Robust Semi-Infinite Interval Equilibrium Problem Involving Data Uncertainty: Optimality Conditions and Duality

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the present paper, the authors studied the semi-infinite equilibrium problems which are consisted of infinite constraints. First, in order to deal with imprecision and uncertainty the authors studied the conditions of optimization about the presented robust semi-infinite equilibrium problems. The authors also analyzed in detail the previous mentioned conditions for the mathematical programming problem (RSIPU). Then, they presented the weak and strong duality theorems, which are Theorems 4 and 5. Finally, the authors provided two examples.

The paper provides some interesting results. I recommend to be accepted after minor revisions.

 

In the introduction section, the authors stated that the results in [1-7] could be considered as particular cases of their provided results. I recommend the authors to be more specific in the presentation of the relationship of their findings with the existing results of the references.

References

[1] Tung, L.T. Karush-Kuhn-Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions, Journal of Applied Mathematics and Computing 2020, 62, 67–91. DOI: 10.1007/s12190-019-01274-x

[2] Wei, Z.F., Gong, X.H. Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems. Journal of Inequalities and Applications, 2010, Article ID 842715. DOI:10.1155/2010/842715.

[3] Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A. et al. Semi-infinite interval equilibrium problems: optimality conditions and existence results. Comp. Appl. Math. 2023, 42, 248. https://doi.org/10.1007/s40314-023-02378-8.

[4] Tripathi, I.P., Arora, M.A. Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty. J. Appl. Math. Comput. 2024, 70, 2641–2664. https://doi.org/10.1007/s12190-024-02067-7

[5] Jayswal, A., Ahmad, I. and Banerjee, J. Nonsmooth Interval-Valued Optimization and Saddle-Point Optimality Criteria. Bull. Malays. Math. Sci. Soc. 2016, 39, 1391–1411. https://doi.org/10.1007/s40840-015-0237-7.

[6] Ahmad, I., Kaur, A. and Sharma, M. Robust optimality conditions and duality in semi-infinite multiobjective programming. Acta Mathematica Universitatis Comenianae, 2022, 91 (1), 87-99.

[7] Jaichander, R.R., Ahmad, I. and Kummari, K. Robust semi-infinite interval-valued optimization problem with uncertain inequality constraints. Korean J. Math. Vol. 2022, 30 No. 3, pp.475-489. DOI: 10.11568/kjm.2022.30.3.475.

 

Strengths of the manuscript: The paper provides a generalization of other scholars work. (optimality necessary and sufficient conditions for the robust semi infinite interval equilibrium problem, weak and strong duality,)

 

Weaknesses of the manuscript:  The presentation of the relationship of their findings with the existing results of the references does not exist.

Author Response

Response to Reviewer 1 Comments
1. Summary
Thank you very much for taking the time to review this manuscript and recommend its acceptance after minor revisions.
2. Response to Comments and Suggestions for Authors
In the present paper, the authors studied the semi-infinite equilibrium problems which are consisted of infinite constraints. First, in order to deal with imprecision and uncertainty the authors studied the conditions of optimization about the presented robust semi-infinite equilibrium problems. The authors also analyzed in detail the previous mentioned conditions for the mathematical programming problem (RSIPU). Then, they presented the weak and strong duality theorems, which are Theorems 4 and 5. Finally, the authors provided two examples. The paper provides some interesting results. I recommend to be accepted after minor revisions.   In the introduction section, the authors stated that the results in [1-7] could be considered as particular cases of their provided results. I recommend the authors to be more specific in the presentation of the relationship of their findings with the existing results of the references. References [1] Tung, L.T. Karush-Kuhn-Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions, Journal of Applied Mathematics and Computing 2020, 62, 67–91. DOI: 10.1007/s12190-019-01274-x [2] Wei, Z.F., Gong, X.H. Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems. Journal of Inequalities and Applications, 2010, Article ID 842715. DOI:10.1155/2010/842715. [3] Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A. et al. Semi-infinite interval equilibrium problems: optimality conditions and existence results. Comp. Appl. Math. 2023, 42, 248. https://doi.org/10.1007/s40314-023-02378-8. [4] Tripathi, I.P., Arora, M.A. Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty. J. Appl. Math. Comput. 2024, 70, 2641–2664. https://doi.org/10.1007/s12190-024-02067-7 [5] Jayswal, A., Ahmad, I. and Banerjee, J. Nonsmooth Interval-Valued Optimization and Saddle-Point Optimality Criteria. Bull. Malays. Math. Sci. Soc. 2016, 39, 1391–1411. https://doi.org/10.1007/s40840-015-0237-7. [6] Ahmad, I., Kaur, A. and Sharma, M. Robust optimality conditions and duality in semi-infinite multiobjective programming. Acta Mathematica Universitatis Comenianae, 2022, 91 (1), 87-99. [7] Jaichander, R.R., Ahmad, I. and Kummari, K. Robust semi-infinite interval-valued optimization problem with uncertain inequality constraints. Korean J. Math. Vol. 2022, 30 No. 3, pp.475-489. DOI: 10.11568/kjm.2022.30.3.475.   Strengths of the manuscript: The paper provides a generalization of other scholars work. (optimality necessary and sufficient conditions for the robust semi infinite interval equilibrium problem, weak and strong duality,)   Weaknesses of the manuscript:  The presentation of the relationship of their findings with the existing results of the references does not exist.

 

Response: We have modified the introduction showing the progress made with respect to the references cited. We have added sentences on page 4, lines 66, 67, 73, 74, 75, 84, 85, in addition to remark 3, page 10, remark 5, page 11, remark 6 and 7, page 13. In summary, we have made all the suggested changes.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The similarty measure is very high apprx 60%, so please look this matter to reduce it.  The paper mentions modeling uncertainty in both the objective function and the constraints for a "robust semi-infinite interval equilibrium problem." However, the introduction does not clearly define what this problem entails or why this approach is necessary. A more detailed explanation of the problem's significance and real-world applicability would improve clarity.  The paper introduces several concepts (robust semi-infinite problems, duality, interval-valued functions) without sufficient context or background. Reorganizing the introduction to include a brief review of relevant literature and key definitions could make the paper more accessible to readers not deeply familiar with this field.  The contribution lacks sufficient scientific soundness and is quite limited. The paper does not appear to discuss any limitations or challenges associated with the proposed model. For a balanced and comprehensive study, acknowledging potential drawbacks or constraints in using the dual robust version of the problem would be valuable. 

Author Response

For research article

 

 

Response to Reviewer 2 Comments
1. Summary
Thank you very much for taking the time to review this manuscript.
2. Comments and Suggestions for Authors
Comments : The similarty measure is very high apprx 60%, so please look this matter to reduce it.  The paper mentions modeling uncertainty in both the objective function and the constraints for a "robust semi-infinite interval equilibrium problem." However, the introduction does not clearly define what this problem entails or why this approach is necessary. A more detailed explanation of the problem's significance and real-world applicability would improve clarity.  The paper introduces several concepts (robust semi-infinite problems, duality, interval-valued functions) without sufficient context or background. Reorganizing the introduction to include a brief review of relevant literature and key definitions could make the paper more accessible to readers not deeply familiar with this field.  The contribution lacks sufficient scientific soundness and is quite limited. The paper does not appear to discuss any limitations or challenges associated with the proposed model. For a balanced and comprehensive study, acknowledging potential drawbacks or constraints in using the dual robust version of the problem would be valuable.

 

Response:

• We have paraphrased throughout the article, mainly in the introduction, since it is difficult elsewhere to reduce the similarity.
• We have clarified the significance of this problem, see page 4, lines 98-101 of the Introduction.
• We have added the context of interval analysis contexts, see page 5, lines 136-137, which complete the 4-page Introduction.
• We have shown the limitations of the dual version, see page 13, Remark 8, lines 295-298.

 

In summary, we have made all the suggested changes.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Report on the paper

Robust semi-infinite interval equilibrium problem involving data uncertainty: Optimality conditions and duality by

Gabriel Ruiz-Garzón  , Rafaela Osuna-Gómez  , Antonio Rufián-Lizana  and Antonio Beato-Moreno

 

In this paper, the authors model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. They particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending results from the literature. They introduce the dual robust version of the above problem, prove the Mond-Weir type weak and strong duality theorems, and illustrate with an example.

 

The paper is well organized and the results presented are correct and interesting.Therefore, I recommend the paper for publication in Axioms.

Following are my comments:

1. R17: instead ''where f'' write ''and f''

2. Pag. 1 row 6 below:   ''<   >'' is not defined

3. R20 is unclear

4. Pag. 2 row 2 below: delete ''is said''

5. R52: instead'', Ghosh and Treanta'' write ''et al.''

6. R59: delete ''Ramon'' and ''Weldon''

7. R62: delete ''In 2001,'' . In general to give up such expressions.

8. R63: instead ''canelo'' write ''canello''

9. R68: delete ''in 2007''

10. Rs.76-77: instead ''In 2016, Jayswal, Ahmad, and Banerjee [29 ]'' write ''Jayswal et al. [29]''

11. R80: instead ''Jaichander, Ahmad, and Kummari '' write ''Jaichander et al.''

12. R149: incomplete sentence

13. R152: instead ''definition'' write ''Definition''

14. R173: instead ''then'' write ''such that''

15. R190: instead ''that'' write ''the''

16. R191 and R208: instead ''hald then'' write ''hold. Then''

17. R209: instead ''and'' write ''and using''

18. R257: instead ''then'' write ''. Then''

19. R272: instead ''This weak duality theorem (4)'' write ''Weak duality theorem [4]'' Idem R287

20. R273: instead ''proposition'' write ''Proposition'' Idem R288

21. Refs. 8, 27, 29: Write the beginning of the words in lowercase letters in the title of the article

22. Add to the References the following:

a) Jayswal, Anurag ; Stancu-Minasian, I. M. ; Jonaki Banerjee and Andreea Mădălina Stancu : Sufficiency and duality for optimization problems involving interval-valued invex function in parametric form. Operational Research-An International Journal (ORIJ) 15(2015)1, 137-161.

 

Author Response

For research article

 

 

Response to Reviewer 3 Comments
1. Summary
Thank you very much for taking the time to review this manuscript and recommend its acceptance for publication in Axioms.
2. Response to Comments and Suggestions for Authors
Comments: In this paper, the authors model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. They particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending results from the literature. They introduce the dual robust version of the above problem, prove the Mond-Weir type weak and strong duality theorems, and illustrate with an example. The paper is well organized and the results presented are correct and interesting.Therefore, I recommend the paper for publication in Axioms. Following are my comments: 1. R17: instead ''where f'' write ''and f'' 2. Pag. 1 row 6 below:   ''<   >'' is not defined 3. R20 is unclear 4. Pag. 2 row 2 below: delete ''is said'' 5. R52: instead'', Ghosh and Treanta'' write ''et al.'' 6. R59: delete ''Ramon'' and ''Weldon'' 7. R62: delete ''In 2001,'' . In general to give up such expressions. 8. R63: instead ''canelo'' write ''canello'' 9. R68: delete ''in 2007'' 10. Rs.76-77: instead ''In 2016, Jayswal, Ahmad, and Banerjee [29 ]'' write ''Jayswal et al. [29]'' 11. R80: instead ''Jaichander, Ahmad, and Kummari '' write ''Jaichander et al.'' 12. R149: incomplete sentence 13. R152: instead ''definition'' write ''Definition'' 14. R173: instead ''then'' write ''such that'' 15. R190: instead ''that'' write ''the'' 16. R191 and R208: instead ''hald then'' write ''hold. Then'' 17. R209: instead ''and'' write ''and using'' 18. R257: instead ''then'' write ''. Then'' 19. R272: instead ''This weak duality theorem (4)'' write ''Weak duality theorem [4]'' Idem R287 20. R273: instead ''proposition'' write ''Proposition'' Idem R288 21. Refs. 8, 27, 29: Write the beginning of the words in lowercase letters in the title of the article 22. Add to the References the following: a) Jayswal, Anurag ; Stancu-Minasian, I. M. ; Jonaki Banerjee and Andreea Mădălina Stancu : Sufficiency and duality for optimization problems involving interval-valued invex function in parametric form. Operational Research-An International Journal (ORIJ) 15(2015)1, 137-161.
Response: We've made all 22 minor changes received. In particular, we have added the suggested reference, which in the new version of the manuscript is reference 21 and page 4, line 55. To sum up, all the changes have been made.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The work has a high level of plagiarism and needs to be completely rewritten. Once that’s done, I’ll provide my feedback.

Comments on the Quality of English Language

English needs to improve.

Author Response

Response to Reviewer 2 Comments
1. Summary
Thank you very much for taking the time to review this manuscript.
2. Comments and Suggestions for Authors
Comments : The work has a high level of plagiarism and needs to be completely rewritten. Once that’s done, I’ll provide my feedback.

 

Response:

Dear Professor,

We have redoubled our efforts and have proceeded to revise the entire manuscript for the second time. From the submitted report, the most cited source is the repository of our university, where it is mandatory to deposit our articles, Rodin. This source contributes 19\% of the total, the other sources have percentages of 1\%.  In addition, many expressions are part of the usual mathematical vocabulary. However, we have worked hard to reduce the level of plagiarism without losing the accuracy and clarity of the work.

We would like to inform you that all the suggestions suggestions made by the referee have been addressed and solved.

 Therefore, I am resubmitting the manuscript to you in the hope that it can be published quickly.

 

Author Response File: Author Response.docx

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Author address all the comments.