This paper addresses a novel variant of Multi-Depot Electric Vehicle Routing Problem in which a finite number of EVs with limited capacity distribute orders to designated customers. Each EV may depart from and return to different depots, can be used multiple times, and be recharged at either recharging stations or depots. EVs operate under adjustable and discrete speed modes, where lower speeds result in lower CO_2 emissions—modeled as a function of both speed and vehicle load. Customers place their orders with specific due dates, failing to meet leads to tardiness penalty. Higher speed modes can be chosen to avoid tardiness, though at the cost of increased CO_2 emissions. Therefore, we simultaneously consider two contradicting objectives, minimizing total CO_2 emission and total tardiness penalty. Operational decisions involve determining both the customer visitation sequence and the travel speed for each segment. Since the objectives are incommensurable or prioritizing them is challenging, we introduce bi-objective optimization techniques to provide pareto (non-dominated) solutions. To this end, we formulate the problem as a bi-objective mixed-integer linear program and implement Augmented ε-Constraint (Augmecon) method. Due to problem intractability, we develop Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Improved Multi-Directional Local Search (IMDLS). Augmecon exhibits better performance versus metaheuristics in terms of Quality Metric and Hypervolume Index for test instances involving 5–7 customers, derived from recent literature. Considering all small-sized instances, IMDLS outperforms NSGA-II. In all large-sized instances, while NSGA-II shows favorable performance against its counterpart, IMDLS beats NSGA-II for instances in which customers are geographically clustered.
Paired comparison methods play a fundamental role in decision-making, preference elicitation, and data evaluation. While the concept of consistency is well established in pairwise comparison matrices-based models, the stochastic models, including Thurstone-motivated models, have remained less explored in this regard. This paper extends the notion of consistency in a groundbreaking way to two- and three-option Thurstone-motivated models, as well as Davidson’s model, establishing connections between the two-option Bradley-Terry, the three-option Davidson’s model, and pairwise comparison matrices-based models through equivalence statements. These statements are the basis for an alternative definition of consistency, valid for general two- and three-option stochastic paired comparison models. Based on this definition we also define a data inconsistency index to quantify deviations from ideal consistency because the consistent data sets are very rare. We formulated a set of axioms for data inconsistency indices and proved that they are independent and form a consistent axiomatic system. We also showed that the proposed inconsistency index satisfies all of these axioms. Based on random simulations, we illustrate its further properties. These results provide a unified framework for the assessment of consistency, paving the way for future developments of formal inconsistency indices.
This study develops a many-objective portfolio optimization framework based on complex network topology to improve portfolio construction in the presence of multiple objective conflicts. By utilizing correlations in stock returns, a financial network is constructed, from which topological features, including centrality, clustering coefficient, and community structure, are extracted. These features are incorporated into the portfolio optimization problem alongside expected return and variance, yielding a five-objective formulation. To address this high-dimensional optimization problem, this study improves upon NSGA-III. It proposes a network-aware evolutionary algorithm with an adaptive mutation mechanism that adjusts the distribution of candidate solutions based on the structural importance of nodes and intercommunity relationships. Using standard performance indicators such as hypervolume, inverted generational distance, and generational distance, along with out-of-sample performance measures, the proposed framework is compared with several representative many-objective evolutionary algorithms. The empirical results indicate that the proposed method achieves comparatively favorable convergence, solution diversity, and risk-adjusted performance.
Emergency decision-making in safety-critical situations is constrained by time pressure, incomplete information, and differences in expert judgment and social influence. These conditions make timely agreement difficult while repeated preference revision may impose a burden. This study develops a personalized consensus model based on distributed linguistic preference relations and personalized numerical scales. The model represents human factors through hesitant judgments, expert-specific numerical semantics, preference distribution uncertainty, evaluative extremity, judgment consistency, and dynamic relational influence. A three-way mechanism assigns state-dependent feedback according to each expert’s current consensus state. The three treatments are retention of the current personalized numerical scale and its induced additive preference relation, eligibility for an update to the personalized numerical scale, and withdrawal with influence delegation. The procedure gives priority to the first feedback round in which the consensus target is reached and then selects the feedback result with the smallest measure of preference revision in that round. In a computational application set in the context of a publicly reported port accident, the consensus target is reached after one feedback round. Sensitivity analysis shows that the lower individual consensus threshold affects expert participation, whereas the group consensus target affects the required preference revision and may change the leading alternative. The final result remains unchanged across the tested values of the upper individual consensus threshold. A structural comparison distinguishes the proposed feedback structure from related consensus approaches.
This work considers a production inventory system where the state equations involve two types of continuous dynamics, one for the supplier and the other for the buyer. The objective is to minimize the total cost resulting from the squares of the inventory of the supplier, the buyer and the production rate. As this is a multi-dimensional system, we express the model in the form of a linear quadratic regulator in matrix form, which ensures closed-form solutions to the problem. By applying Pontryagin’s maximum principle, analytical solutions are obtained and expressed as the solution of a Hamiltonian Jacobi matrix differential equation with time-varying coefficients. The final time is taken as finite, and the corresponding inventory level is taken as free. Under these assumptions, a change of the co-state variable is designed so that the new co-state variable is obtained as the solution of a matrix form of the Riccati differential equation. This can be solved analytically by following a suitable technique to determine the new costate variable. The optimal values of the state and control variables are then obtained with the help of this costate variable. Finally, the proposed model is illustrated through a numerical example.
