A Joint Optimization of Maintenance and Scheduling for Unrelated Parallel Machine Problem Based on Hybrid Discrete Spider Monkey Optimization Algorithm ^†

Abstract

In general, the parallel machine scheduling problem that minimizes maximum completion time is NP-hard in a strong sense; a lot of heuristics have been proposed for this kind of problem. In this paper, the unrelated parallel machine scheduling problem with maintainability (UPMSPM) is studied, in which the reliability of machines obeys exponential distribution. A hybrid algorithm HDSMO, which combines the discrete spider monkey algorithm (SMO) with the crossover and mutation operation, is proposed to solve UPMSPM. In view of the lack of local search capability in the later iteration of the traditional SMO algorithm, inertial weights are introduced to update the local leader and the global leader. Computational experiments with randomly generated instances demonstrate that the proposed HDSMO algorithm can obtain significantly better solutions in a shorter time than GA and SMO algorithms.

1. Introduction

UPMSP is an important branch of production scheduling. In the real-world production system, long-term running wear and performance degradation of the machines can easily lead to production interruptions, requiring preventive maintenance (PM) to keep machines running [1]. Therefore, it is of great significance to consider the joint optimization of maintenance and scheduling for UPMSP [2]. UPMSP studies considering maintenance are relatively rare, and several classic studies are as follows [3].

Cheng et al. studied UPMSP with degradation and maintenance and proved that the problem could be optimally solved in polynomial time [4]. Avalos-Rosales et al. studied unrelated parallel machines and considered preventive maintenance activities and setup times by order and by machine [5]. Luo J et al. proposed a predictable scheduling and rescheduling and accounting for machine failures and consistency in unrelated machine environments, where work separations include printed circuit boards (PCB) [6].

Comparatively speaking, the research on UPMSP based on the Spider Monkey Optimization (SMO) algorithm is rare. Aiming at the optimization problem of unrelated parallel machine maintenance and scheduling integration, this paper proposes a hybrid spider monkey algorithm, and compares it with classical algorithms to provide the foundation for solving UPMSP [7].

2. Problem Formulation

The problem studies in this paper can be described as follows: $n$ jobs are to be processed on $m$ unrelated parallel machines; in most situations, we assume m is less than n, and these jobs are non-preemptive and can all be processed at time 0. Maintenance performed on the machine may depend on the state of the machine (e.g., running time). The state of a machine is determined by reliability, which decreases with the cumulative processing time of the workpiece or degradation of the machine. Once the reliability of the machine falls below the threshold $r_{th}$, PM must be implemented. The reliability of the machine does not change during operation.

Using the three-field notation α|β|γ for describing scheduling problems, we denote our problem by $\mathrm{Rm}/\mathrm{nr},\mathrm{VPM}/{\mathrm{C}}_{\max }$, where $nr$ denotes those jobs are non-resumable; “$\mathrm{VPM}$” denotes variable PM; the objective is to minimize the maximum completion time. The decision is to determine the allocation and sequence of $n$ jobs on $m$ machines and the maintenance time of the machines. Since problem $\mathrm{Rm}//{\mathrm{C}}_{\max }$ has been proved to be an NP-Hard problem, it can be concluded that problem $\mathrm{Rm}/\mathrm{nr},\mathrm{VPM}/{\mathrm{C}}_{\max }$ is an NP-Hard problem by comparison. Thus, the approximate methods are needed to solve real-size instances.

3. HDSMO Algorithm

3.1. Basic Flow of the HDSMO Algorithm

SMO is a proposed global optimization algorithm; the main feature is that it can improve the ability to search for optimal solutions. However, in the traditional SMO algorithm, the spider monkey individual $S{M}_h$ completely inherits the old location information of the individual in the updating process, which makes the algorithm lack the local search ability in the late iteration. An HDSMO algorithm considering inertia weight aims at the above problems and shortcomings. $n_{llc}$ and $n_{lll}$ represent the local leader counter and limit, respectively, while $n_{glc}$ and $n_{gll}$ represents the global leader counter and limit. The process of the proposed HDSMO algorithm is shown in Figure 1.

3.2. Local Leader Phase (LLP) Update with the Inertia Weight

$$S{M}_{ne{w}_h}=p_1\otimes f\left(p_r\otimes g\left(p_w\otimes v(S{M}_h\right),L{L}_l\right),S{M}_r)$$

(1)

The position update process in the local leader stage of the SMO algorithm is shown in Equation (1): the population is first divided into different groups, $v(S{M}_h)$ is the mutation operation added to enhance the local search ability according to inertia weight $P_w$. For the individuals of the first 50% generation population and the last 50% generation population, the mutation operation methods of reverse order and two-point exchange can be used respectively, which can effectively improve the diversity of the population and further improve the local search ability of the algorithm. The mutation method is shown in Figure 2, where 0 represents the machine, and the remaining numbers represent the job.

$g\left(p_w\otimes v(S{M}_h\right),L{L}_l)$ and $f\left(p_r\otimes g\left(p_w\otimes v(S{M}_h\right),L{L}_l\right),S{M}_r)$ represent crossover operations. The mutant individuals cross with $LL$ according to crossover rate $P_r$, and the generated new individuals cross with random individuals according to crossover rate $P_1$. In this paper, two crossover methods are designed based on whether there are identical parts between individuals, as shown in Figure 3.

3.3. Global Leader Phase (GLP) Update with the Inertia Weight

At this phase, individual $S{M}_h$ mutates according to crossover rate inertia weight $P_w$, and then crosses with $GL$ according to crossover rate $P_r$, and the generated new individuals cross with random individuals according to crossover rate $P_2$. The same method is shown in Section 3.2.

4. Numerical Example and Analysis

4.1. Parameters Setting

The experimental data include the number of machines $m$, the number of jobs $n$, the processing time $p_{ij}$, the PM parameters including the threshold $UT$, and the maintenance time $t^{PM}$. For each combination of problem instance size, Generate 10 random problem instances. The instances and the range of experimental parameters are shown in

Table 1. Experimental problem scale and parameter range.

Size	$\mathit{m}$	$\mathit{n}$	${\mathit{p}}_{\mathit{i}\mathit{j}}$	${\mathit{t}}^{\mathit{P}\mathit{M}}$
Small	2,3	6,8	$U\left[1, 100\right]$	10
Medium	4,5	30,50	$U\left[1, 100\right]$	20
Large	10,20	100,200	$U\left[1, 100\right]$	20

, the parameters of the GA algorithm and the DSMO algorithm are experimentally analyzed, and the algorithm parameter values under different problem scales are determined as shown in

Table 2. Parameter values for algorithms.

Parameter	GA			DSMO			HDSMO
	Small	Medium	Large	Small	Medium	Large	Small	Medium	Large
N	100	350	500	100	300	450	100	350	200
$Maxt$	200	400	500	100	400	500	100	400	500
$p_1$	0.4	0.3	0.5	0.3	0.6	0.8	0.2	0.3	0.5
$p_2$	0.15	0.2	0.2	0.5	0.6	0.8	0.4	0.3	0.5
$p_w$	/	/	/	/	/	/	0.05	0.25	0.35

.

4.2. Computational Experiments and Discussion

The computational experiments result for the different algorithms are given in

Table 3. The performance of the algorithms.

Size	n*m*	GA		DSMO		HDSMO
		CT	PD	CT	PD	CT	PD
Small	n6m2	1.741	0	0.904	0	3.237	0
	n6m3	1.619	0	1.027	0.027	4.226	0
	n8m2	1.417	0	0.982	0.012	3.45	0
	n8m3	2.239	0.008	1.139	0.031	4.394	0.002
Medium	n30m4	19.499	0.461	43.592	0.436	50.455	0
	n30m5	18.533	0.764	45.173	0.711	52.502	0
	n50m4	25.016	0.899	67.643	0.92	72.982	0.007
	n50m5	24.789	1.137	68.049	1.175	71.131	0
Large	n100m10	121.056	3.843	289.134	3.926	138.101	0
	n100m20	179.374	7.294	329.983	7.1	169.284	0
	n200m10	215.577	4.709	546.221	4.633	247.991	0.002
	n200m20	255.377	10.451	587.604	9.618	265.533	0.005

. Each algorithm calculates the average relative percentage deviation ($PD$) from the optimal $C_{max}$ solution, i.e., the value $PD=\frac{C_{max}-C_{max}^{*}}{C_{max}^{*}}$. There is also the average computed time in seconds ($CT$).

It can be concluded from Table 3 that HDSMO is superior to DSMO and GA in average relative percentage deviation for three scale problems. However, in terms of computation time, the DSMO algorithm outperforms GA and HDSMO for the small problems, and the needed computation time of HDSMO is decreased with the increase in the problem size. The HDSMO algorithm is a recommended method for solving large and medium-sized problems because it can give approximate optimal solutions in a short computing time.

5. Conclusions

According to the property of the addressed problem and the decision-making method of “job-grouping batch and allocating”, a hybrid discrete SMO algorithm is proposed in this paper. Experimental results demonstrate that HDSMO is superior to GA and DSMO in solving quality and effectiveness.