A Manufacturing SCOS Model (MSCOS) Based on the Similarity of Parameter Sequences Between Tasks and Service Composition

Service composition and optimal selection (SCOS) is a core issue in cloud manufacturing (CMfg) when integrating distributed manufacturing services for complex manufacturing tasks. Generally, a set of recommended task parameter sequences (Tps) will be given when publishing manufacturing tasks. The similarity between the service composition parameter sequence (SCps) and Tps also reflects the rationality of the service composition. However, various evaluation models based on QoS have been proposed, ignoring the rationality between the Tps and SCps. Considering the similarity of the Tps and SCps in an evaluation model, we propose a manufacturing SCOS framework called MSCOS. The framework includes two parts: an evaluation model and an algorithm for both optimization and selection. In the evaluation model, based on the numerical proximity and geometric similarity between the Tps and SCps, improving the technique for order preference by similarity to an ideal solution (TOPSIS) with the grey correlation degree (GC), we propose the GC&TOPSIS (GTOPSIS). In the optimization and selection algorithm, an improved flower pollination algorithm (IFPA) is proposed to achieve optimization and selection based on polyline characteristics between the fitness values in the population. Experiments show that the MSCOS evaluation effect and optimal selection offer better performance than commonly used algorithms.


Introduction
Cloud manufacturing (CMfg) is an advanced manufacturing model that aims to provide on-demand manufacturing services to consumers over the Internet [1]. However, a complex manufacturing task usually contains multiple processes with high complexity, and no single CMfg service can complete such a complex task alone. Therefore, service composition and optimal selection (SCOS) becomes a key for CMfg to efficiently combine various services to fulfil complex manufacturing tasks [2]. Through SCOS, manufacturing enterprises can reduce resource consumption, enhance competitiveness, improve efficiency and achieve complementary advantages in manufacturing service resources. The two key components of SCOS are an evaluation model and an algorithm for optimization and selection [3].
The evaluation model supplies a measurement indicator to estimate the service effect of the composition. Current evaluation models contain many economic indicators, such as usability [4], conflicting criteria [5], economic value [6], and energy consumption [7]. A better correlation between services determines the rationality of service composition. Wang et al. [8] designed a correlation-aware service composition model by linking the services correlation and the quality correlation between services to the feasibility of the composition. Xie et al. [9] considered the interrelations among various services and changes in QoS and proposed an efficient two-phase approach to solving the problem of unstable QoS affecting the reliability of service composition. Wang et al. [10] proposed a service-composition adaptive adjustment model to realize the exception-handling of service composition with a strict deadline or time constraints. However, few scholars have used the task parameters given in actual processing tasks as the evaluation basis. They only used them as the screening basis for the service, ignoring the reasonableness of the service parameter colocation affecting the final service level. In an actual machining process, task parameters are correlated to each other. Similar to cutting tools, the metal removal rate during processing and the three elements of the cutting process (cutting depth, feed rate, and cutting speed) are linearly related. Due to the limitation of the tool life, an increase in any one parameter can lead to decreases in the other two parameters. Therefore, it is necessary to use the correlation between task parameter sequence (Tps) as an evaluation indicator.
The technique for order preference by similarity to an ideal solution (TOPSIS) determines the evaluation value by the Euclidean distance between the positive ideal solution (PIS) and negative ideal solution (NIS) and has been successfully applied in evaluation criteria for optimal selection [11,12]. However, the TOPSIS only evaluates the Euclidean distance between each scheme to PIS and NIS, but does not consider the geometric similarity between the service composition parameter sequence (SCps) and the ideal solution [13]. The parameter curve shape can show the implicit correlation between the task parameters. The SCps should have higher numerical proximity and geometric similarity with Tps. Therefore, this article utilizes the TOP-SIS and improves it so that the evaluation model evaluates both numerical proximity and geometric similarity.
SCOS is a typical NP-hard problem. The main purpose of an evolutionary algorithm is to optimize the population and select the best service composition with the evaluation model. There are many evolutionary algorithms used to solve such problems, such as the genetic algorithm (GA) [14,15], differential evolution (DE) [16,17], and cuckoo search (CS) [18]. These evolutionary algorithms start from an initial state and initial input proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state [19]. However, these basic algorithms, aimed at various uncertainties in the real world, can significantly impact the smooth execution of a task. Therefore, appropriate improvements need to be made according to the specific actual situation. Yang et al. [20] proposed a guiding artificial bee colony grey wolf optimization (gABC-GWO) algorithm to solve the robust service composition and optimal selection model efficiently. Dedicated to exploring possible applications of deep reinforcement learning (DRL) in solving cloud manufacturing service composition (CMFg-sc), Liang et al. [21] designed a duelling deep Q-network (DQN) with prioritized replay named PD-DQN as a DRL algorithm. The effectiveness, robustness, adaptability, and scalability of the PD-DQN were investigated and compared with those of the basic DQN and Q-learning. Discussed from the perspective of sustainability, Wu et al. [22] proposed a comprehensive method to assess the sustainability of cloud manufacturing (SoM) in terms of economic, environmental, and social aspects. Furthermore, they also designed a hybrid particle swarm optimization algorithm to solve the proposed multiobjective integer bilevel multi-follower programming model. Nevertheless, these methods still exhibit some shortcomings for solving the polyline characteristic, such as poor searchability and premature convergence.
The flower pollination algorithm (FPA) is an evolutionary algorithm proposed by Yang to simulate flower pollination in nature [23]. Its feature is that Lévy flight is adopted in the search strategy. The FPA has powerful search capabilities and avoids premature convergence. Therefore, it has been used to solve the SCOS problem. However, the FPA algorithm has inherent defects. As seen from the FPA process in Algorithm 1 (to be described in the following section), the algorithm contains cross-pollination (80%) and self-pollination (20%). During cross-pollination, the best gene is searched globally first, and then the Lévy flight search strategy is used to search for cross genes. Since the optimization process only uses the best gene globally to influence the population, it reduces the influence of the excellent gene locally. Moreover, the genetic diversity of the population depends on the population itself. In the continuous optimization process, the population's genes are fixed or even reduced. Therefore, improving the FPA to increase the influence of optimal local composition and maintain the diversity of the population in continuous iterations is necessary.
Therefore, there are two essential research components to consider in the SCOS process: (1) an evaluation model to assess the similarity between the Tps and SCps; (2) an optimization and selection algorithm based on the data feature. Hence, we propose a manufacturing SCOS (MSCOS) framework to solve SCOS in cloud manufacturing. The major contributions of our work are summarized as follows: 1. Design an MSCOS framework from the perspective of the actual manufacturing process with the objectives of service composition with maximum applicability. Specifically, the framework contains two parts. One is the GC&TOPSIS (GTOPSIS) evaluation model that evaluates service composition based on rationality. The other is the improved FPA (IFPA) that determines the optimal service composition. 2. By analysing the actual characteristics of task parameters, we consider the numerical proximity and geometric similarity between a Tps and SCps to evaluate the rationality of a service composition. Based on this result, the GTOPSIS evaluation model is proposed. 3. We propose the IFPA, which expands the influence of excellent local genes and continuously introduces new genes to maintain the diversity of the population. It is more suitable for handling optimization and selection problems with polyline characteristics.
The rest of this article is organized as follows. Section 2 reviews the SCOS related works and the related basic theories. Section 3 illustrates the MSCOS framework. Section 3.1 describes the MSCOS process; Sect. 3.2 describes the GTOP-SIS for calculating the fitness value of service compositions;

Related Work
This section offers a brief introduction to current SCOS works and basic theories that the article needs to use.

SCOS
With the emergence of CMfg, an increasing number of manufacturing services are available to complete a complex task. Thus, the study of the SCOS problem has attracted increasing attention. Bi et al. [5], aiming to address several conflicting QoS criteria that should be optimized as trade-offs during SCOS in cloud manufacturing, proposed an improved nondominated sorting genetic algorithm III. They conducted several test cases to validate the performance of the proposed algorithm. Li et al. [14] established a multiobjective service composition optimization mathematical model in cloud manufacturing. Moreover, they designed a fitness function of the improved genetic algorithm based on entropy by combining Euclidean deviation with angular deviation. Yang et al. [7] proposed an energy-aware service composition and optimal selection (EA-SCOS) model based on QoS to ensure high quality and low energy consumption during the tasks. Li et al. [24] developed an SDF-oriented genetic algorithm, which uses finegrained SDF definitions to divide the service space and adopts SDF-based optimization strategies. The effectiveness of the proposed algorithm was validated by solving three real-world SCOS problems in a private CMfg setting. Bouzzary et al. [25] proposed a new hybrid approach based on the recently developed grey wolf optimizer algorithm and evolutionary operators of the genetic algorithm for composite service to meet user requirements while maintaining optimal service quality. Then, they designed and conducted a series of experiments to verify the effectiveness of the proposed algorithm. Xiao et al. [26], aiming at the multitask scheduling problem, proposed a new cloud manufacturing multitask scheduling model based on game theory from the customer perspective. Additionally, they proposed an extended biogeography-based optimization algorithm that embeds three improvements to solve the multitask scheduling model. According to the existing related studies concerning manufacturing services, although researchers have conducted comprehensive studies on the SCOS problem, few scholars have solved the problems of evaluation models and optimal selection algorithms simultaneously. In terms of the overall process of SCOS, the evaluation model and evolutionary algorithm are not separate entities. The evaluation model plays a role in evaluating the pros and cons of the optimization effect. The evolutionary algorithm finally achieves the maximization of the evaluation value. So the evaluation models and optimal selection algorithms should be studied together. Moreover, the similarity between SCps and Tps determines the rationality of service composition. The service composition distribution has polyline characteristics, which should be considered in the optimization process. Hence, in this paper, the MSCOS model is proposed that not only includes an evaluation model based on the similarity of a Tps and SCps but also includes an evolutionary algorithm to solve the multiobjective optimization problem with the polyline character.

TOPSIS
Mathematical analysis has been widely explored in recent years and has risen as an available asset for the scientific demonstration of engineering and scientific phenomena [27]. The TOPSIS, as a mathematical analysis method, was first proposed by Ching [28]. It ranks a limited number of evaluation objects according to their proximity to an idealized goal and evaluates the relative merits of existing objects [12]. The TOPSIS has been successfully applied to evaluation systems in various industries, such as engineer management [29] and online shopping websites [30].
The principle of the TOPSIS is that the scheme is sorted by detecting the distance of the PIS and NIS. If a scheme is closest to the PIS and furthest from the NIS, it is considered to be the perfect scheme. The PIS is the composition of the best values of each evaluation indicator. The NIS is the composition of the worst values of each evaluation index.
The specific implementation process is as follows: Step 1 There are n decision schemes, and each scheme has m indicators. Construct an initial matrix = a ij n×m . Step 2 Calculate the normalized decision matrix = b ij n×m . Since the indicators can be divided into efficiency indicators and cost indicators, the standardization methods must be determined separately. The normalization method is as follows [31].
• For an efficiency indicator, such as profit, when the attribute value is more significant, the property value is better.
• For a cost indicator, a smaller value corresponds to better attributes. The normalization method is: Step 3: Construct the weighted standardized matrix W is the weight matrix, and i is the weight of each index.
J + is associated with the benefit criteria and J − is associated with the cost criteria.
Step 5 Calculate the separation measures using the Euclidean distance.

D +
i is the separation of each alternative from the PIS, and D − i is the separation of each alternative from the NIS.
Step 6 Calculate the relative closeness to the PIS of each scheme:

FPA
The FPA is a metaheuristic algorithm that originates from the principle of natural pollination based on the metaphor of the proliferation of flowers in plants. The pseudocode of the FPA is presented in Algorithm 1, and the switching probability (p) is always set to 0.8 [32]: It can be seen from the above algorithm flow that the 80% probability is cross-pollination ( CROSS-POLLINATION() ), and the 20% probability of program iteration is self-pollination ( SELF-POLLINATION() ). When all iterations are complete, the best composition is output ( v best ). Taking the tth iteration as an example, the descriptions of self-pollination and cross-pollination are as follows: The optimization method is shown in Line 7. V t best is the best solution, V t b represents the old solution, V t+1 b represents the new solution, and b is the position in the population. If the fitness of V t b is better than V t+1 b , then the new solution should be discarded; conversely, the new solution replaces the old solution. The formula of Levy( ) is as follows: The optimization method is shown in Line 10. is a random number drawn from a uniform distribution in [0, 1], and V t x and V t y are different flowers from the same plants. If the fitness of V t b is better than V t+1 b , then the new solution should be discarded; otherwise, the new solution replaces the old solution.

MSCOS Framework
The MSCOS model refines the task information and service information, and it mainly contains the GTOPSIS evaluation model and the IFPA for optimization and selection. The symbols that appear in this section are described in Table 2.

MSCOS Process
The SCOS model usually includes four basic structures: sequence structure, conditional structure, parallel structure, and loop structure [33]. In the published task sequence, the composition rationality is not directly displayed but implied in the given parameters when the task is published. Therefore, this article only considers the Tps and does not consider the structure. The MSCOS framework is composed of task publishing, service selection, service composition, optimization and selection. The detailed process solution is shown in Fig. 1. 1. Task publishing It is stipulated here that the total task (T) has been decomposed into several tasks (t) when it is released.
Each published task (t) must contain the following information.
3. Service composition According to the task execution process, the services in their corresponding candidate service pools are combined into a service composition (sc). There are n tasks. 4. Optimization and selection The optimization of service composition must satisfy all task needs and maximize the overall task satisfaction.
Where BSC represents the best service composition.

Evaluation Method-GTOPSIS
The evaluation model plays an important role in applied sciences and engineering [34]. This section describes the implementation process of the GTOPSIS evaluation model. To facilitate the understanding of the following subsection, Fig. 2 illustrates the constants and variables.

PIS and NIS
The PIS and NIS are comparison standards for completing the evaluation. It is necessary to obtain them to form the ideal standard and the least ideal standard. The process of obtaining the PIS and NIS is as follows: 1. First, the service process matrix (SPM) is established by the Tps and SCps. 2. Second, the parameter conversion matrix (PCM) is constructed from the two indices of Euclidean distance and cosine similarity, combining the QoS attributes.  The remaining rows represent the service composition.
There are x service compositions in total. Service ( s i l ) is the ith service for lth service composition. Each element is represented as follows.
Service ( s i l ) has a function parameter list fun

Definition 2 Parameter conversion matrix (PCM)
There are x service compositions in total. In Eq. (20), each element represents a list containing parameter data distance, shape similarity, and QoS. Taking the l th service composition for the ith task ( t i ) as an example, in s i � l (⋅) m i , each element can be represented as v, and there are m i elements. The matching measure of task and service includes two aspects: shape similarity and data distance.
Cosine similarity distinguishes the difference from the direction. It is used to distinguish the similarity and difference of interest by user ratings on the content. The shape similarity calculation between the function property of s i l and the corresponding task is as follows.
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Fig. 2 Description of constants and variables
However, the cosine similarity is not sensitive to the absolute value; therefore, the data distance needs to be supplemented. The Euclidean distance can reflect the absolute difference of the individual numerical characteristics. It is used more often for analysis that needs to reflect the differences in the numerical magnitude or parameter similarity. The Euclidean distance between the s i l and the corresponding t i function property indicates the closeness of the parameter values of the service and tasks.

Definition 3 Standard evaluation matrix (SEM)
SEM is the matrix standardized for PCM. w is a variable whose value is between (0, p), which represents the wth data point in s i � l (⋅) m i . In PCM, for efficiency indicators, including the cosine similarity, recyclability and reliability, a larger attribute value is better. The sp w l values are calculated as follows: For cost indicators, including the Euclidean distance and cost, a smaller attribute value is better. Their elements are calculated as follows.

Definition 4 Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS)
The PIS represents the sequence closest to the Tps. The NIS represents sequences that (22) are far from the Tps. Each composition has p = ∑ n i=1 m i parameters.
J + is associated with the benefit criteria, such as reliability, recyclability, and cosine similarity. J − is associated with the cost criteria, such as the Euclidean distance and cost.

GTOPSIS Evaluation Model
The grey correlation degree (GC) can evaluate the degree of geometric similarity between factors. The GC between data sequences is used as the measurement indicator by analysing the similar relationship between data sequences [35]. The more similar the geometry, the greater the degree of grey correlation. Otherwise, the degree of grey correlation is lower [36]. To determine both numerical proximity and geometric similarity in the evaluation effect, the GC calculation model is added to the TOPSIS model to form the GTOPSIS. The GTOPSIS evaluation model is illustrated in Eq. (28) as the service composition for sc l .
1. Calculation for dp l and dn l The calculation formula for the PIS and NIS is illustrated in Eq. (29).
2. Calculation for gcp l and gcn l The GC of each composition for the PIS and NIS is calculated to reflect the geometric similarity between the composition ( sc l ) and the PIS. The calculation formula is Eq. (30).

Composition Optimization and Selection Algorithm-IFPA
To realize the optimal selection of MSCOS, the IFPA is proposed. The IFPA is a method for implementing optimization and selection in MSCOS, which uses the GTOPSIS evaluation model.

Features of IFPA
The SCOS fitness value of the service composition distribution is shown in Fig. 3. The distribution of population fitness is in polyline form, with many local extreme values. There is no obvious difference between the local extremum and the best value, so the influence of the local optima should not be ignored.
To realize optimization and selection and make the algorithm suitable for the characteristics of polyline form, the implementation process of the IFPA mainly includes two parts. The specific realization concepts and properties are as follows.
• In cross-pollination, global optimization is replaced with regional optimization in each iteration. The influence of excellent local genes is expanded as the region of the iteration moves. Based on the continuous overall optimization of the population, the fitness value of the optimal composition is also improved. • In self-pollination, a new service is selected from the candidate service pool to replace the service corresponding to the task in the composition, as shown in Fig. 4. In the iterative process, the genetic diversity of the population is maintained.

IFPA Process
The purpose of the IFPA is to continuously optimize the population and recommend a service composition with the largest fitness value. The evaluation of the evolution and (30) selection of the IFPA is based on the GTOPSIS, so the optimal model is as follows: f(.) indicates the evaluation model, as shown in Eq. (28). BSC is the optimal composition, the symbol l represents the composition position in the current population, and initpop is the initial population number. The flow of the IFPA algorithm is shown as Algorithm 2, and the switching probability (p) is always set to 0.8.
The explanation of the key process is as follows.  replaces the old solution; otherwise, the new solution will be discarded. 5. Termination condition Suppose the algorithm reaches termination criteria (exceeding the preset maximum fitness, reaching the set number of iterations, or no optimization effect within the specified number of iterations), the experiment stops and produces the best service composition BSC. Otherwise, the experiment will carry out the next iteration until the termination condition is satisfied.

Experiments
This section presents experiments conducted with the GTOPSIS matching degree test and the IFPA optimal selection performance to verify the effectiveness and feasibility of MSCOS. The manufacturing prototype and the experimental case are described in Sect. 4.1, the GTOPSIS fitness degree experiment is described in Sect. 4.2, and the IFPA optimization experiment is described in Sect. 4.3. The experiments are conducted on the Windows 7 platform (64 bit) with an Intel Core i7-4510U 2.00 GHz with 8 GB of RAM. The operating environment is PyCharm 2019.2, and the programming language is Python 3.7.

The Experimental Case
A reducer casing is a component used to install a transmission shaft and is an indispensable basic component in mechanical manufacturing. This article takes the task flow of a factory to use the reducer casing as an example of experimental verification. 'HT200' is used as a material sample, and a part of the task flow is shown in Table 3. The attribute information of the task mainly includes functional property parameters and QoS property parameters. Functional parameters include cutting speed (cs, mm/min), feed rate (fr, mm/r), and cutting depth (cd, mm/min). QoS attributes include reliability (relia) and recyclability (recyc). Table 4 shows some tools that can be used as services. The main attributes are consistent with the tasks. The experiment combines the services according to the task sequence to form an initial population. The GTOPSIS is used to evaluate the fitness of each composition in the population. The IFPA is used to achieve optimization and selection.

Composition Evaluation-GTOPSIS
The TOPSIS indicator was the compared object for evaluation. For the initial input conditions, the span ratio was set as 0.2; the initial population was set as 300; the iteration number was set as 300. We obtain the optimal composition using the GTOPSIS and TOPSIS as the evaluation indicators. Here, the TOPSIS is based on Euclidean distance, cosine similarity, and both parameters. Taking Tps as a benchmark and comparing it with the recommended SCps, the results are shown in Fig. 5. Table 5 compares the parameter values.
The results show the geometric and data distances between the first 50 parameters for the Tps and SCps. The blue diamond symbol is the task parameter and is the comparison standard. The GTOPSIS is closer to the Tps in terms of the value of geometric trend. Therefore, the GTOP-SIS evaluation can evaluate the rationale of the service composition from two aspects: parameter numerical proximity and geometric similarity.

Optimal Selection-IFPA
In this section, the IFPA is compared with four algorithms (FPA, GA, DE, CS) to verify, analyse, and discuss the advantages and problems in dealing with the SCOS problem. For parameter settings, the IFPA switching probability sw = 0.8 , = 1.5 , and the span ratio was 0.2. Basic FPA switching probability sw = 0.8 and = 1.5 . The crossover rate of the GA was 0.9, the mutation rate was 0.05, the DE crossover rate was 0.1, and the CS identification probability was 0.5. When all algorithms reached their maximum number of iterations, they terminated.

Practicality
To analyse the IFPA practicality, we recorded the maximum fitness value of each algorithm iteratively. To ensure the  validity of the comparative experiment, the experiment was carried out three times. These three experiments were based on the same population (population number of 100, initial best value of 4996) and iterated 100 times, 300 times, and 500 times. The representative results of each algorithm are shown in Fig. 6 and Table 6.
The optimization speed for every algorithm is not the same every time. The reason is that the search strategy of the algorithms has a random feature. The IFPA, FPA, and CS algorithms are significantly superior to the GA and DE algorithms. Because the IFPA, FPA, and CS all use the Lévy flight search strategy, this search strategy is suitable for the polyline  character. Moreover, the IFPA and FPA are significantly superior to the CS algorithm. The convergence rate is higher for the IFPA and FPA than for CS due to the influence of optimal genes and the continuous effect of population diversity. Initially, the convergence rate of the FPA is higher than that of the IFPA.
With an increasing number of iterations, the influence of the optimal local gene of the IFPA becomes pronounced, achieving better optimization results than the FPA.

Effectiveness
To test the effectiveness of the IFPA, the average population fitness values in the process of continuous iteration were recorded. The optimization process of average fitness when the initial population is 100 and the number of iterations is 100 is shown in Fig. 7a and Table 7. The population's fitness distribution map after 100 iterations is shown in Fig. 7b. In almost the whole process, the IFPA and FPA are superior to GA, DE, and CS. In the beginning, the optimization speed of the FPA is better than that of the IFPA. With an increasing number of iterations, the average fitness growth speed of the IFPA population is better than that of the FPA. The reason is that (1) the FPA and IFPF use the Lévy flight search strategy to avoid falling into the local optimum, thus avoiding premature convergence. (2) At the beginning of the algorithm, the FPA influences the population by the best gene globally. At the same time, the local optimum of the IFPA affects the population, so the FPA population value  increases faster than that of the IFPA. However, as the iterations increase, the influence of the best local gene continues to increase, while the influence of the best global gene decreases. Therefore, the optimization effect of the IFPA gradually became prominent.

Parameter Influence
To test the parameter influence, three representative groups of experiments with initial populations of 100, 200, and 300 were selected. The optimization is achieved through 300 iterations, and different span ratios (0.5, 0.2, 0.1) are used. The fitness trajectory of the best composition is shown in Fig. 8 and Table 8. When the initial population is 100, the effect of span on optimization is not obvious. When the initial population is 200, the optimized effect of the span ratio (sp) is significantly different. In this case, sp = 0.5 or 0.2, the optimization effect is better. When the initial population is 300, a narrow span, such as sp = 0.1 , the optimization effect is obvious. In contrast, when sp = 0.5 , the optimization effect drops.
When the population is small, the span ratio has no obvious effect. With the increase in the population, the impact of the span ratio is strengthened. The reason is that as the population increases, the number of local excellent genes increases. With continuous iterations, the influence of superior genes expands, and the optimization effect is significantly strengthened. When the parameter is 0.5, the optimization effect is unstable. That is, when the parameter is 0.5 or larger, the composition with the highest fitness in the span has an overly significant impact, may have premature phenomena, or the optimization effect may be more obvious than other parameters. When the parameter is 0.2 or 0.1, the optimization effect is ideal. The reason is the influence of excellent local genes in the population expands, and as the iteration continues, the overall optimization effect is strengthened.

Efficiency Verification
There are n tasks and an x population number. The number of iterations is n iter . The time complexity of DE is O(n × x × n iter + 2) . The time complexity of the GA is (n × x × (n iter + r)) (r is the temporary population number). The time complexity of the CS algorithm is O(n × x × (n iter + 0.5)) . The time complexity of the FPA is O((n × x) × (n iter + 1)) . The time complexity of the IFPA is O((n × x) × (n iter + s + 1)) (s is the time calculation for scope). Clearly, the time complexity is in the order of IFPA > GA > FPA > DE > CS. The initial setting for the initial population is 100, the number of iterations is 100, and the span ratio = 0.1. After ten tests, the time efficiency results are shown in Fig. 9. The average time consumption of the IFPA is higher than that of the other algorithms. The reason Table 7 Comparison of mean  fitness value  Iterations   0  20  40  60  80  100   IFPA  4817  4835  4939  5102  5245  5387  FPA  4817  4887  4948  5079  5217  5387  GA  4817  4897  4916  4956  5002  5010  CS  4817  4813  4831  4841  4868  4869  DE  4817  4844  4846  4856 4864 4865

Fig. 8 Comparison of fitness values under different parameters
is that each iteration requires an additional process to calculate scope, so the overall calculation time of the IFPA is longer than that of the other algorithms.

Conclusion
In this paper, we propose an MSCOS framework by integrating the GTOPSIS evaluation model and the optimal selection algorithm-IFPA to address the problems in actual manufacturing processes with the objective of maximum service composition rationality. The GTOPSIS model uses the perspectives of geometric similarity and numerical proximity between the Tps and SCps to evaluate the rationality of service composition. The IFPA, according to the polyline characteristics of the service composition fitness distribution, uses a regional optimization method to achieve optimization and selection. The experimental results show that the GTOPSIS can effectively evaluate the similarity between SCps and Tps. Compared with other algorithms, the convergence speed and effectiveness of the IFPA are significantly improved. Hence, MSCOS can effectively and feasibly obtain a better service composition.
This work is scalable. On the one hand, the GTOPSIS assessment of relationships between services is not comprehensive. In future evaluation modelling, the dynamic service characteristics of the processing cycle should be considered. On the other hand, the parameter settings of the IFPA, such as the span ratio, should be thoroughly studied to fully improve optimization performance.
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