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This study develops a single-machine manufacturing system for multi-product with defective items and delayed payment policy. Contradictory to the literature limited production capacity and partial backlogging are considered for more realistic result. The objective of this research is to obtain the optimal cycle length, optimal production quantity, and optimal backorder quantity of each product such that the expected total cost is minimum. The model is solved analytically. Three efficient lemmas are developed to obtain the global optimum solution of the model. An improved algorithm is designed to obtain the numerical solution of the model. An illustrative numerical example and sensitivity analysis are provided to show the practical usage of proposed method.

We make an explicit calculation of Casimir effect through an alternative approach of field quantization for a spherical shell in which only the radial part of fluctuation has been taken into account. In this method, negative norm states have been utilized to obtain a natural renormalized theory with the condition that they cannot be affected by the physical boundary conditions. It is shown that by using this method, the exact result is obtained without any appearance of infinity.