The widespread implementation of the internet in recent times has resulted in the integration of Information and Communication Technology (ICT) into the everyday activities of educational institutions, research centres, organizations, government agencies, and individuals (Natarajan, 2015). The growth in ICT and the internet has enabled the development of new learning environments and techniques (Albarrak et al., 2010), providing educational institutions with an innovative educational framework called e-learning. E-learning utilizes multimedia, such as audio, video, animations, and text illustrations, to deliver education and knowledge to students without any location barriers, thus offering an alternative to traditional classroom instruction. The most common form of e-learning is a software application called a learning management system (LMS). An LMS is a digital platform that provides a variety of tools and resources to facilitate and manage online learning. It allows instructors to create and deliver course content, track and assess student progress, and communicate with learners. Similarly, learners can access course materials, participate in discussions, complete assignments, and receive feedback (Ramesh and Ramanathan, 2013). LMSs are commonly used in educational institutions, corporations, and other organizations that offer training and development programs. However, the implementation of an LMS requires significant financial and institutional commitment (Edrees, 2013). According to Caminero et al. (2013), an LMS is also a software system comprised of various tools that support teaching and learning activities. Because of the rapid development of computer and internet-based technologies, LMS has become a critical component in the advancement of educational systems. As a result, many LMSs are now available online, including both authorized and free versions (Cavus, 2007). Many educational institutions spend a significant amount of time and money to implement LMSs (Edrees, 2013). Several LMSs have been presented as open-source software (OSS) licenses, including Moodle, Sakai, ATutor, Dokeos, and eFront, which are especially useful for e-learning (Awang and Darus, 2012). OSS is software that does not require a license and comes with the source code. It is intended to address the rising costs of campus-wide software while allowing the development of learner-centred structures (Abdullateef et al., 2015, 2016b; van Rooij, 2011, 2012).

Selecting the wrong OSS-LMS package can have a negative impact on an organization’s business processes and roles. If an organization selects the wrong OSS-LMS package, it may not meet its requirements and expectations, resulting in inadequate training, lower employee productivity, and decreased organizational performance. An unsuitable OSS-LMS package can result in increased costs and time associated with system modifications or replacements. It is thus crucial for organizations to carefully evaluate and select an appropriate OSS-LMS package that aligns with their specific needs and objectives to ensure efficient and effective management of their e-learning initiatives. However, due to the wide variety of available options, lack of user experience and knowledge, and constant development of information technology, the process of selecting the most suitable OSS-LMS has become increasingly complex (Jadav and Sonar, 2011; Zaidan et al., 2015). Since multi-criteria decision-making (MCDM) methods have wide range of applications (Ulutas˘ et al., 2021; Semenas et al., 2021; Filip, 2021; Krishankumar et al., 2021; Ivanović et al., 2022; Deveci et al., 2022; Hezam et al., 2023a; Deveci et al., 2023; Gökmener et al., 2023; Gokasar et al., 2023a, 2023b; Petrovas et al., 2023), MCDM methods can be used to assess and select the best OSS-LMS package by taking into account multiple conflicting criteria. MCDM methods consider multiple criteria and weigh their relative importance to arrive at a decision that best aligns with the organization’s goals and objectives. The evaluation of OSS-LMS packages requires a thorough examination and investigation of various factors, such as usability, functionality, reliability, security, and compatibility with other systems. Organizations must carefully evaluate each criterion to ensure that the selected OSS-LMS package meets their specific needs and requirements. The application of MCDM methods enables organizations to streamline their decision-making process and make well-informed decisions regarding the selection of an OSS-LMS package.

Literature research has uncovered that various scholars have utilized diverse strategies for the assessment and determination of LMSs. The investigation carried out by Waynet Inc. (2007) depicted an overview style assessment of open-source LMSs aiming to suggest an LMS that could be applied by the Commonwealth of Learning. Hultin’s (2007) studied LMSs and how to assess them, contingent upon the learning condition and the customers’ needs. Graf and List (2005) introduced an appraisal of open-source e-learning stands by concentrating on the adaptation competences of the framework. Another assessment of LMSs was described by Wyles (2007). It was divided into two parts: 1st part portrayed the outcomes of an underlying assessment of open-source LMS software and the 2nd part described the assessment technique applied to choose the best LMS as a feature of the overall platform architecture. Kljun et al. (2007) aimed to assist the individuals who are engaged in e-learning to assess optimal LMS to suit them. The writers categorized the operators of LMSs into three clusters: learners, tutors, and administrators. Arh and Blazic (2007) devised an MCDM model that employs an expert structure to choose the most appropriate and effective LMS from Blackboard 6, Moodle 1.5.2, and CLIX 5.0. Machado and Tao (2007) examined the client experience of two competing LMSs, Moodle and Blackboard, based on their ease of use and viability. Çetin et al. (2010) used Analytical Hierarchy Process (AHP) method to address the problem of LMS evaluation using nine evaluation criteria. Albarrak et al. (2010) evaluated three OSS-LMSs: Jusur, Moodle, and Sakai. Srdevic et al. (2012) used AHP to select the most reasonable LMSs as well. Caminero et al. (2013) utilized a performance assessment technique for three OSS-LMSs, namely dotLRN, Moodle, and Sakai. Edrees (2013) assessed two LMSs, Blackboard, and Moodle, based on their readiness to support Web 2.0. Ramesh and Ramanathan (2013) developed a tool to assess LMSs based on six categories of criteria. Işik et al. (2015) applied fuzzy AHP for choosing the best LMS based on nine considered criteria. Hock et al. (2015) assessed three OSS-LMSs, namely Atutor, Ilias, and Moodle, based on the dependence on the convenience and utilized acknowledgment of the systems. Abdullateef et al. (2016a) introduced the assessment and determination of three OSS-LMSs based on the three directions, namely collection of available three OSS-LMSs, detail of the assessment criteria, and capability of the selection techniques. Karagöz et al. (2017) built up a mobile app for finding analogy of two open-source LMSs and two commercial LMS dependent on some specific criteria. Adewumi et al. (2019) tried to evaluate the LMS software selection using questionnaires for experts and their suggestions. Al Amoush and Sandhu (2020) has focused on the instructor’s perspective for evaluating LMS uses. Santiago et al. (2020) tried to determine the academic efficiency performance by evaluating different LMS systems. Alturki and Aldraiweesh (2021) proposed a model which shows the efficiency of LMS model during covid-19 period. A summary of these works is presented below (see Table 1).

The weighted sum model (WSM) and weighted product model (WPM) were combined by Zavadskas et al. (2012) to develop a unique utility degree-based MCDM method referred to as WASPAS. This methodology was designed to deal with a variety of realistic decision-making concerns. WASPAS allows decision-makers to flexibly assign weights to criteria based on their relative importance, reflecting the preferences and priorities of the decision-maker. It considers multiple criteria simultaneously, enabling a comprehensive assessment of alternatives based on different dimensions or factors. The method aggregates the performance scores using the weighted sum product approach, which takes into account the interdependencies among criteria and the performance of alternatives. WASPAS method provides a transparent decision-making process, as it allows decision-makers to clearly understand how the final scores are calculated and how each criterion contributes to the overall evaluation. Unfortunately, WASPAS model fails to deal with the “consensus-reaching process” for experts. To tackle this, we present a consensus WASPAS methodology with IF data. The procedural steps of the proposed consensus-based decision-making model are as follows:

Step 1: Construct the initial IF decision matrices.

Assume that m is the number of alternatives Q k ( k = 1 , 2 , … , p ) and n is the number of criteria T t ( t = 1 , 2 , … , q ) connected with a group decision-making issue in which each alternative is evaluated by the decision-makers E r ( r = 1 , 2 , … , l ) under the IF environment. Consider that the initial findings examined by the decision-makers are depicted as the IF decision matrices M r = [ φ r ( k t ) ] p × q = [ ⟨ α r ( k t ) , β r ( k t ) ⟩ ] p × q .

Step 2: Obtain the aggregated IF decision matrix by employing the IFWA (or IFWG) operator.

The aggregated IF decision matrix is [ φ ( k t ) ] p × q = [ ⟨ α ( k t ) , β ( k t ) ⟩ ] p × q , where: (4) φ ( k t ) = IFWA ( φ 1 ( k t ) , φ 2 ( k t ) , … , φ l ( k t ) ) = ⨁ r = 1 l ( δ r φ r ( k t ) ) or φ ( k t ) = IFWG ( φ 1 ( k t ) , φ 2 ( k t ) , … , φ l ( k t ) ) = ⨂ r = 1 l ( φ r ( k t ) ) δ r , where δ r is the weight of E r .

Step 3: Find the consensus degree of each decision-maker.

Utilizing the fact that the correlation measure is capable of describing the similarity degree between various opinions, we define the correlation measure ψ t ( r ) of the decision-maker E r under the criterion T t in this way: (5) ψ t ( r ) = ∑ k = 1 p ( Dist k t ( r ) Dist t ( r ) − 1 m ∑ k = 1 p Dist k t ( r ) Dist t ( r ) ) × ( Dist k t Dist t − 1 m ∑ k = 1 p Dist k t Dist t ) ∑ k = 1 p ( Dist k t ( r ) Dist t ( r ) − 1 m ∑ k = 1 p Dist k t ( r ) Dist t ( r ) ) 2 × ∑ k = 1 p ( Dist k t Dist t − 1 m ∑ k = 1 p Dist k t Dist t ) 2 ( t = 1 , 2 , … , q ; r = 1 , 2 , … , l ) , where φ r ( k t ) ( + ) = ⟨ max k α r ( k t ) , min k β r ( k t ) ⟩ , φ r ( k t ) ( − ) = ⟨ min k α r ( k t ) , max k β r ( k t ) ⟩ , φ ( k t ) ( + ) = ⟨ max k α ( k t ) , min k β ( k t ) ⟩ , φ ( k t ) ( − ) = ⟨ min k α ( k t ) , max k β ( k t ) ⟩ , Dist k t ( r ) = Dist ( φ r ( k t ) , φ r ( k t ) ( + ) ) , Dist t ( r ) = Dist ( φ r ( k t ) ( + ) , φ r ( k t ) ( − ) ) , Dist k t = Dist ( φ ( k t ) ( + ) , φ ( k t ) ) , Dist t = Dist ( φ ( k t ) ( + ) , φ ( k t ) ( − ) ) .

Next, the consensus degree ρ ( r ) of the decision-maker E r can be defined as: (6) ρ ( r ) = 1 q ∑ t = 1 q ψ t ( r ) ( r = 1 , 2 , … , l ) .

It can be verified that − 1 ⩽ ρ ( r ) ⩽ 1. The greater value ρ ( r ) means the stronger consensus degree of the decision-maker E r in the group. If ρ denotes the minimum consensus degree, then ρ ( r ) ⩾ ρ needs to be attained. When ρ ( r ) < ρ, the FF decision matrices from Step 1 should be modified until ρ ( r ) ⩾ ρ is obtained for all decision-makers.

Step 4: Normalize the aggregated IF decision matrix.

Suppose that the normalized aggregated IF decision matrix is [ φ ˜ ( k t ) ] p × q = [ ⟨ α ˜ ( k t ) , β ˜ ( k t ) ⟩ ] p × q , where: (7) φ ˜ ( k t ) = ⟨ α ( k t ) , β ( k t ) ⟩ , if C t is beneficial criterion , ⟨ β ( k t ) , α ( k t ) ⟩ , if C t is non-beneficial criterion .

Step 5: Estimate the IF “relative significance degree” (RSD) for every alternative.

Suppose ϑ t is the weight of T t ( t = 1 , 2 , … , q) with ∑ t = 1 q ϑ t = 1 and 0 ⩽ ϑ t ⩽ 1.

Step 5.1: The IF-RSD of Q k using WSM is calculated as: (8) RSD ‾ ( Q k ) = ⨁ t = 1 q ( ϑ t φ ˜ ( k t ) ) .

The IF-RSD of Q k using WPM is calculated as: (9) RSD ‾ ‾ ( Q k ) = ⨂ t = 1 q ( φ ˜ ( k t ) ) ϑ t .

Step 5.2: The overall IF significance degree of Q k is calculated by: (10) η k = ( w RSD ‾ ( Q k ) ) ⊕ ˜ ( ( 1 − w ) RSD ‾ ‾ ( Q k ) ) ( k = 1 , 2 , … , p ) , or (11) η k = ( RSD ‾ ( Q k ) ) w ⊗ ˜ ( RSD ‾ ‾ ( Q k ) ) ( 1 − w ) ( k = 1 , 2 , … , p ) . Here, w ∈ [ 0 , 1 ]. For w = 1, and w = 0, WASPAS reduces to WSM and WPM, respectively.

Step 6: Compute the scores of the IFNs η k ( k = 1 , 2 , … , p ).

Step 7: Generate the ranking order of alternatives and chose the best option.

A. Problem Description

In order to avoid face-to-face interactions, the COVID-19 pandemic has forced many educational institutions to quickly move from traditional attendance-based education to online distance learning. Online distance learning can be either synchronous or asynchronous, depending on the mechanism of delivery. During any pandemic situation like Covid-19, it is very essential for an educational institution to select an appropriate OSS-LMS package to manage administration, monitoring, reporting of online classes and training programs, create a virtual classroom where teachers can interact with their students and conduct learning activities online. Suppose an educational institution wants to select an efficient OSS-LMS package out of four OSS-LMS packages: ATutor (Q1) (Graf and List, 2005; Abdullateef et al., 2016a), eFront (Q3) (Abdullateef et al., 2016b), Moodle (Q2) (Caminero et al., 2013; van Rooij, 2011; Abdullateef et al., 2016a, 2016b; Graf and List, 2005 ), and Sakai (Q4) (Caminero et al., 2013; van Rooij, 2011; Abdullateef et al., 2016a, 2016b; Graf and List, 2005). Their information is given in Table 2.

In this study, four OSS-LMS are being considered as alternatives and will be evaluated based on fifteen criteria. The criteria and their corresponding citations, arranged by publication year within each criterion, are as follows: Activity Tracking (T1): (Graf and List, 2005; Arh and Blazic, 2007; Srdevic et al., 2012; Abdullateef et al., 2016a); Reliability (T2): (Jadav and Sonar, 2011; Srdevic et al., 2012; Abdullateef et al., 2016b); Course Development (T3): (Graf and List, 2005; Arh and Blazic, 2007; Srdevic et al., 2012; Abdullateef et al., 2016a); Assessment (T4): (Arh and Blazic, 2007; Graf and List, 2005; Srdevic et al., 2012; Abdullateef et al., 2016a); Backup and Recovery (T5): (Arh and Blazic, 2007; Srdevic et al., 2012; Abdullateef et al., 2016a); Error Reporting (T6): (Jadav and Sonar, 2011; Abdullateef et al., 2016a); Efficiency (T7): (Abdullateef et al., 2016b); DBMS standards (T8) (Jadav and Sonar, 2011; Abdullateef et al., 2016b); OS compatibility (T9) (Jadav and Sonar, 2011; Abdullateef et al., 2016b); IMS LIP (T10) (Arh and Blazic, 2007; Srdevic et al., 2012; Abdullateef et al., 2016a); AICC Computer managed Instruction (T11) (Arh and Blazic, 2007; Srdevic et al., 2012; Abdullateef et al., 2016a); Authentication (T12) (Arh and Blazic, 2007; Graf and List, 2005; Srdevic et al., 2012; Abdullateef et al., 2016b); Authorization (T13) (Arh and Blazic, 2007; Srdevic et al., 2012; Abdullateef et al., 2016a); Troubleshooting, maintenance and upgrading (T14) (Caminero et al., 2013; Abdullateef et al., 2016b); and Communication synchronous and asynchronous (T15) (Arh and Blazic, 2007; Graf and List, 2005; Srdevic et al., 2012; Abdullateef et al., 2016a).

Details of the criteria based on which OSS-LMS packages are to be evaluated are:

1) Activity Tracking (T1):

Monitoring students’ learning activities is a part of activity tracking in the classroom. The reports are meant to offer the instructor a sense of what occurs in a pedagogically important course. Progress reports are part of course analysis, which often also includes time stamps for the activities’ occurrences. The handling level of the course and participants may be checked by the tutor. The first and last login dates and times are also visible to the course tutor. A tutor can view the amount of time spent on the course or other activities for specific pupils.

2) Reliability (T2):

The software can function continuously without crashing. Software package reliability refers to its capacity to operate consistently under particular circumstances without crashing. The degree of fault tolerance for the software is evaluated using consistency. The number of crashes during a certain task’s execution can also be monitored to gauge dependability.

3) Course Development (T3):

The organization of the course using a web interface satisfies modifying the course outline, the curriculum, the inclusion of customization of the student tools, the communication tools, etc. The course’s content and structure are both easily editable by the author. Content navigation tools are generated automatically by the system. A single zip package may be used to upload and download HTML pages, pictures, and Flash videos. Links must be built between content pages, between courses, and to student tools. The content and course format may both be readily changed by the instructor. The content navigation can be generated automatically by the system.

4) Assessment (T4:)

The test’s questions and format may be simply created or modified by the tutor. The system enables the learner to evaluate themselves. This feature allows the student to evaluate themselves. The method offers online evaluation. The mechanism makes student transcripts available. A quiz editor is provided by the system. With assessment, the tutor has the option to put the student to the test in a variety of ways. several testable possibilities. The test’s questions and content format may simply be edited by the author. Features automated evaluation of the reliability of the test questions and the capacity to import tests from other programs and systems.

5) Backup and recovery (T5):

The software package is capable of providing backup and recovery features. The DBMS backup and recovery subsystem is in charge of recovery. For example, if the computer system breaks in the middle of a complicated update transaction, the recovery subsystem is in charge of restoring the database to the condition it was in before the transaction began. The recovery time target is the maximum amount of time required to get your learning management system back up and running after a failure. When it comes to judging how secure your computer systems are, the RTO is by far the most disregarded criterion. “Are we backed up?” business owners frequently inquire. Usually, the response is “yes”.

6) Error reporting (T6):

One of the most significant requirements is the software package’s error reporting and messaging capability. Sometimes software suffers from various errors or flaws, and prompt reporting of those errors is critical for further resolution in the run time. Error reporting always makes a system possible for smooth processing, which is extremely important for an LMS system since at the time of online assessment, such reporting and rapid remedy are always useful for students and teachers.

7) Efficiency (T7):

Since everyone learns differently, an effective LMS should provide choices for configuring accessibility, display settings, and demonstrating methods in a reasonable period to meet a wide range of courses, learning styles, and accessibility demands. Keep an eye out for an LMS that can be easily utilized for training, learning, and evaluation all at once. The major factor enabling the software package to deliver results in a suitable time is data size.

8) DBMS standards (T8):

A learning management system (LMS) is software used to offer and administer educational courses. It is a client-server system that is often web-based and used to manage student enrollment, course content distribution, test and assignment administration, and associated record keeping. It keeps a record of all information pertaining to students, including their tuition and financial obligations, academic performance, use of school-provided transportation, and frequent attendance at libraries, labs, computer labs, and other facilities. Database applications that are often used include MS-Access, MS-SQL, MS-Excel, Oracle, DB2, Informix, Sybase, MySQL, and Ingrace.

9) OS compatibility (T9):

When using the LMS system, package compatibility with the operating systems MS Windows, Novell, Unix, Linux, and MAC is crucial. When switching from one OS to another, it is necessary to also switch modules, quizzes, courses, etc. OS compatibility is crucial in this circumstance.

10) IMS LIP (T10):

A specification for a common method of storing data on learners is the IMS Learner Information Package (LIP). LIP is made to make it possible to transfer learner data between different software programs, including their current progress and rewards. The Centre for Recording Achievement and CETIS have since modified the LIP standard for usage within the UK HE Sector, resulting in a mapping of the UK HE Transcript to LIP so that crucial student data may be transmitted electronically.

11) AICC Computer managed Instruction (T11):

“Computer Managed Instruction” is referred to as CMI. CMI is a general abbreviation that may be used to describe any type of computer-based learning in that environment. eLearning developers providing less support: AICC is still supported at a basic level by the majority, of course, authoring tools and learning management systems, although instructional designers and course developers are increasingly adopting more recent e-learning standards. The mobility of a course across different CMI learning environments and the communication between a lesson and the learning environment are all covered in the CMI standard.

12) Authentication (T12):

To prevent replay attacks, common security procedures concentrate on how login credentials and subsequent tokens are handled. Application security includes controls over user behaviour and data privacy. The controlling of verification credentials and successive tokens is the main focus of standard security procedures to stop replay attacks.

13) Authorization (T13):

Following successful authentication, authorization processes determine what the user is permitted to do. The majority of web application logging and monitoring is handled by the application framework. Systems that allow anonymous users must be strengthened to validate every user input.

14) Troubleshooting, maintenance and upgrading (T14):

Average independent code module sizes are usually advantageous. The level of module independence can be determined by specifying whether groups or sub-modules must be installed together, even if only a portion is required. The software package’s maximum number of concurrent users it can support should also be noted. Additionally, the software should have the capability to divide into multiple application tiers that can be distributed across multiple servers, as well as the ability to distribute modules across these servers. The software’s ability to be altered is referred to as maintainability, and modifications may include corrections, enhancements, or adaptations of the program to accommodate changes in the environment, requirements, and functional requirements. Measuring maintainability measures in a constrained experimental environment is challenging; they require extensive real-world testing.

15) Communication synchronous and asynchronous (T15):

The LMS emphasizes asynchronous and synchronous communication, mostly in the form of chat rooms and threaded discussion boards. Discussion forums are the major threaders of asynchronous communication. Email communication is crucial in a learning setting. Creators can converse with and observe who is within. Students have access to several discussion platforms for information sharing. Chat rooms, audio conferences, and/or video conferencing are the principal uses of synchronized communications. Wherever uncertainty has to be cleared up or for any other reason, online dialogue between students and instructors is always a smart alternative. The technology allows for the download of all chatroom statistics. Through this device, audio and video conferencing are also possible.

B. Problem Solution

A team of four decision-making specialists was constituted to select the best option among the considered OSS-LMS packages. The linguistic variables and their accompanying IFNs were defined by experts in Table 3. Table 4 gives the IF linguistic decision matrix.

To obtain a reasonable result, we implement the proposed consensus-based IF-WASPAS model to prioritize the considered options. Assume that DMEs’ weights are respectively 0.20, 0.27, 0.30, and 0.23. Then, the aggregated IF decision matrix (Table 5) is obtained by using the IFWA operator. Assume that the minimum consensus degree is ρ = 0.25. The consensus degree of each expert is calculated based on Eqs. (6) and (7) as: ρ ( 1 ) = 0.253, ρ ( 2 ) = 0.496, ρ ( 3 ) = 0.451, and ρ ( 4 ) = 0.210. Since ρ ( 4 ) < 0.25, the initial assessments for 4th expert should be modified. In the revised assessment matrix, for the 4th expert, the updated entries are: (Q1, T1): H, (Q4, T1): VVH, (Q3, T12): VVL. Then, the revised aggregated IF decision matrix is constructed with the help of IFWA operator. The consensus degrees are recalculated with the help of Eqs. (5) and (6) as: ρ ( 1 ) = 0.252, ρ ( 2 ) = 0.496, ρ ( 3 ) = 0.446, and ρ ( 4 ) = 0.260. Since ρ ( r ) ⩾ 0.25 ( r = 1 , 2 , 3 , 4 ), desired consensus reaching process has been done.

Since all the considered criteria are of benefit type, normalization is not required. Suppose the weights of criteria are: ϑ 1 = 0.06, ϑ 2 = 0.1, ϑ 3 = 0.2, ϑ 4 = 0.1, ϑ 5 = 0.01, ϑ 6 = 0.05, ϑ 7 = 0.1, ϑ 8 = 0.05, ϑ 9 = 0.1, ϑ 10 = 0.05, ϑ 11 = 0.06, ϑ 12 = 0.05, ϑ 13 = 0.01, ϑ 14 = 0.05, and ϑ 15 = 0.01. The IF-RSD of all alternatives using WSM and WPM are then calculated using Eqs. (8) and (9), respectively. The overall IF significance degrees of alternatives are calculated by Eq. (10) (taking w = 0.5) and are given as: η 1 = ⟨ 0.8411 , 0.1305 ⟩ , η 2 = ⟨ 0.8900 , 0.0887 ⟩ , η 3 = ⟨ 0.8595 , 0.1206 ⟩ , η 4 = ⟨ 0.8787 , 0.1012 ⟩ . The scores of these FFNs are respectively 0.8553, 0.9006, 0.8695, 0.8887 according to which Q 2 ≻ Q 4 ≻ Q 3 ≻ Q 1 (“≻” means “superior to”) as preference order with Q 2 as the most suitable option.

The discussion section is divided into two parts: (A) sensitivity investigation of criteria weights, and (B) comparison of the suggested approach to currently used methods.

A. Sensitivity analysis of criteria weights

In this section, sensitivity analysis is used to assess the impact of a suitable criterion on the results of the model that has been provided. The term “most significant criterion” is used to denote a criterion with the highest weight value. Saha et al. (2021a, 2023a) used Eq. (12) to calculate the weight ratio. (12) ϑ c = ( 1 − ϑ s ) × ϑ c 0 Θ c 0 = ϑ c 0 − α c × Δ x , where ϑ c – variation in criteria weights,

ϑ s – weight of the most prominent criteria,

ϑ c 0 – original values of criteria weights,

Θ c 0 – sum of actual values of modified criteria weights,

α c – weight coefficient of elasticity.

The relative significance of the various values of the criteria weights is demonstrated by α c when we associate the variations with the most pertinent criterion weight. It is possible to calculate α c (Kirkwood, 1997) using the following formula: (13) α c = ϑ c 0 Θ c 0 . Hypotheses of the adopted sensitivity analysis are as follows Kahraman (2002):

The fluctuation degree applied to a set of weight coefficients is given by a parameter Δ x that is specified in terms of the corresponding weighted elastic coefficient, as indicated in Eq. (14). If the weight of the most important characteristic varies, a constraint should be in place. If this is not done, related weights may turn negative and the restrictions on proportionate weights may be broken. Positive and negative values of the parameter could signify an increase or reduction in the degree of importance, respectively. From the following, we may infer the limit values of Δ x. (14) − ϑ s 0 ⩽ Δ x ⩽ min { ϑ c 0 α c } .

The boundaries and original weights of criteria are established and estimated using the pre-defined parameters. The values of a group of weight coefficients are determined by applying Eqs. (15) and (16): (15) ϑ s = ϑ s 0 + α s × Δ x , (16) ϑ c = ϑ c 0 − α c × Δ x , where ϑ s 0 – given weight of the most significant criteria,

  ϑ c 0 – given value of changeable weights.

It should be kept in mind that the revised criteria set mentioned above satisfies the equation ∑ ϑ s + ∑ ϑ c = 1, which is thought of as the fundamental requirement of the percentage of weight coefficients. The rankings of the alternatives are established taking into consideration the updated criteria weight values. T3 has the most significant weight coefficient, which was determined from the analysis, of ϑ 3 = 0.2, making it one of the most significant criteria in this study. After that, the weight elasticity values are assessed, and it is found that the weight coefficient’s ( Δ x) fluctuation bounds fall within the range of − 0.2 ⩽ Δ x ⩽ 0.8. Several criteria weight sets (G1, G2, … , G15) are then formed based on restrictions for the change of weight coefficient values of criteria.

The weight sets are split into fifteen groups for the range − 0.2 ⩽ Δ x ⩽ 0.8. The weight coefficients are viewed from various perspectives for each set using Eqs. (15) and (16), and these values are shown in Fig. 1. As a result, several criteria weight sets are applied to determine the alternatives’ final scores, which are shown in Fig. 2. The results of this study showed that alternative Q2 is the best choice. After that, we used the results from various weight sets taken into account by various criteria to determine the SRCC values (Saha et al., 2021a). A “high correlation” between alternative ranks is observed, as indicated by the average SRCC value of 0.9 (Saha et al., 2021b). Therefore, stability of the model is established.

B. Comparative Investigation

This part aims to provide a comparative analysis of the developed consensus-based IF decision support model with the existing IF decision-making methods, namely IF-TOPSIS (Rouyendegh et al., 2020), IF-MULTIMOORA (Garg and Rani, 2022), IF-EDAS (Mishra et al., 2020), IF-COPRAS (Kumari and Mishra, 2020), and IF-MARCOS (Deb et al., 2022). These methods are applied to solve the addressed selection issue of OSS-LMS package selection. According to the comparison results, the ranking order obtained by these existing methods is Q 2 ≻ Q 1 ≻ Q 3 ≻ Q 4 . On the other hand, the IF-c-WASPAS method generates the order Q 2 ≻ Q 1 ≻ Q 3 ≻ Q 4 which is exactly the same.

Some advantages of IF-c-WASPAS are as follows:

Adaptive online educational practices are a constantly evolving area of study, notably in academic contexts. It is a unique way for people and organizations to acquire knowledge and to fulfill their demands. These demands can be reached by the use of LMSs that are elements of e-learning. LMSs are software-based applications that incorporate a package of schemes for learning and training online. This LMS has now enhanced a top priority and major projects in enterprises and educational organizations. The number of accessible OSS-LMSs is constantly booming and getting significant projection. OSS is enhancing a choice for every institution, as they are advantageous to users in enabling platforms to be transformed according to user specifications, and because of the minimum costs charged to get a more reliable service, compared to other software like commercial ones that need licensing fees to run, with an added subscription and also a maintenance fee to make the LMS software up-to-date. In answer to mounting demands, software enterprises have been recommending a different types of software packages that can be adapted and fulfill the precise needs of an establishment as well as handmade. The evaluation and adoption of an improper OSS-LMS package unfavourably affect the marketing procedures and duties of the institution. This paper focuses on consensus based IF-WASPAS methodology to help a decision-maker or administrators in the education environment to compare and appraise the OSS-LMS packages and choose the suitable one over certain criteria. The comparison is presented between four OSS-LMSs, namely Moodle, Sakai, ATutor, and eFront. The outcomes show that MOODLE is best for the present study compared to other four OSS-LMSs. As potential future research directions, a combination of subjective and objective weights for expert weight determination can be utilized, taking into account assessments of expert similarity (Saha et al., 2021a, 2021b). To enhance the significance of the model, incorporating measures such as dispersion, uncertainty, and cross-entropy can be explored (Saha et al., 2023a, 2023b). The developed consensus approach can also be integrated with MULTIMOORA, MARCOS, DNMA, COPRAS, and VIKOR methodologies to develop novel techniques. Furthermore, the developed model can be extended to encompass dual hesitant fuzzy sets and probabilistic dual hesitant fuzzy sets.