This paper presents a comprehensive review of existing techniques of k-means clustering algorithms made at various times. The kmeans algorithm is aimed at partitioning objects or points to be analyzed into well-separated clusters. There are different algorithms for k-means clustering of objects such as traditional k-means algorithm, standard k-means algorithm, basic k-means algorithm, and the conventional k-means algorithm, this is perhaps the most widely used version of the k-means algorithms. These algorithms use the Euclidean distance as their metric and minimum distance rule approach by assigning each data point (objects) to its closest centroids.

Several researches have been done to provide better alternative to the existing replacement models, but the research works did not adequately address the replacement problem for items that fail suddenly. Hence, a modified replacement model for items that fail suddenly has been proposed using the knowledge of probability distribution of failure times as well as that of variable replacement cost. The modified cost functions for implementing both individual and group replacements were derived. The modified cost functions were minimized using the principle of classical optimization in order to find the age at which replacement of items would be appropriate. Conditions under which the individual and group replacement policies should be adopted were derived. Two real data sets on failure time of LED bulbs and their replacement costs were used to validate the theoretical claims of this work. In essence, goodness-of-fit test was used to select appropriate probability distribution of failure times as well as that of replacement costs for data sets I and II respectively. The goodness-of-fit results showed that failure times of LED bulbs follow the Smallest Extreme Value and Laplace distributions for data sets I and II respectively. Similarly, it was observed that individual replacement cost followed the two-parameter Gamma and Largest Extreme Value distributions for data sets I and II respectively. Further, the group replacement cost was found to follow the log-normal and two-parameter Weibull distributions for data sets I and II respectively. Based on the empirical study, we observed that individual replacement policy is better than group replacement policy in terms of cost minimization for both existing model and the proposed model. In view of the results, the proposed replacement policy was recommended over the existing one because it yielded lower replacement costs than the existing replacement model.

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This paper introduces an inverse power Akash distribution as a generalization of the Akash distribution to provide better fits than the Akash distribution and some of its known extensions. The fundamental properties of the proposed distribution such as the shapes of the distribution, moments, mean, variance, coefficient of variation, skewness, kurtosis, moment generating function, quantile function, Rényi entropy, stochastic ordering and the distribution of order statistics have been derived. The proposed distribution is observed to be a heavy-tailed distribution and can also be used to model data with upside-down bathtub shape for its hazard rate function. The maximum likelihood estimators of the unknown parameters of the proposed distribution have been obtained. Two numerical examples are given to demonstrate the applicability of the proposed distribution and for the two real data sets, the proposed distribution is found to be superior in its ability to sufficiently model heavy-tailed data than Akash, inverse Akash and power Akash distributions respectively.

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