1 Introduction

Multi-criteria decision-making (MCDM) is about structuring and solving decision-making problems involving multiple criteria [1]. Structing implies clarifying the problem, including identifying available solutions (alternatives) to the problem, and criteria defining those alternatives. Solving indicates ranking alternatives or choosing the best (or most preferred) alternative from the fixed set of alternatives. A decision maker (DM) uses appropriate tools to give evaluation opinions for each alternative under each criterion. At last, the DM makes a decision based on the performance of alternatives defined by criteria. MCDM problems are pervasive in the government, private sector, and even in the lives of individuals [2].

In recent years, with the advancement of big data and Internet technologies, scholars and practitioners, who try to express and process information in various ways to improve the efficiency of ranking alternatives, have conducted more and more in-depth and diverse researches on MCDM problems [3]. In a decision-making process, due to the complexity of the decision-making environment and the limitation of individual cognition, it is often difficult for DMs to give accurate evaluation values, which aggravates the uncertainty and difficulty of MCDM. To efficiently address the ambiguity frequently arising in available information and do more justice to the essential fuzziness in human judgement, the fuzzy set (FS) theory [4] proposed by Zadeh has been used to represent uncertain information in MCDM problems. Many fuzzy MCDM methods have been developed [5].

However, there are still some sources of uncertainty in FSs. It is often difficult for a DM to exactly quantify his/her opinion as a number in interval [0, 1]. Many approaches treating imprecision and uncertainty of FSs were proposed [6]. A well-known generalization is interval arithmetic-based FSs, i.e., interval-valued fuzzy sets (IVFSs) [7]. Moore [8] first proposed the theory of interval arithmetic in his book Interval Analysis. He took into account the data error, truncation error and rounding error in the calculation process, and obtained an interval containing the exact result as the calculation output. Interval arithmetic can effectively solve the distortion problem of the calculation output due to the accumulation of calculation errors in numerical analysis. In 2000, Alefeld and Mayer [9] summarized theories and applications about interval analysis.

Although IVFSs can represent uncertain information, the result is still inaccurate due to the accumulation of calculation errors in the calculation process. On the basis of the interval operation theory, many scholars have enriched and developed interval techniques in the fuzzy MCDM environment, such as interval-valued intuitionistic fuzzy sets (IVIFSs) [10], interval-valued hesitant fuzzy sets (IVHFSs) [11] and interval-valued probabilistic linguistic term sets (IVPLTSs) [12]. Table 10 in Appendix shows the history of different types of representations. Table 11 in Appendix lists acronyms and corresponding full names used in this paper.

Although interval techniques have been frequently used by researchers to solve MCDM problems, there is a lack of a systematic literature review of combining interval techniques and MCDM methodologies. In stark contrast, there have been many literature reviews on fuzzy MCDM [5, 49, 50]. Most of the reviews summarized MCDM methods from the perspective of a specific application filed. Regarding interval fuzzy theories in MCDM, Celik et al. [51] collected 82 papers and summarized the applications of interval type-2 FSs in MCDM problems. It is necessary to conduct a review regarding the combination of MCDM methods with interval theories and their various extension formats to leverage the potential from their synergy. Furthermore, no strategy is given for reviewing practical applications of combining MCDM methods and interval theories. Based on these knowledge gaps, the primary concerns of our study are:

  • What are the overall development status and research hotpots of interval MCDM technologies?

  • What kinds of interval techniques have been used in conjunction with MCDM methods?

  • What application domains used interval MCDM theories?

  • What are lessons learned and future directions of interval MCDM?

This paper aims to summarize the applications of interval techniques used in the MCDM literature, identify current research status, find the research focus, and explore future development directions. To do this, in this study, first, we use the following retrieval strategy: TS = (“interval analysis” OR “interval value” OR “interval number” OR “interval-valued”) AND TS = (“multi-criteria decision-making” OR “multi-attribute decision-making” OR “MCDM” OR “MADM”) AND TS = (“ranking method”) (here “TS” stands for topics) in the Web of Science Core Collection database to search the Science Citation Index Expanded literature. The retrieval strategy returned 211 records from 2007 to 2022 (by January 13, 2022). This number reduces to 209 if we only consider articles. For these 209 articles, to perform bibliometric analyses, we downloaded them as tab-separated text files containing the title, author(s), keywords, source journal, abstract, references and other data. To ensure the reliability of the analysis results, we manually checked keywords and abstract of each article, and then excluded articles with low relevance to the topic of this study. Finally, 195 articles on the applications of interval techniques to MCDM problems are worth discussing.

This study has the following four main contributions. First, we adopt bibliometrics analysis to study publication and citation trends, productive countries/regions and institutes, highly cited papers and keyword co-occurrence analysis. Bibliometric analysis can help us master an overall understanding of the development status and research hotpots for interval MCDM theories. Second, this study focuses on reviewing theories and methods regarding the combination between interval techniques and MCDM methods. In this way, the development of information representation has been preliminarily summarized. Previous researches can not only provide suggestions for combining existing expressions to obtain new expressions with interval form, but also inspire scholars to study the usefulness of different MCDM methods with interval techniques. Third, we review real-world applications of interval MCDM methods in order to discover areas where these methods are commonly used. The summary of applications also inspires scholars to study potentials in other areas of these methods. Finally, we provide learned lessons and future directions. The progress map of this study is shown in Fig. 1. Compared with papers that only involve MCDM research in a specific field or a specific kind of MCDM method, we summarize interval MCDM studies from a more macro perspective.

Fig. 1
figure 1

The progress map of this study

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The structure of this paper is as follows. Section 2 describes main interval techniques and MCDM methods in our study. Section 3 is the results of bibliometric analysis, including publication trends and citation structure, most productive countries/regions and institutes, top 10 highly cited papers, keyword co-occurrence analysis. Section 4 reviews the combination between interval techniques and MCDM methods. Section 5 introduces the applications of specific interval techniques in MCDM methods and classifies their applicable industries. Section 6 proposes future research directions in this research field. Section 7 concludes the paper.

2 Preliminaries

In this section, we display the historical sequence of interval number and different types of interval fuzzy sets. Then, we describe main MCDM methods reviewed in this study.

2.1 Different Types of IFSs

An old example of the interval number was developed because of Archimedes. To obtain the value of \(\pi\), he adopted circumscribing polygons and inscribed polygons of a circle with radius 1. He obtained an increasing sequence of the lower bound and a decreasing sequence of the upper bound for the aera of the circle. The lower and the upper bounds can form an interval containing \(\pi\). Grell et al. [52] used interval arithmetic in numerical computing. Sunaga [53] introduced algebraic rules for the basic operations of interval numbers. The book of Moore [8] about interval analysis made interval analysis get people’s attention. In 1975, Sambuc [7] introduced the concept of IVFS. Zadeh also mentioned IVFS in [54] (page 242).

We denote by X a non-empty universe, either finite or infinite, and an IVFS A. For an IVFS, the membership degree of \(x \in X\) to A is given by \(A(x) = [\underline {A} (x),\overline{A}(x)] \in L[(0,1)]\), where the mappings \(\underline {A} :X \to [0,1]\) and \(\overline{A}:X \to [0,1]\) correspond to the lower and upper bounds of the membership interval \(A(x)\), respectively. \(L[(0,1)]\) is the set of all closed subintervals of \([0,1]\).

Since the proposal of IVFS, different types of IVFS have been developed. Table 1 summarizes the main types of IVFS. Certainty, there are also different extension formats based on the IVFSs in Table 1, and a snapshot on the relationships among the extensions is represented in Fig. 2. It can be found from Table 1 that the different forms of IVFSs are similar. The development idea of their extended form is to directly replace the numerical value of the general fuzzy set with the interval form. Extensions based on linguistic terms can also replace linguistic terms with interval values.

Table 1 Main types of IVFS
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Fig. 2
figure 2

Representation of the inclusion relationships between different IVFSs

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2.2 MCDM Methods

MCDM can be divided into two categories: continuous MCDM, also known as Multiple Objective Decision-Making (MODM), and discrete MCDM, also known as Multiple Attribute Decision-Making (MADM) [55]. In this study, we focus on MADM and use the item “MCDM” to represent “discrete MCDM”, i.e., MADM. No matter what categories, alternatives and their criteria are two important elements of them. Alternatives represent the different choices of action available to the decision maker, and criteria represent the different dimensions from which the alternatives can be viewed. With the two elements, an MCDM problem can be easily solved by a matrix format. Let \(E = \{ E_{i} ,i = 1,2,3, \ldots m\}\) be a set of decision alternatives and \(C = \{ c_{i} ,j =\) \(1,2,3, \ldots ,n\}\) be a set of criteria. A decision matrix \(E\) is an (\(m \times n\)) matrix with element \(e_{ij}\). \(e_{ij}\) represents the performance of alternative \(E_{i}\) evaluated in terms of criterion \(c_{j}\). In dealing with the matrix, the weights of criteria are necessary and can be denoted as \(W = \{ w_{j} ,\)\(j = 1,2,3, \ldots n\}\). A decision matrix \(E\) can de expressed as:

Criteria

Alternatives

\(C_{1}\)

\(C_{2}\)

\(C_{3}\)

\(\cdots\)

\(C_{n}\)

(\(w_{1}\)

\(w_{2}\)

\(w_{3}\)

\(\cdots\)

\(w_{n}\))

\(E_{1}\)

\(e_{11}\)

\(e_{12}\)

\(e_{13}\)

\(\cdots\)

\(e_{1n}\)

\(E_{2}\)

\(e_{21}\)

\(e_{22}\)

\(e_{23}\)

\(\cdots\)

\(e_{2n}\)

\(\vdots\)

\(\vdots\)

\(\vdots\)

\(\vdots\)

\(\vdots\)

\(\vdots\)

\(E_{m}\)

\(e_{m1}\)

\(e_{m2}\)

\(e_{m3}\)

\(\cdots\)

\(e_{mn}\)

Through the matrix, the optimal alternative \(O^{*}\) with the highest desirability under all criteria can be obtained. In this process, important theorems need to be pay attention to. Firstly, criteria may conflict with each other because of their different dimensions. For instance, an ideal alternative generally pursues low cost and high profit. Thus, in the decision matrix, performances of alternatives under conflict criteria can be represented by their reciprocals. Secondly, different criteria may be connected with different units of measure. For example, when measuring the price and mileage of a car, units of the two criteria are different. Therefore, the unification of different units of measure is also a key step. Thirdly, weights of criteria influence the decision. They can either be given by experts or obtained from alternative performances according to decision goals. Usually, these weights are normalized to add up to one.

Studies have widely used interval techniques as opinion expression tools of DMs in MCDM problems. To tackle practical decision-making problems, various MCDM methods have been developed by researchers. Table 2 lists main MCDM methods that combined with interval techniques. It can be found that the computational complexity of TOPSIS and VIKOR is relatively low, which explains the reason for their wide application in Sect. 5.

Table 2 Description of MCDM methods
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3 Bibliometrics

Bibliometrics is a quantitative analysis method that takes the external characteristics of various bibliographic materials as research object [68]. It uses statistical and visualization methods to describe, evaluate and predict the status quo and development trend of research fields with the help of various quantitative characteristics of literature [69]. Bibliometrics has been widely used to identify the development status of a specific research domain or disciplinary. In this section, we use the bibliometric software VOSviewer [70] to analyze the research status and identify the development trend of interval techniques in solving MCDM problems.

3.1 Publication and Citation Trends

Based on the number of publications and the number of citations, it is possible to predict the development status and trends of a research field. The annual trends of three performance indicators, i.e., the number of publications, the total number of citations, and the average number of citations per paper, are shown in Fig. 3. Table 3 shows details about these three performance indicators as well as the number of highly cited papers that obtained at least 150, 100, 50, 25, and 1 citation(s), respectively.

Fig. 3
figure 3

Publication and citation trends from 2007 to 2022 (by January, 2022)

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Table 3 Publication trend and citation structure from 2007 to January 2022
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As can be seen from Fig. 3, from 2007 to 2022, the number of published papers maintained a low growth rate. Although the number of publications in 2012, 2017, and 2021 had a decline compared with the previous year, the number of publications had an overall growing trend, indicating that interval techniques were more and more popular in MCDM problems. However, the number of total citations fluctuated greatly. Figure 3 shows that total citations are not directly proportional to the number of publications. This phenomenon can be explained by several highly cited papers. For instance, the paper with the highest number of citations was published in 2016 and the paper with the second highest number of citations was published in 2013. Moreover, the number of publications has increased a lot since 2016, but the number of citations has declined in recent years. This is a normal phenomenon because a paper needs 3 to 7 years to achieve its majority of citations [71]. The annual trend of the average number of citations per paper also verified this phenomenon. The year of 2020 has the largest number of publications (39), followed by 2021 with 33 publications. However, the highest number of citations is 1312 in 2016, followed by 846 in 2011. In 2009, the average number of citations is 125. In 2010 and 2011, the number exceeds 90, and then gradually goes down. This shows that early articles laid a solid foundation for this research direction, and were valued and developed by later scholars. Thus, it is worth noting that scholars should not only increase the number of publications, but pay more attention to the quality of their papers, so that they can be recognized by more other scholars. Table 3 also reflects the citation structure of published papers. Up to January 13, 2022, there were 8 articles with more than 150 citations, 9 articles between 100 and 150 citations, 13 articles between 50 and 100 citations, 40 articles between 25 and 50 citations, and 112 articles below 25 citations.

3.2 The Most Productive Countries/Regions and Institutions

By studying publication and citation distribution status of different countries/regions and institutions, we can identify the most active locations, which can help scholars seek collaboration or academic exchanges in the future. Table 4 presents top 10 productive countries/regions with the largest number of publications.

Table 4 The most productive countries/regions
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As can be seen from Table 4, China (100) is the largest source of publications, far ahead of other countries/regions, accounting for 51.28% of all publications, followed by India (22), Iran (22), Taiwan (20) and Turkey (14). It can be found that among the top 10, China, India, Iran, Turkey, Malaysia, and Pakistan are developing countries/regions, which to a certain extent shows that developing countries/regions have made more progress in this research field, especially Chinese researchers. Regarding the average number of citations, Iran has the best performance.

Regarding institutes, Table 5 lists the top 20 most productive institutions. Furthermore, we also provide the world ranks of these institutes according to the Academic Ranking of World Universities (ARWU) and the Quacquarelli Symonds (QS) World University Rankings. As can be seen from Table 5, Islamic Azad University (14) from Iran is the largest source of publications, followed by Chang Gung University (11) and Vilnius Gediminas Technical University (11). Combined with the number of publications of Iran in Table 5, it shows that Islamic Azad University is Iran’s main research force in this field. Among these 20 institutes, 12 were from China (mainland). These 12 institutes are the major reason why China ranks first in Table 4. Furthermore, we can also find that the most institutes are not in the top 100 according to the ARWU and QS rankings. Thus, from this perspective, this field is diverse and has influence in non-leading institutes across the world. Figure 4 reflects the collaboration network for institutions. In Fig. 4, a node represents an institute and a tie linking two nodes represent the collaboration relationship. As shown by Fig. 4, some active collaboration relations include the link between Sichuan University and Isiamic Azad University, between Tongji University and Shanghai University, and between Isiamic Azad University and Vilnius Gediminas Technical University.

Table 5 The most productive institutions
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Fig. 4
figure 4

Collaboration network between institutions

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3.3 Top 10 Highly Cited Publications

To some extent, the more times a paper is cited, the higher the quality of the paper is likely to be [72]. A high number of citations indicates that the research of this paper has been recognized by many other scholars. By analyzing highly cited papers, we can identify the topics that scholars in a certain field focused on. Table 6 presents the details of the top 10 highly cited papers with their title, author(s), published journal, published year, and the number of citations.

Table 6 Top 10 highly cited papers of reviewed publications
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From Table 6, we can find that the top 10 highly cited papers were published between 2009 and 2016. Although there were many publications in the past five years, none of them appeared in Table 6. These publications need more time to reach their majority of citations. Still, we can infer that this research field has not made great breakthroughs in recent years, or needs more time to attract more attention. Most of these highly cited papers proposed new theories and were recognized by later scholars. Thus, they can receive a high number of citations. Among those 10 papers, 5 are researches on IVIFSs [75, 77,78,79, 81], and 2 are researches on IVHFSs [74, 80]. The topics of these highly cited papers suggest that IVIFS is a hot topic, and the applications of interval techniques in MCDM problems were mainly realized through IVIFSs. By analyzing these highly cited papers, we can find that although there were many ways of expressing interval information in MCDM problems, IVIFS and IVHFS were relatively widely used and recognized by most scholars. TOPSIS, VIKOR and TODIM, as traditional MCDM methods, were also widely recognized, and new methods still need more time to prove their feasibility.

4 Keyword Co-Occurrence Network

Next, we analyze the literature from the perspective of keywords. Through keywords, we can quickly read the research scale and topic of a paper. Within a research field, some keywords are used frequently and some have a low frequency. Those keywords with high-frequency are usually related to a popular research topic or an important research approach [68]. Figure 5 shows the keyword co-occurrence network of reviewed publications. Each circle represents a keyword, and the larger the circle is, the higher the frequency of the keyword is. The lines between circles indicate that keywords have appeared in some papers at the same time. The warmer the color of the circle is, the closer the time to the present is.

Fig. 5
figure 5

Keyword co-occurrence network of publications

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From Fig. 5, we can find out the keywords with high frequency, such as “ranking”, “model” and “aggregation operators”, which indicate that a large number of reviewed studies established decision-making models to solve ranking problems or introducing aggregation operators to aggregate evaluation information and derive priority vectors in MCDM. We can also find that the IVIFS is a high-frequency keyword, which echoes the content of Table 6. In addition, MCDM methods mainly include TOPSIS, VIKOR, and AHP, which show the feasibility of applying interval techniques in these methods. From the perspective of time evolution, most of early research topics were about basic theories, such as distance measures and aggregation operators, which were explored step by step in an uncertain environment, and continued to extend in the later period. Due to the increasing pressure of resource conversation and environmental protection, scholars gradually focused on “sustainability assessment” and “renewable energy”. Thus, through the keyword co-occurrence analysis, we can see the past hotspots and future possible development directions of interval techniques in solving MCDM problems.

5 Interval Techniques in MCDM

Scholars have used interval techniques in conjunction with MCDM methods. MCDM evaluates and selects alternatives from the best to the worst according to multiple criteria. When making decisions, DMs need to describe the performance of alternatives. Furthermore, they also need to provide ratings about criteria weights. It is common that interval variables are used to assess the ratings of various alternatives over criteria in real-world decision-making problems. In this section, we make a classification for the use of interval techniques in conjunction with MCDM methods. Then, we follow this classification and review-related literatures.

First, we provide an overall quantity overview for application proportion of interval techniques in MCDM. Figure 6 shows the application proportion of different interval techniques. As shown by Fig. 6, about three-fourths of studies combined intervals with other kinds of information representation formats, such as FSs, LTSs, and SSs. Only about one-fourth (25.1%) used interval numbers separately in MCDM. Regarding FSs, interval techniques were mainly used in combination with IFSs [16], HFSs [20], PFSs [23], and TFSs [83].

Fig. 6
figure 6

Application proportion of interval techniques

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We classified all studies into five categories according to the types of information representation formats in conjunction with interval techniques: interval numbers, FSs, LTSs, other representation formats, and heterogeneous representation formats. Table 7 provides the classification result containing details about the categories, extensions and applications in conjunction with interval techniques. Note that some formats can be classified into two or more categories. For instance, IV Pythagorean HFS can be classified into “HFS” or “PFS”. We put these kinds of interval methods into only one category in Table 7.

Table 7 Applications of interval techniques in information processing
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5.1 Interval Numbers (INs)

In this part, a majority of studies used INs to evaluate the performance of alternatives. For instance, Saffarzadeh et al. [84] used INs in linear programming to solve MCDM problems. Luo et al. [85] deduced an equivalent function of the risk appetite index and the target IN after expressing the IN in the rectangular coordinate system. There were also studies that used INs to determine the importance of criteria. For example, Fan et al. [86] adopted INs to represent the aspiration-levels of criteria. Rezaei [73] used interval analysis as a way to determine the ranking of criteria for BWM. Hafezalkotob et al. [87] used interval values throughout the structure of the paper, including the evaluation of importance of experts, the relative importance of criteria, and assessment values of alternatives.

5.2 Interval-Valued Fuzzy Set

Interval techniques have been applied by combining FSs and their extensions, including IFSs, HFSs, PFSs, TFs and other formats. Statistically, there are altogether 120 papers that combined interval techniques with FSs in our collection. As shown in Table 8, some of these reviewed papers were categorized into single and hybrid MCDM approaches based on whether the paper used a single or a hybrid of multiple MCDM methods.

  • IVFS Certainly, there are studies that directly applied IVFSs in MCDM problems. For example, Sharaf [88], Mohagheghi [89], and Wang et al. [90] proposed interval-valued fuzzy TOPSIS, TODIM, and DEMATEL method, respectively. Pires et al. [91] and You et al. [92] combined two kinds of MCDM methods under IVFS environment.

  • IVIFS The literature about IVIFS can be classified into three categories: direct application of IVIFS, extensions of IVIFS, and combination of IVIFS with other formats. The IVIFS has been widely used to express the evaluation information of alternatives and criteria in MCDM problems. For example, regarding alternatives, Nayagam et al. [79] introduced an accuracy function for ranking IVIFSs and used it in MCDM. Chen [115] developed a likelihood-based outranking method for handling MCDM problems based on IVIFSs. To validate the effectiveness of his method, he chose some MCDM methods including simple additive weighting, TOPSIS, and ELECTRE I and ELECTRE II methods to make comparative analyses. Garg and Kumar [138] proposed a novel connection numbers-based likelihood measure to rank interval-valued intuitionistic fuzzy numbers. Li [139] developed an MADM method with both ratings of alternatives on attributes and weights being expressed by IVIFSs. Some authors also proposed operation rules for IVIFSs [140,141,142]. In addition to the above literature, many scholars developed IVIFS-based MCDM theories, as shown in Table 8. Table 8 suggests that TOPSIS is the most widely used MCDM method. TOPSIS was proposed to prioritize solutions and identify the best alternative based on its distance from ideal solutions. Some scholars combined TOPSIS with other MCDM methods. For instance, Dogan et al. [109] coupled AHP and TOPSIS methods to rank potential alternatives. The AHP can systematically determine the weights of criteria and TOPSIS can list alternatives according to the reality situation. It can be found that in other environments, there were also studies that adopted AHP and TOPSIS simultaneously. The combination of AHP and TOPSIS can assist DMs to build a robust evaluation foundation. Several papers also combined three kinds of MCDM methods [116, 133, 137]. Different MCDM methods have their own advantages. The combination of MCDM methods can make full use of the advantages of them. However, combining different MCDM methods also increases the complexity of the decision-making process. Whether the value brought by the combination of different MCDM methods can make up for the loss brought by the increase of complexity of the decision-making process remains to be further investigated.

Table 8 Combination of IVFSs and MCDM methods
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Interval techniques have also been applied to other extension forms of IVIFS, such as CIVIFSs, IVTrFSs, IVITrFSs and IVIHFSs. Liu et al. [121] combined quality function deployment (QFD) with IVITrFSs, to address the ambiguity in human thought, and transform the customer needs into engineering characteristics. Seker [122] combined the newly developed combinative distance-based assessment (CODAS) method with IVITrFS, Du and Liu [108] also conducted a ranking study of MCDM based on IVITrFS.

  • IVHFS Zhang and Xu [80] developed two metric functions to compare the magnitudes of hesitant fuzzy elements and interval-valued hesitant fuzzy elements. Niu et al. [124] applied the IVHFSs with ELECTRE III to rank renewable energy alternatives. Zhang [125] proposed two interval-valued hesitant fuzzy QUALIFLEX ranking methods to deal with MCMD problems. The combination of IVHFS and these MCDM methods can eliminate the limitation of traditional methods that has difficulty in making judgments only depending on crisp values.

  • IVPFS There are both single and hybrid studies under IVPFS setting. Al-Samarraay et al. [127] developed a Fuzzy Decision by Opinion Score Method (FDOSM) for evaluating and benchmarking recognition systems with IVPFSs. MULTIMOORA, EDAS, and COPRAS methods have also been used under the IVPFS environment [128,129,130]. Hybrid studies used TOPSIS and other kinds of methods such as AHP and ANP to weight criteria and rank alternatives [131,132,133].

  • Other extension formats of IVFSs The applications of interval techniques in other expressions of FSs includes IVTFSs, IVTrFSs, interval Gaussian FSs, interval spherical FSs, and interval Fermatean FSs. IVTFSs have been used to express the vague evaluation information of DMs for VIKOR, TOPSIS, AHP, and BWM methods [134,135,136,137]. Wang et al. [143] defined three-trapezoidal-fuzzy-number-bounded type-2 fuzzy numbers on the basis of interval type-2 trapezoidal fuzzy numbers. Tolga, Parlak and Castillo [43] proposed finite interval-valued type-2 Gaussian fuzzy numbers. Jeevaraj [47] introduced the concept of interval-valued Fermat FSs, and Farrokhizadeh et al. [46] studied interval-valued spherical FSs.

5.3 Interval-Valued Linguistic Term Set

The linguistic approach considers the variables in a decision-making problem assessed by linguistic terms instead of numerical values. This approach is appropriate for many problems, since it allows the representation of DMs’ information in a direct way whether they are unable of expressing that of precision. On the basis of LTS, Xu [19] defined the IVLTS. The applications of interval techniques in linguistic approach mainly include IV2TLTS, IVILTS, and IVHFLTS.

  • IV2TLTS The 2-tuple linguistic representation model was developed by Herrera and Martinez [144] based on the concept of symbolic transformation. A linguistic 2-tuple consists of a pair of values: a linguistic term and a numerical value representing the symbolic translation of the linguistic term. It can avoid information distortion in the linguistic information processing. Beg and Rashid [145] suggested that the numerical value can be represented as an interval and developed the IV2TLTS in MCDM. IV2TLTS has been studied and applied in MCDM problems, such as PROMETHEE-II [146], TOPSIS [147], and the combination of BWM with CODAS [148].

  • IVILTS IVILTS was proposed by Wang et al. [39] on the basis of replacing intuitionistic fuzzy numbers by IVILTSs. Wang et al. [149, 150] defined the conversion function, aggregation operations, and a ranking method for IVILTS. Liu et al. [151] combined BWM with the alternative queuing method within the IVILTS setting.

  • IVHFLTS HFLTS was proposed by Rodríguez et al. [22] to allow DMs to express their evaluations towards a linguistic variable flexibly by a set of ordered finite set of linguistic terms. Zhu and Xu [38] introduced the IVHFLTS by assigning interval-valued linguistic terms to the HFLTS. Wang et al. [152] proposed two aggregation operators for IVHFLTSs and used them in MCDM problems. Feng et al. [153] combined the interval-valued q-rung dual HFS with LTS, and proposed the interval-valued q-rung dual hesitant linguistic sets (IVq-RDHLSs). Wang, Gou and Xu [154] constructed the gained and lost dominance score (GLDS) method into the continuous interval-valued double hierarchy linguistic environment. Liao et al. [155] used continuous IVLTS to represent decision matrices for aggregation and ranking. PLTS is also an extension form of HFLTS, which is composed of linguistic terms with assigned probabilities [24]. The IVPLTS changed the crisp probabilities into interval probabilities, which was used by Bai et al. [12] and Sivagami et al. [156] in multi-criteria group decision-making (MCGDM) problems.

There are also other extensions of the IVLTS. For example, Xian et al. [157] proposed an interval-valued Pythagorean fuzzy linguistic approach and combine it with VIKOR method. Liu et al. [158] proposed a model for robot selection in the context of interval-valued Pythagorean linguistic uncertain environment.

5.4 Other Representation Formats

Other information representation methods mainly include neutrosophic sets (NSs), soft sets (SSs), intuitionistic multiplicative sets (IMSs), rough sets (RSs), and belief structure. Thong et al. [159] introduced generalized dynamic IVNSs. Semenas and Bausys [160] extended the WASPAS method into the IVNS environment. Karasan and Kahraman [161] employed the ELECTRE I method with IVNSs to select municipal renewable energy alternatives. Ali [162] proposed interval-valued q-rung orthopair fuzzy soft sets (IVqROFSSs). Wang et al. [41] employed interval-valued rough random variables and interval-valued rough numbers to process decision-making information. Garg [163], and Zhang and Pedrycz [164] applied interval values to IMSs. Zhou et al. [48] applied interval values to evidence reasoning by extending the belief structure to an interval form.

5.5 Heterogeneous Representation Formats

In MCGDM problems with a group of DMs, a critical issue is that DMs usually tend to provide various forms of evaluation information because of their varied backgrounds and cognitive habits. The formats of evaluation information may be given as quantitative or qualitative natures in the group. Heterogeneous MCGDM problems are common in real-life situations. As an important expression format, interval technique is usually incorporated in heterogeneous information. Zheng [165] considered five kinds of evaluation, i.e., crisp numbers, interval numbers, triangular fuzzy numbers, linguistic variables, and intuitionistic fuzzy numbers, in MAGDM problems. Zhao et al. [166], and Zhang et al. [167] applied a heterogeneous model based on TOPSIS and MABAC methods, respectively. One limitation of heterogeneous MCGDM models is that, it is necessary to use a transformation function to transform heterogeneous information into a unified form. However, it is hard to develop different functions to unify heterogeneous information. Furthermore, the original information may be lost in the transforming process, which will further lead to unreasonable decision-making result. Considering these points, Tang et al. [168] did not use a transformation function but used ordinal rankings of DMs to calculate the consensus of a group directly.

To sum up, interval techniques have been widely used to deal with MCDM problems, and played a vital role in the information representation of individuals. Whether the interval technique was directly applied or was further extended and combined with other expressions, the results showed that the use of interval techniques can effectively reduce judgment biases in the information processing process, and laid a solid foundation for the next ranking process.

6 Applications Related to Interval MCDM Problems

In this section, the applications regarding interval MCDM are reviewed. For the convenience of review, we first make a classification for these reviewed papers. As shown by Fig. 7, a large part of these studies focused on the scoring function and aggregation operators, which are the basis of constructing MCDM models. Regarding MCDM methods, TOPSIS, VIKOR, and TODIM are most popular for scholars. Table 9 details the applications of interval techniques in each MCDM method and the applicable industries.

Fig. 7
figure 7

Proportion of interval techniques applied in main ranking methods

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Table 9 Interval-valued MCDM methods with application fields
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In Fig. 8, we present the major application fields concerning interval MCDM problems. It can be found from Fig. 8 that supplier selection, investment management, energy and resource management, and site selection account for a considerable proportion in all application fields. These topics are classic MCDM problems. In addition, scholars also focused on healthcare management, enterprises’ development, risk and reliability analysis, equipment selection, service evaluation, material selection, civil engineering management, product design, waste proposal, education, online consumption, and bank ranking. It seems that any kind of evaluation problems with quantitative and qualitative evaluation information can be solved by interval-based MCDM approaches.

Fig. 8
figure 8

Application fields related to interval MCDM

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7 Lessons Learned and Future Directions

Interval techniques have been applied to MCDM problems by numerous scholars, which is of great significance to today’s society and helps to improve the accuracy of information processing in decision-making processes. Based on our review work, we can learn many lessons and explore future directions.

From the perspective of bibliometric analyses: Based on the results of bibliometric analyses, we can draw the following findings and suggestions:

  1. (1)

    Given the increase in the number of publications in recent years, we believe that research on the applications of interval techniques to MCDM problems will continue to develop in the coming years. Compared with papers published in early years, recent papers have been cited less frequently. Although these recent papers may not have been recognized by other scholars due to time, scholars should focus more on increasing the quality of their studies instead of just extending the work of predecessors.

  2. (2)

    According to the publication distribution status regarding countries/regions and institutions, we find that Asian scholars have made major contributions, which to a certain extent also shows that Asia is in a leading position in this research field. Therefore, in the future, institutions in other continental countries can cooperate with institutions in Asia and learn from each other to achieve breakthroughs in this filed.

  3. (3)

    By analyzing the characteristics of highly cited papers, we can confirm that less great progress has been made in the last half decade. Compared with other types of IVFSs, IVIFSs are more widely recognized by scholars. Therefore, in the future, on the basis of improving the research quality, scholars can mainly conduct in-depth research in the direction of interval-valued qualitative representation formats.

  4. (4)

    From the keyword co-occurrence graph, it can be seen that aggregated operators, score functions and TOPSIS are the main combination ideas of interval techniques in different periods in MCDM problems.

From the perspective of information representation formats; Through the analysis of the combination of MCDM methods and interval techniques, future researchers need to pay attention to the following points:

  1. (5)

    FSs were popular in MCDM problems, and more than 60% of reviewed papers applied interval fuzzy MCDM methods to solve ranking problems. However, we found that some papers have applied interval techniques to SSs [162] and RSs [41] to express uncertain information. However, compared with FSs, the applications of SSs and RSs are still lacking. As a result, future research can consider applying interval techniques to these less-used information expression formats, and further develop theories to support MCDM.

  2. (6)

    Interval techniques have been applied to various ways of information representation, but there were few studies comparing the effectiveness of different methods. According to the mathematical forms of different extensions of IVFSs, most of them simply replaced existing basic fuzzy sets with intervals. To find the most effective and efficient tool, it is necessary to conduct comparative studies on the applications of various interval techniques. In this way, the most suitable information expression method of interval techniques in MCDM problem can be found.

  3. (7)

    Almost all studies established decision-making models and adopted interval techniques as mathematical representation tools. In these models, DMs directly used interval techniques to express their opinions. However, less is known about how to explore the association between the willingness to participate in the decision-making process and the expression representation way based on empirical analysis. Furthermore, it also seems of interest to investigate the association between expression representation way and the individual and group decision-making performance.

From the perspective of MCDM in combination with interval techniques: Future development directions of MCDM methods in combination with interval techniques are as follows:

  1. (8)

    To apply interval MCDM methods to real-life, other outranking-based MCDM methods under interval setting can be further investigated, such as Measuring Attractiveness by a Categorical-Based Evaluation TecHnique (MACBETH) [225], Superiority and Inferiority Ranking (SIR) [226], and Treatment of the Alternatives according to the Importance of Criteria (TACTIC) [227].

  2. (9)

    Nowadays, with the widespread utilization of social media platforms and wearable technologies, there has been an exponential growth of available data [228]. The era of big data have arrived. As a result, for an MCDM problem, there may be a large number of alternatives. How to develop efficient decision-making models to address such problems is also an interesting research issue with challenges.

From the perspective of applications: There are many interesting topics about the applications of interval techniques.

  1. (10)

    According to the observation of industry application, it is found that in the field of interval techniques, the research on MCDM still stays in classic decision-making problems, such as supplier selection and investment. In the realistic context, there are already more complex MCDM problems especially after the outbreak of COVID-19. It is necessary to apply interval techniques, so that changes caused by uncontrollable factors can be taken into account in practical problems. Furthermore, given that the evaluation information is usually provided by DMs, in the future, big data-support decision-making should be further explored through crawling data of some open websites.

8 Conclusions

This paper presented a review of 195 publications from 2007 to 2022 about interval techniques in MCDM problems. First, we performed a bibliometric analysis of the literature, including publication trends, citation structure, countries/regions and institutions distribution status, highly cited papers and keywords. According to statistical charts and data, we can know that from 2007 to 2022, the number of articles published in this field showed an increasing trend, while the number of citations per year had generally shown a downward trend. From a locational and institutional perspective, China, India and Iran were the backbone of researches in this field. Among the top 10 highly cited papers, IVIFS is the most recognized expression. We also presented results for keyword co-occurrence analysis and the evolution of hot topics in each time period intuitively and clearly. Regarding the information representation way in the literature, scholars mainly combined interval techniques with FSs and used them in MCDM problems. Among all MCDM methods, TOPSIS, VIKOR, and TODIM methods were most popular. These methods were mainly used to supplier selection, investment management, energy and resource management, and site selection problems. Finally, based on the above analyses, future development directions of interval techniques in MCDM problems were provided.

This study also has limitations. Only articles written in English were included in our work. Our retrieval strategy may limit the breadth of the sample, leading to some omissions. In addition, due to the limited space, this study only briefly described the content of reviewed publications from a specific perspective. Therefore, it is necessary for readers to read and summarize the relevant literature carefully if they are interested in. In any case, it is hoped that our work will help scholars in the decision-making field to understand the current state of the application of interval techniques in MCDM ranking problems and find future possible directions.