The Profitable Tour Problem with Simultaneous Pickup and Delivery (PTPSPD) is a practical and challenging variant of the Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD), in which not all customers need to be served. This reflects real-world scenarios where fleet size is often limited and incurs significant costs. This study addresses the PTPSPD under realistic constraints and proposes an effective iterated local search algorithm that integrates complementary components to balance exploration and exploitation. A cluster-first, route-second heuristic is employed to generate high-quality initial solutions. The search process is intensified via a randomized variable neighborhood descent strategy. To enhance global exploration, a combination of random perturbation mechanisms and adaptive threshold acceptance criteria is employed. A dynamic tabu list further promotes diversification and prevents premature convergence. The proposed method is evaluated on a benchmark set of 117 PTPSPD instances involving 50 to 199 customers. Results highlight the effectiveness and robustness of the approach, establishing new best-known lower bounds for 64 instances. Additionally, the algorithm is tested on classical VRPSPD instances from the literature, further confirming its ability to consistently provide high-quality solutions within reasonable computational time.

In this paper, a population-based method, along with various local operators, is proposed to tackle the set-union knapsack problem. The devised approach integrates diverse features to form a customized cooperative method. It includes a greedy procedure for generating an initial set of solutions, an optimized greedy repair and optimization operator to handle infeasibility, a tailored local operator for exploring the search space, and the integration of a multi-neighborhood tabu search operator to enhance the global best solution. These operators are integrated into an iterative search based on the whale optimization procedure, maintaining a focus on solution quality throughout the process. Finally, an experimental study is presented to evaluate the proposed method’s performance on benchmark instances extracted from the literature. The provided results are compared with those achieved by more recent methods available in the literature, highlighting the efficiency of the method and revealing several new solutions.

This study introduces and discusses a novel hybrid discrete particle swarm optimization method tailored specifically for addressing the generalized max-mean dispersion problem. The method begins with a random initial population, which is then refined using a constructive operator. This operator iteratively selects elements based on their density, measured as the ratio of contribution to weight, thereby providing high-quality solutions. Subsequently, an adaptation of swarm optimization is employed within variable neighborhood search to perform local optimization. Within such a framework, the variable neighborhood descent method incorporates diverse neighborhood operators, including one-flip and two-flip operations, along with tabu search strategies. Moreover, to counter premature convergence, a drop and rebuild shaking strategy is integrated to explore unvisited subspaces. Empirical evidence illustrates that the designed method not only establishes new lower bounds but also matches the best-known results for other instances within reasonable time limits, highlighting its strong performance and robustness.