An integrated production-inventory system with lot streaming and complete backordering in a three-stage multi-firm supply chain solved algebraically

We formulate an integrated model of a three-stage multi-firm supply chain based on an integer multiplier at each stage, lot streaming allowed for all suppliers and manufacturers, and complete backordering allowed for some/all retailers. 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 Ben-Daya and Al-Nassar's (2008) models and remark on extending to a higher stage multi-firm supply chain from the integrated model.