In this article, our main aim is to suggest enhanced families of estimators for estimating the population distribution function (DF) using twofold auxiliary evidence within the framework of simple random sampling. Numerical analysis is performed on four different actual data sets. The precision of the estimators is further investigated exhausting a simulation study. As equated with existing estimators, the suggested families of estimators have minimum mean square error (MSE) and higher percentage relative efficiency (PRE). The succeeding recommended family of estimators outperforms the first family of estimators across all data sets. These are positive indicators of its performance. The theoretical result shows that the recommended family of estimators performs better than the existing estimators. The extent of improvement in efficiency is noteworthy, indicating the superiority of the suggested estimators in terms of minimum MSE.

In this article, we suggest an enhanced estimator for the estimation of finite population variance using twofold auxiliary variable under stratified random sampling. The numerical expressions for the bias and MSE are determined up to the first order of approximation. In order to effectively validate the theoretical findings, three actual data sets are included. Additionally, the application of the suggested estimators is demonstrated using a simulation study. Results of an empirical comparison among the suggested and existing estimators were investigated. To determine how good the suggested estimator, in comparison to the preliminary estimators, the MSE criterion is used. The suggested estimator has a smaller MSE and better PRE than existing estimators, according to numerical results utilizing actual data sets and a simulation analysis.

In this paper, an emission- and reliability-aware stochastic optimization model is proposed for the economic planning of electric bus parking lots (EBPLs) with photovoltaic (PV) and wind-turbine (WT) resources in an 85-bus radial distribution network. The model simultaneously minimizes operating, emission, and energy-loss costs while increasing system reliability, measured by energy not supplied (ENS), and uses a fuzzy decision-making approach to determine the final solution. To address optimization challenges, a new multi-objective entropy-guided Sinh–Cosh Optimizer (MO-ESCHO) is proposed to efficiently mitigate premature convergence and produce a well-distributed Pareto front. Also, a hybrid forecasting architecture that combines MO-ESCHO and artificial neural networks (ANN) is proposed for accurate prediction of PV and WT power and network loading. The framework is tested across five cases, progressively incorporating EBPL, demand response (DR), forecast information, and stochastic simulation of uncertainties using a new hybrid Unscented Transformation–Cubature Quadrature Rule (UT-CQR) method. Comparative analyses against conventional methods confirm superior performance in achieving better objective values and ensuring computational efficiency. The outcomes indicate that the combination of EBPL with RES reduces operating costs by 5.23%, emission costs by 27.39%, and ENS by 11.48% compared with the base case with RES alone. Moreover, incorporating the stochastic model increases operating costs by 6.03%, emission costs by 5.05%, and ENS by 7.94% over the deterministic forecast case, reflecting the added complexity of uncertainty. The main contributions lie in coupling EBPLs and RES under uncertainty and proposing UT-CQR, which exhibits robust system performance with reduced variance and lower computational effort compared with Monte Carlo and cloud-model approaches.