Design of component reliability test plan for a series system having time dependent testing cost with the presence of covariates

Consider a series system with n different components. Assume that the lifetime of $$i\text {-}th$$ i - t h component follows exponential distribution with parameter $$ \lambda _{i}(\mathbf {x}), \ \mathbf {x} \in \mathbb {R}^{k} $$ λ i ( x ) , x ∈ R k is a covariate vector, $$ 1\le i \le n $$ 1 ≤ i ≤ n . For example, $$ \mathbf {x} $$ x may be (temperature, pressure, humidity), so that $$ \mathbf {x} \in \mathbb {R}^{3} $$ x ∈ R 3 represents a three dimensional covariate vector. Assume that each $$ \lambda _{i}(\mathbf {x}),1\le i \le n $$ λ i ( x ) , 1 ≤ i ≤ n , is distinct and depends upon $$ \mathbf {x} $$ x through linear relationship. Prior information available in the form of upper bounds on $$ \lambda _{i}(\mathbf {x}) $$ λ i ( x ) are also incorporated in the design. We propose to obtain optimal reliability test plan based on maximum likelihood estimator of system reliability. A non-linear integer optimization problem is formulated for minimizing the maximum of total expected cost involved in testing satisfying usual probability requirements (Type-I and Type-II error constraints). In addition, it is also established through simulation that the derived sampling plan meets the specified producer’s and consumer’s risks as well. A sensitivity analysis is carried out to study the effect of various input parameters on maximum total expected testing cost. Finally, a qualitative analysis is presented at the end to discuss the nature of sampling plan derived. Several numerical examples are discussed to illustrate our test plan, and it is observed that the proposed plan has significant potential to reduce the total number of components to be tested for failure. It is noted that the number of components to be tested for failure is reduced by about 96% as compared to the existing test plans in the literature.