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This article considers the two-level supply chain model incorporating an imperfect production process under a variable lead time. The cost of producing a unit item is calculated as a function of the production rate. In addition, two alternative production functions (linear and quadratic functions) are used to relate process quality and production rate. Lead time demand follows two different distributions, based on which two mathematical formulations are described in this paper. In the first model, the lead time demand follows a normal distribution. In the second model, the lead time demand doesn't follow any particular distribution, but the mean and the standard deviation are known. The lead time length is minimized by incorporating the lead time crashing cost. This research aims to analyze the optimized total cost of the supply chain under two different distributions.

A two-tier supply chain model under two distributions with MTTF, rework, variable production rate and lead time