Modified Harris Hawks Optimizer for Solving Machine Scheduling Problems

Abstract

Scheduling can be described as a decision-making process. It is applied in various applications, such as manufacturing, airports, and information processing systems. More so, the presence of symmetry is common in certain types of scheduling problems. There are three types of parallel machine scheduling problems (PMSP): uniform, identical, and unrelated parallel machine scheduling problems (UPMSPs). Recently, UPMSPs with setup time had attracted more attention due to its applications in different industries and services. In this study, we present an efficient method to address the UPMSPs while using a modified harris hawks optimizer (HHO). The new method, called MHHO, uses the salp swarm algorithm (SSA) as a local search for HHO in order to enhance its performance and to decrease its computation time. To test the performance of MHHO, several experiments are implemented using small and large problem instances. Moreover, the proposed method is compared to several state-of-art approaches used for UPMSPs. The MHHO shows better performance in both small and large problem cases.

1. Introduction

Parallel machine scheduling only has an important role not in manufacturing (e.g., electronic and chemical industries), but in many scheduling processes, including transport scheduling like train schedules, airport management like landing scheduling, and hospital management, such as surgery scheduling. Symmetry also has a presence in different scheduling problems [1,2]. Parallel machine scheduling can be considered as complex artificial systems that have the characteristics of randomness, multi constraint, complexity, and multi-objective, etc. Job scheduling problems are known as NP-hard problems, due to their complexity, randomness, multi-objectives, and multi-variables [3]. Therefore, machine job scheduling problems are very important and they have gained a huge influence to increase manufacturing productivity and to improve service operations. More so, they designate limited resources to jobs with reference to operational constraints (i.e., to obtain the typical usage of available resources) [4,5]. Such problems are complex to solve with a specific algorithm during an exact time. Therefore, many studies had been presented while using various heuristic algorithms to propose solutions that may solve machine scheduling problems [6,7,8].

In traditional parallel machine scheduling problems (PMSPs), independent n jobs are to be processed on a number of identical m machines. Accordingly, each job must be executed on one of the m machines within a fixed t time. There are three well-known types of PMSPs, uniform, identical, and unrelated machines. The identical machines process jobs within the same time in all machines. In uniform machines, job processing implemented jobs at a regular rate at different speeds. Where UPMSPs different jobs are processed in machines at various rates, and a machine can implement different jobs at various rates [9].

UPMSPs are employed in various applications, such as various manufacturing industries, including food processing plants, and car factories [10], semiconductor [11,12], tobacco [13], textile [14], petroleum [15], and tire [16]. Additionally, they have been applied in multiprocessor computer [17,18,19], for multithreading [20], in hospital operating rooms [21], human resources management [22], mail facilities [23], printing [24], pharmacy automation [25], vehicular networks [26], and heterogeneous systems that include GPU and CPU [27].

In previous literature, numerous meta-heuristic methods were utilized to address scheduling problems in unrelated parallel machines such as ant colony optimization (ACO) [28,29], genetic algorithm (GA) [15], simulated annealing (SA) [30], tabu search (TS) algorithm [31], firefly algorithm (FA) [32], particle swarm optimization (PSO) [33], artificial bee colony (ABC) [34], or their combinations, such as hybridization of GA and SA [35].

In this paper, we design a new model to solve the problems of unrelated parallel machine scheduling that us based on improved harris hawks optimizer (HHO) by salp swarm algorithm (SSA). HHO is a new nature-inspired optimizer presented by Heidari et al. [36]. It is influenced by the cooperative behavior of Harris’ Hawks that hunt escaping rabbits. Heidari et al. [36] demonstrated that HHO outperforms nature-inspired techniques in 29 engineering problems. It had been used in various applications, such as feature selection [37], engineering problems [38,39,40,41,42,43], satellite image processing [44], prediction models [45], and scheduling tasks in cloud computing [46]. SSA is also a nature-inspired method that simulates the behavior of Salpidae’s family. It was employed in different fields, such as node localization in wireless sensor network [47], feature selection [48,49,50], prediction models [51], cloud computing [52], and image segmentation [53,54]. From these applications for HHO and SSA, it has been observed the high ability of both of them to find the optimal solution for different optimization problems. As well as, in most cases, they provided results better than the comparative algorithms. However, each of them still has its limitations that motivated us to benefit from the advantages of both of them.

The modified version of HHO is called MHHO, in which the SSA is used to enhance the performance of HHO to be applied for solving different UPMSPs problems. SSA works as a local search for HHO to improve its performance and decrease its computational time.

This paper provides the following contributions:

• The improved HHO with SSA is adopted to solve the UPMSPS problem for small and large jobs on different machines with setup time.
• The results of numeric experiments is presented to validate the proposed method with different conditions.
• We compare the MHHO method with known methods in order to demonstrate the superiority of the MHHO over existing methods.

The rest parts of this study are arranged, as follows. Section 2 presents a review of previous works on UPMSPs. Section 3 presents UPMSPS preliminaries. Next, the proposed algorithm is described in Section 4, while in Section 5, experiments and results are given to assess the performance of the proposed algorithm. Finally, we conclude our paper and remark future research in Section 6.

2. Literature Review

In this section, a number of previous published literature on UPMSPs is described. We encourage the reader to see the comprehensive surveys [6,7,8] to obtain more knowledge about the earlier works on this research direction. Arnaout et al. [28] applied the ACO algorithm to address UPMSPs. The performance of the introduced algorithm outperforms the TS algorithm and other meta-heuristic methods. In [17], another metaheuristic method used to solve UPMSPs based on simple-iterated greedy local search. The proposed method produced efficient solutions. In [55], the authors improved the GA by integrating the dominant propensities to solve UPMSPs. The proposed hybrid GADP algorithm reached the optimal solution and compared with previous heuristics. In [56], another algorithm, called GRASP, is presented to deal with the scheduling problems of unrelated parallel machine. They compared GRASP with existed methods and approved its superiority over existed UPMSPs methods. Rodriguez et al. [57] proposed an effective algorithm, namely iterative greedy (IG), to solve a large-scale UPMSPs. They compared their with several previous UMPSPs methods, and showed that IG outperforms previous methods. In [29], a two-stage ACO scheme is presented to solve the UPSMPs and minimize the scheduling makespan on UPM with sequence-dependent set-up time of 120 jobs and 10 machines.

Bitar et al. [58] presented a memetic algorithm to deal with the large-scale of UPMSPs in semiconductor-manufacturing. The proposed method aims to minimize the flow time and maximize the number of processed products. Moreover, a memetic algorithm was employed to solve various scheduling problems, such as [59,60,61,62,63]. In [31], a minimization of the makespan of a number of j jobs and m machines of UPMSPs is considered. A hybridization of two methods, namely truncated branch-and-bound and TS, is proposed and compared to some of the metaheuristic algorithms from the exited approaches. The authors of [64] addressed UPMSPs by employing the power-down scheme in order to minimize total tardiness and total energy consumption. A mathematical model is formulated, then, Ant Optimization algorithm is proposed with Apparent Tardiness Cost (ATC) heuristic rule.

Pacheco et al. [65] presented a metaheuristic algorithm, called Variable Neighborhood Search, to solve the problems of UPMS. They addressed the problem of job scheduling in one machine and regarded two problems, called sequence-dependent setup times and preventive maintenance. Li et al. [66] suggested a polynomial-time approximation algorithm to solve UPMSPs of green manufacturing companies by including the sum of energy cost. In [67], the authors presented a TS algorithm to solve parallel machines scheduling problems. They developed two mathematical models for the problems of a number of parallel machines scheduling of operations that operated together and to minimize the makespan. In [68], an efficient mathematical model and a number of heuristic algorithms are presented in order to minimize the energy consumption and the tardiness of the unrelated parallel machine scheduling problems.

Fanjul-Peyro et al. [69] addressed UPMSPs and presents two models called a mathematical programming and a mixed integer linear program algorithm. The proposed models reached optimal solutions for UPMSPs of 400 jobs and four machines. Wang el al. [70] addressed the large-scale UPMSPs and presented a solution model by applying the TS algorithm components that embedded with multiple-jump strategy. This model solves the on-preemptive scheduling problems and reached the local optimal solution. In [33], the authors presented a hybrid GA and PSO method, called HPSOGA, to address the UPMSPs. They used the Taguchi method to determine and calibrate the optimal parameters. The proposed method outperformed previous solution methods. Ding et al. [71] combined mixed integer linear programming (MILP) with a column generation heuristic to address the UPMSPs under the time-of-use pricing. The proposed scheme was used in order to minimize the total electricity cost with a bounded makespan restriction.

Wu et al. [72] presented an effective method to minimize and reduce the energy consumption and the makespan to resolve unrelated parallel machine scheduling problems. They proposed a memetic differential evolution algorithm. Additionally, they used a meta-Lamarckian learning strategy to enhance their method, which, when compared to two existed methods, demonstrated better performance. Bektur and Tugba [72] considered UPMS problems with a common server. They presented a MILP scheme to solve UPMSPs. More so, they proposed both SA and TS to address the problem. Arroyo et al. [73] proposed an IG algorithm and a MILP model for unrelated parallel machine scheduling. The proposed solution model had been tested on different size of benchmark and compared to some of the existed models, such as ACO, SA, and discrete differential evolution. The IG algorithm outperformed the other three methods in all test benchmarks.

3. Preliminaries

Here, we present a brief definition of the problem, harris hawks optimizer (HHO), and salp swarm algorithm (SSA).

3.1. Problem Definition

Following [74], the problem of the UPMSP is mathematically formulated as a Mixed Integer Programming (MIP) model, as in these equations:

$$Min{C}_{max}$$

(1)

where, $C_{max}$ indicates the max time of completion (makespan). Equation (1) is the fitness function of UPMS problems that requires minimizing the value of $C_{max}$. More details can be found in [74,75].

3.2. Harris Hawks Optimization

The HHO is a new meta-heuristic technique proposed by [36]. It mimics the natural behaviors of Harris’ hawks during searching and catching the rabbits. HHO performs the optimization process while using three stages, i.e., exploration step, transition step from exploration to exploitation, and exploitation step. More details can be found in [36].

3.3. Salp Swarm Algorithm

The SSA is an optimization method that mimics the natural behavior of the salps. The salp chain was mathematically modeled to solve different kinds of global optimization problems [76]. SSA begins by splitting the populations into two categories (i.e., leader and followers). The first category (i.e., leader) is at the front of the salps chain whereas, the other salps called followers. More details can be found in [76].

4. Proposed Method

The proposed method, called Modified HHO (MHHO), to solve UPMSPs with setup times is presented in this section. It applies the SSA as a local search for the HHO algorithm, as shown in Figure 1.

The MHHO algorithm begins by creating random integer numbers to represent the initial solution of UPMSPs solutions (X), whereas the solution includes a set of individuals (N) for a set of n jobs that listed to be processed over m machines. The initial makespan value is calculated for each individual by the fitness function and the best solution is selected based on the smallest fitness value.

Subsequently, the solution is updated using either the HHO or the SSA based on a probability. This probability is computed while using the fitness value for each solution. These steps are repeated until the stop condition is met. Here, it is set to the max number of fitness evaluations, which is set to 100,000.

The description of the initial and updating solutions are explained as follows.

4.1. Solution Representations

The UPMSP solution contains $N_J$ dimension with values in the range $[1,m]$ (i.e., $X\left(1\right)=[x_1,x_2,\dots ,x_n]$). For example, if there are three machines and five jobs, the representation of one individual can be like this: $P=\left[21213\right]$, which means the first and third jobs will be scheduled on machine number 2, then the second and fourth jobs will be scheduled on machine number 1, etc.

Thereafter, the sequence of the jobs for each machine is determined. For example, if there is a T matrix with n jobs and m machines, the $T_{ij}$ indicates the sequence-dependent setup times that are needed to set up for j-th job after i-th job on the m-th machine.

$$T=\left[\begin{array}{ccccc}5& 2& 6& 4& 13\\ 7& 1& 10& 11& 15\\ 3& 8& 9& 12& 14\end{array}\right]$$

Subsequently, for machine number 1, the sequence-dependent setup times are represented by the first row as following (5, 2, 6, 4, and 13) jobs. The other rows represent the jobs for the remaining machines. Zero values in the machine’s row indicate no jobs will be processed by this machine. In this study, the makespan value is determined using the Adjusted Processing Times Matrix ${\left[SP\right]}_k$ [30] to linear combine the previous P and S. It is formulated as in the following equation:

$$SP=S+P$$

(2)

where S is $n\times m$ and represents the processing times. P is $n\times n$ and represents the setup times for each machine. Therefore, this matrix represents the initial solution for the optimization process.

4.2. Updating Stage

The MHHO evaluates the makespan for each solution ($X_i,i=1,2,\dots ,N$) based on the fitness value for each one, as in Equation (1). The smallest makespan $C_{max}$ determines the best solution. After that, the MHHO updates the solution by switching between HHO and SSA algorithms based on a computed probability ($pro{b}_i$), as in Equation (3).

$$pro{b}_i=\frac{fi{t}_i}{{\sum }_{k=1}^{N}fi{t}_k},$$

(3)

where $fi{t}_i$ is the current fitness. If $pro{b}_i>0.5$, then the individual $X_i$ is updated by HHO, or else the SSA is used. These steps are repeated until meeting the stop condition. Finally, the best scheduling based on the minimum $C_{max}$ value is outputted.

5. Experiments and Results

A set of the UPMSP benchmark dataset is used to assess the performance of the MHHO. In addition, its results are compared with the traditional HHO and SSA methods, and state-of-the-art methods with different numbers of machines and job sequences.

5.1. Dataset Description

In general, this UPMSP dataset has different number of instance problems, and each of them has its processing time and setup time. In addition, these instances are randomly constructed according to the discrete uniform distribution from U$[50,100]$.

In this study, the number of machines is set to 2, 4, 6, 8, 10, and 12. While, the number of jobs are 6, 7, 8, 9, 10, 11, 20, 40, 60, 80, 100, and 120. Whereas, the jobs of is classified into two groups, the small group which contains 6, 7, 8, 9, 10, and 11 jobs. The second group (i.e., large group) contains 20, 40, 60, 80, 100, and 120 jobs. The dataset used in this paper is publicly available in the following website http://www.schedulingresearch.com.

For each problem, the algorithm is performed 30 times to enable fair comparison and further statistical analysis of the results.

5.2. Performance Measure

In this section, the metrics used to evaluate the performance of the algorithm are discussed. These metrics, including the relative percentage deviation and improvement ratio measure and the definition of each of them, is given, as follows:

• Relative percentage deviation ($\rho$): It is defined in Equation (4) to compare the results of all methods.

  $$\rho =\frac{C_{max}-LB}{LB}\times 100$$

  (4)

  where $C_{max}$ represents the average of makespan. $LB$ is the lower boundary, which represents the average of the processing time for each job and it is defined in Equation (5).

  $$LB=max(L{B}_{min},L{B}_{max})$$

  (5)

  where $L{B}_{min}$ and $L{B}_{max}$ are the minimum and maximum time of processing, respectively, for each job among the machines and they are defined as:

  $$L{B}_{min}=\frac{{\sum }_{j=1}^{N_J}({min}_{\begin{array}{c}k=1,\dots ,N_m\\ i=1,\dots ,N_J\end{array}}\left[P{R}_{i,j,k}\right])}{N_m},$$

  (6)

  $$L{B}_{max}=\underset{j=1,\dots ,n}{max}\left\{\begin{array}{c}\underset{\begin{array}{c}k=1,\dots ,m\\ i=1,\dots ,n\end{array}}{min}\left[P{R}_{i,j,k}\right]\hfill \end{array}\right.$$

  (7)

  where $PR$ is the adjusted processing time, $N_m$ and $N_J$ are the numbers of machines and jobs, respectively.
• Improvement ratio ($\delta$): It measures the ratio of enhancement of the proposed algorithm against the others and it is defined as:

  $$\delta =\frac{C_{max}\left(method\right)-C_{max}\left(MHHO\right)}{C_{max}\left(MHHO\right)}\times 100$$

  (8)

  where $C_{max}\left(method\right)$ is the average of the makespan values of the compared method.

5.3. Parameter Settings

The proposed MHHO is compared with the traditional HHO and SSA and

Table 1. Parameter settings for the algorithms.

Parameter	Values
SSA	$C_2$$\in [0,1]$, $C_3$$\in [0,1]$
HHO	$E\in [0,2]$
MHHO	$C_2$$\in [0,1]$, $C_3$$\in [0,1]$, $E\in [0,2]$

shows the parameters for each algorithm. These parameters were tuned and selected experimentally.

The experiments were performed using MATLAB R2014b (MathWorks Inc., Natick, MA, USA) run over a computer with CPU Core i7 and 8 GB RAM working with Windows 10 (64 bit). The average of $C_{max}$ of different 15 instances was computed for each job in each problem. The common parameter, such as the maximum number of fitness evaluations, is set to 100,000 for all algorithms, whereas the population size depends on the problem size.

5.4. Comparison with Basic HHO and SSA

In this experiment, we compare the performance of the proposed MHHO with the traditional SSA and HHO algorithm, as given in

Table 2. Average results of $\rho$ values, CPU time, and $STD$ for SSA, HHO, and MHHO methods, as well as the improvement.

m	Job	SSA			HHO			MHHO			Improvement
		$\rho$	${\sigma }_{C_{\mathit{max}}}$	Time	$\rho$	${\sigma }_{C_{\mathit{max}}}$	Time	$\rho$	${\sigma }_{C_{\mathit{max}}}$	Time	${\delta }_{\mathit{SSA}}$	${\delta }_{\mathit{SSA}}$
2	20	1.27	28.44	6.11	2.70	109.23	14.11	0.34	15.50	9.96	2.75	7.00
	40	2.68	37.01	17.57	4.82	38.82	44.79	0.48	17.80	29.96	4.61	9.10
	60	3.18	51.32	35.26	6.90	238.64	97.38	1.32	15.50	58.34	1.41	4.23
	80	3.48	45.87	59.12	4.73	395.73	178.21	0.86	32.30	101.24	3.03	4.48
	100	4.39	69.76	88.67	8.24	88.46	238.95	0.02	31.95	149.14	208.91	393.22
	120	3.34	56.10	126.00	6.60	176.09	350.64	0.04	54.01	208.26	79.14	157.41
4	20	6.39	71.97	6.88	5.33	134.79	17.34	0.40	69.27	11.68	15.04	12.40
	40	8.39	24.54	20.00	6.58	479.95	47.68	0.84	24.33	31.94	8.99	6.83
	60	7.87	18.80	37.07	8.31	704.47	138.16	1.09	15.18	61.37	6.22	6.62
	80	7.00	22.99	61.27	20.38	333.94	580.06	2.55	22.99	317.26	1.74	6.98
	100	9.77	32.66	106.28	19.88	35.08	880.00	0.82	26.47	474.33	10.96	23.33
	120	8.26	8.49	180.86	3.54	101.79	1206.66	2.49	18.37	652.04	2.32	0.42
6	20	25.00	141.87	7.66	6.83	152.95	56.04	2.27	5.20	41.81	10.02	2.01
	40	24.34	122.15	20.43	7.04	127.96	151.05	0.70	7.05	103.42	33.58	9.00
	60	24.28	116.25	39.11	9.79	364.91	342.87	0.79	6.90	189.15	29.89	11.45
	80	13.83	92.21	63.80	22.31	36.06	194.24	3.34	7.60	131.58	3.14	5.68
	100	16.91	129.52	93.72	14.80	192.00	292.54	2.80	24.91	166.74	5.04	4.29
	120	14.58	7.37	130.63	13.75	164.65	407.38	3.64	19.96	230.84	3.00	2.78
8	20	42.08	85.79	8.66	14.88	74.77	22.91	0.73	31.12	15.05	56.62	19.38
	40	40.21	184.82	21.89	6.27	68.77	60.02	0.65	17.20	38.58	60.76	8.64
	60	23.65	29.28	41.00	4.92	89.07	112.34	3.30	27.42	73.38	6.17	0.49
	80	25.10	99.37	66.15	7.02	98.45	183.88	1.73	66.67	116.28	13.52	3.06
	100	21.94	58.53	97.32	10.41	308.32	261.27	3.20	57.05	168.23	5.85	2.25
	120	16.52	31.75	133.67	35.20	97.15	431.87	5.00	25.49	245.40	2.30	6.04
10	20	182.66	220.69	9.55	9.20	53.57	66.84	2.16	26.53	48.76	83.40	3.25
	40	81.80	314.56	23.38	3.72	58.00	178.11	0.21	25.27	111.81	396.83	17.09
	60	28.36	221.70	43.30	13.34	68.00	323.09	0.92	41.02	257.12	29.94	13.55
	80	42.93	102.97	68.88	13.10	142.50	674.73	1.489	46.99	359.23	27.83	7.80
	100	28.97	164.06	100.13	4.88	57.97	267.53	0.77	29.51	175.23	36.69	5.34
	120	21.21	53.67	138.14	5.60	63.62	377.34	5.06	40.67	237.81	3.19	0.11
12	20	189.68	75.35	10.61	19.42	16.99	25.13	15.89	9.74	17.90	10.94	0.22
	40	185.00	229.31	25.19	8.31	77.41	61.38	6.90	16.00	43.02	25.80	0.20
	60	78.85	346.51	45.61	10.94	56.59	111.72	0.73	4.58	78.69	106.89	13.97
	80	35.13	362.36	71.41	13.59	68.43	191.42	2.15	10.50	124.31	15.37	5.33
	100	48.98	154.08	101.16	27.11	78.55	284.76	6.09	26.31	177.38	7.05	3.45
	120	39.48	121.48	134.24	6.24	54.90	360.79	3.92	25.77	248.83	9.08	0.59

. It can be noticed from this table that the high performance of the proposed MHHO according to different metrics. For example, the proposed MHHO has the smallest $\delta$ among all of the tested instances (either in small or large datasets). Moreover, it can be observed the high improvement of the MHHO than two SSA, and HHO with average overall instances nearly 36.890 % and 21.611%, respectively. Figure 2 depicts the average of the improvement along with each machine and job. Note that high improvement of MHHO over SSA is achieved especially at the number of jobs 2, 10, and 12, whereas, at the machines 40 then 100. By comparing the results of HHO and MHHO, it can be noticed the high difference in the improvement is achieved at the number of jobs 2, while, when the number of machines becomes high, like 100 and 120, the proposed MHHO outperforms HHO with high improvements.

According to the value of ${\sigma }_{C_{max}}$, note that the proposed MHHO has the smallest value among all of the tested instances, followed by the SSA, then the HHO. This indicates the high stability of the proposed MHHO algorithm.

In terms of CPU time, the traditional SSA algorithm is the faster algorithm, followed by the MHHO; however, HHO takes a long time until it reached the stop conditions. The main reason that MHHO takes that time results from the operator of traditional HHO that needs more CPU time.

Figure 3 depicts the CPU time that is required by the proposed MHHO and the other two methods among each tested number of machines. From these results, it can be observed that HHO consumes more time than the other two methods, while the faster algorithm is SSA, followed by MHHO. The mean reason for the longer time of MHHO results is because of the drawbacks of the traditional HHO that depends on more operators.

5.5. Results of $\delta$ over Small-Size Problems

In this experiment, we compare the results of $\delta$ of MHHO with other methods using small-size problems as in

Table 3. Results of $\delta$ between MHHO and the other comparison methods in small-size problems.

m	Jobs	HHO	SSA	SA	T9	T8
2	6	0.130	0.253	0.045	0.144	0.139
	7	0.012	0.234	0.165	0.244	0.226
	8	0.026	0.100	0.130	0.149	0.147
	9	0.014	0.006	0.141	0.123	0.110
	10	0.005	0.045	0.215	0.140	0.133
	11	0.020	0.024	0.217	0.119	0.116
4	6	0.009	0.116	0.184	0.192	0.203
	7	0.008	0.099	0.329	0.294	0.318
	8	0.015	0.500	0.748	0.437	0.445
	9	0.007	0.278	0.394	0.287	0.282
	10	0.016	0.108	0.289	0.175	0.165
	11	0.025	0.154	0.347	0.164	0.165
6	8	0.004	0.253	0.189	0.144	0.167
	9	0.028	0.359	0.223	0.116	0.139
	10	0.012	0.304	0.368	0.180	0.201
	11	0.011	0.291	0.238	0.030	0.060
8	10	0.002	0.305	0.113	0.016	0.024
	11	0.018	0.326	0.416	0.006	0.016

. These methods including the traditional HHO, and SSA, SA [75], T9 [74], T8 [74]. From these results, it can be observed that the MHHO outperforms other methods, especially the SA, T9, and T8. However, by comparing the proposed MHHO with SSA, it can be noticed the high performance of the MHHO overall the small-size problems except at machine 2 and job 9, there are small improvements. Additionally, by comparing the proposed MHHO with HHO, the MHHO stills provides better results than the HHO among all the tested problems.

5.6. Comparison with Other MH Methods Using Large-Size Problems

We compare the performance of MHHO with a set of method, including invasive weeds optimization (IWO) [32], Artificial Bee Colony (ABC) [77], metaheuristic for randomized priority search (Meta-RaPS) [32], improved firefly (IFA) [32], Hybrid ABC (HABC) [77], Tabu Search (TS) [32], RSA [78], partition heuristic (PH) [32], Genetic Algorithm (GA) [32], Ant Colony Optimization (ACO) [32], extended ACO (ACOII) [32], GADP [55], basic Simulated Annealing (SA) [78], FA [32], SOS with the longest processing time first (LPT) rule (SOS-LPT) [79], symbiotic organisms search (SOS) [79], HSOSSA [30], ESA [30], and ESOS [30].

The comparison is performed based on the value of $\rho$ and $\delta$ as given in

Table 4. Comparison between MHHO and the state-of-the-art methods while using the $\rho$.

m	Job	MHHO	FA	SOS-LPT	SOS	HSOSSA	ESA	ESOS	ACO	GA	IWO	RSA	HABC
2	20	0.166	1.319	1.267	3.229	0.250	3.505	1.667	13.580	3.719	1.976	4.450	4.140
	40	0.478	2.185	1.346	1.414	0.542	3.365	1.574	20.860	3.789	3.035	2.840	2.370
	60	1.147	2.901	1.325	1.579	1.323	4.434	1.670	23.153	5.480	4.143	2.530	2.100
	80	0.863	3.155	1.294	1.501	0.960	6.113	1.440	24.828	5.270	4.397	2.390	1.940
	100	0.021	3.499	0.175	0.337	0.907	6.055	1.811	25.524	5.416	4.803	2.030	1.800
	120	0.042	3.909	0.193	0.599	1.287	7.208	2.688	25.954	5.233	4.806	1.950	1.690
4	20	0.398	0.944	0.445	5.604	−5.890	0.814	0.814	27.904	5.473	2.717	8.740	8.480
	40	0.840	4.155	2.492	2.745	−0.498	5.365	3.114	39.658	9.129	7.827	5.740	5.130
	60	1.090	4.269	3.331	4.116	1.361	8.158	3.287	45.495	22.523	8.796	4.750	4.080
	80	2.554	6.565	3.772	4.317	2.703	9.926	4.212	48.055	9.823	8.887	4.590	3.880
	100	0.817	5.331	2.932	3.654	3.922	11.191	4.878	50.013	12.750	9.977	4.080	3.450
	120	2.489	5.963	3.216	3.450	3.591	12.233	4.369	50.606	12.190	10.221	3.620	3.330
6	20	2.270	12.288	6.570	6.584	4.857	11.755	7.009	44.445	17.163	15.582	23.120	23.050
	40	0.704	7.050	3.450	4.733	0.732	7.179	3.411	58.658	12.566	11.412	9.370	8.770
	60	0.786	4.989	3.223	4.347	0.963	9.923	3.409	65.317	12.536	9.668	6.510	5.790
	80	3.342	9.970	5.164	5.412	3.476	13.202	5.393	70.939	14.280	11.084	6.910	5.950
	100	2.800	8.972	5.115	5.434	4.227	14.771	5.236	71.113	15.526	13.611	5.630	4.840
	120	3.642	6.677	5.000	5.532	3.874	15.148	5.492	77.632	16.948	12.930	5.370	4.570
8	20	0.730	12.463	3.235	5.596	1.712	8.897	4.706	64.831	13.163	15.034	27.140	27.040
	40	0.651	2.324	1.629	5.967	−2.824	5.709	−0.181	77.577	14.492	12.768	9.030	8.100
	60	3.301	11.223	3.852	5.202	3.569	13.774	5.034	88.479	15.593	13.724	10.490	9.620
	80	1.729	8.633	3.712	6.037	3.411	14.654	5.823	85.424	15.255	11.736	6.900	6.000
	100	3.202	14.269	3.500	4.783	4.228	17.091	5.110	91.589	16.549	15.059	7.530	6.720
	120	5.000	13.988	5.026	6.420	5.364	17.679	6.364	94.596	18.046	16.215	6.660	5.330
10	20	2.164	−9.828	6.790	9.612	−19.660	−10.554	−13.779	78.038	10.157	−5.399	15.790	15.130
	40	0.206	2.963	1.423	6.572	−5.403	5.124	−0.306	90.544	16.477	11.697	10.550	9.350
	60	0.917	7.477	1.084	3.611	0.294	11.356	3.779	98.990	15.364	11.272	9.070	7.620
	80	1.489	14.762	3.258	4.704	2.069	16.750	5.135	111.036	21.404	16.669	7.950	6.960
	100	0.769	12.225	3.462	7.680	4.710	18.286	7.514	116.667	21.692	17.698	7.070	6.350
	120	5.063	9.938	5.219	6.904	6.223	19.865	6.879	116.214	19.635	17.757	6.660	6.230
12	20	15.891	7.836	3.804	31.876	0.923	8.025	3.099	96.691	11.387	6.003	32.430	32.310
	40	6.649	28.823	9.776	25.047	8.436	16.045	9.762	106.705	24.865	21.156	24.940	24.270
	60	0.244	31.407	3.491	9.396	1.078	12.711	3.427	115.136	22.456	21.301	9.850	8.220
	80	2.192	10.842	5.032	10.824	3.061	17.833	5.030	121.426	21.783	14.657	10.980	10.400
	100	6.088	13.665	7.400	11.222	6.470	19.324	7.397	124.443	23.184	22.636	11.670	11.330
	120	3.918	14.045	6.522	8.447	5.123	20.372	8.447	130.772	24.560	18.891	7.290	6.870
Avg.		2.380	8.367	3.570	6.514	1.594	10.647	3.742	72.025	14.330	11.243	9.073	8.423

and

Table 5. Results of $\delta$ between MHHO and the other comparison methods.

m	Job	FA	SOS-LPT	SOS	HSOSSA	ESA	ESOS	ACO	GA	IWO	RSA	HABC
2	20	6.95	6.64	18.46	0.51	20.13	9.05	80.87	21.42	10.92	25.83	23.96
	40	3.57	1.82	1.96	0.13	6.04	2.29	42.67	6.93	5.35	4.95	3.96
	60	1.53	0.15	0.38	0.15	2.86	0.46	19.18	3.78	2.61	1.20	0.83
	80	2.66	0.50	0.74	0.11	6.08	0.67	27.77	5.11	4.10	1.77	1.25
	100	166.33	7.35	15.14	42.36	288.54	85.58	1219.46	257.97	228.66	96.07	85.07
	120	92.84	3.62	13.39	29.89	172.04	63.52	622.02	124.61	114.38	45.81	39.57
4	20	1.37	0.12	13.07	−15.79	1.04	1.04	69.07	12.74	5.82	20.95	20.29
	40	3.95	1.97	2.27	−1.59	5.39	2.71	46.24	9.87	8.32	5.84	5.11
	60	2.92	2.06	2.78	0.25	6.48	2.02	40.74	19.66	7.07	3.36	2.74
	80	1.57	0.48	0.69	0.06	2.89	0.65	17.82	2.85	2.48	0.80	0.52
	100	5.52	2.59	3.47	3.80	12.70	4.97	60.22	14.61	11.21	3.99	3.22
	120	1.40	0.29	0.39	0.44	3.91	0.76	19.33	3.90	3.11	0.45	0.34
6	20	4.41	1.89	1.90	1.14	4.18	2.09	18.58	6.56	5.87	9.19	9.16
	40	9.01	3.90	5.72	0.04	9.20	3.85	82.32	16.85	15.21	12.31	11.46
	60	5.35	3.10	4.53	0.23	11.62	3.34	82.10	14.95	11.30	7.28	6.37
	80	1.98	0.55	0.62	0.04	2.95	0.61	20.23	3.27	2.32	1.07	0.78
	100	2.20	0.83	0.94	0.51	4.28	0.87	24.40	4.55	3.86	1.01	0.73
	120	0.83	0.37	0.52	0.06	3.16	0.51	20.32	3.65	2.55	0.47	0.25
8	20	16.07	3.43	6.66	1.34	11.18	5.44	87.78	17.02	19.59	36.16	36.03
	40	2.57	1.50	8.17	−5.34	7.77	−1.28	118.17	21.26	18.61	12.87	11.44
	60	2.40	0.17	0.58	0.08	3.17	0.53	25.81	3.72	3.16	2.18	1.91
	80	3.99	1.15	2.49	0.97	7.48	2.37	48.42	7.83	5.79	2.99	2.47
	100	3.46	0.09	0.49	0.32	4.34	0.60	27.60	4.17	3.70	1.35	1.10
	120	1.80	0.01	0.28	0.07	2.54	0.27	17.92	2.61	2.24	0.33	0.07
10	20	−5.54	2.14	3.44	−10.08	−5.88	−7.37	35.06	3.69	−3.49	6.30	5.99
	40	13.41	5.92	30.96	−27.28	23.92	−2.49	439.38	79.14	55.89	50.31	44.48
	60	7.16	0.18	2.94	−0.68	11.39	3.12	106.99	15.76	11.30	8.89	7.31
	80	8.91	1.19	2.16	0.39	10.25	2.45	73.57	13.37	10.19	4.34	3.67
	100	14.91	3.50	8.99	5.13	22.79	8.78	150.79	27.22	22.03	8.20	7.26
	120	0.96	0.03	0.36	0.23	2.92	0.36	21.96	2.88	2.51	0.32	0.23
12	20	−0.51	−0.76	1.01	−0.94	−0.49	−0.80	5.08	−0.28	−0.62	1.04	1.03
	40	3.34	0.47	2.77	0.27	1.41	0.47	15.05	2.74	2.18	2.75	2.65
	60	127.90	13.33	37.56	3.42	51.17	13.07	471.53	91.16	86.42	39.43	32.74
	80	3.95	1.30	3.94	0.40	7.14	1.30	54.40	8.94	5.69	4.01	3.75
	100	1.24	0.22	0.84	0.06	2.17	0.22	19.44	2.81	2.72	0.92	0.86
	120	2.58	0.66	1.16	0.31	4.20	1.16	32.38	5.27	3.82	0.86	0.75

.

Table 6. Parameter settings for all algorithms.

Parameter	Values
ACO	Pheromone evaporation = 0.01, Global update rate = 0.081, NP = 60, Pheromone amounts = 1, LocalIter = 31
SADP	Initial temperature = LB + (UB − LB)/10, final temperature = LB, temperature reduction rate = 0.99
GADP2	Crossover rate = 0.6, NP = 50, Mutation rate = 0.5
ESOS	Number of random move = [0.5 m, 0.9 m], NP = 50
GADP	Crossover rate = 0.6, NP = 50, Mutation rate = 0.5
FA	Selection pressure = 3, NP = 100, Max generation = 1000, Uniform mutation rate = 3, $\alpha$ = 0.2, Light absorption = 1
SOS	Number of random move = [0.5 m, 0.9 m], NP = 50
IWO	Min seeds = 0, Max seeds = 5, NP = 15, Initial Standard deviation = 1, Final standard deviation = 0.001, Variance reduction exponent = 2
GA	Crossover rate = 0.6, Mutation rate = 0.5, Np = 15, Selection pressure = 5
ESA	Initial temperature = 0.025, Temperature reduction rate = 0.99
SOS-LPT	Number of random move = [0.5 m, 0.9 m], NP = 50
HSOSSA	Number of random move = [0.5 m, 0.9 m], NP = 50, Initial temperature = 0.025, Temperature reduction rate = 0.99
RSA	Initial temperature = 1.5, final temperature = 0.02, Iter = (jobs + machines−1) × 1500, Cooling schedule = 0.98, Boltzmann constant = 0.1, Non-improving = 28
HABC	Initial temperature = 0.25 × (sum of processing time and average setup time for each job on each machine)/(jobs × machines), NP = 10, limits = 1000, $T_{max}$ = (jobs × machines × 35)/100, Cooling schedule = 0.90

lists the parameter settings for these algorithms.

From Table 4, one can observe that MHHO has the best $\rho$ values at 28 instances from 36 instances (as in boldface), followed by HSOSSA, which achieves the best $\rho$ value, at eight instances.

Moreover, Table 5 shows the performance of the algorithms in terms of $\delta$. Note that MHHO has high improvements over the other approaches in most of the tested instances except at some jobs. For example, in machine 2 and job 20, the HSOSSA provides better results than MHHO. The same observation is noticed at machine 4 and jobs 20 and jobs 40. While, at the machine 8 and job 40, it can be seen that HSOSSA and ESOS have better value than MHHO. As well as, the FA, ESA, and ESOS provide better results than MHHO at machine 10 and job 20, in addition, at machine 10 and job 40, the HSOSSA and ESOS are better than MHHO. Finally, at the machine 20 and job 20, the performance of the MHHO is less than FAm, SOS-LPT, and HSOSSA.

Because there are several state-of-the-art methods that have been developed but used different measures to assess their performance. Accordingly, we compared our proposed MHHO with those methods at machines 2, 6, and 12 with jobs 20, 40, 60, and 80, the results are given in

Table 7. Comparison of the results of MHHO and other state-of-the-art algorithms. The best values are shown in bold.

		ACO [32]				SADP [55]				GADP2 [55]				ESOS [30]				GADP [55]
m	n	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$
2	20	1283	1347	1348	13.6	1196	1255	1338	5.8	1242	1254	1266	5.7	1190	1213	1236	2.3	1242	1255	1267	5.8
	40	2772	2834	2889	20.8	2371	2462	2550	5.0	2441	2459	2474	4.9	2363	2378	2393	1.4	2446	2469	2487	5.3
	60	4223	4323	4404	23.2	3588	3680	3764	4.8	3652	3675	3695	4.7	3519	3575	3631	1.9	3664	3695	3734	5.3
	80	5701	5823	5911	24.8	4753	4879	5045	4.6	4846	4872	4896	4.4	4705	4730	4754	1.4	4859	4892	4923	4.9
6	20	466	523	561	44.6	441	455	481	25.7	448	454	459	25.4	381	387	393	6.9	448	456	463	26.0
	40	992	1137	1269	58.6	796	841	892	17.3	809	831	853	15.9	734	742	750	3.5	815	838	861	16.9
	60	1171	1771	1910	65.4	1210	1259	1295	17.6	1219	1246	1270	16.3	1100	1107	1113	3.3	1225	1252	1277	16.9
	80	2200	2443	2609	71.0	1606	1662	1705	16.3	1622	1648	1672	15.3	1500	1507	1514	5.5	1623	1655	1684	15.8
12	20	266	343	356	96.2	229	241	265	37.7	235	239	244	36.6	177	183	189	4.6	234	240	247	37.1
	40	643	717	796	106.7	440	466	493	34.3	444	455	468	31.1	377	382	386	9.9	444	456	471	31.4
	60	1047	1117	1217	115.2	628	669	715	28.9	809	649	667	25.1	534	538	542	3.7	629	652	673	25.6
	80	1352	1528	1639	121.4	819	891	959	29.1	819	849	884	23.0	713	723	733	4.8	821	852	889	23.5
		FA [32]				IWO [32]				GA [32]				ESA [30]				MHHO
m	n	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$	Min	Avg.	Max	$\mathbf{\delta }$
2	20	1156	1201	1256	1.3	1152	1209	1270	2.0	1176	1224	1331	3.2	1202	1235	1268	4.1	1162	1188	1211	0.2
	40	2330	2396	2462	2.2	2343	2416	2500	3.0	2364	2434	2507	3.8	2416	2428	2440	3.5	2346	2355	2400	0.4
	60	3523	3612	3676	2.9	3543	3656	3739	4.1	3608	3703	3814	5.5	3601	3667	3733	4.5	3533	3556	3582	1.3
	80	4738	4813	4965	3.2	4769	4870	5008	4.4	4796	4911	5049	5.3	4905	4959	5013	6.3	4690	4705	4790	0.9
6	20	386	407	435	12.4	392	419	453	15.7	390	425	486	17.3	397	406	414	12.0	369	371	380	2.4
	40	752	767	1192	7.0	765	796	825	11.1	681	807	891	12.5	754	765	776	6.7	720	722	733	0.6
	60	1111	1125	1142	5.0	1120	1175	1208	9.7	1131	1206	1284	12.6	1163	1177	1190	9.9	1078	1080	1094	0.8
	80	1552	2346	2388	64.2	1533	1588	1661	11.1	1597	1633	1672	14.3	1613	1619	1624	13.3	1474	1477	1505	3.4
12	20	176	188	205	7.6	171	185	206	5.8	181	194	214	11.1	187	189	191	8.0	189	202	212	15.6
	40	365	447	538	28.8	382	420	467	21.1	371	433	488	24.8	394	404	413	16.3	368	371	402	6.9
	60	624	682	775	31.5	569	640	848	23.3	597	636	719	22.5	578	590	601	13.6	520	523	530	0.8
	80	695	765	793	10.9	719	792	846	14.7	723	841	991	21.9	808	813	818	17.8	701	706	731	2.3

, which uses the minimum, average, and maximum of the $C_{max}$. From this table, it can be concluded that the proposed MHHO has better performance than others in terms of average (Avg.) overall the instances, except at the machine 12 and job 20, where the ESOS is better. In addition, in terms of the best $C_{max}$ value, the proposed MHHO provides the smallest value overall the instances and jobs followed by FA. Moreover, it can be seen that the $\delta$ value for the proposed MHHO is better than the $\delta$ for others.

6. Conclusions

We presented a new hybrid meta-heuristic approach to address unrelated parallel machine scheduling problems (UPMSPs) with sequence-dependent setup times. The proposed MHHO approach is a hybrid of Harris Hawks optimizer (HHO) and the salp swarm algorithm (SSA). MHHO allows enhancing the performance of HHO based on SSA, and also avoid premature convergence. Therefore, the SSA is applied as a local operator to improve the performance of HHO to find optimal solutions and avoid local optima. Moreover, we considered the minimization of makespan as a performance evaluation of the proposed approach. The proposed MHHO had been evaluated while using a set of instances, which contains small and large job sizes over six machines during a series of experiments. Furthermore, to approve the superiority of the MHHO, we compared MHHO with the original SSA, HHO, and other well-known methods that had been applied to solve UPMSPs. The comparison results showed that MHHO outperforms both SSA and HHO. Additionally, MHHO outperforms other methods, additionally, it showed better convergence to the optimal solution.

For future work, the MHHO could be employed in various optimization applications, such as data clustering, cloud computing scheduling, image processing, forecasting applications, and others.