Improved Chimp Optimization Algorithm for Matching Combinations of Machine Tool Supply and Demand in Cloud Manufacturing

Abstract

Cloud manufacturing is a current trend in traditional manufacturing enterprises. In this environment, manufacturing resources and manufacturing capabilities are allocated to corresponding services through appropriate scheduling, while research on the production shop floor focuses on realizing a basic cloud manufacturing model. However, the complexity and diversity of tasks in the shop floor supply and demand matching environment can lead to difficulties in finding the optimal solution within a reasonable time period. To address this problem, a basic model for dynamic scheduling and allocation of workshop production resources in a cloud-oriented environment is established, and an improved Chimp optimization algorithm is proposed. To ensure the accuracy of the solution, two key improvements to the ChOA are proposed to solve the problem of efficient and accurate matching combinations of tasks and resources in the cloud manufacturing environment. The experimental results verify the effectiveness and feasibility of the improved ChOA (SDChOA) using a comparative study with various algorithms and show that it can solve the workshop supply and demand matching combination problem and obtain the optimal solution quickly.

1. Introduction

The new global technological revolution and industrial changes have a huge impact on the manufacturing industry, which is developing in the direction of “digitalization, networking, personalization, collaboration, agility, service, green and intelligence” [1]. In this context, academician Li Bohu and his team started to further develop the manufacturing industry in 2009 [2]. His team proposed the concept, mode, technical means, and industry of “cloud manufacturing” and started researching the cloud manufacturing mode with networking and service as the main features. In the manufacturing industry, manufacturing resources and manufacturing capabilities are managed in a unified manner, and the demander can perform the full-process response manufacturing tasks in sequence from the cloud platform according to diverse needs at any time and place [3]. Due to the unified management and allocation of manufacturing resources and manufacturing capabilities, the utilization of resources increases, and the usage of different resources differs, so the rational scheduling of resource use has also become a hot topic of research. Due to the numerous resources and services in the cloud platform, it is relatively difficult to obtain an efficient, fast, and accurate solution. In view of the production scheduling requirements of the cloud manufacturing workshop layer, this paper further studies the flexible job shop scheduling problem, which is an extension of the traditional job shop scheduling problem [4]. Allocating limited manufacturing resources to tasks with specific performance metrics ensures that the correct resources can be efficiently allocated to execute work items/tasks with special needs in a reasonable time [5,6], ensuring a balance between demand for manufacturing capacity and resource availability [2]. On a specific machine, corresponding tasks can be processed in a specific order/sequence. This increases the flexibility of workshop scheduling, which is more in line with actual production and is a type of scheduling problem that urgently needs to be solved. Furthermore, we will delve into the latest applications of the brainstorming optimization (BSO) algorithm, the dual-objective SA-LP (IMOSSA/D) algorithm, and the improved genetic algorithm (IGA) [7,8,9,10,11].

The particle swarm intelligence and genetic algorithm SPGA [12], improved hybrid genetic algorithm and variable neighbor search (HGA-VNS) [13], multi-search algorithm genetic algorithm, novel energy-aware estimation model, improved Dragonfly algorithm (DOLDA), multi-agent reinforcement learning (MARL) [7,8,9,14], multi-objective evolutionary algorithm (EMOEA), improved Antlion optimization algorithm (DALO), improved artificial jellyfish search algorithm (JS), and marine predator algorithm (MPA) [15,16] can be utilized to solve complex large-scale flexible workshop job scheduling problems.

Compared with the traditional job shop scheduling problem, the flexible job shop scheduling problem reduces machine constraints, increases the uncertainty of a specific machine, and expands the search range of feasible solutions. It is a more complex NP-Hard problem. At present, intelligent algorithms such as genetic algorithms (GAs) and artificial bee colonies have been widely used to solve the component optimization problem algorithm (ABC). Simulated annealing (SA), taboo search (TS), and the particle swarm algorithm (PSO) are used in the cloud environment due to their fast search speed [6,17,18,19]. An algorithm called the Chimp optimizer (ChOA) was proposed by Khishe et al. [20]. Inspired by the predatory behavior of chimps, the ChOA has been successfully applied to other fields due to its significant advantages, such as strong convergence performance, few parameters, and easy implementation. Due to the large and complex search space of ChOA, its advantages are of great importance for task scheduling in CMFG. In order to solve the task assignment problem effectively and accurately, this paper makes two improvements to the ChOA to further improve the algorithm [13,21,22] development and apply the improved SDChOA to the task scheduling process.

In summary, in this paper, we propose a new Chimp optimization algorithm based on the hybrid improvement strategy with the characteristics of FJSP and design an improved Chimp optimization algorithm to solve the production resource scheduling model. The results show that the improvement strategy is effective, and SDChOA is superior to the comparison algorithm in terms of optimization accuracy and stability. The contributions and innovations of this study are summarized as follows:

• An improved Chimp optimization algorithm is proposed, which uses an adaptive position update and dynamic weight strategy to enhance the solution speed and solution quality of the improved algorithm.
• The SDChOA algorithm was experimentally compared with five well-known algorithms on 13 benchmark functions. The results show that SDChOA is better than all comparison algorithms in terms of comprehensive performance.
• An SDChOA algorithm was developed and applied to the job shop scheduling problem for the first time and achieved good results.

The remainder of this article is organized as follows. Section 2 introduces the original Chimp algorithm; Section 3 introduces the improvement strategy and implementation process of SDChOA; Section 4 verifies the effectiveness of the algorithm; Section 5 analyzes the established mathematical model and conducts comparative experiments and analysis; and Section 6 provides the conclusion of this article and future research plans.

2. Chimp Optimization Algorithm

This section briefly describes the origin of ChOA, including the behavioral patterns of chimpanzees and the mathematical modeling of ChOA.

2.1. Origins: Social Status and Hunting Behavior of Chimpanzees

In a traditional group of chimpanzees, there are four categories: expeller, obstacle, chaser, and aggressor. They have different abilities necessary for hunting [23]. The chaser stalks the prey, the obstacle blocks the prey, the chaser chases the prey quickly, and finally, the attacker predicts the prey’s breakout route, returning the prey to the stalker or into a lower range. The hunting process is shown in Figure 1. The attacker needs more skills to predict the escape trajectory of the prey, so after successful hunting, it will be paid. This effect is positively correlated with age, IQ, and physical fitness. In addition, chimpanzees changed duties during the same hunt or remained the same. In general, the hunting process of chimpanzees is mainly divided into two stages: the exploration stage, which includes driving, blocking, and chasing prey; and the pursuit stage, which includes attacking prey. In the following sections, all these ChOA concepts are expressed in mathematical form.

2.2. Mathematical Model: Chimp Optimization

ChoA mimics the siege and hunting behavior of chimpanzees. The enclosing behavior can be modeled mathematically in the following way. The mathematical model of chimpanzee eviction and prey pursuit are shown in Equations (1) and (2):

$$d=|C·X_{prey}(t)-m·X_{chimp}(t\left)\right|$$

(1)

$$X_{chimp}\left(t+1\right)=X_{prey}\left(t\right)-ad$$

(2)

where t denotes the number of current iterations; $X_{prey}$ is the prey position vector; $X_{chimp}$ is the current chimpanzee position vector; and a, m, and C are the coefficient vectors, which are calculated as shown in Equations (3)–(5).

$$a=2f·r1-f$$

(3)

$$m=Chaoticvalue$$

(4)

$$C=2·r2$$

(5)

where $r1$ and $r2$ are random vectors in the range of [0, 1], respectively; f is a convergence factor whose value decreases nonlinearly from 2.5 to 0 with an increasing number of iterations; a is a random vector determining the distance between chimpanzees and prey with a random number in the range of [−f, f]; m is a chaotic mapping vector representing the effect of neutral motivation of chimpanzees during hunting; C is a chimpanzee control coefficient of prey expulsion and chasing, whose value is a random number in the range of [0, 2]. After population initialization, the four optimal solutions are selected as the positions of attacker, obstacle, repeller, and chaser, in turn, and the other chimpanzee positions in the population are updated around the following four chimpanzee positions, which are described by the mathematical model shown in Equations (6)–(9):

$$X_1=X_{attacker}-a1·|C1·X_{attacker}-m1·X|$$

(6)

$$X_2=X_{barrier}-a2·|C2·X_{barrier}-m2·X|$$

(7)

$$X_3=X_{chaser}-a3·|C3·X_{chaser}-m3·X|$$

(8)

$$X_4=X_{driver}-a4·|C4·X_{driver}-m4·X|$$

(9)

$$X_{(t+1)}=(X_1+X_2+X_3+X_4)/4$$

(10)

where X denotes the position vector of the current chimpanzee; $X_{attacker}$ denotes the position vector of the attacker; $X_{barrier}$ denotes the position vector of the obstacle; $X_{chaser}$ denotes the position vector of the chaser; $X_{driver}$ denotes the position vector of the repeller; and $X_{(t+1)}$ denotes the updated position vector of the current chimpanzee. Algorithm 1 shows the pseudo-code of the Chimp optimization algorithm.

Algorithm 1: ChOA

$\mathrm{Initialize}\mathrm{the}\mathrm{chimp}\mathrm{population}{x}_i$$(i=1,2,\dots ,n$) Initialize f, m, a and c Calculate the position of each chimp Divide chimps randomly into independent groups Until stopping condition is satisfied Calculate the fitness of each chimp $X_{attacker}$ = the best search agent $X_{chaser}$ = the second-best search agent $X_{barrier}$ = the third best search agent $X_{driver}$ = the fourth best search agent  while (t < maximum number of iterations)    for each chimp:    Extract the chimp’s group    Use its group strategy to update f, m, and c   Use f, m, and c to calculate a and then d  end for   for each search chimp     if (u < 0.5)       if (|a| < 1)   Update the position of the current search agent with Equation (2)        Else if (|a| > 1)        Select a random search agent     end if     else if (u > 0.5) Update the position of the current search with Equation (9)    end if

end for Update f, m, a, and c $\mathrm{Update}{X}_{attacker}$$,X_{chaser}$$,X_{barrier}$, and $X_{driver}$   t = t + 1   end while return $X_{attacker}$

3. Improved Chimp Optimization Algorithm

3.1. Nonlinear Adjustment of Convergence Factor Based on a Logarithmic Function

Inspired by the inertia weight setting in the particle swarm optimization algorithm, this paper proposes a new non-linear strategy for adjusting the convergence factor a. The calculation formula is as follows:

$$a\left(t\right)=a_{initial}-\left(a_{initial}-a_{final}\right)\log (1+\left(e-1\right)\frac{t}{Ma{x}_{iter}})$$

(11)

where t is the current number of iterations and Max_iter is the maximum number of iterations. a_initial and a_final are the initial and final values of a, respectively, and the values in this paper are 2 and 0, respectively. Figure 2 presents a curve comparison between the linear decreasing strategy of parameter a and the proposed nonlinear decreasing strategy.

It can be clearly seen from Figure 1 that this nonlinear transition parameter is more focused on local development in more iterations than the original linear decreasing strategy. It can be clearly seen from Figure 2 that compared with the original linear decreasing strategy, this nonlinear transition parameter focuses on local development in more iterations. The value of the proposed nonlinear parameter is smaller in the middle and later stages of the iteration, which indicates that it facilitates local development over a long period of time compared with global exploration. The figure also shows that during the search process, the proposed nonlinear parameter strategy is beneficial to global exploration only in a small number of iterations.

3.2. Adaptive Location Update Strategy

In ChOA, the solutions of the initial attacker, barrier, chaser, and driver are recorded and retained until they are replaced by individuals with better fitness values in the iterative process. That is, if in the t-th generation, there is no better solution than the recorded population, the current population still updates the position toward these four chimpanzees. However, when all four fall into a local optimum, it is difficult for the entire population to seek a better solution. This can be understood as follows: when the decision maker of the chimpanzee group misjudged where the prey appeared, then all the encircling actions of the chimpanzees would be ineffective, and it would be difficult for them to find the prey in the wrong place.

This paper proposes a new definition of barrier, chaser, and driver to strengthen the role of the current generation of optimal individuals, thereby improving the global search ability of the algorithm. The implementation of the algorithm is the same as ChOA but barrier, chaser, and driver are defined as local variables; they are the best and second fitness values in the t-th (t = 1, 2, 3, …T) generation, the third individual chimpanzee. At the same time, in order to better balance the global search and local development process of the algorithm, this paper proposes a new adaptive walking strategy, which is mathematically expressed as:

$$X\left(t+1\right)=\frac{\left(X_1+X_2+X_3+X_4\right)}{4}\left(1-\frac{t}{T}\right)+X_1\frac{t}{T}$$

(12)

3.3. Revised Dynamic Weighting Strategy

The disadvantage of the position update formula in ChoA is that the average value of X₁, X₂, X₃, and X₄ cannot highlight the importance of the four. For this reason, the authors propose two new proportional weighting strategies, namely, the weighted average and the proportional weight strategy based on the fitness value are experimentally verified. A dynamic proportional weight based on the guiding position vector weight is proposed, different weighting strategies are analyzed and experimentally studied, and it is theoretically proven that the dynamic weighting strategy can be optimized efficiently.

The weight of the distance between the current chimpanzee individual and the attacker, barrier, chaser, and driver are calculated as

$$W_1=\frac{\left|X_1\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|}$$

(13)

$$W_2=\frac{\left|X_2\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|}$$

(14)

$$W_3=\frac{\left|X_3\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|}$$

(15)

$$W_4=\frac{\left|X_4\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|}$$

(16)

However, in practical applications, the denominator of the above equations is likely to be 0. Therefore, a small constant ε needs to be added, with a value of 10⁻¹⁶. The weights are modified to

$$W_1=\frac{\left|X_1\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+\left|X_4\right|+\epsilon }$$

(17)

$$W_2=\frac{\left|X_2\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|+\epsilon }$$

(18)

$$W_3=\frac{\left|X_3\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|+\epsilon }$$

(19)

$$W_4=\frac{\left|X_4\right|}{\left|X_1\right|+\left|X_2\right|+{|X}_3|+|X_4|+\epsilon }$$

(20)

Combined with the previous adaptive location update strategy, the final location update method can be expressed as

$$X\left(t+1\right)=\frac{\left({W_{1}X}_1+{W_{2}X}_2+W_3{X}_3+W_4{X}_4\right)}{4}\left(1-\frac{t}{T}\right)+X_1\frac{t}{T}$$

(21)

3.4. SDChOA Implementation Steps

Combining the above improvement methods, the SDChOA implementation proposed in this paper is described as follows.

The steps are shown as follows.

Step 1: initialize the parameters related to the algorithm: population size N, spatial dimension dim, searchable space lb, ub of the population, and the maximum number of iterations $T_{max}$.

Step 2: initialize the population.

Step 3: calculate the fitness value of each chimpanzee individual and select the top four individual positions with the smallest fitness, recorded as $X_{attacker}$, $X_{barrier}$, $X_{chaser}$, and $X_{driver}$, respectively.

Step 4: calculate the value of convergence factor f according to Equation (11).

Step 4: calculate the values of parameters a, m, and C according to Equations (3)–(5).

Step 5: follow Equation (21) for $X_{attacker}$ into dynamic weighting strategy selection.

Step 6: update $X_{attacker},X_{barrier},X_{chaser},\mathrm{a}\mathrm{n}\mathrm{d}{X}_{driver}$ according to Equations (17)–(20), and further update the chimpanzee population location according to Equation (21) by combining the adaptive location update strategy.

Step 7: determine whether the iteration termination condition is satisfied. If so, output the global optimal chimpanzee location $X_{attacker}$r; otherwise, return to step 3 and continue the execution, that is, calculate the maximum number of iterations $T_{max}$.

Step 8: output the position of the optimal solution $X_{attacker}.$

The specific operation is shown in Figure 3.

4. Simulation Experiment and Result Analysis

In this section, the performance of SDChOA is evaluated using standard benchmark functions reasonably, and the obtained experimental results are compared with five other state-of-the-art algorithms. Finally, we verify the validity and reliability of the algorithm.

4.1. Experimental Design and Test Functions

We use 13 standard benchmark functions commonly used in the literature to test the performance of SDChOA including seven unimodal functions and six multimodal functions, all of which are single objective functions. The unimodal functions F1–F7 have only one global optimal value to evaluate the local search ability of the algorithm, while the multimodal functions F8–F13, with more than two optimal values, are used to evaluate the global search ability of the algorithm [4]. During the experiment, we set the initial parameter settings of the comparison algorithm according to the recommendations, as shown in

Table 1. Parameter setting table.

Algorithm	Parameters
PSO	$C_1=2,C_2=2,W_{Max}=0.9,W_{Min}=0.2$
GWO	$a=2$
SSA	—
ChOA	$m=chao\left(\mathrm{3,1},1\right)$
IChOA	$m=chao\left(\mathrm{3,1},1\right)$
SDChOA	$m=chao\left(\mathrm{3,1},1\right),\epsilon =4,S=2$

.

Table 2. Benchmarking functions.

ID	Function	$\mathbf{d}\mathbf{i}\mathbf{m}$	$\mathbf{R}\mathbf{a}\mathbf{n}\mathbf{g}\mathbf{e}$	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$	Type
F1	$F_1\left(x\right)={\sum }_{i=1}^n{x}_i^2$	30/50/100	[−100, 100]	0	Unimodal
F2	$F_2\left(x\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	30/50/100	[−10, 10]	0	Unimodal
F3	$F_3(x)={\sum }_{i=1}^n{\left({\sum }_{\mathrm{j}-1}^i{\sum }_{i=1}^n\right)}^2$	30/50/100	[−100, 100]	0	Unimodal
F4	$F_4\left(x\right)=ma{x}_i\left\{\left|x_i\right|,1\le i\le n\right\}$	30/50/100	[−100, 100]	0	Unimodal
F5	$F_5\left(x\right)={\sum }_{i=1}^n\left[\left[100{\left(x_{i+1}-x_i^2\right)}^2\right]+{\left(x_i-1\right)}^2\right]$	30/50/100	[−30, 30]	0	Unimodal
F6	$F_6\left(x\right)={\sum }_{i=1}^n{\left(\left[x_i+0.5\right]\right)}^2$	30/50/100	[−100, 100]	0	Unimodal
F7	$F_7\left(x\right)={\sum }_{i=1}^n{i{x}_i}^4+random\left[\mathrm{0,1}\right]$	30/50/100	[−1.28, 1.28]	0	Unimodal
F8	$F_8\left(x\right)={\sum }_{i=1}^n-x_i\mathrm{s}\mathrm{i}\mathrm{n}\left(\sqrt{\left|x_i\right|}\right)$	30/50/100	[−500, 500]	−418.9829 × d	Multimodal
F9	$F_9\left(x\right)={\sum }_{i=1}^n\left[x_i^2-10\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi x_i\right)+10\right]$	30/50/100	[−5.12, 5.12]	0	Multimodal
F10	$F_{10}(x)=-20\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{1}{n}{\sum }_{i=1}^n\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi x_i\right)\right)$	30/50/100	[−32, 32]	0	Multimodal
F11	$F_{11}(x)=\frac{1}{4000}{\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^n\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{x_i}{\sqrt{i}}\right)+1$	30/50/100	[−600, 600]	0	Multimodal
F12	$F_{12}\left(x\right)=\frac{\pi }{n}\left\{10\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi y_1\right)+{\sum }_{i=1}^{n-1}{\left(y_i-1\right)}^2\left[1+10{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2\right\}$ $+{\sum }_{i=1}^{n}u\left(x_i,\mathrm{10,100,4}\right)$	30/50/100	[−50, 50]	0	Multimodal
F13	$F_{13}\left(x\right)=0.1\left\{{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left(3\pi x_1\right)+{\sum }_{i=1}^n{\left(x_i-1\right)}^2\left[1+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left(3\pi x_i+1\right)\right]+{\left(x_n-1\right)}^2\left[1+{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left(1\pi x_{\mathrm{n}}\right)\right]\right\}+{\sum }_{i=1}^{n}u\left(x_i,\mathrm{5,100,4}\right)$	30/50/1200	[−50, 50]	0	Multimodal

lists the benchmark functions.

The population size of each algorithm was set to 30, and the number of iterations was set to 1000. In order to make the experimental results more convincing, under the same experimental environment, when Dim = 30/50/100, each algorithm was run 30 times for functions F1–F13 and recorded the best (best), average (mean), and standard deviation (STD). When solving the minimum problem, the smaller the mean, the better the performance of the algorithm, and the smaller the standard deviation, the more stable the algorithm. Therefore, we used the mean and standard deviation of the best values to evaluate the performance of the 13 algorithms.

4.2. Analysis of the Influence of Different Improvement Strategies on Algorithm Performance

The optimal value and the mean value are used to reflect the ability and convergence accuracy of the algorithm in finding the best performance. First, SDChOA obtains the optimal theoretical values when solving the F9 and F11 functions. For functions F3~F5, its shape is similar to a parabola, and there are a large number of local optimal values. At the same time, F7 is a multimodal function with a valley shape, and its global optimal value is located at the lower end of the mountain, which is difficult to find. As shown in

Table 3. Comparison of the improved algorithm with other algorithms.

Function	Algorithm	30 dim			50 dim			100 dim
		Best	Mean	Std	Best	Mean	Std	Best	Mean	Std
F1	SSA	9.1421 × 10−8	−3.005 × 10−6	4.3087 × 10−5	7.3587 × 10−8	6.5734 × 10−6	3.818 × 10−5	5.5981 × 10−8	4.1605 × 10−6	3.3538 × 10−5
	GWO	1.5466 × 10−43	−1.313 × 10−23	5.4590 × 10−23	1.5098 × 10−43	1.3611 × 10−24	5.5493 × 10−23	6.4721 × 10−44	8.9142 × 10−24	3.5231 × 10−23
	PSO	4.2621 × 10−14	6.0340 × 10−11	1.1311 × 10−1	2.1022 × 10−4	4.5313 × 10−3	4.1124 × 10−3	3.6998 × 10−4	2.3901 × 10−2	9.3198 × 10−2
F2	ChOA	1.4268 × 10−1	1.0546 × 10−6	1.3332 × 10−6	6.4084 × 10−14	1.7854 × 10−8	3.1346 × 10−8	2.7554 × 10−12	1.3215 × 10−7	1.9599 × 10−7
	IChOA	1.2961 × 10−33	2.9527 × 10−18	4.1897 × 10−18	5.8119 × 10−35	6.7475 × 10−19	8.4943 × 10−19	2.8247 × 10−38	1.4105 × 10−2	1.9325 × 10−2
	SDChOA	1.7075 × 10−84	1.3506 × 10−43	1.2741 × 10−43	2.7221 × 10−84	1.7507 × 10−43	1.5582 × 10−43	3.2364 × 10−84	1.7661 × 10−43	1.8499 × 10−43
	SSA	3.7017 × 10−2	8.7012 × 10−2	2.4717 × 10−2	1.7355 × 10−3	1.4546 × 10−2	1.3296 × 10−2	4.3765 × 10−3	9.5751 × 10−2	2.5382 × 10−1
	GWO	2.7808 × 10−3	2.6752 × 10−29	5.6921 × 10−28	1.7637 × 10−26	9.1872 × 10−29	3.4931 × 10−28	5.9162 × 10−27	4.9111 × 10−25	1.2176 × 10−25
	PSO	1.5940 × 10−7	2.2003 × 10−5	9.4840 × 10−5	1.0396 × 10−1	2.9181 × 10−8	3.8751 × 10−8	3.6032 × 10−7	2.3931 × 10−5	1.3810 × 10−5
	ChOA	9.5106 × 10−1	1.9003 × 10−11	3.5603 × 10−11	1.1199 × 10−16	2.2371 × 10−11	4.1212 × 10−11	6.811 × 10−13	1.3604 × 10−12	3.2561 × 10−12
F3	IChOA	2.0463 × 10−24	4.0925 × 10−23	6.8694 × 10−23	1.1423 × 10−26	2.2225 × 10−24	3.4947 × 10−24	5.6336 × 10−24	1.1217 × 10−21	1.9387 × 10−21
	SDChOA	3.5334 × 10−47	7.0668 × 10−49	7.7394 × 10−49	5.1552 × 10−5	1.031 × 10−48	9.8854 × 10−49	1.6527 × 10−53	3.3054 × 10−49	2.9279 × 10−49
	SSA	4.5981 × 103	5.3291 × 103	3.6552 × 102	2.7256 × 103	4.5526 × 103	1.2148 × 102	3.9081 × 102	1.5011 × 102	1.2905 × 101
	GWO	1.1005 × 10−1	3.4324 × 10−8	2.0524 × 10−6	1.7311 × 10−8	1.8933 × 10−7	3.1957 × 10−5	4.5572 × 10−8	5.8381 × 10−7	4.3366 × 10−5
	PSO	2.1130	3.6781	4.2234	5.7381 × 102	3.1312 × 102	8.9761 × 101	3.9081 × 102	4.3094 × 102	9.8371 × 102
	ChOA	2.2215 × 10−3	8.4123 × 10−3	1.5941 × 10−1	4.0205 × 101	7.4268 × 102	7.6071 × 102	1.1590 × 10−5	6.2109 × 10−3	4.9211 × 10−3
F4	IChOA	7.3289 × 10−2	1.4715 × 10−1	3.1918 × 10−2	2.7551 × 101	4.8522 × 102	5.2611 × 102	1.6051 × 102	3.5816 × 101	1.3924 × 101
	SDChOA	9.8578 × 10−39	1.1842 × 10−21	8.8719 × 10−21	2.0991 × 10−48	1.4957 × 10−26	1.5657 × 10−25	1.4992 × 10−54	1.0296 × 10−29	7.7713 × 10−29
	SSA	1.5245	4.0233	2.5733	1.5442 × 101	3.6823 × 102	1.1988 × 101	1.9237	7.2392	1.1239
	GWO	6.7580 × 10−12	5.3937 × 10−11	6.1497 × 10−1	1.6612 × 10−12	3.6931 × 10−1	1.5181 × 10−9	3.5314 × 10−12	9.7243 × 10−11	3.2447 × 10−9
	PSO	1.9582	2.1231	1.1413 × 101	1.9939	3.2910 × 102	1.2319 × 101	2.2321	9.3091	4.1292
	ChOA	1.6521 × 10−3	5.2166 × 10−2	7.6681 × 10−2	3.9418 × 10−1	1.2481 × 10−1	1.5319 × 10−1	1.3672 × 10−1	3.8851 × 10−1	3.3092 × 10−1
	IChOA	4.8536 × 10−2	1.2130 × 101	2.2931 × 101	5.0151 × 10−2	8.3918	1.3142 × 10−1	7.8977	8.3918	9.8390 × 10−1
F5	SDChOA	6.5083 × 10−31	5.0899 × 10−27	6.1499 × 10−27	4.8153 × 10−34	1.7075 × 10−34	1.8815 × 10−34	2.3464 × 10−29	6.6837 × 10−26	8.7751 × 10−26
	SSA	1.0817 × 102	4.1738 × 102	2.0261 × 102	2.0619 × 102	1.9304 × 102	4.5221 × 102	1.3912 × 102	2.1159 × 102	2.8044 × 102
	GWO	3.7099 × 101	4.5387 × 101	2.6333 × 101	4.7061 × 101	5.9464 × 101	1.1675 × 101	4.7364 × 101	6.8091 × 101	4.1707 × 101
	PSO	1.0818 × 101	3.0124 × 101	9.8322 × 101	1.7206 × 102	8.7611 × 102	7.1289 × 101	1.7127 × 101	8.1938	9.8131
	ChOA	4.8895 × 101	2.9394 × 102	5.4152 × 102	4.8922 × 101	1.8778 × 102	4.2324 × 102	4.8902 × 101	66.626 × 101	5.0596 × 101
	IChOA	4.6830	4.1903 × 101	2.2998 × 101	4.0024 × 101	4.0169 × 101	1.2945 × 101	4.6844 × 101	4.7107 × 101	1.5229 × 101
F6	SDChOA	4.8970 × 101	7.3407 × 101	3.8833 × 101	4.8970 × 101	5.9406 × 101	2.3752 × 101	4.8656 × 101	7.5244 × 101	2.4219 × 101
	SSA	4.4727 × 10−1	5.0001 × 10−4	3.9001 × 10−5	8.4195 × 10−8	5.3141 × 10−5	4.1131 × 10−5	7.2298 × 10−8	5.9831 × 10−5	3.8412 × 10−5
	GWO	9.9157 × 10−1	4.2843 × 104	3.6238 × 103	2.5112 × 101	4/0008 × 101	2.0263 × 101	3.6111	5.3524	2.2685
	PSO	4.0808 × 10−9	5.4493 × 10−7	2.6933 × 10−5	1.6921 × 10−8	3.1293 × 10−5	9.3193 × 10−4	1.8383 × 10−5	8.1938 × 10−3	9.8471 × 10−3
	ChOA	8.3511	4.2119	1.8553 × 10−1	7.4971	1.8148	2.2243	6.7683	9.0032	2.1560
	IChOA	2.2992	3.9445 × 10−1	2.4884 × 10−1	4.4019	7.8545	2.0704	3.3711	4.8321	2.3426
F7	SDChOA	2.7994 × 10−1	2.3981	2.1311	5.8675	9.0538	1.7656	5.4412	9.3205	1.9437
	SSA	5.5097 × 10−2	4.8749 × 10−3	1.3062 × 10−3	2.3115 × 10−2	7.2317 × 10−1	1.1639 × 10−1	2.6984 × 10−2	1.1865 × 10−1	1.2011 × 10−1
	GWO	1.6327 × 10−2	4.3618 × 10−2	3.1059 × 10−2	1.0336 × 10−3	2.5274 × 10−3	2.5555 × 10−2	1.3616 × 10−3	1.8751 × 10−3	2.5205 × 10−2
	PSO	2.6521 × 10−2	1.6013 × 10−2	1.5391 × 10−2	3.1921 × 10−1	6.9731 × 10−1	1.3874 × 10−1	3.2627 × 10−1	6.4021 × 10−1	2.3019 × 10−1
	ChOA	1.9691 × 10−3	4.6498 × 10−2	2.4133 × 10−2	4.4695 × 10−3	3.6156 × 10−3	3.3152 × 10−3	2.0794 × 10−3	2.1954 × 10−2	2.0912 × 10−2
	IChOA	5.7946 × 10−3	1.3528 × 10−2	4.7679 × 10−2	3.7255 × 10−2	1.4229 × 10−2	5.6435 × 10−2	3.6226 × 10−3	1.6594 × 10−2	4.6131 × 10−3
F8	SDChOA	1.3653 × 10−5	4.1788 × 10−3	2.9092 × 10−3	1.0592 × 10−5	1.6312 × 10−3	4.3758 × 10−3	1.5669 × 10−4	1.6443 × 10−2	7.8203 × 10−2
	SSA	−2.071 × 102	−2.1622 × 102	3.6626 × 102	−1.217 × 103	−2.564 × 101	−3.3691 × 101	−1.217 × 103	−3.783 × 102	−3.6772 × 101
	GWO	−5.510 × 103	−2.3174 × 102	1.9690 × 102	−1.102 × 104	−1.571 × 103	2.71251 × 10−2	−8.024 × 103	−3.419 × 102	−2.278 × 102
	PSO	−9.281 × 103	−6.1231 × 103	6.6785 × 102	−9.852 × 102	−2.311 × 101	−3.3111 × 101	−4.233 × 103	−7.310 × 102	−9/123 × 102
	ChOA	−5.653 × 103	−4.9221 × 102	3.8515 × 101	−9.1943 × 103	−4.894 × 102	−4.2336 × 101	−9.301 × 103	−4.601 × 102	−1.104 × 102
	IChOA	−5.748 × 103	−3.8410 × 102	2.1347 × 102	−8.0467 × 103	−3.697 × 102	−2.5479 × 102	−8.611 × 103	−4.071 × 102	−1.892 × 102
F9	SDChOA	−2.996 × 103	−2.4842 × 102	1.8739 × 102	−4.9604 × 103	−2.540 × 102	−193572 × 102	−5.680 × 103	−9.765 × 102	−2.580 × 102
	SSA	3.3828 × 101	1.79092 × 101	1.6292	5.7171 × 101	1.5919 × 102	1.0514 × 101	1.0646 × 102	6.1687 × 102	1.3317 × 102
	GWO	5.6843 × 10−14	−7.4994 × 10−1	5.3493 × 10−9	0.0000	0.0000	0.0000	5.6843 × 10−14	2.4307 × 10−1	5.6961 × 10−9
	PSO	1.2323 × 102	3.2211 × 102	7.0898 × 101	1.0869 × 102	1.4191 × 103	2.3910 × 102	1.2642 × 102	8.9282 × 103	8.3918 × 103
	ChOA	5.2953	6.3455 × 10−1	9.4836 × 10−2	2.8246	3.6123	3.6411	7.3962 × 10−4	1.7444 × 10−3	2.1221 × 10−3
	IChOA	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	1.5829×10−12	5.5478×10−12
F10	SDChOA	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	9.5455×10−15	1.2212×10−14
	SSA	8.8522 × 10−16	1.7189 × 10−15	8.5152 × 10−15	2.7720 × 10−1	1.7925 × 10−2	7.3652 × 10−3	3.4391	5.1611 × 10−	1.2905 × 101
	GWO	8.6153 × 10−14	1.7189 × 10−12	8.5152 × 10−12	3.2863 × 10−16	1.4571 × 10−15	8.5302 × 10−15	2.2721 × 10−2	4.0391 × 10−1	3.9310 × 10−1
	PSO	2.4461 × 10−7	1.2391 × 10−6	1.0012 × 10−6	1.1175 × 10−6	3.9812 × 10−4	9.3871 × 10−4	2.9312 × 10−14	3.0285 × 10−16	7.5579 × 10−15
	ChOA	1.9648 × 10−4	3.1341 × 10−2	2.7935 × 10−2	1.9964 × 10−1	3.1299	4.9502	1.9964 × 101	3.1381 × 1011	4.3828
	IChOA	4.4388 × 10−7	6.4165 × 10−6	9.1436 × 10−6	2.1033 × 10−5	1.9196 × 10−3	3.3647 × 10−2	7.4157 × 10−11	1.0952 × 10−11	1.5112 × 10−11
F11	SDChOA	4.4409 × 10−17	5.9777 × 10−16	1.2452 × 10−15	4.4409 × 10−17	5.5904 × 10−16	1.2181 × 10−15	4.4409 × 10−17	2.4452 × 10−16	1.3471 × 10−15
	SSA	1.4266 × 10−5	1.4195 × 10−3	8.8952 × 10−3	1.7822 × 10−3	4.0812 × 10−2	5.1994 × 10−2	2..4327 × 10−3	2.2466 × 10−2	1.3232 × 10−2
	GWO	2.7578 × 10−12	8.0196 × 10−7	1.4373 × 10−6	0.0000	0.0000	0.0002	0.0000	0.0000	0.0000
	PSO	4.5859 × 10−6	1.3112 × 10−2	2.1123 × 10−3	2.2053 × 10−5	8.1931 × 10−4	9.3893 × 10−4	5.8666 × 10−5	7.9831 × 10−3	8.8918 × 10−3
	ChOA	2.7578 × 10−12	8.0196 × 10−7	1.4373 × 10−6	1.197 × 10−11	1.8889 × 10−6	2.9242 × 10−6	4.2002 × 10−12	1.0536 × 10−6	1.7802 × 10−6
	IChOA	0.0000	0.0000	0.0000	0.0000	1.5548×10−1	0.0000	0.0000	1.2245	0.0000
F12	SDChOA	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	SSA	1.6744 × 10−11	5.1334 × 10−8	7.1471 × 10−7	1.3040 × 10−9	6.8916 × 10−6	7.0481 × 10−6	5.7663 × 10−1	1.3542 × 10−8	5.3169 × 10−8
	GWO	2.6549 × 10−2	1.7567 × 10−1	4.2256 × 10−1	7.0587 × 10−2	7.2706 × 10−1	4.3912 × 10−1	6.1556 × 10−2	6.8485 × 10−1	4.6742 × 10−1
	PSO	4.5444 × 10−6	9.3221 × 10−4	4.4422 × 10−4	9.7429 × 10−6	8.0981 × 10−4	9.8419 × 10−3	3.1036 × 10−5	2.1931 × 10−3	3.0912 × 10−2
	ChOA	1.3261 × 10−3	3.5095 × 10−2	4.2457 × 10−2	7.9894 × 10−2	3.5004 × 10−1	4.9141 × 10−1	4.5609 × 10−2	3.0108 × 10−1	3.4785 × 10−1
	IChOA	5.3791 × 10−2	6.4244 × 10−2	5.8355 × 10−2	2.3259 × 10−2	6.3954 × 10−1	6.8171 × 10−1	1.9274 × 10−1	5.8969 × 10−1	6.8137 × 10−1
F13	SDChOA	7.1331 × 10−2	4.9039 × 10−2	3.8346 × 10−2	2.9466 × 10−2	4.1402 × 10−1	3.4397 × 10−1	3.0341 × 10−2	4.8625 × 10−1	4.9012 × 10−1
	SSA	3.8180 × 10−1	9.5551	3.1473	1.1482 × 101	2.4736 × 101	2.6426 × 10−1	5.1226 × 10−1	9.4799 × 10−1	3.0955 × 10−1
	GWO	1.6315 × 10−1	2.9555 × 10−1	3.1473 × 10−1	1.3412 × 10−2	6.7918 × 10−1	4.0911 × 10−1	1.4153 × 10−2	6.7989 × 10−1	4.2659 × 10−1
	PSO	1.1289 × 10−4	3.6831 × 10−3	9.1975 × 10−3	4.4597 × 10−2	9.3901 × 10−1	9.8734 × 10−2	4.5558 × 10−5	9.6471 × 10−4	8.7361 × 10−4
	ChOA	2.7111	6.2852 × 101	3.1859 × 101	4.6371	1.0278 × 10−1	3.4465 × 10−1	4.7994 × 10−3	4.1353 × 10−2	1.9851 × 10−2
	IChOA	2.9575 × 10−2	8.2463 × 10−1	6.1701 × 10−1	2.6111 × 10−4	6.5508 × 10−1	5.2053 × 10−1	2.8211 × 10−6	8.1001 × 10−2	6.2418 × 10−2
	SDChOA	2.9866 × 10−7	1.3423 × 10−3	4.0581 × 10−1	4.9966 × 10−6	6.7420 × 10−3	2.6941 × 10−3	4.9872 × 10−5	4.9312 × 10−3	7.7304 × 10−3

, SDChOA outperforms other algorithms. Peak functions F1–F7 can be used to research and develop capability algorithms. According to the results in Table 3, six out of seven cases of the proposed SDChOA have the best results, with obvious advantages over competitors. Multimodal functions F8–F13 have a large number of local optima, and the global optimum becomes more difficult to obtain as the search space dimension increases. The results in Table 3 show that the proposed SDChOA achieved the optimal solution in six cases and achieved the best effect in nine cases. This shows that the proposed improvement strategy has a significant optimization effect on the standard ChOA. Integrating these strategies into the SDChOA improves its performance in terms of search accuracy and stability, showing certain advantages. In order to further compare the convergence of various algorithms in the optimization process, we drew the optimized iteration curves of the 13 algorithm processes. The convergence curves of functions F1–F13 are shown in Figure 4, which shows that SDChOA performs well in most of the following aspects.

4.3. SDChOA Convergence Analysis

To reflect the dynamic convergence characteristics of SDChOA, as shown in Figure 4, both ChOA and IChOA were used to solve the single-peaked function when F1–F4. Its search accuracy is at least 100 orders of magnitude and 30 orders of magnitude higher than the standard ChOA, respectively. This is attributed to the adaptive location update and dynamic weighting strategy that produce a more optimal solution to guide the population to find the global optimal solution stably, improving the convergence accuracy of the algorithm. At the same time, the “attacker” individuals lead the population to a larger prey search range to find the optimal solution, which helps the algorithm to jump out of the local optimum while maintaining the population diversity, and the nonlinear adjustment of the convergence factor based on the logarithmic function balances the local and global exploration capabilities. Specifically, SDChOA outperforms other comparative algorithms in search performance at the beginning of the iterations and exploration performance at the end of the iterations, with higher solution accuracy and faster convergence speed under the same number of iterations. This indicates that SDChOA can fully guarantee exploration ability without losing population diversity and stability of prey search while ensuring exploration ability.

4.4. Comparison of SDChOA, Swarm Intelligence Algorithms, and Improved Algorithms

To further verify its effectiveness, SDChOA was compared with PSO, SSA, GWO [16,24,25,26], ChOA, and IChOA, citing experimental data from the literature and reproducing experiments from the literature [23,27]. To reflect the fairness of the comparison, the same experimental parameter settings as in the literature (population size of 30, maximum iteration number of 1000) are used. The optimization results of each algorithm for different types of benchmark test functions are listed, where the single-peak and multi-peak function search dimensions are set to 30/50/100, respectively. Each algorithm is run 30 times independently, and its optimal value, mean and standard deviation are recorded.

Also, in all cases, SDChOA outperforms the original ChOA and its improved versions, including the newly proposed IChOA, which indicates that the proposed improvements can improve the performance of ChOA and make it competitive with other comparative algorithms. To further compare the convergence of various algorithms in the optimization process, we plotted the iteration curves of the optimization process for 13 algorithms. The convergence curves of functions F1–F13 are shown in Figure 4, which indicates that SDChOA performs well for most of the functions. ChOA focuses on global search in the early and middle stages and converges slowly [28]. With time, SDChOA converges quickly and eventually converges to good results. Compared with the original ChOA, SDChOA accelerates the convergence speed and proves that the proposed nonlinear control parameter strategy is effective [29,30,31]. In summary, the proposed improvement strategy has an obvious optimization effect on the standard ChOA, and the SDChOA that incorporates the three strategies shows certain advantages in both the accuracy and stability of the optimization search.

5. Flexible Job Shop Scheduling Problem

5.1. Problem Description and Model Building

The cloud manufacturing flexible job shop scheduling problem is described as follows: n workpieces {J₁, ……, J_n} are to be processed on m machines {M₁, ……, M_m}. Each workpiece contains one or more processes in a predetermined order, and each process can be processed on multiple different machines, with the processing time of the process varying with the machine. The scheduling goal is to select the most suitable machine for each process, determine the best process sequence and start time for each workpiece process on each machine [32,33], and optimize certain performance indicators of the system. Thus, the flexible job shop scheduling problem consists of two subproblems: determining the machines for each workpiece and determining the sequence of processing on each machine [18]. Scheduling in which each process can be processed on any of the optional machines is called fully flexible job shop scheduling; conversely, scheduling in which each process can only be processed on some of the optional machines is called partially flexible job shop scheduling, as shown in

Table 4. Example of a partially flexible job shop scheduling problem.

Workpiece	Process	Optional Processing Machines
		${\mathit{M}}_1$	${\mathit{M}}_2$	${\mathit{M}}_3$	${\mathit{M}}_4$	${\mathit{M}}_5$
$J_1$	$O_{11}$	2	6	5	3	4
	$O_{12}$	—	8	—	4	1
	$O_{13}$	4	—	3	6	2
$J_2$	$O_{21}$	3	—	6	—	5
	$O_{31}$	4	6	5	—	6
	$O_{41}$	—	7	11	5	8

Note: J₁ represents workpiece 1, $O_{12}$ represents the second process of the first workpiece, and the rest are similar.

.

In addition, the following constraints need to be met during the machining process [5]

(1)

Only one workpiece can be machined on the same machine at the same time.

(2)

The same process of the same workpiece can be processed by only one machine at the same time.

(3)

Each process of each workpiece cannot be interrupted once it has started.

(4)

Different workpieces have the same priority as each other.

(5)

There are no sequential constraints between processes of different workpieces and sequential constraints between processes of the same workpiece.

(6)

All workpieces can be machined at zero time.

In this paper, three performance metrics are considered simultaneously: minimum maximum completion time, minimum maximum machine load, and minimum total load on all machines.

The objective functions of these three performance indicators are as follows.

(1)

Maximum completion time $C_M$

$$min{C}_M=min\left(max\left(C_K\right)\right)1\le k\le \mathrm{m}$$

(22)

where $C_M$ is the completion time of machine $M_K.$

(2)

Maximum load machine load $W_M$

$$min{W}_M=min\left(max\left(W_M\right)\right)1\le k\le \mathrm{m}$$

(23)

where $W_k$ is the workload of machine $M_K.$

(3)

Total load $W_T$ of all machines

$$min{W}_T=min{\sum }_{K=1}^M{W}_T$$

(24)

Table 4 shows the machining machines and machining schedules for a flexible job shop scheduling problem with two workpieces and five machines. Among them, “—” indicates that the corresponding machine cannot be selected for processing in that process. The problem listed in Table 4 is a partially flexible job shop scheduling problem. If all “—” correspond to processing time, it means that each process of each workpiece can be selectively processed using all machines, which is a fully flexible operation. Workshop scheduling problem.

5.2. Computational Results and Analysis

To verify the performance of the comparison algorithm, this paper validates three flexible job shop problem instances consisting of 8 workpieces with four processes, each on eight machines for the 8 × 4 × 8 partially flexible job shop scheduling problem; 10 workpieces with four processes, each on 10 machines for the 10 × 4 × 10 fully flexible job shop scheduling problem; and 15 workpieces with four processes, each on 10 machines for the 15 × 4 × 10 fully flexible job shop scheduling problem.

The Chimp optimization algorithm with a modified hybrid improvement strategy is compared with several example tests, and the FJSP is not only to solve the machine selection problem but also to solve the all-process sequencing problem. The computational results of the three objective values for the three flexible job shop example problems: the 8×8 problem, 10 × 10 problem, and 15 × 10 problem, are given in

Table 5. The convergence curves of SDChOA.

Example Problem (n × m)	Objective Function	Methods Such as KACEM		Methods Such as XIA		Zhang Chaoyong			SDChOA
						Best	Population	Average Time t/s	Best	Computing Time
8 × 8	$C_M$	15	16	15	16	14	200	16.5	14	1.4
	$W_M$	N/A	N/A	12	13	11		11.2	12
	$W_{\mathrm{T}}$	79	75	75	73	N/A		N/A	75
10 × 10	$C_M$	7		7		7	200	14.5	7	3.2
	$W_M$	5		6		5		9.0	5
	$W_{\mathrm{T}}$	45		44		N/A		N/A	44
15 × 10	$C_M$	24		12		N/A	N/A	0	12	10.3
	$W_M$	11		11				0	11
	$W_{\mathrm{T}}$	91		91				0.44	91

. Also given is a comparison of the test results of the Chimp optimization algorithm using the hybrid improvement strategy in this paper with the KACEM [33] method and the XIA [34] method, in the case of different performance indicator functions, which include the minimum maximum completion time $C_M$, the minimum maximum load machine load $W_M,$ and the minimum total workload $W_T$ on all machines.

As can be seen from Table 5, for the three test instances, the Chimp optimization algorithm with the improved hybrid improvement strategy achieved better test results on all three objective functions, with each objective value equal to or better than the optimal solution obtained using the other three algorithms [35]. For example, when solving the 8 × 8 problem, better results were obtained for both $C_M$ and $W_T$ compared with the results of KACEM and other methods. When solving the 10 × 10 problem, compared with the results of XIA and other methods, better results are obtained for both $W_M$ and $W_T$ in the case of equal $C_M.$ Solving the 15 × 10 problem, compared with the results of methods such as XIA, the maximum completion time was reduced by 1 time unit for the same value of the other two objectives. Among the three example problems, the improved Chimp algorithm achieved the best results in both the 10 × 10 problem and the 15 × 10 problem, and the calculation time was also significantly reduced.

Figure 5, Figure 6 and Figure 7 show the Gantt charts for obtaining better solutions for the three example problems listed in Table 5, respectively. As shown in Figure 8, the convergence speed and parameters of the new SDChOA proposed in this paper are tested for the 8 × 8 flexible job shop problem examples with one objective value of maximum completion time minimization and the same parameters as those set earlier. The test results show that the new improved hybrid improvement strategy of the Chimp optimization algorithm proposed in this paper to solve the flexible job shop scheduling problem significantly improved the quality of the initial solution and the quality of the solution, and the convergence speed also improved significantly.

6. Conclusions

To sum up, intra-shop task distribution in a manufacturing enterprise ensures that the right resources are efficiently assigned to perform work items/tasks with special consequences at the right time. This paper proposes an improved Chimp optimization algorithm based on adaptive position updating and dynamic weighting strategy enhancement and subsequently uses various standard benchmark functions to evaluate the optimization performance. Experimental results show that SDChOA has excellent performance in terms of convergence speed and accuracy compared with many population-based algorithms. The SDChOA search space has high accuracy even in complex environments. In addition, the flexible job shop scheduling problem under three performance objectives (minimum maximum completion time, minimum maximum machine load, and minimum total load of all machines) was studied using the same example test in the literature, and SDChOA was compared with other algorithms in the literature. Comparing the results, the calculation results were further improved, and the calculation time was shortened to a certain extent, which verifies the feasibility and effectiveness of the proposed method. It provides valuable reference and guidance for practical applications and further research. Based on the good performance of SDChOA, we plan to solve other practical problems. At the same time, the flexible workshop scheduling model needs to be adaptive and scalable to adapt to more realistic scenarios. From an algorithmic perspective, algorithm performance is closely related to key parameters such as coefficient vectors and convergence factors, and adaptive adjustment methods need to be developed to further improve the performance of the algorithm. Optimization algorithms still have a long way to go in the field of scheduling and require continuous improvement and development, which will eventually provide huge research value.