Analytical evaluation of lead-time demand in polytree supply chains with uncertain demand, lead-time and inter-demand time

In a stochastic supply chain network knowing the exact distribution of lead-time demand (LTD) for each node allows determining inventory policy parameters, service levels, and optimal costs. To determine the LTD distributions a stochastic analytical model has been developed for a multi-echelon supply chain organised into an arbitrary polytree network, where the variables: demand quantity, lead-time, and inter-demand time are uncertain and can follow any probability distribution. The model is based on a conventional nQ batch policy for each node with discrete time and continuous demand. Any network can be broken down into four basic elements which represent root and leaf node topologies. The model derives key convolutions for the four basic elements and then follows the same technique for evaluating resulting LTD distributions for each node in the network. The paper presents the formulation in seven steps that can be used to obtain exact LTD distributions either in the two dimensional form or as a compound one-dimensional LTD distribution per variable lead-time for each node in any polytree supply chain. The LTD distributions can be calculated either analytically for smaller networks or using numerical integration for larger networks. The validation of the developed framework has been conducted by comparing the obtained formalism-based LTD distribution for two test networks against simulation. The pairs of simulated and formalism-based distributions have been plotted against each other as well as subjected to various goodness-of-fit tests and error analysis. The results suggest that the formalism-based and simulated probability distributions are consistent.