1 Introduction

Evaluating and selecting innovative project ideas presents a challenging task complicated by several factors, such as the uncertainty of the innovation environment, the qualification level of decision makers, the presence of many ideas, and the complexity of cognitive processes [1]. As the number of ideas increases, considering them becomes more challenging, and generating more creative ideas does not necessarily lead to better idea selection performance [2], given the complexity of the cognitive processes. Moreover, only a small percentage of project ideas ever achieve commercial success in an innovation and new product development process, making guidance essential for decision makers during design [3]. The importance and difficulty of the evaluation and selection phase are thus evident [3, 4]. To address these challenges in the current business environment, a systematic approach to select the best new project ideas is required to enhance the effectiveness and efficiency of the decision process [5].

Multi-criteria decision-making (MCDM) techniques are among the most popular methods used in previous research on corporate sustainability for innovation, and they are effective techniques for investigating, evaluating, and ranking projects [6]. All MCDM problems have common characteristics, such as multiple criteria, a conflict among criteria, unmeasurable units, alternatives, and preference decisions [7, 8]. In sophisticated MCDM scenarios, knowledge, and judgments about the criteria, alternatives and the set of alternatives/criteria may change over time. Alkan and Kahraman [9] defined a dynamic decision environment as one that involves changes in the judgments of criteria and alternatives depending on the conditions that arise in the future. The definition of a dynamic decision environment can also be extended to circumstances in which the sets of alternatives and criteria are subject to change over time. In a dynamic decision environment, a complete change in the relative ranking or ordering of alternatives after adding or removing one or more alternatives is referred to as the rank reversal problem in the literature [10].

The alternative-by-alternative comparison-based (ABAC) method is used to evaluate innovative project ideas in this study. ABAC, as proposed by Biswas et al. [10], is based on the idea that comparison-based functions and dominance relationships can be used to solve complex problems under conflicting criteria. The ABAC method has been applied in different fields, such as disaster vulnerability assessment and failure mode and effects analysis [11, 12]. In addition, the main motivation for using ABAC is that it can effectively manage the rank reversal problem [10]. Most potential innovative ideas are eliminated during the innovation funnel by being subjected to pre-assessments such as crowd voting, proof of concept, and minimum viable product (MVP) tests, which are a natural and inevitable roadmap of the innovation process. Therefore, the ranking of the superiority of the idea alternatives relative to each other might change in other well-known MCDM methods. From this viewpoint, ABAC is preferred in the evaluation phase of this study.

ABAC is a novel technique in the field of MCDM, making its application in the selection of innovation projects a distinctive aspect of this study. In addition, we leverage fuzzy sets to address the inherent uncertainty in the innovation environment. In decision-making environments, especially in real-world situations such as project selection, imprecision and uncertainty are common and cannot be precisely described by crisp or deterministic models [9]. The concept of fuzzy set theory, first introduced by Zadeh [13], has been widely used in various applications to address these issues [14, 15]. To solve more complicated decision-making problems, different variations of ordinary fuzzy sets have been proposed in the literature, such as type-2 fuzzy sets [16], hesitant fuzzy sets [17], and intuitionistic fuzzy sets (IFSs) [18]. IFS was implemented to enhance the modeling capabilities of fuzzy sets. It employs a set of values within a closed interval to indicate the degrees of membership and non-membership, ensuring that (i) their combined sum is equal to or less than one, and (ii) there is the potential for a third parameter known as the hesitation margin [19, 20]. IFSs are one of the most widely used types of fuzzy sets in the literature, offering a high degree of flexibility in handling uncertain data by incorporating the degree of membership and non-membership [21]. Compared with ordinary fuzzy sets, IFS provides more information by incorporating negative information alongside positive information. Various information measures, such as distance operator [22, 23], similarity operator [24, 25], and ranking methods related to IFS, have been proposed to gain insights into the structure of intuitionistic fuzzy information [26,27,28]. By using IFSs, data can be more accurately represented, and uncertainties inherent in decision-making problems can be better managed. These advancements suggest a trend in knowledge representation where intuitionistic fuzzy sets are preferred to deal with uncertainty.

This study introduces the intuitionistic fuzzy alternative-by-alternative comparison-based (IF-ABAC) method developed by extending the ABAC method to an intuitionistic fuzzy (IF) environment. The developed method is used in the evaluation phase. Additionally, the best–worst method (BWM) is utilized to determine the criteria weights, where decision makers also present their evaluations using intuitionistic fuzzy values (IFVs). Furthermore, it is essential to note that many decision-making problems involve multiple decision makers; thus, many methods have been extended to group decision-making environments to adapt to this requirement. Based on this, the classical ABAC method is transformed into a process involving more than one decision maker to improve decision efficiency and robustness. As such, an integrated group decision-making methodology fully utilizing IFVs is presented to accomplish the main objective and address the outlined research gaps.

In the case study, the potential innovative ideas of entrepreneurs are gathered through an entrepreneurship acceleration program organized through the innovation platform offered by InnoCentrum, a Turkey-based corporate innovation management software company. This research contributes to the literature by introducing a novel integrated framework and enhancing the evaluation process of innovative project ideas. In addition, this framework is tested and validated through its application to a real-world problem.

In summary, based on the above characteristics, we can underline the following contributions of the proposed methodology:

  • This paper introduces a systematic approach for evaluating innovative project ideas within a limited research area.

  • The IF-ABAC method is developed, which enables more robust and efficient results by using uncertain information in the decision-making process and effectively addresses the rank reversal problem.

  • An integrated methodology is presented that combines IF-BWM and IF-ABAC, where expert opinions are expressed in intuitionistic fuzzy numbers.

  • The applicability and effectiveness of the developed group decision-making methodology are demonstrated for an innovative project idea evaluation problem in which the set of alternatives is subject to change.

The remaining of this study is organized as follows. Section 2 reviews the related literature on project evaluation and selection in the innovation context. In Sect. 3, the principles of IFSs are discussed. Moreover, this section presents the steps of the IF-BWM and details of the proposed IF-ABAC method. Section 4 applies the proposed methodology to a case study of an innovative project idea evaluation problem. Section 5 discusses a comprehensive sensitivity and comparative analysis to demonstrate the validity of the results. Finally, Sect. 6 provides the conclusion, limitations, and suggestions for further research.

2 Literature review

The literature on evaluating and selecting innovative project ideas can be divided into two main categories: qualitative dimensions and decision-support tools with quantitative techniques. The first category focuses on qualitative factors that influence the idea generation and evaluation process, such as the cognitive level [1], behavioral level of participants [4], or other factors that affect the idea generation and evaluation phase [29, 30]. The second category mainly employs decision-support tools with quantitative techniques, such as voting, ranking, or scoring, to support decision making [2, 3, 5]. For instance, Klein and Garcia [31] suggested an approach for using the crowd to filter the ideas submitted in open innovation engagements. They provide a taxonomy of idea filtering techniques: author and content-based filtering. Author-based filtering filters ideas according to the contributor’s experience and level of competence. Content-based filtering distinguishes ideas based on their content. This approach has two sub-dimensions. One is algorithmic, where decision makers use software to derive idea quality metrics based on text-mining techniques. The other is crowd-based filtering, which is the most common practice in crowdsourcing and open innovation engagement. Author-based filtering or algorithmic techniques may contribute to the decision-making process; however, offering a final assessment with these techniques is challenging. Therefore, the authors mainly focus on crowd-based filtering systems such as voting, rating, ranking, and prediction markets.

Despite not as much as idea generation, studies on idea evaluation have begun to receive attention [2], but there is no consensus on how to evaluate innovative project ideas. This is quite intelligible because of the uncertainty of innovation environments. It is unclear who will conduct the evaluations as well as the evaluation method. Runco and colleagues found that people are mostly entirely accurate in evaluating others’ ideas, as indicated by a significant positive correlation between their evaluations and those of experts [32,33,34]. Çubukcu et al. [35] analyzed that while harnessing the power of groups is essential and valuable, it may make sense to assess the average scores of the groups to provide insight instead of the primary evaluation technique in the selection of innovative projects. However, some results indicate that crowds can fall behind in evaluating innovative and creative ideas [36,37,38].

The evaluation process requires two approaches: top-down and bottom-up [39]. Bottom-up approaches based on a community-based evaluation that uses the wisdom of crowds may be essential for pre-assessment ideas. Simple quantitative techniques such as commenting, voting, ranking, or rating can be used in the bottom-up system. Nevertheless, top-down approaches are needed to achieve high decision quality. The evaluation phase should include experts or a cross-functional group of senior managers. In addition, idea selection decisions within the innovation process are usually based on a set of criteria that the innovation projects must fulfill [39]. In this context, MCDM methods constitute an essential tool that guides experts and managers in evaluating innovative project ideas [5].

Due to limited resources, organizations require effective methods for selecting profitable projects and innovative ideas. In this regard, MCDM approaches become preferable to project selection [40]. Research and Development (R&D) projects have high commercialization potential, but their commercialization indicators are not obvious. In the study [41], criteria were obtained from 272 successful entrepreneurs and researchers. Then, structural equation modeling was used to analyze and weight the obtained criteria. Six criteria were evaluated: marketing, technology, finance, non-financial impact, intellectual property, and human resources. An experimental evaluation is provided in their study to validate the method. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is used to rank alternatives provided in the experimental evaluation. The R-number-based BWM method is proposed for R&D project selection for a medical device company. R numbers analyze the effects of uncertainty and risk [42]. Likewise, Li et al. [5] used the fuzzy set theory and BWM, which are integrated into the fuzzy BWM to structure the new product idea selection process. Six Sigma projects have a complex nature; therefore, the selection process for these projects is crucial. In the study [43], the projects are weighted using the analytical hierarchy process (AHP), and then, prioritization is performed by combinative distance-based assessment (CODAS).

Similarly, the AHP-TOPSIS method is used to prioritize business projects [44]. In another study [45], the fuzzy AHP-VIKOR method was used to select innovation projects. The ELECTRE method is recommended to public administrators to choose the best innovation strategy to reduce greenhouse gas emissions [46]. The main reason for choosing ELECTRE is under certain conditions, for the two options to be classified as incomparable. The second reason is that the method allows decision makers to consider uncertainty through an indifference threshold for each criterion.

Determining a set of evaluation criteria is another critical step in selecting innovative project ideas. These criteria in Table 1 are adopted from Eisenreich et al. [39] and by the authors with opinions of decision makers and experience, and they examine circular project selection by investigating selection processes and evaluation criteria for circular innovation management. This is one of the most comprehensive studies concerned with project selection in circular innovation management and putting forward evaluation criteria for circular innovation. While the criteria included in the study are more suitable for corporate firm structure, the criteria that can be adapted to an entrepreneurship acceleration program by using the criteria in the study have been chosen for this study.

Table 1 Evaluation Criteria for Innovative Project Idea Selection (adopted from [39])
Full size table

3 Methodology

This section presents the methodology used in this study. First, the preliminaries of intuitionistic fuzzy sets are given. A hybrid decision-making framework is introduced based on the combination of the IF-BWM and extended IF-ABAC methods (see Fig. 1).

Fig. 1
figure 1

The methodology framework

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3.1 Preliminaries of intuitionistic fuzzy sets

Definition 1

Intuitionistic fuzzy set

For a set X, A is defined as IFS in the sense of Atanassov [18] as follows:

$$A = \left\{ {\left\langle {x,\mu_{A} \left( x \right),\nu_{A} \left( x \right)} \right\rangle \left| {x \in X} \right.} \right\}$$
(1)

where the functions are \(\mu_{A} :X \to \left[ {0,1} \right]\) and \(\nu_{A} :X \to \left[ {0,1} \right]\), with the condition \(0 \le \mu_{A} \left( x \right) + \nu_{A} \left( x \right) \le 1,\,\forall x \in X\). The parameter \(\mu_{A} \left( x \right)\) indicates the degree of membership of the element x to set A. \(\nu_{A} \left( x \right)\) shows the degree of non-membership, the parameter of \(\pi_{A} \left( x \right) = 1 - \left( {\mu_{A} \left( x \right) + \nu_{A} \left( x \right)} \right)\)\(,\) is named the degree of indeterminacy (or hesitation margin).

For two intuitionistic fuzzy numbers (IFNs) \(a = \left\langle {\mu_{1} ,\nu_{1} } \right\rangle\) and \(b = \left\langle {\mu_{2} ,\nu_{2} } \right\rangle\), the basic algebraic operations between the IFNs are defined as follows [47]:

$$a \oplus b = \left\langle {\mu_{1} + \mu_{2} - \mu_{1} \mu_{2} ,\nu_{1} \nu_{2} } \right\rangle$$
(2)
$$a \otimes b = \left\langle {\mu_{1} \mu_{2} ,\nu_{1} + \nu_{2} - \nu_{1} \nu_{2} } \right\rangle$$
(3)
$$\lambda a = \left\langle {1 - \left( {1 - \mu_{1} } \right)^{\lambda } ,\nu_{1}^{\lambda } } \right\rangle$$
(4)
$$a^{\lambda } = \left\langle {\mu_{1}^{\lambda } ,1 - \left( {1 - \nu_{1} } \right)^{\lambda } } \right\rangle$$
(5)

The subtraction and division operations on IFNs are defined in [48] as follows:

$$a \ominus b = \left\{ {\begin{array}{*{20}l} {\left\langle {0,\frac{{\nu_{1} }}{{\nu_{2} }}} \right\rangle ,} \hfill & {\,\,\,if\,\,\,\mu_{1} \le \mu_{2} ,\nu_{1} \le \nu_{2} } \hfill \\ {\left\langle {\frac{{\mu_{1} - \mu_{2} }}{{1 - \mu_{2} }},\frac{{\nu_{1} }}{{\nu_{2} }}} \right\rangle ,} \hfill & {\,\,\,if\,\,\,0 \le \frac{{\nu_{1} }}{{\nu_{2} }} \le \frac{{1 - \mu_{1} }}{{1 - \mu_{2} }} < 1} \hfill \\ {\left\langle {\frac{{\mu_{1} - \mu_{2} }}{{1 - \mu_{2} }},\frac{{1 - \mu_{1} }}{{1 - \mu_{2} }}} \right\rangle ,} \hfill & {\,\,\,if\,\,\,\mu_{1} > \mu_{2} ,\frac{{1 - \mu_{1} }}{{1 - \mu_{2} }} < \frac{{\nu_{1} }}{{\nu_{2} }}} \hfill \\ {\left\langle {0,1} \right\rangle ,} \hfill & {\,\,\,if\,\,\,\mu_{1} \le \mu_{2} ,\nu_{1} > \nu_{2} } \hfill \\ \end{array} } \right.$$
(6)
$$a{\mathbf{ \oslash }}b = \left\{ {\begin{array}{*{20}l} {\left\langle {\frac{{\mu_{1} }}{{\mu_{2} }},0} \right\rangle ,} \hfill & {\,\,\,if\,\,\,\mu_{1} \le \mu_{2} ,\,\,\nu_{1} \le \nu_{2} } \hfill \\ {\left\langle {\frac{{\mu_{1} }}{{\mu_{2} }},\frac{{\nu_{1} - \nu_{2} }}{{1 - \nu_{2} }}} \right\rangle ,} \hfill & {\,\,\,if\,\,\,0 \le \frac{{\mu_{1} }}{{\mu_{2} }} \le \frac{{1 - \nu_{1} }}{{1 - \nu_{2} }} < 1} \hfill \\ {\left\langle {\frac{{1 - \nu_{1} }}{{1 - \nu_{2} }},\frac{{\nu_{1} - \nu_{2} }}{{1 - \nu_{2} }}} \right\rangle ,} \hfill & {\,\,\,if\,\,\,\frac{{1 - \nu_{1} }}{{1 - \nu_{2} }} < \frac{{\mu_{1} }}{{\mu_{2} }},\,\,\nu_{1} > \nu_{2} } \hfill \\ {\left\langle {1,0} \right\rangle ,} \hfill & {\,\,\,if\,\,\,\mu_{1} > \mu_{2} ,\,\,\nu_{1} \le \nu_{2} } \hfill \\ \end{array} } \right.$$
(7)

Definition 2

To aggregate IFNs, a well-known aggregation method introduced by Xu [47]. Let \(\alpha_{i} = \left\langle {\mu_{{\alpha_{i} }} ,\nu_{{\alpha_{i} }} } \right\rangle \left( {i = 1,2, \ldots ,m} \right)\) be a collection of IFNs, and \(\omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{m} } \right){}^{T}\) be the weight vector of \(\alpha_{i} \left( {i = 1,2, \ldots ,m} \right)\), which satisfies \(\sum\nolimits_{i = 1}^{m} {\omega_{i} = 1}\) and \(\omega_{i} \ge 0\,\,\left( {i = 1,2, \ldots ,m} \right)\). Then,

$$IFWA(\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{m} ) = \omega_{1} \alpha_{1}^{{}} \oplus \omega_{2} \alpha_{2}^{{}} \oplus \cdots \oplus \omega_{m} \alpha_{m}^{{}}$$
(8)
$$IFWA\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{m} } \right) = \left\langle {1 - \prod\limits_{i = 1}^{m} {\left( {1 - \mu_{{\alpha_{i} }} } \right)^{{\omega_{i} }} ,\prod\limits_{i = 1}^{m} {\left( {\nu_{{\alpha_{i} }} } \right)^{{\omega_{i} }} } } } \right\rangle$$
(9)

Definition 3

Let \(\alpha_{i} = \left\langle {\mu_{{\alpha_{i} }} ,\nu_{{\alpha_{i} }} } \right\rangle \left( {i = 1,2, \ldots ,m} \right)\) be a collection of IFNs, and the IIFWG operator for the IFVs αi is defined as follows [49]:

$$IIFWG(\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{m} ) = \alpha_{1}^{{\omega_{1} }} \otimes \alpha_{2}^{{\omega_{2} }} \otimes \cdots \otimes \alpha_{m}^{{\omega_{m} }}$$
(10)
$$IIFWG\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{m} } \right) = \left\langle {1 - \frac{1}{\lambda }\left( {1 - \prod\limits_{i = 1}^{m} {\left( {1 - \lambda \left( {1 - \mu_{{\alpha_{i} }} } \right)} \right)^{{\omega_{i} }} } } \right),1 - \frac{1}{\lambda }\left( {1 - \prod\limits_{i = 1}^{m} {\left( {1 - \lambda \left( {1 - \nu_{{\alpha_{i} }} } \right)} \right)^{{\omega_{i} }} } } \right)} \right\rangle$$
(11)

where ωi is the weight of αi, \(\sum\nolimits_{i = 1}^{m} {\omega_{i} = 1}\), \(\omega_{i} \ge 0\,\,\left( {i = 1,2, \ldots ,m} \right)\) and \(0 < \lambda < 1\).

Definition 4

For two intuitionistic fuzzy alternatives \(a = \left\langle {\mu_{1} ,\nu_{1} } \right\rangle\) \(b = \left\langle {\mu_{2} ,\nu_{2} } \right\rangle\), if \(\mu_{1} \le \mu_{2} \,\,\,\,and\,\,\,\,\nu_{1} \ge \nu_{2}\) conditions are fulfilled, then a is smaller than b, denoted by a < b [18].

According to the detailed critical review presented by Szmidt et al. [26, 50], if the conditions of Definition 4 are not satisfied, the following measure, R, is used to rank alternatives. Let \(\alpha = \left\langle {\mu ,\nu } \right\rangle\) be an intuitionistic fuzzy alternative, then:

$$R\left( \alpha \right) = 0.5 \times \left( {1 + (1 - \mu - \nu )} \right) \times d_{IFS} \left( {M,\alpha } \right)$$
(12)

where dIFS (M,α) is the Hamming distance α from the ideal-positive alternative \(M = \left\langle {1,0} \right\rangle\) and calculated as follows:

$$d_{IFS} \left( {M,\alpha } \right) = \frac{1}{2}\left( {\left| {1 - \mu } \right| + \left| {0 - \nu } \right| + \left| {0 - (1 - \mu - \nu } \right|} \right) = 1 - \mu$$
(13)

For the two intuitionistic fuzzy alternatives \(a = \mu_{1} ,\nu_{1}\) and \(b = \mu_{2} ,\nu_{2}\), if \(\mu_{1} \le \mu_{2}\) and \(\nu_{1} \ge \nu_{2}\) conditions are not met, \(R\left( a \right)\) and \(R\left( b \right)\) are calculated using Eq. (12); if \(R\left( a \right) < R\left( b \right)\), the result \(a > b\) is reached. The defined comparison scheme is displayed in Fig. 2.

Fig. 2
figure 2

Comparison procedure for two intuitionistic fuzzy alternatives

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3.2 IF-BWM method

Considering the subjective nature of project selection factors, to calculate the weight of the criteria, the IF-BWM introduced by Wan and Dong [51] is applied in this study. To apply the IF-BWM method after identifying the best and worst criterion, each criterion is compared to the best and worst criterion using an intuitionistic fuzzy preference relation (IFPR). Assume there are n criteria to evaluate the alternatives. In this status, total (2n − 3) pairwise comparisons should be performed to calculate the weight of each criterion.

The procedure of the IF-BWM methodology proposed by Wan and Dong [51] is as follows:

Step 1 Determine a set of decision criteria C = {C1, C2,…,Cn}.

Step 2 Identify the best criterion CB and the worst criterion CW.

Step 3 Provide preference for the best criterion over all other criteria.

Let \(\tilde{w}_{B} = \left( {a_{B}^{u} ,a_{B}^{\nu } } \right),\,\tilde{w}_{W} = \left( {a_{W}^{u} ,a_{W}^{\nu } } \right)\) and \(\tilde{w}_{j} = \left( {a_{j}^{u} ,a_{j}^{\nu } } \right)\) be the intuitionistic fuzzy (IF) weights for the best criterion, the worst criterion, and a criterion j, respectively. Let \(\tilde{a}_{Bj} = \left( {a_{Bj}^{u} ,a_{Bj}^{\nu } } \right)\) be the IF evaluation of the preference of the best criterion CB over criterion Cj, satisfying \(a_{Bj}^{u} \in \left[ {0,1} \right],\,a_{Bj}^{\nu } \in \left[ {0,1} \right],\,a_{Bj}^{u} + a_{Bj}^{\nu } \le 1\) and \(\tilde{a}_{BB} = \left( {0.5,\,0.5} \right)\). The best-to-others vector can be obtained as follows:

$$\tilde{A}_{B} = \left[ {\tilde{a}_{B1}^{{}} ,\tilde{a}_{B2}^{{}} , \ldots ,\tilde{a}_{Bn}^{{}} } \right]$$
(14)

Step 4 Provide preference for all criteria over the worst criterion.

Let \(\tilde{a}_{jW} = \left( {a_{jW}^{u} ,a_{jW}^{\nu } } \right)\) be the IF evaluation of the preference of a criterion Cj over the worst criterion CW, satisfying \(a_{jW}^{u} \in \left[ {0,1} \right],\,a_{jW}^{\nu } \in \left[ {0,1} \right],\,a_{jW}^{u} + a_{jW}^{\nu } \le 1\) and \(\tilde{a}_{WW} = \left( {0.5,\,0.5} \right)\). The others-to-worst vector can be obtained as follows:

$$\tilde{A}_{W} = \left[ {\tilde{a}_{1W}^{{}} ,\tilde{a}_{2W}^{{}} , \ldots ,\tilde{a}_{nW}^{{}} } \right]$$
(15)

Step 5 Derive the optimal weight vector \(\tilde{w}^{*} = \left[ {\tilde{w}_{1}^{*} ,\tilde{w}_{2}^{*} , \ldots ,\tilde{w}_{n}^{*} } \right]\) by building the following programming model.

$$\begin{aligned} & \max \left\{ {\beta - \gamma } \right\} \\ & s.t.\left\{ \begin{gathered} \mu \left( {R^{t} \left( {\tilde{w}_{j} } \right)} \right) \ge \beta \,\,\,\left( {j = 1,2, \ldots ,n;\,\,t = 1,2,3,4} \right) \hfill \\ \nu \left( {R^{t} \left( {\tilde{w}_{j} } \right)} \right) \le \gamma \,\,\,\,\,\left( {j = 1,2, \ldots ,n;\,\,t = 1,2,3,4} \right) \hfill \\ \beta + \gamma \le 1,\,\,\,\,\beta \ge 0,\,\,\,\,\gamma \ge 0 \hfill \\ \sum\nolimits_{j = 1,j \ne i}^{n} {w_{j}^{u} \le w_{i}^{\nu } ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(i = 1,2, \ldots ,n)} \hfill \\ w_{i}^{u} + n - 2 \ge \sum\nolimits_{j = 1,j \ne i}^{n} {w_{j}^{\nu } \,\,\,\,\,\,\,\,\,(i = 1,2, \ldots ,n)} \hfill \\ w_{i}^{u} + w_{i}^{\nu } \le 1,\,w_{i}^{u} \ge 0,\,w_{i}^{\nu } \ge 0\,\,\,(i = 1,2, \ldots ,n) \hfill \\ \end{gathered} \right. \\ \end{aligned}$$
(16)

where β denotes the minimal acceptance degree of the IF constraints, and γ indicates the maximal rejection degree of the IF constraints.

For a neutral decision maker, by putting \(\mu \left( {R^{t} \left( {\tilde{w}_{j} } \right)} \right) = 1 - \left( {{{R^{t} \left( {\tilde{w}_{j} } \right)} \mathord{\left/ {\vphantom {{R^{t} \left( {\tilde{w}_{j} } \right)} {d_{j}^{t} }}} \right. \kern-0pt} {d_{j}^{t} }}} \right)\) and \(\nu \left( {R^{t} \left( {\tilde{w}_{j} } \right)} \right) = - \left( {{{R^{t} \left( {\tilde{w}_{j} } \right)} \mathord{\left/ {\vphantom {{R^{t} \left( {\tilde{w}_{j} } \right)} {d_{j}^{t} }}} \right. \kern-0pt} {d_{j}^{t} }}} \right)\) into Eq. (16), the model is rewritten as a linear programming model. Here, \(d_{j}^{t} \,\,\left( {j = 1,2, \ldots ,n;t = 1,2,3,4} \right)\) are the values of the tolerance parameters. Generally, take any value from 1 to 9 to obtain a unique optimal solution. In this study, all the \(d_{j}^{t}\) are considered as 1. Also, \(R^{t} \left( {\tilde{w}_{j} } \right)\,\,(t = 1,2,3,4)\) values are obtained using the following equations:

$$\begin{aligned} R^{1} \left( {w_{j}^{u} } \right) & = a_{Bj}^{u} \left( {w_{B}^{u} - w_{B}^{\nu } + w_{j}^{u} - w_{j}^{\nu } + 2} \right) - 2w_{B}^{u} \cong 0 \\ R^{2} \left( {w_{j}^{\nu } } \right) & = a_{Bj}^{\nu } \left( {w_{B}^{u} - w_{B}^{\nu } + w_{j}^{u} - w_{j}^{\nu } + 2} \right) - 2w_{j}^{u} \cong 0 \\ R^{3} \left( {w_{j}^{u} } \right) & = a_{jW}^{u} \left( {w_{j}^{u} - w_{j}^{\nu } + w_{W}^{u} - w_{W}^{\nu } + 2} \right) - 2w_{j}^{u} \cong 0 \\ R^{4} \left( {w_{j}^{\nu } } \right) & = a_{jW}^{\nu } \left( {w_{j}^{u} - w_{j}^{\nu } + w_{W}^{u} - w_{W}^{\nu } + 2} \right) - 2w_{W}^{u} \cong 0 \\ \end{aligned}$$
(17)

Step 6 Check the consistency of the pairwise comparisons from the fuzzy deviation of the comparisons (ξ∗) using Eq. (18).

$$\xi^{*} = \min \mathop {\max }\limits_{j} \left\{ {\left| {\frac{{a_{Bj}^{u} }}{{a_{Bj}^{\nu } }} - \frac{{w_{B}^{u * } }}{{w_{j}^{u * } }}} \right|,\left| {\frac{{a_{jW}^{u} }}{{a_{jW}^{\nu } }} - \frac{{w_{j}^{u * } }}{{w_{W}^{u * } }}} \right|} \right\}$$
(18)

Step 7 Obtain the consistency index (CI), ζ, by Table 2 (or by solving Eq. (19)) and calculate the consistency ratio (CR) using Eq. (20).

$$\begin{aligned} & \rho_{BW} = {{a_{BW}^{u} } \mathord{\left/ {\vphantom {{a_{BW}^{u} } {a_{BW}^{\nu } }}} \right. \kern-0pt} {a_{BW}^{\nu } }} \\ & \zeta^{2} - \left( {1 + 2\rho_{BW} } \right)\zeta + \left( {\rho_{BW}^{2} - \rho_{BW} } \right) = 0 \\ \end{aligned}$$
(19)
$$CR = {\raise0.7ex\hbox{${\xi^{ * } }$} \!\mathord{\left/ {\vphantom {{\xi^{ * } } {CI}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${CI}$}}$$
(20)
Table 2 Consistency index for some IF preference relations
Full size table

Step 8 Control consistency with the following rules:

  1. i.

    If the obtained optimal objective values are β∗  = 1 and γ∗  = 0 or CR = 0, then all IF reference comparisons for the worst criterion CW and the best criterion CB are entirely consistent.

  2. ii.

    If β∗  < 1, γ∗  > 0, and CR ≤ 0.1, all IF reference comparisons are acceptably consistent; otherwise, comparisons are inconsistent.

Step 9 Obtain the crisp weight using the similarity function improved by Zhang and Xu [52].

$$L\left( \alpha \right) = \frac{{1 - \nu_{\alpha } }}{{1 + (1 - \mu_{\alpha } - \nu_{\alpha } )}}$$
(21)

3.3 Extended ABAC Method based on intuitionistic fuzzy sets

The conventional ABAC method was recently proposed by Biswas et al. [10] in 2022 to deal with the rank reversal problem of MCDM methods. This method is based on analyzing the dominance of the alternatives with each other by using the given decision matrix and the weight vector to select the most suitable one among a set of alternatives.

This section presents the IF-ABAC methodology for resolving a group decision-making problem in a fuzzy environment where all information is represented by intuitionistic fuzzy numbers.

Consider an MCDM problem in which the evaluations of experts on alternatives according to each criterion can be expressed in the form of IFN. Let \(\tilde{X}^{\left( e \right)}\) be the decision matrix given by expert e under an intuitionistic fuzzy environment as follows:

$$\tilde{X}^{(e)} = \left[ {\tilde{x}_{i,j}^{e} } \right]_{\,m \times n} = \left[ {\left\langle {\mu_{i,j}^{e} ,\nu_{i,j}^{e} } \right\rangle } \right]_{m \times n} \,\,\,\,\,\,\,i = 1,2, \ldots ,m\,,j = 1,2, \ldots ,n\,\,and\,\,e = 1,2, \ldots ,h$$
(22)

where \(\tilde{x}_{i,j}^{e}\) indicates the performance evaluation of expert e (e = 1,2,…,h) to alternative i (Ai, i = 1,2,…,m) on criterion j (Cj, j = 1,2,…,n), which is an IFN including membership degree \(\mu_{i,j}^{e}\) and non-membership degree \(\nu_{i,j}^{e}\). Based on the above information, the steps are elaborated as follows.

Step 1 Describe the problem and collect preference information using IF values based on linguistic terms for each expert \(\left( {\tilde{X}^{\left( e \right)} } \right)\). Experts use the linguistic terms illustrated in Table 3 to score the ratings of alternatives regarding each criterion.

Table 3 Linguistic term scale for rating alternatives
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Step 2 Calculate the aggregated group preferences using the individual information provided by all experts. Assume that each expert has an importance degree, given as \(\omega_{e}\) (ensures that \(\omega_{e} \in \left( {0,1} \right)\) and \(\mathop \sum \limits_{e = 1}^{h} \omega_{e} = 1\)), according to his/her expertise and experience. Here, we calculate the aggregated group decision values \(\tilde{X} = \left[ {\tilde{x}_{i,j} } \right]_{m \times n} = \left[ {\mu_{i,j} ,\nu_{i,j} } \right]_{m \times n}\) using standard intuitionistic fuzzy aggregation operators.

Aggregate the decision values using the intuitionistic fuzzy weighted averaging (IFWA) operator and weights \(\omega_{e}\) of experts as follows:

$$\begin{aligned} \tilde{x}_{{_{i,j} }}^{{}} & = IFWA_{{\omega_{e} }} (\tilde{x}_{{_{i,j} }}^{1} ,\tilde{x}_{{_{i,j} }}^{2} , \ldots ,\tilde{x}_{{_{i,j} }}^{h} ) \\ & = \sum\nolimits_{e = 1}^{h} {\omega_{e} \,\tilde{x}_{{_{i,j} }}^{e} } = \omega_{1} \tilde{x}_{{_{i,j} }}^{1} \oplus \omega_{2} \tilde{x}_{{_{i,j} }}^{2} \oplus \cdots \oplus \omega_{h} \tilde{x}_{{_{i,j} }}^{h} \\ & = \left\langle {1 - \prod\limits_{e = 1}^{h} {\left( {1 - \mu_{{\tilde{x}_{{_{i,j} }}^{e} }} } \right)^{{\omega_{e} }} ,\prod\limits_{e = 1}^{h} {\left( {\nu_{{\tilde{x}_{{_{i,j} }}^{e} }} } \right)^{{\omega_{e} }} } } } \right\rangle \\ \end{aligned}$$
(23)

Step 3 Compare and rank the alternatives in an intuitionistic fuzzy environment.

This step aims to obtain a final ranking by comparing the alternatives. The main idea involves selecting a pivot alternative and assigning its correct order among the other alternatives by dividing the decision matrix into two parts. This selection and assignment procedure is given in Algorithms 1 and 2.

Step 3.1 The overall relative goodness \(\left( {\tilde{G}} \right)\) of pair of alternatives (assume Ai and Ak, i ≠ k) is calculated using Eq. (24).

$$\begin{aligned} \tilde{G}_{i,k} & = IIFWG_{{w_{j} }} (\tilde{\delta }_{i,k}^{1} ,\tilde{\delta }_{i,k}^{2} , \ldots ,\tilde{\delta }_{i,k}^{n} ) \\ & = \prod\limits_{j = 1}^{n} {\left( {\tilde{\delta }_{i,k}^{j} } \right)^{{w_{j} }} } = \left( {\tilde{\delta }_{i,k}^{1} } \right)^{{w_{1} }} \otimes \left( {\tilde{\delta }_{i,k}^{2} } \right)^{{w_{2} }} \otimes ... \otimes \left( {\tilde{\delta }_{i,k}^{n} } \right)^{{w_{n} }} \\ & = \left\langle {1 - \frac{1}{\lambda }\left( {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \lambda \left( {1 - \mu_{{\tilde{\delta }_{i,k}^{j} }} } \right)} \right)^{{w_{j} }} } } \right),1 - \frac{1}{\lambda }\left( {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \lambda \left( {1 - \nu_{{\tilde{\delta }_{i,k}^{j} }} } \right)} \right)^{{w_{j} }} } } \right)} \right\rangle \\ \end{aligned}$$
(24)

where \(w_{j}\) is the weight of the criterion j and \(\tilde{\delta }_{i,k}^{j}\) denotes the relative goodness of Ai with respect to Ak regarding criterion j, which is obtained using the following equations:

$$\tilde{\delta }_{i,k}^{j} = \frac{{\tilde{x}_{k,j} }}{{\tilde{x}_{i,j} }}\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for}}\,{\text{cost}}\,{\text{criteria}}$$
(25)
$$\tilde{\delta }_{i,k}^{j} = \frac{{\tilde{x}_{i,j} }}{{\tilde{x}_{k,j} }}\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{for}}\,{\text{benefit}}\,{\text{criteria}}$$
(26)

Note that, division operations on IFNs are employed in Eq. (7).

Step 3.2 The relative goodness values are compared as given in Definition 4. For two relative goodness values \(\tilde{G}_{i,k} = \left\langle {\mu_{{\tilde{G}_{i,k} }} ,\nu_{{\tilde{G}_{i,k} }} } \right\rangle\) and \(\tilde{G}_{k,i} = \left\langle {\mu_{{\tilde{G}_{k,i} }} ,\nu_{{\tilde{G}_{k,i} }} } \right\rangle\), if \(\mu_{{\tilde{G}_{i,k} }} \le \mu_{{\tilde{G}_{k,i} }} \,\,\,\,and\,\,\,\,\nu_{{\tilde{G}_{i,k} }} \ge \nu_{{\tilde{G}_{k,i} }}\), then \(\tilde{G}_{i,k} < \tilde{G}_{k,i}\), else calculate \(R\left( {\tilde{G}_{i,k} } \right)\) and \(R\left( {\tilde{G}_{k,i} } \right)\) using Eqs. (12)–(13), if \(R\left( {\tilde{G}_{i,k} } \right) < R\left( {\tilde{G}_{k,i} } \right)\), then \(\tilde{G}_{i,k} > \tilde{G}_{k,i}\) else \(\tilde{G}_{i,k} < \tilde{G}_{k,i}\).

When comparing the \(\tilde{G}\) values, the following three cases can occur:

  1. (i)

    \(\tilde{G}_{i,k} > \tilde{G}_{k,i}\) means Ai is superior to Ak,

  2. (ii)

    \(\tilde{G}_{i,k} < \tilde{G}_{k,i}\) means Ai is inferior to Ak,

  3. (iii)

    \(\tilde{G}_{i,k} = \tilde{G}_{k,i}\) means Ai is indifferent to Ak.

figure a

Algorithm 1. PartIF-ABAC algorithm

figure b

Algorithm 2. IF-ABAC algorithm

4 A real-world case study

This section presents a case study that evaluates an entrepreneurship acceleration program organized by InnoCentrum, a company that provides innovation management software for businesses. The program aims to bring together entrepreneurs, mentors, and relevant institutions and organizations to accelerate the maturation of business ideas and implement projects in the shortest time possible. Through this platform, entrepreneurs receive online training and mentoring and can view and support each other’s ideas.

The program received 176 project ideas, of which 102 progressed to the mentor stage, and 38 were submitted to the expert jury team. The submitted ideas are evaluated by a jury team consisting of three experts (E1–E3), who are senior managers of the stakeholder institutions. The experts assessed the 38 project ideas (alternatives A1–A38) using a set of six criteria: (C1) the effect, benefit, and urgency of the project, (C2) the feasibility of the project, (C3) the time required to implement the project, (C4) the commercial potential of the project and its scalability, (C5) the competence level of the team, and (C6) the innovation level of the project. The criteria and their definitions are presented in Table 1. Tables 2 and 3 provide linguistic terms used for comparing the criteria and rating ideas, respectively.

4.1 Calculating the weights of the criteria using an intuitionistic fuzzy BWM

After selecting the relevant criteria, the weight of the criteria is evaluated by a group of three experts using the new IF-BWM. An overview of the implementation of the IF-BWM method proposed by Wan and Dong [51] is given in Sect. 3.2. The optimal global weights of the criteria are calculated using pairwise comparisons from the view of a neutral decision maker, considering the neutral approach-I and all tolerance parameters (\(d_{j}^{t}\)) 1.

During the IF-BWM process, first, experts individually identify the best (e.g., most desirable, most important) and the worst (e.g., least desirable, least important) criteria from a predefined set, guided by their attitudes. No direct comparisons are made during this stage. As an illustrative example, Expert 1 selects C1 as the best criterion and C6 as the worst criterion (refer to Table 4). Following this, the experts are requested to provide pairwise comparisons between the criteria using intuitionistic fuzzy numbers. The comparisons of the best criterion over the other criteria \(\left( {\tilde{a}_{Bj}^{e} } \right)\) and all the criteria over the worst criterion \(\left( {\tilde{a}_{jW}^{e} } \right)\) are displayed in Table 4 for each expert (Table 5).

Table 4 Preferences of experts
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Table 5 Individual and overall weights
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After determining of the best-to-others \(\left( {\tilde{a}_{Bj}^{e} } \right)\) and others-to-worst vectors \(\left( {\tilde{a}_{jW}^{e} } \right)\), the linear programming (LP) model in Eq. (16) is solved using Lingo 19.0 optimization software to derive the optimal weight vector. Table 6 displays the IF weights obtained using the LP model for the three experts. The crisp weights of IFVs are calculated using Eq. (21). Subsequently, an overall weight vector is computed by taking the simple average of the weights for each criterion by the experts, with the importance levels of the experts considered equal.

Table 6 Consistency values
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Additionally, Table 6 provides the minimal acceptance degree (β), maximal rejection degree (γ), fuzzy deviations (ξ∗), and CR for each expert. As observed in Table 6, all CR values are less than 0.1, indicating that the comparisons are reasonably consistent for the experts.

4.2 Ranking of the alternatives using the IF-ABAC method

After calculating the criterion weights, the alternatives are evaluated using the six predefined criteria. Experts employ the linguistic terms illustrated in Table 3 to assign scores to alternatives for each criterion. The expert preference information, by which the alternatives are assessed, is outlined in Table 7.

Table 7 Evaluation of projects using the IF scale—Initial decision matrix
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To implement the IF-ABAC methodology, individual preferences from Table 7 are aggregated into group preferences using the IFWA operator, as defined in Eq. (23) (refer to Table 8). An illustration of the aggregation procedure for C4 based on A1 is provided below.

$$\begin{gathered} \tilde{x}_{{_{1,2} }}^{{}} = \sum\nolimits_{e = 1}^{3} {\omega_{e} \,\tilde{x}_{{_{1,2} }}^{e} } = \frac{1}{3}\tilde{x}_{{_{1,2} }}^{1} \oplus \frac{1}{3}\tilde{x}_{{_{1,2} }}^{2} \oplus \frac{1}{3}\tilde{x}_{{_{1,2} }}^{3} \hfill \\ \,\,\,\,\,\,\,\, = \left\langle {1 - \left( {\left( {1 - 0.5} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} \times \left( {1 - 0.5} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} \times \left( {1 - 0.35} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} } \right),\,\,\left( {0.4} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} \times \,\,\left( {0.4} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} \times \,\,\left( {0.6} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}} } \right\rangle \hfill \\ \,\,\,\,\,\,\,\, = \left\langle {0.45,\,\,0.44} \right\rangle \hfill \\ \end{gathered}$$
Table 8 Aggregated decision matrix
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After aggregating expert preferences, the IF-ABAC method is employed to prioritize innovative project ideas. As detailed in Step 3 of the method, Algorithms 1 and 2 are applied to compare the alternatives and assign their final positions. Initially, Algorithm 2 is executed with the IF-ABAC (\(\tilde{X}\), W, 1, 38) using the aggregated decision matrix \(\left( {\tilde{X}} \right)\) given in Table 8 and the set of criteria weights (W) calculated in Sect. 4.1.

In the initial iteration, IF-ABAC (\(\tilde{X}\), W, 1, 38) selects A38, the alternative in the last row, as a pivot in the partitioning algorithm. Subsequently, each alternative is compared with this pivot. To illustrate the partitioning procedure, consider Alternatives 38 and 32. Initially, the relative goodness of A38 compared to A32 \(\left( {\tilde{\delta }_{i,k}^{j} } \right)\) for each criterion is determined using Eqs. (25) and (26).

$$\begin{aligned} (\tilde{\delta }_{38,32}^{1} ,\tilde{\delta }_{38,32}^{2} , \ldots ,\tilde{\delta }_{38,32}^{6} ) & = \left( {\left\langle {0.915,0.058} \right\rangle ,\left\langle {1,0} \right\rangle ,\left\langle {0.928,0.048} \right\rangle ,\left\langle {1,0} \right\rangle ,\left\langle {0.893,0.079} \right\rangle ,\left\langle {1,0} \right\rangle } \right) \\ (\tilde{\delta }_{32,38}^{1} ,\tilde{\delta }_{32,38}^{2} , \ldots ,\tilde{\delta }_{32,38}^{6} ) & = \left( {\left\langle {1,0} \right\rangle ,\left\langle {0.933,0.054} \right\rangle ,\left\langle {1,0} \right\rangle ,\left\langle {0.892,0.075} \right\rangle ,\left\langle {1,0} \right\rangle ,\left\langle {0.796,0.177} \right\rangle } \right) \\ \end{aligned}$$

After calculating the relative goodness, the overall relative goodness \(\left( {\tilde{G}} \right)\) of the pair of alternatives is calculated using Eq. (24).

$$\begin{aligned} \tilde{G}_{38,32} & = \prod\limits_{j = 1}^{n = 6} {\left( {\tilde{\delta }_{38,32}^{j} } \right)^{{w_{j} }} } = \left( {\left\langle {0.915,0.058} \right\rangle } \right)^{0.23} \otimes \left( {\left\langle {1,0} \right\rangle } \right)^{0.21} \otimes \left( {\left\langle {0.928,0.048} \right\rangle } \right)^{0.21} \otimes \left( {\left\langle {1,0} \right\rangle } \right)^{0.19} \otimes \left( {\left\langle {0.893,0.079} \right\rangle } \right)^{0.11} \otimes \left( {\left\langle {1,0} \right\rangle } \right)^{0.05} \\ & = \left\langle \begin{gathered} 1 - \frac{1}{0.99}\left( \begin{gathered} 1 - \left( {1 - 0.99\left( {1 - 0.915} \right)} \right)^{0.23} \times \left( {1 - 0.99\left( {1 - 1} \right)} \right)^{0.21} \times \left( {1 - 0.99\left( {1 - 0.928} \right)} \right)^{0.21} \hfill \\ \times \left( {1 - 0.99\left( {1 - 1} \right)} \right)^{0.19} \times \left( {1 - 0.99\left( {1 - 0.893} \right)} \right)^{0.11} \times \left( {1 - 0.99\left( {1 - 1} \right)} \right)^{0.05} \hfill \\ \end{gathered} \right), \hfill \\ 1 - \frac{1}{0.99}\left( \begin{gathered} 1 - \left( {1 - 0.99\left( {1 - 0.058} \right)} \right)^{0.23} \times \left( {1 - 0.99\left( {1 - 0} \right)} \right)^{0.21} \times \left( {1 - 0.99\left( {1 - 0.048} \right)} \right)^{0.21} \hfill \\ \times \left( {1 - 0.99\left( {1 - 0} \right)} \right)^{0.19} \times \left( {1 - 0.99\left( {1 - 0.079} \right)} \right)^{0.11} \times \left( {1 - 0.99\left( {1 - 0} \right)} \right)^{0.05} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right\rangle \\ & = \left\langle {0.9532,0.0285} \right\rangle \\ \end{aligned}$$

With similar calculations, \(\tilde{G}_{32,38}\) is found as \(0.9540,0.0255\).

Finally, we rank the alternatives based on the obtained relative goodness values. In this context, considering \(\mu_{{\tilde{G}_{38,32} }} \le \mu_{{\tilde{G}_{32,38} }}\) and \(\nu_{{\tilde{G}_{38,32} }} \ge \nu_{{\tilde{G}_{32,38} }}\), we find that \(\tilde{G}_{38,32} < \tilde{G}_{32,38}\). Consequently, it indicates that A38 is inferior to A32.

The algorithm iteratively runs until all alternatives are accurately positioned by executing the specified steps. In Fig. 3, the highlighted green positions correspond to pivot elements, while the blue highlighted positions represent elements whose positions have been determined due to no further movement in the iteration (refer to the detailed algorithm description in Sect. 3.3). The complete figure, including all iterations, is available in the Appendix. The final ranking is presented in the last row of Fig. 3, where A33 emerges as the most satisfying innovation project, outperforming all other alternatives.

Fig. 3
figure 3

Position changes in alternatives while obtaining the final ranking using the IF-ABAC algorithm

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5 Sensitivity analysis and validation of the results

The applicability of the proposed methodology is demonstrated through a case study in Sect. 4, where the process for conducting innovation project idea selection is presented. The five most suitable new project ideas are selected by a group of decision makers illustrated in the case study by analyzing six major criteria. Thus, the ranking of the innovative project ideas is expressed as {A33, A32, A38, A25, A15}. To further evaluate the proposed methodology, (i) sensitivity analysis of changing the criteria weights, (ii) comparative analysis with existing methods, and (iii) analysis of the changing set of alternatives are discussed.

5.1 Effect of changes in criterion weights on results

To analyze the robustness of the algorithm, we consider the effect of changes in the criteria weights on the ranking. Therefore, sensitivity analysis is performed by changing the criterion weights. As shown in Table 9, the first scenario assumes that all criteria are equally important. In other scenarios, a criterion is ignored, and the other criteria are weighted equally. All scenarios and related weights are given in Table 9.

Table 9 Sensitivity analysis of criteria weights
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Figure 4 displays the changes in rankings for the top 11 alternatives for the sake of clarity. Our findings demonstrate that alternative A33 is the most preferred project proposal across all scenarios when considering the effects of changes in criterion weights on alternative rankings. However, aside from alternative A33, other alternatives are affected by changes in weight. For example, while alternative A30 is currently ranked 7th, it rises to 4th in scenarios SC1, SC4, SC6, and SC7. Meanwhile, the third-best alternative, alternative A38, maintains its position in SC2 and ranks second in the remaining six scenarios. Notably, the observed ranking changes in the scenarios are due to the significant increase in the weights of the two least important criteria (C5 and C6).

Fig. 4
figure 4

Ranking of the top alternatives for each scenario

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Our analysis reveals that alternative A33 is the superior option, despite substantial changes in the criterion weights. Additionally, our results indicate that the obtained solution is relatively stable, as slight changes in the ranking of alternatives, except the first-ranked alternative, are observed. It is therefore concluded that in problems where the number of alternatives is vast, changes in criterion weights may significantly influence the final ranking results.

To establish whether there is a statistically significant correlation between the current ranking and the rankings derived from the scenarios, Spearman’s rank correlation coefficient is used. Our findings demonstrate a strong correlation between the current ranking and all scenarios, with the lowest correlation coefficient value being 0.970 (SC6) and the highest value being 0.991 (SC2). Therefore, we conclude that the current ranking is highly correlated with the rankings derived from the scenarios.

5.2 Comparative analysis with other techniques

To assess the robustness of the proposed IF-ABAC method, we conducted a comparative analysis by applying three additional IF-based MCDM techniques: the intuitionistic fuzzy TOPSIS proposed by Boran et al. [53], the intuitionistic fuzzy CODAS method introduced by Büyüközkan and Göçer [54], and the intuitionistic fuzzy EDAS method suggested by Liang [55]. The analysis was performed using the weights obtained with IF-BWM, and the resulting rankings for innovation project ideas are presented in Table 10 and Fig. 5. Comprehensive rankings are provided in the Appendix, along with the calculation values for each method. The ranks obtained using the four methods exhibit similarity. As depicted in Table 10, project A33 consistently emerges as the most favored option across all methods, whereas A13 consistently ranks as the least favorable choice.

Table 10 Results obtained from different methods
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Fig. 5
figure 5

Comparison of the results with different methods

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Spearman’s correlation coefficient was utilized to evaluate the presence of a statistically significant correlation between the proposed method and the compared methods. Following analysis, a correlation value of 0.9 was identified between the developed method and the IF-TOPSIS, IF-CODAS, and IF-EDAS. Consequently, the comparative analysis substantiates the conclusion that the IF-ABAC method is robust and reliable in evaluating and ranking innovation project ideas.

5.3 Effect of changes in the set of alternatives on results

The issue of rank reversal, which refers to changes in the relative ranking or order of alternatives after adding or removing one or more alternatives, is a major problem commonly encountered in MCDM methods. However, during the evaluation and selection process, some projects may be withdrawn by their owners, eliminated by decision makers during evaluation, or the idea may not pass from the maturation stage to the evaluation stage during tests such as proof of concepts and MVP tests. Similarly, the evaluation phase may last by including new alternative(s) to the system. In this section, the impact of changes in the set of alternatives on ranking is discussed in comparison with similar methods.

5.3.1 Including new alternatives in the analysis

In the evaluation and selection of innovation project ideas, new project ideas may be introduced later in the process. Ensuring that these additions do not alter the rankings of previously assessed alternatives is crucial. This issue is particularly pronounced in certain methods found in the literature, especially those that rely on distance-based calculations. To investigate this, we hypothetically introduced three new alternatives into the analysis: the first alternative (A39) was considered superior to the existing alternatives, the second (A40) was rated moderate, and the third (A41) was deemed inferior. This approach was adopted to better capture the potential impact of these changes on the overall evaluation. The aggregated scores of these newly added alternatives are presented in Table 11.

Table 11 Aggregated scores of the new alternatives
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The new results obtained from the IF-ABAC and IF-TOPSIS methods with the added new alternatives are presented in Tables 12 and 13, respectively. As shown in Table 12, the ranking of other alternatives remains unchanged in subsequent rankings after adding new alternatives, as obtained from the IF-ABAC method. However, when examining the results of the IF-TOPSIS method, it becomes evident that the ranking of other alternatives is altered by introducing new alternatives. For instance, with the addition of the A39 alternative, the positions of 12 out of the 38 alternatives (approximately 32 percent) experience changes compared with the current ranking. Similarly, significant alterations in rankings were observed in the IF-CODAS and IF-EDAS methods. These alterations are indicated by highlighting in gray (refer to Table 13). The rankings derived with IF-EDAS and IF-CODAS with the new alternatives added are also provided in Table 14 (see Appendix 2).

Table 12 Effect of adding new alternatives on ranking: IF-ABAC
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Table 13 Effect of adding new alternatives on ranking: IF-TOPSIS
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Table 14 Comparative analysis of adding new alternatives
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5.3.2 Excluding an existing alternative from the analysis

During the project selection process, some projects may be withdrawn by their owners or eliminated by experts. Additionally, the idea may not progress from the maturation stage to the evaluation stage during the evaluation process. Therefore, it is essential that the applied model yields robust results in this dynamic environment. The impact of removing particular project ideas from a dataset on the ranking of remaining ideas was investigated using comparative results obtained through the IF-ABAC, IF-TOPSIS, IF-CODAS, and IF-EDAS methods, as shown in Tables 15, 16, 17, and 18 in the Appendix. Specifically, 38 project ideas were removed from the selection process, and the columns of the tables denote the eliminated project ideas and the ranking of the remaining ideas. The rows indicate the rank of the project ideas in terms of priority. The analysis revealed that removing specific project ideas can change the priority order of the remaining ideas using methods other than IF-ABAC. For example, when A1 was removed in the IF-TOPSIS, the priority order of A37, A14, A5, A35, and A11 changed, resulting in a rank reversal problem. This issue is observed in most MCDM methods. However, the IF-ABAC method overcomes this limitation by using pairwise comparison, in which all project ideas are removed and then ranked based on their relative importance. Table 15 demonstrates that IF-ABAC effectively addresses the rank reversal problem. To investigate the effect of eliminating certain project ideas from the evaluation on the ranking of the ideas, the results obtained using all methods are displayed in Fig. 6 for clarity.

Fig. 6
figure 6

Effect of excluding an alternative on rankings of a IF-ABAC, b IF-TOPSIS, c IF-CODAS, and d IF-EDAS

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5.4 Discussion

A dynamic decision environment, such as selecting the best innovative ideas in innovation contests or intrapreneurship programs, requires multiple evaluation phases. Potential ideas undergo pre-assessments, including crowd voting, mentor or expert evaluation, proof of concept tests, and MVP tests, leading to the elimination of many proposals. In quantitative techniques used in these phases, the number of project idea alternatives may vary across different evaluation layers.

While many methods have proven effective in single-stage evaluations, the literature and study results reveal a potential issue known as the rank reversal problem. This problem arises when the ranking of alternatives changes with repeated MCDM evaluations, either by adding or removing alternatives in the same decision environment. This issue significantly reduces the trustworthiness of the method and, at times, leads to inaccurate rankings, requiring substantial compromises in the accuracy of the results [56]. In response to this challenge, we propose the IF-ABAC method, which is a systematic approach for evaluating innovative project ideas. IF-ABAC utilizes uncertain information in the decision-making process and effectively addresses the rank reversal problem, providing more robust and efficient results.

In this article, we have performed comparative analysis with existing approaches, in Sects. 5.2 and 5.3, to highlight the effectiveness and adequacy of the proposed approach. Our findings indicate that IF-ABAC ranks alternatives similarly to other IF-based MCDM techniques (IF-TOPSIS [53], IF-CODAS [54], and IF-EDAS [55]). The ranking results are summarized in Table 10. Besides, the ranking of alternatives in IF-ABAC remains unchanged after adding new alternatives, as long as the ranking of the new alternatives is ignored as given in Table 12. This consistency is evident not only when the analysis is conducted by including new alternatives but also when each alternative is systematically excluded (refer to Fig. 6). However, changes to the set of alternatives significantly affect the compared methods. The results are graphically illustrated in Fig. 6 and presented in Appendix 2. Specifically, distance-based methods like TOPSIS, which rank alternatives based on their similarity to the ideal solution, are highly sensitive to including or excluding an extreme alternative (positive or negative ideals) and such modifications can drastically change the overall ranking.

In addition, we have conducted a sensitivity analysis to assess the robustness of the algorithm used for ranking project ideas. The analysis involves changing the criteria weights and observing the impact on rankings. The developed scenarios are provided in Table 9. The solution is deemed stable, with only slight alterations in rankings, as shown in Fig. 4. Spearman’s rank correlation coefficient confirms a strong correlation between the current ranking and all scenarios, affirming the reliability of the algorithm in diverse weight settings.

These findings affirm the robustness of the IF-ABAC methodology despite changes in the set of alternatives and weights. The methodology’s ability to maintain a stable ranking under both scenarios underscores its reliability and suitability for decision-making processes in dynamic environments. These results demonstrate that the IF-ABAC offers robust and accurate method during the evaluation of innovation projects.

6 Conclusion

Innovation is a crucial aspect of business, and the ability to identify and select the best new project ideas is vital for success. Innovation project ideas involve high levels of risk and uncertainty, making the selection of the best ideas a complex and time-consuming process. It is essential to treat this process as iterative and continuous, where each stage aids in developing potential ideas and quickly eliminating unsuitable ones from the innovation funnel. The innovation process can be likened to a funnel, where ideas are developed and evaluated throughout the journey. In this process, it is crucial to involve diverse experts with varying perspectives and expertise to reduce potential bias in the assessment process. In addition, it is essential to address the inherent uncertainties associated with subjective judgments in expert opinions.

The IF-ABAC method is developed to use a systematic approach to select innovative business ideas. This approach involves extending the ABAC method to the intuitionistic fuzzy environment, combining fuzzy sets and ABAC under a group decision-making environment for the first time. The BWM is used to determine the criteria weights, and the approach is tested and validated through its application to a real-world problem involving the collection of entrepreneurial business ideas through an entrepreneurship acceleration program organized by InnoCentrum, a Turkey-based corporate innovation management software company. The IF-ABAC approach is capable of managing the problem of changes in the set of alternatives after the decision process, which is a type of dynamic decision environment. This method effectively manages the rank reversal problem, where a complete change in the relative ranking or ordering of alternatives occurs after adding or removing one or more alternatives. IFSs enable a more accurate representation of expert evaluations and better manage the uncertainties inherent in innovation management and decision-making problems. The approach is extended to group decision-making environments, allowing for the participation of multiple decision makers, and improving decision efficiency and robustness.

The feasibility and efficiency of the proposed novel approach are demonstrated through a sensitivity analysis. Finally, the results are compared with three other IF-based MCDM methods in the literature, highlighting the superiority of the proposed approach. The IF-ABAC approach offers decision makers an effective tool for evaluating and selecting innovative project ideas in a dynamic decision environment. By using fuzzy sets and ABAC in a group decision-making environment, the approach provides a more accurate representation of expert evaluations and better management of uncertainties, enabling decision makers to make more informed decisions.

To further advance the proposed approach, future research can extend its application to other features of dynamic decision-making environments. In addition, the proposed method can be enhanced by incorporating hesitant, neutrosophic, Pythagorean, picture, spherical, or q-rung fuzzy sets and their interval types. Also, the proposed methodology can be applied to various complex decision-making problems, such as supplier selection, performance evaluation, and portfolio selection, to test its generalizability and effectiveness.