Coordinated optimization of production scheduling and maintenance activities with machine reliability deterioration

Chaoming Hu; Xiaofei Qian; Shaojun Lu; Xinbao Liu; Panos M Pardalos

doi:10.3934/jimo.2021142

Article Contents

2022, Volume 18, Issue 6: 3953-3973. Doi: 10.3934/jimo.2021142

This issue Previous Article Design of an environmental contract under trade credits and carbon emission reduction Next Article Nonmonotone spectral gradient method based on memoryless symmetric rank-one update for large-scale unconstrained optimization

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
^* Corresponding author: Qian Xiaofei, Lu Shaojun

Received: March 2021

Revised: May 2021

Early access: August 2021

Published: September 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Failure rate adjustment model

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Figure 2. Schematic diagram of the machine age

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Figure 3. An example to illustrate the encoding method

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Figure 4. A comparison of two functions

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Figure 5. Flow chart of IGWO algorithm

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Figure 6. (a), (t) Convergence curvers for different instances.

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Figure 7(a). Relative percentage deviations of AVE

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Figure 7(b) . Relative percentage deviations of VAR

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Table 1. 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

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Table 2. 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.

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Table 3. 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^)<f(X) $ is true, if so, let $ X \leftarrow X^{'} $ and stop the search, otherwise, go to step 6
Step 6	Judge whether $ s\leq or \le s_{max} $, if so, turn to Step 2, otherwise keep $ X $ and stop search

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Table 4. 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<t_{max} $)
6	*for* each search agent
7	generate a random probability $ p $
8	if $ p\geq \varepsilon $
9	generate two random numbers $ r_1 $, $ r_2 $ in $ (0,1) $
10	calculate the coefficients $ A $, $ C $ by formula (4.3)-(4.4)
11	update the position of the current search agent by formula (4.5)-(4.7)
12	*else if*
13	update the position of the current search agent by Local search strategy
14	*end if*
15	update the population
16	calculate the fitness of all individuals
17	update $ X_\alpha $, $ X_\beta $, and $ X_\gamma $
18	update $ a $ by formula(4.8)
19	$ k = k+1 $
20	*end while*
21	rerurn $ X_\alpha $

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Table 5. 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

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Table 6. 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

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