Optimization Models for the Vehicle Routing Problem under Disruptions

Round 1

Reviewer 1 Report

The authors study a vehicle routing problem in which part of the route can be disrupted, which is relevant in case of war, conflicts or natural disasters. They prove some interesting properties of the problem, formulate it as a mathematical program and conclude that, in case the network is large, the program cannot be solved in a reasonable time. Therefore they use of the option of heuristics. They test the heuristics and conclude that the optimization gap is expected to be low.

The document is written very clearly and the mathematics are sound. I see no fundamental objection to publishing the paper as it is.

A small suggestion could be to add a few more references from the literature on VRP, where uncertainties are involved. The authors mention a few articles in which probabilities are used to express the uncertainty (probably because they also use probabilities in their model). But fuzzy numbers have also been used in literature to express uncertainty in the VRP. There also exists a number of articles on robust optimization of the VRP to handle the uncerainties. A few more references on the fuzzy and robust matters could be useful tot he reader.

Small remarks:

p. 19, section 5, line 4: 'Propoerty 3 and 4' replace by 'Properties 3 and 4'

p. 19, section Cost minimization, line 2: because, in many cases, (add comma behind 'because')

p. 21, line 13: 'pieces' replace by 'parts'

p. 21, line 35: solution, which maximizes our objective function, (add two comma's)

p. 23, section 6, line 1: drop '.' (dot) behind '1)'

same line, replace 'We' by 'we'

p. 14, line 14: replace 'amount' by 'number'

p. 31, line 13: replace 'perform' by 'performs'

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The paper concerns the vehicle routing problem with a non-zero probability of the vehicle losing. The authors consider two optimization criteria: to minimize cost and to maximize demand fulfillment. The problem is relevant since there are many cases when the route passability cannot be guaranteed.

The authors propose two heuristic algorithms and assess their effectiveness. The results of the numerical experiment are presented. 

The paper is well-organized and well-written. I suppose it can be accepted. 

 I have just some minor comments and questions.

- The authors should avoid the term “less optimal” since it looks strange. What does it mean? Optimal, not optimal, close to be optimal?

- For the small network it would be helpful and more illustrative to show a plot with the optimal routes for the both criteria and compare them.

- If we set the disruption probability equal to zero, will we get the SDVRP? Are the algorithms proposed appropriate in this case?

- Is it possible to consider the problem with the linear combination of the criteria?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

In this paper, the authors study the role of disruptions in the Multi-Period Vehicle Routing Problem (VRPMD). The main objective is to minimize the total travel cost or to maximize the demand fulfillment, depending on the supply quantity. VRPMD does not deal with disruptions in real time and is more focused on the long-term performance of a single routing plan. The authors first prove that the proposed VRPMD problems are NP-hard. Then they show some analytical properties about the optimal solutions to these problems. Finally, the authors present efficient heuristic algorithms to solve these problems and show the effectiveness of the proposed models and algorithms through numerical studies.

The paper’s scope is within the scope of the journal, and it presents an original contribution. The abstract is sufficient to give useful information about the paper’s topic. The proposed algorithms are described and thoroughly illustrated. The paper is well-structured and written, and the text is clear and easy to read. However, there are some comments we recommend the authors to do:

Comment-1: In the introduction section or where appropriate, you may need to cite and add the following references regarding Vehicle Routing Problem (VRP) and robotic intelligent wheelchair systems.

Muñoz-Herrera, S.; Suchan, K. Local Optima Network Analysis of Multi-Attribute Vehicle Routing Problems. Mathematics 2022, 10, 4644. https://doi.org/10.3390/math10244644

Alshraideh, M.; Mahafzah, B.; Al-Sharaeh, S.; Hawamdeh, Z. A Robotic Intelligent Wheelchair System Based on Obstacle Avoidance and Navigation Functions. Journal of Experimental & Theoretical Artificial Intelligence 2015, 27, 471–482. https://doi.org/10.1080/0952813X.2014.971441

Comment-2: Also, in the introduction or where appropriate, since the routing problem is a special case of the traveling salesman problem, and since you did use the tabu search algorithm, then you must cite and add the following references in the introduction section or where appropriate:

Perera, J.; Liu, S.-H.; Mernik, M.; Črepinšek, M.; Ravber, M. A Graph Pointer Network-Based Multi-Objective Deep Reinforcement Learning Algorithm for Solving the Traveling Salesman Problem. Mathematics 2023, 11, 437. https://doi.org/10.3390/math11020437

Bany Doumi, A.; Mahafzah, B.; Hiary, H. Solving Traveling Salesman Problem Using Genetic Algorithm Based on Efficient Mutation Operator. Journal of Theoretical and Applied Information Technology 2021, 99(15), 3768–3781. http://www.jatit.org/volumes/Vol99No15/9Vol99No15.pdf

Gendreau, M.; Hertz, A.; Laporte, G. A Tabu Search Heuristic for the Vehicle Routing Problem. Management Science 1994, 40(10), 1276–1290. https://doi.org/10.1287/mnsc.40.10.1276                                                                                                                                      

Comment-3: The figures' titles must be placed under the figures, not above them.

Comment-4: The obtained results in Tables 4–7 need to be explained and justified in more detail. That is, the results must be explained from the algorithmic design point of view.

Comment-5: It is more appropriate and if possible to compare your algorithms’ results with other state-of-art algorithms for solving the vehicle routing problem.

English Language: The quality of the English language is good. The authors may need to check the whole manuscript for grammar, spelling, and formatting issues in general.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf