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The run length distribution of progressive mean (PM) control charts is highly skewed when the process is in-control or there is small shift in the process, so the interpretation based on average run length (ARL) may not be suitable. Furthermore, skewness varies at different shifts which cause difficulty in its interpretation on the basis of ARL. In the presence of skewness in run length, median run length (MRL) is the best approach for interpretation and accuracy in monitoring the process. The discussion of run length on the base of MRL is quite easy and readily understood. In this article, performance of PM control chart is evaluated in MRL and it is compared with ARL at the same shifts. This article indicates that MRL is more versatile average for understanding and explaining the distribution of run length because it is quick detector in case of small and moderate shift (highly and moderately skewed run length distribution) and gives good results in case of large shifts (symmetrical run length distribution). Along with standard deviation of run length (SDRL) is computed which is showing higher variability in process when shift is small but in case of moderate and large shifts smaller values of (SDRL). The performance of proposed control is compared with existing optimal EWMA control charts based on MRL and optimal CUSUM control charts based on MRL. The proposed control chart found to be more efficient than competitors.

In this paper, the classical notions of m-convex and m-concave sets are characterized under soft set environment and their important aggregation operations are discussed. Moreover, certain classical approaches (i.e. first and second senses) are employed on m-convex and m-concave soft sets to get more generalized results for uncertain scenarios.

The s-convex functions of first kind and of second kind are well known functions. We would use recently introduced notion of s-convex functions of mixed kind and we would call it (s; r)-convex function. It would generalize the notion of first kind and second kind convexity in the sense that both kinds of convexities would be obtained easily by imposing certain specific conditions on it. We would state generalization of Hermite-Hadamard type inequalities by using our new notion of (s; r)-convex function via Riemann-Lioville fractional integrals.

Home automation is one of the fields working on automating home or household activities. The core objectives of the domotics technology are to save time and effort. Home automation can be implemented in various ways to achieve desire results. Currently, no home automation system can track the location of the person to perform some task at a particular time. We can make home automation more effective and efficient if we can implement a combination of techniques. This research aims to design and implement a hybrid technique based on the Global Position System and hand gestures in the field of home automation. Global Position System works when a person is outside the home, and the hand gesture technique is implemented when a person is present at home. By implementing a hybrid technique, we can overcome the limitations of the Global Position System and hand gesture technique. It is designed to replace sensors used in home automation system with another state-of-the-art technology, the Global Position System. Global Position System is a satellite-based technology that provides location and time information in all weather, anywhere on or near the Earth and hand gestures are used for making it intelligent by setting the trend. Idea is to integrate these two technologies and develop an automatic home automation system using Cellular Global Position System and moves a step forward in the field of automation. This research is useful for the disabled and elderly people for convenience, energy efficiency, and safety benefits and the general population as well.

Through this research article we prove parameterized inequality of fractional Hermite-Hadamard type. So, we acquired many results of Hermite-Hadamard type of inequalities.

This paper includes new versions of generalized Fejer-Hadamard (FH) type fractional integral inequalities (FIIs) for exponentially (p,h)-convex functions. We utilize an extracted generalized fractional integral operator (FIO) involving Mittag-Leffler function (MLF) which contains a monotone increasing function. Some special already published results are deduced from the presented results.

Education is a very important medium for obtaining skills and knowledge. The world is continually changing so it is significant to teach and bring up those peoples who can understand and solve modern social problems. Forecasting techniques are necessary to find out the future trends that are meet the upcoming problems. For the desired purpose, different time series models are applied and selected the best one for better-forecasted values of the gross domestic product (GDP) percentage share on education. The best model is determined based on the lowest value of the Akaike information criterion (AIC) and Bayesian information criterion. Moreover, the quality and predictability power of the selected models are measured based on the minimum value of the mean error, root mean square error, mean percentage error, and mean percentage absolute error. The AIC criteria showed that the best time series model is the autoregressive integrated moving average (ARIMA) (2, 1, 1) to forecast the (GDP) percentage share on the education of Pakistan. Furthermore, the assumptions of the fitted model such as independence, no heteroscedasticity, and normality are also evaluated. The selected model showed that the (GDP) percentage share on education is decreasing slowly, which is not good for the education sector and development of the country.