# Prognosis of product take-back for enhanced remanufacturing

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## Abstract

Remanufacturing reduces final wastes to sinks, extraction of virgin materials and pollution from production processes by reinstating products taken back by end-users to satisfy part of overall demand. Product returns are delayed and possibly limited in periods of fast growth and excessive in the aftermath. Varying growth/demand and volatile take back by consumers and industrial end-users introduce uncertainty, regarding quantity and quality of returns. As remanufacturing expands, escalating competition for acquisition of high quality returns exacerbates uncertainty. Production planning and control for efficient remanufacturing depends on reliable prediction of quantity and quality of returns. A method is developed for prognosis of product return quantity and quality grades, as reflected by vintage flows. It is anchored on a law relating stock and end-of-life level, under random losses and arbitrary end-of-life distribution. Efficacy is tested via a model that describes stock and flows in reuse/remanufacturing, allowing for varying demand, random stock losses, random product returns with time-varying distributions and time-varying utilisation of product returns. Realisations are obtained by Marko-chain Monte-Carlo simulation. Inherently integral in nature, using scaled data and founded on rigorous balances, the method enables prognosis of returns and age-vintage flows, under realistic conditions, including unknown nonlinearities and non-stationarities. It features improved performance (mean absolute error less than one half) compared to leading methods in-use that employ black-box models with error-driven parameter adaptation (e.g. regression). Efficacy is particularly high at crucial peaks and lows (shortage or surplus periods) enabling resourceful planning of acquisition and inventory control of product returns towards sustainability.

## Keywords

Remanufacturing Reuse Product take-back Forecasting of product returns Closed loop supply chain Circular economy## Nomenclature

*a*_{t}=

*P*_{t}+*I*_{net},_{t}, Product inflow in period*t*, e.g. year*t*, (tons/period, e.g.*t*/y),*a*= steady state level- ARMA
Autoregressive moving average, ARIMA AR integral MA

- C
_{f,t} Overall sales, originals+remanufactured, consumption (

*t*/y)*D*(*x*)Polynomial of degree 2(N-κ) + μ + 1

*d*_{k}Coefficient of order-k term in the polynomial

*D*(*x*)- EoL
End-of-life (no further reusable product returns)

- EoU
End-of-use (reusable product returns)

*E*_{t}EoL flow (EoL product returns) or EoL exit in period

*t*, e.g. year*t*, (tons/period)*G*'_{c,k}= 1-

*g'*_{1}-*g'*_{2}-…-*g'*_{k}, complementary cumulative distribution of the EoL exit distribution,*g*_{i},*g*_{i, t},*i*= 1, 2, ...,*ν*Reusable product return distribution (

*g*_{i, t}= fraction returned in period*t*=*t** + jκ-*μ*+*i*,*i*= 1,2,..,*ν*,*j*= 1,2,…,*Ν*-1,of an originally manufactured product in time period*t**)*g*_{i}Expected value of the stochastic process

*g*_{i, t},*i*= 1,2,..,*ν*,*h*_{i}Entries of vector \( \underset{\_}{h} \) given by eqs. 4–6 (or coefficients of polynomial eq. A3), Appendix A

*I*_{net}Flow of original net imported products = imports -exports =

*I*_{prod,t}–*Ex*_{prod,t}(*t*/period)*k*_{Q}Maximum age in the reusable product return sample

*k*_{U}Maximum age in the stock sample

- MAPE
Mean absolute percentage error

- MRT
Mean residence time = mean lifespan, time periods, e.g. years

*m*_{E}Minimum age in the EoL sample

*m*_{Q}Minimum age in the reusable product return sample

*N*Number of manufacturing cycles (original plus

*N*-1 remanufacturing cycles)*P*_{t}Original production flow (

*t*/period), (original items made from virgin or recycled material)*P*(*x*)Polynomial defined in eq. A4, Appendix A,

*p*_{k}Coefficients of order-k term in the polynomial

*P*(*x*) found from eqs. 7*Q*_{t}Reusable return flow,

*Q*= steady state value*Q*_{s, t}Size (mass) of the reusable product return sample at time

*t**Q*_{s, i, t}Size (mass) of vintage of age

*i*in the reusable return sample at time*t**q*Steady state product return flow rate with respect to inflow of original products =

*Q/a**RU*_{t}Actually reused/remanufactured product flow, (

*t*/period)*s*_{t}Early loss ratio =

*Ω*_{t}/(*U*_{t}+*Ω*_{t}) = probability of early loss (prior to EoL exit) in period*t**T*Time periods form production to centre axis of EoL exit (

*T*= maximum lifetime for. products with non-distributed, deterministic exit in a single time period)*U*_{t}Product accumulation: quantity of product stock present at the end of time period

*t*(tons)*x*_{t}=1-

*s*_{t}= retention ratio = probability of remaining in the reuse/remanufacturing cycle in period*t**y*_{i, t}Mass fraction of vintage of age

*i*in the reusable return flow in time period*t**y**_{,t}Ratio of mass of the vintage of age

*i*in the reusable return flow in time period*t*over the mass of the same vintage in the corresponding product stock

## Greek

- ε
=

*E/a*= steady state EoL flow ratio with respect to inflow of originals = EoL rate or yield*η*_{t}=Stock mean age at time period

*t**θ*_{t}=EoL flow mean age at time period

*t**κ*=Mean cycle duration, time periods

*μ*Half spread of the take-back/EoL distribution, time periods

*ν*=2 μ + 1 = spread of the take-back/EoL distribution, time periods

*Π*(*x*)Polynomial in

*x*defined in Appendix A, eqs. A5, A6*π*_{i}- τ
=

*U/a*= mean residence time or mean product lifespan = MRT*φ*Mean take-back fraction of reusable products with respect to reusable product stock

*Ω*_{t}=Early loss flow (

*t*/period)

## Symbols

- =:
Equal by definition.

- < >
Mean sample path value (MSPV)

## Subscripts

_{t}*t*is discrete time,*t*= 1: first time a product under consideration is launched in the market._{s}Sample.

## Notes

## References

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