Approximations to production lot sizing with machine breakdowns

In general, the study of convexity (concavity) of the total annual cost function (the annual net profit function) should be one of the main research topics about the inventory model. This paper first shows that the long-run average cost function per unit of time for the case of exponential failure is unimodal. However, it is neither convex nor concave. Second, the better lower bound Q𝓁^* and upper bound Q𝓊^* can be obtained to improve some existing results. Finally, numerical examples reveal the lower bound Q𝓁^* for the optimal lot size is a rather good approximation to the optimal lot size.