Indistinguishability Operators via Yager t-norms and Their Applications to Swarm Multi-Agent Task Allocation
Round 1
Reviewer 1 Report
This paper is very difficult to review due to english language. It requires extensive editing of english. See more comments:
Abstract: Re-write
Introduction: Failed to reach out the level of merit publication. This is not the way to present, I am afraid. Re-write
Methods: No assumptions and limitations are expalined.
Conclusion: Re-write!
Author Response
We have rewritten many parts of the text in order to improve the English language. All changes have been written in red, in Abstract, Introduction, Methods and Conclusion. We hope the new version of the text is adequate and it facilitates the reviewer task. Many thanks for your remarks because they have allowed us to improve a lot the text.
Reviewer 2 Report
The presented paper describes a response function for swarm multi-agent systems. In terms of applicability these functions are very important for the MAS environments.
The paper is well written and organized but it contains a lot of figures that can be on the appendix section.
Regarding the results Table 1 is very strange because the YPRF requires more percentage of experiments to converge and more steps and the authors did not commented this aspect.
Another limitation of this work is the usage of only one restriction the power level of the robots to limit the action radius. In my personal opinion this work requires further investigation on this aspect to make it more robust.
A real case study if it is possible it is a plus for the experimentation section.
Author Response
We deeply appreciate the invaluable comments to our manuscript and we would like to thank the reviewer for the constructive comments. We have highlighted the changes made in response to the specific comments of the Reviewer.
* Comment 1: The paper is well written and organized but it contains a lot of figures that can be on the appendix section.
- Response 1: As the figures are very related to the text we have considered include them in the main body. If the manuscript were accepted for publication, before editing the text with all the changes suggested by the reviewers we will move the placement of some figures.
* Comment 2: Regarding the results Table 1 is very strange because the YPRF requires more percentage of experiments to converge and more steps and the authors did not commented this aspect.
- Response 2: The theoretical reason of this behaviour are still under research. It seems that a probabilistic matrix with values equal to zero increases the number of steps required to converge. In any case, Fuzzy Markov chains show more robust behaviour with always the same number of steps needed to converge In order to clarify this point the following lines have been added to the manuscript: lines 375-379
* Comment 3: Another limitation of this work is the usage of only one restriction the power level of the robots to limit the action radius. In my personal opinion this work requires further investigation on this aspect to make it more robust.
- Response 3: We only have considered the power level of a robot as an example of application. The Markov chains induced by a YRPF are able to model any kind of movement restriction with is reflect via the stimulus. Moreover, we use robots as an example but any other agent with restriction, such as tourists, air planes or insects, can be modelled with our system. In order to clarify this point the following lines have been added to the manuscript: line 39.
* Comment 4: A real case study if it is possible it is a plus for the experimentation section.
- Response 4: As a future work we plan to implement this system on 5 real robots. In order to clarify this point the following lines have been added to the manuscript: line 508.
Author Response File: 
Author Response.pdf
Reviewer 3 Report
Thank you for an interesting manuscript. I have a few major and some minor points.
Major:
Line 134 - 136: This sentence sound like the agent starts a specific task with certainity, but from the rest of the paper it is clear, that there is some randomness involved, so please clarify, how the agent chooses the task. I assume by some weihghted choices between task with s_r,t > 0.
Line 147-148: This sound like there always is an agent at each task, which can not be true. Is the interpretation of p_i,j the probability of a robot leaving task t_i towards t_k, conditional on a robot being at task t_i?
Line 311-313: Please give a reference to the statement that using auction algorithms give equivalent settings if possible.
Line 328: Having only one d_max of 800.5 seems like it is the overall maximum over all simulations and experiments. Isn't it a bit uncommon to have a model parameter, that depends on all data sets combined?
Figure 1: Clearly the cluster have different distances from each other, you should discuss to which degree this is visible in the simulation results.
Line 332-334: Here it sounds like probabilistic Markov chains ever will reach their stady state. Doesn't this only hold when accepting small differences? Full convergence will typically only be achived in the (infinite) limit.
Line 349-354: Was it expected that the fuzzy Markov chains use the same mean and sd number of steps for all scenarios. It seems surprising, that they give precisely the same results.
Figure 2: Some measure of variation on the graph would be helpfull, e.g. sd as uncertainty bars.
Figure 5,6,7,10: Wouldn't one for the probabilistic transition matrices expect that P^500 again is a transistion matrix, with columns and rows summing to 1? This is inconsistent with the vertical line structure of the figures.
Section 444-461: It seems weird to compare nTH with numbers of clusters, as nTH only seems to enter the model via theta, and hence is dependent on the scale (via d_max) of the distances measured, but this should not have influence on the number of clusters.
Making the Matlab code used for the simulations avaible as supplementary files would be highly desirable.
Minor:
Line 32: You are a bit inconsistent in using "agent" or "robot". I would suggest using either "agent" or "robot" everywhere, apart from the descriptions of concrete applications to avoid confusion.
Line 177: You use "simply power convergent" here, but do not explain what this is, or why it is relevant.
Line 308: What is a "fool disaster"?
Line 344: I do not understand the statement "very similar in the probabilistic case"
Table 1: Having the same number of decimals for all numbers in a column, would make the table more readable.
Line 367: When you write "probablistic Markov chain only converges for 4 and 10 clusters" does that mean for some of the 500 simulations or for all of them?
Line 431-432: I think this sentence is missing a "not". ("does not move" instead of "moves" or similar.
Furthermore, there are a number of gramatical and spelling erros in the manuscript, so I would suggest an additional round of checking the language.
Author Response
We deeply appreciate the invaluable comments to our manuscript and we would like to thank the reviewer for the constructive comments. We have highlighted the changes made in response to the specific comments of the Reviewer.
Major:
* Comment 1: Line 134 - 136: This sentence sound like the agent starts a specific task with certainity, but from the rest of the paper it is clear, that there is some randomness involved, so please clarify, how the agent chooses the task. I assume by some weihghted choices between task with s_r,t > 0.
- Response 1: The transition from the current task (task assigned to the robot) to the next one follows the probability/possibility given by p_{r_i,j}. Thus, the p_{r_i,j} provides the "weighted choices between task" mentioned by the reviewer. In order to clarify this point the following lines have been added: 158-159
* Comment 2: Line 147-148: This sound like there always is an agent at each task, which can not be true. Is the interpretation of p_i,j the probability of a robot leaving task t_i towards t_k, conditional on a robot being at task t_i?
- Response 2: The interpretation given by the reviewer is correct. We assume that each robot is always allocated to a specific task.The robots only make the decision of leaving a task after reaching its current task. Through the path form one task to the next one the robots can not change its decision. In order to clarify this point we have changed the following lines: 159-160
* Comment 3: Line 311-313: Please give a reference to the statement that using auction algorithms give equivalent settings if possible.
- Response 3: In order to address this question, the following references have been added: [24] [25] and [26]. Moreover, we have changed the following lines: 324-325
Comment 4: Line 328: Having only one d_max of 800.5 seems like it is the overall maximum over all simulations and experiments. Isn't it a bit uncommon to have a model parameter, that depends on all data sets combined?
- Response 4: The d_max parameter stands for the maximum distance between any two point in the environment, instead of the maximum distance between two tasks. This sentence has been modified in the manuscript in order to clarify this concept (line 340). Thus, the d_max value is always the same for all the experiments as all the environments have the same dimensions. Here it must recall that for the YRPF the threshold value (\theta) is the maximum distance that an agent is able to reach (see Equation (6)). In Equation (8), d_max value is divided by nTH to obtain the threshold, thus the parameter nTH is the portion of the space reachable by an agent. For example, when nTH=1, the agents can reach all the tasks and there is no restrictions. Or for example, when nTH=2 the agents only can reach half of the total space. In order to clarify this point the following lines have been added or modified: lines 342-346.
* Comment 5: Figure 1: Clearly the cluster have different distances from each other, you should discuss to which degree this is visible in the simulation results.
- Response 5: The distance mentioned by the reviewer is visible in Figure 3 and subsequent, where the possibilistic or probabilistic matrix values are visualized. The clusters are colored in yellow tones (possibilistic values near to one). In contrast, the farther away the groups are from each other, the darker the colours of the figure (dark blue colours) are. In these figures dark blue colours represent low possibilities of transition from one task to another. In order to clarify this point the following lines have been added: lines 417-422.
* Comment 6: Line 332-334: Here it sounds like probabilistic Markov chains ever will reach their stady state. Doesn't this only hold when accepting small differences? Full convergence will typically only be achived in the (infinite) limit.
-Response 6: May be the probabilistic Markov chains will reach with a greater number of iteration or in the limit, but a real system requires a convergence in a finite number of steps in order to make on-line decisions. Thanks to Duan's theorem (Theorem (1) in the manuscript) we can guarantee the convergence of our Fuzzy Markov chains always in a finite time.
* Comment 7: Line 349-354: Was it expected that the fuzzy Markov chains use the same mean and sd number of steps for all scenarios. It seems surprising, that they give precisely the same results.
-Response 7: The convergence conditions for fuzzy Markov chains do not depend on the parameters of the Markov chain (in this case nTH), only depends on the position of the objects, as can be seen in the conditions of Duan's Theorem (see Theorem (1)). Clearly, the values of the transition possibility matrix depends on the system's parameters (in our case nTH), as can be seen in Figure 3 and subsequent. Actually, this is one of the strength points of fuzzy Markov chains, the number of steps to converge does not depend on the system parameters. To stress this point the following lines have been added: lines 373-374.
* Comment 8: Figure 2: Some measure of variation on the graph would be helpfull, e.g. sd as uncertainty bars.
-Response 8: The tasks in the clusters are very near between them and therefore the variability between scenarios is very low, for a given number of clusters. As the number of steps does not depend on the nTH parameter (see the answer of the previous question), the standard deviation for the experiments is too low for being shown in the figure.
* Comment 9: Figure 5,6,7,10: Wouldn't one for the probabilistic transition matrices expect that P^500 again is a transistion matrix, with columns and rows summing to 1? This is inconsistent with the vertical line structure of the figures.
-Response 9: As the reviewer states, if P is a probabilistic transition matrix P^500
is transition probabilistic matrix too. In all the above-mentioned figures the transition probability is given by the colour (darker colours represent lower probabilities; yellow colours represent probabilities near to 1). Moreover, the X and the Y axes of those figures are the number of tasks, in this case there are 120 tasks and therefore the values of the axes go from 1 to 120. The 120x120 numerical values of a matrix can not be showed in the manuscript due to its huge size, but they verify all the axioms of probabilistic distribution.
* Comment 10: Section 444-461: It seems weird to compare nTH with numbers of clusters, as nTH only seems to enter the model via theta, and hence is dependent on the scale (via d_max) of the distances measured, but this should not have influence on the number of clusters.
-Response 10 The d_max value is the same for all environments and therefore the theta value only depends on nTh parameter. This is the reason for which we can compare the nTH parameter and the number of clusters. Moreover, with these experiments we can see the impact on the system performance that was also commented for the "Comment 4", where we explained what the nTH means. In order to clarify this point we have added the following lines: lines 477-481.
* Comment 11: Making the Matlab code used for the simulations avaible as supplementary files would be highly desirable.
- Response 11: the Matlab source code is available at the following github repository: https://github.com/joseGuerreroUIB/MDPIMathematics2021.git. This code has been tested and executed under the version 2016b of Matlab. In order to add this information, line 335 has been modified.
Minor:
* Comment 12: Line 32: You are a bit inconsistent in using "agent" or "robot". I would suggest using either "agent" or "robot" everywhere, apart from the descriptions of concrete applications to avoid confusion.
- Response 12 Most of the references to "robots" in the manuscript has been replaced by "agent".
* Comment 13: Line 177: You use "simply power convergent" here, but do notexplain what this is, or why it is relevant.
- Response 13: In order to clarify the sentence we have been deleted “ (or simply power-convergent)”.
* Comment 14: Line 308: What is a "fool disaster"?
- Response 14: Sorry for the typo, it should bee "flood disaster".
* Comment 15: Line 344: I do not understand the statement "very similar in the probabilistic case"
- Response 15: This sentence has been rewritten as can be seen in lines 361-363 compared to possibilistic case.
* Comment 16: Table 1: Having the same number of decimals for all numbers in a column, would make the table more readable.
- Response 16: The values of Table 1 has been modified in order to improve its readability.
* Comment 17: Line 367: When you write "probablistic Markov chain only converges for 4 and 10 clusters" does that mean for some of the 500 simulations or for all of them?
- Response 17: In this case, the convergence is for all simulations. In order to clarify this point we have modified the line 392 in the manuscript.
* Comment 18: Line 431-432: I think this sentence is missing a "not". ("does not move" instead of "moves" or similar.
- Response 18: The reviewer is right, the sentence has been rewritten.
* Comment 19: Furthermore, there are a number of gramatical and spelling errors in the manuscript, so I would suggest an additional round of checking the language.
- Response 19: We have tried our best to revise the whole manuscript regarding the spelling problems. All changes have been written in red.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The paper still requires english editing. Reading by native speaker or using english service is recommended.
Author Response
Thank you so much for your comment. We would like to apologize the spelling and grammar errors in the text. As the reviewer pointed out, these errors made the document difficult to read. Thus, according to the comments in the review report, we have done a thorough review of spelling, grammar and typographical errors. Furthermore, several sections of the manuscript have been modified in order to improve the style of the paper. All the changes have been highlighted in green in the text. We hope that this new version will facilitate the task of the reviewer.
