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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2021.015314

ARTICLE

A Homogeneous Cloud Task Distribution Method Based on an Improved Leapfrog Algorithm

Yunliang Huo1, Ji Xiong1,*, Zhixing Guo1, Qianbing You1 and Yi Peng2

1School of Mechanical Engineering, Sichuan University, Chengdu, 610065, China
2Chengdu Yigao Intelligent Technology Co., Ltd., Chengdu, 610065, China
*Corresponding Author: Ji Xiong. Email: 13668149296@163.com
Received: 08 December 2020; Accepted: 08 April 2021

Abstract: Cloud manufacturing is a new manufacturing model with crowd-sourcing characteristics, where a cloud alliance composed of multiple enterprises, completes tasks that a single enterprise cannot accomplish by itself. However, compared with heterogeneous cloud tasks, there are relatively few studies on cloud alliance formation for homogeneous tasks. To bridge this gap, a novel method is presented in this paper. First, a homogeneous cloud task distribution model under cloud environment was constructed, where services description, selection and combination were modeled. An improved leapfrog algorithm for cloud task distribution (ILA-CTD) was designed to solve the proposed model. Different from the current alternatives, the initialization operator and the leapfrog operator in ILA-CTD can ensure that the algorithm always searches the optimal solution in the feasible space. Finally, the processing of task allocation for 1000 pieces of medical labeling machine bottom plates was studied as a case to show the feasibility of the proposed method. The superiority of ILA-CTD was also proven based on more optimal solutions found, compared with the three other methods.

Keywords: Cloud manufacturing; service composition; tasks distribution; intelligent optimization; leapfrog algorithm

1  Introduction

The application of modern technologies (the Internet of Things [1,2], service-oriented technology [3], cloud computing [4], big data [5], etc.) had a profound impact on manufacturing methods. Cloud manufacturing (CMfg) is a network-based and knowledge-enhanced manufacturing model, it is a specific form of the service-oriented manufacturing paradigm [6]. Generally, there are two types of clouds: private clouds in large enterprises [6,7] and public clouds for small and medium enterprises (SMEs) [8]. The application of CMfg in SMEs would better reflect the characteristics of CMfg, such as centralized management of distributed resources, crowd-sourcing manufacturing, and highly shared manufacturing resources [9].

There is consistent requirement in CMfg in SMEs, namely that a large order needs to be fulfilled in a short time, this is undertaken by an alliance of multiple SMEs [10]. A single SME is unable to complete such a large order in a timely manner. A reasonable distribution of a large order to multiple SMEs can drastically reduce the time required to fulfill the order, because the order can be executed parallelly. The global optimization for this activity can be achieved through reasonable task allocation according to the capabilities of the alliance members. Two crucial steps are needed to form a cloud alliance: selection of members (services selection) and reasonable task distribution (services composition) [11].

Selection of alliance members, namely services selection (SS) can be achieved through methods such as semantic similarity [12], QoS (quality of services) [13] and rough-fuzzy approach [14]. For task distribution, cloud manufacturing relates to two types: heterogeneous tasks that require different services [15], and homogeneous tasks that require the same services [16]. The difference is that the latter selects services based only on QoS, and even if the production capacity of an individual service is insufficient, a large number of available services will still be obtained. However, not every available service can be assigned a task because there is the constraint of starting quantity. Therefore, how to reasonably allocate mass homogeneous tasks to such services is a complicated question with many constraints such as production capacity, QoS, and starting quantity. To propose a solution to this problem, an improved leapfrog algorithm for cloud task distribution (ILA-CTD) is presented in this study. A novel initialization operator was designed in this method to avoid the solution modification caused by the restriction of the starting quantity. The solution obtained by the leap operator satisfies the restriction of the starting quantity. Pareto optimal theory was applied to leapfrog algorithm, so that the proposed algorithm has the ability to deal with multi-objective optimization problems.

This article is organized as follows: Section 2 reviews related works; Section 3 presents the proposed model; Section 4 introduces the novel task distribution method; Section 5 describes a case study with the results discussions; and Section 6 concludes the study with remarks and suggestions for future work.

2  Related Works

2.1 Service Selection (SS)

Manufacturing services are usually designed to be user-friendly (i.e., services can be easily identified through accurate description of QoS). After alternative services are found, discrimination of the services with overlapping or identical functionalities based on QoS can be achieved [17,18]. However, things become more complex when a task consists of several subtasks, that need multiple services to accomplish collaboratively [16].

Many methods have been developed to rank and select services. Zhao et al. [19] proposed an optimal service selection approach using crowd-based cooperative computing, whose main contribution was optimally balancing the QoS and the synergy effect. Eisa et al. [18] presented a Multi-Criteria Decision-Making model to rank services based on various QoS attributes; unlike other approaches, their work was based on a real cloud provider (Amazon). Hussain et al. [20] presented a novel customer-centric Methodology for Optimal Service Selection (MOSS) in a cloud environment. Bouzary et al. [21] used TF-IDF (term frequency-inverse document frequency) to identify services that satisfy QoS. A modified interval DEA model with undesirable outputs has been adopted to achieve more accurate web service selection [22]. In addition, the dynamic change of consumer requirements was also considered by Devi et al. [23], they proposed a Linear Programming model to rank and select services dynamically.

Obviously, QoS plays an important role in cloud services selection. To build upon previous research, a selection method for homogeneous services based on QoS is proposed in this work.

2.2 Task Distribution and Services Composition

From ants to human beings, animals have the ability to cooperate, communicate and divide labor among individuals, which is inspiring collaborative manufacturing. Chen et al. [16] proposed an improved multi-objective evolutionary algorithm based on the decomposition-particle swarm optimization (MOEA/D-PSO) to obtain the optimal combination of services. Aimed at minimizing the making span, monetary and energy costs of tasks in the cloud-fog paradigm, a two-tier bipartite graph task allocation approach was presented by Gad-Elrab et al. [24] based on fuzzy set theory. Gigliotta et al. [25] examined the issues in task allocation in homogeneous communicating robots using the evolutionary algorithm. Sharma et al. [26] proposed an improved cloud task allocation strategy using a modified K-means clustering technique. A hybrid approach combining the features of genetic algorithm and the analytical hierarchy process, was implemented by Mostafa et al. [27] to distribute tasks to service suppliers. A multi-objective genetic algorithm was used by Jiang et al. [28] to allocate the disassembly tasks in the cloud environment. Jatoth et al. [29] presented an Optimal Fitness Aware Cloud Service Composition (OFASC) to balance multiple parameters of QoS. Zhou et al. [30] used evolutionary algorithms for many-objective cloud services composition. Somasundaram et al. [31] designed and developed a cloud resources broker (CLOUDRB), which integrated CLOUDRB with deadline-based job scheduling. They also developed a particle swarm optimization (PSO)-based resource allocation mechanism, to allocate users’ requirements to cloud resources in a near-optimal manner.

In summary, task distribution (services combination) is a complex problem, and intelligent evolution algorithms (such as GA, EA and PSO) have made substantial contribution to address the complexities. Inspired by the previous efforts, ILA-CTD is proposed in this study to address the homogeneous cloud task distribution problem.

3  Homogeneous Task Distribution Model

3.1 Problem Statement

The manufacturing resources in CMfg are encapsulated as manufacturing service in a cloud resources pool. When the manufacturing tasks published in CMfg, three stages follow, namely, task decomposition, services selection, and services composition. The homogeneous cloud task distribution model is shown in Fig. 1.

Manufacturing resources of factories are virtualized to various cloud services in a cloud service pool. When the tasks are uploaded, the service pool will be searched and available services will be selected. Then, the optimal service combination will be determined to identify an optimized cloud manufacturing alliance. The factories in the alliance will be notified to perform manufacturing tasks. The focus of our work is services selection and combination optimization, specifically for homogeneous tasks. This is expressed in Eq. (1):

{T(n)p={P1,P2,…,Pi,…,Pn}S(f)c={S1,S2,…,Sj,…,Sf}Mj≤sj (1)

Tp(n) indicates cloud tasks with n products, Pj is the ith product; Sc(f) is the set of available services; Sj is the jth available service; sj denotes the manufacturing capacity of Sj; and Mj is the starting quantity of Sj. Services selection can be achieved based on QoS, and then, the rest of the problem converts to the distribution of n products to f services with both QoS and cloud alliance quality considered.

Figure 1: The model of homogeneous cloud task distribution

3.2 Service Description and Selection

3.2.1 Manufacturing Service Description in the Cloud

Offline services selection mainly focuses on QoS, which involves cost, time, and quality. However, for cloud services, the attributes that affect the quality of the cloud manufacturing alliance also need to be considered. Evaluation indexes proposed by Chen give a comprehensive description of cloud manufacturing services; cloud manufacturing services (C-QoS) were expressed as 6 tuples (C-QoS = {SI, G, C0, T, C, Q}) [16].

1) SI shown in Eq. (2) is the quality consistency:

{SIi=∑j=1f∑k=1q(Qjk.Sgn(Sj)−µk)2k.∑j=1fSgn(Sj)µk=∑j=1fQjk.Sgn(Sj)∑j=1fSgn(Sj) (2)

SIi is the quality consistency of ith cloud alliance; Qjk is the evaluation value of kth quality index of Sj in its history; Sj is jth available service; Sgn (Sj ) is the status of service Sj. If Sj is selected, Sgn(Sj ) = 1; else Sgn(Sj ) = 0. f is the number of available services, and q is the total number of quality indexes.

2) G shown in Eq. (3) is the composability:

{Gi=∑j=1fSgn(Sj).Gj∑j=1fSgn(Sj)Gj=GojGnj (3)

Gi is the composability of the ith cloud alliance; Goj is the number of times that Sj has been used in a combination in its history; Gnj is the number of times that Sj has been used in its history.

3) C0 shown in Eq. (4) is communication ability:

{Coi=∑j=1fSgn(Sj).Coj∑j=1fSgn(Sj)Coj=∑h=1pCojh/p (4)

Coi is the communication ability of the ith cloud alliance; Cojh is the evaluation value of communication ability given by the hth service provider who has cooperated with Sj; p is the number of services that have collaborated with Sj.

4) T shown in Eq. (5) is the time consumed by service:

{Ti=max(Tmj+Ttj)Tmj=sij.TmajTtj=Ttaj (5)

Ti is the consumed time of the ith cloud alliance; Tmaj is the time consumed by Sj for producing a unit product; Ttaj is the time consumed when the unit product is transported by Sj; and sj is the number of products undertaken by Sj.

5) C shown in Eq. (6) is the cost:

{Ci=∑j=1f(Cmj+Ctj)Cmj=sj.CmajCtj=sj.Ctaj (6)

Ci is the cost of the ith cloud alliance; Cmaj is the cost consumed by Sj for producing a unit product; Ctaj is the cost consumed by Sj when the unit product is transported.

6) Q shown in Eq. (7) is the quality:

{Qi=∑j=1fSgn(Sj).Qj∑j=1fSgn(Sj)Qj=∑k=1qQjk/nQ (7)

Qjk is the evaluation value of the kth quality index of Sj in its history; and nQ is the total number of evaluations of Qjk in its history.

3.2.2 Services Selection

For r candidate services in the cloud pool, incapable services should first be eliminated from selection. Then, f (\ttextitf<r) available services will exist. As shown in Eq. (8), a selection method based on QoS was designed to filter incapable services; the task requirements are expressed as R = [r1, r2, …, rk, …, rc], and the attributes of jth service are expressed as Sja=[a1,a2,…,ak,…,ac].

{Ej=[e1,e2,…,ek,…,ec]ek=rk−ak∀ek∈Ejif ek<0;Sj is incapableelse;Sj is capable (8)

rk is the kth requirement, and ak is the value of the service attribute corresponding to rk.

3.3 Distribution Model

The distribution model of n products to f available services is shown in Eqs. (9)–(11). As sj≥Mj must be satisfied, a situation may arise, namely, that not every available service can be assigned with tasks if n is relatively small or f is relatively large.

M=[S1Tm1Tt1Cm1Ct1Q11⋯Q1qGn1Go1Co1⋮⋱⋮⋮⋮⋮⋯⋮⋮⋮⋱SjTmjTtjCmjCtjQj1⋯QjqGnjGojCoj⋮⋱⋮⋮⋮⋮⋯⋮⋮⋮⋱SfTmfTtfCmfCtfQf1⋯QfqGnfGofCof| M1s1⋮⋮Mjsj⋮⋮Mfsf] (9)

Therefore, a complex problem arises, because product quantities undertaken by each service and different service combinations will constitute different cloud alliance options. We assign a reasonable quantity and sj (shown in Eq. (9)) and optimize objectives (shown in Eq. (10)) under constraints (shown in Eq. (11)):

F=Max(Q,G,C0)&&Min(SI,T,C) (10)

∑j=1fsj=nsj≥Mjorsj=0Ti≤TrCi≤Cr (11)

Tr and Cr represent consumer expectations of time and cost, respectively.

4  Solution Methods

The leapfrog method is an efficient algorithm with both hereditary and group behavior [32–34]. However, the classic leapfrog algorithm is incapable of solving the above model because it is a multi-objective optimization problem, and many infeasible solutions will be generated because of the constraint of starting quality. To solve the presented model, an improved leapfrog algorithm for cloud task distribution (ILA-CTD) is proposed.

4.1 Basic Definitions

Pareto domination: Assuming that a and b are two solutions of a multi-objective function F with m benefit-oriented targets and n cost-oriented targets, if ∀ i, Fi(a) ≤ Fi(b) (i = 1, 2,…, m) and Fj(a) ≥ Fj(b) (j = 1, 2,…, n), where Fi is the benefit-oriented target and Fj is the cost-oriented target, and at least one strict inequality holds, then b dominates a.

Non-dominated solution: Assuming that c is a feasible solution of a multi-objective function F with m targets, if there is no other solution d that satisfies Fi(c) ≤ Fi(d) (i = 1, 2,…, m) and Fj(a) ≥ Fj(b) (j = 1, 2,…, n), then c is a non-dominated solution.

Elite archives: Set of non-dominated solutions obtained in the process of searching for the best solution.

4.2 ILA-CTD Operators

4.2.1 Initialization of Frog Population

A solution of the model is mapped to a frog Ui = (si1,…,sij,…,sif ) in ILA-CTD, and p frogs form a population P. Different from the classic leapfrog algorithm in which the initialization of the population is achieved via the rand methods, in this work, a novel initialization method shown in Eq. (12), was designed to satisfy the constraint in Eq. (13).

{j=unidrnd(f)k=(a∗unidrnd(b))∗(n−Mj)if r≤(c+Mj/T);sj=0else sj=Mj+unidrnd(k) (12)

unidrnd(f) is a function that produces an integer from 0 to f; a and b are the initial speed factors; c is the diversity factor; and r is a random number between 0 and 1.

∑j=1fsj=n (13)

n is the total number required to be distributed. Obviously, all solutions obtained through the designed initialization method satisfy the constraint of starting quality, and the diversity of population P is also guaranteed via factor c.

4.2.2 Fitness Function

A fitness function shown in Eq. (14) was designed to rank frogs, where fi is the fitness value of the ith frog in a population.

fi={1Ui is non−dominated solution0else (14)

4.2.3 Grouping

Dividing p frogs into q groups, where q<p. The ith group is called memeplex(i). The detailed grouping rules are as follows: (1) sorting frogs by fitness value. (2) assigning the sorted frogs to each group in turn. For example, q=3; 1th frog enters memeplex(1); 2th frog enters memeplex(2); 3th frog enters memeplex(3); and 4th frog enters memeplex(1) again.

4.2.4 Leap Operator τ(Ui)

The search process constantly makes the worst frog in each memeplex leap to a better position. The leap operator τ(Ub) is shown in Eq. (15):

Uw=(sw1,…,swj,…,swf)Ub=(sb1,…,sbj,…,sbf)(Uw=Uw+Ub−Uw) (15)

Uw is the worst frog in a memeplex, and Ub is the best frog. It is easy to prove that the solution obtained via the leap operator also satisfies the constraint of starting quality, which is more efficient than the other methods that consume time to amend infeasible solutions.

Proof. If Ui = (si1,…sij,…sif ) is a solution that satisfies the constraint of starting quality, then sij>Mj and ∑j=1fsij=n. Assuming that Uw=(sw1,…,swj,…,swf ) and Ub=(sb1,…,sbj,…,sbf ) are two solutions that satisfy the constraint of starting quality, then Eq. (16) follows:

Ub−Uw=∑j=1fsbj−∑j=1fswj=0 (16)

Hence, Uw = Uw +(Ub − Uw) satisfies constraint that ∑j=1fswj=n, for swj of the new Uw, the Eq. (17) is always existence.

snwj=swj+sbj−swj=sbj (17)

Ub is feasible and satisfies the constraint of starting quality. Therefore, sbj>Mj is also satisfied. Thus, the feasibility of the solution obtained using the leap operator is proved.

4.2.5 Local Search

Local search replaces the worst frog with a better one in each memeplex. The flowchart is shown in Fig. 2.

Figure 2: Flowchart of local search

Step 1: Find the local worst frog Uw, the local best frog Ub, and the global best frog Ugb.

Step 2: Update Uw by using the leap operator τ(Ub).

Step 3: Is the new Uw better than the old one? If yes, go to Step 4; else, go to Step 5.

Step 4: Replace the Uw with the new one.

Step 5: Update the Uw by using the leap operator τ(Ugb).

Step 6: Is the new Uw better than the existing Uw? If yes, go to Step 4; else, go to Step 7.

Step 7: Randomly generate a frog to replace Uw.

4.3 Process of ILA-CTD

Based on the above operators, the flowchart of ILA-CTD is shown in Fig. 3. The detailed steps are as follows:

Figure 3: Flowchart of ILA-CTD

Step 1: Determine the population size p, number of group q, and termination condition T.

Step 2: Initialize population with the proposed method.

Step 3: Calculate the fitness value of each frog in the population.

Step 4: Sort frogs by fitness value.

Step 5: Group p frogs into q groups.

Step 6: Perform local search for each group (update the worst one in each group).

Step 7: Converge frogs in each group, and sort frogs by fitness value again.

Step 8: Whether termination condition is met? If yes, go to Step 9; else, go to Step 10.

Step 9: Are the elite archives full? If the elite archives are not full, then fill elite archives with non-dominated solutions obtained in this generation; else, discard dominated solutions in the elite archives and fill with non-dominated solutions. Then, go to Step 5.

Step 10: Out put the elite archives.

5  A Case Study

The allocation of processing 1,000 pieces of medical labeling machine bottom plates shown in Fig. 4, was used as a case to demonstrate the feasibility of the proposed method. The performance of the plate mainly depends on the dimensional accuracy Q1, position accuracy Q2 of the holes, and flatness Q3 of the plate. The specific accuracy requirements and delivery time constraint is shown in Tab. 1; 10 available services shown in Tab. 2 were obtained after services selection.

Figure 4: Medical labeling machine bottom plate

A personalized solution is always required in actual application, and a comprehensive evaluation based on consumer preferences for targets is necessary. Therefore, the function shown in Eq. (18) was adopted to normalize the attributes values.

∙faij=faij−fjaminfjamax−fjaminbenefit−oriented∙faij=fjamin−faijfjamax−fjamincost−oriented (18)

∙faij is the normalized value of faij; faij is the jth attribute value of the ith available service; famaxj is the maximum of the jth attribute of available services; faminj is the minimum of the jth attribute of available services. In this case, Cmaj, Ctaj, Qj1, Qj2, Qj3 and Coj were normalized first to address their large data fluctuations (Tab. 3).

Then, the experiment was conducted under the environment of Windows10 and MATLAB 2016a. The parameters of ILA-CTD were set as population size P=100, elite archives = 100, number of groups q=5, evolutionary generation T=2,000, speed factors a=0.1 and b=10, and diversity factor c=0.4. The solution set obtained via ILA-CTD is shown in Tab. 4 and Fig. 5.

Figure 5: Performances of solutions obtained through ILA-CTD

5.1 Feasibility Analysis

The number of non-dominated solutions in the elite archives of each generation (Edi shown in Fig. 6) can reflect the dynamic process of the generated non-dominated solutions. As the generations increase, Eai grows rapidly at first, and then fluctuates dynamically. This means that better solutions have been found and filled into the elite archives with the increase of generations.

Figure 6: Non-dominated solutions in the elite archives of each generation

The IGD (inverted generational distance) is a simple way to evaluate the convergence of the algorithm based on the optimal solutions set. However, it is impossible for this problem. Because the optimal solutions cannot be obtained at first. Therefore, a new measure was constructed to evaluate the performance of the method proposed in this article. The model proposed in this work is a six-targets problem, and it is difficult to directly display the Pareto frontier. If the algorithm converges, then it means the Pareto frontier converges to a certain area. Thus, the average distance between adjacent generations should become steady. Hence, the average distance between adjacent generations (Da) was adopted to evaluate the convergence of the proposed method (Eq. (19)).

Da=∑i=1n∑j=1n1D(si∗,sj)/n1nD(s∗i,sj)=∑s=1k(fis−fjs)2 (19)

D(s*i, sj) is the distance between the ith solution in the kth generation and the jth solution in the (k + 1)th generation; n1 and n is the number of non-dominated solutions in the elite archives of the kth and the (k + 1)th generation, respectively. The evolution process of Da along generations is shown in Fig. 7.

Figure 7: Average distance between adjacent generations

As shown in Fig. 7, Da rises sharply at first, then falls, and finally stabilizes, which is consistent with the process predicted by the algorithm. The phenomenon results because of three reasons: (1) The non-dominated solutions with different search directions increase dramatically as the generations increase, (2) the optimal search direction is determined progressively and the non-dominated solutions tend to the definite search direction; so, Da decrease, and (3) the Pareto frontier is found, Da becomes steady, which validates that the proposed method converges.

5.2 Performance Comparison

To assess the performance of the method proposed in this work, the results obtained via ILA-CTD were compared with MOEA/D-PSO, MOEA/D-GA [11], and NSGA-Π under the same conditions. The solutions set obtained via ILA-CTD, MOEA/D-PSO, MOEA/D-GA and NSGA-Π are shown in Tabs. 5–7, respectively, and their performances are shown in Figs. 8–10, respectively.

Figure 8: Performances of solutions obtained through MOEA/D-PSO

Figure 9: Performances of solutions obtained through MOEA/D-GA

Because MOEA/D-PSO, MOEA/D-GA, and NSGA-Π are methods that have been proven to be feasible in previous studies, they can be used as comparisons to evaluate the performance of ILA-CTD. Two indicators were constructed investigate the relative performance of the proposed method.

1) Proportion of non-dominated solutions in mixed Pareto set ta

Selecting m solutions from each Pareto set that was finally obtained via MOEA/D-PSO, MOEA/D-GA, NSGA-Π, and ILA-CTD, respectively, and dominance relationships shown in Eq. (20) for the selected 4 m solutions were judged.

ta=D(ma)PDPD=4m (20)

D(ma) is the number of non-dominated solutions obtained through method a (a ∈ MOEA/D-PSO, MOEA/D-GA, NSGA-Π, ILA-CTD), and PD is total number of solutions in the mixed Pareto set. The comparison results are shown in Fig. 11 (m = 10).

Figure 10: Performances of solutions obtained through NSGA-Π

Figure 11: Performance of ILA-CTD in each generation

As the generations increased, more and more non-dominated solutions in PD were obtained via ILA-CTD. From the perspective of the number of non-dominated solutions in PD under the same conditions (T = 2000 and m = 10), ILA-CTD performed best, followed by NSGA-Π, MOEA/D-PSO, and MOEA/D-GA.

2) Proportion of feasible solutions Pf

In practical applications, the obtained solutions must be feasible. Most evolutionary algorithms deal with infeasible solutions through a penalty function. The feasibility of all final solutions cannot be guaranteed if the penalty function is unreasonable; therefore, Pf can express the reliability and practicability of a method. The Pf of ILA-CTD, NSGA-Π, MOEA/D-PSO, and MOEA/D-GA is shown in Fig. 12.

Figure 12: Pf of each method

From the perspective of Pf in the final output solutions, ILA-CTD performed best, followed by MOEA/D-PSO, NSGA-Π, and MOEA/D-GA. It can be concluded that ILA-CTD is more practical compared with the other methods.

6  Conclusions

As one of the research hotspots of CMfg, more and more studies are being undertaken on services selection and their combination especially the combination of heterogeneous services. However, the works related to homogeneous cloud services combinations are relatively few. After reviewing the existing works and noting certain limitations, a homogeneous cloud task distribution model was proposed, and an improved leapfrog algorithm for cloud task distribution (ILA-CTD) was designed to solve the model. The contributions of the paper are summarized as follows:

1) Compared to heterogeneous task distribution, there is relatively little research on homogeneous cloud tasks. A strategy of dealing with homogeneous tasks in the cloud environment was presented to bridge this gap.

2) A specific model was proposed to describe homogeneous cloud task distribution more precisely compared with alternative methods. An improved leapfrog algorithm was designed, which was proven to be more suitable for solving the proposed model.

3) The distribution of real manufacturing tasks of medical labeling machine bottom plates was presented as a case. It was shown that the proposed method provided a combination of factories with a high degree of quality similarity and high informationization under the constraints of factory capacity, task duration, and cost.

To promote the implementation of CMfg platforms with higher agility, future works should focus on the development of efficient demand-service matching algorithm. Personalized service customization should also be studied, because personalized demands from consumers are increasing.

Funding Statement: The research was financially supported by the National Science and Technology Major Project of China (No. 2019ZX04007001), the Science and Technology Major Project of Sichuan Province (No. 2020ZDZX0022).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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