A novel method for MCDM and evaluation of manufacturing services using collaborative filtering and IVIF theory

Abstract

The new era of global cooperation and competition has created challenges for manufacturing enterprises in selecting optimal services and collaboration enterprises. Multi-criteria decision making (MCDM) and evaluation has been applied in the domain of manufacturing service to address these issues. This study proposes a novel method for MCDM and evaluation of manufacturing services using collaborative filtering and interval-valued intuitionistic fuzzy (IVIF) theory. Quality of service (QoS)-aware collaborative filtering predicts missing QoS values in an IVIF rating matrix that is normalized from initial matrices that employ mixed numerical terms. An interval-valued intuitionistic fuzzy weighted arithmetic (IIFWA) aggregation operator derived from IVIF theory is utilized to evaluate and select the optimal service(s) or supplier(s). An illustrative example of manufacturing service evaluation and comparison are provided to validate the effectiveness and practicability of the proposed method.

Introduction

With the rapid development of advanced network technology, manufacturing service enterprises have entered an era of global cooperation and competition. In distributed manufacturing environments, an enterprise usually does not manufacture all required products or services alone because of high cost and poor efficiency. Cross-enterprise manufacturing collaboration, ¹ which provides effective supply-chain deployment, is an obvious trend. A more suitable supply chain of manufacturing services could provide robust shared resources and mutual interests. However, manufacturing enterprises may not be apt to select optimal services from among several functionally equivalent manufacturing services supplied by cooperative partners to form a supply-chain network. This is because of the critical predicament of evaluation and decision making in a complex, competitive, and fuzzy environment.

Nowadays, most researchers put their focus on multi-criteria decision making (MCDM) and evaluation recommendation systems, especially personalized recommendation systems, to resolve the aforementioned problems. A recommendation system is an intelligent platform that offers recommendations to users for decision-making purposes. ² Collaborative filtering (CF) ³ is a widely used recommendation algorithm and plays a crucial role in the field of personalized service evaluation and recommendation. Numerous algorithms have been applied to the manufacturing of MCDM problem to improve the quality of decision making and evaluation. These include the analytic hierarchy process (AHP), technique for order preference by similarity to an ideal solution (TOPSIS), and CF, etc. However, some issues must be addressed to deal with MCDM and CF. First, representation of evaluation information in an initial matrix is relatively monotonous, such as crisp values, ⁴ fuzzy values, ⁵ intuitionistic fuzzy numbers, ⁶ and interval-valued intuitionistic fuzzy numbers (IVIFNs). ⁷ This often results in the loss of information concerning the evaluated object, which further affects the accuracy of evaluation and decision making. Second, manufacturing enterprises usually focus on the functional requirements of services but ignore service criteria related to non-functionalization such as quality of service (QoS). Third, providing additional information to a decision maker after poor evaluation results have been obtained is difficult, because such results are typically monotonous and low dimensional.

Thus, in this study, we propose a novel manufacturing service MCDM and evaluation model to explore first the initial user-item evaluation matrix represented by hybrid numerical forms (e.g. crisp and linguistic values, intuitionistic fuzzy numbers, and IVIFNs). We then normalize the initial matrix into a new matrix that is expressed by IVIFNs, which in turn provides a more detailed description of the fuzzy object. We finally propose a QoS-aware CF algorithm (based on Ding et al. ⁸ ) with a hybrid-rating matrix in an IVIF environment. In our calculations, we utilize IVIFNs, operational rules, and an interval-valued intuitionistic fuzzy weighted arithmetic (IIFWA) aggregation operator to maintain rating information and extend the interpretative dimension of the evaluation results.

The remainder of this paper is organized as follows. The next section introduces related works. The proposed method section describes the overview of the study’s proposed method, which is further divided into four parts each of which is discussed in greater detail. In An illustrative example and evaluation section, an illustrative example and comparison are presented to verify the performance of our approach. We also discuss the results. The last section offers a conclusion and suggests topics and issues to address in future research.

Related works

Manufacturing service production environments have developed a distributed, synergistic, and cooperative characteristic, which is affected by advances in Internet and information technology. Supply-chain deployment enables manufacturing enterprises to acclimatize themselves to this changing environment. Meanwhile, MCDM, evaluation, and recommendation systems provide assistance to enterprises in selecting optimal services that will meet their personalized needs.

MCDM and evaluation

In the past few years, MCDM has been studied extensively. MCDM is primarily used to work out complex decisions and evaluations. In the domain of manufacturing services, many researchers have expended considerable effort in analyzing supplier evaluation and selection. Specifically, Talluri and Narasimhan ⁴ proposed two linear programming models to tackle performance variability measures in evaluating alternative suppliers. Akarte et al. ⁹ proposed a casting suppliers evaluation model by using an AHP algorithm that employs 18 weighted criteria for MCDM.

Zadeh ¹⁰ first proposed the theory of fuzzy sets (FS), which brought the objective and precise world into a vague and fuzzy field, and has been widely applied in MCDM. Kahraman et al. ⁵ integrated FS and AHP algorithms in order to select an optimal manufacturing supplier. Awasthi et al. ¹¹ applied the TOPSIS algorithm to the environmental performance MCDM problem under trapezoidal fuzzy theory. Khaleie and Fasanghari ⁶ combined an association coefficient and intuitionistic fuzzy entropy for group decision-making. Li et al. ⁷ proposed a method for group decision-making by using grey relation analysis (GRA) in an IVIF environment. Li and Peng ¹² used the interval-valued hesitant fuzzy theory to select shale gas. To identify locations for the selection of transshipment ports, Ding and Chou ¹³ proposed a fuzzy MCDM and evaluation model.

However, as previously mentioned, methods such as crisp values or IVIFNs do not support the use of a mixed numerical-terms rating matrix as an initial matrix nor maintain sufficient information to improve the evaluation accuracy. In addition, the data sparsity of the rating matrix is a major problem that must be solved.

In this study, we propose a novel method to tackle the initial rating matrices that involve mixed numerical terms. These matrices are then transformed into IVIFNs. A QoS-aware CF algorithm is then applied to predict missing values for the purpose of alleviating the data sparsity.

CF algorithm

CF as an effective and serviceable algorithm has been widely adopted in recommendation systems, including in manufacturing domains.^1,14 The main two categories of CF algorithms are user- and item-based algorithms.^15–17 These are heuristic-based algorithms, which refer to methods that search a data warehouse for user records that are similar to those of an active user who requires a recommendation. ¹⁸ The fundamental differences between user- and item-based CFs are the objects for calculating similarities. The objects of the first are users, whereas those of the second are items. Moreover, the Pearson correlation coefficient (PCC) and cosine-based coefficient (COS) are two critical methods that should be frequently used to calculate similarities between users and items. PCC is defined mathematically as follows

$\text{PCC}(\text{a},\text{b})=\frac{\sum _{u\in Ua\cap Ub}(r\mathrm{ua}-\overline{r}a)(r\mathrm{ub}-\overline{r}b)}{\sqrt{\sum _{u\in Ua\cap U\text{b}}(r\mathrm{ua}-\overline{r}a)2}\sqrt{\sum _{u\in Ua\cap U\text{b}}(r\mathrm{ub}-\overline{r}b)2}}$

(1) where $r\mathrm{ua}$ and $r\mathrm{ub}$ denote the assessments of items a and b, respectively, by user u, and $\overline{r}a$ and $\overline{r}b$ express the average assessment values of items a and b, respectively, by all users. COS is defined mathematically as follows

$\text{COS}(\overrightarrow{a},\overrightarrow{b})=\frac{\overrightarrow{a}·\overrightarrow{b}}{\Vert \overrightarrow{a}\Vert *\Vert \overrightarrow{b}\Vert }$

(2) where vectors a and b denote the assessments of users a and b, respectively, in n-dimensional space.

However, a problem exists in the conventional CF and must not be neglected. This concerns the means by which to improve effectively and efficiently the accuracy of a recommendation. The following studies have addressed these issues. Vozalis and Margaritis ¹⁹ proposed a method that combines singular value decomposition (SVD) and demographic information to improve CF accuracy. In addition, social networks have become increasingly popular and are a regular feature of people’s daily lives. These include such popular applications as Twitter, Facebook, LinkedIn, and Instagram. As a result, Javari and Jalili ²⁰ dug and analyzed people’s links (positive and negative) from social network applications to produce clusters for use in user-based CF. Chelmis and Prasanna ²¹ used social tags or tagging behaviors to improve the quality of recommendations. Gao et al. ²² applied used relations between users to item-based CF for recommendation.

In our previous work, ¹ we combined not only social networks but also QoS values of manufacturing services with traditional CF to produce a personalized manufacturing service recommendation. In previous studies, researchers focus on the functional evaluation criterion, whereas manufacturing services should be evaluated based on two aspects. In fact, another is a nonfunctional criterion, e.g. QoS (reliability, availability, and performance, etc.). Many studies^8,23,24 have also proposed nonfunctional QoS-aware CF for comprehensive and practical personalized recommendations.

However, even though innovative and critical research has been devoted to developing the CF algorithm which supports personalized recommendations, data sparsity issue remains in the user-item (or criterion) matrix, in which rating data possesses a high degree of dispersion and fuzziness. Specifically, on the one hand, users or enterprises that are restricted by domain knowledge or evaluation ability cannot readily provide a target service with an extremely precise and objective evaluation. On the other hand, under the circumstance of multi-criteria QoS, people are prone to explicitly judge the QoS by using crisp values because of the natural characteristics of criteria. Thus, Rodriguez et al. ²⁵ used linguistic values and a linguistic 2-tuple model to manage uncertain information caused by human cognitive processes. Zhang et al. ²⁶ proposed a novel method using CF and a Bayesian approach for an optimal procurement scheme in which criteria are expressed as trapezoidal fuzzy numbers. However, they replaced CF’s similarity calculating method with a fuzzy number similarity algorithm. Son and Thong ²⁷ presented the service criteria in the form of intuitionistic fuzzy sets generated from a recommendation system.

Although these previous studies have been revolving around replacing certain values with linguistic values, IVIFNs, and trapezoidal fuzzy numbers, they are converted into numerical crisp-values, which result in a considerable loss of information. Therefore, in this study, we apply a QoS-aware hybrid user-item (criteria) rating matrix containing crisp, linguistic, intuitionistic fuzzy, and IVIF values to an item-based CF algorithm in order to personalize manufacturing service evaluations and recommendations in an IVIF environment. In addition, the proposed method possesses a maximum measure to protect information integrity and multi-dimensional results.

Interval-valued intuitionistic fuzzy set

In 1965, Zadeh ¹⁰ first proposed the theory of fuzzy sets (FS), which has been applied to numerous research and application fields such as MCDM. An example of fuzzy set is a 50-year-old man. We regard him as neither an old nor a young man. We can assert that the man is the old man with a membership degree of 0.6 (a membership degree belongs to [0, 1]). The membership degree represents an interpretation of the vague and subjective world provided by the FS founder Zadeh.

Definition 2.1

The fuzzy set A on X is defined as follows

$\text{A}=\{〈x,\mu A(x)〉|x\in \text{X}\}$

(3) where μ_A(x) denotes the degrees of membership and satisfies the following condition: 0 ≤ μ_A(x) ≤ 1.

However, Atanassov ²⁸ showed that the membership degree is not sufficient to indicate the fuzzy object. Therefore, the intuitionistic fuzzy set (IFS) ²⁸ theory was developed as an extension of FS. IFS theory combines a non-membership with a membership degree to define an element in a set.

Definition 2.2

The concept of intuitionistic fuzzy set A on X is given by

$\text{A}=\{〈x,\mu A(x),\nu A(x)〉|x\in \text{X}\}$

(4) where μ_A(x) and ν_A(x) denote the degrees of membership and non-membership, respectively, of the element $x\in \text{X}$ to the set A and based on the condition:

$0\le \mu A(x)+vA(x)\le 1,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\forall x\in X$

For every IFS A in X

$\pi A(x)=1-\mu A(x)-\nu A(x),\forall x\in \text{X}$

(5) Next, π_A(x) is named the hesitation degree of element x to set A.

Atanassov and Gargov ²⁹ then proposed the concept of IVIFS to further promote IFS.

Definition 2.3

The concept of IVIFS A on X is an object that assumes the following mathematical form

$\text{A}=\{〈x,\tilde{\mu }A(x),\tilde{\nu }A(x)〉|x\in \text{X}\}$

(6) where $0\le \tilde{\mu }A(x)\le 1\text{and}0\le \tilde{\nu }A(x)\le 1$ denote the degrees of membership and non-membership, respectively, of the element $x\in \text{X}$ to the set A and based on the condition

$0\le \text{sup}\tilde{\mu }A(x)+\text{sup}\nu ~A(x)\le 1,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\forall x\in X$

(7) Xu ³⁰ then proposed the basic forms and operations of IVIFNs. Its mathematical form is $\alpha =([a,b],[c,d])$ , which is an IVIFN, where $[a,b]$ and $[c,d]$ denote the degrees of membership and non-membership, respectively. In addition, $\pi =[1-b-d,1-a-c]$ is given as a form to denote the hesitant degree of the IVIFN.

Definition 2.4

For three IVIFNs α = ([a, b], [c, d]), α₁ = ([a₁, b₁], [c₁, d₁]) and α₂ = ([a₂, b₂], [c₂, d₂]), a few operations are presented as follows.

$\alpha 1\oplus \alpha 2=([a1+a2-a1a2,b1+b2-b1b2],[c1c2,d1d2])$

(8)

$\alpha 1\otimes \alpha 2=([a1a2,b1b2],[c1+c2-c1c2,d1+d2-d1d2])$

(9)

$\lambda \alpha =([1-(1-a)\lambda ,1-(1-b)\lambda ],[c,d]),\lambda >0$

(10)

$\alpha \lambda =([a\lambda ,b\lambda ],[1-(1-c)\lambda ,1-(1-d)\lambda ]),\lambda >0$

(11) Based on the definitions of the addition and multiplication operations of IVIFNs, ³⁰ we expand the definitions of the subtraction and division operations as follows

$\alpha 1\ominus \alpha 2=([a1c2,b1d2],[c1+a2-c1a2,d1+b2-d1b2])$

(12)

$\alpha 1ø\alpha 2=([a1+c2-a1c2,b1+d2-b1d2],[c1a2,d1b2])$

(13) Next, we introduce the score and accuracy functions proposed by Xu ³⁰ to rank two IVIFNs.

Definition 2.5

Let α = ([a, b], [c, d]) be an IVIFN. The score and accuracy functions can be given by the following two equations (14) and (15), respectively

$\text{S}(\alpha )=\frac{1}{2}(a-c+b-d),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{S}(\alpha )\in [-1,1]$

(14)

$\text{H}(\alpha )=\frac{1}{2}(a+b+c+d),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{H}(\alpha )\in [-1,1]$

(15) where the greater the $\text{S}(\alpha )(\text{H}(\alpha ))$ , the greater is α.

Xu ³⁰ then proposed an order relation between two IVIFNs by using the two complementary and effective functions of score and accuracy, which is represented as follows:

Definition 2.6

Let $\alpha i=([ai,bi],[ci,di])(i=1,2)$ be any two IVIFNs. Then, if S (α₁) < S (α₂), then α₁ < α₂; if S (α₁) = S (α₂), then

If H (α₁) = H (α₂), then α₁ = α₂;

If H (α₁) < H (α₂), then α₁ < α₂.

In this study, the initial matrix having hybrid numeric forms will be transformed into IVIFNs.

The proposed method

In this section, we present an overview of our proposed method. Our method employs a QoS-aware item-based CF algorithm on hybrid evaluation user-item matrix with hybrid numerical forms (crisp and linguistic values, intuitionistic fuzzy numbers (IFNs), and IVIFNs). The hybrid numerical forms help to maintain the integrity of assessment information as much as possible, and are transformed finally into IVIFNs. We then combine IVIF theory, including operational rules, order relation rules, and one aggregation operator, with a modified item-based CF algorithm to recommend the Top-N services for manufacturing enterprise decision-making and evaluation. The novel approach is composed of four phases: (1) assessment data preparation; (2) service similarity computation; (3) missing QoS value prediction; and (4) evaluation and decision-making. The architecture of our proposed method is shown in Figure 1.

Figure 1.

The architecture of our method.

Assessing data preparation

In the process of assessment data preparation, we explain QoS criteria selection in a detailed manner, and then display the method for transforming multiform rating values into IVIFNs.

QoS criteria selection

As a type of nonfunctional service indicator, QoS can be used as a set of important reference indices for users or enterprises to select an optimal service. Zhou et al. ³¹ used four common QoS criteria, including transaction delay, reliability, reputation, and price, for service composition in a service-oriented network. These criteria were assessed by different numerical terms. In the manufacturing service domain, our previous work ³² proposed a set of QoS properties used for optimal manufacturing service recommendation, such as service and time cost, availability, performance, integrity, interoperability, security, etc. Considering the actual situation, we should select different QoS parameters according to different evaluation objects.

Multiple rating values of matrix normalization

As previously mentioned, certain manufacturing QoS properties can be represented by diverse forms, as enterprise users, service providers, or experts cannot always provide accurate and objective assessments. This is because of conflict of interest or limited domain knowledge. Therefore, in this study, we choose four expression forms to construct a hybrid user-item rating matrix: crisp values, linguistic values, IFNs, and IVIFNs.

After obtaining hybrid user-item rating matrices, we should transform the hybrid numerical forms into IVIFNs.

1. Transforming crisp values into IVIFNs Before the fuzzy set was proposed by Zadeh, ¹⁰ in most research on MCDM and CF, user-item rating matrix was represented by a crisp value. However, a crisp value has extremely poor evaluation ability. Here we define a method to transform a crisp value into an IVIFN. A matrix may possess some crisp values, expressed as $V=v1,v2,\dots ,vn$ . The process of transforming crisp values into IVIFNs is as follows. Select the maximum value $v\text{max}$ to be a benchmark. Then compute $vi/v\text{max}(i=1,\dots ,n)$ with a range of 0–1. If the percentile of $vi/v\text{max}$ is 5 or greater, round to the greater tenth. If the percentile of $vi/v\text{max}$ is less than 5, round to the smaller tenth. For example, suppose $V=15,16,17,18,19,20$ is a set of crisp values. After computation, the values will be $0.8,0.8,0.9,0.9,1,1$ . Then transform these values into IVIFNs, which are shown in Table 1.

   Table 1.

   Transforming crisp values into IVIFNs.

   

   Normalized crisp numbers	IVIFNs
   1	$([0.875,0.975][0.005,0.025])$
   0.9	$([0.785,0.885][0.095,0.115])$
   0.8	$([0.695,0.795][0.185,0.205])$
   0.7	$([0.605,0.705][0.275,0.295])$
   0.6	$([0.515,0.615][0.365,0.385])$
   0.5	$([0.425,0.525][0.455,0.475])$
   0.4	$([0.335,0.435][0.545,0.565])$
   0.3	$([0.245,0.345][0.635,0.655])$
   0.2	$([0.155,0.255][0.725,0.745])$
   0.1	$([0.065,0.165][0.815,0.835])$

Transforming linguistic values into IVIFNs Linguistic variables assume values defined in a term set, i.e. the set of linguistic terms, which are subjective categories for linguistic variables. ³³ In this study, we use nine linguistic terms (inspired by Sun ³³ ) to evaluate the manufacturing service as “Perfect,” “Very good,” “Fairly good,” “Good,” “Fair,” “Bad,” “Fairly bad,” “Very bad,” and “Terrible.” Table 2 that follows shows the defined method to transform linguistic values into IVIFNs.

Table 2.

Transforming linguistic values into IVIFNs.

Linguistic numbers	IVIFNs
Perfect	$([0.85,0.95][0.01,0.05])$
Very good	$([0.75,0.85][0.11,0.15])$
Fairly good	$([0.65,0.75][0.21,0.25])$
Good	$([0.55,0.65][0.31,0.35])$
Fair	$([0.45,0.55][0.41,0.45])$
Bad	$([0.35,0.45][0.51,0.55])$
Fairly bad	$([0.25,0.35][0.61,0.65])$
Very bad	$([0.15,0.25][0.71,0.75])$
Terrible	$([0.05,0.15][0.81,0.85])$

Transforming intuitionistic fuzzy values into IVIFNs The concept of IVIFS was expanded from IFS by Atanassov and Gargov. ²⁹ We then define a method to transform IFNs into IVIFNs.

Definition 3.1 Let $\alpha =(\mu ,\epsilon )$ be an IFN, $\alpha ~=\{[a,b],[c,d]\}$ be an IVIFN, then $\alpha \Rightarrow \alpha ~$ is given by

$\alpha ~=\{[\mu -\beta ,\mu +\beta ],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}[\epsilon -\beta ,\epsilon +\beta ]\}$

(16) where β denotes a free variable (in this study, β is set to 0.05).

Missing QoS value prediction

The phase of missing QoS value prediction consists of three parts: (a) service similarity computation; (b) Top-N services selection and (c) QoS prediction.

Service similarity computations

In the phase of service similarity computation, a modified item-based CF proposed by Ding et al. ⁸ is adopted to calculate the similarity between two manufacturing services.

First, we construct a model to represent a multiple-criteria decision-making and evaluation-CF (MCDME-CF) problem. Suppose a MCDME-CF model has M users, N services, and K criteria. The typical model is then expressed by

$Xi=(x_{\mathrm{mn}}^i)\mathrm{MN}=(x_{11}^i\dots x_{1N}^i::x_{M1}^i\dots x_{\mathrm{MN}}^i),i\in K$

where $x_{\mathrm{mn}}^i$ (m = 1, 2,…M, n = 1, 2,…N and i = 1, 2,…K) denotes the assessment value against each criterion. If $x_{\mathrm{mn}}^i$  = Null, it means service n has not previously been invoked by user m.

In Ding et al., ⁸ the Pearson correlation coefficient (PCC) was applied to calculate the similarity between two manufacturing services. The particular form of the formula is as follows

$\text{Sim}\_\text{PCC}(\text{S}\text{n},\text{S}\text{y})=\frac{\sum _{m\in U}(xm,n-\overline{x}n)(xm,y-\overline{x}y)}{\sqrt{\sum _{m\in U}(xm,n-\overline{x}n)2}\sqrt{\sum _{m\in U}(xm,y-\overline{x}y)2}}$

(17) where S_n and S_y are two manufacturing services, Sim_PCC(S_n, S_y) is the expected similarity degree between S_n and S_y, and U represents the set of users who have invoked both S_n and S_y. In addition, m denotes one user in U. Specifically, because each $xm,n$ ( $xm,y$ ) that denotes the assessment value by user m for service S_n (S_y) is expressed with IVIFN, we employ an interval-valued intuitionistic fuzzy weighted arithmetic (IIFWA) aggregation operator proposed by Xu ³⁰ to substitute for the calculation of the average QoS value $\overline{x}n(\overline{x}y)$ for service S_n (S_y). In recent years, the IIFWA aggregation operator has been used in many fields in an IVIF environment such as MCDM. ³⁰

Definition 3.2

Let $\alpha j=\{[aj,bj],[cj,dj]\}(j=1,2,\dots ,n)$ be a set of IVIFNs. The IIFWA aggregation operator is then calculated as follows

$\text{IIFWA}(\alpha 1,\alpha 2,\dots ,\alpha n)={\otimes }_{j=1}^n{\alpha }_j^{\omega j}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=([1-\underset{j=1}{\overset{n}{\Pi }}(1-aj)\omega j,1-\underset{j=1}{\overset{n}{\Pi }}(1-bj)\omega j],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}[\underset{j=1}{\overset{n}{\Pi }}(cj)\omega j,\underset{j=1}{\overset{n}{\Pi }}(dj)\omega j])$

(18) where $\omega =(\omega 1,\omega 2,\dots ,\omega n)\text{T}$ is the weight vector of the IIFWA aggregation operator, $\omega j\in [0,1]$ , and $\sum _{j=1}^n\omega j=1.$

Nonetheless, Ding et al. ⁸ argued that the similarity computation with PCC is affected by positive and negative services. ⁸ Therefore, they introduced an extended PCC approach called f-PCC that would amend this problem and improve prediction accuracy.

Definition 3.3⁸

Supposing $Sn$ and $Sy$ are two services, the f-PCC approach is defined as

$Cn,y=2(\frac{|Un,y|}{\mathrm{MAX}y\in [1,N],y\ne n(|Un,y|)})-1$

(19)

$\text{Sim}C(Sn,\text{S}y)=Cn,y\text{Sim}(Sn,\text{S}y)$

(20) where $|Un,y|$ is the set of users who have invoked both S_n and S_y, $Cn,y\ge 0$ , when $|Un,y|=\mathrm{MAX}y\in [1,N],y\ne n(|Un,y|)$ , and $Cn,y$ achieves its maximum of 1.

Top-N services selection

According to the computation similarity degrees from f-PCC, it is convenient to filter the Top-N nearest neighbor services from all manufacturing services based on the following condition:

Definition 3.4

Suppose ${\mathrm{Sim}}_i^C(Sn,\text{S}y)=([ai,bi],[ci,di]),(i=1,\dots ,N-1)$ is a set of IVIFNs. The condition (adapted from Ding et al. ⁸ ) is given by

$U(\varepsilon )=\{Sy|S({\mathrm{Sim}}_i^C(Sn,\text{S}y))\ge 0,y\ne n,i=1,\dots ,N-1\}$

(21) where $S({\mathrm{Sim}}_i^C(Sn,\text{S}y))$ denotes the score function (14) of ${\mathrm{Sim}}_i^C(Sn,Sy)$ and $U(\varepsilon )$ represents the set of Top-N services.

QoS prediction

Item-based CF uses services with the highest similarity to predict the missing QoS criterion values in order to decrease the sparsity of the QoS rating matrix and acquire assessments about the unknown service. The missing value $\stackrel{\wedge }{x}m,n$ for service $Sn$ is calculated by Ding et al. ⁸ as

$\stackrel{\wedge }{x}m,n=\stackrel{\wedge }{x}n+\frac{\sum _{Sy\in U(\varepsilon )}\text{Sim}\text{C}(Sn,Sy)\times (xm,y-\overline{x}y)}{\sum _{Sy\in U(\varepsilon )}\text{Sim}\text{C}(Sn,Sy)}$

(22) where $\overline{x}n$ ( $\overline{x}y$ ) is the average value of all assessment QoS values for service $Sn$ ( $Sy$ ). Service $Sy$ is one of $U(\varepsilon ),$ which is the set of Top-N services. In addition, $\text{Sim}\text{C}(Sn,Sy)$ denotes the similarity value between $Sn$ and $Sy$ , and $xm,y$ represents the assessment value of service $\text{S}\text{y}$ .

Evaluation and decision making

Until now, all complete matrices have been obtained. However, these matrices are classified according to QoS criteria, and we have yet to obtain a comprehensive evaluation value of the manufacturing services. We must then apply an approach to integrate the rating scores from assessment values against each QoS criterion. We use an IIFWA aggregation operator to integrate the rating matrices. For every enterprise (decision maker), the preferences of different nonfunctional QoS properties of functionally equivalent manufacturing services are usually different. Therefore, the weights of criteria are set by the enterprise decision maker.

After the integration, we can aggregate all matrices $Xi(i\in K)$ into a comprehensive matrix $X(M\times N)$ by using the IIFWA operator, expressed by IVIFNs for rating scores as follows

$X=(x\mathrm{mn})\mathrm{MN}=(x11\dots x1N::xM1\dots x\mathrm{MN})$

where every row denotes a user, each column represents a service, and $x\mathrm{mn}$ is the aggregated evaluation value for user m who invokes service n.

Finally, we again use the IIFWA aggregation operator to calculate each service’s comprehensive evaluation value into a one-dimensional matrix $Z(1\times N)$ from the matrix $X$ as follows

$Z=(zn)1N=(z1\dots zN)$

where z_n denotes the aggregated comprehensive evaluation value for all users who invokes service n and is represented by IVIFN.

Finally, the score and accuracy functions of IVIFNs are employed to rank $zn(n\epsilon [1,N])$ and to select optimal service(s).

An illustrative example and evaluation

An Illustrative example

In this section, we discuss the effectiveness and practicability of the proposed method by using an actual problem encountered by manufacturing services. A manufacturing enterprise would like to select the best bearings supplier(s) by considering four evaluation criteria: (1) k₁: availability; (2) k₂: performance; (3) $k3$ : service cost; and (4) $k4$ : reliability. Five potential bearing suppliers exist, given as s_i ( $i=1,2,3,4,5$ ), along with six enterprises, given as u_j ( $j=1,2,3,4,5$ ), who may or may not have collaborated with all of the five potential suppliers. The initial user-service rating matrices $\mathrm{tb}l(l=1,2,3,4)$ , as shown in Tables 3–6.

Table 3.

Hybrid-valued user-service rating matrix $\mathrm{tb}1$ for criterion k₁.

	s 1	s 2	s 3	s 4	s 5
u 1	$([0.5,0.6],[0.2,0.3])$	$([0.3,0.4],[0.4,0.6])$	Fair	$([0.8,0.9],[0.05,0.1])$	NULL
u 2	10	NULL	$([0.7,0.8],[0.1,0.2])$	$[0.7,0.2]$	$([0.3,0.4],[0.45,0.55])$
u 3	$[0.6,0.2]$	5	Good	NULL	$[0.2,0.6]$
u 4	$([0.5,0.7],[0.1,0.2])$	NULL	$([0.4,0.6],[0.2,0.3])$	NULL	4
u 5	Very good	$([0.2,0.4],[0.5,0.6])$	7	Bad	$[0.3,0.4]$
u 6	NULL	$[0.8,0.1]$	$([0.8,0.9],[0.05,0.1])$	6	Bad

Table 4.

Hybrid-valued user-service rating matrix $\mathrm{tb}2$ for criterionk₂.

	s 1	s 2	s 3	s 4	s 5
u 1	$([0.7,0.8],[0.1,0.2])$	NULL	$([0.6,0.8],[0.1,0.2])$	$([0.6,0.8],[0.1,0.2])$	NULL
u 2	$[0.4,0.3]$	$([0.6,0.7],[0.1,0.2])$	$([0.4,0.6],[0.3,0.4])$	$7$	Fairly good
u 3	$10$	$[0.5,0.2]$	Fair	NULL	$3$
u 4	Good	NULL	$([0.7,0.8],[0.1,0.2])$	$([0.1,0.2],[0.6,0.8])$	$[0.4,0.2]$
u 5	Very good	$([0.5,0.6],[0.1,0.3])$	$[0.7,0.1]$	Bad	NULL
u 6	NULL	9	$([0.3,0.4],[0.4,0.6])$	$[0.6,0.3]$	$([0.2,0.4],[0.3,0.4])$

Table 5.

Hybrid-valued user-service rating matrix $\mathrm{tb}3$ for criterion k₃.

	s 1	s 2	s 3	s 4	s 5
u 1	Fair	7	NULL	$([0.5,0.7],[0.2,0.3])$	NULL
u 2	$([0.5,0.6],[0.2,0.3])$	Good	$[0.4,0.1]$	$([0.4,0.5],[0.2,0.4])$	Very good
u 3	$([0.3,0.5],[0.3,0.4])$	$([0.6,0.8],[0.1,0.2])$	3	NULL	$([0.5,0.8],[0.1,0.2])$
u 4	$[0.3,0.4]$	NULL	Bad	$([0.5,0.7],[0.1,0.3])$	10
u 5	4	$[0.7,0.2]$	$([0.2,0.4],[0.4,0.5])$	6	$([0.4,0.6],[0.2,0.3])$
u 6	NULL	NULL	$[0.6,0.2]$	Fair	$[0.4,0.1]$

Table 6.

Hybrid-valued user-service rating matrix $\mathrm{tb}4$ for criterionk₄.

	s 1	s 2	s 3	s 4	s 5
u 1	5	Bad	10	$[0.4,0.4]$	NULL
u 2	Good	$[0.4,0.3]$	Good	6	$([0.6,0.8],[0.1,0.2])$
u 3	$\text{Fair}$	$[0.5,0.2]$	$[0.8,0.1]$	NULL	$([0.5,0.7],[0.1,0.3])$
u 4	$([0.5,0.7],[0.1,0.2])$	NULL	NULL	$([0.6,0.7],[0.1,0.2])$	7
u 5	$[0.6,0.1]$	5	Fair	$([0.4,0.6],[0.2,0.3])$	Fairly good
u 6	NULL	$([0.4,0.6],[0.2,0.3])$	7	Fair	$[0.8,0.1]$

Based on the detailed analysis provided in the The proposed method section, we next analyze the problem to evaluate and select the optimal bearing supplier(s):

Step 1. Utilize the three proposed approaches to transform the hybrid-valued rating matrices (including the crisp and linguistic values, and IFNs) to IVIF-rating matrices. For conciseness, following Step 3, the results are shown together with the prediction of missing QoS values in Tables 8–11.

Step 2. Apply the IIFWA aggregation operator to calculate the average assessments of services and similarity degrees of target and other services:

Table 7.

The similarities between s₁ and s_i(i = 2, 3, 4, 5) in matrix tb₁.

	s2	s3	s4	s5
s1	([0.90,0.98],[0.007,0.0103])	([0.88,0.98],[0.0002,0.0060])	([0.89,0.98],[0.0016,0.0145])	([0.82,0.97],[0.0007,0.0108])

Table 8

IVIFNs in user-service rating matrix tf₁ for criterion k₁.

	s 1	s 2	s 3	s 4	s 5
u 1	([0.5,0.6],[0.2,0.3])	([0.3,0.4],[0.4,0.6])	([0.45,0.55],[0.41,0.55])	([0.8,0.9],[0.05,0.1])	([0.47,0.68],[0.12,0.27])
u 2	([0.875,0.975],[0.005,0.025])	([0.65,0.82],[0.05,0.15])	([0.7,0.8],[0.1,0.2])	([0.65,0.75],[0.15,0.25])	([0.3,0.4],[0.45,0.55])
u 3	([0.55,0.65],[0.15,0.25])	([0.425,0.525],[0.455,0.475])	([0.55,0.65],[0.31,0.35])	([0.71,0.86],[0.05,0.12])	([0.15,0.25],[0.55,0.65])
u 4	([0.5,0.7],[0.1,0.2])	([0.58,0.76],[0.09,0.21])	([0.4,0.6],[0.2,0.3])	([0.70,0.84],[0.06,0.14])	([0.335,0.435],[0.545,0.565])
u 5	([0.75,0.85],[0.11,0.15])	([0.2,0.4],[0.5,0.6])	([0.605,0.705],[0.275,0.295])	([0.35,0.45],[0.51,0.55])	([0.25,0.35],[0.35,0.45])
u 6	([0.82,0.94],[0.01,0.04])	([0.75,0.85],[0.05,0.15])	([0.8,0.9],[0.05,0.10])	([0.515,0.615],[0.365,0.385])	([0.35,0.45],[0.51,0.55])

Table 9.

IVIFNs in user-service rating matrix tf₂ for criterion k₂.

	s 1	s 2	s 3	s 4	s 5
u 1	([0.7,0.8],[0.1,0.2])	([0.71,0.87],[0.02,0.09])	([0.6,0.8],[0.1,0.2])	([0.6,0.8],[0.1,0.2])	([0.54,0.77],[0.06,0.16])
u 2	([0.35,0.45],[0.25,0.35])	([0.6,0.7],[0.1,0.2])	([0.4,0.6],[0.3,0.4])	([0.605,0.705],[0.275,0.295])	([0.65,0.75],[0.21,0.25])
u 3	([0.875,0.975],[0.005,0.025])	([0.45,0.55],[0.15,0.25])	([0.45,0.55],[0.41,0.55])	([0.58,0.78],[0.08,0.18])	([0.245,0.345],[0.635,0.655])
u 4	([0.55,0.65],[0.31,0.35])	([0.70,0.85],[0.02,0.08])	([0.7,0.8],[0.1,0.2])	([0.1,0.2],[0.6,0.8])	([0.35,0.45],[0.15,0.25])
u 5	([0.75,0.85],[0.11,0.15])	([0.5,0.6],[0.1,0.3])	([0.65,0.75],[0.05,0.15])	([0.35,0.45],[0.51,0.55])	([0.54,0.76],[0.05,0.15])
u 6	([0.78,0.92],[0.01,0.05])	([0.79,0.89],[0.10,0.12])	([0.3,0.4],[0.4,0.6])	([0.55,0.65],[0.25,0.35])	([0.2,0.4],[0.3,0.4])

Table 10.

IVIFNs in user-service rating matrix tf₃ for criterion k₃.

	s 1	s 2	s 3	s 4	s 5
u 1	([0.45,0.55],[0.41,0.55])	([0.605,0.705],[0.275,0.295])	([0.51,0.73],[0.06,0.16])	([0.5,0.7],[0.2,0.3])	([0.74,0.92],[0.01,0.06])
u 2	([0.5,0.6],[0.2,0.3])	([0.55,0.65],[0.31,0.35])	([0.35,0.45],[0.05,0.15])	([0.4,0.05],[0.2,0.4])	([0.75,0.85],[0.11,0.15])
u 3	([0.3,0.5],[0.3,0.4])	([0.6,0.8],[0.1,0.2])	([0.245,0.345],[0.635,0.655])	([0.59,0.81],[0.04,0.14])	([0.5,0.8],[0.1,0.2])
u 4	([0.25,0.35],[0.35,0.45])	([0.70,0.87],[0.03,0.09])	([0.35,0.45],[0.51,0.55])	([0.5,0.7],[0.1,0.3])	([0.875,0.975],[0.005,0.025])
u 5	([0.335,0.435],[0.545,0.565])	([0.65,0.75],[0.15,0.25])	([0.2,0.4],[0.4,0.5])	([0.515,0.615],[0.365,0.385])	([0.4,0.6],[0.2,0.3])
u 6	([0.49,0.70],[0.07,0.17])	([0.68,0.84],[0.04,0.11])	([0.55,0.65],[0.15,0.25])	([0.45,0.55],[0.41,0.45])	([0.35,0.45],[0.05,0.15])

Table 11.

IVIFNs in user-service rating matrix tf₄ for criterion k₄.

	s 1	s 2	s 3	s 4	s 5
u 1	([0.425,0.525],[0.455,0.475])	([0.35,0.45],[0.51,0.55])	([0.875,0.975],[0.005,0.025])	([0.35,0.45],[0.35,0.45])	([0.73,0.88],[0.03,0.09])
u 2	([0.55,0.65],[0.31,0.35])	([0.35,0.45],[0.25,0.35])	([0.55,0.65],[0.31,0.35])	([0.515,0.615],[0.365,0.385])	([0.6,0.8],[0.1,0.2])
u 3	([0.45,0.55],[0.41,0.45])	([0.45,0.5],[0.15,0.25])	([0.75,0.85],[0.05,0.15])	([0.61,0.80],[0.04,0.12])	([0.5,0.7],[0.1,0.3])
u 4	([0.5,0.7],[0.1,0.2])	([0.53,0.74],[0.06,0.15])	([0.77,0.91],[0.02,0.07])	([0.6,0.7],[0.1,0.2])	([0.605,0.705],[0.275,0.295])
u 5	([0.55,0.65],[0.05,0.15])	([0.425,0.525],[0.455,0.475])	([0.45,0.55],[0.41,0.45])	([0.4,0.6],[0.2,0.3])	([0.65,0.75],[0.21,0.25])
u 6	([0.64,0.83],[0.03,0.11])	([0.4,0.6],[0.2,0.3])	([0.605,0.705],[0.275,0.295])	([0.45,0.55],[0.41,0.45])	([0.75,0.85],[0.05,0.15])

We show partial calculation results, that is, the similarities between s₁ and $si(i=2,3,4,5)$ in matrix $\mathrm{tb}1$ , as shown in Table 7.

We then utilize the score and accuracy functions to compare the order relation of the similarities

$\text{S}(s12)=0.9361,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{S}(s13)=0.9302,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{S}(s14)=0.9241,\text{S}(s15)=0.8893.$

Therefore, the order relation is

$\text{S}(s12)>S(s13)>S(s14)>S(s15).$

Service $s2$ is the most similar to $\text{s}1$ .

Step 3. Use Definition 3.4 to select Top-N nearest neighbor services from all manufacturing services. In addition, use equation (22) to predict the overall missing QoS values. Tables 8–11 reveal the results of transformation and prediction.

Step 4. Utilize the IIFWA operator, whose weight is set as $\omega =0.4,0.3,0.2,0.1$ , to integrate all QoS IVIF rating matrices into a comprehensive IVIF rating matrix $X$ , which is given in Table 12.

Step 5. Utilize the IIFWA aggregation operator to calculate comprehensive evaluation values for all services into a one-dimensional matrix $Z(1\times N)$ from matrix $\text{X}$ . This is shown as follows (Table 13).

Step 6. Use the score and accuracy functions of IVIF theory to rank the services and select the best one(s). The scores of $zi(i=1,2,3,4,5)$ are computed as

$\text{S}(z1)=0.5986,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{S}(z2)=0.4805,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{S}(z3)=0.4153,\text{S}(z4)=0.4145,\text{S}(z5)=0.3422.$

Therefore, the comprehensive ranking of the services is

$s1>s2>s3>s4>s5.$

In addition, we analyze the results that come from another three dimensionalities as follows

Table 12.

IVIFNs in comprehensive user-service rating matrix X.

	s 1	s 2	s 3	s 4	s 5
u 1	([0.55,0.65],[0.20,0.30])	([0.53,0.67],[0.16,0.29])	([0.58,0.76],[0.12,0.22])	([0.67,0.82],[0.10,0.18])	([0.59,0.80],[0.06,0.15])
u 2	([0.69,0.86],[0.05,0.12])	([0.59,0.73],[0.10,0.21])	([0.55,0.68],[0.14,0.25])	([0.58,0.68],[0.21,0.30])	([0.56,0.69],[0.23,0.30])
u 3	([0.66,0.83],[0.07,0.15])	([0.47,0.61],[0.22,0.31])	([0.50,0.61],[0.32,0.39])	([0.64,0.82],[0.05,0.14])	([0.30,0.50],[0.34,0.48])
u 4	([0.47,0.63],[0.18,0.28])	([0.64,0.82],[0.04,0.13])	([0.55,0.70],[0.16,0.26])	([0.52,0.69],[0.14,0.28])	([0.55,0.72],[0.14,0.22])
u 5	([0.68,0.79],[0.14,0.20])	([0.43,0.56],[0.24,0.40])	([0.55,0.66],[0.19,0.28])	([0.39,0.50],[0.43,0.48])	([0.43,0.60],[0.17,0.28])
u 6	([0.75,0.90],[0.02,0.07])	([0.73,0.85],[0.07,0.14])	([0.63,0.76],[0.14,0.23])	([0.51,0.61],[0.34,0.39])	([0.37,0.50],[0.22,0.34])

Table 13.

Final comprehensive evaluation matrix Z.

	s 1	s 2	s 3	s 4	s 5
z	([0.64,0.80], [0.08,0.16])	([0.58,0.73], [0.12,0.23])	([0.56,0.70], [0.17,0.26])	([0.56,0.71], [0.17,0.27])	([0.48,0.65], [0.17,0.28])

For s₁, the membership, non-membership, and hesitation degrees are given as

$m(s1)=[0.64,0.80],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{nm}(s1)=[0.08,0.16],\pi (s1)=[0.04,0.28].$

We can use the score and accuracy functions of IFNs and their order relations ³⁴ to measure the three degrees of $\text{s}1$ as

$S\mathrm{IFN}(\text{m}(s1))=0.16,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}S\mathrm{IFN}(\text{nm}(s1))=0.08,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}S\mathrm{IFN}(\pi (s1))=0.23.$

Then, for all services

$\text{m}(\text{s}5)>m(\text{s}1)>m(\text{s}2)>m(\text{s}4)>m(\text{s}3)\text{nm}(\text{s}5)>\mathrm{nm}(\text{s}2)>\mathrm{nm}(\text{s}4)>\mathrm{nm}(\text{s}3)>\mathrm{nm}(\text{s}1)\pi (\text{s}5)>\pi (\text{s}2)>\pi (\text{s}4)>\pi (\text{s}3)=\pi (\text{s}1).$

We can determine that the order relations of the three degrees are different from the comprehensive ranking of the services. This means the decision-making enterprise can select the optimal bearing service(s) from different perspectives.

Evaluation results

In order to validate the effectiveness of the method, we compare the proposed and previous methods in this section. In previous studies, researchers simply utilize the IIFWA aggregation operator to predict missing values. We use the IIFWA as the benchmark method to compare and the predicted results (using the example of predicting missing values from Table 6) of the two methods are shown in Table 14. At the end of the process of missing value prediction, we display the ranking results for the two methods in Table 15.

Table 14.

Comparison of the prediction of missing values.

	missing value1	missing value2	missing value3	missing value4	missing value5
IIFWA	([0.50,0.62], [0.20,0.30])	([0.40,0.52], [0.28,0.37])	([0.69,0.82], [0.10,0.18])	([0.47,0.59], [0.25,0.34])	([0.63,0.77], [0.12,0.23])
QoS-CF	([0.64,0.83], [0.03,0.11])	([0.53,0.74], [0.06,0.15])	([0.77,0.91], [0.02,0.07])	([0.61,0.80], [0.04,0.12])	([0.73,0.88], [0.03,0.09])

Table 15.

Comparison of the final evaluation.

	comprehensive ranking	membership degree	non-membership degree	hesitation degree
IIFWA	s1 > s3 > s2 > s4 > s5	s5 > s1 > s3 > s4 > s2	s2 > s5 > s3 > s4 > s1	s5 > s2 > s3 > s1 > s4
QoS-CF	s1 > s2 > s3 > s4 > s5	s5 > s1 > s2 > s4 > s3	s5 > s2 > s4 > s3 > s1	s5 > s2 > s4 > s3 > s1

In Tables 14 and 15, we can clearly see that the results for the benchmark method are significantly different from those for the method we applied in this study. This is because, in the process of prediction, the IIFWA aggregation operator only considers longitudinal information (that is, the other users of the same service using the same criterion). The method (QoS-CF) used in this study develops this process further by considering the transverse information (i.e. the current user evaluation for other services). Thus, using only the IIFWA aggregation operator to predict the missing value, which causes a considerable loss of information, is not comprehensive and effective.

Conclusion

We presented a method for multi-criteria evaluation and decision making for manufacturing services by combining CF and IVIF theories.

The main contributions of our proposed method are: (1) we utilize the most sufficient amount of and reasonable evaluation information to predict missing QoS values and evaluate the alternative services. In addition, we employ the QoS-aware item-based CF and IVIF theories to resolve the problem of data sparsity; (2) we extend the explanation of evaluation results to multiple dimensions to select the optimal service from additional perspectives.

However, some limitations remain that must be overcome in our future research. Specifically, in our proposed method, the weights of users and evaluation criteria are manually set, which is a subjective process. In a future study, we will use more reasonable and objective methods to address this and other issues.