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Buffer allocation and preventive maintenance optimization in unreliable production lines

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Abstract

In this paper, we consider a serial production line consisting of \(n\) unreliable machines with \(n-1\) buffers. The objective is to determine the optimal preventive maintenance policy and the optimal buffer allocation that will minimize the total system cost subject to a given system throughput level. We assume that the mean time between failure of all machines will be increased after performing periodic preventive maintenance. An analytical decomposition-type approximation is used to estimate the production line throughput. The optimal design problem is formulated as a combinatorial optimization one where the decision variables are buffer levels and times between preventive maintenance. To solve this problem, the extended great deluge algorithm is proposed. Illustrative numerical examples are presented to illustrate the model.

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Acknowledgments

The author would like to knowledge the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through project No. FT-131006.

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Correspondence to Nabil Nahas.

Appendix

Appendix

Additional notations

\(p_{s}(i)\) :

Probability of machine \(M_{d}(i)\) being starved in line \(i\).

\(p_{b}(i)\) :

Probability of machine \(M_{u}(i)\) being blocked in line \(i\).

\(E(i) \) :

Efficiency of line \(i\)

\(e_{i }\) :

The isolated efficiency of machine \(M_{i}\)

DDX Algorithm

Step 1: Initialize

$$\begin{aligned} \begin{array}{lll} \lambda _{u}(1)&{}=&{} \lambda _{1}, \mu _{u}(1)= \mu _{1}\\ \lambda _{d}(i)&{}=&{} \lambda _{i+1}, \mu _{d}(i)= \mu _{i+1} \quad \hbox { For }i =1,\ldots ,n \end{array} \end{aligned}$$

Step 2: For \(\hbox {i} = 2, 3, \ldots , n -1\), calculate \(I_u (i), \mu _{u}(i) \,{and} \lambda _{u}(i)\) using the following equations:

$$\begin{aligned} \begin{array}{l} I_u (i)=\displaystyle \frac{1}{E(i-1)}+\frac{1}{e_i }-I_d (i-1)-2 \\ \quad \hbox {for any } {i} = 2,\ldots , n-1\\ \mu _u (i)=X \cdot \mu _u (i-1)+(1-X) \cdot \mu _i \\ \quad \hbox {for any } i = 2,\ldots , n-1 \end{array} \end{aligned}$$

where

$$\begin{aligned} X&= \frac{p_s (i-1)}{I_u (i)\cdot E(i-1)}\\ I_u (i)&= \frac{\lambda _u (i)}{\mu _u (i)} \qquad and \qquad {e_u (i)=\frac{\mu _u (i)}{\lambda _u (i)+\mu _u (i)}} \end{aligned}$$

Step 3: For \(\hbox {i} = \hbox {n}- 2,\hbox {n}-3,\ldots ,1\), calculate \(I_d (i), \mu _{d}(i) \,{ and } \lambda _{d}(i)\) using the following equations:

$$\begin{aligned} \begin{array}{l} I_d (i)=\displaystyle \frac{1}{E(i+1)}+\frac{1}{e_{i+1} }-I_u (i+1)-2 \\ \quad \hbox {for any }\,i= 1,\ldots , n-2\\ \mu _d (i)=Y.\mu _d (i+1)+(1-Y).\mu _{i+1}\\ \quad \hbox {for any }\,i = 1,\ldots , n-2 \end{array} \end{aligned}$$

where

$$\begin{aligned} Y&= \frac{p_b (i+1)}{I_d (i)\cdot E(i+1)}\\ I_d (i)&= \frac{\lambda _d (i)}{\mu _d (i)}\quad and \quad {e_d (i)=\frac{\mu _d (i)}{\lambda _d (i)+\mu _d (i)}} \end{aligned}$$

Go to step 2 until convergence of the unknown parameters.

For given values of \(\lambda _{u}(i), \mu _{u}(i),\lambda _{d}(i)\) and \(\mu _{d}(i)\), the performance parameters \(E(i)\), \(p_{s}(i)\) and \(p_{b}(i)\) and the average buffer level \(Q(i)\) can be obtained from Dubois and Forestier (1982).

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Nahas, N. Buffer allocation and preventive maintenance optimization in unreliable production lines. J Intell Manuf 28, 85–93 (2017). https://doi.org/10.1007/s10845-014-0963-y

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