Two general forms of approximation for determining near optimal inspection intervals with non-zero inspection and replacement times in a deteriorating production system

The construction and solution of the next simplest but more realistic profit model for an exponential shift (or failure) distribution were presented in Ben-Daya and Hariga (B&H) [Ben-Daya, M., & Hariga, M. (1998). A maintenance inspection model: Optimal and heuristic solutions. International Journal of Quality and Reliability Management, 15, 481–488], in which the model presented looks much more complicated than the simplest profit model proposed by Baker [Baker, M. J. C. (1990). How often should a machine be inspected? International Journal of Quality and Reliability Management, 7, 14–18]. However, once we confer economic meanings to some mathematical expressions and rewrite the whole model, the revised model not only becomes less complicated and more comprehensible than the original one but also draws a closer analogy with Baker’s model and the associated solution algorithms.We also develop two distinct sets of general explicit formulae and present two corresponding heuristic algorithms for solving B&H’s model. The examples show that either of these new algorithms for determining a near optimal inspection interval for a deteriorating (or an imperfect) production system is not only more close to the optimum but also more efficient computationally than the one suggested by B&H (1998), and can give the value to any required degree of accuracy.The main purpose of this article is threefold: (1) to confer economic meanings to some expressions, (2) to rewrite the model and the Proof of Theorem 1, and (3) to introduce general exponential and logarithmic forms of approximation for further approximating optimal inspection intervals and to give merits and demerits of the two algorithms.