By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.

Based on uncertainty theory, this paper studies the problem of unmanned aerial vehicle (UAV) combat mission assignment under an uncertain environment. First, considering both the target value, which is the combat mission benefit gained from attacking the target, and the unit fuel consumption of UAV as uncertain variables, an uncertain UAV combat mission assignment model is established. And according to decisions under the realization of uncertain variables, the first stage generates an initial mission allocation scheme corresponding to the realization of target value, while the second stage dynamically adjusts the scheme according to the realization of unit fuel consumption; a two-stage uncertain UAV combat mission assignment (TUCMA) model is obtained. Then, because of the difficulty of obtaining analytical solutions due to uncertainty and the complexity of solving the second stage, the TUCMA model is transformed into an expected value-effective deterministic model of the two-stage uncertain UAV combat mission assignment (ETUCMA). A modified particle swarm optimization (PSO) algorithm is designed to solve the ETUCMA model to get the expected value-effective solution of the TUCMA model. Finally, experimental simulations of multiple UAV combat task allocation scenarios demonstrate that the proposed modified PSO algorithm yields an optimal decision with maximum combat mission benefits under a maximum iteration limit, which are significantly greater benefits than those for the mission assignment achieved by the original PSO algorithm. The proposed modified PSO exhibits superior performance compared with the ant colony optimization algorithm, enabling the acquisition of an optimal allocation scheme with greater benefits. This verifies the effectiveness and superiority of the proposed model and algorithm in maximizing combat mission benefits.

The cooperative task assignment problem with time windows for heterogeneous multiple unmanned aerial vehicles is an attractive complex combinatorial optimization problem. In reality, unmanned aerial vehicles’ fuel consumption exhibits uncertainty due to environmental factors or operational maneuvers, and accurately determining the probability distributions for these uncertainties remains challenging. This paper investigates the heterogeneous multiple unmanned aerial vehicle cooperative task assignment model that incorporates time window constraints under uncertain environments. To model the time window constraints, we employ the big-M method. To address the uncertainty in fuel consumption, we apply an adjustable robust optimization approach combined with duality theory, which allows us to derive the robust equivalent form and transform the model into a deterministic mixed-integer linear programming problem. We conduct a series of numerical experiments to compare the optimization results across different objectives, including maximizing task profit, minimizing total distance, minimizing makespan, and incorporating three different time window constraints. The numerical results demonstrate that the robust optimization-based heterogeneous multiple unmanned aerial vehicle cooperative task assignment model effectively mitigates the impact of parameter uncertainty, while achieving a balanced trade-off between robustness and the optimality of task assignment objectives.

Multi-objective optimization problems (MOPs) constitute a vital component in the field of mathematical optimization and operations research. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) decomposes a MOP into a set of single-objective subproblems and approximates the true Pareto front (PF) by optimizing these subproblems in a collaborative manner. However, most existing MOEA/Ds maintain population diversity by limiting the replacement region or scale, which come at the cost of decreasing convergence. To better balance convergence and diversity, we introduce auction theory into algorithm design and propose an auction-based matching (ABM) mechanism to coordinate the replacement procedure in MOEA/D. In the ABM mechanism, each subproblem can be associated with its preferred individual in a competitive manner by simulating the auction process in economic activities. The integration of ABM into MOEA/D forms the proposed MOEA/D-ABM. Furthermore, to make the appropriate distribution of weight vectors, a modified adjustment strategy is utilized to adaptively adjust the weight vectors during the evolution process, where the trigger timing is determined by the convergence activity of the population. Finally, MOEA/D-ABM is compared with six state-of-the-art multi-objective evolutionary algorithms (MOEAs) on some benchmark problems with two to ten objectives. The experimental results show the competitiveness of MOEA/D-ABM in the performance of diversity and convergence. They also demonstrate that the use of the ABM mechanism can greatly improve the convergence rate of the algorithm.