Cost assessment of complex repairable system consisting two subsystems in series configuration using Gumbel Hougaard family copula
Authors: Lado Abdul Kareem , Singh V.V. 
Published: July 2019

Purpose
The purpose of this paper is to covenant with the cost assessment of a complex repairable system, consisting of two subsystems (Subsystem 1 and Subsystem 2) connected in series configuration and being operated by a human operator. Each subsystem has two identical units in parallel configuration and has different types of failure and two types of repairs (general repair and copula repair). Through the transition diagram, the system of first-order partial differential equations is derived and solved using a supplementary variable technique, Laplace transforms. All failures are assumed to follow exponential distribution, whereas repairs follow two types of distributions that are general and Gumbel–Hougaard family copula. In this paper, explicit expressions for reliability, availability, mean time to failure (MTTF) and cost analysis functions have been obtained. In this paper, two types of repairs (copula repair and general repair) have been studied, and it has been concluded that copula repair is more reliable as compared to general repair. Some computations are taken as particular case by evaluating: reliability, availability, MTTF and cost analysis, so as to capture the effect of both failure and repair rates to reliability measures. The results have been shown in tables and graphs. The convincing part has been discussed in last section of this study.

Design/methodology/approach
This paper is focused on the cost assessment of a system consisting two subsystem series configuration. Each subsystem has two identical units in parallel configuration. The performance of the system has been analyzed by supplementary variable techniques and Laplace transforms. Various measures of the reliability have been discussed by evaluations. Software called Maple 13 is used for computations.

Findings
In this research paper, the authors have evaluated the operational cost and incurred profit of the system together with other reliability measures for various situations and different types of failures and two types of repairs using Gumbel–Hougaard family copula distribution.

Research limitations/implications
The present research focuses on the series and parallel configured complex systems that is used everywhere in industry and other sectors. The authors main aim is to claim that repair through the joint probability distribution copula is far better than general repair. Copula repair for a completely failed system is more beneficial for industrial system operations that will increase profit to the industrial sector.

Practical implications
The authors have observed that when repair follows general distribution the values of reliability obtained of the system are less compared to the those obtained when the authors apply copula repair, a joint probability distribution. It is a clear implication for industrial sector and organization to use the policy for a better generate revenue.

Social implications
According to the best of authors’ knowledge, there is no social implication as this study is meant for reliability section. The study in management and case study matters is considered to have social implication.

Originality/value
This research is the original work of authors. Nothing has been copied from any paper or book. The references are cited according to the relevance of study.