1. Introduction
The unprecedented growth of the Internet and social media platforms has led to the emergence of specialized websites, such as those dedicated to books and music, as well as e-commerce sites. As a result, user reviews have burgeoned as a primary conduit for information dissemination. A significant portion of the public now gravitates towards online platforms to voice their opinions, with experiential products like movies particularly benefiting from this trend [
1]. Websites like Douban and Rotten Tomatoes have evolved into primary sources for audiences for movie details and reviews. Concurrently, there has been a noticeable uptick in the quality, expertise, and social interactivity of these reviews. The emotional slant of these reviews wields considerable influence over potential consumers’ decisions [
2]. Factors such as online reviews, ratings, and extensive feedback play a pivotal role in influencing consumer decisions.
However, the deluge of reviews brings its own challenges, notably the phenomenon of information overload. Current recommendation systems falter in tailoring suggestions to individual genre preferences, highlighting the pressing need to deftly extract and harness the emotional nuances from multitudes of online reviews [
3]. This extraction process aims to guide consumers towards more informed decisions.
While some review platforms showcase aggregate product ratings to circumvent the necessity of sifting through individual reviews, the text within reviews remains invaluable for potential consumers [
4]. Therefore, efficiently pinpointing crucial information within these reviews is essential for refining consumer decision making. Notably, given that movies are experiential products, there is a surprising dearth of research on movie sorting. This paper seeks to bridge this gap and validate the proposed method by using movies as an example.
Historically, ranking methodologies grounded in online reviews have predominantly zeroed in on positive and negative sentiments, overlooking the nuances of neutral emotions [
5,
6,
7,
8,
9,
10]. Such oversights can lead to the omission of crucial information. Recent academic endeavors have harnessed multi-attribute decision making (MADM) methodologies for online product categorization. For instance, Fan et al. [
11] derived a comparative superiority degree for various alternatives using the distribution percentages of specific features across distinct commodities, employing the PROMETHEE II (Preference Ranking Organization Method for Enrichment of Evaluations) method for comprehensive evaluation. In another study, Fan et al. [
12] leveraged user ratings from online reviews to formulate two utility functions, with the subsequent application of the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method for holistic evaluation and ranking. Other notable works include Lee et al.’s [
13] utilization of hierarchical deep neural networks (DNNs) for product ranking, and Wang et al.’s [
14] user-centric commodity recommendation model. As a result of our research, we found that many extant studies exhibit a propensity to view products monolithically, often sidelining detailed attributes and features. Some product ranking methods based on online reviews only take into account the positive and negative affective tendencies of online reviews, ignoring the fact that the affective tendencies in the reviews can be neutral, i.e., ambiguous information, which can lead to a loss of information in the decision making process. Established models like TOPSIS and PROMETHEE II, premised on the notion of the decision maker’s absolute rationality, disregard the psychological intricacies underpinning the decision making process [
11,
12]. The TODIM method is appropriate for illustrating the psychological behavior of consumers during the product prioritization process [
15,
16]. Its central concept involves determining gain and loss values by comparing the characteristic values of each alternative product, and then calculating the dominance degree between every pair of alternatives and the overall prospect value of each product [
17,
18]. The alternative products are ranked based on their overall prospect value.
To address these lacunae, this paper embarks on an exploration of product ranking, leveraging online review data within an intuitionistic fuzzy framework. We introduce an intuitionistic fuzzy TODIM methodology, predicated on the multifaceted aspects of product reviews, particularly emphasizing the quantification of emotional tendencies across varying product attributes. This method encompasses the following:
- (1)
The creation of a quantitative model for product attribute sentiment within an intuitionistic fuzzy paradigm. Recognizing the diverse preferences among consumers, we harness the Cemotion library, a sentiment analysis tool rooted in Bert, to discern nuanced emotional cues from reviews. Subsequently, we introduce the emotional intuition fuzzy value (E-IFV), which intuitively demonstrates the level of support for an attribute by integrating multidimensional eigenvalues of a product attribute with its emotional tendency.
- (2)
An enhancement of the current intuitionistic fuzzy score function is presented. We analyze existing research to discern gaps in the current model. An augmented intuitionistic fuzzy score function, coupled with an exact function, is proposed, aiming to streamline decision making. This enhancement amalgamates concepts from hesitation allocation and voting models.
- (3)
The integration of the TODIM approach to classify alternative products. The correlation among attributes is acknowledged, with the DEMATEL (Decision Making Trial and Evaluation Laboratory) method determining attribute weights for distinct product genres. These weights, in tandem with the loss aversion coefficients within the TODIM model, ascertain the relative prominence of alternative products. The final result is a personalized ranking, derived from consumer preferences, and a loss aversion risk factor.
By synthesizing information from online comments, likes, and comment volumes into intuitive fuzzy values, and aligning products with consumer attribute preferences, we aim to significantly augment the consumers’ decision making efficiency. In essence, our refined intuitionistic fuzzy TODIM product ranking method, rooted in multidimensional product review features, integrates diverse data sources, providing consumers with a tailored decision making guide.
2. Materials and Methods
This article introduces a novel methodology for product ranking by integrating insights from online reviews with inherent product attributes. In the intuitionistic fuzzy environment, the emotional tendency of product attributes is transformed into an emotional intuitionistic fuzzy value (E-IFV); in order to effectively compare the magnitude of intuitionistic fuzzy numbers, an intuitionistic fuzzy score function and an intuitionistic fuzzy exact function are proposed. Finally, a product ranking model based on online reviews is established by integrating the TODIM method. As this paper is based on the study of product ranking in an intuitionistic fuzzy environment, the intuitionistic fuzzy score function and the exact function are proposed, and the advantages of the intuitionistic fuzzy value (IFV) and intuitionistic fuzzy set (IFS) in terms of representing the affective tendencies of product features are considered. This section presents the literature related to intuitionistic fuzzy sets.
Zadeh [
19], in 1965, introduced the fuzzy set (FS) theory. However, as fuzzy multi-attribute decision making paradigms evolved, it became evident that fuzzy sets were insufficient in capturing decision makers’ uncertainty comprehensively [
20]. Addressing this, Atanassov [
21] unveiled the intuitionistic fuzzy sets theory. This enhanced approach emphasized both membership and non-membership degrees, providing a richer representation of a decision maker’s hesitations. Later on, Liu et al. [
22,
23] built upon this theory by incorporating sentiment analysis, allowing online product reviews to be represented by intuitionistic fuzzy numbers. Furthermore, Roszkowska et al. [
24,
25,
26] introduced a composite measure. This measure was tailored to the evaluation of complex social phenomena using questionnaires, refining the fuzzy set based on “objective” data. Specifically, research [
24] advocated for the use of interval intuitionistic fuzzy sets (I-VIFS) to articulate the data from questionnaires. They crafted the I-VIFS composite measure and utilized the outcomes to stipulate the optimistic coefficients, thereby defining the bounds of the interval for the I-VIFS parameters. Meanwhile, other reserach [
25] employed the intuitionistic fuzzy synthesis measure (IFSM) based on pattern object distance. They suggested translating ordered data using intuitionistic fuzzy sets and juxtaposed the results with traditional methods. Çalı and Balaman [
27] used IFS to represent online ratings of hotel customers and used IF-ELECTRE to rank alternative hotels integrated with VIKOR. In this paper, we draw inspiration from the literature [
24] and aim to augment the precision of product attributes tied to intuitionistic fuzzy values by formulating multidimensional eigenvalues for these attributes.
Upon this proposition, our subsequent focus revolved around refining the intuitionistic fuzzy multi-attribute decision theory to bolster decision making efficacy. Of paramount importance here is the intuitionistic fuzzy score function, pivotal for comparing and ranking intuitionistic fuzzy numbers. Despite significant contributions from scholars like Chen and HONG [
28,
29], certain aspects of these functions, such as the hesitancy degree’s impact on scheme ranking, remain under-explored.
While significant strides have been made in areas like text sentiment analysis and multi-attribute decision making using online reviews, specific challenges persist:
- (1)
The existing research often overlooks neutral and ambiguous sentiments in reviews.
- (2)
Most research presumes attribute independence, neglecting potential inter-attribute correlations.
- (3)
Models like TOPSIS tend to assume complete decision maker rationality, sidelining psychological influences.
- (4)
Research surrounding intuitive fuzzy score functions requires bolstering to achieve improved accuracy in intuitive fuzzy number discrimination.
Addressing these challenges, this paper presents a product-ranking model anchored in consumer preferences. By amalgamating sentiment analysis with intuitionistic fuzzy sets and integrating the TODIM multi-attribute decision making method, we aim to elevate decision making efficiency for consumers, as exemplified using data from the Douban (
https://movie.douban.com (accessed on 13 May 2023)) platform.
3. Problem Description
Prior to finalizing a product selection, consumers frequently consult various online indicators including reviews, ratings, aggregate review counts, and other pertinent metrics. A substantial body of research corroborates the utility of these scoring data in facilitating informed decisions [
30,
31,
32,
33].
The primary challenge addressed in this study is to formulate a product ranking system that serves as a robust decision making tool for consumers. This ranking integrates data from online reviews and the associated engagement metrics—such as “likes” and comment counts—and takes into consideration consumers’ preferences related to specific product attributes. The overarching goal is to improve the decision making efficiency for potential consumers. The subsequent sections detail the representations and precise definitions of the sets and variables relevant to this problem.
Let $A=\{{A}_{1},{A}_{2},\cdots ,{A}_{m}\}$ be the set of products that the consumer is interested in choosing, where ${A}_{i}$ denotes the ith product, $i=1,2,\cdots ,m$.
Let $F=\{{f}_{1},{f}_{2},\cdots ,{f}_{n}\}$ be the set of n attributes of the alternative product, where ${f}_{j}$ denotes the jth attribute, $j=1,2,\cdots ,n$.
Let $w=\left[{w}_{1},{w}_{2},\cdots ,{w}_{n}\right]$ be a vector of product attribute weights, where ${w}_{j}$ is the weight corresponding to attribute ${f}_{j}$. ${w}_{j}\ge 0$, $\sum _{j=1}^{n}{w}_{j}}=1$. The weights represent the differences in consumer preferences for product attributes.
Let $Q=\{{q}_{1},{q}_{2},\cdots ,{q}_{n}\}$ be the number of online reviews for the alternative product, where ${q}_{i}$ is the number of online reviews for product ${A}_{i}$, $i=1,2,\cdots ,n$.
The following collections will be described below:
Suppose ${z}_{p,{q}_{i}}^{j}$ denotes the number of “likes” for the qth comment under the jth attribute of product ${A}_{i}$. If the sentiment tendency of the qth comment is positive, then ${z}_{p,{q}_{i}}^{j}\in {z}_{pos}^{{f}_{j}}$, otherwise ${z}_{p,{q}_{i}}^{j}=0$;
Suppose ${z}_{n,{q}_{i}}^{j}$ denotes the number of “likes” for the qth comment under the jth attribute of product ${A}_{i}$. If the sentiment tendency of the qth comment is negative, then ${z}_{n,{q}_{i}}^{j}\in {z}_{neg}^{{f}_{j}}$, otherwise ${z}_{n,{q}_{i}}^{j}=0$;
Suppose ${t}_{p,{q}_{i}}^{j}$ denotes the number of words of the qth comment under the jth attribute of product ${A}_{i}$. If the sentiment tendency of the qth comment is positive, then ${t}_{p,{q}_{i}}^{j}\in {t}_{pos}^{{f}_{j}}$, otherwise ${t}_{p,{q}_{i}}^{j}=0$;
Suppose ${t}_{n,{q}_{i}}^{j}$ denotes the number of words of the qth comment under the jth attribute of product ${A}_{i}$. If the sentiment tendency of the qth comment is negative, then ${t}_{n,{q}_{i}}^{j}\in {t}_{neg}^{{f}_{j}}$, otherwise ${t}_{n,{q}_{i}}^{j}=0$;
Let ${z}_{pos}^{{f}_{j}}=\{{z}_{p,1}^{j},{z}_{p,2}^{j},\cdots ,{z}_{p,{q}_{i}}^{j}\}$ be the set consisting of the number of “likes” by consumers for the comments of product ${A}_{i}$ on attribute ${f}_{j}$ with a positive emotional tendency, where ${z}_{p,k}^{j}$ denotes the data of likes for the kth comment on product ${A}_{i}$ on attribute ${f}_{j}$, $k=1,2,\cdots ,{q}_{i}$, $i=1,2,\cdots ,m$.
Let ${z}_{neg}^{{f}_{j}}=\{{z}_{n,1}^{j},{z}_{n,2}^{j},\cdots ,{z}_{n,{q}_{i}}^{j}\}$ be the set consisting of the number of “likes” by consumers for the comments of product ${A}_{i}$ on attribute ${f}_{j}$ with a negative emotional tendency, where ${z}_{n,k}^{j}$ denotes the data of likes for the kth comment on product ${A}_{i}$ on attribute ${f}_{j}$, $k=1,2,\cdots ,{q}_{i}$, $i=1,2,\cdots ,m$.
Let ${t}_{pos}^{{f}_{j}}=\{{t}_{p,1}^{j},{t}_{p,2}^{j},\cdots ,{t}_{p,{q}_{i}}^{j}\}$ be the set consisting of the word counts of the consumer comments on the product ${A}_{i}$ on attribute ${f}_{j}$ with a positive affective tendency, where ${t}_{p,k}^{j}$ denotes the word count of the kth comment on the product ${A}_{i}$ on attribute ${f}_{j}$,$k=1,2,\cdots ,{q}_{i}$, $i=1,2,\cdots ,m$.
Let ${t}_{neg}^{{f}_{j}}=\{{t}_{n,1}^{j},{t}_{n,2}^{j},\cdots ,{t}_{n,{q}_{i}}^{j}\}$ be the set consisting of the word counts of the consumer’s comments on the product ${A}_{i}$ on attribute ${f}_{j}$ with a negative affective tendency, where ${t}_{n,k}^{j}$ denotes the word count of the kth comment on the product ${A}_{i}$ on attribute ${f}_{j}$, $k=1,2,\cdots ,{q}_{i}$, $i=1,2,\cdots ,m$.
The process of problem solving in this research is divided into two primary segments:
Firstly, this investigation introduces a model tailored to the quantification of emotions within the realm of intuitionistic fuzzy contexts. The preliminary step involves the transformation of online product reviews into intuitionistic fuzzy numbers. Subsequently, using these comments, we propose an improved intuitionistic method for fuzzy and exact functions. This leads to the derivation of intuitionistic fuzzy exact functions anchored to specific product attributes.
Secondly, the ranking of alternative products is executed via the enhanced intuitionistic fuzzy TODIM methodology, emphasizing attribute associations. Acknowledging the interdependencies among attributes, the DEMATEL approach is deployed to ascertain attribute weights. The dominance hierarchy amongst products is determined utilizing the intuitionistic fuzzy TODIM methodology. This hierarchy, when integrated with the risk tolerance parameters of the consumer, culminates in a bespoke product ranking schema.
The flowchart for solving the problem is shown in
Figure 1.
4. A Quantitative Model of Product Attribute Sentiment in an Intuitively Ambiguous Environment
In order to solve the above problems, this section proposes a quantitative model of emotion based on product attributes in an intuitionistic fuzzy environment. The methodology comprises three parts:
Identification of emotional tendencies in online product reviews;
Multidimensional eigenvalue computation based on product attributes;
Calculation of emotional intuition fuzzy values based on product attributes.
4.1. Identification of Emotional Tendencies in Online Product Reviews
This study uses Cemotion, a Chinese sentiment tendency analysis library based on Bert, to identify positive, neutral, and negative sentiment tendencies regarding alternative product attributes in online product reviews. Cemotion’s model is trained by a recurrent neural network, which returns a confidence level between 0 and 1 for the sentiment tendency of Chinese text and can accurately identify the sentiment tendency of online product reviews by linking them to the context of the online reviews.
Using the movie
Titanic as an example,
Table 1 presents some of the identified movie review data. In
Figure 2, (a) illustrates the distribution of various attribute comments among online reviews of the movie
Titanic, while (b) shows the proportion of the emotional tendency of the “Frame” attribute in those reviews.
4.2. Multidimensional Eigenvalue Computation Based on Product Attributes
Consumers usually refer to existing online reviews, ratings, the number of likes, the number of comments, and other information to make a comparative choice of products before making a decision [
34,
35]. Therefore, in this paper, multidimensional eigenvalues of products are introduced to improve the reasonableness of the ranking [
36,
37]. Since the credibility of a comment can be determined by the number of likes it receives [
31], the length of the text [
36,
38,
39], and the emotion it conveys, this study used the number of likes and the number of words in the text as indicators to evaluate the satisfaction of consumers with the product characteristics. The indicators will be described in more detail in the following sections.
Suppose there are m alternative products A_{i} and n decision attributes F = {f_{1}, f_{2}, ⋯, f_{n}}.
Definition 1. “Positive Likes” is the ratio of the number of likes in the comments with positive emotional tendencies to the total number of likes in the comments of the corresponding attribute ${f}_{j}$ of the alternative product ${A}_{i}$.
where
${Z}_{pos}^{{f}_{j}}$ is the positive liking rate of the corresponding attribute
${f}_{j}$ of product
${A}_{i}$, and
${z}_{{f}_{j}}$ is the total number of likes of the alternative product
${A}_{i}$ on attribute
${f}_{j}$.
Definition 2. “Negative Likes” is the ratio of the number of likes in the comments with negative emotional tendencies to the total number of likes in the comments of the corresponding attribute ${f}_{j}$ of the alternative product ${A}_{i}$.
where
${Z}_{neg}^{{f}_{j}}$ is the negative liking rate of the corresponding attribute
${f}_{j}$ of product
${A}_{i}$, and
${z}_{{f}_{j}}$ is the total number of likes of the alternative product
${A}_{i}$ on attribute
${f}_{j}$.
Definition 3. “The Positive Text Rate” is the ratio of the number of words of text in the comments with positive emotional tendencies to the total number of words of text in the comments corresponding to attribute ${f}_{j}$ of alternative product ${A}_{i}$.
where
${T}_{pos}^{{f}_{j}}$ is the positive text rate of the corresponding attribute
${f}_{j}$ of product
${A}_{i}$, and
${t}_{{f}_{j}}$ is the total number of words for the alternative product
${A}_{i}$ on attribute
${f}_{j}$.
Definition 4. “The Negative Text Rate” is the ratio of the number of words of text in the comments with negative emotional tendencies to the total number of words of text in the comments corresponding to attribute ${f}_{j}$ of alternative product ${A}_{i}$.
where
${T}_{neg}^{{f}_{j}}$ is the negative text rate of the corresponding attribute
${f}_{j}$ of product
${A}_{i}$, and
${t}_{{f}_{j}}$ is the total number of words for the alternative product
${A}_{i}$ on attribute
${f}_{j}$.
From Equations (1)–(4), we can obtain the text rate and like rate of the product ${A}_{i}$ corresponding to the attribute ${f}_{j}$. The multidimensional eigenvalue for the product attribute ${f}_{j}$ is obtained through further calculation:
Definition 5. “The Positive Multidimensional Eigenvalue”
${D}_{pos}^{{f}_{j}}$ is the average of “The Positive Liking Rate” ${Z}_{pos}^{{f}_{j}}$ and “The Positive Text Rate” ${T}_{pos}^{{f}_{j}}$ of the attribute ${f}_{j}$ corresponding to product ${A}_{i}$.
Definition 6. “The Negative Multidimensional Eigenvalue”
${D}_{neg}^{{f}_{j}}$ is the average of “The Negative Liking Rate” ${Z}_{neg}^{{f}_{j}}$ and “The Negative Text Rate” ${T}_{neg}^{{f}_{j}}$ of the attribute ${f}_{j}$ corresponding to product ${A}_{i}$.
4.3. Calculation of Sentiment Means for Product Attributes
The identification of affective tendencies for the products’ online product reviews through
Section 4.1 yields positive, neutral, and negative affective tendencies regarding the attributes of the alternative products. The identified affective tendencies are further explained below.
Let ${\alpha}_{\mathrm{ik}}^{\mathrm{j}}$, ${\beta}_{\mathrm{ik}}^{\mathrm{j}}$, and ${\nu}_{\mathrm{ik}}^{\mathrm{j}}$ be the positive, negative, and neutral sentiment strengths of the kth comment by the consumer on attribute ${f}_{j}$ of the alternative product ${A}_{i}$, respectively. Since this study uses Cemotion, a Bert-based Chinese affective tendency analysis library, for identification, Cemotion returns an affective tendency confidence level between 0 and 1 for the Chinese text, so ${\alpha}_{\mathrm{ik}}^{\mathrm{j}},{\beta}_{\mathrm{ik}}^{\mathrm{j}},{\nu}_{\mathrm{ik}}^{\mathrm{j}}\in \left[0,1\right]$, and ${\alpha}_{\mathrm{ik}}^{\mathrm{j}}+{\beta}_{\mathrm{ik}}^{\mathrm{j}}+{\nu}_{\mathrm{ik}}^{\mathrm{j}}\in \left[0,1\right]$, $k=1,2,\cdots ,{q}_{i}$, $i=1,2,\cdots ,m$.
In the following, the mean values of positive, negative, and neutral emotions corresponding to attribute
${f}_{j}$ of alternative product
${A}_{i}$ will be calculated as follows:
where
${q}_{{f}_{i}}$ is the total number of comments on attribute
${f}_{j}$ for alternative product
${A}_{i}$;
${p}_{ij}^{pos}$ is the mean value of positive sentiment about attribute ${f}_{j}$ for alternative product ${A}_{i}$;
${p}_{ij}^{neg}$ is the mean value of negative sentiment about attribute ${f}_{j}$ for alternative product ${A}_{i}$;
and ${p}_{ij}^{neu}$ is the mean value of neutral sentiment about attribute ${f}_{j}$ for alternative product ${A}_{i}$.
4.4. Calculation of Intuitional Fuzzy Values
In this section, leveraging the unique characteristics of product reviews and integrating the foundational principles of IFV, we propose the emotional intuitionistic fuzzy value (E-IFV) model for product attributes. The E-IFV serves as an intuitive representation of the emotional tendencies associated with attributes in product reviews.
The concept of the emotional intuitionistic fuzzy value introduced in this paper is rooted in the theory of intuitionistic fuzzy value, which will be elaborated upon in subsequent sections.
Definition 7. Suppose $X$ is an argument. If there are two mappings ${\mu}_{A}:X\to \left[0,1\right]$ and ${\nu}_{A}:X\to \left[0,1\right]$ above $X$ so that andsimultaneously satisfy condition${\mu}_{A}$
and
${\nu}_{A}$
are said to determine an intuitionistic fuzzy set on
$X$, which can be denoted as where
${\mu}_{A}\left(x\right)$
and
${\nu}_{A}\left(x\right)$ are referred to as the degree of affiliation and non-affiliation of x. The degree of hesitancy is defined as the following equation: For any $x\in X$, there is $0\le {\pi}_{A}\le 1$. To facilitate the understanding and application of intuitionistic fuzzy sets, Xu [
40] defines
$\alpha =\left(\mu ,\nu \right)$ as an intuitionistic fuzzy number, where
$\mu \ge 0$,
$\nu \ge 0$, and
$\mu +\nu \le 1$, and the degree of hesitation of intuitionistic fuzzy number
$\alpha $ is
${\pi}_{\alpha}=1-\mu -\nu $.
From the aforementioned definition, it is evident that the intuitionistic fuzzy set (IFS) can encapsulate affirmative, negative, and hesitant attitudes simultaneously, minimizing the loss of emotional information. This makes it a robust tool for representing ambiguous and uncertain data. Given the intricate and subjective nature of emotions in online product reviews, factors such as online reviews, ratings, and particularly the emotional sentiments within these reviews, significantly influence consumers’ inclination to select alternative products. To enhance the applicability of the intuitionistic fuzzy value, this paper introduces the emotional intuitionistic fuzzy value (E-IFV). This is achieved by integrating the emotional mean with the multidimensional eigenvalue of product attributes, building upon the foundational intuitionistic fuzzy value. The subsequent sections detail the calculation methodology for E-IFV.
${x}_{ij}=\left({\mu}_{ij},{\nu}_{ij}\right)$ denotes the E-IFV of alternative product ${A}_{i}$ on the attribute word ${f}_{j}$.
Where ${\mu}_{ij}$, ${\nu}_{ij}$, and ${\pi}_{ij}$ are the degree of affiliation, non-affiliation, and hesitation of the alternative product ${A}_{i}$ on attribute word ${f}_{j}$, i.e., the consumers’ support, opposition, and neutrality to attribute ${f}_{j}$ to alternative product ${A}_{i}$.
5. A Study on the Improved Intuitionistic Fuzzy TODIM Model Based on Attribute Association for Product Ranking
This paper proposes s an improved intuitionistic fuzzy TODIM model for the correlation of attributes to rank alternative products, taking into account that consumer attributes vary person-to-person and that correlations exist between them. The model comprises two primary segments:
An improved intuitionistic fuzzy score function is proposed, based on which an exact function is determined to improve the accuracy of comparing intuitionistic fuzzy numbers.
Since the attributes are correlated, instead of being independent of each other, the weights for different product types’ attributes are determined using the DEMATEL method. This study employs the TODIM method to determine the superiority of alternative products. The loss aversion coefficient $\theta $, which reflects the consumer’s risk appetite, is combined with the ranking to provide personalized decision making suggestions for the consumer.
5.1. The Available Intuitionistic Fuzzy Score Function
Establishing relative dominance (superiority and inferiority relationships) between intuitionistic fuzzy numbers is pivotal to intuitionistic fuzzy multi-attribute decision making. Existing research offers multiple methodologies for computing the score function of intuitionistic fuzzy values [
17,
18,
28,
29,
41,
42,
43,
44,
45,
46], such as
etc, where Equations (18)–(20) belong to the literature [
44,
45,
46] respectively.
These studies provide new ideas for the improvement of intuitionistic fuzzy functions, but at the same time, there are shortcomings.
Counterexample 1. Taking Equation (18) [44] as an example, suppose $\alpha =\left({\mu}_{\alpha},{\nu}_{\alpha}\right)$ is an intuitionistic fuzzy number, then let be the score value of $\alpha $ and $S\left(\alpha \right)$ be the score function of $\alpha $. For any two intuitionistic fuzzy numbers ${\alpha}_{1}$ and ${\alpha}_{2}$, there is then If $S\left({\alpha}_{1}\right)\uff1cS\left({\alpha}_{2}\right)$, ${\alpha}_{1}\prec {\alpha}_{2}$;
If
$S\left({\alpha}_{1}\right)\uff1eS\left({\alpha}_{2}\right)$,
${\alpha}_{1}\succ {\alpha}_{2}$;
If $S\left({\alpha}_{1}\right)=S\left({\alpha}_{2}\right)$, ${\alpha}_{1}\sim {\alpha}_{2}$.
Suppose two intuitionistic fuzzy numbers
${\alpha}_{1}=\left(0,0.2\right)$ and
${\alpha}_{2}=\left(0\uff0c0.3\right)$ exist, which can be obtained using Equation (18),
$S\left({\alpha}_{1}\right)=-0.7$,
$S\left({\alpha}_{2}\right)=-0.55$. Obviously,
$S\left({\alpha}_{1}\right)\uff1cS\left({\alpha}_{2}\right)$, i.e.,
${\alpha}_{1}\prec {\alpha}_{2}$. However, in the actual decision making process, people tend to choose a small degree of opposition to
${\alpha}_{1}$. At this time, Equation (18) cannot judge the size of the two intuitionistic fuzzy numbers. (For a comparison of the arithmetic examples of different intuitionistic fuzzy score function ranking methods, see
Table 2 in Summary 5.2).
From this, it can be found that the existing intuitionistic fuzzy score function has the following problems:
- (1)
The same result is obtained when calculating two different intuitionistic fuzzy numbers, and it is impossible to judge the size of two intuitionistic fuzzy numbers.
- (2)
Owing to the inherent constraints of the score function, the derived results occasionally contradict real-world decision making scenarios.
To enhance the comparative efficacy of intuitionistic fuzzy numbers and, in turn, refine the product ranking model, this study introduces a refined intuitionistic fuzzy score function and an exact function. Subsequently, we provide proof for the formula of this newly formulated intuitionistic fuzzy score function.
5.2. Improved Intuitionistic Fuzzy Score Function
In this paper, we introduce the concept of the improved intuitionistic fuzzy score function using a voting model as an illustrative example.
In real life, when faced with indecision, people often choose to wait and see what others do before making a final decision. Thus, for the intuitionistic fuzzy number
$\alpha =({\mu}_{\alpha},{\nu}_{\alpha})$, suppose that during the first vote,
$\mu $ represents the proportion of those who voted in favor,
$\nu $ represents the proportion of those who voted against, and
$\pi $ represents the proportion of those who voted neutrally; and during the second vote, since the hesitant part of the first vote is affected by the first vote, that during the second vote
${\pi}_{\mu}$ represents the part of the vote that voted in favor,
${\pi}_{\nu}$ is the part of the vote that voted against, and
${\pi}^{2}$ is the part of the vote that continued to be neutral. The cycle continues in rounds, calculating the sum of the proportional parts in favor. The intuitionistic fuzzy score function considered in this paper is the sum of the final proportions in favor:
The following equation is obtained by taking the limit of the formula:
Therefore, this paper proposes an improved intuitionistic fuzzy score function as shown in the following equation:
Theorem 1. Suppose $\alpha =({\mu}_{\alpha},{\nu}_{\alpha})$ is an intuitionistic fuzzy number, then is the score function of the intuitionistic fuzzy number $\alpha =({\mu}_{\alpha},{\nu}_{\alpha})$. In particular, when $\mu =\nu =0$, we define $S\left(\alpha \right)=0$. The equation above reveals that the intuitionistic fuzzy score function proposed in this paper has limitations, since the degree of affiliation of the intuitionistic fuzzy number cannot be 0. To address this issue, the paper introduces the intuitionistic fuzzy exact function. This paper presents a study on intuitionistic fuzzy exact functions as follows: for an intuitionistic fuzzy number, a larger degree of affiliation is considered better while a smaller degree of non-affiliation is preferred. If the hesitation degree is taken into account, the smaller the hesitation degree, the better. Building on the above concepts, this paper puts forward the subsequent equation:
where, to avoid the case of
$\pi =0$, the denominator of the formula is taken as
$\pi +1$.
Further simplifying the formula, the new intuitionistic fuzzy exact function given in this paper is defined as follows:
Theorem 2. Suppose $\alpha =(\mu ,\nu )$ is an intuitionistic fuzzy number, then is said to be an exact function of the intuitionistic fuzzy number $\alpha =(\mu ,\nu )$. After proposing the intuitionistic fuzzy score function and intuitionistic fuzzy exact function, this paper proposes the following new ranking method for intuitionistic fuzzy numbers:
Definition 8. Suppose that for any two intuitionistic fuzzy numbers ${\alpha}_{1}=({\mu}_{{\alpha}_{1}},{\nu}_{{\alpha}_{1}})$ and ${\alpha}_{2}=({\mu}_{{\alpha}_{2}},{\nu}_{{\alpha}_{2}})$, ${S}_{E}\left({\alpha}_{1}\right)$ and ${h}_{E}\left({\alpha}_{1}\right)$ are the values of the score function and the exact function for ${\alpha}_{1}$, and ${S}_{E}\left({\alpha}_{2}\right)$ and ${h}_{E}\left({\alpha}_{2}\right)$ are the values of the score function and the exact function for ${\alpha}_{2}$. Then
If
${S}_{E}\left({\alpha}_{1}\right)\uff1c{S}_{E}\left({\alpha}_{2}\right)$, then
${\alpha}_{1}\prec {\alpha}_{2}$;
If
${S}_{E}\left({\alpha}_{1}\right)\uff1e{S}_{E}\left({\alpha}_{2}\right)$, then
${\alpha}_{1}\succ {\alpha}_{2}$;
If
${S}_{E}\left({\alpha}_{1}\right)\uff1d{S}_{E}\left({\alpha}_{2}\right)$, then
when
${h}_{E}\left({\alpha}_{1}\right)\uff1c{h}_{E}\left({\alpha}_{2}\right)$,
${\alpha}_{1}\prec {\alpha}_{2}$,
when
${h}_{E}\left({\alpha}_{1}\right)\uff1e{h}_{E}\left({\alpha}_{2}\right)$,
${\alpha}_{1}\succ {\alpha}_{2}$,
when ${h}_{E}\left({\alpha}_{1}\right)\uff1d{h}_{E}\left({\alpha}_{2}\right)$, ${\alpha}_{1}\sim {\alpha}_{2}$.
The properties of the intuitionistic fuzzy score function proposed in this paper are described and formulas are proved in the following:
Property 1. The score function ${S}_{E}\left(\alpha \right)$ of an intuitionistic fuzzy number $\alpha =(\mu ,\nu )$ is monotonically increasing with respect to the degree of affiliation $\mu $ and monotonically decreasing with respect to the degree of non-affiliation $\nu $.
Proof of Property 1. Since
and
$0\le \mu \uff0b\nu \le 1$, so
$\frac{\nu}{{\left(\mu \uff0b\nu \right)}^{2}}\ge 0$, the score function of the intuitionistic fuzzy number
$\alpha =(\mu ,\nu )$ is monotonically increasing with respect to the degree of affiliation
$\mu $;
similarly
and
$0\le \mu \le 1$ and
$0\le \mu \uff0b\nu \le 1$, so that
$\frac{-\mu}{{\left(\mu \uff0b\nu \right)}^{2}}\le 0$, so the score function of the intuitionistic fuzzy number
$\alpha =(\mu ,\nu )$ is monotonically decreasing with respect to the unaffiliated degree
$\nu $, which is proved. □
Property 2. Intuitionistic fuzzy score function ${S}_{E}\left(\alpha \right)\in \left[0,1\right]$.
Proof of Property 2. Since Formula (24), and $0\le \mu \le \mu +\nu \le 1$, $0\le \frac{\mu}{\mu +\nu}\le 1$.
- (1)
when $\mu =0$, then ${S}_{E}\left(\alpha \right)=0$;
- (2)
when $\nu =0$, then ${S}_{E}\left(\alpha \right)=1$;
- (3)
when $\mu =\nu =0$, then ${S}_{E}\left(\alpha \right)=0$.
Proof is completed. □
Property 3. Suppose two intuitionistic fuzzy numbers ${\alpha}_{1}=({\mu}_{1},{\nu}_{1})$ and ${\alpha}_{2}=({\mu}_{2},{\nu}_{2})$. If ${\mu}_{1}\uff1e{\mu}_{2}$ and ${\nu}_{1}\uff1c{\nu}_{2}$, then ${S}_{E}\left({\alpha}_{1}\right)\uff1e{S}_{E}\left({\alpha}_{2}\right)$.
Proof of Property 3. Because
${S}_{E}\left({\alpha}_{1}\right)-{S}_{E}\left({\alpha}_{2}\right)=\frac{{\mu}_{1}}{{\mu}_{1}+{\nu}_{1}}-\frac{{\mu}_{2}}{{\mu}_{2}+{\nu}_{2}}=\frac{{\mu}_{1}\left({\mu}_{2}+{\nu}_{2}\right)-{\mu}_{2}\left({\mu}_{1}+{\nu}_{1}\right)}{\left({\mu}_{1}+{\nu}_{1}\right)\left({\mu}_{2}+{\nu}_{2}\right)}=\frac{{\mu}_{1}{\nu}_{2}-{\mu}_{2}{\nu}_{1}}{\left({\mu}_{1}+{\nu}_{1}\right)\left({\mu}_{2}+{\nu}_{2}\right)}$
and ${\mu}_{1}\uff1e{\mu}_{2}$ and ${\nu}_{1}\uff1c{\nu}_{2}$, then ${\mu}_{1}{\nu}_{2}-{\mu}_{2}{\nu}_{1}\uff1e0$, $\left({\mu}_{1}+{\nu}_{1}\right)\left({\mu}_{2}+{\nu}_{2}\right)\uff1e0$; then, $\frac{{\mu}_{1}{\nu}_{2}-{\mu}_{2}{\nu}_{1}}{\left({\mu}_{1}+{\nu}_{1}\right)\left({\mu}_{2}+{\nu}_{2}\right)}\uff1e0$, i.e., ${S}_{E}\left({\alpha}_{1}\right)-{S}_{E}\left({\alpha}_{2}\right)\uff1e0$.
It is clear that ${S}_{E}\left({\alpha}_{1}\right)\uff1e{S}_{E}\left({\alpha}_{2}\right)$. Proof is completed. □
The proposed intuitionistic fuzzy function and intuitionistic fuzzy exact function are compared and analyzed with the existing methods in the following, and the results of the comparative analysis are shown in
Table 2.
5.3. DEMATEL Determines Attribute Weights
In problems involving multi-attribute decision making, calculating indicator weights through traditional methods is often based on subjective or objective criteria to reflect the attributes’ characteristics, but this ignores the correlation between them. This paper utilizes the DEMATEL method to analyze the mutual influence relationship between attributes and determine their respective weights. The traditional multi-attribute decision making approach, which does not consider attribute weights in relation to each other, is addressed. Because attribute weights vary between different types of products, experts are invited to score the attributes of each product type. The average of these scores is then calculated to obtain the relative weights of the attributes for each particular product type.
Assuming that the alternative product to be evaluated
${A}_{i}$$(i=1,2,\cdots ,m)$ belongs to a certain category of products, the evaluation of the product attribute is
${f}_{j}$$(j=1,2,\cdots ,n)$ and that the category of the product is
t, and
r experts are invited to use the linguistic scale evaluation for evaluation and scoring on a five-level scale, then, the DEMATEL method is used to calculate the attribute weight matrix
${W}_{t,n}^{r}$ given by the
r experts under the different categories of the product, and the specific representation is as follows:
where
t denotes the number of product types,
n is the number of attributes of the product, and
r is the number of experts.
Finally, the weights obtained by
r experts are averaged to obtain the attribute weight matrix for different types of products:
where
r denotes the number of experts.
The alternative product average weight vector is calculated as follows:
where
$\overline{W}$ denotes the vector of average weights of alternative products and
t is the number of categories of products.
5.4. A Product Ranking Method Based on the Intuitionistic Fuzzy TODIM Model
Multi-attribute decision making refers to a process where a decision maker identifies the best solution among various alternatives based on selected attributes; the method ranks and selects solutions by calculating their perceived superiority relative to each other. When selecting a product, consumers often consider various attributes to make alternative choices. This paper presents an emotion quantification model for online reviews of products, constructed in an intuitionistic fuzzy environment. Product attributes are transformed into intuitionistic fuzzy values. The differences between attributes of different types of products and the correlation relationship between them are considered. The DEMATEL method is used to obtain the attribute weights of the correlations. Finally, identifying the degree of superiority of alternative products based on consumer preferences is achieved using the intuitionistic fuzzy TODIM model. A detailed description of the decision making steps is provided below.
Assume that
$A=\{{A}_{i}|i\in M\}$ is a limited number of alternative product scenarios and that
$f=\{{f}_{j}|j\in N\}$ is a finite set of attributes, where
$N=\left\{1,2,3,\cdots ,n\right\}$ and the weight of each attribute is
$w={({w}_{1},{w}_{2},\cdots ,{w}_{n})}^{T}$, where
$\sum _{i=1}^{n}{w}_{i}}=1$. The alternative product
${A}_{i}$ has an evaluation value of
${I}_{ij}$ under the product attribute
${f}_{j}$, where
${I}_{ij}$ expresses the intuitionistic fuzzy set. This results in a decision matrix of
$m$ decision scenarios under
n decision attributes.
the specific decision making steps are as follows:
normalize the original decision matrix $D={\left[{I}_{ij}\right]}_{m\times n}$ to obtain $X={\left[{x}_{ij}\right]}_{n\times n}$, where $M=\left\{1,2,3,\cdots ,m\right\}$, $N=\left\{1,2,3,\cdots ,n\right\}$, and $i\in M$, $j\in N$.
Determine the attribute with the largest value as the reference attribute
${f}_{j}$ and calculate the ratio of each attribute relative to the reference attribute
${w}_{jr}$. The formula is
where
${w}_{jr}=\mathrm{max}\left\{{w}_{j}|j\in N\right\}$.
The degree of dominance of Scenario
${A}_{i}$ over Scenario
${A}_{k}$ when the attribute is
${f}_{j}$ is calculated. The formula is
Loss aversion coefficient $\theta $ can reflect the psychological behavior of decision makers; the smaller the value of $\theta $, the higher the risk tolerance of decision makers, and the larger the value of $\theta $, the lower the risk tolerance of decision makers. In the subsequent experiments, this paper will analyze the value of $\theta $ to confirm whether the size of $\theta $ will have an impact on the final ranking results.
Calculate the degree of dominance of Scenario
${A}_{i}$ over Scenario
${A}_{k}$ for all attributes; the formula is
Calculate the combined degree of dominance of all alternatives
${A}_{i}$ over the other alternatives. The formula is
According to the above equation, the $\xi \left({A}_{i}\right)$ of each scenario can be obtained, and the scenarios are ranked according to the size relationship of the $\xi \left({A}_{i}\right)$. A larger value of $\xi \left({A}_{i}\right)$ indicates a better scenario ${A}_{i}$.
7. Discussion
In this study, movies are systematically ranked while considering diverse weightage criteria, which is followed by a comparative analysis of the experimental outcomes. Notably, when movies are ranked using uniform and objective weights, the results display consistency. However, alterations in movie attribute weights $\omega $ lead to discernible variations in rankings. This accentuates the pivotal role that attribute weight preferences play in influencing ranking outcomes.
Our experimental framework provides a salient methodology that empowers viewers to obtain recommendations that are meticulously tailored to their individualistic preferences concerning diverse movie attributes. Through parameter sensitivity analysis, it has been discerned that the parameter value $\theta $ oscillates between 0.2 and 3. The chosen parameter value $\theta $ plays a pivotal role, inducing specific shifts in the final ranking algorithm.
The efficacy of the decision making approach developed in this paper is not merely theoretical; it finds practical resonance in real-world decision making paradigms. The underpinning rationale is that the parameter $\theta $ encapsulates the spectrum of risk preferences among decision makers, and such heterogeneity directly modulates the final ranking. This paradigm can be analogously understood in the context of audience movie preferences: those with a more eclectic taste, displaying an openness to diverse movie genres and attributes, exhibit higher risk tolerance. They are, in essence, more resilient to potential misalignments between movie preferences and actual viewings. Consequently, for such an audience demographic, a lower parameter value $\theta $ is apt, as it resonates with their higher risk tolerance threshold, suggesting they are more amenable to movie selections that might not align perfectly with their expectations. Conversely, an audience cohort with a pronounced predilection for movie personalization demonstrates reduced resilience to discrepancies in attribute preferences. Their diminished risk tolerance suggests the advisability of a higher parameter value $\theta $ when leveraging the model proposed for movie selections.
Furthermore, our empirical observations emphasize the inherent sensitivity of the ranking model to the parameter value. Variability in parameter preferences and weightage criteria culminates in diverse ranking outcomes. This underscores the burgeoning demand for bespoke movie recommendations, aligning seamlessly with nuanced viewer predilections.