# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021142
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## Coordinated optimization of production scheduling and maintenance activities with machine reliability deterioration

 1 School of Management, Hefei University of Technology, 193 Tunxi Road, Hefei City, Anhui Province, China 2 Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, USA

* Corresponding author: Qian Xiaofei, Lu Shaojun

Received  March 2021 Revised  May 2021 Early access August 2021

Fund Project: The first author is supported by the National Key Research and Development Program of China (2019YFB1705300)

In this paper, we investigate a coordinated optimization problem of production and maintenance where the machine reliability decreases with the use of the machine. Lower reliability means the machine is more likely to fail during the production stage. In the event of a machine failure, corrective maintenance (CM) of the machine is required, and the CM of the machine will cause a certain cost. Preventive maintenance (PM) can improve machine reliability and reduce machine failures during the production stage, but it will also cause a certain cost. To minimize the total maintenance cost, we must determine an appropriate PM plan to balance these two types of maintenance. In addition, the tardiness cost of jobs is also considered, which is affected not only by the processing sequence of jobs but also by the PM decision. The objective is to find the optimal job processing sequence and the optimal PM plan to minimize the total expected cost. To solve the proposed problem, an improved grey wolf optimizer (IGWO) algorithm is proposed. Experimental results show that the IGWO algorithm outperforms GA, VNS, TS, and standard GWO in optimization and computational stability.

Citation: Chaoming Hu, Xiaofei Qian, Shaojun Lu, Xinbao Liu, Panos M Pardalos. Coordinated optimization of production scheduling and maintenance activities with machine reliability deterioration. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021142
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##### References:
Schematic diagram of the machine age
An example to illustrate the encoding method
A comparison of two functions
Flow chart of IGWO algorithm
(a), (t) Convergence curvers for different instances.
Relative percentage deviations of AVE
Relative percentage deviations of VAR
Main features of previous studies and ours
 Authors Production system Unexpected failures PM Objectives Solutions Berrichi et al. (2010) Parallel machine - Perfect Minimize $C_{max}$ as well as the system unavailability Multi-Objective Ant Colony Optimization approach Mokhtari et al. (2012) Parallel machine - Multiple-level Minimize the total system unavailability Population-based variable neighborhood search Liu et al.(2018) Single machine - Multiple-level Minimize total cost Genetic algorithm Hu et al.(2017) Single machine - Imperfect Minimize the total cost Enumeration method Lu et al. (2018) Parallel machine - Perfect Minimize makespan A hybrid ABC-TS algorithm Wong et al.(2013) Parallel machine - Multiple-level Minimize the makespan Genetic algorithm Kumar et al.(2017) Parallel machine - Perfect Minimize overall operations cost Simulation-based Genetic Algorithm Feng et al.(2018) Flexible flowshop ✓ Imperfect Minimize the total cost Simulated annealing embedded genetic algorithm Lu et al. (2015) Single machine ✓ Perfect System robustness and stability Genetic algorithm Benmansour et al.(2011) Single machine ✓ Perfect Minimize the cost related to production and maintenance Heuristic algorithm An et al.(2020) Flexible job-shop - Imperfect Minimize the makespan and total cost Evolutionary algorithm This paper Single machine ✓ Imperfect Minimize the total expected cost Improved grey wolf optimizer algorithm
 Authors Production system Unexpected failures PM Objectives Solutions Berrichi et al. (2010) Parallel machine - Perfect Minimize $C_{max}$ as well as the system unavailability Multi-Objective Ant Colony Optimization approach Mokhtari et al. (2012) Parallel machine - Multiple-level Minimize the total system unavailability Population-based variable neighborhood search Liu et al.(2018) Single machine - Multiple-level Minimize total cost Genetic algorithm Hu et al.(2017) Single machine - Imperfect Minimize the total cost Enumeration method Lu et al. (2018) Parallel machine - Perfect Minimize makespan A hybrid ABC-TS algorithm Wong et al.(2013) Parallel machine - Multiple-level Minimize the makespan Genetic algorithm Kumar et al.(2017) Parallel machine - Perfect Minimize overall operations cost Simulation-based Genetic Algorithm Feng et al.(2018) Flexible flowshop ✓ Imperfect Minimize the total cost Simulated annealing embedded genetic algorithm Lu et al. (2015) Single machine ✓ Perfect System robustness and stability Genetic algorithm Benmansour et al.(2011) Single machine ✓ Perfect Minimize the cost related to production and maintenance Heuristic algorithm An et al.(2020) Flexible job-shop - Imperfect Minimize the makespan and total cost Evolutionary algorithm This paper Single machine ✓ Imperfect Minimize the total expected cost Improved grey wolf optimizer algorithm
Notations and their explanations
 $N$ Number of jobs $p_j$ The processing time of job $j$, $j = 1,…,N$ $d_j$ The due date of job $j$, $j = 1,…,N$ $w_j$ Tardiness cost per unit of time of job $j$, $j = 1,…,N$ $c_l$ The unit basic cost of the factory. $t_r$ The time required for performing corrective maintenance (CM). $c_r$ The cost of a CM activity. $t_p$ The time required for performing preventive maintenance (PM). $c_p$ The cost of a PM activity. $x_{[i]j}$ Job sequencing decision variable, and it is a zero and one variable $y_{[i]}$ PM decision variable, and it is a zero and one variable $p_{[i]}$ The processing time of the $i$th job in the sequence $f_{[i]}(t)$ The failure rate function during the processing of the $i$th job and $t$ is the machine age (i.e. the time since the last PM) $m$ The number of PMs performed on the machine before time 0. $Z_{[0]}^{f}$ The machine age at time 0. $Z_{[i]}^{s}$ The machine age when starting the processing of the $i$th job $Z_{[i]}^{f}$ The machine age when finishing the processing of the $i$th job $E(N_{[i]})$ The expected number of failures during the processing of the $i$th job $E(C_{[i]})$ The expected completion time of the $i$th job $E(T_{[i]})$ The expected tardiness time of the $i$th job $E(TC)$ The expected total cost.
 $N$ Number of jobs $p_j$ The processing time of job $j$, $j = 1,…,N$ $d_j$ The due date of job $j$, $j = 1,…,N$ $w_j$ Tardiness cost per unit of time of job $j$, $j = 1,…,N$ $c_l$ The unit basic cost of the factory. $t_r$ The time required for performing corrective maintenance (CM). $c_r$ The cost of a CM activity. $t_p$ The time required for performing preventive maintenance (PM). $c_p$ The cost of a PM activity. $x_{[i]j}$ Job sequencing decision variable, and it is a zero and one variable $y_{[i]}$ PM decision variable, and it is a zero and one variable $p_{[i]}$ The processing time of the $i$th job in the sequence $f_{[i]}(t)$ The failure rate function during the processing of the $i$th job and $t$ is the machine age (i.e. the time since the last PM) $m$ The number of PMs performed on the machine before time 0. $Z_{[0]}^{f}$ The machine age at time 0. $Z_{[i]}^{s}$ The machine age when starting the processing of the $i$th job $Z_{[i]}^{f}$ The machine age when finishing the processing of the $i$th job $E(N_{[i]})$ The expected number of failures during the processing of the $i$th job $E(C_{[i]})$ The expected completion time of the $i$th job $E(T_{[i]})$ The expected tardiness time of the $i$th job $E(TC)$ The expected total cost.
Local search strategy
 Step 1 Set the maximum number of searches $s_{max}$ and let $s = 0$ Step 2 Randomly generate three unequal integer $a,b,c$ in $[1,N]$ Step 3 Let $t = x_a, x_a = x_b, x_b = t$, and $x_c = - x_c$ and then obtain a new solution $X^{'}$ Step 4 Calculate the fitness of $X^{'}$, and let $s = s+1$ Step 5 Judge whether $f(X^)  Step 1 Set the maximum number of searches$ s_{max} $and let$ s = 0 $Step 2 Randomly generate three unequal integer$ a,b,c $in$ [1,N] $Step 3 Let$ t = x_a, x_a = x_b, x_b = t $, and$ x_c = - x_c $and then obtain a new solution$ X^{'} $Step 4 Calculate the fitness of$ X^{'} $, and let$ s = s+1 $Step 5 Judge whether$ f(X^)
Pseudocode of IGWO algorithm
 1 Initializes the grey wolf population 2 Initializes the max number of iterations $t_{max}$ and the local search probability $\varepsilon$, and set $a = 2$, $k = 1$ 3 Calculate the fitness of all individuals 4 Set the three individuals with the best fitness as $X_\alpha$, $X_\beta$, and $X_\gamma$ 5 while($k  1 Initializes the grey wolf population 2 Initializes the max number of iterations$ t_{max} $and the local search probability$ \varepsilon $, and set$ a = 2 $,$ k = 1 $3 Calculate the fitness of all individuals 4 Set the three individuals with the best fitness as$ X_\alpha $,$ X_\beta $, and$ X_\gamma $5 while($ k
Experimental factors and their levels
 The total number of jobs, $N$ 10, 20, 30, …, 180,190,200 The failure rate adjustment parameter, $\theta$ 1.05 The failure rate parameter, $\beta$ 2 The failure rate parameter, $\eta$ 50 The processing time of job $j$, $p_j$ Uniform$[1,10]$ The due date of job $j$, $d_j$ Uniform $[1,100]$ The tardiness cost per unit of time cost of job $j$, $w_j$ Uniform $[0, 1]$ The unit production cost, $c_l$ 1 The time required for performing corrective maintenance, $t_r$ 1 The cost of a corrective maintenance activity, $c_r$ 1 The time required for performing preventive maintenance, $t_p$ 2 The cost of a preventive maintenance activity, $c_p$ 2
 The total number of jobs, $N$ 10, 20, 30, …, 180,190,200 The failure rate adjustment parameter, $\theta$ 1.05 The failure rate parameter, $\beta$ 2 The failure rate parameter, $\eta$ 50 The processing time of job $j$, $p_j$ Uniform$[1,10]$ The due date of job $j$, $d_j$ Uniform $[1,100]$ The tardiness cost per unit of time cost of job $j$, $w_j$ Uniform $[0, 1]$ The unit production cost, $c_l$ 1 The time required for performing corrective maintenance, $t_r$ 1 The cost of a corrective maintenance activity, $c_r$ 1 The time required for performing preventive maintenance, $t_p$ 2 The cost of a preventive maintenance activity, $c_p$ 2
Comparison of GWO and IGWO
 GWO IGWO Instance MIN MAX AVE VAR MIN MAX AVE VAR N = 10 72 78 76 0.87 65 67 66 0.28 N = 20 315 355 338 124 244 262 253 28 N = 30 595 641 615 263 420 479 442 198 N = 40 1595 1680 1649 3668 1330 1489 1424 11075 N = 50 2462 2683 2610 2375 2311 2518 2405 2231 N = 60 3608 4046 3849 14612 3171 3336 3266 2206 N = 70 4888 5268 5057 17758 4481 4722 4583 5833 N = 80 6760 7107 6929 20810 5909 6339 6089 9859 N = 90 7037 7328 7174 13697 6179 6544 6357 9622 N = 100 10781 11429 11147 35652 9768 10393 10101 33610 N = 110 13256 14317 13811 103514 12511 13184 12788 53694 N = 120 17281 18388 17783 106493 15711 16953 16331 104363 N = 130 20102 21580 20758 214099 18108 19249 18573 63512 N = 140 20919 22114 21594 112116 18600 19460 19020 20809 N = 150 23002 24554 23793 246926 19369 20752 19891 56352 N = 160 29341 30523 29801 157775 25098 25768 25436 61450 N = 170 34321 36133 35142 420426 28998 30418 29632 134003 N = 180 38393 40478 39255 390366 32914 34377 33557 177787 N = 190 49190 51420 50013 395090 42772 44139 43557 177696 N = 200 50416 52568 51498 575500 43267 44589 44003 102377
 GWO IGWO Instance MIN MAX AVE VAR MIN MAX AVE VAR N = 10 72 78 76 0.87 65 67 66 0.28 N = 20 315 355 338 124 244 262 253 28 N = 30 595 641 615 263 420 479 442 198 N = 40 1595 1680 1649 3668 1330 1489 1424 11075 N = 50 2462 2683 2610 2375 2311 2518 2405 2231 N = 60 3608 4046 3849 14612 3171 3336 3266 2206 N = 70 4888 5268 5057 17758 4481 4722 4583 5833 N = 80 6760 7107 6929 20810 5909 6339 6089 9859 N = 90 7037 7328 7174 13697 6179 6544 6357 9622 N = 100 10781 11429 11147 35652 9768 10393 10101 33610 N = 110 13256 14317 13811 103514 12511 13184 12788 53694 N = 120 17281 18388 17783 106493 15711 16953 16331 104363 N = 130 20102 21580 20758 214099 18108 19249 18573 63512 N = 140 20919 22114 21594 112116 18600 19460 19020 20809 N = 150 23002 24554 23793 246926 19369 20752 19891 56352 N = 160 29341 30523 29801 157775 25098 25768 25436 61450 N = 170 34321 36133 35142 420426 28998 30418 29632 134003 N = 180 38393 40478 39255 390366 32914 34377 33557 177787 N = 190 49190 51420 50013 395090 42772 44139 43557 177696 N = 200 50416 52568 51498 575500 43267 44589 44003 102377
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