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You have solved the issues raised by the reviewers.
No problems observed here.
No problems observed here.
No problems observed here.
The author has successfully answered and explained all issues I commented in the initial review. After considering these comments, and the others reviewers comments, this manuscript is in my opinion valid for publication.
You should restructure and extend your manuscript regarding to the reviews.
No comment
No comment
Validity of the findings seems right.
In the future work:
Authors should choose the data set that has the following properties:
1- The benchmark problems must be varied according to size
2- The benchmark problems must be varied according to the sparsity because most of the used problems are dense problems.
3- The data set of the benchmark problems must has a large number of problems.
On the other hand, the CPU time of the proposed algorithm must be manipulates.
Line 128: The meaning of the sentence “…should the initial direction be feasible, can be applied directly on the original problem, without having to first compute an initial feasible basic solution thus completely avoiding Phase I.” is not clear to me. I do not understand what the meaning of feasible direction is. The condition of feasibility applies to points, not directions. Perhaps this sentence should be rewritten.
More on the paragraphs from line 124 to line 150: the intention of these paragraphs is valid. Here the authors establish a quick comparison among the EPSA, the iEPSA and the PDIPSA. The reading is attractive and invites for details. But the writing is not clear. I think these three paragraphs are crucial to begin the explanations of the proposed algorithm. In line 127, for example, the word “another“ suggests a previous advantage was mentioned. It was not. The sentence starting in line 137” The iEPSA is initialized with an infeasible basic solution and with an interior point” seems to be a contradiction as it is written. Or maybe I do not understand the exact meaning the authors want to convey. If there are two point to initialize this algorithm it should be clearly stated. If there is an error in the sentence it should be corrected.
The terms αα and ββ are extensively used, but nowhere is indicates their meaning.
The authors try to explain the method using an example as if an example guaranteed the functioning of a mathematical procedure. They even inappropriately use the term “demonstrate”. This would not be a major problem if changing some words could fix it. But the explanation using this example is poorly organized. I just cannot follow it. Much less understand the method and its rationale. It may even have serious algebraic errors! If you compute A_B ∙ 〖A_B〗^(-1) using the values presented in the example, it does not yield the Identity matrix I. There are things here that not corresponding to the textual explanation. I suggest explaining the method in a logical and general manner, and then, illustrating it with an example. Moreover, why using eight constraints? Is not it unnecessarily “large” example?
In line 165 you say: “The N set of indexes actually represents the current basic solution.”. But previously you used the letter N precisely to refer to NON-basic solutions. This is impossible to understand.
There seems to be an interesting idea about exploiting the behavior of linear programming in the feasible and unfeasible spaces, to accelerate the convergence of the method towards the optimal solution. However, the explanation offered in this manuscript lacks of the organization needed to evaluate the plausibility of the method. In the conclusions the authors signal as the advantage of the method the reduction of the number of iterations of their method, not mentioning the fact that, according to the algorithm presented, they need to compute the insertion of matrix A_B at every iteration. Thus, it is difficult to believe the offered method actually represents important advantages.
This is not to disqualify the study. I insist, the idea of combining interior and exterior point methods could be promising. But it has to be better organized, evaluated and explained.
As authors state, the proposed algorithm is comprised of three different parts : i) interior Exterior Primal Simplex Algorithm (iEPSA), ii) Exterior Point Simplex Algorithm (EPSA) and iii) Primal-Dual Interior Point Simplex Algorithm (PDIPSA). The diagram of the proposed algoriithm (iEPSA) is provided in Figure 1 ad Algorithm 1. This algorithm applies EPSA after finding a primal feasible basic solution and applies PDIPSA when moves to a dual feasible partition.
The language is clear and unambiguous, but there are some partial sentences such as ; at line 246 where the sentence lacks some reference to a section. Authors provide sufficient lemmas/theorems. However, numbering of theorems and lemmas should be separated from each other.
The structure of the paper should be improved. It is hard to follow the paper;
The diagram of the iEPSA is given on page 8 and afterwards an example is provided to explain the inner wokings of the algorithm, which is nicely presented. However, the formal definition of the algorithm is given on page 15. This makes it very difficult to follow the algorithm. The diagram in figure 1 and the algorithm 1 should be given close to each other. Also, before moving to the example, the authors should explain the formal definition of the algorithm.
The authors sate that the proposed approach aims to reduce the number of iterations. They also compare it with CPLEX and Gurobi in terms of number of iterations and CPU time. They benchmark it on netlib, Kennington, Mesz´aros.
Hwever, the proposed algorithm is implemented in Matlab. Others are implemented in C. This makes the CPU time comparison not very meaningful.
- Since the proposed algorithm is implemented ina different language, then authors should comment on the iteration times of CPLEX, Gurobi and the proposed algorithms. Since, it is not possible to comment on the superiority of the algorithm in terms of the runtime.
- After reporting the number of iterations and CPU time, they only provide the percentage of the experiments the algorithm is better in terms of number of iterations. Some more deeper analysis should be provided such as; the effect of the size or the sparsity of the problem.
The structure of the paper should be improved in order to follow work. Also the algorithm should be explained line by line if possible.
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