Algebraically derive optimal inventory decisions in a multi-stage multi-firm supply chain based on an integer multiplier at each stage and complete backorders allowed for downstream firms

We formulate an integrated model of a three-stage multi-firm supply chain based on an integer multiplier at each stage and complete backorders allowed for downstream firms. Then we derive the optimal solution to the integrated model using the methods of complete squares and perfect squares. These are simple algebraic approaches so that ordinary readers unfamiliar with differential calculus can understand the optimal solution procedure with ease. For our model, we also need check that the optimal solution, which is algebraically derived, is a global one. We solve a numerical example to illustrate the procedure. We finally deduce two special cases and remark on extending to a higher stage multi-firm supply chain from the integrated model.