Clearing function in the context of the invariant manifold method

TL;DR

This paper introduces a mechanistic model for clearing functions in production systems, linking them to invariant manifold theory and analyzing their validity in unsteady conditions through a slow-fast differential equation system.

Contribution

It proposes a novel mechanistic model based on enzyme-machine analogy and invariant manifold theory, extending the understanding of clearing functions beyond steady-state assumptions.

Findings

CFs are derived from the asymptotic expansion of the slow invariant manifold.

The validity of CF approximation depends on the smallness of the characteristic ratio.

Sufficiently small ratio of working machines to WIP ensures CF applicability in unsteady states.

Abstract

Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-in-progress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio 'working machines : WIP' guarantees the applicability of CF approximation in unsteady-state operation.