributions of this work, therefore could you provide more details on how you calculated them, especially KSA and MKSA?

We added this sentence into the introduction

“The complexity value of an algorithm obtains by calculating the worst time translated into the pseudo-code algorithm. For KSA, the complexity is  and for MKSA, the complexity is .”

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Section 2.4:

Eq (1): Could you explain the meaning of matrix A?

In the railroad network,  is a matrix in which its elements represent travel time between two stations.  is travel time from station  to station .

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Theorem 6: Does matrix A always only have one eigenvalue? If yes, could you explain it or provide a reference? If no, does the lambda in Theorem 6 mean the smallest or largest eigenvalue or something else?

This is explained by adding a theorem from Bacelli 1992 whose contents are :

If  is irreducible, or equivalently if graph  is strongly connected, there

exists one and only one eigenvalue (but possibly several eigenvectors).

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Table 1: It’s very well-known that n! grows much faster than n^4. If you want to demonstrate this, it’s better to draw a figure instead of listing all values.

We have added a KSA and MKSA comparison chart based on table 1.

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There are many English grammar and wording issues as well as typos. Please fix them. Just name a few:

Line 29-30: It is be given in terms of ……

Line 41: …… are important discussions …… ==> topics

Line 76: two binary operation ==> operations

Line 102: di

Line 355: time complexity of ((n!)) ==> O(n(n!))

And many more…

We have corrected typos and imperfect writing according to the reviewer's suggestions.

REVIEWER 3

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In this article, the authors is considered the application of max-plus algebra on the scheduling of the railroad network system. I think the article will be of interest for researchers involved in the study of complexity theory of algorithms

Thank you for the comments

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I have some comment on this article. From the description and application of max-plus algebra in the article, I didn't understand until the end the meaning of using "-infinity". How do the authors interpret this element? Is the set of real numbers (R) not enough? I think the authors should give an example.

“-infinity” is treated as the identity element of the maximum operation in max plus algebra, since the maximum of any element of Real numbers and -infinity is the element itself. Real numbers set is not enough, since any element in it is finite and there are other elements that are smaller than the element itself.

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I recommend publishing the article "Modified Kleene Star Algorithm on Max-Plus Algebra and Its Application on Railroad Scheduling Graphical User Interface" when correcting this question.

Thank you for the suggestion of for publishing the revised version of the article

REVIEWER 2

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The authors addressed most of my comments.

Thank you for the comments

Figure 1: It’s much better to use log scale on Y-axis. Currently the last point (input=40) makes the Y values of other points invisible.

We have changed the Y axis to log scale as suggested

Please proof-read and fix the English grammar and wording issues. The followings are just a few examples, not a complete list:

Line 29-30: “It is be given in terms of ……” should be “It is given in terms of ……”

Line 50-51: “While the KSA has existed before 1992 ([11]), and is still being discussed and applied in several studies.” should be “While the KSA was created before 1992 ([11]), it is still being discussed and applied in several studies.”

Line 75-76: “Resulting in quick railroad network scheduling and easy to use with an intuitive interface.” should be “It leads to a fast and easy-to-use railroad network scheduling system with an intuitive interface.”

Line 178: redundant space.

Line 353: “periode” should be “period”

We have corrected it as the reviewer suggested and checked the grammar used again.