A new inclusion measure-based clustering method and its application to product classification

Abstract

Appropriate product classification can not only promote the sale of products but also highly improve the profits of manufacturers and retailers. To derive a desirable clustering of products, we need to consider qualitative (e.g., customers’ evaluation) and quantitative (e.g., product sales) information simultaneously. However, traditional methods such as K-means clustering, hierarchical clustering, and density-based clustering fail to depict the fuzzy cognitions of decision makers (DMs) in real cases. This article investigates clustering methods to aid DMs in their classification management of products. To do so, an intuitionistic multiplicative set (IMS), a new form of information expression, is applied to express the DMs’ evaluations of products due to its strong ability to handle unbalanced or asymmetric information. This study includes an in-depth analysis of inclusion measures of IMSs to measure the inclusion relationships between products under the given evaluation criteria. We propose two types of clustering techniques—the intuitionistic multiplicative (IM) transitive closure clustering method and the IM netting clustering method—to tackle the clustering problem with IMS information. Lastly, a case study of product classification management is presented to illustrate the applicability of the proposed methods. Furthermore, the validity and superiority of these clustering methods are verified by theoretical and practical comparisons.

Introduction

Retailers usually classify products in terms of qualitative characteristics (e.g., retailers’ or customers’ evaluations) or quantitative characteristics (e.g., product sales). Product classification can be used to select alternatives for sold-out products, but it also has wide applications in marketing, such as the development of new products, optimal layout of retail products on shelves, and storage strategy of retail products [13]. In this regard, appropriate product classification positively impacts both product sales and profit growth for retailers.

In practice, the implementation of product classification by retailers mainly goes through two stages: one is the collection of product information, and the second is product clustering and result exporting. That is, according to the consumers’ after-sale evaluation information (i.e., their evaluation) or the retailers’ (i.e., self-evaluation), decision makers (DMs) can derive the clusters of products using clustering techniques. A typical clustering problem integrates multiple criteria evaluation, in which both qualitative and quantitative information need to be considered at the same time. Traditional methods such as K-means clustering, hierarchical clustering, and density-based clustering fail to address this kind of problem that includes uncertainty [49]. This is because, in most cases, many objects have no rigid restrictions, and therefore, the DMs cannot directly point out which class they should belong to in practice. Zadeh’s fuzzy set provides a tool to classify the objects softly, beginning research on fuzzy clustering analysis. Previously, on the basis of Zadeh’s fuzzy set, many forms of uncertainty information expression have been discussed, such as intuitionistic fuzzy set (IFS) [38], hesitant fuzzy set (HFS) [44], hesitant fuzzy linguistic term set (HFLTS) [31], Pythagorean fuzzy set (PFS) [47], q-rung orthopair fuzzy set (q-ROFS) [3], and three-way decision [34]. However, all these methods have difficulty handling asymmetric or unbalanced information; for an example, consider the law of diminishing marginal utility in economics [36]. The intuitionistic multiplicative set (IMS), a new form of information expression, can depict experts’ evaluations from both positive and negative aspects [36]. Meanwhile, with the 1/9-9 scale, the IMS technique can be used to tackle multiple-criteria decision-making (MCDM) problems with unbalanced or asymmetric information, and therefore, it has been widely applied in medical diagnosis [28], hospital management [21], [22], [25], engineering management [8], human resource management [40], partner selection [27], and logistics transfer station selection [33]. In view of the advantages of IMS in depicting uncertainty and its rich theoretical basis, this article uses IMS to model the DMs’ complex evaluation information on products. This is the first research motivation of the article.

DMs can apply appropriate clustering techniques to classify products after obtaining product evaluation information. Clustering analysis, a meta-learning tool to cope with the given dataset effectively, can classify the data objects with similar features and observations into the same cluster by means of (i) hierarchical and partitioning methods or (ii) model-based, grid-based, and density-based methods [2], [17], [19], [29]. In recent years, a series of classical clustering algorithms have been extended to analyze fuzzy datasets to address the clustering problem with imprecise evaluation information, including the graph theory-based clustering technique in the IFS and interval-valued IFS settings [49], the agglomerative hierarchical clustering method in contexts of both PFS [47] and linear ordinal ranking theory [24], the hybrid fuzzy interval type-2 multidimensional decision-making approach [16], the consensus-based clustering method within contexts of hesitant qualitative information [5] and heterogeneous information [18], and correlation-based and distance-based transitive closure clustering based on HFLTSs [31].

Up to now, however, little attention has been paid to clustering techniques using IMS information. More importantly, apart from aggregation operators, few methods can be applied to deal with the product clustering analysis with IMS information. Inspired by this, the article explores clustering methods in the IMS setting based on our newly proposed intuitionistic multiplicative inclusion measure. Using our method, DMs can carry out clustering analysis of products by measuring inclusion relationships under the given evaluation criteria. In this way, the clustering result integrating inclusion relationships can reveal the substitutability among products and provide the DMs effective guidance in product classification. Hence, exploring and using intuitionistic multiplicative clustering methods effectively are the second motivation of the article.

In summary, the primary contributions of the article can be highlighted as follows:

• We define the intuitionistic multiplicative (IM) inclusion measure, by which two types of inclusion measures are proposed: inclusion measure between intuitionistic multiplicative values (IMVs) and inclusion measure between IMSs. Some desirable properties are discussed with examples.

• Based on the explored IM inclusion measures, two types of clustering methods are investigated: the intuitionistic multiplicative transitive closure clustering method (IMTCM) and the intuitionistic multiplicative netting clustering method (IMNCM).

• We apply the proposed methods to a case study of product classification management to explore its applicability and feasibility in practice. The usefulness of the proposed clustering methods is further demonstrated by theoretical and practical comparisons.

The remainder of this paper is as follows: We introduce basic knowledge of IMS and inclusion measures in Section 2. The IM inclusion measures and their desirable properties are investigated in Section 3. On this basis, new clustering techniques are explored in Section 4. In 5 Case study: Classification management of new energy vehicles, 6 Discussion and comparative analyses, the applicability and usefulness of the developed clustering methods are illustrated by a case study and further demonstrated by comparative analyses. Section 7 concludes the paper. All proofs are provided in Appendix B.