Improved Spider Monkey Optimization Algorithm for Hybrid Flow Shop Scheduling Problem with Lot Streaming ^†

Abstract

This paper investigates the hybrid flow shop scheduling problem with lot streaming, which integrates the order lot problem (OLP), order sequence problem (OSP), and lots assignment problem (LAP), with the objective of minimizing both the maximum completion time ($C_{max}$) and the total tardiness (TT) simultaneously. An improved spider monkey optimization (I-SMO) algorithm is proposed by combining the advantages of crossover and mutation operations of a genetic algorithm (GA) with the spider monkey optimization algorithm. The contribution value method is employed to select both global and local leaders. Experimental comparisons with classical optimization algorithms, including particle swarm optimization (PSO) and differential evolution (DE), were conducted to demonstrate the superiority of the proposed I-SMO algorithm.

1. Introduction

The Hybrid Flow Shop Scheduling Problem (HFSP), also known as the Flexible Flow Shop Scheduling Problem (FFSP), is an NP-hard problem. In the context of fast response requirements in the multi-variety small-batch customization production mode, the HFSP with Lot Streaming (LS) can effectively improve production efficiency, reduce production cycle, and enhance on-time delivery rate.

Zaky et al. address an integrated HFSP with LS for the objective of minimizing $C_{max}$. They formulate two mixed-integer nonlinear programming models to tackle this problem [1]. Danial and Fantahun proposed a two-stage GA for the HFSP with LS [2]. Beren and Ömer employed a GA to solve the HFSP with constraints on machine capabilities and limited waiting times [3]. However, there is limited research on HFSP with LS utilizing the Spider Monkey Optimization algorithm. In this paper, we propose an I-SMO algorithm for the HFSP with LS.

2. Problem Formulation

This study focuses on the scheduling problem of LS in a three-stage HFSP with different numbers of machines in each stage. Additionally, at least one of the stages has more than one machine. The objective is to minimize $C_{max}$ and TT simultaneously. This problem mainly includes three sub-problems: OLP, OSP, and LAP. The corresponding decisions include for LSP, dividing the order into lots; for OSP, sorting of $N^P$; for LAP, determining each lot of machines that are processed in each process. In this case, the quantity of each order $c(c=\mathrm{1,2},\dots ,N^P),\mathrm{i}.\mathrm{e}.$, ${CN}_c$, delivery time $d_c$, number of lots in each order $n_c^{lots}$, and size of the lot ${size}_c^{lots}$.

The basic assumptions include the following: (1) All machines can be used at zero time; (2) A machine can only process one lot at the same time; (3) All lots can be processed at zero time, and any lot can enter the next process only after the previous stage is completed; (4) The mix-flow production is not considered; (5) There is a setup time before lots processing; (6) There is a buffer area between different stages, and the transportation time need to be considered; (7) The size of the lot is a decision variable, and the size of lot remains unchanged in all stages; (8) The sequence of lots of an order remains unchanged on all stages.

3. I-SMO Algorithm

3.1. Basic Flow of the I-SMO Algorithm

The SMO algorithm, introduced by Bansal et al. in 2014, is a novel swarm intelligence optimization algorithm that simulates the splitting and merging behavior of spider monkeys during their foraging process [4]. Mumtaz et al. proposed a new hybrid spider monkey optimization (HSMO) algorithm for PCB assembly scheduling problem [5]. In this study, the SMO algorithm is discretized using the crossover and mutation operations from the GA. Furthermore, the contribution value method is employed to select the global leader and local leaders. The I-SMO algorithm process is shown in Figure 1.

3.2. Solution Selection Based on Contribution Value

In this study, the hyper-volume performance indicator and the contribution value method proposed in the literature are used for selecting local and global leaders. This is because the greater the contribution values of the solution, the larger the area of the independent dominance of the solution, the more conducive to the distribution of the solution set. Specific steps are as follows: (1) Normalize the target value; (2) Pareto solution after the normalization is arranged in order of one of the goals; (3) According to Equation (1), the reference point $z^{*}(z_1^{*},z_2^{*})$, where the $z_r^{*}$ indicates the value of the reference point on the target $r$, and $\sigma$ is obtained by the test; (4) The dominance area of the calculation solution and its two adjacent solutions; (5) The area of the independent dominance of the calculation and solution is the contribution value of the solution. The contribution value is calculated based on Equation (2) at the edge or Pareto front end.

$$z_r^{*}=f_r^{max}+\sigma (f_r^{max}-f_r^{min})$$

(1)

$${CV}_A=(Z_1^{*}-f_1^A)(Z_2^{*}-f_2^A)-(Z_1^{*}-\mathrm{m}\mathrm{a}\mathrm{x}(f_1^A,f_1^B)\left)\right(Z_2^{*}-f_2^A)$$

(2)

3.3. Encoding

The hierarchical encoding scheme is utilized in this study. The first layer consists of a $2\times N^P$ dimensional matrix, where the first row represents the lots quantity of $N^P$ orders, and the second row represents the processing sequence of the orders. The second layer is a $4\times N^{lot}$ dimensional matrix, where the first row represents the processing sequence of the lots, and the remaining layers represent the machine sequence to which the lots are allocated for each stage.

3.4. Local Leader Phase and Global Leader Phase (LLP and GLP)

During the LLP stage, members of each group will update their position by approaching their respective local leaders, while during the GLP stage, all members will update their position by approaching the global leader. Cross operation is divided into three layers: (1) Crossing the processing order by using the two-point crossing method. (2) According to the order sorting, the lots sorting is obtained; (3) For lots assignment, the partially matched crossover method, similar to the two-point cross method, is used to cross each process in order. An example of crossover is shown in Figure 2.

Randomly select order c, and its lots coding sorting for reverse sequence processing. For the variation of the distribution and encoding of the process machine, the two batches of the order c randomly select 0 or 1 for each process. If it is 1, the corresponding gene is exchanged, otherwise it will not be exchanged. An example of mutation is shown in Figure 3.

4. Numerical Example and Analysis

4.1. Parameter Settings

The experimental data includes the number of orders ${CN}_C$, the number of orders $N^P$, the delivery date dc, the processing time of a sub-batch of orders on machine M in each stage $P_{stage}^{mc}$, number of machines in each stage $n_{stage}^m$, the setup time for the first processing of an order on machine M in each stage ${st}_{stage}^{moc}$, the setup time related to the sequence between different batches of orders ${st}_{stage}^{mcc’}$, and the transportation time during the order batch transition stage ${tt}_{stage}$. The range of batch quantities for orders is [1, 12]. The parameter ranges for different scale problems, based on production data from a specific company, are shown in

Table 1. Scale of experimental problems and parameter.

Parameter	Value			Parameter	Value	Parameter	Value	Parameter	Value
$N^P$	U [2, 3]	U [4, 6]	U [8, 12]	${CN}_C$	100 × U [3, 10]	${st}_{stage}^{moc}$	U [15, 20]	${tt}_{stage}$	U [20, 30]
$n_{stage}^m$	U [1, 2]	U [1, 3]	U [2, 3]	$P_{stage}^{mc}$	U [50, 100]	${st}_{stage}^{mcc’}$	U [10, 15]	dc	$\mathrm{U} \left[0, \rho \widehat{C_{max}}\right]$

. The parameter values for I-SMO and the comparative algorithms are obtained through Taguchi experiments, as shown in

Table 2. Parameter values for different algorithms.

Parameter	Small			Medium			Large
	PSO	DE	I-SMO	PSO	DE	I-SMO	PSO	DE	I-SMO
$popsize$	200	200	200	200	200	200	200	200	200
$maxgen$	200	100	200	200	150	200	200	200	200
$p_c/p_m$	/	/	0.85/0.1	/	/	0.85/0.25	/	/	0.85/0.25

.

4.2. Computational Experiments and Discussion

The computational experimental results for various algorithms are presented in

Table 3. Performance results of different algorithms.

Size	PSO		DE		I-SMO
	IGD	NR	IGD	NR	IGD	NR
Small	20,416.8006	0.0277	23,514.9404	0.0000	110.6899	0.9722
Medium	67,529.3626	0.0370	71,691.7599	0.0370	664.9987	0.9259
$\mathrm{Large}$	218,301.1570	0.0000	244,254.7911	0.0000	0.0000	1.0000

. Each problem instance was independently run 10 times, where IGD stands for Inverted Generational Distance and NR represents Non-Dominated Rate.

IGD denotes the minimum Euclidean distance from the individual $S^u$ to the optimal Pareto optimal solution, and the smaller this value is, the closer to the optimal solution the $C_{max}$ and TT obtained by the algorithm are. In addition, NR denotes the proportion of solutions in the solution set of the current method that are also in the Pareto optimal frontier solution set; the larger this value is, the higher the proportion of the target solution obtained by the algorithm is in the Paret optimal frontier solution set. Therefore, I-SMO algorithm outperforms traditional population-based optimization algorithms in terms of IGD and NR.

5. Conclusions

This paper mainly includes the OLP, OSP, and LAP problems of HFSP with LS, and the experimental results show that the proposed I-SMO algorithm is superior to traditional algorithms and can obtain effective solutions. To study the HFSP with mixed-flow using the proposed ISMO should be further considered in the future.