An Overview of Interval Analysis Techniques and Their Fuzzy Extensions in Multi-Criteria Decision-Making: What’s Going on and What’s Next?

Abstract

Due to the diversity of information, many decision-making problems cannot be solved based on a single criterion. The complexity of assessment objects and the limitations of individual cognition cause the opinions given by experts uncertain, which further aggravates decision-making difficulties. Although fuzzy sets and intuitionistic fuzzy sets are proposed to express vague information, they still have the problem of losing information. As a tool to express uncertain information, interval techniques can effectively prevent the loss of information and improve the accuracy of decision-making. In this regard, many scholars applied interval analysis techniques and their fuzzy extensions to solve multi-criteria decision-making (MCDM) problems. This study reviews 195-related articles published from 2007 to 2022, and analyzes the research progress of interval analysis techniques and their fuzzy extensions in MCDM problems. Through bibliometrics analyses, publication and citation trends as well as productive countries can be intuitively and quantitatively obtained. Then, we review theories and methods regarding the combination of interval techniques and MCDM methods, and find that interval-valued fuzzy sets are extensively discussed and combined with TOPSIS. Next, real-world applications of these publications are reviewed, and we obtain that interval techniques are mainly used in supply chain selection. Finally, we propose future directions regarding interval techniques in MCDM problems. It is hoped that this study would be helpful for scholars and practitioners to carry out further research.

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.

Table 10.

History of different types of representations

General representation		Representation with interval
Year	Event	Year	Event
1965	Zadeh [4] introduced the fuzzy set theory	1966	Moore [8] proposed the theory of interval arithmetic
1971	Zadeh [13] introduced the idea of T2FS	1975	Sambuc [7] proposed the concept of IVFS under the name of $\mathrm{\Phi }$-Flou sets. Gorzalczany [29] and Turksen [30] fixed the name of IVFS in the 1980s
1982	Pawlak [14] developed the concept of RS theory	1989	Atanassov and Gargov [10] presented the notion of IVIFS
1985	Chen [15] proposed the concept of generalized TrFS	1999	Liang and Mendel [31] introduced the idea of IVT2FS
1986	Atanassov [16] presented the definition of IFS	2000	Alefeld and Mayer [9] summarized theories and applications about interval analysis
1994	Yang and Singh [17] proposed the ER approach	2005	Wang et al. [32] introduced the definition of IVNS
2002	Smaradache [18] introduced the concept of NS	2006	Chen and Chen [33] presented the concept of generalized IVTrFS
2004	Xu [19] introduced the concept of LTS	2009	Yang et al. [34] combined IVFS and SS into IVSS
2010	Torra [20] introduced the notion of HFS	2011	Zhang, Liu and Zhai [35] introduced IVTFS into MCDM problems
	Wang and Li [21] proposed the concept of ILTS	2012	Zhang [36] proposed IV2TLTS in MCDM problems
2012	Rodríguez et al. [22] proposed the HFLTS	2013	Chen, Xu and Xia [11] introduced the concept of IVHFS
2013	Yager [23] gave the idea of PFS		Zhang [37] extended the HFS to IVIFS environments and proposed the concept of IVIHFS
2016	Pang et al. [24] introduced the concept of PLTS	2014	Zhu and Xu [38] developed the concept of IVHFLTS
		2015	Wang et al. [39] extented into the concept of IVILTS
	Peng and Yang [25] proposed PFLTS	2016	Peng and Yang [40] introduced fundamental properties of aggregation operators of IVPFSs. Wang et al. [41] employed IVRSs to process decision information
2018	Wei et al. [26] definited the concept of PHFS
2019	Gundogdu and Kahraman [27] developed the SFS. Senapati and Yager [28] introduced the concept of FFS	2017	Du et al. [42] proposed IVPFLTS
		2018	Bai et al. [12] presented the definition of IVPLTS
		2020	Tolga, Parlak and Castillo [43] introduced IVGFS. Wu, Liao and Pedrycz [44] enriched the concept of IUPLTS. Zhang et al. [45] propose the concept of IVPHFS
		2021	Farrokhizadeh et al. [46] introduced the IVSFS. Jeevaraj [47] proposed the concept of IVFFS. Zhou et al. [48] used IVER approach in MCDM problems

Table 11.

Acronyms and full names

Acronym	Full name	Acronym	Full name
FS	Fuzzy set	PFLTS	Pythagorean fuzzy LTS
T2FS	Type-2 fuzzy set	PHFS	Pythagorean hesitant fuzzy set
RS	Rough set	SFS	Spherical fuzzy sets
TrFS	Trapezoidal fuzzy set	FFS	Fermatean fuzzy set
IFS	Intuitionistic fuzzy set	GFS	Gaussian fuzzy set
ER	Evidential reasoning	IMS	Intuitionistic multiplicative set
NS	Neutrosophic set	DHHFLTS	Double-hierarchy hesitant fuzzy LTS
LTS	Linguistic term set	IV	Interval-valued
HFS	Hesitant fuzzy set	IVSS	Interval-valued soft set
ILTS	Intuitionistic LTS	IVTFS	Interval-valued triangular fuzzy set
HFLTS	Hesitant fuzzy LTS	IV2TLTS	Interval-valued 2-tuple linguistic term set
PFS	Pythagorean fuzzy set	IVIHFS	Interval-valued intuitionistic hesitant fuzzy set
PLTS	Probabilistic LTS	IUPLTS	Interval uncertainty PLTS
IN	Interval number	IVIMS	Interval-valued intuitionistic multiplicative set

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.

The progress map of this study

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.

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.

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 $A(x)=[\underset{\_}{A}(x),\overline{A}(x)]\in L[(0,1)]$, where the mappings $\underset{\_}{A}:X\to [0,1]$ 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

Model	Year	Mathematical form	Example
IVFS	1975	$\{<x,[\underset{\_}{A}(x),\overline{A}(x)]>|x\in X,\underset{\_}{A}(x)\le \overline{A}(x)\}$	$\{[0.1,0.4]\}$
IVIFS	1989	$\begin{array}{c}\{<x,[{\mu }_A^{-}(x),{\mu }_A^{+}(x)],[v_A^{-}(x),v_A^{+}(x)]>|\\ x\in X,0\le {\mu }_A^{+}(x)+v_A^{+}(x)\le 1\}\\ \end{array}$	$\{[0.1,0.2],[0.3,0.4]\}$
IVTrFS	2006	$\{[T{r}^{-},T{r}^{+}]=[(a_1^{-},a_2^{-},a_3^{-},a_4^{-};h_A^{-}),(a_1^{+},a_2^{+},a_3^{+},a_4^{+};h_A^{+})]\}$	$\begin{array}{c}\{[(0.01,0.2,0.4,0.5;0.7),\\ (0.1,0.2,0.5,0.6;0.9)]\}\\ \end{array}$
IVTFS	2011	$\{(a_1,a_2,a_3)=[(a_1^{-},a_1^{+});a_2;(a_3^{-},a_3^{+})]\}$	$[(0.45,0.55);0.7;(0.8,0.95)]$
IV2TLTS	2012	$\{[(s_i,{\alpha }_1),(s_j,{\alpha }_2)]|i\le j\}$	$[(s_1,0.1),(s_3,0.4)]$
IVHFS	2013	$\begin{array}{c}\{<x_i,h_A(x_i)>|x_i\in X,i=1,2,\dots ,n,\\ h_A(x_i)=\{[a_1^{-},a_1^{+}],[a_2^{-},a_2^{+}],\dots ,[a_n^{-},a_n^{+}]\}\}\\ \end{array}$	$\begin{array}{c}\{[0.2,0.3],[0.4,0.5],\\ [0.4,0.7]\}\\ \end{array}$
IVHFLTS	2014	$\begin{array}{c}\{[{(b_S^l)}^{-},{(b_S^l)}^{+}]|l=1,2,\dots ,\#b_S^l,\\ {(b_S^l)}^{-}\le {(b_S^l)}^{+},{(b_S^l)}^{-},{(b_S^l)}^{+}\in S\}\\ \end{array}$	$\{[s_{1.25},s_2],[s_{1.75},s_3]\}$
IVILTS	2015	$\begin{array}{c}\{<x,s_t,[{\mu }_A^{-}(x),{\mu }_A^{+}(x)],[v_A^{-}(x),v_A^{+}(x)]>|\\ x\in X,0\le {\mu }_A^{+}(x)+v_A^{+}(x)\le 1\}\\ \end{array}$	$\{s_2,[0.2,0.3],[0.6,0.7]\}$
IVPFS	2016	$\begin{array}{c}\{<x,[{\mu }_{\tilde{A}}^{-}(x),{\mu }_{\tilde{A}}^{+}(x)],[v_{\tilde{A}}^{-}(x),v_{\tilde{A}}^{+}(x)]>|\\ x\in X,0\le {({\mu }_{\tilde{A}}^{+}(x))}^2+{(v_{\tilde{A}}^{+}(x))}^2\le 1\}\\ \end{array}$	$\{[0.1,0.6],[0.3,0.7]\}$
IVPLTS	2018	$\begin{array}{c}\{L^k([{(p^k)}^{-},{(p^k)}^{+}])|L^k\in S,0\le {(p^k)}^{-}\le {(p^k)}^{+}\le 1,\\ \sum _{k=1}^{\#IL(p)}{(p^k)}^{+}\le 1,k=1,2,\dots ,\#IL(p)\}\\ \end{array}$	$\{s_1(0.2,0.3),s_2(0.4,0.6)\}$

The interval ${\mu }_A(x)$ and $v_A(x)$ denote the degree of belongingness and the nonbelongingness of the element $x$ to the set $A$. $\mathit{Tr}$ is a trapezoidal fuzzy number, and $a_1^{-}<a_2^{-}<a_3^{-}<a_4^{-}$, $a_1^{+}<a_2^{+}<a_3^{+}<a_4^{+}$, $h\in [0,1]$ is the height of $x$. $(a_1,a_2,a_3)$ is a triangular fuzzy number with $a_1<a_2<a_3$. $S=\{s_t|t=0,1,2,,g\}$ is a linguistic term set, $i=round(\beta )$ with $\beta$ representing the result of an aggregation of the indices of $S$. $\alpha =\beta -i$ so that $\alpha \in [-0.5,0.5)$. $h_A(x_i)$ denotes the possible membership degrees of $x$. $\#b_S^l$ is the number of linguistic terms. $L^k([{(p^k)}^{-},{(p^k)}^{+}])$ represents the linguistic term $L^k$ associated with its interval-valued probability $[{(p^k)}^{-},{(p^k)}^{+}]$, $\#IL(p)$ is the number of all different linguistic terms.

Fig. 2.

Representation of the inclusion relationships between different IVFSs

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 $E=\{E_i,i=1,2,3,\dots m\}$ be a set of decision alternatives and $C=\{c_i,j=$ $1,2,3,\dots ,n\}$ be a set of criteria. A decision matrix $E$ is an ($m\times n$) matrix with element $e_{\mathit{ij}}$. $e_{\mathit{ij}}$ 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,\dots n\}$. A decision matrix $E$ can de expressed as:

Criteria
Alternatives	$C_1$	$C_2$	$C_3$	$\cdots$	$C_n$
	($w_1$	$w_2$	$w_3$	$\cdots$	$w_n$)
$E_1$	$e_{11}$	$e_{12}$	$e_{13}$	$\cdots$	$e_{1n}$
$E_2$	$e_{21}$	$e_{22}$	$e_{23}$	$\cdots$	$e_{2n}$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$E_m$	$e_{m1}$	$e_{m2}$	$e_{m3}$	$\cdots$	$e_{\mathit{mn}}$

Through the matrix, the optimal alternative $O^{\ast }$ 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

MCDM methods	Acronym	Description	Complexity	References
Analytic Hierarchy Process	AHP	A structured method for organizing and analyzing complex decisions	$O(min\{m{n}^2,n{m}^2\})$	Saaty [56]
Analytic Network Process	ANP	Generalization of the AHP method which enables the existence of interdependencies among criteria	$O(min\{m{n}^2,n{m}^2\})$	Saaty [57]
Technique for order of preference by similarity to ideal solution	TOPSIS	A method of compensatory aggregation	$O(mn)$	Yoon and Hwang [58]
VlseKriterijumska Optimizacija I Kompromisno Resenje	VIKOR	A compromise approach of MCDM, determines the best alternative by ranking the closeness to the ideal solution	$O(mn)$	Opricovic [59]
An acronym in Portuguese for Interactive and Multicriteria Decision-Making	TODIM	A method by measuring the dominance degree of each alternative over the other alternatives using the overall value	$O(m^{2}n)$	Gomes and Lima [60]
Preference ranking organization method for enrichment of evaluations	PROMETHEE	Family of outranking methods based on the selection of a preference function for each criterion	$O(m^{2}n)$	Brans and Vincke [61]
Multi-objective optimization ratio analysis	MOORA	An objective method based on comprehensive values of alternatives on all criteria and the most regret value on one criterion	$O(mn)$	Brauers and Zavadskas [62]
Elimination and Choice Translating Reality	ELECTRE	An outranking method to sort out the best alternative by comparing each pair of actions based on several outranking relations	$O(m^{2}n)$	Benayoun et al. [63]
DEcision-MAking Trial and Evaluation Laboratory	DEMATEL	A method used to construct a network relation in order to examine the internal relation among attributes	$O(n^2)$	Yamazaki et al. [64]
Best–Worst Method	BWM	An MCDM method based on pairwise comparisons between the best or the worst criterion and other criteria	$O(n^2)$	Rezaei [65]
COmplex PRoportional ASsessment	COPRAS	A method used to assess the maximum and minimum index values. The effect of maximum and minimum indexes of attributes on the assessment results is considered separately	$O(mn)$	Zavadskas et al. [66]
Weighted Aggregates Sum Product Assessment	WASPAS	A combination of weighted sum model and weighted product model	$O(mn)$	Zavadskas et al. [67]

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.

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.

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

Table 3.

Publication trend and citation structure from 2007 to January 2022

Year	TP	TC	TC/TP	Publications with citations
				>  = 150	>  = 100	>  = 50	>  = 25	>  = 1	Total
2007	1	10	10.00	0	0	0	0	1	1
2008	2	140	70.00	0	0	2	0	0	2
2009	2	250	125.00	1	0	0	0	1	2
2010	5	474	94.80	1	1	0	1	2	5
2011	9	846	94.00	2	0	5	2	0	9
2012	3	133	44.33	0	1	0	0	2	3
2013	9	743	82.56	1	1	2	1	4	9
2014	11	604	54.91	1	1	2	3	4	11
2015	11	351	31.91	0	2	0	2	7	11
2016	20	1312	65.60	2	3	1	6	8	20
2017	15	289	19.27	0	0	0	5	10	15
2018	16	435	27.19	0	0	0	8	8	16
2019	18	322	17.89	0	0	1	5	12	18
2020	39	452	11.59	0	0	0	7	31	38
2021	33	69	2.09	0	0	0	0	22	22
2022	1	0	0.00	0	0	0	0	0	0
Total	195	6430	751.13	8	9	13	40	112	182

TP total number of publications, TC total number of citations, TC/TP average citations per publication

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.

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

Country/Region	Publications	Proportion (%)	Citations	Average citations
China	100	51.28	3219	32.19
India	22	11.28	708	32.18
Iran	22	11.28	934	42.45
Taiwan	20	10.26	476	23.8
Turkey	14	7.18	211	15.07
USA	13	6.67	550	42.31
Lithuania	10	5.13	390	39
Spain	7	3.59	173	24.71
Malaysia	6	3.08	156	26
Pakistan	4	2.05	34	8.5

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

Institution	Country/Region	Publications	ARWU	QS
Islamic Azad University	Iran	14	–	–
Chang Gung University	Taiwan	11	401–500	480
Vilnius Gediminas Technical University	Lithuania	11	901–1000	751–800
Hong Kong Polytechnic University	China	9	151–200	66
National Institute of Technology Nit System	Bangladesh	9	–	–
Shanghai University	China	9	201–300	436
Sichuan University	China	9	151–200	451
Central South University	China	8	151–200	521–530
Fuzhou University	China	7	301–400	–
Istanbul Technical University	Turkey	6	901–1000	701–750
National Taiwan University of Science Technology	Taiwan	6	901–1000	314
Xidian University	China	6	401–500	-
Chongqing University	China	5	201–300	651–700
National Institute of Technology Tiruchirappalli	India	5	–	–
North China Electric Power University	China	5	–	–
Tongji University	China	5	151–200	211
China Jiliang University	China	4	–	–
Henan Agricultural University	China	4	–	–
King Abdulaziz University	Saudi Arabia	4	101–150	109
Northeastern University China	China	4	301–400	–

ARWU Academic Ranking of World Universities, QS Quacquarelli Symonds World University Rankings

Fig. 4.

Collaboration network between institutions

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

Title	Author(s)	Journal	Year	Citations
Best–worst multi-criteria decision-making method: Some properties and a linear model [73]	Rezaei	Omega	2016	471
Hesitant fuzzy multi-attribute decision-making based on TOPSIS with incomplete weight information [74]	Xu and Zhang	Knowledge-Based Systems	2013	379
TOPSIS-based nonlinear-programming methodology for multiattribute decision-making with interval-valued intuitionistic fuzzy sets [75]	Li	IEEE Transactions on Fuzzy Systems	2010	284
Extension of VIKOR method for decision-making problem with interval numbers [76]	Sayadi, Heydari and Shahanaghi	Applied Mathematical Modelling	2009	230
Extension of the TOPSIS method for decision-making problems under interval-valued intuitionistic fuzzy environment [77]	Park et al.	Applied Mathematical Modelling	2011	228
A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems [78]	Garg	Applied Soft Computing	2016	213
Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets [79]	Nayagam, Muralikrishnan and Sivaraman	Expert Systems with Applications	2011	180
The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment [80]	Zhang and Xu	Knowledge-Based Systems	2014	160
Extension of weighted aggregated sum product assessment with interval-valued intuitionistic fuzzy numbers (WASPAS-IVIF) [81]	Zavadskas et al.	Applied Soft Computing	2014	147
Extended TODIM method for hybrid multiple attribute decision-making problems [82]	Fan et al	Knowledge-Based Systems	2013	144

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, 77–79, 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.

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.

Keyword co-occurrence network of publications

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.

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.

Application proportion of interval techniques

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

Categories	Extensions	Applications in conjunction with interval techniques
Interval numbers
Fuzzy sets	FS	IVFS
	IFS	IVIFS, IV intuitionistic TrFS (IVITrFS), IVIHFS, Complex IVIFS (CIFIFS)
	HFS	IVHFS, IVPHFS, Probabilistic dual HFS
	PFS	IVPFS
	TFS	IVTFS, Three trapezoidal fuzzy number bounded type-2 FS
	TrFS	IVTrFS, Generalized IVTrFS
	GFS	IV Gaussian type-2 FS
	SFS	IVSFS
	FFS	IVFFS
Linguistic sets	LTS	IVLTS
	PLTS	IVPLTS
	ILTS	IVILTS, IV intuitionistic uncertain LTS
	HFLTS	IVHFLTS, IVDHHFLTS, IV hesitant uncertain LTS
	PFLTS	IVPFLTS, IV Pythagorean uncertain LTS
Others	NS	Interval NS
	SS	IVSS
	IMS	IVIMS
	RS	IVRS
	ER	Interval ER

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.

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 [140–142]. 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

Environment	Category	MCDM methods	References
IVFS	Single	TOPSIS	Sharaf [88]
		TODIM	Mohagheghi [89]
		DEMATEL	Wang et al. [90]
	Hybrid	AHP + TOPSIS	Pires et al. [91]
		ELECTRE III + BWM	You et al. [92]
IVIFS	Single	TODIM	Zindani et al. [93]
		TOPSIS	Wang et al. [94]; Aikhuele and Turan [95]; Bu et al. [96]; Liu et al. [97]; Zhang and Yu [98]; Li et al. [99]; Hajek and Froelich [100]
		AHP	Zheng et al. [101]; Abdullah and Najib [102]
		WASPAS	Gireesha et al. [103]; Zavadskas et al. [81]
		MOORA	Thao [104]
		MULTIMOORA	Zavadskas et al. [105]
		MABAC	Xue et al. [106]
		VIKOR	Krishankumar et al. [107]; Du and Liu [108];
	Hybrid	AHP + TOPSIS	Dogan et al. [109]
		AHP + COPRAS	Perçin [110]
		AHP + WASPAS	Alimohammadlou and Khoshsepehr [111]
		ANP + COPRAS	Wang et al. [112]
		ANP + TODIM	Li et al. [113]
		TOPSIS + VIKOR	Dammak et al. [114]
		PROMETHEE + ELECTRE	Chen [115]
		TOPSIS + TODIM + WASPAS	Davoudabadi et al. [116]
CIVIFS	Single	TODIM	Zindani et al. [117]
IVTrFS	Single	VIKOR	Vahabzadeh et al. [118]
		BWM	Ali and Rashid [119]
	Hybrid	ANP + TOPSIS	Skondras et al. [120]
IVITrFS	Single	VIKOR	Liu et al. [121]; Du and Liu [108]
		CODAS	Seker [122]
IVIHFS	Single	VIKOR	Wang et al. [123]
IVHFS	Single	ELECTRE III	Niu et al. [124]
		QUALIFLEX	Zhang [125]
		CODAS	Karasan et al. [126]
		TODIM	Zhang and Xu [80]
IVPFS	Single	FDOSM	Al-Samarraay et al. [127]
		MULTIMOORA	Liang et al. [128]
		EDAS	Yanmaz et al. [129]
		COPRAS	Haktanir [130]
	Hybrid	AHP + TOPSIS	Boyaci and Sisman [131]
		ANP + TOPSIS	Mohandes and Zhang [132]
		AHP + TOPSIS + VIKOR	Aslantas et al. [133]
IVTFS	Single	VIKOR	Wang and Wang [134]
	Hybrid	AHP + VIKOR	Singh et al. [135]
		BWM + TOPSIS	Mohandes et al. [136]
		ANP + TOPSIS + VIKOR	Vahdani et al. [137]
IV type-2 GFS	Single	TODIM	Tolga et al. [43]
IVFFS	Single	TOPSIS	Jeevaraj [47]
IVSFS	Single	TOPSIS	Farrokhizadeh [46]

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 [128–130]. Hybrid studies used TOPSIS and other kinds of methods such as AHP and ANP to weight criteria and rank alternatives [131–133].

• 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 [134–137]. 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.

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.

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 (IV^qROFSSs). 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.

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.

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.

Proportion of interval techniques applied in main ranking methods

Table 9.

Interval-valued MCDM methods with application fields

MCDM method	Information representation	Categories	Industry of applications
AHP	FS	FS	Waste management system selection [91]
		IVIFS	Enterprise ranking [111], Supplier selection [110]
		IVPFS	Risk assessment in health sector [133]
		IVTFS	Strategy selection for sustainable manufacturing [135]
	IN	IN	Hydrogen technology competitiveness [169], Supplier selection [170], Desalination system selection [171], Earth dam risk identification [172], Open pit equipment selection [173]
ARAS\PIPRCIA	FS	IVTFS	E-course selection [174]
COCOSO	IN	IN	Material selection [175], Supplier selection [176]
CODAS	FS	IVIFS	Investment [122], Hospital fire risk assessment [177]
		IVHFS	Residential location site selection [99]
COPRAS	FS	IVIFS	Hospital service evaluation [112], Supplier selection [110]
		IVPFS	Supplier selection [130]
	IN	IN	Material selection [178]
EDAS	FS	IVPFS	Car selection [129]
	SS	IVSS	Economic policy selection [179]
	IN	IN	Bank ranking [180], Ranking of hydrogen production paths [181]
ELECTRE	IN	IN	Hydrogen production paths [182], Structural system selection [183]
	NS	INS	Selection of renewable energy alternatives [161]
	RS	IVRRV	Site selection [41]
	FS	IVFS	Cultural facility location [92]
		IVHFS	Selection of renewable energy alternatives [124]
PROMETHEE	LS	IVPLTS	Wind farm site selection [184]
	IN	IN	Selection of travel plan [185], Supplier selection [170], Reliability allocation of mechanical products [186]
GLDS	LS	IVDHHFLTS	Corporate performance [154]
GRA	FS	IVTFS	Evaluation of system analysis engineers [35]
		IVFS	Garbage disposal solution [90], Site selection [187]
		IVPFS	Safety risk assessment for a sustainable construction project [132]
	IN	IN	Selection of energy systems [188], sustainability assessment of Polygeneration systems [189], Project assessment [190]
MABAC	IN	IN	Evaluation of construction waste utilization schemes [167]
	FS	IVIFS	Material selection [106]
MULTIMOORA	FS	IVPFS	Selection of hospital open-source electronic health record systems in Ghana [128]
		IVIFS	Revitalization of derelict and mismanaged buildings in Lithuanian rural areas [105]
	IN	IN	Material selection [191], Hybrid vehicle engine selection [87]
ORESTE	LS	IVLTS	Selection of innovative sharing bike design for “Mobike” sharing bikes [155]
QUALIFLEX	FS	IVHFS	Green supplier selection [125]
		IVIFS	Selection of bridge construction method [192]
	LS	IVPULTS	Robot selection [158]
SPA	FS	IVIFS	Enterprise selection for corporate union [193]
TODIM	FS	IVHFS	Evaluation of the service quality among domestic airlines [80]
		IVIFS	Supplier selection [116], Design of a large tonnage crawler crane [99], Selection of new machine [117], evaluation of community question-answering websites [113]
		IVGFS	Healthcare device selection [43]
	IN	IN	Location of urban photovoltaic charging station [194], supplier selection [82], Emergency decision of gas explosion [165], residential properties evaluation [195]
WASPAS	FS	IVIFS	Investment [81], Supplier selection [103, 116], Enterprise selection [111]
		IVHFS	Thermal energy storage technology [196]
	NS	INS	Robot decision-making [160]
TOPSIS	FS	IVIFS	Investment [75, 97, 98, 114]
			Supplier selection [94, 100, 197]
			Reliability evaluation of complex systems [96], selection of air conditioning system in a library [77], selection of shipping partner [95]
		IVTFN	Assessment of property responsibility insurance companies [137], Assessment of construction labours’ safety level [136]
		IVFFS	Investment [47]
		IVSFS	Advertising strategy choices [46]
		IVFS	Network selection [88], Waste management system selection [91]
		IVTrFS	Robot selection [198], Network access selection [120]
		IVHFS	Energy policy selection [74]
	IN	IN	Supplier selection [199]
			Desalination system selection [171, 200], Polygeneration system ranking [189]
			Hydropower project risk identification [172]
	LS	IVILTS	Hot dry rock exploration and development location [151]
	SF	IVPFS	Health sector risk assessment [133], Construction worker safety [132], Hospital site selection [131]
	NS	INS	Student assessment [159], Investment [201]
VIKOR	FS	IVIFS	Investment [114], Personnel assessment [107]
			Supplier selection [121, 202]
		IVTFN	Sustainable strategic selection [135], Insurance corporate performance [137], Mechanical failure detection [134]
		IVPFS	Health sector risk assessment [133]
		IVTrFS	Robot selection [198]
	LS	IVPFLTS	Site selection problem for a manufacturer [157]
	IN	IN	Weapon system [203, 204], Energy system selection [205], Machine selection [206], Open pit equipment selection [173], Material selection [207], Bank performance [208]
Programming	FS	IVIFS	Investment [139, 209]
			Supplier selection [94, 210]
			Bridge construction method [211], Telephone selection [212]
		IVFS	Venture capital [213]
	ER	IVER	Diesel generator performance evaluation [48]
Score function and aggregation operators	FS	IVIFS	Supplier selection [214], Car selection [215], Investment [69, 79, 216, 217], Partner selection with collaborative innovation [218], Smash circuit design selection [219], Watershed site selection [220], Platform entrepreneurial orientation assessment [221]
		IVFS	Medical decision-making [222]
	IN	IN	Investment strategy selection for a bank [223], military UAV selection [224]
	LS	IVPLTS	Assessment of cloud vendors [156]
		IVILTS	Supplier selection [149], Investment [150]
		IVHLS	Medical device selection [153]
		IVPFLTS	Manufacturer site selection [157]

Additive Ratio ASsessment (ARAS), PIvot Pairwise Relative Criteria Importance Assessment (PIPRECIA), COmbined COmpromise SOlution (CoCoSo), Evaluation based on Distance from the Average Solution (EDAS), Grey Relational Analysis (GRA), Data Envelopment Analysis (DEA), ORESTE (organísation, rangement et Synthèse de données relarionnelles, in French), QUALItative FLEXible multiple criteria method (QUALIFLEX), Set Pair Analysis (SPA); Interval-valued rough random variables (IVRRV), Interval-valued Pythagorean uncertain linguistic term set (IVPULTS)

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.

Application fields related to interval MCDM

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. 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. 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. 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. 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:

• (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.
• (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.
• (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:

• (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].
• (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.

• (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.

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.

Biographies

Huchang Liao

is a Research Fellow at the Business School, Sichuan University, Chengdu, China. He received his PhD degree in Management Science and Engineering from the Shanghai Jiao Tong University, Shanghai, China, in 2015. He has published four monographs, and more than 320 peer-reviewed papers, many in high-quality international journals including European Journal of Operational Research, Omega, IEEE Transactions on Fuzzy Systems, IEEE Transactions on Engineering Management, IEEE Transactions on Systems, Man, and Cybernetics: Systems, IEEE Transaction on Cybernetics, etc. His publications have been cited over 15,000 times, and his h-index is 67. He has been a Highly Cited Researcher in Computer Science (2019–2022), and a Highly Cited Chinese Researchers in Management Science (2020–2021). He ranked within the top 2% Ranking of Scientists in the World by Stanford University (2020–2022). His main research interests include multiple criteria decision analysis, group decision analysis, fuzzy decision analysis, mechine learning-based decision analysis and medical decision analysis. Prof. Liao has been elected to be the Fellow of IET (The Institution of Engineering and Technology) and the Fellow of BCS (British Computer Society). He is the Area Editor of International Journal of Information Technology & Decision-Making (SCI), Associate Editor, Guest Editor or Editorial Board Member for many top-tier international journals, including IEEE Transactions on Fuzzy Systems (SCI), Information Fusion (SCI), Applied Soft Computing (SCI), Omega (SSCI/SCI), Technological and Economic Development of Economy (SSCI), International Journal of Strategic Property Management (SSCI), Engineering Applications of Artificial Intelligence (SCI), International Journal of Fuzzy Systems (SCI) and Journal of Intelligent and Fuzzy Systems (SCI).

Jiayi Wang

is now pursuing her Master Degree in Management Science and Engineering at Sichuan University, Chengdu, China. Her current research interests include multiple criteria decision-making, fuzzy information and online consumer reviews.

Ming Tang

is currently pursuing his Ph.D. degree in Management Science and Engineering with the Sichuan University, Chengdu, China. He has published more than 30 peer-reviewed papers, many in high-quality international journals, including European Journal of Operational Research, Omega, IEEE Transactions on Cybernetics, IEEE Transactions on Systems, Man, and Cybernetics: Systems, Reliability Engineering and System Safety, Technological Forecasting & Social Change, Journal of the Operational Research Society, Fuzzy Optimization and Decision-Making, Information Fusion, Information Sciences, and Knowledge-Based Systems, etc. His publications have been cited over 950 times, and his h-index is 15. His current research interests include collective intelligence and group decision-making. Mr. Tang is an Editorial Board Member of the ICSES Transactions on Neural and Fuzzy Computing, Journal of Artificial Intelligence Evolution and Researches in Science and Technology: Open Access. He was a recipient of the National scholarship for graduates of China Awards in 2018, 2019 and 2021. He won the Baosteel Scholarship in 2019 and 2021, and Tang Lixin scholarship in 2020. He was the "Top Ten Academic Stars" of graduate student of Sichuan University in 2019.

Abdullah Al-Barakati

is a Professor at King Abdulaziz University, Saudi Arabia. Prior to that, he was the head of the Information Systems Department at King Abdulaziz University, Saudi Arabia. He was also appointed as the general supervisor of the IT department of Makkah Province in Saudi Arabia in the period 2015–2021. He received a B.Sc. (Hons) degree in Computer Science in 2004, a M.Sc. degree in Software Engineering and a Ph.D. degree in Computer Science in 2012, all from University of Sussex, United Kingdom. His research interests revolve on the utilization of Big Data and Web technologies for innovative solutions and services. He has published more than 50 peer-reviewed research papers on Fuzzy System, Data Mining, social media analytics, big data and digital heritage.

Appendix

See Tables 10, 11.

Funding

The work was supported by the National Natural Science Foundation of China (72171158, 71971145, 71771156).

Data Availability

The authors declare that there is no any issue about data availability.

Declarations

Conflict of interest

The authors have no competing financial, professional, or personal interests from other parties that are related to this paper.

Ethical Approval

There is not any ethical issue for this paper.