# Interval valued reliability indices assessment of multi-state system using interval $$L_{z}$$-transform

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## Abstract

This research introduces a new method, called Interval $${L}_{z}$$-transform (ILz), designed to estimate the reliability indices of Multi-State systems (MSS) even when data is uncertain or insufficient. Traditionally, precise values of state probabilities and performance metrics for each component were required, which could be challenging when data is lacking. To address this, the Interval $${L}_{z}$$ function is proposed, along with corresponding operators, enabling the calculation of interval-valued reliability indices for MSS. To demonstrate the effectiveness of the proposed method, it is applied to a numerical example of a series–parallel system. In this example, we determine interval-valued reliability indices such as reliability, availability, mean expected performance, and expected profit, considering uncertain values for the performance and failure rates of each multi-state component.

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## Data availability

No specific data has been used in the preparation of manuscript.

## Abbreviations

M :

The system’s total number of conceivable states

$$g_{ij}$$ :

Performance level of the element i in state j

$$\mathop {\underline {g} }\nolimits_{ij}$$ :

Element i’s performance level lower bound in state j

$$\mathop {\overline{g}}\nolimits_{ij}$$ :

Element i’s performance level upper bound in state j

$$\mathop \phi \nolimits_{ij}$$ :

Probability of the element i in state j

$$\mathop {\underline {\phi } }\nolimits_{ij}$$ :

Lower bound of probability of component i which is in state j

$$\mathop {\overline{\phi }}\nolimits_{ij}$$ :

Upper bound of probability of component i which is in state j

$$\mathop \lambda \nolimits_{ij}^{k}$$ :

Failure rate, i.e. rate by which component k is transitioned from one level/state i to j

$$\mathop \mu \nolimits_{ji}^{k}$$ :

Repair rate, i.e. rate by which component k is transitioned from level/state j to i

$$\left[ w \right]$$ :

Required performance level

$$\left[ R \right]$$ :

Interval-valued reliability of the system

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## Acknowledgements

The first author gratefully acknowledges the Department of Science and Technology (India) for providing INSPIRE fellowship (DST/INSPIRE/03/2022/005092) and financial support the course of this study.

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Correspondence to Vaibhav Bisht.

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Bisht, V., Singh, S.B. Interval valued reliability indices assessment of multi-state system using interval $$L_{z}$$-transform. Int J Syst Assur Eng Manag 15, 3293–3305 (2024). https://doi.org/10.1007/s13198-024-02337-4

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• DOI: https://doi.org/10.1007/s13198-024-02337-4