Control policies of changeable manufacturing-remanufacturing systems using two failure-prone production facilities

Abstract

This paper deals with the problem of production planning and control (PPC) within hybrid manufacturing-remanufacturing systems (HMRSs), in which demand is fulfilled by either remanufacturing of returned items or manufacturing of new ones. It investigates how companies could benefit from changeable system settings with two production facilities operating in a stochastic and dynamic context. The rationale of such a system arises from its convertibility features that help to adapt the production process to internal (e.g., failures) and external (e.g., limited returned items) constraints. In this respect, we propose an integrated control policy that minimizes the long-term total cost by determining production and disposal rates as well as the switching strategy between manufacturing and remanufacturing modes. To tackle the problem, numerical techniques and optimal control theory are combined to characterize the control policy. A simulation-based approach is then used to optimize the resultant control parameters minimizing the total cost and to carry out sensitivity and comparative studies. The results illustrate the robustness of our approach showing its capability to assess the dynamic behaviour of integrated decisions and show that our proposal is more cost-effective compared to existing policies applied for different system settings with two facilities.

Appendix

1.1 Appendix 1. Mathematical Formulation

1.1.1 Stochastic process

The stochastic process state $\alpha \left(t\right)=\left({\alpha }_{1}\left(t\right),{\alpha }_{2}\left(t\right)\right)$ describe the operational mode of the entire production system composed of both facilities ${F}_{1}$ and ${F}_{2}$. Their operational mode at t could be given by the variables ${\alpha }_{1}\left(\mathrm{t}\right)$ and ${\alpha }_{2}\left(\mathrm{t}\right)$ taking values in ${M}_{1}=\left\{\mathrm{1,2}\right\}$ and ${M}_{2}=\left\{\mathrm{1,2}\right\}$, with:

$$\begin{array}{cc}{\alpha }_{1}\left(\mathrm{t}\right)=\left\{ \begin{array}{cc}1& {\mathrm{F}}_{1}\mathrm{ is available}/\mathrm{up }\\ 2& {\mathrm{F}}_{1}\mathrm{ is unavailable}/\mathrm{down}\end{array}\right.,& {\alpha }_{2}\left(\mathrm{t}\right)=\left\{ \begin{array}{cc}1& {\mathrm{F}}_{2}\mathrm{ is available}/\mathrm{up }\\ 2& {\mathrm{F}}_{2}\mathrm{ is unavailable}/\mathrm{down}\end{array}\right.\end{array}$$

From the definition of continuous-time discrete-state Markov chain [34], the stochastic processes ${\alpha }_{1}\left(\mathrm{t}\right)$ and ${\alpha }_{2}\left(\mathrm{t}\right)$ can be defined by the transition rates matrix denoted by ${\mathrm{T}}_{1}$ and ${\mathrm{T}}_{2}$, where ${\mathrm{T}}_{\mathrm{i}}=\left\{{q}_{\delta \vartheta }^{i}\right\}$, with ${q}_{\delta \vartheta }^{i}\ge 0$ if $\vartheta \ne \delta$ and ${q}_{\delta \delta }^{i}=-\sum_{\delta \ne \vartheta }{q}_{\delta \vartheta }^{i}$, where $\delta ,\vartheta \in {M}_{i}$.

$${T}_{i}=\left|\begin{array}{cc}-{q}_{12}^{i}& {q}_{12}^{i}\\ {q}_{21}^{i}& -{q}_{21}^{i}\end{array}\right|$$

We can derive the transition rate matrix of $\alpha \left(\mathrm{t}\right)$ ($T=\left\{{q}_{\delta \vartheta }\right\},\delta ,\vartheta \in M$) from ${\mathrm{T}}_{\mathrm{i}}=\left\{{q}_{\delta \vartheta }^{i}\right\}$ as follows:

$$\mathrm{T}=\left[\begin{array}{cccc}{q}_{11}& {q}_{12}& {q}_{13}& {q}_{14}\\ {q}_{21}& {q}_{22}& {q}_{23}& {q}_{24}\\ {q}_{31}& {q}_{32}& {q}_{33}& {q}_{34}\\ {q}_{41}& {q}_{42}& {q}_{43}& {q}_{44}\end{array}\right]=\left[\begin{array}{cccc}-\left({\mathrm{q}}_{12}^{1}+{\mathrm{q}}_{12}^{2}\right)& {\mathrm{q}}_{12}^{2}& {\mathrm{q}}_{12}^{1}& 0\\ {\mathrm{q}}_{21}^{2}& -\left({\mathrm{q}}_{12}^{1}+{\mathrm{q}}_{21}^{2}\right)& 0& {\mathrm{q}}_{12}^{1}\\ {\mathrm{q}}_{21}^{1}& 0& -\left({\mathrm{q}}_{21}^{1}+{\mathrm{q}}_{12}^{2}\right)& {\mathrm{q}}_{12}^{2}\\ 0& {\mathrm{q}}_{21}^{1}& {\mathrm{q}}_{21}^{2}& -\left({\mathrm{q}}_{21}^{1}+{\mathrm{q}}_{21}^{2}\right)\end{array}\right]$$

For the HMRS under study, the continuous and discrete state of the system is given by the three components vector $\left({x}_{R},{x}_{F},\alpha \right)$, each component is subject to some constraints such that: ${x}_{R}\in \left[0,{x}_{R}^{esp}\right]$, ${x}_{F}\in R$ and $\alpha \in M$. Let $0\le {x}_{R}\left(t\right)\le {x}_{R}^{esp}$ the constraint of the storage space capacity, $S=\left[0,{x}_{R}^{esp}\right]$ and $\partial S=\left\{0,{x}_{R}^{esp}\right\}$, and let ${S}^{0}=\left]0,{x}_{R}^{esp}\right[$ be the interior of S.

1.1.2 Switching mode

The switching mode of the whole system can be described by the vector I taking values in $M=\left\{\mathrm{1,2},\mathrm{3,4}\right\}$, where:

$$\mathrm{I}=\left\{ \begin{array}{c}1 \;Both\; {\mathrm{F}}_{1} \;and\; {\mathrm{F}}_{2} \;are \;in \;mode \;1\; \left(\mathrm{1,1}\right) \\ 2\; {\mathrm{F}}_{1} \;is \;in \;mode \;1 \;and\; {\mathrm{F}}_{2} \;is \;in \;mode \;2\; \left(\mathrm{1,2}\right) \\ 3\; {\mathrm{F}}_{1} \;is \;in \;mode \;2 \;and\; {\mathrm{F}}_{2} \;is \;in \;mode \;1\; \left(\mathrm{2,1}\right) \\ 4\; Both\; {\mathrm{F}}_{1} \;and\; {\mathrm{F}}_{2} \;are \;in \;mode \;2\; \left(\mathrm{2,2}\right)\end{array}\right.$$

1.1.3 Admissible solution

Let A and ${\mathrm{A}}_{p}$ define the admissible decision sets $\left({\Omega }_{1}, {\Omega }_{2},p\right)$:

$$A=\left\{\begin{array}{c}\left({\Omega }_{1}, {\Omega }_{2},p\right),0\le {u}_{kman}\left(t\right)\le {U}_{kman}^{max} ,0\le {u}_{krem}\left(t\right)\le {U}_{krem}^{max}, \\ {u}_{dis}\left(t\right)\ge 0 , \forall t\ge 0, k\in \left\{\mathrm{1,2}\right\}\\ \end{array}\right\}$$

$${A}_{p}=\left\{\begin{array}{c}p,0\le {u}_{kman}\left(t\right)\le {U}_{kman}^{max} ,0\le {u}_{krem}\left(t\right)\le {U}_{krem}^{max},\\ {u}_{dis}\left(t\right)\ge 0 , \forall t\ge 0, k\in \left\{\mathrm{1,2}\right\}\\ \end{array}\right\}$$

1.1.4 Properties of the value function

By applying the dynamic programming principle as in [38], it can be shown that in the interior space ${S}^{0}$, the value function is the unique solution of the following optimality conditions (a set of coupled partial derivatives equations).

$$min\left\{\begin{array}{c}\underset{p\in {A}_{p}}{min}\left\{\begin{array}{c}\left({u}_{R}-{u}_{1rem}-{u}_{2rem}-{u}_{dis}\right).{\left({v}_{i}\right)}_{{x}_{R}}\\ +\left({u}_{1man}+{u}_{1rem}+{u}_{2man}+{u}_{2rem}-d\right).{\left({v}_{i}\right)}_{{x}_{F}}\\ +g\left({x}_{R},{x}_{F},p\right)\\ +\sum_{\beta \ne \alpha }{q}_{\alpha \beta }\left(\left({v}_{i}\right)\left({x}_{R},{x}_{F},\beta \right)-\left({v}_{i}\right)\left({x}_{R},{x}_{F},\alpha \right)\right)\end{array}\right\}\\ -\rho .\left({v}_{i}\right)\left({x}_{R},{x}_{F},\alpha \right) ;\\ \underset{j\ne i,k}{min}\left\{\begin{array}{c}{R}_{ij}\left({x}_{R},{x}_{F},p,{T}_{{S}_{{F}_{k}}}^{ij}\right)\\ +{e}^{-\rho {T}_{S\_{F}_{k}}^{ij}}\left({v}_{j}\right)\left(\begin{array}{c}{x}_{R}+{u}_{R}{T}_{S\_{F}_{k}}^{ij},{x}_{F}\\ +\left(\begin{array}{c}{U}_{\left(2-k+1\right)man}^{max}.{S}_{11}^{Fk}\\ +{U}_{\left(2-k+1\right)rem}^{max}.{S}_{22}^{Fk}-d\end{array}\right){T}_{{S}_{{F}_{k}}}^{ij},\\ 1\end{array}\right)\end{array}\right\}\\ -\left({v}_{i}\right)\left({x}_{R},{x}_{F},\alpha \right) \end{array}\right\}=0$$

(18)

where ${v}_{x}\left(.\right)$, denotes the gradient of ${v}_{x}\left(.\right)$ With respect to $x$.

1.2 Appendix 2. Numerical resolution

Following [39], we can apply the successive approximation algorithm or policy improvement techniques to solve the numerical approximation of (17).

Let ${\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}}$ and ${\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}}$ define the finite difference interval lengths of ${\mathrm{x}}_{\mathrm{R}}$ and ${\mathrm{x}}_{\mathrm{F}}$, respectively. Using the finite difference approximation, ${\mathrm{v}}_{\mathrm{i}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right)$ could be given by ${\left({\mathrm{v}}_{\mathrm{i}}\right)}^{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right)$, ${\left({\mathrm{v}}_{\mathrm{i}}\right)}^{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right)$ and the gradients ${\left({\mathrm{v}}_{\mathrm{i}}\right)}_{{\mathrm{x}}_{\mathrm{R}}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right)$ and ${\left({\mathrm{v}}_{\mathrm{i}}\right)}_{{\mathrm{x}}_{\mathrm{F}}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right)$ by:

$${\left({v}_{i}\right)}_{{x}_{R}}\left({x}_{R},{x}_{F},\alpha \right)=\left\{\begin{array}{cc}\frac{1}{{h}_{{x}_{R}}}\left({\left({v}_{i}\right)}^{{h}_{{x}_{R}}}\left({x}_{R}+{h}_{{x}_{R}},{x}_{F},\alpha \right)-{\left({v}_{i}\right)}^{{h}_{{x}_{R}}}\left({x}_{R},{x}_{F},\alpha \right)\right)& if\; {u}_{R}-{u}_{1rem}-{u}_{2rem}-{u}_{dis}\ge 0\\ \frac{1}{{h}_{{x}_{R}}}\left({\left({v}_{i}\right)}^{{h}_{{x}_{R}}}\left({x}_{R},{x}_{F},\alpha \right)-{\left({v}_{i}\right)}^{{h}_{{x}_{R}}}\left({x}_{R}-{h}_{{x}_{R}},{x}_{F},\alpha \right)\right)& if\; {u}_{R}-{u}_{1rem}-{u}_{2rem}-{u}_{dis}<0\end{array}\right.$$

$${\left({v}_{i}\right)}_{{x}_{F}}\left({x}_{R},{x}_{F},\alpha \right)=\left\{\begin{array}{cc}\frac{1}{{h}_{{x}_{F}}}\left({\left({v}_{i}\right)}^{{h}_{{x}_{F}}}\left({x}_{R},{x}_{F}+{h}_{{x}_{F}},\alpha \right)-{\left({v}_{i}\right)}^{{h}_{{x}_{F}}}\left({x}_{R},{x}_{F},\alpha \right)\right)& if\; {u}_{1man}+{u}_{1rem}+{u}_{2man}+{u}_{2rem}-d\ge 0\\ \frac{1}{{h}_{{x}_{F}}}\left({\left({v}_{i}\right)}^{{h}_{{x}_{F}}}\left({x}_{R},{x}_{F},\alpha \right)-{\left({v}_{i}\right)}^{{h}_{{x}_{F}}}\left({x}_{R},{x}_{F}-{h}_{{x}_{F}},\alpha \right)\right)& if\; {u}_{1man}+{u}_{1rem}+{u}_{2man}+{u}_{2rem}-d<0\end{array}\right.$$

Let: ${\uppsi }^{\mathrm{h}}\left({\mathrm{p}}_{1},{\mathrm{p}}_{2},\mathrm{\alpha },\uprho \right)=\uprho +\left|{\mathrm{q}}_{\mathrm{\alpha \alpha }}\right|+\frac{\left|{\mathrm{p}}_{1}\right|}{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}}}+\frac{\left|{\mathrm{p}}_{2}\right|}{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}}}$,

With, ${\mathrm{p}}_{1}={\mathrm{u}}_{\mathrm{R}}-{\mathrm{u}}_{2\mathrm{rem}}-{\mathrm{u}}_{1\mathrm{rem}}-{\mathrm{u}}_{\mathrm{dis}}$ and ${\mathrm{p}}_{2}={\mathrm{u}}_{1\mathrm{rem}}+{\mathrm{u}}_{1\mathrm{man}}+{\mathrm{u}}_{2\mathrm{rem}}+{\mathrm{u}}_{2\mathrm{man}}-\mathrm{d}$.

$${\left({\Gamma }_{\mathrm{i}}\right)}^{{\mathrm{x}}_{\mathrm{R}}}\left({\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}},{\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},{\mathrm{p}}_{1},\mathrm{\alpha }\right)=\frac{\left|{\mathrm{p}}_{1}\right|}{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}}}\left(\begin{array}{c}{\left({\mathrm{v}}_{\mathrm{i}}\right)}^{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}}}\left({\mathrm{x}}_{\mathrm{R}}+{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right) I\left({\mathrm{p}}_{1}\ge 0\right)\\ +{\left({\mathrm{v}}_{\mathrm{i}}\right)}^{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}}}\left({\mathrm{x}}_{\mathrm{R}}-{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{R}}},{\mathrm{x}}_{\mathrm{F}},\mathrm{\alpha }\right) I\left({\mathrm{p}}_{1}<0\right)\end{array}\right)$$

$${\left({\Gamma }_{\mathrm{i}}\right)}^{{\mathrm{x}}_{\mathrm{F}}}\left({\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}},{\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}},{\mathrm{p}}_{2},\mathrm{\alpha }\right)=\frac{\left|{\mathrm{p}}_{2}\right|}{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}}}\left(\begin{array}{c}{\left({\mathrm{v}}_{\mathrm{i}}\right)}^{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}}+{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}},\mathrm{\alpha }\right) I\left({\mathrm{p}}_{2}\ge 0\right)\\ +{\left({\mathrm{v}}_{\mathrm{i}}\right)}^{{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}}}\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}}-{\mathrm{h}}_{{\mathrm{x}}_{\mathrm{F}}},\mathrm{\alpha }\right) I\left({\mathrm{p}}_{2}<0\right)\end{array}\right)$$

To simplify the notation, let ${h}_{{x}_{R}}={h}_{{x}_{F}}=h.$ Equation (A.1) can be expressed, in terms of ${\left({v}_{i}\right)}^{h}\left({x}_{R},{x}_{F},\alpha \right)$, as follows:

$${\left({v}_{i}\right)}^{h}\left({x}_{R},{x}_{F},\alpha \right)=min\left\{\begin{array}{c}\underset{p\in {A}_{p}}{\mathit{min}}\left\{\begin{array}{c}{\left({\psi }^{h}\left({p}_{1},{p}_{2},\alpha ,\rho \right)\right)}^{-1}\\ \times \left(\begin{array}{c}{\left({\Gamma }_{i}\right)}^{{x}_{R}}\left({h}_{{x}_{R}},{x}_{R},{x}_{F},{p}_{1},\alpha \right)+{\left({\Gamma }_{i}\right)}^{{x}_{F}}\left({h}_{{x}_{F}},{x}_{R},{x}_{F},{p}_{2},\alpha \right)\\ +g\left({x}_{R},{x}_{F},p\right)+\sum\limits_{\beta \ne \alpha }{q}_{\alpha \beta }\left({\left({v}_{i}\right)}^{h}\left({x}_{R},{x}_{F},\beta \right)\right)\end{array}\right)\end{array}\right\};\\ \underset{j\ne i,k}{min}\left\{\begin{array}{c}{R}_{ij}\left({x}_{R},{x}_{F},p,{T}_{S\_{F}_{k}}^{ij}\right)+\\ {e}^{-\rho {T}_{S\_{F}_{k}}^{ij}}{\left({v}_{j}\right)}^{h}\left(\begin{array}{c}{x}_{R}+{u}_{R}{T}_{{S}_{{F}_{k}}}^{ij},{x}_{F}\\ +\left(\begin{array}{c}{U}_{\left(2-k+1\right)man}^{max}.{S}_{11}^{Fk}\\ +{U}_{\left(2-k+1\right)rem}^{max}.{S}_{22}^{Fk}-d\end{array}\right){T}_{S\_{F}_{k}}^{ij},1\end{array}\right)\end{array}\right\}\end{array}\right\}$$

The use of a finite grid denoted by the computational domain CD is needed for the implementation of the successive approximation technique.

$$CD=\left\{\left({\mathrm{x}}_{\mathrm{R}},{\mathrm{x}}_{\mathrm{F}}\right):{0\le \mathrm{x}}_{\mathrm{R}}\le {\mathrm{x}}_{\mathrm{R}}^{\mathrm{sup}},{-{\mathrm{x}}_{\mathrm{F}}^{\mathrm{sup}}\le \mathrm{x}}_{\mathrm{F}}\le {\mathrm{x}}_{\mathrm{F}}^{\mathrm{sup}}\right\}$$

With ${x}_{R}^{sup}$ and ${x}_{F}^{sup}$ are given positive constants. For more details on the application of the successive approximation algorithm, we refer the reader to [40].