Production, maintenance and quality inspection planning of a hybrid manufacturing/remanufacturing system under production rate-dependent deterioration

Abstract

This paper presents a study on a hybrid manufacturing and remanufacturing system that degrades according to its production rate. The system consists of two failures and repairs prone machines that produce a single type of product. Both machines degrade according to their production rates, which affects their availability and the quality of the products. The main objective of this study is to develop optimal joint manufacturing, remanufacturing, maintenance and quality control policies for a deteriorating production system. A stochastic dynamic programming approach is used to develop the Hamilton–Jacobi–Bellman (HJB)-type optimality conditions. Subsequently, we used the numerical methods to solve its obtained HJB equations in order to determine the optimal manufacturing and remanufacturing thresholds, the optimal fractions of products to be controlled and the optimal conditions to start preventive maintenance operations. To illustrate this work, we have simulated a numerical example of a hybrid production line (manufacturing/remanufacturing). The obtained results allowed us to develop simultaneously a critical threshold production policy, a sampling inspection policy and an opportunistic maintenance policy. Next, we performed a sensitivity analysis of our models to show their robustness. Finally, we compared our policy with policies adapted from the literature. This comparison allowed us to highlight the gains generated by the proposed control policies.

Appendices

Appendix A

The solution of Eq. (28) by the analytical method is very complex [8]. Therefore, we will use the numerical method of Kushner. Therefore, an approximation of the derivative of the value function by a finite difference is defined by the following Eq. (36).

$$\frac{\partial \vartheta \left(x,\alpha \right)}{\partial x}=\left\{\begin{array}{c}\frac{{ \vartheta }^{h}\left(x+h,\alpha \right)-{ \vartheta }^{h}\left(x,\alpha \right)}{h}\;if\;\frac{dx(t)}{dt}\ge 0\\ \frac{{ \vartheta }^{h}\left(x,\alpha \right)-{ \vartheta }^{h}\left(x-h,\alpha \right)}{h}\;if\;not\end{array}\right.$$

(36)

By injecting the last equation in Hamilton–Jacobi–Bellman (HJB) and developing, we obtain:

$$\rho {\vartheta }^{h}\left(x,\alpha \right)=\underset{({{[u}_{1}\left(\cdot \right),u}_{2}\left(\cdot \right),f\left(\cdot \right)]\mathit{\epsilon \Gamma }\left(\alpha \right)}{\mathit{min}}\left[g\left(.\right)+{q}_{\mathit{\alpha \alpha }}{\vartheta }^{h}\left(x,\alpha \right)+\sum_{j\ne \alpha }{q}_{\alpha j}{ \vartheta }^{h}\left(x,j\right)+\frac{\mathit{dx}(t)}{\mathit{dt}}*\left\{\begin{array}{c}\frac{{ \vartheta }^{h}\left(x+h,\alpha \right)-{ \vartheta }^{h}\left(x,\alpha \right)}{h}ind(\dot{x})\ge 0\\ \frac{{ \vartheta }^{h}\left(x,\alpha \right)-{ \vartheta }^{h}\left(x-h,\alpha \right)}{h}ind(\dot{x})<0\end{array}\right.\right]$$

(37)

Isolating all ϑ (x, α) members in the left-hand member, we have:

$$(\rho +\frac{\left|\dot{x}\right|}{h}+\left|{q}_{\alpha \alpha }\right|){ \vartheta }^{h}\left(x,\alpha \right)=\underset{{{[u}_{1}\left(\cdot \right),u}_{2}\left(\cdot \right),f\left(\cdot \right)]\mathrm{\epsilon \Gamma }\left(\mathrm{\alpha }\right)}{\mathrm{min}}\left[g(.)+\sum_{j\ne \alpha }{q}_{\mathit{\alpha j}}{ \vartheta }^{h}\left(x,j\right)+\frac{\dot{x} }{h}\left\{\begin{array}{c}{ \vartheta }^{h}\left(x+h,\alpha \right)ind(u-d)\ge 0\\ { \vartheta }^{h}\left(x-h,\alpha \right)ind(u-d)<0\end{array}\right.\right]$$

(38)

Hence:

$${\vartheta }^{h}\left(x,\alpha \right)=\underset{{{[u}_{1}\left(\cdot \right),u}_{2}\left(\cdot \right),f\left(\cdot \right)]\mathit{\epsilon \Gamma }\left(\alpha \right)}{\mathit{min}}\left[\frac{g(.)+\sum_{j\ne \alpha }{q}_{\mathit{\alpha j}}{ \vartheta }^{h}\left(x,j\right)+\frac{\dot{x} }{h}\left\{\begin{array}{c}{ \vartheta }^{h}\left(x+h,\alpha \right)ind(u-d)\ge 0\\ { \vartheta }^{h}\left(x-h,\alpha \right)ind(u-d)<0\end{array}\right.}{(\rho +\frac{\left|\dot{x} \right|}{h}+\left|{q}_{\mathit{\alpha \alpha }}\right|)}\right]$$

(39)

$${\Omega }_{h}^{\alpha }=\frac{\left|\dot{x}\right|}{h}+\left|{q}_{\alpha \alpha }\right| \quad {\mathrm{P}}_{x}^{+}\left(\alpha \right)=\left\{\begin{array}{lc}\frac{\dot{x}}{h{\Omega }_{h}^{\alpha }} & if\;\dot{x}>0\\ 0 & if\;not\end{array}\right.$$

(40)

$${\mathrm{P}}_{x}^{-}\left(\alpha \right)=\left\{\begin{array}{lc}\frac{\dot{x} }{h{\Omega }_{h}^{\alpha }} & if\;\dot{x} \le 0\\ 0 & if\;not\end{array} \quad {P}^{j}\left(\alpha \right)=\frac{{q}_{\alpha j}}{{\Omega }_{h}^{\alpha }}\right.$$

(41)

Now we have:

$${\vartheta }^{h}\left(x,\alpha \right)=\underset{(u,f)\mathit{\epsilon \Gamma }\left(\alpha \right)}{\mathit{min}}\left[\frac{g\left(x,\alpha \right)}{{\Omega }_{h}^{\alpha }\left(1+\frac{\rho }{{\Omega }_{h}^{\alpha }}\right)}+\frac{1}{1+\frac{\rho }{{\Omega }_{h}^{\alpha }}}({P}_{x}^{\pm }\left(\alpha \right){ \vartheta }^{h}\left(x\pm h,\alpha \right)+{\sum}_{j\ne \alpha }{P}^{j}\left(\alpha \right){ \vartheta }^{h}\left(x,\alpha \right)\right]$$

(42)

Mode 1

Both machines are in operation. The rejection rate of the main machine is $\beta$, and the control fraction is ${f}_{1}$.

$$\frac{dx\left(t\right)}{dt}=\dot{x}={u}_{1}^{r}(t)+{u}_{2}^{r}\left(t\right)-{d}^{^{\prime}}$$

(43)

$$\begin{array}{c}{\vartheta }^{h}\left(x,1\right)=\underset{(u,f)\mathit{\epsilon \Gamma }\left(1\right)}{\mathit{min}}\left[\frac{g\left(x,1\right)}{{\Omega }_{h}^{1}(1+\rho /{\Omega }_{h}^{1})}+\frac{1}{1+\frac{\rho }{{\Omega }_{h}^{1}}}({P}_{x}^{\pm }\left(1\right){ \vartheta }^{h}\left(x\pm h,1\right)+\frac{{q}_{12}}{{\Omega }_{h}^{1}}{ \vartheta }^{h}\left(x,2\right)+\frac{{q}_{13}}{{\Omega }_{h}^{1}}{ \vartheta }^{h}\left(x,3\right)\right]\\ \mathrm {and} \\ \mathrm{AOQL}\le {\mathrm{AOQ}}_{\mathrm{max}}\end{array}$$

(44)

With: ${\Omega }_{h}^{1}=\left|{q}_{11}\right|+\frac{\left|\dot{x}\right|}{h}$ ${\mathrm{P}}_{x}^{+}\left(1\right)=\left\{\begin{array}{lc}\frac{\dot{x}}{h{\Omega }_{h}^{1}} & if\;\dot{x}>0\\ 0 & if\;not\end{array}\right.$

$${\mathrm{P}}_{x}^{-}\left(1\right)=\left\{\begin{array}{lc}\frac{-\dot{x}}{h{\Omega }_{h}^{1}} & if\;\dot{x}\le 0\\ 0&if \;not\end{array} ;\quad \right.\dot{x}={u}_{s}-d/(1-AOQ\left(1\right)$$

Mode 2

Only the main machine M1 is in operation. The rejection rate of the main machine is $\beta$, and the control fraction is ${f}_{2}.$

$$\frac{dx\left(t\right)}{dt}=\dot{x}={u}_{1}^{r}(t)-{d}^{^{\prime}}$$

(45)

$$\begin{array}{c}{\vartheta }^{h}\left(x,2\right)=\underset{(u,f)\mathit{\epsilon \Gamma }\left(2\right)}{\mathit{min}}\left[\frac{g\left(x,2\right)}{{\Omega }_{h}^{2}(1+\rho /{\Omega }_{h}^{2})}+\frac{1}{1+\frac{\rho }{{\Omega }_{h}^{2}}}({P}_{x}^{\pm }\left(2\right){ \vartheta }^{h}\left(x\pm h,2\right)+\frac{{q}_{24}}{{\Omega }_{h}^{2}}{ \vartheta }^{h}\left(x,4\right)+\frac{{q}_{25}}{{\Omega }_{h}^{2}}{ \vartheta }^{h}\left(x,5\right)\right] \\ \mathrm{and} \\ AOQL\le AO{Q}_{max}\end{array}$$

(46)

With: ${\Omega }_{h}^{1}=\left|{q}_{11}\right|+\frac{\left|\dot{x}\right|}{h}$ ${\mathrm{P}}_{x}^{+}\left(1\right)=\left\{\begin{array}{lc}\frac{\dot{x}}{h{\Omega }_{h}^{1}} & if\;\dot{x}>0\\ 0 & if\;not\end{array}\right.$

$${\mathrm{P}}_{x}^{-}\left(1\right)=\left\{\begin{array}{lc}\frac{-\dot{x}}{h{\Omega }_{h}^{1}} & if\;\dot{x}\le 0\\ 0 & if\;not\end{array} ; \right.\dot{x}={u}_{1}^{r}\left(t\right)-{d}^{^{\prime}}$$

Mode 3

Only the second M2 machine is in operation. No fraction to control in this case.

$$\frac{dx\left(t\right)}{dt}=\dot{x}={u}_{2}^{r}(t)-{d}^{^{\prime}}$$

(47)

$$\begin{array}{c}{\vartheta }^{h}\left(x,3\right)=\underset{\mathit{f\epsilon \Gamma }\left(1\right)}{\mathit{min}}\left[\frac{g\left(x,3\right)}{{\Omega }_{h}^{3}(1+\rho /{\Omega }_{h}^{3})}+\frac{1}{1+\frac{\rho }{{\Omega }_{h}^{3}}}({P}_{x}^{-}\left(3\right){ \vartheta }^{h}\left(x-h,3\right)+\frac{{q}_{31}}{{\Omega }_{h}^{3}}{ \vartheta }^{h}\left(x,1\right)\right]\\ \mathrm {and} \\ \mathrm{AOQL}\le {\mathrm{AOQ}}_{\mathrm{max}}\end{array}{c}$$

(48)

with: ${\Omega }_{h}^{3}=\left|{q}_{33}\right|+\frac{\left|\dot{x}\right|}{h}$

$${\mathrm{P}}_{x}^{-}\left(1\right)=\left\{\begin{array}{lc}\frac{-\dot{x}}{h{\Omega }_{h}^{1}} & if\;\dot{x}\le 0\\ 0 & if\;not\end{array}\right.$$

Mode 4: Both machines are out of order.

$$\dot{x}=-{d}^{^{\prime}}$$

(49)

$$\begin{array}{c}{\vartheta }^{h}\left(x,4\right)=\underset{\mathit{f\epsilon \Gamma }\left(2\right)}{\mathit{min}}\left[\frac{g\left(x,4\right)}{{\Omega }_{h}^{4}(1+\rho /{\Omega }_{h}^{4})}+\frac{1}{1+\frac{\rho }{{\Omega }_{h}^{4}}}({P}_{x}^{-}\left(4\right){ \vartheta }^{h}\left(x-h,4\right)+\frac{{q}_{42}}{{\Omega }_{h}^{4}}{ \vartheta }^{h}\left(x,2\right)\right]\\ \mathrm {and} \\AOQL\le AO{Q}_{max}\end{array}$$

(50)

with

$${\Omega }_{h}^{4}=\left|{q}_{44}\right|;$$

and

$${\mathrm{P}}_{x}^{-}\left(4\right)=\frac{-\dot{x}}{h{\Omega }_{h}^{4}}$$

Appendix B

Table 5 Optimal parameters depending on the variation of the system parameters