Tri-objective Optimization Models For Solving Capacitated Walking School Bus And Bike Train Problems:Oriented Next-generation School Transportation

Unregulated and unsupervised walking（normal walking stride）and bicycling convention-ally to and from school are experiencing increased child pedestrian casualties,unusual delays,supervision,and travel time challenges.These related problems have partly led to a signifi-cant plummet in the utilization and sustainability of bicycle and walking mobility for school transportation,as well as decreased active lifestyles among schoolchildren.Next-generation walking school buses（WSB）and cycling trains（BT）are environmentally-friendly school transport modes and have significant potential to address most of the issues as-sociated with child pedestrian casualties,non-sustainable car-based transport,and travel time constraints.However,in-depth models and algorithms to successfully implement such sustain-able transport and stand-alone initiatives are scarcely limited.This study,therefore,introduced new state-of-the-art algorithms to minimize total route costs,total walking/cycling risk,and adult supervisors for all children en route to/from school.Our main objectives are the following:（1）minimization of the number of accompany-ing persons;（2）minimization of total route risk,and（3）minimization of total travel route cost.These overarching modeling objectives are more conflictive.Meaning multiobjective mathematical optimization models are suitable for addressing such a tri-objective problem na-ture.Therefore,tri-objective optimization techniques were successfully deployed in Python and Julia to solve these capacitated WSB and BT routing problems with time windows（i.e.,CWSBRPTW and CBTRPTW）.Exact methods and approximation algorithms were mathematically formulated to address CWSBRPTW and CBTRPTW for next-generation school transportation.Four nature-inspired and evolutionary algorithms were introduced to solve the above-mentioned problems.These innovative metaheuristics include the Multiobjective Max-Min Ant System（M3AS）,Non-dominated Sorting Genetic Algorithms（NSGA-II）,Strength Pareto Evolutionary Algorithms（SPEA-II）,and Ant Colony Optimization-Multi-Agent System（ACO-MAS）.In the exact methods,four approaches labeled₀（base case）without walkability and risk factors and₂,₂,and₃that incorporated randomly generated neighborhood walkability scores and risk factors were introduced to solve the CWSBRPTW problem.In addition,dis-cretized walking and cycling distances were created through a uniform distribution function to measure how distance thresholds may affect both WSB and BT utilization.A novel model with three Scenarios was proposed to solve the CBTRPTW problem.These scenarios constructed during the BT optimization were utilized to evaluate the solutions’qual-ity.Scenario one（₁）was formulated as an easy version to minimize the total route cost and the starting service time of each itinerary while satisfying all underlying constraints.In Scenario Two（₂）,the problem was made more complicated by applying the model to sparsely popu-lated neighborhoods where bike trains must deal with high distance variations in low/sparsed demand communities.We permitted the violation of time windows by introducing a penalty coefficient in Scenario Three（₃）.Our mathematically formulated CBTRPTW main model and submodels were tested on 15newly created realistic-set problems（created using the US and Canada WSB datasets）,while56 modified Solomon’s vehicle routing problem with time windows（VRPTW）benchmark instances were used to test the performance of the novel CWSBRPTW models.Scenario analyses and internal solution benchmarking were performed amongst the exact solution methods.Again,using statistical tools in Minitab 19,we analyzed whether the optimal solutions（i.e.,mean values）differ significantly through one-way ANOVA and Tukey and Fisher pairwise comparisons.Moreover,comparative analyses between generated solutions from the intractable exact methods and four metaheuristic algorithms were done.Numerical experiments were also carried out to identify the best-performing metaheuristic algorithm through the generated Pareto solutions.Solomon’s VRPTW problem sets c102 and rc101 were employed here.Out of the 56 Solomon’s problem sets,our experimental results from the proposed CWS-BRPTW models indicate that these exact techniques can completely solve nine benchmark instances.Among the three newly proposed CWSBRPTW solution approaches,₁outper-formed other sub-models（₂and₃）.Scenario three was the best performer in the CB-TRPTW modeling with assumptions that allow arriving outside of time windows but with penalty payment.It also performed better on larger instances but worst on smaller instances.Based on Pareto optimization,the trade-off solutions show that the hybrid ACO-MAS outperformed the other three algorithms.Also,solutions obtained from exact and approximate methods compare favorably with exact algorithms as best performers in all considered cases.Therefore,an effective BT size of 10 bikers-to-a-leader and an optimal WSB size of 12children per adult leader are suggested for these stand-alone school travel modes.These estab-lished thresholds have relevancy for participants’safety,overall cost-effectiveness,and opera-tionalization of schemes.Conclusively,simulation results from the introduced algorithms have exhibited a high proven-capability to solve the proposed CWSBRPTW and CBTRPTW problems both exactly and approximately.Our study also contributed immensely to extending and improving the existing few WSB and BT models to include time windows for their development and deploy-ment as school transportation options in building sustainable and livable societies（walkable and cyclable）.It also adds to the growing efforts to improve children’s school travel safety and augments growing calls to adopt active lifestyle behavior changes.Consequently,useful policy recommendations are outlined to design,implement and sus-tain the two important mobility schemes at the local,national,and international levels.Finally,plausible research directions are suggested to drive various implementation phases of such fun and non-infrastructure alternatives to normal walking and cycling to school.