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In this paper, we have suggested a penalty method to modify the combinatorial optimization problem with the linear constraints to a global optimization problem with linear constraints. It also deals with a topic of vital significance of pump operation optimization in a water system. In this connection we have done a lot of work to formulate a model based on a simplified flow volume balance to resolve the problem of optimal pump operation settings of switching "ON" and "OFF" with the reduced gradient method. This global solution approach incorporates some benefits for practical application to a real system as is shown in the case study.

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To obtain the best estimates of the unknown population parameters have been the key theme of the statisticians. In the present paper we have suggested some estimators which estimate the population parameters efficiently. In short we propose a ratio, product, and regression estimators using two auxiliary variables, when there are some maximum and minimum values of the study and auxiliary variables, respectively. The properties of the proposed strategies in terms of mean square errors (variances) are derived up to first order of approximation. Also the performance of the proposed estimators have shown theoretically and these theoretical conditions are verified numerically by taking four real data sets under which the proposed class of estimators performed better than the other previous works.

In this manuscript, we have proposed a difference-type estimator for population mean under two-phase sampling scheme using two auxiliary variables. The properties and the mean square error of the proposed estimator are derived up to first order of approximation; we have also found some efficiency comparison conditions for the proposed estimator in comparison with the other existing estimators under which the proposed estimator performed better than the other relevant existing estimators. We show that the proposed estimator is more efficient than other available estimators under the two phase sampling scheme for this one example; however, further study is needed to establish the superiority of the proposed estimator for other populations.

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In this paper we study the distribution of order statistics of the two parametric Lomax distribution. We consider the single and product moment of order statistics from Lomax distribution. Also, we establish some recurrence relation for single moments of order statistics. The exact analytical expressions of entropy, residual entropy and past residual entropy for order statistics of Lomax distribution is derived.