A Unique Bifuzzy Manufacturing Service Composition Model Using an Extended Teaching-Learning-Based Optimization Algorithm

Abstract

In today’s competitive and rapidly evolving manufacturing environment, optimizing the composition of manufacturing services is critical for effective supply chain deployment. Since the manufacturing environment involves many two-fold uncertainties, there are limited studies that have specifically tackled these two-fold uncertainties. Based on bifuzzy theory, we put forward a unique bifuzzy manufacturing service portfolio model. Through the application of the fuzzy variable to express quality of service (QoS) value of manufacturing services, this model also accounts for the preferences of manufacturing firms by allocating various weights to different sub-tasks. Next, we address the multi-objective optimization issue through the application of extended teaching-learning-based optimization (ETLBO) algorithm. The improvements of the ETLBO algorithm include utilizing the adaptive parameters and introducing a local search strategy combined with a genetic algorithm (GA). Finally, we conduct simulation experiments to show off the efficacy and efficiency of the suggested approach in comparison to six other benchmark algorithms.

1. Introduction

Since cloud computing technologies have become more prevalent, many manufacturing services that are functionally similar but non-functionally different have emerged in cloud manufacturing systems. Customer requirements have become increasingly complex and manufacturing services have grown in scale. Because of this, selecting and combining the appropriate services to form a manufacturing service has become a dominant concern to serve the complex needs of customers [1]. At the same time, how to evaluate the final service composition solution has become an increasingly concerned issue for many experts and scholars [2].

Many prior academic and industrial studies have proposed various approaches and various perspectives to address the issue of service composition [3]. At present, the most popular approach, which has been well researched, is a workflow-based service portfolio [4]. Sequence structure, choice structure, parallel structure, and loop structure are the four fundamental representative structures of the manufacturing service composition workflow [5]. The most widely used evaluation standard for producing service portfolio solutions is the overall quality of service (QoS). However, most studies [6] assume that each service of the manufacturing service composition solution possesses the same weight for each QoS property, but that view ignores the different preferences of manufacturing firms.

Our previous work [7] and that of others [8] mainly concentrated on the deterministic manufacturing environment and avoided the fact that manufacturing activities are often subject to double uncertainty. The real manufacturing environment is complex and dynamic, and it is full of uncertainties and fuzziness. In fact, there are many uncertainties and examples of vague data in practical manufacturing activity. For example, the requirements and preferences of different users are uncertain. Additionally, it is challenging to effectively assess the quality of manufacturing services under the scope of complex human activities and the environment. The uncertainty of the machine state further increases the ambiguity of the manufacturing service quality with two-fold uncertainty [9]. Indeed, previous studies [10] have shown that the difficulty and complexity of the manufacturing service composition increase because of two-fold uncertainty.

Although some scholars have used traditional fuzzy theory to address the uncertainty issue under a manufacturing environment, many adopted triangular fuzzy numbers to describe the exact value of the QoS [11,12]. Conventional fuzzy theories typically address simple uncertainties originating from a single source, yet they fall short in handling complex issues with dual uncertainties [13]. As far as the authors are aware, no existing research utilizes bifuzzy theory to tackle service composition challenges within the manufacturing sector. Motivated by these considerations, this study introduces an innovative bifuzzy manufacturing service composition model rooted in bifuzzy theory to address dual uncertainty issues in cloud manufacturing. To better meet the manufacturing environment’s requirements, the proposed model applies bifuzzy variables to represent the dual uncertainty QoS values, which may include execution time, execution cost, availability, and reputation.

In the supply chain, when the quantity of candidate manufacturing services and sub-tasks rises, this multi-objective service combination optimization problem becomes a typical NP-hard issue [14]. Service portfolio issue has been addressed by various meta-heuristic techniques algorithms, for instance the differential evolution algorithm (DE), flower pollination algorithm (FPA), particle swarm optimization (PSO), etc. For the purpose of addressing large-scale unconstrained and constrained optimization issues, Rao and Vakharia [15] have presented the teaching-learning-based optimization (TLBO) method. Drawing inspiration from the traditional human teaching and learning process that happens in the schoolroom, the TLBO algorithm mimics the teacher–student teaching and learning process. In addition, because the TLBO algorithm has fewer parameters and stronger convergence ability, the TLBO algorithm outperforms other meta-heuristic algorithms in addressing large-scale optimization problems. Few researchers have utilized TLBO algorithms in the context of service combination optimization in a bifuzzy manufacturing setting, despite their widespread use in other domains.

In order to address dual uncertainties in cloud manufacturing, we initially create an innovative bifuzzy manufacturing service composition model rooted in bifuzzy theory. The dual uncertainty values of QoS qualities, including execution time, execution cost, availability, and reputation, are described by the proposed model using bifuzzy variables. Afterward, to resolve the proposed bifuzzy model with greater efficiency and effectiveness, we utilize a unique method referred to as extended teaching-and-learning-based optimization (ETLBO). Lastly, we carry out three sets of simulation experiments to exhibit the efficacy and efficiency of the suggested ETLBO method in comparison to PSO, GA, FPA, the basic TLBO algorithm, EGAPSO, and IDFPA.

The main contributions and innovations of this study include the following three aspects. Firstly, the bifuzzy model we have mentioned incorporates bifuzzy theory and employs bifuzzy variables to describe double uncertainties in the manufacturing environment. Secondly, the proposed bifuzzy model endows different weights to different sub-tasks to represent the preferences of manufacturing enterprises. Finally, we enhance the fundamental TLBO algorithm through two primary modifications. On the one hand, the parameters involved in the heuristic learning step length and the teaching factor are transformed from random values into dynamic adaptive parameters. On the other hand, the local search strategy for the fundamental TLBO algorithm includes the crossover operator and mutation operator of the GA. The remaining content of this study is arranged in the following manner. In Section 2, the relevant research on the evolution of the manufacturing service portfolio, the usage of bifuzzy theory, and the implementation of meta-heuristic algorithms to settle optimization issues are presented. The proposed manufacturing service combination model based on bifuzzy theory is explained in Section 3. Both the ETLBO algorithm and the fundamental TLBO algorithm are examined in Section 4. For the purpose of displaying the applicability and efficiency of the ETLBO algorithm in resolving the suggested composition model, Section 5 contains the experimental details. Finally, Section 6 presents the conclusions and recommendations for additional study.

2. Related Work

2.1. Manufacturing Service Portfolio Optimization Problem

Cloud manufacturing systems expand constantly, and problems of service composition have become more complicated with the emergence of cloud computing. Researchers investigate how to obtain appropriate solutions for manufacturing service composition by addressing trade-offs between multiple objectives such as QoS attributes (Tao et al. 2010) [16]. Therefore, a large number of approaches with different aspects have been applied in addressing the service composition and optimization problems.

A unique manufacturing service portfolio model was put forward in the authors’ prior study [17]; it took the environmental index into account and utilized an extended FPA in addressing the QoS-based manufacturing service composition issue. Additionally, the authors [18] employed a matrix-based representation scheme that concurrently addressed service scheduling, selection, and aggregation in the context of the three-dimensional manufacturing service composition issue.

For the purpose of acquiring an efficient service composition solution with high levels of QoS, Souri et al. [19] developed a mixed-form validation method to evaluate service composition, service selection, and composition procedures. Liu et al. [20] devised the cooperative service concept and a cloud-platform-based cooperation approach, integrating various dimensions and stakeholders for collaboration between multiple stakeholders and processes in service value chains. In addition to the usual QoS properties, Yang et al. [21] looked at energy usage regarding the multi-objective issue of service options and optimization from the perspective of sustainable manufacturing.

However, the previous studies expressed the value of QoS attributes in exact numbers and did not take into account the uncertainties of the manufacturing environment. In fact, due to the complexity and fuzziness of cloud manufacturing, it is challenging to precisely evaluate the QoS properties’ values of manufacturing services. Moreover, the above related research assumed that each service reflects the same weight for each of the QoS properties. The different preferences of manufacturing firms were overlooked, and the fact that different sub-tasks may have different weights for each of the QoS property was also ignored. Therefore, in this study we put forward a novel bifuzzy model for the composition of manufacturing services, which applies a bifuzzy variable to describe the values of QoS properties, and it considers the preferences of the manufacturing enterprises.

2.2. Bifuzzy Theory

In the existing studies about service composition, the values of QoS properties are usually expressed by precise numbers. However, many scholars neglect the complexity and fuzziness of manufacturing service composition optimization, leading to a deviation between actual value and the theoretical QoS values.

Liu [22] proposed bifuzzy theory, which is derived from classic fuzzy theory. Because bifuzzy theory can successfully handle the double uncertainty that classical fuzzy theory is unable to answer, it has found widespread use in a variety of domains [23]. In the evaluation process of data envelopment analysis, Paryab et al. [24] employed a bifuzzy variable to express the input data and output data observed in practical performance assessment issues. Deng and Qiu [25] proposed a novel bifuzzy discrete event system to solve a supervisory control problem. Xu et al. [24] employed complex bifuzzy variables to model the design of a dangerous goods transport network, taking emergency response into account. QoS values are better described as bifuzzy variables because of the actual production environment’s complexity and unpredictability. The basic definitions and theorems about bifuzzy variables are introduced below.

Definition 1 [22].

Let Θ be a non-empty set, and let P(Θ) be the capacity of set Θ. For every set A∈P(Θ), there exists a non-negative number Pos{A}, known as its possibility, if all the following conditions are satisfied:

(i)

Pos{0} = 0, Pos{Θ} = 1;

(ii)

for any arbitrary collection {A_k} in P(Θ), Pos{∪_k A_k} = sup_kPos{∪_k A_k}.

Then, the triplet (Θ, P(Θ), Pos) is referred to as a possibility space, and the function Pos is regarded as a possibility criterion.

Definition 2 [22].

If $\stackrel{\approx }{\xi }$ is a function of the possibility space (Θ, P(Θ), Pos) for a set of fuzzy variables, then $\stackrel{\approx }{\xi }$ is called a bifuzzy variable, and it is a fuzzy variable that takes the value of “fuzzy variable”.

Definition 3 [22].

Let ${\tilde{\eta }}_1,{\tilde{\eta }}_2,\dots ,{\tilde{\eta }}_{\mathrm{n}}$ be fuzzy numbers, and $po{s}_1,po{s}_2,\dots ,po{s}_n$ be real numbers in the range [0, 1], while satisfying that $po{s}_1\vee po{s}_2\vee \dots \vee po{s}_n=1$ . In that way,

$$\stackrel{\approx }{\xi }=\left\{\begin{array}{ll}{\tilde{\eta }}_1& \mathrm{with} \mathrm{possibility} po{s}_1\\ {\tilde{\eta }}_2& \mathrm{with} \mathrm{possibility} po{s}_2\\ & \dots \\ {\tilde{\eta }}_{\mathrm{n}}& \mathrm{with} \mathrm{possibility} po{s}_n\end{array}\right.$$

(1)

is a bifuzzy variable.

Example [22].

The forecast quantity of grain yield can be regarded as a bifuzzy variable. For instance,

$$\stackrel{\approx }{\xi }=\left\{\begin{array}{l}“\mathrm{about} \mathrm{10,000} \mathrm{ton}” \mathrm{with} \mathrm{possibility} 0.6\\ “\mathrm{about} \mathrm{10,500} \mathrm{ton}” \mathrm{with} \mathrm{possibility} 0.8\\ “\mathrm{about} \mathrm{11,200} \mathrm{ton}” \mathrm{with} \mathrm{possibility} 1\\ “\mathrm{about} \mathrm{12,000} \mathrm{ton}” \mathrm{with} \mathrm{possibility} 0.7\end{array}\right.$$

(2)

Theorem 1 [26].

Suppose that $\Re$ is a set of real numbers, and $\stackrel{\approx }{\xi }$ is a bifuzzy variable defined over the possibility space (Θ, P(Θ), Pos). Then, for any set of B of $\Re$ , the following assertions are founded:

(a)

The possibility $Pos\{\stackrel{\approx }{\xi }(\theta )\in B\}$ is a fuzzy variable;

(b)

The necessity $Nec\{\stackrel{\approx }{\xi }(\theta )\in B\}$ is a fuzzy variable;

(c)

The necessity $Cr\{\stackrel{\approx }{\xi }(\theta )\in B\}$ is a fuzzy variable.

Theorem 2 [26].

Let $\stackrel{\approx }{\xi }$ be an n-dimensional bifuzzy vector defined over the possibility space (Θ, P(Θ), Pos). If the function $f:{\Re }^n\to \Re$ is fathomable, then $f(\stackrel{\approx }{\xi })$ is a bifuzzy variable, which is defined over the possibility space (Θ, P(Θ), Pos).

Definition 4 [26].

(Bifuzzy arithmetic on a single space). Let $f:{\Re }^n\to \Re$ be a function, and $\stackrel{\approx }{{\xi }_1},\stackrel{\approx }{{\xi }_2},\dots ,\stackrel{\approx }{{\xi }_n}$ are bifuzzy variables, which are defined over the possibility space (Θ, P(Θ), Pos). In that way, $\stackrel{\approx }{\xi }=f(\stackrel{\approx }{{\xi }_1},\stackrel{\approx }{{\xi }_1},\dots \stackrel{\approx }{{\xi }_n})$ is a bifuzzy variable defined as

$$\stackrel{\approx }{\xi }(\theta )=f(\stackrel{\approx }{{\xi }_1}(\theta ),\stackrel{\approx }{{\xi }_1}(\theta ),\dots ,\stackrel{\approx }{{\xi }_n}(\theta )),\forall \theta \in \Theta .$$

(3)

Definition 5 [26].

(Bifuzzy arithmetic on different spaces). Let $f:{\Re }^n\to \Re$ be a function, and $\stackrel{\approx }{{\xi }_1},\stackrel{\approx }{{\xi }_2},\dots ,\stackrel{\approx }{{\xi }_n}$ are bifuzzy variables defined over the possibility space (Θ, P(Θ), Pos). Then, for any $({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\in \Theta$, $\stackrel{\approx }{\xi }=f(\stackrel{\approx }{{\xi }_1},\stackrel{\approx }{{\xi }_1},\dots ,\stackrel{\approx }{{\xi }_n})$ is a bifuzzy variable, which is defined over the possibility space (Θ, P(Θ), Pos), as shown by

$$\stackrel{\approx }{\xi }({\theta }_1,{\theta }_2,\dots ,{\theta }_n)=f(\stackrel{\approx }{{\xi }_1}({\theta }_1),\stackrel{\approx }{{\xi }_2}({\theta }_2),\dots ,\stackrel{\approx }{{\xi }_n}({\theta }_n))$$

(4)

In cloud manufacturing systems, to provide a more reasonable explanation of the intricacy and fuzziness of service portfolio optimization, we use bifuzzy theory to illustrate the two-fold uncertainty. Under the influence of uncertain user demands and product quality, there exists two-fold uncertainty, and the QoS values of manufacturing service including time, cost, availability, and reputation can be described by using bifuzzy variables. An example is given as follows: In the case of a manufacturing service, the processing cost is approximately $15 (with probability 0.8) under normal working conditions, $18 (with probability 1) under full-load working conditions, and $20 (with probability 0.5) under overload working conditions. Therefore, the processing cost of manufacturing service can be characterized by the following bifuzzy variable.

$$\stackrel{\approx }{C}=\left\{\begin{array}{l}“\mathrm{about} 15 \mathrm{dollars}” \mathrm{with} \mathrm{possibility} 0.8\\ “\mathrm{about} 18 \mathrm{dollars}” \mathrm{with} \mathrm{possibility} 1\\ “\mathrm{about} 20 \mathrm{dollars}” \mathrm{with} \mathrm{possibility} 0.5\end{array}\right.$$

(5)

2.3. The Implementation of Meta-Heuristic Algorithms in Manufacturing Service Portfolio

In cloud manufacturing, multi-objective service combination problems are highly dimensional and complex, making standard techniques like integer programming unsuitable for handling these kinds of optimization problems. Consequently, for the purpose of addressing the issue of allocating manufacturing resources in the supply chain, a number of meta-heuristic evolutionary algorithms, consisting of GA, PSO, and TLBO algorithms, have been proposed and studied to tackle NP-hard issues in vast search spaces [19].

For instance, we applied the flower pollination algorithm (FPA) to address various manufacturing service portfolio issues in cloud manufacturing in our earlier work [7,17]. A two-phase method combining clustering and particle swarm optimization (PSO) was put forward by Xie et al. [27] to address the issue of unstable QoS in cloud manufacturing. Seghir and Kouachi [28] utilized interval numbers to represent QoS imprecision and proposed a genetic algorithm (GA)-based optimization method to address the interval multiple criteria optimization issue in cloud manufacturing. Shi [29] put forward a manufacturing service recommendation model and utilizing ant colony optimization (ACO) in conjunction with genetic algorithms (GAs) for the selection and portfolio optimization of cloud manufacturing services. To guarantee model accuracy for energy-efficient service composition and optimal selection, Yang et al. [21] created an extended grey wolf approach, taking into account the high flexibility and complexity of cloud manufacturing. In order to minimize service time, service cost, and energy consumption, Natesha et al. [30] employed a unique meta-heuristic-based hybrid algorithm, EGAPSO, which was developed by combining the GA and PSO. Seghir and Khababa [31] introduced the generalized trapezoidal fuzzy number to represent the ambiguity of the QoS parameters and employed an improved discrete flower pollination algorithm (IDFPA) to solve the QoS-aware service composition problem.

Apart from the previously described evolutionary methods utilized in composition optimization, Rao et al. [15] developed an innovative effectively population-based optimization technique known as the TLBO algorithm in 2012. This approach has found widespread application across multiple industrial domains. In the case of considering multi-level loads, Kanwar [32] employed an enhanced TLBO algorithm to address the issue of concurrently allocating dispersed power resources in a radial distribution network. To deal with the new multi-objective, multi-skill resource limited task organizing issue, Zabihi et al. [33] put forward a mixed multi-objective TLBO algorithm to satisfy all constraints and to produce feasible project scheduling solutions. Turgut and Coban [34] suggested combining a differential evolutionary algorithm with a mixed TLBO method to effectively estimate the model parameters of fuel cells with unknown proton exchange membranes. Meena et al. [35] applied an extended TLBO to optimize trust-enforced cloud service composition for enhanced trust and QoS in cloud service composition. We used an enhanced version of the TLBO method to address the parallel optimal distribution issue associated with distributed manufacturing resources in the authors’ prior work [36].

Previous studies have shown that the TLBO algorithm can be used in various engineering fields. However, most scholars have used the TLBO algorithm to study optimization problems in deterministic environments. Few scholars have used the TLBO algorithm to solve the manufacturing service combination problem in a bifuzzy environment. In this work, building on our earlier research [36], we demonstrate an enhanced TLBO algorithm to settle the manufacturing service portfolio issue under the two-fold uncertain environment. We enhance the fundamental TLBO algorithm through two primary modifications. Initially, the parameters involved in the heuristic learning step length and the teaching factor are transformed from random values to dynamic adaptive parameters. Furthermore, the local search strategy for the fundamental TLBO algorithm includes the crossover operator and mutation operator of the GA.

3. Mathematical Model for Manufacturing Service Portfolio Optimization Problem

Many academics refer to QoS values as precise values in their earlier research. Nevertheless, because of the two-fold ambiguity in the cloud manufacturing environment, attribute values should neither be represented as exact values nor as straightforward fuzzy values. In this study, we introduce bifuzzy theory into the proposed model to represent the doubly uncertain attribute values under the manufacturing environment. Furthermore, given the diversity of preferences among manufacturing organizations, distinct sub-tasks may have differing weights assigned to each QoS property. In this study, we present a unique bifuzzy QoS-based model to address the manufacturing service portfolio optimization issue under the two-fold uncertainty environment. The evaluation method details for the mentioned model in this paper are as follows.

The manufacturing service portfolio model framework under the two-fold uncertainty environment is illustrated in Figure 1. The supply chain’s composite structure of sub-tasks is described in the first layer. The candidate service set that corresponds to various sub-tasks is shown in the second layer. The ultimate manufacturing service composition solution is created by combining the selected service candidates for each sub-task, which are described in the third layer.

3.1. Symbols Utilized in Mathematical Model for Manufacturing Service Portfolio

We need to utilize the following mathematical symbols for the purpose of representing the manufacturing service portfolio optimization issue in the mathematical model, which are shown in

Table 1. The mathematical symbols.

Symbol	Definition
STi	the ith sub-task, where i = 1, …, N, within a manufacturing service composition solution
$M{S}_j^i$	the jth candidate manufacturing service in ith sub-task, where j = 1, …, Ji
Ji	the quantity of candidate manufacturing services in ith sub-task
$\stackrel{\approx }{C}(M{S}_j^i)$	the bifuzzy execution cost for MSij to fulfil ith sub-task
$\stackrel{\approx }{T}(M{S}_j^i)$	the bifuzzy execution time for MSij to fulfil ith sub-task
$\stackrel{\approx }{Ava}(M{S}_j^i)$	the bifuzzy availability for MSij to fulfil ith sub-task
$\stackrel{\approx }{Rep}(M{S}_j^i)$	the bifuzzy reputation for MSij to fulfil ith sub-task
${\alpha }_i^T$	the ith sub-task’s time property weight, ${\displaystyle \sum _{i=1}^N{\alpha }_i^T}=1$
${\alpha }_i^C$	the ith sub-task’s cost property weight, ${\displaystyle \sum _{i=1}^N{\alpha }_i^C}=1$
${\alpha }_i^A$	the ith sub-task’s availability property weight, ${\displaystyle \sum _{i=1}^N{\alpha }_i^A}=1$
${\alpha }_i^R$	the ith sub-task’s reputation property weight, ${\displaystyle \sum _{i=1}^N{\alpha }_i^R}=1$
${\theta }_j^i$	binary variable, where 1 denotes the selection of MSij to fulfil ith sub-task, and 0 otherwise
$\stackrel{\approx }{TC}$	the bifuzzy entire cost of manufacturing service composition scheme
$\stackrel{\approx }{TT}$	the bifuzzy entire time of manufacturing service composition scheme
$\stackrel{\approx }{TA}va$	the bifuzzy entire availability of manufacturing service composition scheme
$\stackrel{\approx }{TR}ep$	the bifuzzy whole reputation of manufacturing service composition scheme
$E[\stackrel{\approx }{C}(M{S}_j^i)]$	the expected value of bifuzzy execution cost for MSij
$E[\stackrel{\approx }{T}(M{S}_j^i)]$	the expected value of bifuzzy execution time for MSij
$E[\stackrel{\approx }{Ava}(M{S}_j^i)]$	the expected value of bifuzzy availability for MSij
$E[\stackrel{\approx }{Rep}(M{S}_j^i)]$	the expected value of bifuzzy reputation for MSij

.

3.2. The Calculation Methods of QoS Properties with Bifuzzy Variables

Sequence structure, choice structure, parallel structure, and loop structure are four primary architectures used in creating service composition solutions. Figure 2 displays the four fundamental manufacturing service composition structures.

There are four normal QoS attributes in this study, consisting of execution time (T), execution cost (C), reputation (Rep), and availability (Ava) (Tao et al. 2010). Additionally, they are used to assess how well manufacturing services function. Different from the basic aggregation formulas [16], we employ the bifuzzy variable to express the QoS values rather than real numbers. In addition, we consider the preference of manufacturing firms for each of the QoS properties, and we employ a new weight coefficient α to describe the different weights for different sub-tasks for each of the QoS properties.

The calculation methods of bifuzzy QoS properties in different structures are shown as follows. Because the loop structure can be converted into the sequence structure, the calculation method of QoS in the loop structure is omitted here.

(1)

Sequence structure.

$$\left\{\begin{array}{l}\stackrel{\approx }{TT}(\mathrm{seq})={\displaystyle \sum _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^T{\theta }_j^i\stackrel{\approx }{T}(M{S}_j^i)}}\\ \stackrel{\approx }{TC}(\mathrm{seq})={\displaystyle \sum _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^C{\theta }_j^i\stackrel{\approx }{C}(M{S}_j^i)}}\\ \stackrel{\approx }{TA}va(\mathrm{seq})={\displaystyle \prod _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{[{\theta }_j^i\stackrel{\approx }{Ava}(M{S}_j^i)]}^{{\alpha }_i^A}}}\\ \stackrel{\approx }{TR}ep(\mathrm{seq})={\displaystyle \prod _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{[{\theta }_j^i\stackrel{\approx }{Rep}(M{S}_j^i)]}^{{\alpha }_i^R}}}\end{array}\right.$$

(6)

where $\stackrel{\approx }{TT}(\mathrm{seq})$, $\stackrel{\approx }{TC}(\mathrm{seq})$, $\stackrel{\approx }{TA}va(\mathrm{seq})$, and $\stackrel{\approx }{TR}ep(\mathrm{seq})$ are bifuzzy variables that represent the total execution cost property value, the entire execution time property value, the entire availability property value, and the entire reputation property value of all sub-tasks in the sequence structure, respectively. The S denotes the quantity of sub-tasks in the sequence structure.

(2)

Choice structure.

$$\left\{\begin{array}{l}\stackrel{\approx }{TT}(\mathrm{cho})={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^T{p}_i{\theta }_j^i\stackrel{\approx }{T}(M{S}_j^i)}}\\ \stackrel{\approx }{TC}(\mathrm{cho})={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^C{p}_i{\theta }_j^i\stackrel{\approx }{C}(M{S}_j^i)}}\\ \stackrel{\approx }{TA}va(\mathrm{cho})={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{p}_i{[{\theta }_j^i\stackrel{\approx }{Ava}(M{S}_j^i)]}^{{\alpha }_i^A}}}\\ \stackrel{\approx }{TR}ep(\mathrm{cho})={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{p}_i{[{\theta }_j^i\stackrel{\approx }{Rep}(M{S}_j^i)]}^{{\alpha }_i^R}}}\end{array}\right.$$

(7)

where $\stackrel{\approx }{TT}(\mathrm{cho})$, $\stackrel{\approx }{TC}(\mathrm{cho})$, $\stackrel{\approx }{TA}va(\mathrm{cho})$, and $\stackrel{\approx }{TR}ep(\mathrm{cho})$ are bifuzzy variables that represent the total execution cost property value, the entire execution time property value, the entire availability property value, and the entire reputation property value of all sub-tasks in the choice structure, respectively. The C denotes the quantity of sub-tasks in the choice structure. $p_i$ denotes the likelihood that the ith sub-task in the decision structure will be chosen.

(3)

Parallel structure.

$$\left\{\begin{array}{l}\stackrel{\approx }{TT}(\mathrm{par})=\underset{i=1,\dots ,P}{\max }\left\{{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^T{\theta }_j^i\stackrel{\approx }{T}(M{S}_j^i)}\right\}\\ \stackrel{\approx }{TC}(\mathrm{par})={\displaystyle \sum _{i=1}^P{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^C{\theta }_j^i\stackrel{\approx }{C}(M{S}_j^i)}}\\ \stackrel{\approx }{TA}va(\mathrm{par})={\displaystyle \prod _{i=1}^P{\displaystyle \sum _{j=1}^{J_i}{[{\theta }_j^i\stackrel{\approx }{Ava}(M{S}_j^i)]}^{{\alpha }_i^A}}}\\ \stackrel{\approx }{TR}ep(\mathrm{par})={\displaystyle \prod _{i=1}^P{\displaystyle \sum _{j=1}^{J_i}{[{\theta }_j^i\stackrel{\approx }{Rep}(M{S}_j^i)]}^{{\alpha }_i^R}}}\end{array}\right.$$

(8)

where $\stackrel{\approx }{TT}(\mathrm{par})$, $\stackrel{\approx }{TC}(\mathrm{par})$, $\stackrel{\approx }{TA}va(\mathrm{par})$, and $\stackrel{\approx }{TR}ep(\mathrm{par})$ are bifuzzy variables that represent the total execution cost property value, the total execution time property value, the entire availability property value, and the entire reputation property value of all sub-tasks in the parallel structure, respectively. The P represents the number of sub-tasks in the parallel structure.

By giving each QoS characteristic a varied weight based on the various user needs, we transform multiple objectives into a single objective that evaluates the manufacturing service composition solution’s overall performance using a single criterion. For the manufacturing service portfolio solution, its ultimate objective function is calculated using the following formula.

$$f(QoS)=w_T{\displaystyle \frac{{\stackrel{\approx }{TT}}_{\max }-\stackrel{\approx }{TT}}{{\stackrel{\approx }{TT}}_{\max }-{\stackrel{\approx }{TT}}_{\min }}}+w_C{\displaystyle \frac{{\stackrel{\approx }{TC}}_{\max }-\stackrel{\approx }{TC}}{{\stackrel{\approx }{TC}}_{\max }-{\stackrel{\approx }{TC}}_{\min }}}+w_A{\displaystyle \frac{\stackrel{\approx }{TA}va-\stackrel{\approx }{TA}v{a}_{\min }}{\stackrel{\approx }{TA}v{a}_{\max }-\stackrel{\approx }{TA}v{a}_{\min }}}+w_R{\displaystyle \frac{\stackrel{\approx }{TR}ep-\stackrel{\approx }{TR}e{p}_{\min }}{\stackrel{\approx }{TR}e{p}_{\max }-\stackrel{\approx }{TR}e{p}_{\min }}}$$

(9)

The total utility value of a manufacturing service portfolio solution is represented by the equation f(QoS) above. The weights of consumer preferences are denoted by $w_T$, $w_C$, $w_A$, and $w_R$, respectively, in terms of $\stackrel{\approx }{TT}$, $\stackrel{\approx }{TC}$, $\stackrel{\approx }{TA}va$, and $\stackrel{\approx }{TR}ep$. Also, the sum of the four weights is 1. ${\stackrel{\approx }{TT}}_{\max }$, ${\stackrel{\approx }{TC}}_{\max }$, $\stackrel{\approx }{TA}v{a}_{\max }$, and $\stackrel{\approx }{TR}e{p}_{\max }$ are the maximum values of $\stackrel{\approx }{TT}$, $\stackrel{\approx }{TC}$, $\stackrel{\approx }{TA}va$, and $\stackrel{\approx }{TR}ep$, respectively. ${\stackrel{\approx }{TT}}_{\min }$, ${\stackrel{\approx }{TC}}_{\min }$, $\stackrel{\approx }{TA}v{a}_{\min }$, and $\stackrel{\approx }{TR}e{p}_{\min }$ are the minimum values of $\stackrel{\approx }{TT}$, $\stackrel{\approx }{TC}$, $\stackrel{\approx }{TA}va$, and $\stackrel{\approx }{TR}ep$, respectively. Our goal in this model is to maximize the aggregate utility value.

3.3. The Equivalent Crisp Model for Manufacturing Service Composition Issue

Traditional fuzzy methods are not suitable for directly solving combinatorial optimization problems involving uncertainty since the suggested model involves bifuzzy variables. Therefore, the suggested model should be converted into an equivalent crisp model that is comparable [37]. It is common practice to utilize the expectation value operator to convert an uncertain model to another deterministic version. Thus, this study also employs the equivalent crisp model to dispose of the service portfolio issue under an uncertain environment. Definitions of the anticipated value operator for bifuzzy variables are provided below.

Definition 6 [26].

The expected value of a bifuzzy variable $\stackrel{\approx }{\xi }$, which is given by the possibility space (Θ, P(Θ), Pos), can be determined using Equation (10).

$$E[\stackrel{\approx }{\xi }]={\displaystyle {\int }_0^{+\infty }Cr\{\theta \in \Theta |E[\stackrel{\approx }{\xi }(\theta )]\ge r\}dr}-{\displaystyle {\int }_{-\infty }^{0}Cr\{\theta \in \Theta |E[\stackrel{\approx }{\xi }(\theta )]\le r\}dr}$$

(10)

It is assumed in this formulation that there is a finite integral for at least one of the two integrals.

Theorem 3 [26].

Assume that the variables $\stackrel{\approx }{\xi }$ and $\stackrel{\approx }{\eta }$ have finite expected values and are bifuzzy variables. The following equation is established for any real numbers a and b, if the following two assumptions are satisfied: (i) for each $\theta \in \Theta$, $\stackrel{\approx }{\xi }(\theta )$ and $\stackrel{\approx }{\eta }(\theta )$ are independent fuzzy variables; (ii) $E[\stackrel{\approx }{\xi }(\theta )]$ and $E[\stackrel{\approx }{\eta }(\theta )]$ are independent bifuzzy variables.

$$E[a\stackrel{\approx }{\xi }+b\stackrel{\approx }{\eta }]=aE[\stackrel{\approx }{\xi }]+bE[\stackrel{\approx }{\eta }]$$

(11)

Definition 7 [26].

For the possibility space (Θ, P(Θ), Pos), let $\stackrel{\approx }{\xi }$ and $\stackrel{\approx }{\eta }$ be two bifuzzy variables. The recommended ranking approaches are put forward.

It can be argued that $\stackrel{\approx }{\xi }>\stackrel{\approx }{\eta }$ if and only if $E[\stackrel{\approx }{\xi }]>E[\stackrel{\approx }{\eta }]$, where E is the expected value operator of the bifuzzy variable. A fuzzy expected value model is the result of this.

In accordance with the above description, the expectation value operator may be used to transform the suggested composition model in Section 3.2 into a crisp model. According to Theorem 3, Equation (6) can finally be transformed into Equation (12).

(1)

Sequence structure.

$$\left\{\begin{array}{l}E[\stackrel{\approx }{TT}(\mathrm{seq})]={\displaystyle \sum _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^T{\theta }_j^{i}E[\stackrel{\approx }{T}(M{S}_j^i)]}}\\ E[\stackrel{\approx }{TC}(\mathrm{seq})]={\displaystyle \sum _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^C{\theta }_j^{i}E[\stackrel{\approx }{C}(M{S}_j^i)]}}\\ E[\stackrel{\approx }{TA}va(\mathrm{seq})]={\displaystyle \prod _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{\{{\theta }_j^{i}E[\stackrel{\approx }{Ava}(M{S}_j^i)]\}}^{{\alpha }_i^A}}}\\ E[\stackrel{\approx }{TR}ep(\mathrm{seq})]={\displaystyle \prod _{i=1}^S{\displaystyle \sum _{j=1}^{J_i}{\{{\theta }_j^{i}E[\stackrel{\approx }{Rep}(M{S}_j^i)]\}}^{{\alpha }_i^R}}}\end{array}\right.$$

(12)

where $E[\stackrel{\approx }{TT}(\mathrm{seq})]$, $E[\stackrel{\approx }{TC}(\mathrm{seq})]$, $E[\stackrel{\approx }{TA}va(\mathrm{seq})]$, and $E[\stackrel{\approx }{TR}ep(\mathrm{seq})]$ are the expected values of $\stackrel{\approx }{TT}(\mathrm{seq})$, $\stackrel{\approx }{TC}(\mathrm{seq})$, $\stackrel{\approx }{TA}va(\mathrm{seq})$, and $\stackrel{\approx }{TR}ep(\mathrm{seq})$, respectively.

In the same way, the other calculation formulas for execution time, execution cost, availability, and reputation in the choice structure and parallel structure can be converted into the deterministic formulas as follows.

(2)

Choice structure.

$$\left\{\begin{array}{l}E[\stackrel{\approx }{TT}(\mathrm{cho})]={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^T{p}_i{\theta }_j^{i}E[\stackrel{\approx }{T}(M{S}_j^i)]}}\\ E[\stackrel{\approx }{TC}(\mathrm{cho})]={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^C{p}_i{\theta }_j^{i}E[\stackrel{\approx }{C}(M{S}_j^i)]}}\\ E[\stackrel{\approx }{TA}va(\mathrm{cho})]={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{p}_i{\{{\theta }_j^{i}E[\stackrel{\approx }{Ava}(M{S}_j^i)]\}}^{{\alpha }_i^A}}}\\ E[\stackrel{\approx }{TR}ep(\mathrm{cho})]={\displaystyle \sum _{i=1}^C{\displaystyle \sum _{j=1}^{J_i}{p}_i{\{{\theta }_j^{i}E[\stackrel{\approx }{Rep}(M{S}_j^i)]\}}^{{\alpha }_i^R}}}\end{array}\right.$$

(13)

where $E[\stackrel{\approx }{TT}(\mathrm{cho})]$, $E[\stackrel{\approx }{TC}(\mathrm{cho})]$, $E[\stackrel{\approx }{TA}va(\mathrm{cho})]$, and $E[\stackrel{\approx }{TR}ep(\mathrm{cho})]$ are the expected values of $\stackrel{\approx }{TT}(\mathrm{cho})$, $\stackrel{\approx }{TC}(\mathrm{cho})$, $\stackrel{\approx }{TA}va(\mathrm{cho})$, and $\stackrel{\approx }{TR}ep(\mathrm{cho})$, respectively.

(3)

Parallel structure.

$$\left\{\begin{array}{l}E[\stackrel{\approx }{TT}(\mathrm{par})]=\underset{i=1,\dots ,P}{\max }\left\{{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^T{\theta }_j^{i}E[\stackrel{\approx }{T}(M{S}_j^i)]}\right\}\\ E[\stackrel{\approx }{TC}(\mathrm{par})]={\displaystyle \sum _{i=1}^P{\displaystyle \sum _{j=1}^{J_i}{\alpha }_i^C{\theta }_j^{i}E[\stackrel{\approx }{C}(M{S}_j^i)]}}\\ E[\stackrel{\approx }{TA}va(\mathrm{par})]={\displaystyle \prod _{i=1}^P{\displaystyle \sum _{j=1}^{J_i}{\{{\theta }_j^{i}E[\stackrel{\approx }{Ava}(M{S}_j^i)]\}}^{{\alpha }_i^A}}}\\ E[\stackrel{\approx }{TR}ep(\mathrm{par})]={\displaystyle \prod _{i=1}^P{\displaystyle \sum _{j=1}^{J_i}{\{{\theta }_j^{i}E[\stackrel{\approx }{Rep}(M{S}_j^i)]\}}^{{\alpha }_i^R}}}\end{array}\right.$$

(14)

where $E[\stackrel{\approx }{TT}(\mathrm{par})]$, $E[\stackrel{\approx }{TC}(\mathrm{par})]$, $E[\stackrel{\approx }{TA}va(\mathrm{par})]$, and $E[\stackrel{\approx }{TR}ep(\mathrm{par})]$ are the expected values of $\stackrel{\approx }{TT}(\mathrm{par})$, $\stackrel{\approx }{TC}(\mathrm{par})$, $\stackrel{\approx }{TA}va(\mathrm{par})$, and $\stackrel{\approx }{TR}ep(\mathrm{par})$, respectively.

After obtaining the above deterministic formulas, for the manufacturing service portfolio solution, its ultimate objective function is calculated using the following formula.

$$f(QoS)=w_T{\displaystyle \frac{T{T}_{\max }-E[\stackrel{\approx }{TT}]}{T{T}_{\max }-T{T}_{\min }}}+w_C{\displaystyle \frac{T{C}_{\max }-E[\stackrel{\approx }{TC}]}{T{C}_{\max }-T{C}_{\min }}}+w_A{\displaystyle \frac{E[\stackrel{\approx }{TA}va]-TAv{a}_{\min }}{TAv{a}_{\max }-TAv{a}_{\min }}}+w_R{\displaystyle \frac{E[\stackrel{\approx }{TR}ep]-TRe{p}_{\min }}{TRe{p}_{\max }-TRe{p}_{\min }}}$$

(15)

In the above equation, $E[\stackrel{\approx }{TT}]$, $E[\stackrel{\approx }{TC}]$, $E[\stackrel{\approx }{TA}va]$, and $E[\stackrel{\approx }{TR}ep]$ represent the expected values of $\stackrel{\approx }{TT}$, $\stackrel{\approx }{TC}$, $\stackrel{\approx }{TA}va$, and $\stackrel{\approx }{TR}ep$, respectively. TT_max, TC_max, TAva_max, and TRep_max signify the maximal values of $E[\stackrel{\approx }{TT}]$, $E[\stackrel{\approx }{TC}]$, $E[\stackrel{\approx }{TA}va]$, and $E[\stackrel{\approx }{TR}ep]$, respectively. TT_min, TC_min, TAva_min, and TRep_min signify the minimal values of $E[\stackrel{\approx }{TT}]$, $E[\stackrel{\approx }{TC}]$, $E[\stackrel{\approx }{TA}va]$, and $E[\stackrel{\approx }{TR}ep]$, respectively.

4. The Application of ETLBO Algorithm in Manufacturing Service Portfolio

4.1. An Overview of the Fundamental TLBO Algorithm

As a unique group of intelligent optimization algorithms, the TLBO algorithm has been put forward by Rao et al. [15] (2012). The TLBO algorithm has fewer parameters and stronger convergence ability; thus, it has been widely used since being proposed. The algorithm simulates the process of teachers’ teaching and students’ learning, and it emphasizes interaction between teachers and students to improve students’ academic performance.

Teachers and students are treated as distinct individuals in the evolutionary process of the TLBO algorithm, and the goal function value in Section 3.3 determines the fitness value. One of the greatest individual adaptive values is the teacher, and a certain subject learned by the student is comparable to a decision variable. The two stages of the TLBO algorithm’s evolutionary process are the teaching phase and learning phase.

(1)

Teaching phase

In the teaching phase, students acquire fresh information from the teacher and their current status improves. For an optimal problem with a maximal value of the objective function f(x), NP represents the population size. The most outstanding individual is teacher X_teacher, which has the highest fitness value in the current generation, whereas X_mean is the mean condition of the current generation.

The updated location of the ith learner is based on the distinction between the status of the teacher and the average status of the current generation. Based on the previous literature, the evolution formulas for the teaching phase are as follows [15].

$$X_i^{t+1}=X_i^t+r_i(X_{teacher}^t-T{F}_i\cdot X_{mean}^t)$$

(16)

$$X_{mean}^t={\displaystyle \frac{1}{NP}}{\displaystyle \sum _{i=1}^{NP}{X}_i^t}$$

(17)

In the above equations, r_i is the step size of the heuristic learning, and it is a uniform random value that varies between 0 and 1. TF_i represents the teaching factor, and it has an equal likelihood of being 1 or 2. The calculated formula is $T{F}_i=round[1+rand(0,1)]$. $X_i^t$ represents the old state of the ith learner. $X_i^{t+1}$ represents the updated state of the ith learner after going through a teaching cycle. $X_{teacher}^t$ is the state of the teacher in iteration t, and in the entire population teacher is the optimal individual. $X_{mean}^t$ represents the mean state of the current population. If $X_i^{t+1}$ turns out to be better than $X_i^t$, then we need to update the position of the ith learner using $X_i^{t+1}$. If not then, the ith learner retains its old position $X_i^t$.

(2)

Learning phase

During the learning phase, learners attempt to polish up their state by interacting randomly with another learner. Each learner gains from each other by resolving the differences in the states of the two learners to improve their current states. The formula that follows is the associated updating equation [15].

$$X_i^{t+1}=\left\{\begin{array}{l}{X}_i^t+r_i(X_i^t-X_j^t) \mathrm{if} X_i^t \mathrm{is} \mathrm{better} \mathrm{than} X_j^t\\ X_i^t+r_i(X_j^t-X_i^t) \mathrm{if} X_j^t \mathrm{is} \mathrm{better} \mathrm{than} X_i^t\end{array}\right.$$

(18)

In the above equation, $X_i^{t+1}$ and $X_i^t$ represent the state of the ith learner before and after a learning cycle. $X_j^t$ is the jth learner, which is a randomly selected individual in iteration t. If $X_i^{t+1}$ turns out to be better than $X_i^t$, then we need to update the position of the ith learner using $X_i^{t+1}$. If not then, the ith learner retains its old position $X_i^t$.

4.2. The ETLBO Algorithm’s Representation Scheme in the Manufacturing Service Portfolio

To settle the bifuzzy manufacturing service portfolio issue, we employ an actual quantity-encoding technique in the aforementioned ETLBO method to address the bifuzzy manufacturing service portfolio issue. A manufacturing service portfolio scheme is represented by each leaner $X_i=\{x_1, x_2,\dots , x_N\}$, and N denotes the entire quantity of sub-tasks in the supply chain. Each subject of a learner is represented by a vector $x_k=[s_k, p_k, m{s}_k]$, and k = 1, 2, …, N. In this vector, $s_k$ denotes the kth sub-task’s structure, $p_k$ denotes the kth sub-task’s chosen probability of performing the manufacturing service composition solution, and $m{s}_k$ denotes the kth sub-task’s chosen candidate manufacturing service. We put up different numbers to match different structures in this part. In other words, we set $s_k$ as 1, 2, 3, or 4, depending on whether the kth sub-task has a sequence structure, choice structure, parallel structure, or loop structure.

For example, X_i = {[1, 1, 3], [3, 1, 6], [3, 1, 4], [2, 0.3, 6], [2, 0.7, 8], [4, 1, 7]} indicates that the manufacturing service composition solution contains six sub-tasks and four structures. The first subject demonstrates that the initial sub-task has a sequence structure and a 100% selection probability, and the third candidate manufacturing service is picked to perform the first sub-task. The remaining subjects are expressed in a similar manner.

4.3. Improvements of the Proposed ETLBO Algorithm

According to other meta-heuristic algorithms, the basic TBLO algorithm has an early convergence issue and has a propensity to enter local optima when iterating. To overcome these drawbacks and address the two-fold manufacturing service composition issue, we propose an ETLBO algorithm to extend the performances of the basic TLBO algorithm. The improvements to the ETLBO algorithm include these two aspects. Initially, the parameters comprising the heuristic learning step length r_i and teaching factor TF_i are changed from a random number to dynamic adaptive parameters. Second, the crossover operation and mutation operation of the GA are included in the basic TLBO algorithm as a local search strategy.

4.3.1. Dynamic Adaptive Parameters: Heuristic Learning Step Length and Teaching Factor

As for the fundamental TLBO algorithm, the heuristic learning step length r_i is a uniform random value varying between 0 and 1. Additionally, the teaching factor TF_i has an equal likelihood of being 1 or 2. That is to say, the heuristic learning step length r_i and the teaching factor TF_i are randomly generated, and their values have nothing to do with the population’s condition or the process of population evolution. However, an improved algorithm would be one that could be dynamically adaptive with the process of population evolution. Thus, in order to prevent the basic TBLO algorithm from prematurely converging and falling into local optima, we propose two dynamic adaptive parameters, including the heuristic learning step length and teaching factor. The teaching factor TF_i is dynamically generated according to the fitness of the current population. Meanwhile, the heuristic learning step length r_i is dynamically generated based on the number of iterations. The following are the corresponding calculation formulas.

$$T{F}_i=1+{\displaystyle \frac{\max f\left(X^t\right)-\mathrm{avg} f\left(X^t\right)}{\max f\left(X^t\right)}}$$

(19)

$$r_i=\delta {\displaystyle \frac{\max \_iter-t}{\max \_iter}}+0.2$$

(20)

In the above equations, $\max f\left(X^t\right)$ signifies the maximum fitness value of the current iteration t. At the same time, $\mathrm{avg} f\left(X^t\right)$ signifies the average fitness value of the current iteration t. t stands for the current iteration, and $\max \_iter$ stands for the maximum number of iterations. $\delta$ is a random value distributed between 0 and 1.

4.3.2. The Local Search Strategy Combined with GA

Although the global search capability of the basic TLBO approach is very strong, it is weak in terms of local search and often falls into the local optimum dilemma. Therefore, we introduce a new local strategy combined with the GA into the basic TLBO algorithm. Before the teaching process takes place, this new local strategy introduces the crossover operator and the variational operator, which can improve the fundamental TLBO algorithm’s efficiency of convergence and optimization accuracy. For the multi-point crossover operator, we randomly select the positions of some sub-tasks in the supply chain and exchange their selected candidate manufacturing services. In the ETLBO algorithm, we configure the crossover operator’s probability to 0.2; meanwhile, we configure the mutation operator’s probability to 0.3.

In Figure 3, we present the local search strategy combined with the GA in three parts. The first part (a) demonstrates a learner representation scheme example. The second part (b) demonstrates an example of how the multi-point crossover operator works in the ETLBO algorithm. The third part (c) demonstrates an example of the multi-point mutation operator in the ETLBO approach. By introducing a local search strategy combined with the GA, the basic TLBO algorithm is capable of avoiding local optima when addressing the manufacturing service composition issue.

4.3.3. The Specific Operation Flow of the Proposed ETLBO Algorithm

We exhibit the entire flow of the suggested ETLBO algorithm in Figure 4, which helps to illustrate in detail the specific operational procedures for addressing the manufacturing service composition issue.

5. Experimental Example and Comparative Analysis

We simulated an actual experimental example in practical implementation to illustrate the applicability of the suggested method in addressing the manufacturing service composition issue under the two-fold environment. Meanwhile, to exhibit the efficacy and efficiency of the ETLBO algorithm, we designed a series of contrast tests as compared to the GA [38], PSO [39], FPA [40](Yang et al. 2014), basic TLBO algorithm [15], EGAPSO [30], and IDFPA [31] in different experimental conditions. The computer configuration used for the simulation experiments was Windows 10-64 bit, Intel Core 3 GHz and 4 GB RAM, and the programming development language used for this experiment is Python.

5.1. Experiment Design

For the purpose of illustrating the practicability and efficacy of the mentioned ETLBO algorithm in addressing the manufacturing service composition issue in a two-fold environment, we set up a simulation experiment in which the supply chain contains four basic combination structures and six sub-tasks, as indicated in Figure 1 in Section 3.1. In each sub-task, the bifuzzy QoS values of the candidate manufacturing service were stochastically generated within a certain range. The following were the randomly generated ranges for the bifuzzy QoS values: execution time (20–30 h), execution cost (30–40 dollars), reputation (0.8–1), and availability (0.8–1). Meanwhile, to replicate the preference of manufacturing firms for QoS properties, we assigned varying weights at random for each of the QoS properties in each sub-task.

To ensure the fairness and eliminate random errors of the experimental results, all meta-heuristic algorithms were replicated 50 times to solve the proposed model with different numbers of candidate manufacturing services and different sub-tasks. Two terminal conditions existed for these experiments: first, the evolutionary trajectory reached the maximum iteration; second, less than 0.001 separated the QoS fitness difference value across three consecutive iterations [36].

The ETLBO algorithm’s parameter settings were displayed afterward. The teaching factor and the heuristic learning step length were calculated according to Equations (19) and (20) in Section 4.3.1. Through several comparative experiments, we obtained the optimal parameter values for the ETLBO algorithm. The probability of mutation was configured to 0.1. The probability of crossover was configured to 0.8. Furthermore,

Table 2. The other baseline algorithms’ parameter configurations.

Baseline Algorithm	Parameter Configurations
GA	the probability of crossover is 0.75, the probability of mutation is 0.1
PSO	the eggs’ maximum number is 2, the eggs’ minimum number is 1
FPA	the switch probability is 0.8, the scaling factor is 1
basic TLBO	the teaching factor is 1 or 2, the heuristic learning step length = uniform (0, 1)
EGAPSO	the elitism rate is 0.08 the mutation rate is 0.3 the crossover rate is 0,5
IDFPA	the switch probability is 0.8

delivers the parameter configurations for the other baseline algorithms. For these meta-heuristic algorithms, the maximum quantity of iterations was configured to 100, and the initial population scale was configured to 20.

5.2. Comparative Evaluation of the ETLBO Algorithm’s Practicability

To measure the practicability of the suggested ETLBO algorithm, we addressed the manufacturing service composition issue by designing an actual experimental example. The supply chain in the manufacturing service was configured to six sub-tasks: ST¹ = casting, ST² = forging, ST³ = drilling, ST⁴ = stamping, ST⁵ = lacquering, and ST⁶ = assembling. Each sub-task had the same number of potential manufacturing services as those given in Section 5.1. The partial expectations of the bifuzzy QoS values are listed in

Table 3. The partial expectations of the bifuzzy QoS values.

Sub-Task	Manufacturing Service	Cost ($)	Time (h)	Availability	Reliability
ST1: casting	${\mathrm{M}\mathrm{S}}_1^1$	38.66	24.33	0.87	0.87
	${\mathrm{M}\mathrm{S}}_2^1$	35.41	26.73	0.90	0.93
ST2: forging	${\mathrm{M}\mathrm{S}}_1^2$	39.03	26.49	0.98	0.92
	${\mathrm{M}\mathrm{S}}_2^2$	32.16	28.35	0.97	0.97
ST3: drilling	${\mathrm{M}\mathrm{S}}_1^3$	33.77	25.66	0.87	0.86
	${\mathrm{M}\mathrm{S}}_2^3$	35.05	29.48	0.92	0.87
ST4: stamping	${\mathrm{M}\mathrm{S}}_1^4$	35.98	20.84	0.82	0.94
	${\mathrm{M}\mathrm{S}}_2^4$	32.16	25.97	0.93	0.88
ST5: lacquering	${\mathrm{M}\mathrm{S}}_1^5$	36.12	28.74	0.91	0.90
	${\mathrm{M}\mathrm{S}}_2^5$	39.67	25.22	0.85	0.89
ST6: assembling	${\mathrm{M}\mathrm{S}}_1^6$	35.05	23.91	0.81	0.91
	${\mathrm{M}\mathrm{S}}_2^6$	37.18	27.98	0.93	0.96

. The coefficient weights of the four objectives were configured to w_T = 0.2, w_C = 0.3, w_A = 0.2, and w_R = 0.3, respectively. The preference of enterprise for different QoS properties in different sub-tasks is listed in

Table 4. Sub-task weights with respect to QoS properties.

Sub-Task	Cost	Time	Reliability	Availability
ST1: casting	0.21	0.15	0.18	0.09
ST2: forging	0.17	0.18	0.15	0.11
ST3: drilling	0.16	0.19	0.08	0.15
ST4: stampling	0.19	0.18	0.23	0.19
ST5: lacquering	0.17	0.14	0.12	0.14
ST6: assembling	0.10	0.16	0.24	0.32

.

Figure 5 exhibits the evolutionary curves of the most outstanding individual’s fitness value obtained by different algorithms when the population scale was configured to 20. In this simulation experiment, the total number of sub-tasks was set to 30 and the number of candidate services under each sub-task was set to 50, and these two settings were kept constant. From this figure, we can determine that all of the fitness values of the seven algorithms increased with the iteration process, and in the majority of cases, the fitness value of the best individual obtained by the ETLBO was the greatest. The findings demonstrate that the suggested method yielded a better manufacturing service composition solution with a higher fitness value than the other baseline algorithms. Thus, the ETLBO algorithm outperformed the other baseline algorithms, consisting of the GA, PSO, FPA, basic TLBO algorithm, EGAPSO, and IDFPA, in looking for the optimal service composition solution in a bifuzzy environment.

5.3. Comparative Evaluation of the ETLBO Algorithm’s Efficacy

To confirm the ETLBO algorithm’s efficacy in addressing the manufacturing service issue in a bifuzzy environment, we designed a series of contrast tests under different experimental conditions. As the measure indicators, these tests evaluated the best manufacturing service composition solution’s fitness value. The comparison algorithms included the GA, PSO, FPA, basic TLBO algorithm, EGAPSO, and IDFPA. The parameters comprised the quantity of candidate manufacturing services of each sub-task, the beginning population size, and the entire quantity of sub-tasks in the supply chain. Most notably, in the following contrast experiments, various combinations of weights for the QoS are considered.

In the above simulation experiments, the total number of sub-tasks was set to 30 and the number of candidate manufacturing services under each sub-task was set to 50, and these two settings were kept constant. The corresponding initial population size was set from 5 to 45. The experimental results of the seven algorithms with various initial population sizes are revealed in Figure 6. As can be seen from the figure, the fitness values of all seven algorithms become larger as the initial population size increases, and in most cases, the ETLBO algorithm obtains the largest fitness value for the best individual. This indicates that the larger the population size, the better the iteration of each algorithm. The experimental findings demonstrate that when the original population scale varies, the fitness value of the optimal scheme found by the ETLBO algorithm is greater in most cases when compared to the other baseline algorithms.

In the above simulation experiments, the total number of sub-tasks was set to 30 and the initial population size was set to 20, and these two settings were kept constant. The corresponding quantity of the candidate manufacturing services was set from 3 to 51. In order to ensure the stability of each algorithm, we conducted each set of experiments ten times and took the average fitness value of the results of the ten experiments for comparison. The experimental findings of the mentioned ETLBO algorithm in comparison to the other six algorithms when the quantity of candidate manufacturing services varied are displayed in Figure 7. As can be seen from the figures, in most cases, the ETLBO algorithm obtained the largest fitness value for the best individual. The experimental findings demonstrate that when the quantity of sub-tasks in supply chain changed, the optimal scheme’s fitness value discovered by the ETLBO algorithm was larger in most cases when compared to the other baseline algorithms.

In the above simulation experiments, the quantity of candidate manufacturing services was set to50 and the initial population size was set to 20, and these two settings were kept constant. The corresponding quantity of sub-tasks was set from 6 to 24. In order to ensure the stability of each algorithm, we conducted each set of experiments ten times and took the average fitness value of the results of the ten experiments for comparison. The experimental findings of the mentioned ETLBO algorithm in comparison to the other six algorithms when the quantity of candidate manufacturing services varied is displayed in Figure 8. As can be seen from the figures, in most cases, the ETLBO algorithm obtained the largest fitness value for the best individual. The experimental findings demonstrate that when the initial population scale varied, the optimal scheme’s fitness value discovered by ETLBO algorithm was larger in most cases when compared to the other baseline algorithms.

5.4. Comparative Analysis about the Efficiency of ETLBO Algorithm

For the purpose of verifying the efficiency of the ETLBO algorithm in addressing the manufacturing service issue in a bifuzzy environment, we designed a series of contrast tests under different experimental conditions. The measured indicators were determined by averaging the runtime of the best manufacturing service portfolio scheme. The comparison approaches included the GA, PSO, FPA, basic TLBO algorithm, EGAPSO, and IDFPA. The parameters consisted of the quantity of candidate manufacturing services of every sub-task and the quantity of sub-tasks in the supply chain. In order to ensure the stability of each algorithm, each algorithm was iterated 100 times, and each set of experiments was conducted 10 times. Noteworthy is the fact that, in the following contrast experiments, various combinations of weights for QoS were considered.

In the above simulation experiments, the total number of sub-tasks was set to 30 and the initial population size was set to 20, and these two settings were kept constant. The corresponding number of candidate manufacturing services was set from 3 to 51. The experimental findings of the mentioned ETLBO algorithm compared to the other six algorithms for various numbers of candidate manufacturing services are displayed in Figure 9. The experimental results display that regardless of how many manufacturing services were considered as candidates, the running time of all these algorithms remained stable. Therefore, when the quantity of candidate manufacturing services changed, all compared algorithms could address the manufacturing service composition issue efficiently.

In the above simulation experiments, the number of candidate manufacturing services was set to 50 and the initial population size was set to 20, and these two settings were kept constant. The corresponding number of sub-tasks was set from 6 to 24. Figure 10 shows the experimental results of the mentioned ETLBO algorithm compared to the other six algorithms with various numbers of sub-tasks in the supply chain. The experimental results clearly show that, when the quantity of sub-tasks rose, every algorithm’s running time increased linearly. Moreover, the ETLBO algorithm required more running time than the other algorithms. This was largely because the operations of calculating the dynamic adaptive parameters and local search strategy increase computational effort. Thus, to achieve a better service composition solution, the ETLBO algorithm demands more computational effort.

In conclusion, the above experimental findings display that the mentioned method, the ETLBO algorithm, could effectively address the bifuzzy manufacturing service composition problem with better performance in comparison to the other baseline algorithms, consisting of the GA, PSO, FPA, and the basic TLBO algorithm.

6. Conclusions

We put forward a unique manufacturing service composition model based on bifuzzy theory to tackle the multi-objectives service composition optimization issue, successfully addressing the two-fold uncertainties in the manufacturing environment. The service composition solution was found using an ETLBO algorithm, which performed better than existing swarm intelligence-based optimization methods like GA, PSO, FPA, basic TLBO algorithm, EGAPSO and IDFPA.

The main contributions and innovations of this study include the following three aspects. Firstly, the bifuzzy model we have mentioned incorporates the bifuzzy theory and employs bifuzzy variables to descriptive double uncertainties in manufacturing environment. Secondly, the proposed bifuzzy model endows different weights to different sub-tasks to represent the preferences of manufacturing enterprises. Finally, compared with the traditional TLBO algorithm, there are two improvements in the ETLBO algorithm. On the one hand, the parameters comprising the heuristic learning step length and the teaching factor are changed from random numbers to dynamic adaptive parameters. On the other hand, to stay out of local optima, the local search strategy is added to the fundamental TLBO algorithm, which is combined with the crossover operation and mutation operation of the GA.

However, there are some limitations in our study that offer several directions for future research. For instance, the association between manufacturing services and their impact on QoS values is also important. The relationship between manufacturing service providers also affects the final portfolio optimization solution of the supply chain. Furthermore, to enhance the algorithm’s performance even more, the TLBO algorithm can be coupled with other heuristic algorithms to produce manufacturing service composition solutions that are more efficient. Therefore, supply chain managers should consider the manufacturing service portfolio optimization problem from the three perspectives of manufacturing services, suppliers, and customers, and finally obtain the optimal service portfolio solution.