Uncovering the academic evolution of VIKOR method: a comprehensive main path analysis

Abstract

The rapid advancement of the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method has led to a substantial body of related literature. Reviewing and extracting the academic trajectory can be challenging, as conventional literature review studies and bibliometric approaches may prove ineffective. This paper proposes a comprehensive main path analysis (MPA) framework to unveil the knowledge structure within the field of VIKOR. By integrating global MPA, local MPA, and key-route MPA, this study reveals the significant developmental process and pivotal papers in the VIKOR domain. Employing multiple global main paths offers a more holistic understanding of the knowledge evolution network in VIKOR, enabling deeper exploration of research frontiers. To our knowledge, this is the first study to conduct an extensive MPA in the field of VIKOR, providing novel insights for scholars interested in comprehending VIKOR and inspiring future research.

Keywords:

• VIKOR
• main path analysis
• academic evolution
• knowledge structure
• research fronts

1. Introduction

In social and economic activities, decision-making problems often face conflicts between rationality and reality. The early normative decision-making theory, also known as classical statistical decision theory, emphasized the rational individual’s ideal decision-making method to achieve the best solution. In real-world decision-making problems, the conflicts and incommensurability between decision criteria and goals often require subjective intervention by decision-makers, and non-rational factors such as experience and emotions can also affect decision-making (Simon, 1960). Therefore, for decision-makers with limited rationality, decision-making problems do not have optimal solutions, only satisfactory decision-making methods. Based on this, Kahneman and Tversky (1979) introduced uncertainty factors into traditional decision-making processes and studied the subjective decision-making behavior of decision-makers, providing an explanation for their decision-making behavior under uncertain risks: prospect theory. These theories that focus on specific conditions in actual decision-making problems are called descriptive decision-making theories. These two decision-making theories together form the theoretical basis for multi-criteria decision-making methods, which use normative methods to approximate ideal solutions based on real decision-making conditions. Real-world decision problems often face uncertain factors such as multiple decision objectives, incomparability of decision alternatives, conflicting evaluation criteria, and individual subjective preferences. Multi-criteria decision-making (MCDM) methods provide a realistic decision-making solution in a multi-criteria, multi-objective context.

The origin of MCDM theory can be traced back to the concept of Pareto optimal solution (Pareto, 1897). The systematic introduction of MCDM into the decision-making field is represented by goal programming research (Charnes & Cooper, 1962) and the ELECTRE method (Roy 1968). Depending on whether the number of decision alternatives is finite or infinite, MCDM theory can be divided into multi-objective decision-making (infinite continuous) and multi-attribute decision-making (finite discrete) (Hwang & Yoon, 1981). The former is usually analyzed using matrix optimization methods, while Lahdelma et al. (2000) categorizes multi-attribute decision-making methods into two categories: methods based on utility functions values and methods based on advantage ranking. Among the MCDM methods based on utility function values, the most famous include (Rezaei, 2015; Saaty, 1980; 2005): Analytic Hierarchy Process (AHP), Network Analysis Process (ANP) and Best Worst Method (BWM). Although such methods are widely used, they still have the defects of strong subjectivity and inability to provide new solutions for decision makers. Decision-making methods based on advantage ordering can be subdivided into rank ordering method and distance ordering method. The former can determine the ordering of decision schemes by evaluating the superiority between alternative schemes in the context of complex criteria decision-making. The most widely used rank ordering decision-making methods are ELECTRE method proposed by Roy (1966) and PROMETHEE method proposed by Brans et al. (1986). Distance ordering decision-making methods identify the most suitable alternative schemes by the degree of proximity between each scheme and the ideal solution, such as TOPSIS method and VIKOR method (Hwang & Yoon, 1981; Opricovic, 1998).

With the complexity of decision-making problems, such as the subjective preferences of decision makers and the hesitation or fuzziness of decision-making behaviors. Researchers began to introduce fuzzy theory into the field of MCDM (Bellman & Zadeh, 1970) to solve decision-making problems in fuzzy environments, while natural language processing technology provides solutions for analyzing the subjective preferences of decision makers. For example, Gou et al. (2023) introduced the two-level language preference ordering based on ORESTE method to study the medical resource allocation problem in the context of expert opinions. Opricovic (2007) proposed a fuzzy extended model of VIKOR method. Gou et al. (2017) proposed a hesitant fuzzy linguistic alternative queuing method to support the MCDM problems under the subjective cognitive background, and validated the method’s reliability in a medical system management case. Another trend in the development of MCDM theory is the integration of different decision-making methods, such as: Wang and Chen (2017) combined linear programming methods and TOPSIS methods to propose a multiple attribute decision-making method based on interval-valued intuitionistic fuzzy sets. Lin et al. (2021) developed a method that integrates multiplicative analytic hierarchy process and TOPSIS to handle multi-attribute decision-making problems under probabilistic linguistic term sets.

There are many MCDM methods, and their trends and application scenarios also differ significantly. For example, multi-attribute decision-making methods are suitable for evaluation and selection problems, while multi-objective decision-making methods are more suitable for design-type problems. Many scholars have also conducted the bibliometrics analysis to explore the innovative development direction of MCDM methods, such as: Dai et al. (2022) used a knowledge graph analysis to study the application of MCDM methods in the medical field. Zyoud and Fuchs-Hanusch (2017) conducted a bibliometric study on the AHP method and TOPSIS method to investigate the current application status and method innovation trends of these methods. Yu et al. (2023) conduct a bibliometrics research in the field of PROMETHEE method, revealing the knowledge evolution of PROMETHEE in from a longitudinal and dynamic perspective. Chowdhury and Paul (2020) used tradition bibliometrics approaches to analyze the applications of MCDM method in the field of corporate sustainability. Bortoluzzi et al. (2021) combined statistical reasoning method, cluster analysis and keyword co-occurrence analysis to research the application of hybrid MCDM frameworks in the renewable energy technologies field. Morkūnaitė et al. (2019) used VOSview to analyze papers that adopt MCDM methods in heritage building conservation.

Distance-based sorting MCDM methods make full use of data information, and this quantitative MCDM method can more accurately evaluate the gap between the advantages and disadvantages of different decision-making schemes, making it the most popular MCDM method in the decision-making field at present. Compared to the TOPSIS method, the VIKOR method emphasizes a better compromise solution, which is more suitable for the current complex, conflicting, and incommensurable decision-making problems. The method has also become a research focus for relevant scholars, and has been widely adopted in various fields. The flourishing development of VIKOR methodology has led to an increasing number of scholars dedicating their efforts to applying this approach in different industries and developing novel VIKOR methods, resulting in a large number of relevant papers. Tradition literature reviews are inadequate for effectively handling the large volume of research on the VIKOR methodology. Moreover, the most important factor affecting the reliability of a review study is the subjective judgment of the author regarding the VIKOR theory, which can limit the scope of relevant studies. Bibliometrics methods are effective in identifying the development trend, research hotspot, key scholars, cooperative relationship in a specific field. However, traditional bibliometrics methods are limited in their ability to objectively reveal the specific development route of a given academic field and accurately identify key papers. Therefore, this paper introduces the MPA method to research the knowledge development trace of the VIKOR method.

Compared to traditional bibliometrics methods, MPA technology has four advantages. Firstly, the MPA method can intuitively reveals the development track of a certain field. Secondly, it accurately identify important papers, enabling exploration of academic evolution details. Thirdly, the results of the MPA method based on citation relationship is more objective. Thus, we develop a comprehensive main path analysis (MPA) framework and adopt this bibliometric method in VIKOR field to analyze the knowledge diffuse process and provide insight for future.

The paper is organized as following processes: After introduction, the concept of VIKOR, methodology and research framework are introduced in Section 2; In Section 3, the results of four MPA methods are provided; Finally, the discussion and finding are concluded in Section 4.

2. Background

The concept of VIKOR and MPA method are introduced in this section. In addition, the data process and relevant details are provided to ensure reproducibility of experimental results.

2.1. VIKOR

VIKOR proposed by Opricovic and Tzeng is a MCDM techniques that provide a compromise solution for conflicting attributes (Opricovic, 1998; Opricovic & Tzeng, 2002). In reality, tasks or projects often require the selection of a suitable solution from different alternatives with non-commensurable or conflicting attributes (Büyüközkan & Tüfekçi, 2021; Lin et al., 2021; Tavana et al., 2018). Moreover, decision makers are often unable to suggest precise individual preferences that could lead to different outcomes, and VIKOR is effective in solving this problem.

The VIKOR method consists of five steps, as illustrated in Fig. 1. The specific processes involved in the method are described below.

Figure 1. The theoretical framework of VIKOR method.

• Step 1: Constructing the performance matrix with the attributes of research objects; Providing a weight vector according to the preferences of decision makers or the specific decision-making techniques.

• Step 2: Using aggregating function $L_p$-metric to measure the group utility S and the individual regret R.

• Step 3: Computing the compromise decision index Q.

• Step 4: According to the values of S, R and Q, ranking the alternatives; Choosing the alternative $A\prime$ with the minimum Q value; Proposing the compromise solution according to $A\prime$ if the alternative meets the following two conditions:

1. Acceptable advantage: (1) $$Q\left(A″\right)-Q\left(A^{\prime }\right)\ge \mathit{\text{DQ}}$$(1) (2) $$\mathit{\text{DQ}}=1/(m-1)$$(2) $A″$ is the next best object in the ranking list.

2. Acceptable stability in decision-making:

   The alternative $A\prime$ must be the best in S ranking list or/and R ranking list.

2.2. Main path analysis method

Citation network is commonly used to study the development or structure of knowledge in the given field. It consists of nodes representing academic research papers and edges representing citation relationships among them. The links which point from the previous nodes to the later ones can mean the inherited relationship of knowledge among different academic papers (Liu et al., 2013). The flow of knowledge has greatly promoted academic development and innovation. This knowledge inheritance and development also lays the foundation for evaluating the prestige of scholars or academic institutions. Related researchers proposed many evaluation index based on citation. For example, H-index, G-index, PageRank, global citation score (GCS), local citation score (LCS) and so on.

Hummon and Dereian (1989) introduced the MPA in their research of a DNA citation network, using it to uncover the theoretical structure and knowledge evolution path of the DNA field. MPA can extract the most important knowledge evolution path based on the importance of citation links. This method was widely applied in bibliometrics studies to reveal the knowledge structure and the development trace of the given academic field. Hung et al. (2014) used the MPA method to research the technological diffusion trace of the field of lithium iron phosphate (LFP) battery, illustrating that LFP battery technology was in the third technology cycle. Yu and Yan (2024) used main path analysis methods to research the knowledge diffusiontrajectories of PageRank. Xu et al. (2020) used the MPA method to study the knowledge evolution path of the emerging research topics field, and observed some research drifts. Yu and Pan (2021) explored the knowledge evolution of TOPSIS with MPA method. Yu et al. (2022) explored the evolutionary process in intuitionistic fuzzy set theory with a MPA research framework. The process of MPA method can be divided into the following steps: First, creating a network according to the citation information of chosen research papers; Second, weighting the links of the citation network; And third, adopting different pathfinding strategies to extract main path of different levels from the citation network.

More details would be provided to introduce the last two steps. Relevant scholars proposed different algorithms for weighting the citation links. Hummon and Dereian (1989) introduced three indices: NPPC (Node Pair Projection Count), SPNP (Search Path Nodes Pair) and SPLC (Search Path Link Count). Batagelj and Mrvar (1998) introduced a new weight measuring method, i.e., SPC (Search Path Count). Compared to previous weighting methods, the SPC algorithm has better running speed and calculation rate. Therefore, we adopted SPC as weighting algorithm in the study. The SPC weight value of a citation is the times the edge has been traversed from all sources (the start nodes) to all sinks (the end nodes). We use a simple network to explain the process of counting SPC weighting value as shown in Fig. 2(a). This network includes two sources (A and B) and four sinks (E, I, J and K). For example, the weight of link D-G is three because there are three paths contain this link, i.e., B-D-G-H-J, B-D-G-J and B-D-G-H-K.

Figure 2. Different main paths based on a simple citation network.

After the second step, we get a weighted citation network. On this basis, the third step is conducted, namely main path extracting. There are mainly 4 different main path searching algorithms, namely local MPA searching algorithm, global MPA searching algorithm, key-route MPA searching algorithm and multiple MPA searching algorithm. In the same way, these searching algorithms are demonstrated in a simple citation network. Local main path searching algorithm is the original one, it was proposed by Hummon and Dereian (1989). This algorithm starts from all the sources and each subsequent pathfinding selects the link which has the highest weight until pathfinding meets sinks. As shown in Fig. 2(b), paths B-D-G-H-J and B-D-G-H-K are the local main paths (forward) in the network. The other three searching algorithms were introduced in Liu et al. (2012) and Liu et al. (2013). Global main path searching algorithm pays more attention to the holistic weighting value of main paths, it can extract the paths which have the highest weight in the given citation network. As shown in Fig. 2(c), the global main path is composed with A-C-G-H-K, A-C-G-H-J, B-C-G-H-K and B-C-G-H-J. The first two algorithm cannot contain all key-routes (the links which have the highest weight), thus they may miss some important researches. Aiming at this issue, key-route main path searching strategy was proposed in Liu and Lu (2012). It firstly identifies all links with the highest weight, then starting from these key-routes, searching forward and backward as same as local main path or global main path searching algorithms. The key-route main path in the sample citation network is shown in Fig. 2(d).

In order to explore more information of different levels, multiple main path method was developed by Liu et al. (2013). Besides the most important path, this searching method could identify the significant paths in different levels (these paths may not have the highest weight, they are still valued to be researched). In essence, the multiple main path method enables us to observe more important researches from various levels, thus providing insights into the development and structure of a given academic field.

2.3. Data process

The accuracy and reliability of data analysis heavily rely on proper data processing. As the concept of VIKOR was proposed by Opricovic in 1998, we set the time span for this study to cover the period from 1998 to 2021. As shown in Fig. 3, the advanced query is designed as follow: TS= (“VIKOR”) OR TS = (“VlseKriterijumska Optimizacija I Kompromisno Resenje”). The document formats are set to article and review, and databases are limited to SCI-Expanded or SSCI to ensure the reliability of data quality. The original collected dataset includes 1794 papers. After deduplicating in Histcite, there are 1793 papers left. The founding paper is not recorded in WOS because it is a thesis. After adding this paper, the final citation network contains 1794 papers. Histcite is used in this study to structure the citation network, it might fail to identify papers or its citations because of the format of citation information. Therefore, it is necessary to check the dataset manually in histcite and adding relevant papers or citations. Finally, local MPA, global MPA and key-route MPA are applied to trace the evolution trajectory of VIKOR. We also explore the structure of the given field with multiple MPA method.

Figure 3. Flow chart of research process.

3. Result

In this section, four different MPA methods are used to uncover the evolution trajectory of the field of VIKOR. Firstly, we employ a global MPA to capture the most significant developmental trajectory of the VIKOR field from a comprehensive perspective. Essentially, we extract the fundamental essence of the field through this global MPA approach. Secondly, Local MPA provides us more detailed information regarding the evolutive paths of the evolutive details, especially the directions of up-to-date researches. To ensure the inclusion of all key citations in the evolutive paths, key-route MPA is also applied in this study. Thirdly, multiple global main paths method is adopted to explore the academic evolution structure of VIKOR under a broader perspective. In the main path diagrams, sources are denoted in red and sinks in blue to visually represent their significance. The width of the link is proportionate to its strength, specifically measured by the SPC value. Furthermore, papers that deviate from those previously encountered on the main paths are highlighted in yellow.

3.1. Global MPA of VIKOR

As illustrated in Fig. 4, the global main path containing 22 papers represents the most significant evolutive trajectory of VIKOR. Opricovic (1998) was the first to propose the compromise ranking method, VIKOR, for the MCDM optimization of civil engineering system. Later, Opricovic and Tzeng (2002) employed the VIKOR method in conjunction with a fuzzy multicriteria model to propose a comprehensive solution plan for post-disaster recovery efforts, effectively addressing reconstruction challenges in disaster-prone regions and global security concerns. In a study of alternative fuel technologies for buses, Tzeng et al. (2005) used both TOPSIS and VIKOR to identify the best mode of compromise alternative fuel. The distinguishing feature of VIKOR is its ability to maximize group benefit while minimize individual regret over opposing opinions, as well as incorporating individual preferences of decision makers. In summary, VIKOR provides decision makers with a compromise solution that addresses conflicting and non-commensurable criteria, thereby enhancing its practical applicability compared to previous methodologies. Undoubtedly, Opricovic and Tzeng have made significant contributions to the field of decision making by introducing this method and paving the way for further research in MCDM.

Figure 4. Global main path.

The following six papers used the VIKOR method in different application fields. Wu et al. (2009) proposed a comprehensive fuzzy MCDM method combining balanced scorecard theory, FAHP and VIKOR to evaluate the finance performance in the bank field. Through applying VIKOR method with grey relational analysis (GRA) techniques and FAHP, Kuo and Liang (2011) developed a fuzzy MCDM method aiming at the evaluation of service quality in airports. Mousavi et al. (2013) proved the effectiveness of a multi-attribute group decision-making technique consisting of fuzzy set, stochastic approaches, and VIKOR in a risk selection problem of a highway project. Ebrahimnejad et al. (2012) proposed an approach that integrated a modified analytic network process (ANP) and VIKOR method, which was verified to be effective and feasible in a case of construction industry project selection. Liu et al. (2012) applied VIKOR to failure mode and effects analysis (FMEA) and proposed a fuzzy FMEA model, demonstrating its application in assessing general anesthesia process risk. In the environmental field, Kim and Chung (2013) used fuzzy VIKOR to assess how much the climate change and variability may affect the water supply in South Korea.

Given the inherent advantage of linguistic terms in effectively conveying decision makers’ judgments, a novel approach that integrates VIOKR and linguistic variables has been developed for diverse decision-making domains. Liu et al. (2014) introduced an interval 2-tuple linguistic VIKOR model to address the site selection issue in the city waste management field. You et al. (2015) extended the above VIKOR method to supply chain management field, demonstrating its effectiveness in handling supplier selection issue under fuzzy environment. The extended VIKOR method, considering linguistic variables, was also applied to power industry to evaluate candidate power restoration schemes, as shown in the work of Sun et al. (2015).

The VIKOR method has found extensive application in operation and management field. To better understand and interpret the research based on VIKOR, Gul et al. (2016) conducted a literature review, which included classification, analysis, and exploration of the extension of the VIKOR method in a fuzzy environment. Building on this review, Soner et al. (2017) studied the application of VIKOR and AHP methods under type 2 fuzzy environment, demonstrating its effectiveness in a hatch cover design selection issue in the maritime transportation industry. VIKOR method is also applied to risk assessment field in the maritime industry, Gul et al. (2017) proposed a hybrid approach based on the VIKOR and FAHP methods to assess the risk of maritime management. Fattahi and Khalilzadeh (2018) further extended the VIKOR method by combining it with fuzzy MULTIMOORA methods for risk assessment, which was verified in a steel industries factory. Given the wide application of VIKOR and relevant decision methods in risk assessment field, Gul (2018) reviewed the papers based on MCDM methods for risk assessment.

Fuzzy set technology has brought innovation to the MCDM field. Ak and Gul (2019) introduced Pythagorean fuzzy set (PFS) and conducted a case study in risk assessment of corrugated cardboard sector. The Pythagorean fuzzy based VIKOR method also applied to risk assessment in underground mining industry (Gul et al., 2019). Combining PFS, 2-tuple linguistic variables and VIKOR method, He et al. (2020) designed an integrated decision framework to evaluate human factors in construction project management. By introducing intuitionistic fuzzy set (IFS), Zhao et al. (2021) compared the performance of cumulative prospect theory based intuitionistic fuzzy multi-attribute border approximation area comparison (CPT-IF-MABAC) and IF-VIKOR in a case study of cold chain. Based on the interval-valued intuitionistic fuzzy set (IVIF), Xiao et al. (2021) proposed IVIF-Taxonomy method, and compared with other traditional decision methods in a fund selection case.

The extensive application of the VIKOR method in various industries is evident based on the analysis of global main path. The recent innovations in the field of VIKOR primarily focuses on integrating different fuzzy set methods and management decision methods. In addition, as shown in Fig. 4, there are three links having significant SPC weights that signify the impact of the initial article Opricovic (1998), as well as Gul et al. (2016) and Ak and Gul (2019) on the academic development of VIKOR. These reviews highlight the value and importance of research studies in VIKOR field. In the subsequent section, we used more MPA methods to explore further information on the evolution of VIKOR.

3.2. Local MPA and key-route MPA of VIKOR

There are seven papers highlighted in yellow in Fig. 5 that different from those related to global MPA. These papers represent research frontiers in the VIKOR field and can provide valuable insights for future investigations.

Figure 5. Local main path.

In contrast to prior research which combined VIKOR with FAHP or other fuzzy multicriteria models, Opricovic (2011) directly introduced fuzzy set theory into VIKOR, and developed a fuzzy VIKOR method in a case of water resources planning. The same fuzzy VIKOR method was utilized by Girubha and Vinodh (2012) to select alternate materials for the instrument panel of an electric car.

The other five papers could reveal more details about the research fronts of VIKOR. Jun et al. (2021) developed an interval type 2 (IT2) fuzzy set based stochastic hybrid decision-making method to evaluate the innovation performance of financial institution. This same IT2-fuzzy VIKOR method was applied in the assessment of the innovative strategies in the field of renewable energy (Li et al., 2020; Liu et al., 2021). Ding et al. (2021) proposed 2-tuple hesitant interval-valued PFS based TOPSIS and VIKOR methods to analyze the environmental management systems in the renewable energy field. Fang et al. (2021) developed a risk assessment framework for energy safety management field based on an innovative intuitionistic fuzzy VIKOR approach. The extensive utilization of the VIKOR method as a benchmarking technique to validate other MCDM methods in recent years highlights its reliability and widespread acceptance within the field.

In comparison to the previous two MPAs, Key-route main path has three different papers which could reveal different academic evolution traces of VIKOR field as shown in Fig. 6. Tadic et al. (2014) developed a MCDM framework using the decision-making trial and evaluation laboratory model, fuzzy ANP and VIKOR to aid in the selection of city logistics. Liu et al. (2015) applied FAHP, entropy method and VIKOR in FMEA to conduct a risk assessment of the general anesthesia process. Ozdemir et al. (2017) combined FMEA, IT2 fuzzy set, AHP and VIKOR to evaluate the occupational health and safety risks.

Figure 6. Key-route main path.

Several key findings can be derived from these two MPAs. Firstly, Opricovic not only proposed the VIKOR method but also incorporated fuzzy theory to develop the fuzzy VIKOR method. Secondly, VIKOR has been utilized as a benchmarking technique to validate other MCDM decision methods, indicating its maturity and reliability. Thirdly, the advancement of the VIKOR domain has primarily been driven by peer citations and the integration of other decision-making methods, particularly fuzzy set technology.

3.3. Multiple global main paths

In this section, multiple global main paths are analyzed to explore more development trends and details for the current and future research (see Fig. 7). The multiple global main paths are divided into five research subgroups of VIKOR. Some key papers of each subgroup would be selected to conduct MPA.

Figure 7. Multiple global main path.

3.3.1. VIKOR in sustainable field

Base on the review of VIKOR, Kumar et al. (2017) conducted a literature review specially focused on MCDM methods in the field of sustainability. This review had a significant impact on future research, and contributed to a large academic cluster of VIKOR applications in sustainable field.

Several scholars have focused on the strategy research of sustainability field. Ture et al. (2019) used VIKOR and TOPSIS to assess the sustainable performance of EU countries. Suganthi (2018) integrated FAHP, VIKOR and DEA methods to evaluate the sectoral investments for sustainable development. Shen et al. (2018) conducted a review on the application of hybrid MCDM methods to sustainability issues. In addition, VIKOR has been widely applied to various industries to enhance sustainability performance. For example, Zhang et al. (2019) proposed an IFS based MCDM method to assess energy storage technologies, while Moghtadernejad et al. (2018) applied VIKOR and other MCDM approaches to help designers to select sustainable building structures. Elzarka et al. (2017) developed a fuzzy MCGDM model to select renewable energy technology for institutional owners. A similar approach combining AHP and F-VIKOR was also used to support the location selection of a Chinese seawater pumped hydro storage station (Wu et al., 2019).

In addition, some scholars have integrated VIKOR and grey system theory, creating a specialized academic subgroup. Tian et al. (2018) combined AHP, grey-correlation technology and MCDM methods to develop a hybrid framework for selecting green decoration materials. Zhang et al. (2019) explored the optimization problem of train energy-absorbing structure using multi-objective artificial bee colony, best worst method, grey relational analysis and VIKOR. Tian et al. (2019) developed a Fuzzy-VIKOR based on Grey-DEMATEL to handle the selection of vehicle reverse logistics. It is worth noting that most of researches in this academic cluster still focus on the sustainable development field.

3.3.2. The adoption of quantum decision theory

In this academic development trajectory, the VIKOR method has been integrated with various decision technologies. One widely used integration is with hesitant fuzzy set theory, which has been applied to handle supplier selection issues (Çalı & Balaman, 2019; Phochanikorn & Tan, 2019; Wang & Cai, 2017). VIKOR has also been applied in healthcare field, where scholars have incorporated health index into the method to assess medical systems (Huang et al., 2021; Mardani et al., 2019).

Within this academic subgroup, Wu et al. (2021) have presented an insightful paper that integrates quantum decision theory and TODIM to conduct a case study on optimal automobile recommendation. Furthermore, exploring the incorporation of quantum decision theory into VIKOR to address relevant issues would yield significant contributions.

3.3.3. VIKOR and AHP

The AHP theory is usually combined with VIKOR to solve selection or assessment issues across various fields, with three distinct integration frameworks being prevalent in this subgroup.

The first framework involves the integration of VIKOR and tradition AHP methods. Emeç and Akkaya (2018) utilized stochastic AHP and F-VIKOR to address warehouse location selection issues, while Zarei et al. (2021) conducted a fuzzy hybrid MCDM analysis, combining F-AHP and VIKOR, to evaluate the resilience of process systems. This hybrid research framework has also been adopted in retail industry and building trades (Kaushik et al., 2020; Koc & Pelin Gurgun, 2021). The main focus of these studies is to address practical issues of different industries by utilizing hybrid decision-making method that combines AHP and VIKOR.

The other two frameworks emphasize the innovation of the hybrid model that integrates VIKOR, AHP and the other decision-making theories. Gul (2018) proposed a MCDM framework combining Pythagorean fuzzy AHP and VIKOR methods, which was verified in a case study involving occupational health and safety. Meanwhile, Wang et al. (2019) developed an Interval Type-2 Fuzzy VIKOR method and performed a sensitivity analysis to verify its stability. The introduction of Interval Type-2 Fuzzy set and PFS theories into VIKOR in these two studies has stimulated extensive discussions and innovative thinking among scholars regarding novel decision-making frameworks, specifically the integration of VIKOR and AHP under various fuzzy set theories.

3.3.4. VIKOR in risk assessment field

The papers in this subgroup primarily focus on risk assessment, utilizing various hybrid MCDM method to address practical issues.

Failure Mode and Effect Analysis (FMEA) is a widely used tool in these researches, supporting different industries to promote the reliability and safety of products, service or business system (Lo & Liou, 2018). Different MCDM technologies, such as VIKOR, are used to overcome the deficiencies of FMEA and enhance its availability (Liu et al., 2019). The method innovations of related researches are still reflected in the integration with different fuzzy set technologies, for example, IFS, interval type-2 fuzzy set, PFS and picture fuzzy set (Boral et al., 2020; Efe, 2019; Jin et al., 2021; Shahri et al., 2021). Considering the application field, these hybrid MCDM methods mainly focus on the traditional manufacturing and environment sustainability.

It is noteworthy that cloud model theory has been applied in the risk assessment field. This method, based on fuzzy set theory, is an artificial intelligence method that can capture the uncertainties in human thinking (Li, 1995). Yu and Pan (2021) used FMEA and cloud model based VIKOR methods to assess the risk of submarine pipeline leakage. A similar decision framework was used to evaluate the risk of steam valve system failure (Li et al., 2019). However, whether this theory can more effectively handle uncertain data in decision-making problems still requires validation through case analysis.

3.3.5. Pythagorean fuzzy VIKOR

There are two papers that have had a significant impact on the technological innovation of VIKOR method. Chen (2018) proposed a Pythagorean fuzzy VIKOR method, which was applied in real-world scenarios and comparative analyses in different cases. Gul et al. (2019) integrated Pythagorean fuzzy theory and VIKOR to develop a Pythagorean fuzzy VIKOR-based risk assessment approach in the case of underground mining, emphasizing the importance of Pythagorean fuzzy theory in VIKOR application. Due to the advantages of Pythagorean-fuzzy-set in handling vague and uncertain information over tradition fuzzy set method, relevant scholars have conducted case studies and innovations based on this theory, leading to the formation of two academic subgroups. Within these subgroups, several promising innovations related to VIKOR application have emerged.

Spherical Fuzzy Set (SFS), a new extension of Pythagorean Fuzzy theory, provides a larger preference area for decision-makers. Kutlu Gündoğdu and Kahraman (2019) developed a novel VIKOR method integrating spherical fuzzy sets to handle a warehouse location selection issue and conducted a contrast experiment with the spherical fuzzy TOPSIS approach. The same VIKOR framework was also used with FAHP to select location-based advertisement (Oztaysi et al., 2020). Aydoğdu and Gül (2020) introduced SFS into WASPAS (Weighted Aggregated Sum Product Assessment) and compared with spherical fuzzy VIKOR in a case study of 3D printer alternative selection.

The application of linguistic term set in MCDM has gained interest among decision scholars as it can effectively process complex language information in the decision-making process. Various linguistic term set technologies have been adopted in VIKOR method to improve decision-making practicability. For instance, Gou et al. (2021) proposed a probabilistic double hierarchy linguistic VIKOR method that was tested on a smart healthcare. Ding and Liu (2019) introduced 2-dimension uncertain linguistic variable into PT-VIKOR (prospect theory VIKOR) for selecting alternative solutions. Other linguistic term sets used in VIKOR include multigranular probabilistic linguistic term set, proportional hesitant 2‐tuple linguistic term set, linguistic Z-numbers and probabilistic linguistic term set (Du & Liu, 2021; Liu et al., 2021; Wang et al., 2021; Xiong et al., 2022).

The q-rung orthopair fuzzy set (Q-ROFS) which generalizes IFS and PFS has proven to be an effective approach in handling multiple attribute group decision making problem. Wang et al. (2020) proposed a framework integrating Q-ROFS and multi-attributive border approximation area comparison. Q-ROFS theory was similarly adopted in VIKOR method to assess the Sustainability Enterprise Risk (Cheng et al., 2021). Under the Q-ROFS environment, Yang et al. (2022) combined Criteria Importance Through Intercriteria Correlation (CRITIC) and VIKOR methods to analyze the internet of thing in sustainable supply chain management.

Additionally, within this VIKOR subgroup, there exists a dedicated sustainability research cluster that underscores the significance and potential of sustainability in the application field of VIKOR.

Decision-making trial and evaluation laboratory (DEMATEL) methodology is another hotspot in VIKOR application as it can effectively optimize criteria in the VIKOR process and enhance the decision reliability. For example, Jun et al. (2021) adopted DEMATEL-ANP to weight the criteria in MCDM methods and used TOPSIS and VIKOR methods to extract the best alternative. The hybrid IT2 fuzzy- DEMATEL-VIKOR method was also applied in the selection of innovative strategies for different renewable energy alternatives (Li et al., 2020). It is noteworthy that the hybrid fuzzy-DEMATEL-VIKOR methods have been widely applied in sustainable energy field (Ding et al., 2021; Fang et al., 2021; Liu et al., 2021; Wang et al., 2020).

The optimization aimed at PFS is also an innovation direction of VIKOR methods. Ma et al. (2021) developed a complex Pythagorean fuzzy VIKOR method based on the traditional Pythagorean fuzzy theory and verified its validity in a renewable energy project and a logistic village location selection project. Meng et al. (2021) developed a hybrid heterogeneous Pythagorean fuzzy MCDM model and compared it with VIKOR method in a clean energy investment assessment project. Akram et al. (2021) integrated VIKOR and complex spherical fuzzy N-soft sets (a new extension based on Pythagorean Fuzzy theory) to structure a MAGDM framework, its capabilities and validity were proved in a Saudi oil refinery project of Pakistan.

4. Conclusion

The applications and innovations of VIKOR in various fields have significantly advanced the progress of related research. However, traditional literature review and bibliometric methods face challenges in unveiling the knowledge evolution process from a vast number of papers. To address this issue, we propose a comprehensive MPA framework in this study to extract academic development trajectories and explore future research directions.

Several previous studies (Lee & Chang, 2018; Liu et al., 2019; Mardani et al., 2016; Rojas-Zerpa & Yusta, 2015) have validated the findings of this paper, indicating that VIKOR is widely applied in the fields of sustainability, renewable energy, and risk assessment. The advancements in the VIKOR method primarily stem from its extensions based on fuzzy set theory (Gul et al., 2016). However, compared to these traditional review studies, an MPA framework based on citation relationships can objectively extract the knowledge evolution process and provide more comprehensive insights into the development of VIKOR.

The global main trajectory unveils the significant developmental path of the VIKOR domain. The VIKOR methodology is extensively employed in project evaluation and solution selection across diverse industries. In the initial stages, scholars tended to integrate the VIKOR method with the well-established FAHP theory in practical applications. Subsequently, an increasing number of scholars started directly incorporating various fuzzy set technologies into the VIKOR method and conducting empirical research. Meanwhile, various decision theories are employed to enhance the validity and reliability of VIKOR methods, such as the application of interval 2-Tuple linguistic variables theory. The current advancements in VIKOR approaches primarily revolve around integrating advanced fuzzy set theory with management theories from diverse industries. According to the global MPA results, PFS and IFS along with interval 2-Tuple linguistic theory emerge as crucial integration technologies within the VIKOR method, a fact that can be corroborated by local MPA findings. Considering their application domains, risk assessment, alternative plan selection, supply chain management, and sustainable energy stand out as prominent areas of focus. It is noteworthy that the two most influential papers are reviews, underscoring the significance of review studies in a thriving and intricate academic domain like VIKOR. Furthermore, numerous contemporary studies have embraced VIKOR as a benchmark method to validate their MCDM approaches, signifying the recognition of relevant scholars towards the maturity of VIKOR as an established MCDM theory.

The multiple global main path method was employed in this study to enhance the granularity of knowledge evolution analysis, facilitating a more comprehensive extraction of the academic development trajectory within the VIKOR field. The network structure revealed six distinct paper clusters that could be classified into five academic subgroups. These subgroups encompassed research on VIKOR in sustainable fields, applications of quantum decision theory, integration of AHP and VIKOR methodologies, utilization of VIKOR for risk assessment purposes, and innovation based on Pythagorean Fuzzy VIKOR.

Two distinct subgroups were identified, each focusing on specific domains: the sustainable field and the risk assessment field. Research in the sustainable field primarily employed the VIKOR method in energy, material, industrial management, and environmental industries. The hybrid framework that integrated VIKOR and AHP methods emerged as a prevalent multi-criteria decision-making approach within this domain. Notably, the application of VIKOR in risk assessment demonstrated its distinctive feature of integrating FMEA theory. It reflects the importance of FMEA theory in the field of risk assessment. Thus, relevant scholars promote a classical assessment framework which combining FMEA, Fuzzy set and VIKOR. Since the multiple global MPA is based on research frontiers to explore more academic evolution, we have reason to believe that sustainable field and risk assessment field have been the academic hotspots of VIKOR application field.

The other three subgroups contribute to the VIKOR method innovation. AHP theory is more mature compared to the VIKOR method, so some scholars adopted different FAHP methods to VIKOR. The largest VIKOR subgroup was the innovation based on Pythagorean Fuzzy VIKOR, which contained two research clusters. Meanwhile, the PFS theory is widely used in the current studies from an overall temporal perspective. There is no doubt that the PFS theory has attracted great attention from related scholars, and strongly promoted the method innovation in VIKOR field.

According to the analysis results, the following findings and discussions can be concluded:

• The development and application of the VIKOR method is thriving, and it has become a mature MCDM theory.

• The most promising application fields for VIKOR include supply chain management, sustainability development (such as energy, environment and material), risk assessment and healthcare.

• The MCDM framework integrating FAHP and VIKOR methods is the most classic one in the sustainable fields. Relevant scholars should consider adopting more VIKOR frameworks, rather than relying on the FAHP-VIKOR method.

• In practical application, the VIKOR method is commonly integrated with fuzzy set theory, decision-making approaches and different industry criteria.

According to the conclusion of MPA, we propose the following predictions for future research focus: (1) Future hot application fields of VIKOR include: energy, environment, material sustainability research and medical risk assessment; (2) The introduction of innovative fuzzy set theory (such as IFS, PFS, the Q-ROFS and spherical fuzzy set) will become the main driving force for the development of VIKOR method; (3) The integration with other decision making theories will still be an important means for the innovation of VIKOR method. The promising decision-making theories include decision making trial and evaluation laboratory methodology, linguistic term set, ANP, Grey relational analysis and failure mode and effects analysis.

Using the MPA method to conduct a development path study on the VIKOR domain can provide a quantitative and objective review and analysis of the development trajectory of the field. By analyzing the development, application innovations, and usage scenarios of the VIKOR method in important scientific development trajectories. It not only deepens the understanding of the VIKOR method among relevant scholars, but also provide insights for future hot application areas and method innovation routes from the citation path analysis. However, there are two limitations of this paper: First, we collect relevant paper information from WoS database, which may not cover all relevant papers in the VIKOR field. Therefore, future research should consider using more databases to obtain a larger sample size for analysis; Second, this paper only uses the MPA method to study the development path of the VIKOR field. In the future, relevant scholars may consider introducing natural language processing technologies (such as topic modeling methods) to mine textual information from related studies, combined with citation path analysis to extract a better knowledge development trajectory of the field.

Acknowledgements

This manuscript was supported by the Ministry of Education of Humanities and Social Science Project (No. 19YJC630208).

Disclosure Statement

No potential conflict of interest was reported by the author(s).