Scheduling Disjoint Setups in a Single-Server Permutation Flow Shop Manufacturing Process

Round 1

Reviewer 1 Report

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

We would like to thank the reviewer for all their valuable remarks. Below are our responses.

 

Remark 1: Introduction section: The authors could use more reference citations to support their arguments, especially in the first four paragraphs.

Response: Thank you for the suggestion. We added some references to surveys and other papers supporting the claims. Unfortunately, due to confidentiality, we cannot elaborate on the company mentioned in the fourth paragraph. 

 

Remark 2: Some figures, tables, and algorithms appear before being mentioned in the text.

Response: We have re-checked the new version of the paper and ensured all Figures, Tables, and Algorithms appear at the latest on the same page as their first reference.

 

Remark 3: Although the manuscript is well-written, there are a few grammar errors to correct (e.g. sometimes the authors use “then” instead of “than”). An accurate language review should be performed.

Response: Thank you, we have re-checked the paper and fixed several errors.

 

Remark 4: Figure 4: When this figure is printed in grayscale, it is not possible to differentiate some curves.  The authors could use more dotted or dashed lines.

Response: The figure was modified to be more informative in grayscale.

 

Remark 5: Section 5.1: An illustrative example could be presented.

Response: As suggested, we have added an illustrative example. 

 

Remark 6: Section 6.3: The authors evaluate the performance of the algorithms for different running times. Does the initial solution have more influence on the results obtained for smaller running times? I discussionon this  topic would be interesting.

Response: A brief discussion on this topic can be found in Sec. 5.3. The initial experiments indicated an insignificant impact of the initialization procedure on the Tabu Search algorithm performance. Such behavior can be extrapolated from convergence plots in Fig. 8, where rapid improvements can be observed before 1 second of computations (iterations are fast). Therefore, since the shortest experiments were conducted for 1 second, the tests for different initialization procedures were not reported. Such experiments could be more interesting for very large instances or extremely low computation budgets.

 

Remark 7: Abbreviation list: please replace “SDTS” with “SDST”.

Response: Thank you, the mistake was fixed.

 

Remark 8: About 2/3 of the references are older than 5 years. It is not necessarily a problem, but the authors should evaluate the possibility of including more recent research articles in the reference section.

Response: Thank you for the suggestion. Indeed, most of the references are relatively old. It is a direct result of the family of problems considered having been researched for decades. However, the single-server variants of FSSP are not that well-known, so modern literature is sparse. Nevertheless, we added some newer references for related problems, such as open shop with single server (to build a context for the constraint) or reviews on FSSP (to solidify claims in the introduction, addressing Remark 1).

Reviewer 2 Report

The manuscript considers a single server flow shop scheduling problem which involves finding an optimal order of setups to be performed on different machines given the order of jobs are the fixed. The authors have developed theorems for two problem formulation properties, one for elimination of infeasible solution and other for refinement of solutions. These two properties were then included in the solution strategies of two tabu search algorithm implementations which were then compared with a corresponding MILP formulation and solution. The manuscript can be accepted after the following comments are adequately addressed.

Comments:

1) Typo: "is" should be "in" on line 53; "constraint" instead of "constrain" on line 255

2) The authors should mention references by name in addition to by numbers while mentioning them in the text

3) The main objectives of the paper section can be moved to the end of section 2 (just above section 3). In addition, along with the objectives, the novelty of the work compared to those in the literature should be specified as well

4) The lattice path representation in section 4.1 needs to be explained clearly with more examples. The connection/equivalence/translation between the lattice representation and the tau, sigma representation should be shown

5) Is the ordering of the points in the set important in eqn. 12?

6) The authors keep switching from one problem notation representation to the other. The authors should explain the notations clearly with more context. E.g., the notation for x' in the algorithm needs to be explained clearly

7) The authors should explain how eqn. 17 is arrived at.

8) The definition of blocks arrived at in lines 351-353 is unclear. The authors should provide an illustrating example to show that 7 blocks exist.

9) What is the definition of |sigma*| in eqn. 13?

10) Why have the authors not tried another Tabu approach say Tabu-C which uses both the elimination property and the refinement procedure together? This way, would not it be possible that the benefits of both the Tabu-A and Tabu-B approach can be realized?

Author Response

We would like to the reviewer for all of their valuable remarks. Below are our responses.

 

Remark 1: Typo: "is" should be "in" on line 53; "constraint" instead of "constrain" on line 255

Response: Thank you, the typo was fixed.

 

Remark 2: The authors should mention references by name in addition to by numbers while mentioning them in the text

Response: The references were modified to include the authors’ names.

 

Remark 3: The main objectives of the paper section can be moved to the end of section 2 (just above section 3). In addition, along with the objectives, the novelty of the work compared to those in the literature should be specified as well

Response: While we agree that the structure proposed by the Reviewer is sound, we chose a different approach that is also frequently used in the Journal (we analyzed recently published scheduling papers). The novelty of the work is summarized in the contributions and the motivation above: underresearched problem and—consequently—novel problem properties.

 

Remark 4: The lattice path representation in section 4.1 needs to be explained clearly with more examples. The connection/equivalence/translation between the lattice representation and the tau, sigma representation should be shown

Response: Example 3 was extended to show the relationship between both representations.

 

Remark 5: Is the ordering of the points in the set important in eqn. 12?

Response: It is an interesting observation. It should be first noted that lattice paths are formally sequences e.g. a lattice path ((0,0),(0,1),(1,1)) is a 3-element sequence, where each element is, in turn, a 2-element sequence. One can treat the “outer” sequence as a set to obtain {(0,0),(0,1),(1,1)}. The order of elements in such a set does not matter, as there is only one ordering that would result in a feasible lattice path. As such, any ordering can be transformed into the correct one by appropriate sorting of the elements. 

 

Remark 6: The authors keep switching from one problem notation representation to the other. The authors should explain the notations clearly with more context. E.g., the notation for x' in the algorithm needs to be explained clearly

Response: The representation of a solution is chosen based on the use case. Tau notation is used when talking about problems in a broad context, as it can express infeasible solutions or solutions of different problems. Sigma notation is preferred when discussing the problem because it naturally excludes many infeasible solutions representable by tau. Notation using x denotes nodes of lattice paths and not the solutions. x’ is an integer index representation of x. To avoid confusion with other ad-hoc variables using “ ‘ “, we changed the index representation symbol to “i” and added a reference to the definition to the algorithms where the notation is used. The index notation itself is now explained in more detail around the definition, addressing Remark 7.

 

Remark 7: The authors should explain how eqn. 17 is arrived at.

Response: The explanation was added near the equation.

 

Remark 8: The definition of blocks arrived at in lines 351-353 is unclear. The authors should provide an illustrating example to show that 7 blocks exist.

Response: The block definition was changed to a more mathematical formulation and the example explanation was extended.

 

Remark 9: What is the definition of |sigma*| in eqn. 13?

Response: We will address the issue in Eq. 31. This is the number of unique preserved solutions, calculated from Algorithm 3. For clarity, the notation in Algorithm 3 was altered to use |sigma*| and the definition is now more explicit around Equation 33 (previously Equation 31).

 

Remark 10: Why have the authors not tried another Tabu approach say Tabu-C which uses both the elimination property and the refinement procedure together? This way, would not it be possible that the benefits of both the Tabu-A and Tabu-B approach can be realized?

Response: Unfortunately, a direct merge of both approaches is not possible. The neighborhood utilizing the elimination property guarantees that only the solutions satisfying the property are traversed. Such solutions cannot be improved by the refinement procedure. For the algorithm to use both approaches it would have to incorporate multiple neighborhoods or other features, which is beyond the scope of the current paper. However, it is an interesting direction for further research.