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This paper discusses the hazard rate order of the fail-safe systems arising from two sets of independent multiple-outlier scale distributed components. Under certain conditions on scale parameters in the scale model and the submajorization order between the sample size vectors, the hazard rate ordering between the corresponding fail-safe systems from multiple-outlier scale random variables is established. Under certain conditions on the Archimedean copula and scale parameters, we also discuss the usual stochastic order of these systems with dependent components.

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Adding parameters to a known distribution is a valuable way of constructing flexible families of distributions. In this paper, we introduce a new model, the modified additive hazard rate model, by replacing the additive hazard rate distribution in the general proportional add ratio model. Next, when two sets of random variables follow the modified additive hazard model, we establish stochastic comparisons between the series and parallel systems comprising these components.

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This paper investigates the hazard rate order in fail-safe systems comprising multiple-outlier scale components subject to random shocks. Under certain  conditions, we first establish that increasing the survival probabilities of components against shocks improves the system in the hazard rate sense. We then demonstrate that an balanced allocation of component counts, in the sense of majorization of the size vector, can serve as an effective strategy for enhancing the reliability of such systems. Finally, we examine the effect of simultaneous changes in scale parameters and shock probabilities on the hazard rate order within a two-outlier model. Practical interpretations and potential applications of the derived results are also discussed.