Newsboy-style Results for Multi-echelon Inventory Problems: Backorders Optimization with Intermediate Delays

Multi-level inventory problems provide the opportunity for operations researchers to assert simultaneous control over entire distribution and production systems. A backorders minimization problem is considered for a particular multi-level inventory structure for which several finished goods with normally distributed demand are produced from a common component. Complete centralization of all inventory at the component level versus complete decentralization of all inventory to the finished-good level has previously been studied. Inventory may be stored at either level in this approach. A general, non-linear mathematical formulation with the objective of minimizing total penalty costs for expected backorders and a budget constraint upon holding costs is considered to determine amounts to stock at each point. Necessary conditions for optimality are derived and are shown to resemble closely the classical newsboy result for which a proper trade-off must be achieved between the consequences of too much and too little inventory.