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A number of problems in several areas such as power transmission and distribution, communication and transportation can be formulated as a stochastic-flow network (SFN). The system reliability of an SFN can be computed in terms of all the upper boundary points, called d-MinCuts (d-MCs). Several algorithms have been proposed to find all the d-MCs in an SFN. Here, some recent studies in the literature on search for all d-MCs are investigated. We show that some existing results and the corresponding algorithms are incorrect. Then, correct versions of the results are established. By modifying an incorrect algorithm, we also propose an improved algorithm. In addition, complexity results on a number of studies are shown to be erroneous and correct counts are provided. Finally, we present comparative numerical results in the sense of performance profile of Dolan and Moré showing the proposed algorithm to be more efficient than some existing algorithms.

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A multi-periodic, multi-echelon green supply chain network consisting of manufacturing plants, potential distribution centers, and customers is developed. The manufacturing plants can provide the products in three modes including production in regular time, production in over time, or by subcontracting. The problem has three objectives including minimization of the total costs of the green supply chain network, maximization of the average safe inventory levels of the manufacturing plants and the distribution centers and minimization of the environmental impacts of the manufacturing plants in producing, holding and dispatching the products and also the environmental impacts of the distribution centers in holding and dispatching the products. The problem is first formulated as a mixed-integer mathematical model. Then, in order to solve the model, the augmented weighted Tchebycheff method is employed and its performance in producing the Pareto optimal solutions is compared with the goal attainment method.

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The optimal planning of hybrid renewable energy systems has become increasingly important due to rising energy demand, environmental concerns, and the need for reliable electricity supply in remote and isolated regions. This study presents a comprehensive mixed-integer linear programming (MILP) framework for the optimal design of a hybrid solar–wind–battery–fuel cell–diesel microgrid. The proposed model simultaneously determines the optimal type, size, and operation of system components over the project lifetime while minimizing the equivalent annual cost. Unlike many existing optimization approaches, the framework integrates economic, environmental, and reliability considerations into a unified objective function by incorporating carbon emission penalties, load curtailment penalties, and renewable energy incentive policies. Furthermore, replacement costs and component lifetime degradation are reformulated into linear constraints, enabling the problem to be solved efficiently using exact optimization techniques. Hourly meteorological and load data are employed to accurately capture seasonal and daily variations in renewable energy generation and electricity demand. The optimization model is implemented in GAMS and solved using the CPLEX solver. Different system configurations and regulatory scenarios are evaluated to investigate the influence of diesel generators, battery storage, hydrogen technologies, and government support policies on system performance. The results demonstrate that an appropriately designed hybrid microgrid can substantially improve renewable energy penetration while maintaining system reliability and reducing the overall lifecycle cost. Moreover, regulatory incentive mechanisms significantly increase the economic viability of renewable energy resources and decrease dependence on fossil-fuel-based generation. The proposed MILP framework provides an effective decision-support tool for policymakers, system planners, and investors involved in the development of sustainable hybrid microgrids.