Researchers have used analytic methods (calculus) to solve inventory models with fixed and linear backorder costs. They have found conditions to partition the feasible domain into two parts. For one part, the system of the first partial derivatives has a solution. For the other part, the inventory model degenerates to the inventory model without shortages. A scholar tried to use the algebraic method to solve this kind of model. The scholar mentioned the partition of the feasible domain. However, other researchers cannot understand why the partition appears, even though the scholar provided two motivations for his derivations. After two other researchers provided their derivations by algebraic methods, the scholar showed a generalized solution to combine inventory models with and without shortages together. In this paper, we will point out that this generalized solution approach not only did not provide explanations for his previous partition but also contained twelve questionable results. Recently, an expert indicated questionable findings from two other researchers. Hence, we can claim that solving inventory models with fixed and linear backorder costs is still an open problem for future researchers.
This paper is a response to two papers. We improve the lengthy proof for the first paper by an elegant verification. For the second paper, we point out the three-sequence approach will result in different convergent rates such that when the other two sequences are converged, the ordering quantity sequence may still not converge to the optimal solution. We construct a novel iterative method to simplify the previous approach proposed by the three-sequence approach for the optimal solution. By the same numerical examples of three published papers, we demonstrate that we can control our findings to converge more accurately than previous results. Moreover, we show that there are three distinct features of our proposed approach. (i) It converges to the desired solution within the preassigned threshold value. (ii) We estimate the convergent ratio. (iii) We find the dominant factors for our proposed convergent sequence.
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