A decision support framework for optimizing career path prediction and vocational mobility of college graduates

Background

Vocational mobility (VM), which means the freedom of individuals to move easily from one opportunity to another, is an essential factor that enhances personal career development within a volatile job market (Mavlankulovich, 2024). The emergence and continued growth of various industries due to globalization and technological advancements have compounded the need for college graduates to provide reasoned actions while making multi-step career choices. This adaptability enables people to pursue their desired career choices and enhances the global workforce. However, evaluating VM is more complex and involves frameworks that capture objective and perceptual factors on career choices.

Conventional methods of assessing VM sometimes fail to depict the multifaceted systems that characterize career changes adequately. These methods are especially limited by linear models or single evaluation criteria, which are insufficient for complex business environment situations. Furthermore, current tools that use prediction are inadequate in handling situations involving a large amount of ambiguity or missing data in parts. Although more recent studies have contributed to the growth in organizing careers, a common drawback of these studies is their focus on real-world uncertainty and group decision-making. For example, Yamashita et al. (2022) employed a machine learning methodology to predict the careers of individuals, but did not consider MCGDM input. On the same note, Du & Zhao (2025) presented a neutrosophic MCDM approach in vocational assessment; however, it does not provide a thorough modeling of group agreements and struggles with processing incomplete and unclear data. Conversely, the IVSF-MARCOS model, introduced within the MCGDM framework, addresses these gaps and constitutes a combination of MARCOS’s well-established decision-making ability and the model’s capacity to address the ambiguity of the IVSFS, while incorporating the views of multiple experts.

In this context, fuzzy sets (FS) were developed by Zadeh (1965) in 1965 and provide a theoretical foundation for working with fuzziness and uncertainty in decision-making operations. FS allows membership degrees (MD) of components; in this sense, it handles fuzzy, or imprecise, information better than classical sets. Later on, as part of its development, intuitionistic fuzzy sets (IFS), introduced by Atanassov (1986), were enhanced with a non-membership degree (NMD) to assess the degree of uncertainty better. Later, the Pythagorean fuzzy sets (PyFS) (Yager, 2013) and q-rung orthopair fuzzy sets (q-ROFS) (Yager, 2017) gained more flexibility to address the situation when a higher order of uncertainty is required. In the case under consideration, picture fuzzy sets (PFS) (Cuong, 2015) offer the abstinence degree (AD) as a more convenient approach for decisions concerning neutrality or those with some uncertainties. This led to the development of extensions to fuzzy sets, specifically spherical fuzzy sets (SFS) (Mahmood et al., 2019), which find particularly suitable applications in multidimensional settings. A more recent extension of this framework is the introduction of interval-valued spherical fuzzy sets (IVSFS) (Aydoğdu & Gül, 2022), which further expand the scope of this framework by expressing these values as intervals, thereby implying an even higher degree of uncertainty and complexity in decision-making. The use of added complexity is particularly warranted in our study, as it ensures the closer modeling of the vaguely hesitant and fact-opposing judgments provided by various experts assessing VM.

This research employs the developed IVSF-MARCOS approach within an MCGDM environment to measure VM. The proposed model encompasses five decision-makers, fifteen alternatives, and ten criteria, providing a more comprehensive method for assessing vocational mobility and forecasting suitable career patterns. The IVSF-MARCOS approach handles uncertainties more effectively, provides more accurate decisions, and is more adaptable to complex real-world problems than traditional methods. Thus, by analyzing large amounts of data and addressing uncertain situations that frequently arise in decision-making processes, IVSF-MARCOS operates efficiently. Although the suggested framework has not introduced any methods, such as those discussed in social network-based group decision-making approaches (Cao et al., 2025) and trust propagation models (Gai et al., 2025), it could be responsible for overcoming reluctance and conflict by leveraging the expressive power of IVSFS. It enables the decision-makers to present incomplete, ambivalent, or uncertain information and minimizes the intragroup conflict. A clear definition of all abbreviations used in the entire manuscript has been provided in Table 1.

Research gap and motivations

Describing VM as a critical predictor of career success and work adjustment among college graduates, this study seeks to assess how developments such as work, advanced technology, and a changing economy influenced applicants’ scores on both scales in the context of the college-to-work transition. Although the importance of VM in workforce development is continually increasing, the majority of existing studies have examined the subject matter in only slight depth or in an evaluation tool that is already methodically confined, which fail to incorporate multiple evaluative criteria or capture the diversity of stakeholder perspectives. Specifically, these methods are weak when it comes to managing the vagueness, unpredictability, and imprecision that is part of the real-world decision environment.

Based on these research gaps, this study proposes a comprehensive decision-support model for assessing vocational mobility and also considers the benefits of employing IVSFS. By integrating the MARCOS method within an MCGDM framework, the presented approach delivers robust and context-sensitive career choice recommendations. This innovative methodology meets this need and provides graduates, educators, and policymakers with practical solutions to improve VM.

Significance of the study

The significance of this study lies in its ability to address the critical challenges faced by college graduates: ensuring job seekers make sound career decisions within the current and preparing environment, and in a competitive world. Another essential concept is VM, which implies readiness and the possibility of changing from one type of job or career to another; this is important to graduates in search of meaningful and sustainable careers. However, traditional methods of assessing VM and forecasting career development do not always allow for capturing this choice’s multiple and stochastic character. Consequently, this study presents a systematic and comprehensive solution by proposing a decision support model based on the IVSF-MARCOS method within the MCGDM framework. These features enable the consideration of multiple criteria, facilitate the evaluation of decision-makers’ perceptions, account for uncertainty, and deliver accurate and practical results.

Objectives and contributions

The study is informed by the emerging reality that there is a dire need to help college graduates make informed career choices by fixing some uncertainty in VM evaluations. The general purposes of the research are:

• To determine how current theoretical models in career path prediction fail to address imprecision and the lack of information, among other limitations.

• To suggest a unified decision-making system based on the IVSF-MARCOS approach in an MCGDM context.

• To decide on various career options based on the synergy of multiple decision-makers and a wide range of criteria, thereby providing a solid and circumstantial career path application.

• To give good, practical, and on-the-ground information to the graduates, teachers, and policymakers by tapping the power of fuzzy logic in dealing with matters of vagueness and ambiguity.

The key contributions of this study are as follows:

• Integration of VM and career path prediction: The research fills the gap between VM research and career option forecasting and creates a common framework for holistically assessing career options.

• Application of IVSF-MARCOS method: A new development of the IVSF-MARCOS method is proposed due to its effectiveness in dealing with vagueness and fuzziness in decision-making. This enhances the accuracy and reliability of the results, thereby improving the model’s applicability.

• MCGDM framework: The model’ also considers decision-makers’ views by involving five decision-makers in assessing fifteen alternatives based on ten criteria.

• Actionable insights for stakeholders: This study’s implications can be helpful to graduates, educators, and policymakers in making appropriate decisions if needed and planning for the best operations to enhance VM.

• Advancement of decision science methodologies: This study enriches the existing knowledge of decision science by applying IVSFS and the MARCOS method to the domain of VM and concerns for career development.

Through these contributions, this study not only addresses the mentioned drawbacks but also presents a valuable tool to enhance the career choices of college graduates.

Organization of study

The remainder of this article is structured as follows: ‘Literature Review’ presents a literature review that includes VM, career prediction, and the MARCOS method. ‘Preliminaries’ describes the preliminaries of IVSFS, while ‘IVSF-MARCOS Method’ discusses the IVSF-MARCOS methodology. ‘Case Study: Vocational Mobility of College Graduates’ assesses VM with an emphasis on determining the career path and theoretical implications of the study. ‘Comparison Analysis’ presents the comparison findings, performs a sensitivity analysis, and discusses the study’s practical implications, limitations, and advantages. ‘Conclusion’ concludes the article with future research directions.

Studies integrating vocational mobility of college graduates

VM has garnered significant attention due to its implications for careers and economic security. Previous studies focused on occupational and spatial mobility, including graduates from college and vocational schools, as well as vocational employment in specific areas. Hence, Oliveira, Moutinho & Afonso (2024) have drawn attention to the differences in employment conditions between college students and graduates of vocational schools, highlighting the impact of supply and demand factors. Mobility programs, as described by Salukvadze & Golubchikov (2016), aimed at widening global opportunities, appear to foster soft skills and employability among vocational college learners. Furthermore, Yakhina et al. (2016) provide practical recommendations on how to build readiness for occupational mobility based on the results of the education received. Yue and her colleagues present the work on postgraduate choices based on student mobility in Huang & Jung (2024). The present work by Schwartz (2016) describes pathways that enhance equity and occupational mobility, noting that vocational education systems play a role in facilitating upward mobility. Last but not least, the spatial mobility of higher education graduates (Venhorst, Van Dijk & Van Wissen, 2011) raises an essential aspect of relocation among college graduates and their connections between residential and occupational spaces.

Studies integrating career path prediction

Career mapping is the art of predicting or envisioning possible career progress for individuals in light of their skills, interests, and the market. The latest developments in career dynamics have incorporated multiple strategies, including artificial intelligence, machine learning, and theory. For instance, Kosaraju et al. (2025) introduce a new strategy of bending career counseling via machine learning to help students gain insight into the proper choice of careers. In the same regard, Kaddoura, Popescu & Hemanth (2022) provide an extensive analysis of machine learning models in career path forecasting, highlighting their significance in enhancing precision and efficiency in decision-making. The latest research in using large language models, including the study by Adhikary (2016), also presents high levels of confidence in forecasting career readiness. Kokkodis & Ipeirotis (2021) design demand-aware career path recommendations from a reinforcement learning perspective, integrating them to accommodate changing labor market demands. Furthermore, Miller (2008) investigate the moderating effect of career decision self-efficacy, which connects emotional intelligence with career adaptability. At a tactical level, Sydler, Haefliger & Pruksa (2014) examine how planning affects the criterion-related validity of the expected career trajectory. In the article by Güler & Emir (2024), educational data mining is explored to develop models of career prediction that include explain ability for better student navigation. Additionally, Fletcher et al. (2014) investigate career pathways and job satisfaction, providing a better understanding of motivation and its predictors.

Studies integrating the MARCOS method

The Measurement Alternatives and Ranking According to the Compromise Solution (MARCOS) method has become one of the most powerful multi-criteria decision-making (MCDM) techniques in applications due to its provisions for assessing alternatives against ideal and anti-ideal solutions. The MARCOS method has been used and extended with fuzzy logic to solve generalized decision-making problems in various fields. Stanković et al. (2020) proposed a fuzzy MARCOS model to analyze the risk of road traffic in uncertain conditions, aiming to enhance the distinctiveness of fuzzy sets. When applied to assess sustainable suppliers, as done by Boral et al. (2020), this approach was described as having high usefulness for supply chain management. Later, Bakır & Atalık (2021) concurrently applied fuzzy MARCOS with AHP to measure service quality in airline companies. Moreover, Büyüközkan, Havle & Feyzioğlu (2021) proposed an integrated SWOT-based fuzzy MARCOS model that may improve trajectories of sustainable transportation planning. In waste management, Ali (2022) has introduced a promising fuzzy MARCOS application for scoring solid waste systems. Failure modes in engineering systems have been analyzed by Boral et al. (2020) by using the fuzzy MARCOS method, whereas Ali (2021) has developed an advanced CRITIC-MARCOS framework with spherical fuzzy sets. Nila, Pamucar & Roy (2024) noted that the MARCOS method can be employed to analyze drone-based city logistics using spherical fuzzy criteria. In the last case, Wang et al. (2024) employed the Pythagorean fuzzy CRITIC-MARCOS method to select sustainable food suppliers, suggesting that this type of approach may be effective in highly uncertain situations. Recent research developments in the MCGDM literature have introduced advanced fuzzy models. Das & Granados (2023) presented the concept of IFP-intuitionistic multi-fuzzy N-soft sets, whereas Das & Granados (2022) presented a concept of fuzzy parameterized intuitionistic fuzzy soft multisets to the group decision-making problem. Das et al. (2025) made use of a fuzzy MCDM model in environmental analysis, whereas previous studies by Mukherjee & Das (2015a, 2015b) made use of interval-valued intuitionistic fuzzy soft sets in a bid to make investment decisions. The mentioned models, however, are conventionally deficient in representing both topic uncertainty and the subjectivity of groups simultaneously. IVSF-MARCOS is the solution we have proposed to address this issue by combining interval-valued spherical fuzzy information with MARCOS, which presents a more helpful solution to complex MCGDM problems, such as VM assessment. To clearly outline the novelty of the proposed study and determine the motivation, Table 2 includes a comparative summary of the literature reviewed and the related research gaps.

Preliminaries

The concept of FS, initially proposed by Zadeh (1965), is one of the mathematical frameworks for handling uncertainty. Conversely, it is ineffective for NMD or ambiguity in ambiguous environments, while it performs well for MD in such circumstances. Atanassov (1986) proposed a novel IFS method to overcome these limitations. The PFS was developed by Cuong (2015) to address more complicated cases involving indeterminacy and abstinence concerns. To fulfill the goals of FS, PFS includes the AD together with the MD, NMD, and refusal degree (RD). Mahmood et al. (2019) introduced the SFS in 2018. For clarity, a complete list of mathematical notations is provided in Table 3.

Definition 1: (Mahmood et al., 2019) Let $X$ be the universe of discourse involves MD $\left(\mathit{\beta }\right)$, AD $\left(\mathbf{\text{α}}\right)$, and NMD $\left(\mathbf{\text{γ}}\right)$. SFS $\mathit{\delta }$ is defined as:

$$\mathit{\delta }=\left\{\left({\mathit{\beta }}_{\mathit{\delta }}\left(\mathrm{\aleph }\right),{\mathbf{\text{α}}}_{\mathit{\delta }}\left(\mathrm{\aleph }\right),{\mathbf{\text{γ}}}_{\mathit{\delta }}\left(\mathrm{\aleph }\right)\right):\mathrm{\aleph }\in X\right\}$$where:

$${\mathit{\beta }}_{\delta }\left(\mathrm{\aleph }\right),{\mathbf{\text{α}}}_{\mathit{\delta }}\left(\mathrm{\aleph }\right),{\mathbf{\text{γ}}}_{\mathit{\delta }}\left(\mathrm{\aleph }\right)\in \left[0,1\right]$$with the conditions that:

(1) $$0\le {\mathit{\beta }}_{\mathit{\delta }}^2\left(\mathrm{\aleph }\right)+{\mathbf{\text{α}}}_{\mathit{\delta }}^2\left(\mathrm{\aleph }\right)+{\mathbf{\text{γ}}}_{\mathit{\delta }}^2\left(\mathrm{\aleph }\right)\le 1.$$

Also, the RD is determined by:

$${\mathbf{\text{γ}}}_{\mathit{\delta }}\left(\mathrm{\aleph }\right)={\left(1-\left({\mathit{\beta }}_{\mathit{\delta }}^2\left(\mathrm{\aleph }\right)+{\mathbf{\text{α}}}_{\mathit{\delta }}^2\left(\mathrm{\aleph }\right)+{\mathbf{\text{γ}}}_{\mathit{\delta }}^2\left(\mathrm{\aleph }\right)\right)\right)}^{\frac{1}{2}}.$$

Definition 2: (Aydoğdu & Gül, 2022) Let $X$ be the universe of discourse involves MD $\left(\mathit{\beta }\right)$, AD $\left(\mathbf{\text{α}}\right)$, and NMD $\left(\mathbf{\text{γ}}\right)$. IVSFS $\mathit{\delta }$ is defined as:

$$\mathit{\delta }=\left\{\left(\left({\mathit{\beta }}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^L,{\beta }_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right),\left({\mathbf{\text{α}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right),\left({\mathbf{\text{γ}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)\right):\mathrm{\aleph }\in X\right\}$$where:

$$\left({\mathit{\beta }}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right),\left({\mathbf{\text{α}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right),\left({\mathbf{\text{γ}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)\in \left[0,1\right]$$with the conditions that:

(2) $$0\le {\left({\mathit{\beta }}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)}^2+{\left({\mathbf{\text{α}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)}^2+{\left({\mathbf{\text{γ}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)}^2\le 1.$$

Also, the RD is determined by:

$${\mathbf{\text{γ}}}_{\mathit{\delta }}\left(\mathrm{\aleph }\right)={\left(1-\left({\left({\mathit{\beta }}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)}^2+{\left({\mathbf{\text{α}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)}^2+{\left({\mathbf{\text{γ}}}_{\mathit{\delta }}{\left(\mathrm{\aleph }\right)}^U\right)}^2\right)\right)}^{\frac{1}{2}}.$$

Definition 3: (Aydoğdu & Gül, 2022). Let ${\mathit{\delta }}_1=\left(\left[{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right],\left[{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right],\left[{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)$, ${\mathit{\delta }}_2=\left(\left[{\mathit{\beta }}_{{\mathit{\delta }}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathit{\delta }}_2}{\left(\mathrm{\aleph }\right)}^U\right],\left[{\mathbf{\text{α}}}_{{\mathit{\delta }}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathit{\delta }}_2}{\left(\mathrm{\aleph }\right)}^U\right],\left[{\mathbf{\text{γ}}}_{{\mathit{\delta }}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathit{\delta }}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)$ be two interval-valued spherical fuzzy values (IVSFVs) and $\mathit{\tau }>0$, $\mathit{\tau }$ be any scalar number, then it satisfies the following operations:

• ${\mathbf{\text{δ}}}_1\oplus {\mathbf{\text{δ}}}_2=\left(\begin{array}{c}\sqrt{\begin{array}{c}{\left(\left[{\mathit{\beta }}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2+{\left(\left[{\mathit{\beta }}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\\ -{\left(\left[{\mathit{\beta }}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2.{\left(\left[{\mathit{\beta }}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\end{array}},\\ \left(\left[{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right].\left[{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right),\left(\left[{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right].\left[{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)\end{array}\right).$

• ${\mathbf{\text{δ}}}_1\otimes {\mathbf{\text{δ}}}_2=\left(\begin{array}{c}\left(\left[{\mathit{\beta }}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right].\left[{\mathit{\beta }}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right),\sqrt{\begin{array}{c}{\left(\left[{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2+{\left(\left[{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\\ -{\left(\left[{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2.{\left(\left[{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\end{array}},\\ \sqrt{\begin{array}{c}{\left(\left[{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2+{\left(\left[{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\\ -{\left(\left[{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2.{\left(\left[{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathbf{\text{δ}}}_2}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\end{array}}\end{array}\right).$

• $\mathit{\tau }.{\mathit{\delta }}_1=\left(\sqrt{1-{\left(1-{\left(\left[{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\right)}^{\mathit{\tau }}},{\left(\left[{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^{\mathbf{\text{τ}}},{\left(\left[{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^{\mathit{\tau }}\right).$

• ${\mathit{\delta }}_1^{\mathit{\tau }}=\left({\left(\left[{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^{\mathit{\tau }},\sqrt{1-{\left(1-{\left(\left[{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\right)}^{\mathit{\tau }}},\sqrt{1-{\left(1-{\left(\left[{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)}^2\right)}^{\mathit{\tau }}}\right).$

• ${\mathit{\delta }}_1^c=\left(\left[{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{γ}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right],\left[{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathbf{\text{α}}}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right],\left[{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^L,{\mathit{\beta }}_{{\mathit{\delta }}_1}{\left(\mathrm{\aleph }\right)}^U\right]\right)$

  Additionally, the addition and multiplication of IVSFS can be done using the following procedures.

• ${\mathit{\delta }}_1\oplus {\mathit{\delta }}_2={\mathit{\delta }}_2\oplus {\mathit{\delta }}_1$

• ${\mathit{\delta }}_1\otimes {\mathit{\delta }}_2={\mathit{\delta }}_2\otimes {\mathit{\delta }}_1$.

Career path prediction

Career path prediction in VM is a systematic process of identifying suitable career options for graduates in relation to existing choices. To achieve this, five participants with expertise in career counseling, human resources, or education management were included. The diverse perspectives from these decision-makers enhanced the depth of the evaluation methodology.

Criteria for evaluation

The evaluation process involves 10 critical criteria that are essential for determining the VM of college graduates:

• Salary potential ( $C_1$): Cash incentives related to the profession or the industry as a whole.

• Job stability ( $C_2$): Employment security and tenure security.

• Growth opportunities ( $C_3$): Opportunity to grow in one’s career.

• Skill utilization ( $C_4$): Use of skills acquired by graduates.

• Work-life balance ( $C_5$): Relationship between work and family.

• Location flexibility ( $C_6$): Location flexibility of the career.

• Industry demand ( $C_7$): Present and future trends with respect to demand for jobs.

• Educational alignment ( $C_8$): Suitability of academic attainment in relation to the line of work.

• Networking opportunities ( $C_9$): Opportunity to establish business-professional contacts.

• Job satisfaction ( $C_{10}$): Satisfaction corresponding to the job description.

Figure 2 illustrates the detailed flowchart for evaluating alternatives and criteria in vocational mobility career path prediction.

The decision-makers’ evaluations are combined using the IVSF-MARCOS method, which leverages the advantages of IVSFS for handling uncertainties and vagueness in decisions, as shown in Fig. 2. In this method, variations in expert opinions can be captured and integrated to accurately rank the alternatives. This assessment provides information on the desirability ranking of careers in relation to optimizing the VM of college graduates. The approach ensures an accurate, comprehensive, and reliable evaluation, making the IVSF-MARCOS method dependable. This model serves as a strategic tool in career management, offering valuable information to help graduates make necessary career decisions. Tables 6 and 7 below display the linguistic evaluation of criteria and alternatives based on the decision-makers’ judgments.

Table 6:

Evaluation of criteria in linguistics.

	D1	D2	D3	D4	D5
C1	M	I	VI	M	VI
C2	M	I	VI	M	VI
C3	I	I	M	I	VI
C4	I	VU	M	I	M
C5	I	VU	M	I	M
C6	UI	VI	VI	VU	M
C7	UI	VI	VU	VU	I
C8	UI	VI	VU	VU	I
C9	M	M	M	I	I
C10	M	M	M	I	UI

DOI: 10.7717/peerj-cs.3345/table-6

Table 7:

Linguistic assessment of alternatives.

Alternatives	Experts	Criteria
		C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
A1	D1	F	G	VG	F	MG	B	G	EG	MB	MG
	D2	VVG	MB	MG	F	EG	B	G	B	G	VVG
	D3	VVG	MB	MG	F	EG	B	G	B	G	F
	D4	VVG	MB	G	B	EG	B	G	B	G	F
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B
A2	D1	MB	B	MB	EG	MG	EG	VVB	B	VVG	VB
	D2	MB	B	MB	G	MB	G	F	B	EG	VB
	D3	MB	B	MB	G	MB	G	F	B	EG	VB
	D4	MB	B	MB	G	MB	G	F	B	EG	VB
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B
A3	D1	VB	MG	G	VVG	G	MB	VG	MB	MG	MG
	D2	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D3	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D4	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B
A4	D1	MG	MB	EG	VVG	MB	B	G	B	B	B
	D2	MG	MB	VG	VVG	MB	B	G	EG	MB	MG
	D3	MG	MB	VG	VVG	MB	B	G	EG	MB	MG
	D4	F	MB	VG	VVG	MB	B	G	EG	MB	MG
	D5	F	G	VG	F	MG	B	G	EG	MB	MG
A5	D1	VVG	MB	MG	F	EG	B	VG	B	G	VVG
	D2	VVG	MB	MG	F	EG	B	G	B	G	VVG
	D3	VVG	MB	MG	F	EG	B	G	B	G	F
	D4	VVG	MB	G	B	EG	B	G	B	G	F
	D5	MG	MG	G	B	G	VVB	G	VB	VVB	F
A6	D1	MB	B	MB	EG	MG	EG	VVB	B	VVG	VB
	D2	MB	B	MB	G	MB	G	F	B	EG	VB
	D3	MB	B	MB	G	MB	G	F	B	EG	VB
	D4	MB	B	MB	G	MB	G	F	B	EG	VB
	D5	G	G	G	G	MB	G	F	G	EG	G
A7	D1	VB	MG	G	VVG	G	MB	VG	MB	MG	MG
	D2	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D3	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D4	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D5	B	F	MG	VG	MG	B	MG	VVB	G	B
A8	D1	F	G	VG	F	MG	B	G	EG	MB	MG
	D2	MG	MB	VG	VVG	MB	B	G	EG	MB	MG
	D3	MG	MB	VG	VVG	MB	B	G	EG	MB	MG
	D4	F	MB	VG	VVG	MB	B	G	EG	MB	MG
	D5	F	G	VG	F	MG	B	G	EG	MB	MG
A9	D1	VVG	MB	MG	F	EG	B	VG	B	G	VVG
	D2	VVG	MB	MG	F	EG	B	G	B	G	VVG
	D3	VVG	MB	MG	F	EG	B	G	B	G	F
	D4	VVG	MB	G	B	EG	B	G	B	G	F
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B
A10	D1	MB	B	MB	EG	MG	EG	VVB	B	VVG	VB
	D2	MB	B	MB	G	MB	G	F	B	EG	VB
	D3	MB	B	MB	G	MB	G	F	B	EG	VB
	D4	MB	B	MB	G	MB	G	F	B	EG	VB
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B
A11	D1	VVG	MB	G	B	EG	B	G	B	G	F
	D2	MB	B	MB	G	MB	G	F	B	EG	VB
	D3	MB	B	MB	G	MB	G	F	B	EG	VB
	D4	MB	B	MB	G	MB	G	F	B	EG	VB
	D5	G	G	G	G	MB	G	F	G	EG	G
A12	D1	VVG	MB	G	B	EG	B	G	B	G	F
	D2	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D3	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D4	VB	MG	G	VG	G	MB	MG	VVB	F	MB
	D5	B	F	MG	VG	MG	B	MG	VVB	G	B
A13	D1	VVG	MB	G	B	EG	B	G	B	G	F
	D2	MG	MB	VG	VVG	MB	B	G	EG	MB	MG
	D3	MG	MB	VG	VVG	MB	B	G	EG	MB	MG
	D4	F	MB	VG	VVG	MB	B	G	EG	MB	MG
	D5	F	G	VG	F	MG	B	G	EG	MB	MG
A14	D1	VVG	MB	G	B	EG	B	G	B	G	F
	D2	VVG	MB	MG	F	EG	B	G	B	G	VVG
	D3	VVG	MB	MG	F	EG	B	G	B	G	F
	D4	VVG	MB	G	B	EG	B	G	B	G	F
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B
A15	D1	VVG	MB	G	B	EG	B	G	B	G	F
	D2	MB	B	MB	G	MB	G	F	B	EG	VB
	D3	MB	B	MB	G	MB	G	F	B	EG	VB
	D4	MB	B	MB	G	MB	G	F	B	EG	VB
	D5	MG	MB	EG	VVG	MB	B	G	B	B	B

DOI: 10.7717/peerj-cs.3345/table-7

Table 8 presents the weights assigned to the decision-makers. Applying the interval-valued spherical fuzzy weighted averaging (IVSFWA) operator yields the results presented in Table 9.

Table 8:

Decision-makers’ weight.

	D1	D2	D3	D4	D5
Linguistic variables	I	I	VI	M	M
Weight (L)	0.211	0.211	0.225	0.177	0.177
Weight (U)	0.209	0.209	0.223	0.179	0.179

DOI: 10.7717/peerj-cs.3345/table-8

Table 9:

Decision matrix with aggregated criteria.

	${\mathit{\beta }}^{\mathit{L}}$	${\mathbf{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$
C1	0.760	0.877	0.142	0.194	0.108	0.167
C2	0.760	0.877	0.142	0.194	0.108	0.167
C3	0.748	0.858	0.149	0.200	0.109	0.163
C4	0.634	0.741	0.207	0.259	0.198	0.255
C5	0.634	0.741	0.207	0.259	0.198	0.255
C6	0.704	0.838	0.182	0.239	0.148	0.217
C7	0.617	0.751	0.236	0.293	0.206	0.275
C8	0.617	0.751	0.236	0.293	0.206	0.275
C9	0.664	0.768	0.181	0.231	0.181	0.234
C10	0.610	0.714	0.204	0.255	0.226	0.279

DOI: 10.7717/peerj-cs.3345/table-9

After applying Eqs. (5)–(7), we get Table 10. Also, Fig. 3 shows the criteria weights.

Table 10:

Weights of normalized criterion.

	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	CC	Normalized weights
C1	0.299	0.284	1.181	1.225	0.805	0.110
C2	0.299	0.284	1.181	1.225	0.805	0.110
C3	0.312	0.294	1.173	1.216	0.798	0.109
C4	0.465	0.446	1.043	1.082	0.700	0.095
C5	0.465	0.446	1.043	1.082	0.700	0.095
C6	0.377	0.362	1.121	1.171	0.756	0.103
C7	0.495	0.473	1.033	1.085	0.686	0.093
C8	0.495	0.473	1.033	1.085	0.686	0.093
C9	0.422	0.402	1.070	1.109	0.725	0.099
C10	0.495	0.474	1.007	1.046	0.679	0.093

DOI: 10.7717/peerj-cs.3345/table-10

The following Tables 11A and 11B show the aggregated IVSF decision matrix for alternatives.

Table 11:

Decision matrix for alternatives based on aggregated IVSFVs.

A. Decision matrix for alternatives based on aggregated IVSFVs.
	C1						C2						C3
	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$
A1	0.66	0.77	0.09	0.15	0.15	0.20	0.35	0.44	0.26	0.31	0.31	0.36	0.63	0.76	0.00	0.16	0.16	0.22
A2	0.30	0.40	0.28	0.33	0.33	0.38	0.17	0.27	0.34	0.39	0.39	0.44	0.49	0.64	0.00	0.25	0.25	0.31
A3	0.20	0.28	0.35	0.40	0.41	0.46	0.42	0.52	0.21	0.27	0.27	0.32	0.64	0.76	0.00	0.16	0.16	0.21
A4	0.42	0.52	0.22	0.27	0.27	0.32	0.33	0.43	0.27	0.32	0.32	0.37	0.71	0.82	0.00	0.12	0.12	0.17
A5	0.72	0.82	0.06	0.12	0.12	0.17	0.30	0.40	0.28	0.33	0.33	0.38	0.49	0.59	0.18	0.23	0.23	0.28
A6	0.33	0.43	0.27	0.32	0.32	0.37	0.28	0.37	0.30	0.35	0.35	0.40	0.33	0.43	0.27	0.32	0.32	0.37
A7	0.08	0.17	0.39	0.44	0.44	0.49	0.43	0.53	0.21	0.26	0.26	0.31	0.53	0.63	0.16	0.21	0.21	0.26
A8	0.40	0.50	0.23	0.28	0.28	0.33	0.40	0.50	0.23	0.28	0.28	0.33	0.65	0.75	0.10	0.15	0.15	0.20
A9	0.72	0.82	0.06	0.12	0.12	0.17	0.25	0.35	0.30	0.35	0.35	0.40	0.59	0.73	0.00	0.18	0.18	0.24
A10	0.30	0.40	0.28	0.33	0.33	0.38	0.17	0.27	0.34	0.39	0.39	0.44	0.49	0.64	0.00	0.25	0.25	0.31
A11	0.49	0.60	0.18	0.24	0.24	0.30	0.30	0.39	0.29	0.34	0.34	0.40	0.40	0.50	0.23	0.28	0.28	0.33
A12	0.41	0.50	0.25	0.32	0.32	0.38	0.40	0.50	0.23	0.28	0.28	0.33	0.53	0.63	0.16	0.21	0.21	0.26
A13	0.52	0.63	0.16	0.22	0.22	0.27	0.33	0.43	0.27	0.32	0.32	0.37	0.63	0.73	0.11	0.16	0.16	0.21
A14	0.72	0.82	0.06	0.12	0.12	0.17	0.25	0.35	0.30	0.35	0.35	0.40	0.61	0.74	0.00	0.17	0.17	0.23
A15	0.47	0.58	0.19	0.25	0.25	0.31	0.20	0.29	0.33	0.38	0.38	0.43	0.54	0.68	0.00	0.22	0.22	0.28
	C4						C5
	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$
A1	0.46	0.57	0.20	0.39	0.26	0.31	0.76	0.88	0.00	0.10	0.10	0.16
A2	0.69	0.81	0.00	0.20	0.13	0.19	0.31	0.40	0.28	0.33	0.33	0.38
A3	0.69	0.80	0.08	0.19	0.13	0.18	0.51	0.61	0.17	0.22	0.22	0.27
A4	0.71	0.82	0.07	0.15	0.12	0.17	0.30	0.40	0.28	0.33	0.33	0.38
A5	0.30	0.39	0.28	0.39	0.33	0.38	0.82	0.93	0.00	0.06	0.06	0.12
A6	0.65	0.78	0.00	0.20	0.15	0.21	0.31	0.40	0.28	0.33	0.33	0.38
A7	0.67	0.78	0.09	0.19	0.14	0.19	0.53	0.63	0.16	0.21	0.21	0.26
A8	0.65	0.76	0.09	0.19	0.15	0.21	0.35	0.44	0.26	0.31	0.31	0.36
A9	0.46	0.57	0.20	0.39	0.26	0.31	0.81	0.92	0.00	0.07	0.07	0.13
A10	0.69	0.81	0.00	0.20	0.13	0.19	0.31	0.40	0.28	0.33	0.33	0.38
A11	0.50	0.60	0.18	0.31	0.23	0.28	0.52	0.67	0.00	0.23	0.23	0.30
A12	0.60	0.70	0.13	0.26	0.18	0.24	0.64	0.77	0.00	0.16	0.16	0.21
A13	0.64	0.75	0.10	0.20	0.16	0.22	0.54	0.68	0.00	0.22	0.22	0.28
A14	0.44	0.55	0.21	0.42	0.28	0.33	0.81	0.92	0.00	0.07	0.07	0.13
A15	0.56	0.66	0.15	0.31	0.20	0.26	0.52	0.67	0.00	0.23	0.23	0.30

B. Continued decision matrix for alternatives based on aggregated IVSFVs.
	C6						C7						C8
	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$
A1	0.15	0.25	0.35	0.40	0.40	0.45	0.55	0.65	0.15	0.20	0.20	0.25	0.50	0.65	0.00	0.26	0.26	0.33
A2	0.62	0.75	0.00	0.17	0.17	0.23	0.37	0.46	0.26	0.31	0.31	0.36	0.15	0.25	0.35	0.40	0.40	0.45
A3	0.24	0.33	0.31	0.36	0.36	0.41	0.52	0.62	0.16	0.22	0.22	0.27	0.13	0.20	0.17	0.45	0.45	0.50
A4	0.15	0.25	0.35	0.40	0.40	0.45	0.55	0.65	0.15	0.20	0.20	0.25	0.80	0.92	0.00	0.08	0.08	0.14
A5	0.14	0.23	0.37	0.42	0.42	0.47	0.57	0.67	0.14	0.19	0.19	0.24	0.14	0.24	0.30	0.41	0.41	0.46
A6	0.65	0.78	0.00	0.15	0.15	0.21	0.31	0.41	0.28	0.33	0.33	0.38	0.28	0.37	0.31	0.35	0.35	0.40
A7	0.24	0.33	0.31	0.36	0.36	0.41	0.50	0.61	0.17	0.22	0.22	0.28	0.12	0.17	0.17	0.46	0.46	0.51
A8	0.15	0.25	0.35	0.40	0.40	0.45	0.55	0.65	0.15	0.20	0.20	0.25	0.85	0.95	0.00	0.05	0.05	0.10
A9	0.15	0.25	0.35	0.40	0.40	0.45	0.57	0.67	0.14	0.19	0.19	0.24	0.15	0.25	0.34	0.40	0.40	0.45
A10	0.62	0.75	0.00	0.17	0.17	0.23	0.37	0.46	0.26	0.31	0.31	0.36	0.15	0.25	0.29	0.40	0.40	0.45
A11	0.50	0.60	0.18	0.23	0.23	0.28	0.40	0.50	0.22	0.28	0.28	0.33	0.28	0.37	0.33	0.35	0.35	0.40
A12	0.22	0.32	0.32	0.37	0.37	0.42	0.47	0.57	0.19	0.24	0.24	0.29	0.07	0.12	0.20	0.48	0.48	0.53
A13	0.15	0.25	0.35	0.40	0.40	0.45	0.55	0.65	0.15	0.20	0.20	0.25	0.80	0.92	0.00	0.08	0.08	0.14
A14	0.15	0.25	0.35	0.40	0.40	0.45	0.55	0.65	0.15	0.20	0.20	0.25	0.15	0.25	0.34	0.40	0.40	0.45
A15	0.45	0.55	0.21	0.26	0.26	0.31	0.44	0.54	0.21	0.26	0.26	0.31	0.15	0.25	0.30	0.40	0.40	0.45
	C9						C10
	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^{\mathit{U}}$	${\mathit{\beta }}^{\mathit{L}}$	${\mathit{\beta }}^{\mathit{U}}$	${\mathbf{\text{α}}}^{\mathit{L}}$	${\mathbf{\text{α}}}^{\mathit{U}}$	${\mathbf{\text{γ}}}^{\mathit{L}}$	${\mathbf{\text{γ}}}^U$
A1	0.46	0.56	0.20	0.25	0.25	0.31	0.49	0.60	0.18	0.24	0.24	0.30
A2	0.79	0.90	0.00	0.08	0.08	0.14	0.08	0.17	0.39	0.44	0.44	0.49
A3	0.35	0.45	0.25	0.30	0.30	0.35	0.30	0.39	0.28	0.33	0.33	0.38
A4	0.23	0.33	0.31	0.36	0.36	0.41	0.41	0.51	0.23	0.28	0.28	0.33
A5	0.51	0.60	0.18	0.24	0.24	0.29	0.59	0.70	0.13	0.19	0.19	0.25
A6	0.83	0.94	0.00	0.06	0.06	0.11	0.25	0.33	0.34	0.39	0.39	0.44
A7	0.37	0.47	0.24	0.29	0.29	0.34	0.30	0.39	0.28	0.33	0.33	0.38
A8	0.25	0.35	0.30	0.35	0.35	0.40	0.45	0.55	0.20	0.25	0.25	0.30
A9	0.51	0.61	0.17	0.23	0.23	0.28	0.58	0.69	0.13	0.20	0.20	0.26
A10	0.79	0.90	0.00	0.08	0.08	0.14	0.08	0.17	0.39	0.44	0.44	0.49
A11	0.81	0.93	0.00	0.07	0.07	0.12	0.30	0.38	0.30	0.36	0.36	0.41
A12	0.40	0.50	0.22	0.28	0.28	0.33	0.26	0.36	0.30	0.35	0.35	0.40
A13	0.35	0.44	0.26	0.31	0.31	0.36	0.43	0.53	0.21	0.26	0.26	0.31
A14	0.51	0.61	0.17	0.23	0.23	0.28	0.48	0.58	0.19	0.25	0.25	0.31
A15	0.76	0.89	0.00	0.10	0.10	0.16	0.18	0.27	0.35	0.40	0.40	0.46

DOI: 10.7717/peerj-cs.3345/table-11

The following Tables 12A and 12B display the distance measure of alternatives for IVSFPIS and IVSFNIS. Table 13 shows the closeness coefficient of other options.

Table 12:

Distance measure of alternatives for IVSFPIS and IVSFNIS.

A. Closeness coefficient of alternatives.
	C1					C2					C3
	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$
A1	0.38	0.34	1.08	1.12	0.75	0.77	0.73	0.81	0.84	0.52	0.40	0.37	1.05	1.10	0.74
A2	0.82	0.78	0.78	0.81	0.50	0.98	0.94	0.72	0.73	0.43	0.56	0.54	0.90	0.97	0.63
A3	0.96	0.94	0.72	0.73	0.43	0.67	0.63	0.87	0.90	0.58	0.39	0.36	1.06	1.11	0.74
A4	0.68	0.64	0.87	0.90	0.57	0.79	0.75	0.80	0.83	0.52	0.31	0.27	1.13	1.17	0.80
A5	0.31	0.27	1.14	1.17	0.80	0.82	0.78	0.78	0.81	0.50	0.59	0.55	0.93	0.96	0.62
A6	0.79	0.75	0.80	0.83	0.52	0.86	0.82	0.77	0.79	0.48	0.79	0.75	0.80	0.83	0.52
A7	1.09	1.06	0.69	0.70	0.39	0.66	0.62	0.88	0.91	0.59	0.53	0.49	0.97	1.00	0.66
A8	0.70	0.66	0.86	0.88	0.56	0.70	0.66	0.86	0.88	0.56	0.39	0.35	1.07	1.11	0.74
A9	0.31	0.27	1.14	1.17	0.80	0.88	0.84	0.76	0.78	0.47	0.44	0.41	1.01	1.07	0.71
A10	0.82	0.78	0.78	0.81	0.50	0.98	0.94	0.72	0.73	0.43	0.56	0.54	0.90	0.97	0.63
A11	0.59	0.56	0.92	0.95	0.62	0.84	0.80	0.78	0.80	0.49	0.70	0.66	0.86	0.88	0.56
A12	0.72	0.70	0.83	0.86	0.54	0.70	0.66	0.86	0.88	0.56	0.53	0.49	0.97	1.00	0.66
A13	0.55	0.51	0.95	0.99	0.65	0.79	0.75	0.80	0.83	0.52	0.42	0.38	1.06	1.09	0.73
A14	0.31	0.27	1.14	1.17	0.80	0.88	0.84	0.76	0.78	0.47	0.43	0.39	1.03	1.08	0.72
A15	0.62	0.58	0.90	0.94	0.61	0.95	0.91	0.73	0.75	0.44	0.51	0.48	0.95	1.01	0.66
	C4					C5
	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$
A1	0.63	0.66	0.89	0.97	0.59	0.26	0.22	1.18	1.22	0.83
A2	0.34	0.34	1.11	1.16	0.77	0.81	0.78	0.79	0.81	0.50
A3	0.34	0.33	1.12	1.16	0.77	0.56	0.52	0.95	0.98	0.64
A4	0.32	0.30	1.13	1.17	0.79	0.82	0.78	0.78	0.81	0.50
A5	0.83	0.82	0.78	0.83	0.50	0.19	0.15	1.24	1.28	0.88
A6	0.38	0.36	1.07	1.13	0.75	0.81	0.78	0.79	0.81	0.50
A7	0.36	0.35	1.10	1.14	0.76	0.53	0.49	0.97	1.00	0.66
A8	0.39	0.37	1.07	1.12	0.74	0.77	0.73	0.82	0.84	0.53
A9	0.63	0.66	0.89	0.97	0.59	0.20	0.16	1.23	1.27	0.87
A10	0.34	0.34	1.11	1.16	0.77	0.81	0.78	0.79	0.81	0.50
A11	0.58	0.58	0.93	0.99	0.62	0.53	0.50	0.93	1.00	0.65
A12	0.46	0.46	1.02	1.07	0.69	0.39	0.35	1.06	1.11	0.75
A13	0.40	0.39	1.06	1.10	0.73	0.51	0.48	0.95	1.01	0.67
A14	0.66	0.70	0.87	0.96	0.57	0.20	0.16	1.23	1.27	0.87
A15	0.51	0.52	0.98	1.04	0.66	0.53	0.50	0.93	1.00	0.65

B. Continued closeness coefficient of alternatives.
	C6					C7					C8
	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$
A1	1.00	0.96	0.71	0.72	0.42	0.51	0.47	0.98	1.01	0.67	0.56	0.55	0.89	0.97	0.63
A2	0.41	0.38	1.04	1.09	0.73	0.75	0.72	0.82	0.85	0.53	1.00	0.96	0.71	0.72	0.42
A3	0.90	0.86	0.75	0.77	0.46	0.55	0.51	0.96	0.99	0.65	0.99	1.04	0.59	0.70	0.39
A4	1.00	0.96	0.71	0.72	0.42	0.51	0.47	0.98	1.01	0.67	0.22	0.18	1.22	1.26	0.86
A5	1.03	0.99	0.70	0.71	0.41	0.49	0.45	1.00	1.03	0.69	1.00	0.98	0.68	0.72	0.41
A6	0.38	0.34	1.07	1.12	0.75	0.81	0.78	0.79	0.81	0.50	0.86	0.82	0.77	0.79	0.48
A7	0.90	0.86	0.75	0.77	0.46	0.57	0.53	0.94	0.97	0.63	1.01	1.08	0.57	0.69	0.38
A8	1.00	0.96	0.71	0.72	0.42	0.51	0.47	0.98	1.01	0.67	0.16	0.12	1.27	1.31	0.90
A9	1.00	0.96	0.71	0.72	0.42	0.49	0.45	1.00	1.03	0.69	1.00	0.96	0.70	0.72	0.42
A10	0.41	0.38	1.04	1.09	0.73	0.75	0.72	0.82	0.85	0.53	0.98	0.96	0.69	0.72	0.42
A11	0.58	0.54	0.93	0.96	0.63	0.69	0.65	0.86	0.89	0.56	0.87	0.82	0.78	0.79	0.48
A12	0.92	0.88	0.74	0.76	0.45	0.61	0.57	0.92	0.94	0.61	1.07	1.13	0.57	0.68	0.36
A13	1.00	0.96	0.71	0.72	0.42	0.51	0.47	0.98	1.01	0.67	0.22	0.18	1.22	1.26	0.86
A14	1.00	0.96	0.71	0.72	0.42	0.51	0.47	0.98	1.01	0.67	1.00	0.96	0.70	0.72	0.42
A15	0.64	0.61	0.89	0.92	0.59	0.65	0.61	0.89	0.92	0.59	0.99	0.96	0.69	0.72	0.42
	C9					C10
	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{+}^{\mathit{U}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{L}}}$	${\mathbf{₰}}_{\mathit{q}}^{{-}^{\mathit{U}}}$	$\mathit{C}\text{C}|$
A1	0.63	0.59	0.90	0.93	0.60	0.59	0.55	0.92	0.96	0.62
A2	0.23	0.19	1.21	1.25	0.85	1.09	1.06	0.69	0.70	0.39
A3	0.76	0.72	0.82	0.84	0.53	0.83	0.79	0.78	0.80	0.49
A4	0.90	0.86	0.75	0.77	0.46	0.69	0.65	0.86	0.89	0.57
A5	0.58	0.54	0.94	0.96	0.63	0.47	0.43	1.01	1.05	0.69
A6	0.18	0.14	1.26	1.29	0.89	0.91	0.89	0.74	0.76	0.45
A7	0.73	0.69	0.84	0.86	0.55	0.83	0.79	0.78	0.80	0.49
A8	0.88	0.84	0.76	0.78	0.47	0.64	0.60	0.90	0.92	0.60
A9	0.57	0.53	0.94	0.97	0.64	0.49	0.45	1.00	1.03	0.68
A10	0.23	0.19	1.21	1.25	0.85	1.09	1.06	0.69	0.70	0.39
A11	0.20	0.16	1.24	1.28	0.88	0.85	0.82	0.77	0.79	0.48
A12	0.69	0.65	0.86	0.89	0.56	0.87	0.83	0.76	0.78	0.48
A13	0.77	0.73	0.81	0.84	0.52	0.66	0.62	0.88	0.91	0.58
A14	0.57	0.53	0.94	0.97	0.64	0.61	0.57	0.91	0.94	0.61
A15	0.26	0.22	1.18	1.23	0.84	0.98	0.95	0.72	0.73	0.43

DOI: 10.7717/peerj-cs.3345/table-12

Table 13:

Closeness coefficients of alternatives.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
A1	0.75	0.52	0.74	0.59	0.83	0.42	0.67	0.63	0.60	0.62
A2	0.50	0.43	0.63	0.77	0.50	0.73	0.53	0.42	0.85	0.39
A3	0.43	0.58	0.74	0.77	0.64	0.46	0.65	0.39	0.53	0.49
A4	0.57	0.52	0.80	0.79	0.50	0.42	0.67	0.86	0.46	0.57
A5	0.80	0.50	0.62	0.50	0.88	0.41	0.69	0.41	0.63	0.69
A6	0.52	0.48	0.52	0.75	0.50	0.75	0.50	0.48	0.89	0.45
A7	0.39	0.59	0.66	0.76	0.66	0.46	0.63	0.38	0.55	0.49
A8	0.56	0.56	0.74	0.74	0.53	0.42	0.67	0.90	0.47	0.60
A9	0.80	0.47	0.71	0.59	0.87	0.42	0.69	0.42	0.64	0.68
A10	0.50	0.43	0.63	0.77	0.50	0.73	0.53	0.42	0.85	0.39
A11	0.62	0.49	0.56	0.62	0.65	0.63	0.56	0.48	0.88	0.48
A12	0.54	0.56	0.66	0.69	0.75	0.45	0.61	0.36	0.56	0.48
A13	0.65	0.52	0.73	0.73	0.67	0.42	0.67	0.86	0.52	0.58
A14	0.80	0.47	0.72	0.57	0.87	0.42	0.67	0.42	0.64	0.61
A15	0.61	0.44	0.66	0.66	0.65	0.59	0.59	0.42	0.84	0.43
I	0.80	0.59	0.80	0.79	0.88	0.75	0.69	0.36	0.46	0.39
AI	0.39	0.43	0.52	0.50	0.50	0.41	0.50	0.90	0.89	0.69

DOI: 10.7717/peerj-cs.3345/table-13

Tables 14 and 15 display the normalized IVSF decision matrix and the weighted normalized IVSF decision matrix, respectively.

Table 14:

Normalized IVSF decision matrix.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
A1	0.944	0.893	0.923	0.747	0.944	0.560	0.974	1.027	0.579	0.772
A2	0.624	0.737	0.790	0.977	0.569	0.969	0.773	1.293	0.859	0.541
A3	0.543	0.986	0.932	0.978	0.727	0.615	0.943	1.061	0.932	0.874
A4	0.720	0.881	1.000	1.000	0.564	0.560	0.974	1.027	0.420	1.000
A5	1.000	0.849	0.783	0.628	1.000	0.547	1.000	1.000	0.877	0.735
A6	0.647	0.820	0.647	0.948	0.569	1.000	0.728	1.373	0.755	0.519
A7	0.490	1.000	0.824	0.961	0.746	0.615	0.924	1.082	0.962	0.847
A8	0.703	0.959	0.934	0.941	0.596	0.560	0.974	1.027	0.402	0.979
A9	1.000	0.806	0.891	0.747	0.990	0.560	1.000	1.000	0.860	0.727
A10	0.624	0.737	0.790	0.977	0.569	0.969	0.773	1.293	0.863	0.541
A11	0.778	0.836	0.704	0.791	0.738	0.835	0.822	1.216	0.754	0.527
A12	0.681	0.960	0.824	0.878	0.846	0.602	0.893	1.120	1.000	0.818
A13	0.811	0.881	0.916	0.928	0.756	0.560	0.974	1.027	0.420	0.883
A14	1.000	0.806	0.905	0.727	0.990	0.560	0.974	1.027	0.860	0.727
A15	0.761	0.755	0.832	0.839	0.738	0.785	0.861	1.162	0.863	0.553
I	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
AI	0.490	0.737	0.647	0.628	0.564	0.547	0.728	1.373	0.402	0.519

DOI: 10.7717/peerj-cs.3345/table-14

Table 15:

Weighted normalized IVSF decision matrix.

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10
A1	0.104	0.098	0.100	0.071	0.090	0.058	0.091	0.096	0.057	0.071
A2	0.068	0.081	0.086	0.093	0.054	0.100	0.072	0.121	0.085	0.050
A3	0.060	0.108	0.101	0.093	0.069	0.063	0.088	0.099	0.092	0.081
A4	0.079	0.097	0.109	0.095	0.054	0.058	0.091	0.096	0.042	0.093
A5	0.110	0.093	0.085	0.060	0.095	0.056	0.093	0.093	0.087	0.068
A6	0.071	0.090	0.070	0.090	0.054	0.103	0.068	0.128	0.075	0.048
A7	0.054	0.110	0.090	0.092	0.071	0.063	0.086	0.101	0.095	0.078
A8	0.077	0.105	0.102	0.090	0.057	0.058	0.091	0.096	0.040	0.091
A9	0.110	0.088	0.097	0.071	0.094	0.058	0.093	0.093	0.085	0.067
A10	0.068	0.081	0.086	0.093	0.054	0.100	0.072	0.121	0.085	0.050
A11	0.085	0.092	0.077	0.075	0.070	0.086	0.077	0.114	0.075	0.049
A12	0.075	0.105	0.090	0.084	0.081	0.062	0.083	0.105	0.099	0.076
A13	0.089	0.097	0.100	0.088	0.072	0.058	0.091	0.096	0.042	0.082
A14	0.110	0.088	0.098	0.069	0.094	0.058	0.091	0.096	0.085	0.067
A15	0.083	0.083	0.090	0.080	0.070	0.081	0.080	0.109	0.085	0.051
I	0.110	0.110	0.109	0.095	0.095	0.103	0.093	0.093	0.099	0.093
AI	0.054	0.081	0.070	0.060	0.054	0.056	0.068	0.128	0.040	0.048

DOI: 10.7717/peerj-cs.3345/table-15

Table 16 shows the utility degree and utility function of alternatives. Additionally, Fig. 4 presents a graphical representation of the other options.

Table 16:

Utility degree and utility function of alternatives.

	$\mathit{S}\mathcal{i}$	${\stackrel{˘}{\overline{\mathbf{{\rm Y}}}}}_{\mathit{i}}^{\mathbf{-}}$	${\stackrel{˘}{\overline{\mathbf{{\rm Y}}}}}_{\mathit{i}}^{\mathbf{+}}$	$\mathit{f}({\stackrel{˘}{\overline{\mathbf{{\rm Y}}}}}_{\mathit{i}}^{\mathbf{-}}\mathbf{)}$	$\mathit{f}({\stackrel{˘}{\overline{\mathbf{{\rm Y}}}}}_{\mathit{i}}^{\mathbf{+}}\mathbf{)}$	$\mathit{f}({\stackrel{˘}{\overline{\mathbf{{\rm Y}}}}}_{\mathit{i}}\mathbf{)}$	$\mathit{R}\mathit{a}\mathit{n}k$
A1	0.836	1.269	0.836	0.397	0.603	0.6628	7
A2	0.810	1.229	0.810	0.397	0.603	0.6421	12
A3	0.855	1.297	0.855	0.397	0.603	0.6776	4
A4	0.812	1.232	0.812	0.397	0.603	0.6436	10
A5	0.841	1.276	0.841	0.397	0.603	0.6665	5
A6	0.798	1.210	0.798	0.397	0.603	0.6323	15
A7	0.840	1.274	0.840	0.397	0.603	0.6657	6
A8	0.805	1.222	0.805	0.397	0.603	0.6383	13
A9	0.857	1.301	0.857	0.397	0.603	0.6795	2
A10	0.811	1.230	0.811	0.397	0.603	0.6425	11
A11	0.799	1.213	0.799	0.397	0.603	0.6334	14
A12	0.858	1.302	0.858	0.397	0.603	0.6803	1
A13	0.814	1.234	0.814	0.397	0.603	0.6448	8
A14	0.857	1.300	0.857	0.397	0.603	0.6792	3
A15	0.813	1.234	0.813	0.397	0.603	0.6447	9

DOI: 10.7717/peerj-cs.3345/table-16