A Constructive Heuristics and an Iterated Neighborhood Search Procedure to Solve the Cost-Balanced Path Problem
Round 1
Reviewer 1 Report
The paper is clearly written and well organized.
The authors stated that the CBBP has been introduced recently in their previous work [1].
However, in my opinion, the problem is very close to shortest path problem and TSP. I don’t think it is a new novel problem.
Besides, the first stage of their two-stage heuristic is based on Dijkstra’s algorithm.
I believe even this paper cannot be considered after a major revision because the nature of the paper is under question.
Author Response
Thank you for your review. We are pleased that the paper is clear and well structured.
We understand your doubt about the originality of the problem. Surely we are talking about a variant of the shortest path problem (SPP).
The first and most important difference between the classical SPP and our problem is the complexity class. Whereas for the SPP there is an algorithm that identifies the optimal solution in polynomial time, as our problem is NP-Hard the only approaches known to date require exponential time.
Anyway, thanks to your comment, we tried to better explain the differences between the two problems, especially in the introduction of the paper.
We can also state that the scientific literature is replete with variants of the SPP that are both solvable in polynomial time and exponential. These variants often prove to be fundamental for modelling specific application scenarios (see for example, Turner (2011)).
Regarding your remark on the similarity between our technique and Dijkstra's algorithm, it was helpful for both describing better the differences of the constructive step of our proposed approach putting more emphasis on the important part of our approach, based on the re-optimisation of the solution (Step2).
Thank you for your review. It was very helpful in improving the overall quality of our work.
Turner, L. (2011). Variants of the shortest path problem. Algorithmic Operations Research, 6(2), 91-104.
All major changes made to the article have been highlighted in red.
Minor changes (e.g. just updating the name of the algorithms) have not been highlighted.
Except for Tables 1,2 and Figures 1,2,3, all tables and figures have been updated; we have not changed their colour to make them easier to read.
Reviewer 2 Report
Please consider these comments and suggestions for Authors: 1. All the methods applied in this paper are not new. Authors need to justify the novelty of the paper. 2. Authors need to do comparison with other prominent path planning algorithm like GA, PSO etc. 3. Authors need to justify the sentence " the optimal objective function value equal to zero" regarding to cost-balanced path. 4. Authors need to provide the axis names in figure 4.Author Response
Thank you very much for your review, thanks to your comments we have greatly improved the overall quality of our work.
In this article, we mainly propose a construction technique inspired by Dijkstra's algorithm, and an improvement technique.
We have shown that, even for large graphs, the combination of these two approaches is capable of producing many excellent solutions.
Even if, our constructive approach based on Dijkstra's algorithm can be considered a minor contribution, the novelty is certainly given by the improvement step To the authors' knowledge, there are no other similar approaches that exploit the tree produced by Dijkstra's algorithm to re-optimise the solution produced. Thanks to your comment, we have made this innovative aspect clearer in the paper.
A further original aspect of our work is the low computational time typical of a constructive approach. To make a comparison, as you suggested, we implemented an iterated randomised technique. This very simple technique suggests that, even generating very few initial paths (100) typically as many as are needed for an initial population of a genetic algorithm, computational times increase significantly. Surely a more complex approach based on a metaheuristic scheme could achieve better results for the problem. But in this paper we have tried to develop a constructive approach focusing on speed in relation to the quality of the solution (see Table 5 and Figure 5).
Regarding your question about the sentence: "the optimal objective function value equal to zero".
We have tried to clarify the point in the paper.
Obviously the minimum of a sum of absolute values cannot be negative, so zero is a lower bound for the optimal solution. Every admissible solution with value zero has the same value as a lower bound, so it can automatically be considered optimal.
All major changes made to the article have been highlighted in red.
Minor changes (e.g. just updating the name of the algorithms) have not been highlighted.
Except for Tables 1,2 and Figures 1,2,3, all tables and figures have been updated; we have not changed their colour to make them easier to read.
Reviewer 3 Report
This article deals with the Cost Balanced Path Problem (CBPP), that was proven NP-hard and modeled as an Integer Linear Program in a previous paper. A constructive heuristic is designed, adapting the well-known Dijkstra's algorithm for the Shortest Path Problem (SPP), with the same complexity. A local search improving procedure is given, with a cubic complexity, using the stored tables form the constructive phase. Some numerical results are presented. The conclusion overstates the importance of some findings, I will precise it below. If the paper is technically correct, the question of the importance of contributions is the major point reviewing this paper.
As a general comment, one may criticize that this paper is a kind of "iterative research", paper [1] defining the problem was written by 2 co-authors of this paper. These two 13 page papers could be one, with small iterative contributions. For this heuristic paper, there are several way to have something deeper and also more convincing that these are strong results and that it would be not easy to improve these results. The results are not compared with other state-of-the-art meta-heuristics, this would be convincing to state if this simple approach is efficient. Using LocalSolver is an easy way to have a benchmark local search solver. A weakness of the improving procedure is the cubic complexity (O(N^3))), which would be a bottleneck for large graphs with N>1000, 10000 for instance. The conclusion " is particularly suitable for large-sized graphs" does not hold with graphs of size 100, 200. Also related to the "personal iterative research", instances are not made public, so that other researchers cannot compare with other approaches on the same instances, or using the instance generator also for larger sizes of N. It may be mentioned in perspective, it reinforces the conviction of reading incremental research with minimal contributions.
The first question I had naturally for this review: What is the interest of this specific problem ? In conclusion, it is stated "Since the CBPP can be applied in many real-life problems for which only large size instances are required, such as vehicle battery level, altitude change, and cargo problems among others", it is not discussed, even mentioned before. I wanted to be convinced about this point. I had a look to paper [1] for that (and I was not convinced with vehicle battery level that should be always positive in a path), this article should mention it. In the instance generator, real life case studies can inspire the generation for specific instances.
There are very few references for CBBP (only one), and also cost-balanced Traveling Salesman Problem, one may doubt about the interest of these problems for the research community.
With so many doubts, I encourage the authors to strengthen this paper using these points, and not postpone it to a next paper with minimal contributions.
MINOR REMARKS
"Step 1 / step 2" for heuristics was ambiguous in my first reading, I had in mind 2-stage optimization problems and heuristics decomposing the search with this structure. I suggest to replace "step 1 " by "constructive phase" and "step 2 " by "improvement/local post-processing phase" and
In the title, "Dijkstra-based" was ambiguous (and also combined with the ambiguity of 2 stage heuristics), "adaptation of Dijkstra Shortest Path", or "Modified Dijkstra Shortest Path" would be less ambiguous
In Algorithm 1, a high level explanation is missing, just referring to the Algorithm 1 is not easy to read.
Algorithm 3 : problem line 4, something is missing.
line 271: "proposed in [? ]" : problem with the reference. Anyway, the ILP formulation is not used in the paper, direct copy/pasting from the first paper, I suggest to remove it.
Author Response
We thank you for your valuable work. Thanks to your comments, the quality of the paper has increased a lot.
We agree with your observation that it would have been possible to combine the results of our first work with these to have one larger paper. Unfortunately, the idea of realising a heuristic approach for the problem came after the publication of our first paper, and it came substantially through the intervention of the third author.
In our opinion, the most interesting part of this work is how to use the tree resulting from the first iteration of the technique proposed in Step1 to improve the quality of the final solution.
Having focused our attention on a heuristic technique, we did not consider it necessary to compare it with meta-heuristic approaches that would certainly have produced better solutions in exchange for more computational time. Thanks to your observation, we have further compared it with a metaheuristic technique based on the iterative re-optimisation of randomly generated solutions.
We find this comparison interesting especially to evaluate the performance of our technique in terms of the quality of the solution in relation to the computational time used.
Thanks to your observation regarding the complexity of Step2 O(|N|^3), we have included additional tables in the paper to better highlight the growth in computational time. We have shown that the complexity of the worst case O(|N|^3) is far from that of the average case. In fact, in real tests, there is never an improvement for each iteration of the technique that leads to a total reconstruction of the solution.
Experimental results show that even for very large 5000 node graphs, whose input file occupies 200Mb of memory, the computational time is less than half a second. There are other examples in which algorithms whose complexity in the worst case is very high are preferred to algorithms with a better worst case.
For example, for linear programming, the simplex algorithm with exponential bad case is preferred to the ellipsoid algorithm with polynomial bad case precisely because the average computational time of the simplex is significantly lower than the average computational time of the ellipsoid algorithm.
We have also added to the paper (as you suggested) the link to download the instances used to allow other authors to verify and improve the obtained results.
Regarding the importance of CBPP, and the small number of articles in the literature, we have improved the introduction section of our article. Obviously, the CBPP is a new problem, conceived from the work "Cost-Balanced Travelling Salesman Problem". The idea of applying the concept of "Cost-Balanced" to the minimum path problem stems both from a theoretical interest, to establish whether this variant was solvable in polynomial time, and from the observation that the concept of balanced could be useful in modelling path problems in the presence of vehicles with batteries. Noting that for many batteries, a charge level of around 80 per cent guarantees battery longevity. The CBPP (with additional constraints) could be useful to define routes to arrive at the destination with the charge level closest to the desired one (Eg 80%), considering battery consumption on arcs or possible charging due to downhill roads or arcs modelling charging stations. We find this problem interesting, especially considering it as a sub-problem of more complex scenarios. Thank you very much for your review. We are confident that the quality of our paper has grown thanks to your comments.
We sincerely hope that the new version of this work is also, in your opinion, adequate to the quality of the journal.
All major changes made to the article have been highlighted in red.
Minor changes (e.g. just updating the name of the algorithms) have not been highlighted.
Except for Tables 1,2 and Figures 1,2,3, all tables and figures have been updated; we have not changed their colour to make them easier to read.
Round 2
Reviewer 1 Report
The authors have addressed all the concerns I raised. I think the paper can be accepted at current status.
Reviewer 3 Report
My points were answered in a satisfactory way. I also think the paper was improved a lot with this 2nd version, weaknesses were improved and also one point that was not clear for me is now clarified. Hence, I recommend acceptation now.
