Stochastic comparisons of two-series–parallel systems with independent components randomly chosen from two batches

Stochastic comparisons are quite helpful in optimizing system designs by evaluating the performance of different configurations and ensuring that critical systems meet reliability standards under diverse conditions. In this paper, we discuss stochastic comparisons of lifetimes of two-series–parallel systems with 2 n independent components randomly chosen from two different batches. We assume the n components from the first batch is more reliable than the n components from the second batch. Then, in the case of two-series–parallel system, we prove that the reliability of the system increases in terms of the usual stochastic order, as the random number, K , taking values in { 0 , 1 , … , [ n + 1 2 ] } , of components chosen from the first batch increases in increasing concave order or the random number, K , taking values in { [ n + 1 2 ] + 1 , … , n } , of components chosen from the first batch decreases in increasing convex order. We also present some numerical examples to illustrate all the results established here.