A fundamental law relating stock and end-of-life flow in cyclic manufacturing

• End-of-life (EoL) products are inevitable future resources in circular economy.
• Sustainable reuse/remanufacturing/recycle hinges on EoL flow and stock prediction.
• Product stock and EoLF are random, unobservable and affected by random early loss.
• The fundamental law relates mean EoLF, mean product stock and mean ages.
• It is simple, linear algebraic, valid for any EoL exit distribution and early loss.

Cyclic manufacturing (CM) emerges as a sine-qua-non for sustainability. Consumer product stocks of today are the end-of-life flows (EoLF) of tomorrow in circular economy. Enacted legislation fosters reuse/recycle of EoL consumer products, of chemicals, raw materials and hazardous products and components (batteries, brake fluids, printed circuit boards, cellular phones, computers). But when is tomorrow and how much of the stock will appear as EoLF? Efficient CM operations depend on cognizance of EoLF and accumulating stock, the pool from which EoLF emanates. Consumer uncertainty, personal income, economic cycles, advent of technology, social and health reasons, stricter eco-standards and random early product losses during use, render EoLF and product stock uncertain and unobservable. Conventional identification methods based on regression, sequential least squares, or actuary science methods presuming specific residual life distributions, may not provide reliable estimates under uncertainty and non-stationarities. An appealingly simple constitutive law is revealed, relating the mean stock and EoLF in terms of stock mean-age and EoLF mean-age, which are scaled, readily and reliably monitored variables, even from relatively small, decentralized samples. Valid under random lifetime early losses, the law encompasses any EoL exit distribution, enabling stock and EoLF identification, data consolidation and assessment. Being a linear algebraic expression between stock and EoLF, the law reduces computational burden and evades presumption of any survival or exit probability distribution, or cognizance of early loss history, or ex-post fitting of stochastic parameters. The results may prove useful in planning/sizing EoL facilities and recycling operations and in environmental policy or compliance assessment.