An integrated production-inventory system in a multi-stage multi-firm supply chain

We first generalize a number of integrated models with/without lot streaming and with/without complete backorders under the integer–multiplier coordination mechanism, and then individually derive the optimal solution to the three- and four-stage model, using algebraic methods of complete squares and perfect squares. We subsequently deduce optimal expressions for some well-known models. For our model, we check that the optimal solution, which is algebraically derived, is a global one. We present three numerical examples for illustrative purposes. We finally suggest some future research work involving extension or modification of the generalized model.