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Integrated optimization on production scheduling and imperfect preventive maintenance considering multi-degradation and learning-forgetting effects

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Flexible Services and Manufacturing Journal Aims and scope Submit manuscript

Abstract

This paper proposes a multi-objective integrated optimization model of production scheduling and machine maintenance to find the optimal production sequence and preventive maintenance (PM) decisions. This model considers the setup time, the learning-forgetting effects and the multi-degradation effects. The setup time and the learning-forgetting effects are associated with jobs' similarities and PM decisions. The multi-degradation effects including machine deterioration, failure rate and quality characteristic loss determine the stochastic nature of the objectives. A hybrid maintenance strategy combining imperfect PM and minimal repair (MR) is adopted to effectively reduce the failure frequency and improve the processing quality. Then, the multi-objective solution is simplified by normalizing cost, time and utilization. The local search and the elitism strategy are conducted to avoid the solutions falling into local optimum and losing the best chromosome during evolutions. Finally, a case study of automobile engine manufacturing shows that our proposed model can reduce the total maintenance costs by 27%, shorten the total processing time by 3%, as well as improve the machine utilization by 3%.

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Abbreviations

\(G_{i}\) :

\(i{th}\) Group (\(i \in \{ 1,2,...,h\}\)), where \(h\) is the total number of groups

\(n_{i}\) :

Number of job types in \(G_{i}\)

\(J_{i,j}\) :

\(j{th}\) Job type in \(G_{i}\), where \(j \in \{ 1,2, \ldots ,n_{i} \}\)

\(q_{i,j}\) :

Production batch of \(J_{i,j}\)

\(L_{PM}\) :

Machine processing quality characteristic loss threshold

\(\upsilon\),\(\tau\) :

Degradation coefficients of processing quality characteristic distribution

\(l(x)\) :

Machine processing quality characteristic loss function

\(L(x(t))\) :

Machine processing quality characteristic loss at runtime t

\(C_{L}\) :

Scraping loss

\(C_{U}\) :

Reworking loss

\(k_{1}\),\(k_{2}\) :

Processing quality characteristic loss coefficient which bias toward scrapping and reworking

\(\lambda_{PM}\) :

Failure rate threshold

\(\beta\),\(\theta\) :

Shape parameter and size parameter of failure rate function

\(\lambda^{ - }\), \(\lambda^{ + }\) :

Machine failure rate before and after PM

\(\gamma_{0}\), \(\gamma_{1}\) :

Constants related to the difficulty of PM

\(T_{{pm_{{_{i,j} }} }}\) :

Time of PM before processing \(J_{i,j}\)

\(c\) :

PM cost per unit time

\(C_{{pm_{{_{i,j} }} }}\) :

Cost of PM before processing \(J_{i,j}\)

\(a_{i,j}^{ - }\), \(a_{{_{i,j} }}^{ + }\) :

Machine's age before and after processing \(J_{i,j}\)

\(t_{mr}\), \(c_{mr}\) :

Time and cost of MR

\(n_{{mr_{i,j} }}\) :

Expected times of MR while processing \(J_{i,j}\)

\(T_{{mr_{{_{i,j} }} }}\), \(C_{{mr_{{_{i,j} }} }}\) :

Expected time and cost of MR while processing \(J_{i,j}\)

\(t_{i,j}\) :

Standard processing time of \(J_{i,j}\)

\(s\), \(F\) :

Basic setup time of jobs in same group and between groups

\(s_{i,j}^{i}\) :

Actual setup time between \(J_{i,j}\) and \(J_{i,j + 1}\)

\(F_{i,i + 1}\) :

Actual setup time between \(G_{i}\) and \(G_{i + 1}\)

\(r_{j,k}^{i}\) :

Similarity between \(J_{i,j}\) and \(J_{i,k}\)

\(r_{i,l}\) :

Similarity between \(G_{i}\) and \(G_{l}\)

\(p_{i,j,g}\) :

Processing time of the \(g^{th}\) job of \(J_{i,j}\), where \(g \in \{ 1,2,...,q_{ij} \}\)

\(\varepsilon\) :

Fixed learning rate

\(p_{{_{i,j} }}^{ * }\) :

Processing time of \(J_{i,j}\) considering learning-forgetting effects

\(p_{i,j}\) :

Three-effect based processing time of \(J_{i,j}\)

\(f(r_{j - 1,j}^{i} ,s_{j - 1,j}^{i} )\) :

Forgetting rate between jobs in same group

\(\varphi\) :

Forgetting coefficient between jobs in same group

\(f^{^{\prime}} (r_{i - 1,i}^{G} ,F_{i - 1,i} )\) :

Forgetting rate between different groups

\(\phi\) :

Forgetting coefficient between different groups

\(\alpha_{k}\) :

Deteriorating factor

\(\partial\) :

Deteriorating weight

\(a^{*}\) :

Machine's age when the failure rate reaches the threshold

\(a_{0}\) :

Machine's initial age

\(F_{0}\) :

Setup time for \(G_{{1}}\)

\(E\left( {T_{i,j} } \right)\) :

Expected completion time of \(J_{i,j}\)

\(E\left( {C_{i,j} } \right)\) :

Expected total cost after processing \(J_{i,j}\)

\(R\) :

Machine utilization

\(T_{min}\), \(T_{max}\) :

Minimum and maximum completion time of all feasible solutions

\(C_{min}\), \(C_{max}\) :

Minimum and maximum total maintenance cost of all feasible solutions

\(R_{min}\), \(R_{max}\) :

Minimum and maximum machine utilization of all feasible solutions

\(\omega_{t}\), \(\omega_{c}\), \(\omega_{r}\) :

Weight of completion time, maintenance cost and machine utilization

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Acknowledgements

The research is funded by National Natural Science Foundation of China (Grant No.72071127 and 51875359) and Key Laboratory of Quality Infrastructure Efficacy Research of the State Administration for Market Regulation of China (KF20180302).

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, School of Mechanical Engineering, Shanghai Jiao Tong University, No. 800, Dongchuan Road, Shanghai, 200240, China

    Xiufang Zhang, Tangbin Xia, Ershun Pan & Yuqing Li

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  1. Xiufang Zhang
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  2. Tangbin Xia
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  3. Ershun Pan
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Correspondence to Ershun Pan.

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Appendices

Appendix

The derivation of equation (20):

$$ \begin{gathered} p_{{_{\left[ i \right]\left[ j \right]} }}^{ * } = p_{\left[ i \right]\left[ j \right]1} { + }p_{\left[ i \right]\left[ j \right]2} { + }p_{\left[ i \right]\left[ j \right]3} { + }...{ + }p_{{\left[ i \right]\left[ j \right]q_{\left[ i \right]\left[ j \right]} }} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = p_{\left[ i \right]\left[ j \right]1} { + }p_{\left[ i \right]\left[ j \right]1} \cdot 2^{ - \eta } { + }p_{\left[ i \right]\left[ j \right]1} \cdot 3^{ - \eta } { + }...{ + }p_{\left[ i \right]\left[ j \right]1} \cdot q_{\left[ i \right]\left[ j \right]}^{ - \eta } \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = p_{\left[ i \right]\left[ j \right]1} \cdot (1 + 2^{ - \eta } + 3^{ - \eta } { + }... + q_{\left[ i \right]\left[ j \right]}^{ - \eta } ) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = p_{\left[ i \right]\left[ j \right]1} \sum\limits_{g = 1}^{{q_{\left[ i \right]\left[ j \right]} }} {g^{ - \eta } } \hfill \\ \end{gathered} $$

The derivation of equation (21):

$$ \begin{gathered} p_{\left[ i \right]\left[ j \right]1} = t_{\left[ i \right]\left[ j \right]} \times \left( {\frac{{p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }} + f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ) \times \frac{{t_{{\left[ i \right]\left[ {j - 1} \right]}} - p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = t_{\left[ i \right]\left[ j \right]} \times \left( {\frac{{p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }} + f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ) \times (1 - \frac{{p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }})} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = t_{\left[ i \right]\left[ j \right]} \times \left( {\frac{{p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }} + f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ) - f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ) \times \frac{{p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = t_{\left[ i \right]\left[ j \right]} \times \left( {f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ) + (1 - f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} )) \times \frac{{p_{{\left[ i \right]\left[ {j - 1} \right]1}} q_{{\left[ i \right]\left[ {j - 1} \right]}}^{ - \eta } }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }}} \right) \hfill \\ \, = t_{\left[ i \right]\left[ j \right]} \times \left( {f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ) + (1 - f(r_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} ,s_{{\left[ {j - 1} \right]\left[ j \right]}}^{\left[ i \right]} )) \times \frac{{p_{{\left[ i \right]\left[ {j - 1} \right]q_{{\left[ i \right]\left[ {j - 1} \right]}} }} }}{{t_{{\left[ i \right]\left[ {j - 1} \right]}} }}} \right) \hfill \\ \end{gathered} $$

The derivation of equation (23):

$$ \begin{gathered} p_{\left[ i \right]\left[ 1 \right]1} = t_{\left[ i \right]\left[ 1 \right]} \times \left( {\frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }} + f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} ) \times \frac{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} - p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = t_{\left[ i \right]\left[ 1 \right]} \times \left( {\frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }} + f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} ) \times (1 - \frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }})} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = t_{\left[ i \right]\left[ 1 \right]} \times \left( {\frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }} + f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} ) - f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} ) \times \frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = t_{\left[ i \right]\left[ 1 \right]} \times \left( {f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} ) + (1 - f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} )) \times \frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]1}} q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}}^{ - \eta } }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }}} \right) \hfill \\ \, = t_{\left[ i \right]\left[ 1 \right]} \times \left( {f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} ) + (1 - f^{^{\prime}} (r_{{\left[ {i - 1} \right]\left[ i \right]}}^{G} ,F_{{\left[ {i - 1} \right]\left[ i \right]}} )) \times \frac{{p_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]q_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }} }}{{t_{{\left[ {i - 1} \right]\left[ {n_{i - 1} } \right]}} }}} \right) \hfill \\ \end{gathered} $$

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Zhang, X., Xia, T., Pan, E. et al. Integrated optimization on production scheduling and imperfect preventive maintenance considering multi-degradation and learning-forgetting effects. Flex Serv Manuf J 34, 451–482 (2022). https://doi.org/10.1007/s10696-021-09410-1

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