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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.