This work involves a heuristic for solving vehicle routing problems with time windows (VRPTW) with general compatibility-matching between customer/patient and server/caretaker constraints to capture the nature of systems such as caretakers’ home visiting systems or home healthcare (HHC) systems. Since any variation of VRPTW is more complicated than regular VRP, a specific, custom-made heuristic is needed to solve the problem. The heuristic proposed in this work is an efficient hybrid of a novice Local Search (LS), Ruin and Recreate procedure (R&R) and Particle Swarm Optimization (PSO). The proposed LS acts as the initial solution finder as well as the engine for finding a feasible/local optimum. While PSO helps in moving from current best solution to the next best solution, the R&R part allows the solution to be over-optimized and LS moves the solution back on the feasible side. To test our heuristic, we solved 56 benchmark instances of 25, 50, and 100 customers and found that our heuristics can find 52, 21, and 18 optimal cases, respectively. To further investigate the proficiency of our heuristic, we modified the benchmark instances to include compatibility constraints. The results show that our heuristic can reach the optimal solutions in 5 out of 56 instances.

We propose a vehicle routing problem with time windows (VRPTW) with compatibility-matching constraints and total completion time as the objective function, with applications in home healthcare routing and scheduling. Mixed integer linear programming is provided with total completion time minimization as the objective function. The solution approach has two objectives, total completion time (primary objective) and total distance (secondary objective). A heuristic is proposed comprising three phases: initializing to find an initial feasible routing (inserting the procedure with a modified K-means algorithm), swapping and moving the procedure to find a local optimal routing, and shooting the procedure to move away from the local optimum. Proof of feasibility for the inserting procedure is provided to prevent unnecessary insertions. Phases 2 and 3 will be repeated as needed to ensure solution quality. Solving our model with the proposed heuristic algorithm increases the total distance by 90.00% but reduces the total completion time by 25.86%. To test our model and heuristic, we examined a system with 400 home-healthcare cases in Chiang Mai. The heuristic quickly solved the problem. When total completion time is minimized, some caretakers serve up to twice as many patients as their coworkers; when total distance is minimized, workload discrepancies can increase up to seven-fold.

This study presents an iterative method for approximating common fixed points of a finite set of G-nonexpansive mappings within a real Hilbert space with a directed graph. We establish definitions for left and right coordinate convexity and demonstrate both weak and strong convergence results based on reasonable assumptions. Furthermore, our algorithm’s effectiveness in solving the heat equation is highlighted, contributing to energy optimization and sustainable development.

Cost reduction and staff retention are important optimization objectives in home healthcare (HHC) systems. Home healthcare operators need to balance their objectives by optimizing resource use, service delivery and profits. Minimizing total travel distances to control costs is a common routing problem objective while minimizing total finishing time differences is a scheduling objective whose purpose is to enhance staff satisfaction. To optimize routing and scheduling, we propose mixed integer linear programming with a bi-objective function, which is a subset of the vehicle routing problem with time windows (VRPTWs). VRPTWs is a known NP-hard problem, and optimal solutions are very hard to obtain in practice. Metaheuristics offer an alternative solution to this type of problem. Our metaheuristic uses the simulated annealing algorithm and weighted sum approach to convert the problems to single-objective problems and is equipped with operators including swapping, moving, path exchange and ruin and recreate. The results show, firstly, that the algorithm can effectively find the Pareto front, and secondly, that minimizing total finishing time differences to balance the number of jobs per caretaker is an efficient way to tackle HHC scheduling. A statistical test shows that the algorithm can obtain the Pareto front with a lower number of weighted sum problems.

This article introduces an iterative algorithm that is created by integrating the $S^{*}$-iteration process with the inertial forward-backward algorithm. The algorithm is designed to solve optimization problems formulated as variational inclusions in a real Hilbert space. We establish the weak convergence of the algorithm under conventional assumptions. One of the applications of the algorithm is to solve the extreme learning machine, which can be transformed into the variational inclusion problem. Different algorithms, with all parameters set to be identical, are employed to solve the stroke classification problem in order to evaluate the algorithm’s performance. The results indicate that our algorithm converges faster than others and achieves a precision of 93.90%, a recall of 100%, and an F1-score of 96.58%.

Temples are tourist attractions that represent the history and culture of Thailand, especially in Chiang Mai province—a city with a rich history that has become a prominent destination attracting visitors from around the world. Many temples remain undiscovered yet are ready for tourists to visit; however, due to unfamiliarity, tourists tend to visit only the well-known temples, as other visitors do, missing great opportunities to engage with new cultural heritage tourism experiences. To address this issue, we propose a Hybrid Three-Stage Route Planning Recommendation (HTS-RPR), a novel method for tourist route planning that delivers recommended routes based on tourists’ preferred constraints. This model contains three-stage route recommendations providing an optimal single-day route with mandatory and recommended points of interest (POIs) through a metaheuristic integrating Mixed Integer Programming (MIP), heuristic-based POI recommendation filtering, and Genetic Algorithm route optimization with Bayesian reward and peak-time awareness, ensuring that users can effectively travel cultural routes with high popularity and satisfaction while avoiding attractions during periods of high traffic. To validate the efficacy of the proposed model, experiments with three baseline methods were conducted. The results demonstrate that HTS-RPR achieves the best fitness score in 55 out of 60 scenarios and the best reward in 54 out of 60 scenarios, with a median fitness score 28.34% and 103.67% higher than the Genetic Algorithm and Multi-Start Simulated Annealing baselines, respectively, and a median total reward exceeding all three baselines by up to 40.74%. Although HTS-RPR’s median execution time is approximately 2.6 times that of the Genetic Algorithm, it remains 84.5% faster than the Multi-Start Simulated Annealing baseline, offering a favorable trade-off between solution quality and computational cost. Moreover, the framework’s pluggable reward function enables destination managers to configure recommendation priorities, including the promotion of undiscovered tourist attractions, while the peak-time-aware optimization mitigates congestion at specific POIs.