1 Introduction

Circular economy focuses on enhancing the efficiency of the economic cycle through the tenets of resource use reduction, reuse, and recycling (Murray et al. [82], [43]). Compared to the linear economy, which disposes of generated wastes at the end of the production process, circular economy encourages the implementation of approaches that enable the reutilization of these wastes as resource inputs in the upstream production processes [109]. Yadav et al. [122] emphasized that as organizations undergo increasing pressure to implement sustainable practices, circular economy provides a framework for promoting the shift towards cleaner production methods, which can help address environmental issues such as the climate crisis, eutrophication, and habitat loss, among others. These problems are primarily caused by human activities, such as agriculture and deforestation, which have been ongoing since the industrial revolution [73]. Such transition encourages scholars, industrial experts, and policymakers to promote discussions on collectively reducing emissions, optimizing resource utilization, and implementing efficient waste management strategies [25, 127]. The direct consequence of inherently closing production loops associates a circular economy with a supply chain denoted as a circular supply chain. However, Govindan et al. [42] asserted that putting circularity principles into practice, especially in circular supply chains, often encounters several barriers due to the needed technological advancements, hefty investments, and substantial changes in managing human resources, materials, machinery, and production environments. The identification of these barriers has been prominent in the current literature. For instance, Mangla et al. [75] identified 16 barriers to managing circular supply chains in developing economies. Some barriers include a lack of immediate economic advantages, inadequate provision of training, lack of synchronization among supply chains, and development opportunities for members [75]. On the other hand, Khandelwal and Barua [66] determined 26 barriers, including the lack of global standards for evaluating effectiveness, inadequate information systems in monitoring recycled materials, the absence of an integrated system for reverse logistics, and the lack of financial support for implementing initiatives.

Among the potential responses and initiatives in addressing these barriers to circular supply chains, the implementation of digitalization is notable in the literature. Digitalization in supply chain processes, a byproduct of the onset of the Fourth Industrial Revolution, is the utilization and adoption of digital technologies, e.g., machine learning, simulation, and blockchain, which enable and promote new business models and opportunities for organizations [83, 97]. The barriers to implementing circular supply chains and their potential solutions through digital technologies are apparent in the private sector supply chains, leaving public sector supply chains with limited attention. Public sector supply chains are associated with unique barriers as they focus primarily on public welfare rather than profit [62, 65]. For instance, the public sector would invest in various initiatives that enhance public well-being, all within the frames of reference to existing legal measures and regulations [68, 71]. The strict bureaucratic structure of public sector institutions may hinder the implementation of circular supply chains. Thus, the barriers faced by the private sector and strategies that may address these barriers may not apply to public sector supply chains due to their distinct characteristics, making the agenda of investigating their barriers an intriguing point of inquiry. In the current literature, there is a dearth of studies exclusively attending to the topic, with most studies solely focusing on the private sector or without adequate differentiation between the public and private sectors. This topic has become more relevant due to the public sector’s consumption volume and the volume of waste generated amidst the COVID-19 pandemic, where it acquires a huge share of personal protective equipment. Accordingly, the increase in the generation of medical waste amidst the pandemic can be attributed to the linear approach practiced in the supply chain of most public sectors [115]. Thus, understanding public sector supply chains is critical for promoting circularity, especially during the “new normal”, where waste generation has surged due to the pandemic. However, in the current literature, identifying the barriers faced by the public sector in implementing the circularity of supply chains has received minimal attention, creating a considerable gap in knowledge.

Moreover, recent directions in circularity initiatives explore the critical role of digital technologies in accelerating the transitioning of organizations [52]. Several studies contend that digital technologies, such as big data analytics, artificial intelligence (AI), blockchain, and digital twins, support circular supply chains by improving operational efficiency [7, 27, 114, 121], enhancing economic performance [27], promoting worker safety [48], and managing the complexities across the supply chain [118]. However, as the presence of digital technologies is becoming ubiquitous, evaluating how they support circular supply chains is scant, especially in the public sector. As the role of digital technologies in overcoming these barriers has not been explored, evaluating them in addressing circular public sector supply chains would be a meaningful collective departure from the domain literature. Thus, this work offers an analytical evaluation in an integrated manner that captures both complexity and uncertainty. In this work, with the exhaustive list of Aro et al. [5] on circular supply chain barriers and a comprehensive list of Mahroof et al. [73] on digital technologies, a decision-making model that maps the digital technologies according to their relevance in addressing the identified barriers is proposed. In such a case, a multi-attribute decision-making (MADM) problem is generated. In this work, three phases are involved: (1) setting a threshold that determines the list of barriers that are of high relevance in the context of public sector supply chains, (2) obtaining the priority weights of the circular supply chain barriers, and (3) evaluating the digital technologies based on their relevance in addressing the identified barriers. To advance this agenda, two MADM tools, i.e., the Criteria Importance Through Intercriteria Correlation (CRITIC) and the Combinative Distance-based Assessment (CODAS), were integrated. Moreover, to address uncertainties inherent in the decision-making process, the proposed integrated approach adopts the Fermatean fuzzy sets (FFS). In particular, the Fermatian fuzzy (FF) CRITIC assigns the importance weights for the identified barriers, and FF CODAS evaluates the digital technologies in overcoming these barriers.

Unlike other methods for allocating weights (e.g., analytical hierarchy process, best–worst method, full consistency method) of attributes or criteria, the CRITIC offers an effective computational structure that significantly reduces the cognitive workload of decision-makers, particularly when dealing with large-scale MADM problems [33]. The MADM literature offers recent applications of the CRITIC method, including studies by Nguyen et al. [84], Pan et al. [91], and Wang et al. [120]. Reports suggest that CRITIC provides relatively consistent results with recent priority weight allocation methods such as the MEthod based on the Removal Effects of Criteria introduced by [58] and the Stepwise Weight Assessment Ratio Analysis II (SWARA II) proposed by [59]. However, SWARA II requires an additional layer of analysis that increases the decision-makers’ cognitive workload when dealing with a large number of attributes. On the other hand, the CODAS approach is a powerful MADM technique that uses Euclidean distance, which has a similar conceptualization to the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [60]. While TOPSIS has gained more applications due to the concept of distance as a more appealing and straightforward yardstick for ranking alternatives within a MADM framework, it fails to consider the relative importance of the distances from an alternative to the positive-ideal and negative-ideal solutions. Other methods were also recently offered, such as the Weighted Aggregated Sum Product Assessment (WASPAS) [126] and the Evaluation based on Distance from Average Solution (EDAS) [56]. The outputs of these methods were found to be highly correlated with CODAS, as demonstrated by [57]. In addition, although computationally efficient, the Simultaneous Evaluation of Criteria and Alternatives (SECA) method [60] requires solving a non-linear optimization model, which may suffer from trapping in local optima. Nonetheless, SECA shows consistent results with WASPAS and EDAS [60] and, by arguing transitivity, CODAS.

Due to its computational efficiency, there has been an increase in the application of CODAS and its various extensions in the literature, as seen in the studies reported by Pamučar et al. [90], Simic et al. [111], and Lei et al. [70]. However, prompted by the inherent vagueness and uncertainty in any MADM problems, the FFS was developed by Senapati and Yager [105] to address these limitations of the classical fuzzy sets [125], intuitionistic fuzzy sets Atanassov [6], and Pythagorean fuzzy sets [123]. Compared to other extensions of the fuzzy sets, the FFS is more effective in handling high levels of uncertainties and thus handles more complex ambiguous information [29, 95, 107]. Some formulations of the FF CRITIC are available in the literature [41, 81, 103], and so is the integration of FFS within CODAS [107, 110, 111, 128], Barraza et al. [10], the application of an integrated FF CRITIC-CODAS in a MADM problem is only explored by Aro et al. [5] for multicriteria sorting problems. Evidently, the application of FF CRITIC and FF CODAS in actual decision-making problems remains limited, hence, this work advances the literature by demonstrating the efficacy of its integrated framework in handling a novel supply chain circularity problem.

To summarize, the main departure of this work can be outlined in three folds. First, it addresses the gap in the current literature by narrowing the characterization of the barriers to implementing circular supply chains in the public sector. The significance of this focus is attributed to the complexity and distinctiveness of public sector supply chains and their substantial societal and environmental implications. Second, this study examines the efficacy of digital technologies in addressing the barriers to achieving circularity in public sector supply chains. Third, the computational efficiency of integrating CRITIC and CODAS methods under FFS, which remains limited, is demonstrated. The structure of this paper is organized as follows: Sect. 2 discusses the circularity and barriers encountered in implementing circularity concepts in public sector supply chains as well as the role of digital technologies in a circular economy. On the other hand, Sect. 3 details the preliminary concepts of the proposed approaches. Section 4 showcases the application of the integrated approach in evaluating digital technologies for addressing these barriers. Sect. 5 reports sensitivity and comparative analyses to analyze the robustness and efficacy of the proposed methodology. Section 6 presents the results, while Sect. 7 enhances the discussion by outlining important managerial insights. The paper concludes in Sect. 8 by providing the key findings and future research agenda.

2 Literature Review

2.1 The Circularity of Public Sector Supply Chains

Due to the apparent benefits of a circular economy in all aspects of sustainability [23], the popular agenda is to investigate the application of the concept at the supply chain level that leads to the development of the circular supply chain concept [45]. The amalgamation of circularity practices in the supply chain is commonly referred to as a “circular supply chain”, wherein it organizes all aspects of the chain in a manner that supports circularity instead of linearity [12, 45, 119]. Various works in the literature have explored the adoption of circularity concepts to supply chains utilizing MADM tools (see Table 1). Note that Table 1 may not be exhaustive. However, linking firms into a circular supply chain has been the primary challenge to effective circular economy implementation [21]. Accordingly, structuring all aspects of an organization to commit and collaborate in implementing a circular supply chain is considered one of the most critical barriers [2, 32, 37, 54]. Prior research on the circularity concept and its implementation in the supply chain context has discussed some barriers at a macro level [53, 74].

Table 1 Review of the most recent literature on the adoption of MADM methods for circular supply chains
Full size table

Unlike supply chains in the private sector, the public sector supply chain is associated with the activities of the various areas of government, and its primary objective is to maximize public welfare [85]. Scholars assert that the goal of managing a public sector supply chain is to add value to all stages of the supply chain: from the demand for goods and services to their acquisition, managing the logistic process, and finally, the after-use and disposal [79, 86]. Material cycles in the supply chain of the public sector encompass those utilized in the delivery of public goods, from education and health to social housing and transportation [106]. Despite the complexity involved in the public sector following its differences from the private sector in terms of the coordination, communication, and network structure among supply chain members, limited attention is attributed to the literature [46]. Observations in South Africa, which may be deemed relevant in other developing economies, suggest that public sector supply chains within municipalities in the country are facing numerous problems, including the conflict between professional municipal workers and political appointees, poor planning, ineffective internal processes, ineffective leadership, demoralized employees, corruption, and fraud [85]. These conditions are mostly outside the scope of current discussions on circular supply chains and may be complex when the circularity agenda is integrated into public sector supply chains.

2.2 Barriers to Circular Supply Chains

Adopting the concept of a circular supply chain requires firms to make substantial investments and rely on all stakeholders to collaborate in the value chain [34, 129]. The transition towards a circular economy necessitates a fundamental change in supply chain practices related to production, consumption, design, waste management, reuse, recycling, and implications for logistics throughout the chain [15, 98, 129]. Accordingly, this shift in the supply chain practices is disruptive, wherein firms face various barriers throughout their implementation. For instance, the circularity concept leads to a longer end-of-life phase for products that can reduce sales with constant demand [75]. Moreover, organizing institutions to commit and cooperate with circularity initiatives in circular supply chain implementation is a significant and prevalent challenge across institutions [14, 24, 35, 75]. Since the primary motivation of private institutions is profit, the high costs of circularity initiatives implementation discourage institutions from committing to the transition, fearing they will become uncompetitive in the market [35]. The costs of circular economy implementation arise mainly from the need for retrofitting production processes that enable recycling waste materials to become available for remanufacturing [102]. Additionally, one of the major challenges in the implementation of initiatives leading to supply chain circularity is the insufficiency of amenable materials for recycling [37, 55, 112]. Accordingly, organizations are often deterred by the expensive and extensive processes such as collecting, handling, and storing waste, particularly hazardous, volatile, and perishable materials or wastes involved in reverse logistics [26, 44, 92].

Moreover, decision-makers often face organizational barriers while implementing circular supply chain initiatives, which include the lack of adequate planning and management [75], poor leadership support, poor organizational culture and management, and commitment from the management [66]. Evidently, sufficient planning and management are necessary for the successful implementation of managing circularity initiatives, particularly in designing scenarios that maximize resource utilization [75, 127]. Furthermore, transitioning to a circular supply chain presents changes to the current operations, and resisting to these changes would hinder the transition [76]. Barriers such as the lack of a performance evaluation system [113] and traceability [36] impede circular supply chain adoption since they lead to inefficiencies. Implementing circularity initiatives requires intensive labor [55]. The challenge lies in obtaining a workforce with appropriate work experience and necessary scientific skills [54, 55, 87]. An extensive list of barriers to a circular supply chain can be found in Aro et al. [5].

2.3 Digital Technologies and their Roles in the Circular Economy

Digitalization has profoundly affected the way people communicate and interact with their surroundings, thus, altering various operations for the production and delivery of goods and services to consumers [16]. Furthermore, digitalization, operationalized through digital technologies, offers interoperability, neutrality, open software platform, and data owner controls, which reduce the cost, time, and risk of conventional supply chains (Jensen [47]). Moreover, systems that support circular supply chains with interrelated cycles consist of extensive data, and digital technologies provide new ways to access and analyze these data [4]. Hence, Kayikci et al. [52] emphasized the significance of consistently developing a digital foundation in the industry to accelerate the transition of various organizations into a circular economy most efficiently. There are a plethora of studies investigating the role of digital technologies within circular supply chains. For instance, Del Giudice et al. [27] explored the moderating impact of big data in implementing circular supply chain initiatives. They revealed that a big-data-driven supply chain improves the operational and economic performance of the processes involved.

Moreover, Tantalaki et al. [114] highlighted that real-time data, a component of big data, optimize activities and reduce operational costs. Jha et al. [48] further supported the favorable implication of big data, suggesting that big data and AI enhance the safety of products, workers, and the environment, at lower operational costs. Later studies (i.e., [39, 121]) also observed the positive impact of digital technology utilization, specifically the significant benefits (e.g., accelerated product design) of AI across all functions in the supply chain and various tasks in the reverse logistics process. Moreover, some studies investigated the applicability of digital twins in resolving information asymmetry in the remanufacturing supply chain [22], and the potential of blockchain in managing the complexities of a circular supply chain [18, 118], among others. On the other hand, Mahroof et al. [72] comprehensively explored the potential contributions of digital technologies in managing sustainable supply chains. They outlined a list of commonly available digital technologies that can overcome the key barriers to supply chain sustainability. Thus, overcoming the traditional barriers to the implementation of circular supply chains is possible by leveraging the benefits of digital technologies within supply chains.

3 Preliminaries

3.1 Fermatean Fuzzy Set

The use of the fuzzy set theory has been widely recognized as an effective method for handling imprecise information and uncertainties. Initially introduced by Zadeh [125], the theory was primarily developed as a range of multivalued logic that can be applied to explain intricate electrical systems [20]. Atanassov [6] later introduced an extension to the fuzzy set theory known as the intuitionistic fuzzy set (IFS), which involves incorporating a membership function, non-membership function, and the consequent hesitancy degree, respectively representing support, opposition, and neutrality, in obtaining information. The IFS is specified in the following definition.

Definition 1

[6]. Suppose that \(\Delta\) is a non-empty universe of discourse. The general form of IFS \(F\) is expressed as follows.

$$F = \left\{ {\left( {x,\mu_{F} \left( x \right),\nu_{F} \left( x \right)} \right):x \in \Delta } \right\}$$
(1)

where \(\mu_{F} \left( x \right):\Delta \to \left[ {0,1} \right]\) a \(\nu_{F} \left( x \right):\Delta \to \left[ {0,1} \right]\) such that \(0 \le \mu_{F} \left( x \right) + \nu_{F} \left( x \right) \le 1\) for all \(x \in \Delta\). Furthermore, \(\mu_{F} \left( x \right)\) indicates the degree of membership while \(\nu_{F} \left( x \right)\) describes the degree of non-membership of the element \(x \in \Delta\) in set \(F\).

In real-world scenarios, however, decision-makers may elicit their judgment not satisfying the condition for IFS, i.e., \(\mu_{F} \left( x \right) + \nu_{F} \left( x \right) > 1\). Yager [123] proposed the concept of Pythagorean fuzzy set (PFS) to address this limitation, which is illustrated in the following definition.

Definition 2

[123]. Suppose that \(\Delta\) be a non-empty universe of discourse. Then, the PFS \({\mathcal{P}}\) is expressed as.

$${\mathcal{P}} = \left\{ {\left( {x, \mu_{{\mathcal{P}}} \left( x \right),\nu_{{\mathcal{P}}} \left( x \right)} \right):x \in \Delta } \right\}$$
(2)

where \(\mu_{{\mathcal{P}}} \left( x \right):\Delta \to \left[ {0,1} \right]\) and \(\nu_{{\mathcal{P}}} \left( x \right):\Delta \to \left[ {0,1} \right]\) such that \(0 \le \left( { \mu_{{\mathcal{P}}} \left( x \right)} \right)^{2} + \left( {\nu_{{\mathcal{P}}} \left( x \right)} \right)^{2} \le 1\) for all \(x \in \Delta\). Furthermore, \(\mu_{{\mathcal{P}}} \left( x \right)\) and \(\nu_{{\mathcal{P}}} \left( x \right)\) refer to the degree of membership and degree of non-membership of the element \(x \in \Delta\) in the set \({\mathcal{P}}\), respectively. For any \({\mathcal{P}}\) and \(x \in \Delta\), the degree of indeterminacy, \(\pi_{{\mathcal{P}}} \left( x \right)\) can be calculated by

$$\pi_{{\mathcal{P}}} \left( x \right) = \sqrt {1 - \left( {\mu_{{\mathcal{P}}} \left( x \right)} \right)^{2} - \left( {\nu_{{\mathcal{P}}} \left( x \right)} \right)^{2} }$$
(3)

Although the PFS is a step forward in addressing uncertainty, it fails to handle some conditions. For instance, consider \(\mu_{{\mathcal{P}}} = 0.9\) and \(\nu_{{\mathcal{P}}} = 0.6\). The condition suggests that 0.92+0.62 ≰ 1. Thus, to address this limitation, the concept of FFS was introduced, providing a more effective tool for handling uncertain information and more flexibility in capturing uncertainty than both IFS and PFS [105]. The definition and operations within FFS are presented as follows:

Definition 3

[105]. Let \(\Delta\) be a non-empty universe of discourse. The Fermatean fuzzy set \(T\) in \(\Delta\) can be presented as:

$$T = \left\{ {\left( {x, \mu_{T} \left( x \right),\nu_{T} \left( x \right)} \right):x \in s\Delta } \right\}$$
(4)

where \(\mu_{T} \left( x \right):\Delta \to \left[ {0,1} \right]\) and \(\nu_{T} \left( x \right):\Delta \to \left[ {0,1} \right]\) such that \(0 \le \left( {\mu_{T} \left( x \right)} \right)^{3} + \left( {\nu_{T} \left( x \right)} \right)^{3} \le 1\) for all \(x \in \Delta\). Furthermore, \(\mu_{T} \left( x \right)\) and \(\nu_{T} \left( x \right)\) refer to the degree of membership and degree of non-membership of the element \(x\) in the set \(T\), respectively. Consequently, the degree of indeterminacy \(\pi_{T} \left( x \right)\) is defined as:

$${ }\pi_{T} \left( x \right) = \sqrt[3]{{1 - \left( {\mu_{T} \left( x \right)} \right)^{3} - \left( {\nu_{T} \left( x \right)} \right)^{3} }}$$
(5)

The disparity among the intuitionistic membership grades (IMG), Pythagorean membership grades (PMG), and Fermatean membership grades (FMG) spaces is featured in Fig. 1. These fuzzy sets belong to the category of \(q\)-rung orthopair fuzzy sets, in which the sum of the \(q\)th power of the membership function and the \(q\)th power of the non-membership function is bounded by a value of one [124]. For instance, \(q = 1\) implies IFS, \(q = 2\) implies PFS.

Fig. 1
figure 1

Comparison of space of FMG, PMG, and IMG

Full size image

Definition 4

[105]. Suppose that \(T=\left({\mu }_{T},{\nu }_{T}\right)\), \({T}_{1}=\left({\mu }_{{T}_{1}},{\nu }_{{T}_{1}}\right)\), and \({T}_{2}=\left({\mu }_{{T}_{2}},{\nu }_{{T}_{2}}\right)\) are three distinct FFS and \(\alpha >0\), then the following are the operators for the FFS:

$${T}_{1}\cap {T}_{2}=\left(\mathrm{min}\left\{{\mu }_{{T}_{1}},{\mu }_{{T}_{2}}\right\},\mathrm{max}\left\{{\nu }_{{T}_{1}},{\nu }_{{T}_{2}}\right\}\right)$$
(6)
$${T}_{1}\cup {T}_{2}=\left(\mathrm{max}\left\{{\mu }_{{T}_{1}},{\mu }_{{T}_{2}}\right\},\mathrm{min}\left\{{\nu }_{{T}_{1}},{\nu }_{{T}_{2}}\right\}\right)$$
(7)
$${T}^{c}=\left({\nu }_{T},{\mu }_{T}\right)$$
(8)
$$T_{1} \oplus T_{2} = \left( {\sqrt[3]{{\mu_{{T_{1} }}^{3} + \mu_{{T_{2} }}^{3} - \mu_{{T_{1} }}^{3} \mu_{{T_{2} }}^{3} }},\nu_{{T_{1} }} \nu_{{T_{2} }} } \right)$$
(9)
$$T_{1} \otimes T_{2} = \left( {\mu_{{T_{1} }} \mu_{{T_{2} }} ,\sqrt[3]{{\nu_{{T_{1} }}^{3} + \nu_{{T_{2} }}^{3} - \nu_{{T_{1} }}^{3} \nu_{{T_{2} }}^{3} }}} \right)$$
(10)
$$\alpha T=\left(\sqrt[3]{1-{\left(1-{\mu }_{T}^{3}\right)}^{\alpha } }, {\left({\nu }_{T}\right)}^{\alpha }\right)$$
(11)
$${T}^{\alpha }=\left({\left({\mu }_{T}\right)}^{\alpha },\sqrt[3]{1-{\left(1-{\nu }_{T}^{3}\right)}^{\alpha } }\right)$$
(12)

Additional operations on subtraction and division of FFS were introduced by Senapati and Yager [104].

Definition 5

[104]. Assume \({T}_{1}=\left({\mu }_{{T}_{1}},{\nu }_{{T}_{1}}\right)\), and \({T}_{2}=\left({\mu }_{{T}_{2}},{\nu }_{{T}_{2}}\right)\) are two distinct FFNs, then:

(13)
(14)

Definition 6

[105]. Assume \({T}_{i}=\left({\mu }_{{T}_{i}},{\nu }_{{T}_{i}}\right)\) \(\left(i=\mathrm{1,2},\dots ,n\right)\) be a set of FFS, and \(w={\left({w}_{1},{w}_{2},\dots ,{w}_{n}\right)}\) be the corresponding weight vector for \(\sum_{i}{w}_{i}=1\). Then, the Fermatean fuzzy weighted average \(FFWA\) aggregation operator is defined by.

$$FFWA\left({T}_{1},{T}_{2},\dots ,{T}_{n}\right)=\left(\sum_{i=1}^{n}{w}_{i}{\mu }_{{T}_{i}},\sum_{i=1}^{n}{w}_{i}{\nu }_{{T}_{i}}\right)$$
(15)

Senapati and Yager [105] introduced a basis for ranking FFS via a score function S:T→ℝ, where \(T\) is an FFS, along with the accuracy function. However, there are certain cases where the score function cannot precisely rank FFS. As a solution to this issue, Mishra and Rani [80] presented a new Fermatean fuzzy score function, which addresses the shortcomings of the existing score function. The characteristics of this score function are defined as follows:

Definition 7

[80]. Suppose that \(T=\left({\mu }_{T},{\nu }_{T}\right)\) is an FFS, the score function for \(T\) is defined as:

$$S\left(T\right)=\frac{1}{2}\left[\left({\mu }_{T}^{3}-{\nu }_{T}^{3}-ln\left(1+{\pi }_{T}^{3}\right)\right)+1\right]$$
(16)

where \({\pi }_{T}\) is the corresponding indeterminacy degree. For all \({\mu }_{T}\) and \({\nu }_{T}\), provided that \({\mu }_{T}^{3}+{\nu }_{T}^{3}\le 1\), \(S\left(T\right)\in \left[\mathrm{0,1}\right]\).

According to Mishra and Rani [80], \(S\) increases monotonically relative to \({\mu }_{T}\) and decreases monotonically relative to \({\nu }_{T}\). Thus, \(S\) can be used as a basis for ranking FFS.

Theorem 1

[80]. For any two FFS \({T}_{1}=\left({\mu }_{{T}_{1}},{\nu }_{{T}_{1}}\right)\) and \({T}_{2}=\left({\mu }_{{T}_{2}},{\nu }_{{T}_{2}}\right)\), if \({\mu }_{{T}_{1}}>{\mu }_{{T}_{2}}\) and \({\nu }_{{T}_{1}}<{\nu }_{{T}_{2}}\), then \(S\left({T}_{1}\right)>S\left({T}_{2}\right)\).

The Euclidean distance is also defined in FFS.

Definition 8

[105]. Let \({T}_{1}=\left({\mu }_{{T}_{1}},{\nu }_{{T}_{1}}\right)\) and \({T}_{2}=\left({\mu }_{{T}_{2}},{\nu }_{{T}_{2}}\right)\) be two distinct FFS. Then, the Euclidean distance between \({T}_{1}\) and \({T}_{2}\) is.

$$d\left({T}_{1},{T}_{2}\right)=\sqrt{\frac{1}{2}\left[{\left({\mu }_{{T}_{1}}^{3}-{\mu }_{{T}_{2}}^{3}\right)}^{2}+{\left({\nu }_{{T}_{1}}^{3}-{\nu }_{{T}_{2}}^{3}\right)}^{2}+{\left({\pi }_{{T}_{1}}^{3}-{\pi }_{{T}_{2}}^{3}\right)}^{2}\right]}$$
(17)

3.2 The CRITIC Method

The CRITIC method offers a computational framework to determine the importance weights of a specified set of criteria or alternatives wherein an aggregation process defines the overall priorities [33]. Although the CRITIC method is often utilized for prioritizing criteria or attributes set in a hierarchical MADM problem, such as those by Rostamzadeh et al. [100] and Abdel-Basset and Mohamed [1], it emphasizes the importance of the information that can be derived from the criteria set [94]. In this method, the standard deviation of the raw scores for each criterion and the pairwise linear correlation coefficients are calculated, which measures the contrast and conflict concentration within and between criteria under evaluation [33]. Thus, the method analyzes the adequate information present in the intersection of the criteria and alternative sets. The method has been applied in various domains such as healthcare (e.g., [93]), logistics (e.g., [81]), software selection (e.g., Tuş and Adalı [116]), education (e.g., [41]), strategy development (e.g., [77]), risk analysis (e.g., [78]), among others.

Definition 9

[33]. Assume that \(A\) is a finite set consisting of \(m\) alternatives. The MADM problem in its general form for a given system consisting of \(n\) evaluation criteria \({c}_{j}\) \(\left(j=1,\dots ,n\right)\) can be expressed as.

$$\mathrm{max}\left\{{c}_{1}\left(a\right),\dots ,{c}_{n}\left(a\right)|a\in A\right\}$$
(18)

A membership function \({x}_{j}\) is defined for each criterion \({c}_{j}\), which maps the values of \({c}_{j}\) to the interval \(\left[\mathrm{0,1}\right]\), analogous to the concept of an ideal point. Thus, a value near the ideal \({c}_{j}^{*}\) represents the best performance in criterion \(j\), while a value near the anti-ideal value \({c}_{{j}^{*}}\) indicates the worst performance in criterion \(j\). The computational steps of the CRITIC method are detailed as follows.

Step 1. Determine the set of \(n\) criteria and \(m\) alternatives, constructed as a hierarchical MADM problem.

Step 2. Evaluate the performance score \({a}_{ij}\) of the \(i\) th \(\left(i=1,\dots ,m\right)\) alternative with respect to the \(j\) th \(\left(j=1,\dots ,n\right)\) criterion.

Step 3. Compute the normalized matrix \(X={\left({x}_{ij}\right)}_{m\times n}\), where the normalized score \({x}_{ij}\) describes a linear normalization of \({a}_{ij}\) scores and represents:

$${x}_{ij}=\frac{{a}_{ij}-{a}_{{j}^{*}}}{{a}_{j}^{*}-{a}_{{j}^{*}}}\; \forall i,j$$
(19)

where \({a}_{j}^{*}=\underset{i}{\mathrm{max}}{a}_{ij}\) and \({a}_{{j}^{*}}=\underset{i}{\mathrm{min}}{a}_{ij}\).

Step 4. Extract the vector \({x}_{j}\), which denotes the normalized scores of \(m\) alternatives and is represented as follows.

$${x}_{j}=\left({x}_{1j}, {x}_{2j},\dots , {x}_{mj}\right)\; \forall j$$
(20)

Step 5. Calculate the standard deviation of each \({x}_{j}\), denoted as \({\sigma }_{j}\), which is obtained as follows.

$${\sigma }_{j}=\sqrt{\frac{\sum_{i=1}^{m}{\left({x}_{ij}-{\overline{x} }_{j}\right)}^{2}}{m}} ,\mathrm{ where }\;{\overline{x} }_{j}=\frac{\sum_{i=1}^{m}{x}_{ij}}{m}.\; \forall j$$
(21)

Step 6. Construct the symmetric matrix \(R={\left({r}_{jk}\right)}_{n\times n}\),

$${r}_{jk}=\frac{\sum_{i=1}^{m}({x}_{ij-}{\overline{x} }_{j})({x}_{ik-}{\overline{x} }_{k})}{\sqrt{\sum_{i=1}^{m}{\left({x}_{ij}-{\overline{x} }_{j}\right)}^{2}\sum_{i=1}^{m}{\left({x}_{ik}-{\overline{x} }_{k}\right)}^{2}}}\; \forall j,k\in \left\{1,\dots ,n\right\}$$
(22)

where \({r}_{jk}\) represents the linear correlation coefficient of vectors \({x}_{j}\) and \({x}_{k}\), \(j,k\in \left\{1,\dots ,n\right\}\), \({r}_{jk}\in \left[-1, 1\right]\). Obviously, for \(j=k\), \({r}_{jk}=1\).

Step 7. Compute the amount of information \({z}_{j}\) as follows:

$${z}_{j}={\sigma }_{j}\sum_{k=1}^{n}\left(1-{r}_{jk}\right)\; \forall j$$
(23)

The higher the value of \({z}_{j}\) of criterion \(j\), the greater amount of information.

Step 8. Obtain the weights of the criteria \({w}_{j}\) as follows:

$${w}_{j}=\frac{{z}_{j}}{\sum_{k=1}^{n}{z}_{k}}\; \forall j$$
(24)

3.3 The CODAS Method

The CODAS approach assesses the desirability of alternatives relative to \({l}^{1}\)- norm and \({l}^{2}\)-norm indifference spaces for criteria [57]. The method adopts a combinative approach that utilizes both the Euclidean distance and the secondary measure, namely the Taxicab distance, otherwise known as the Manhattan distance, to compute the assessment score of alternatives. Since its introduction, CODAS has been applied in various domains, including selecting alternatives among renewable resources [28], prioritizing and selecting Six Sigma projects in the healthcare industry [17], supplier selection in the manufacturing context (Bolturk [13]), and analyzing workforce attributes for Industry 4.0 (Vinodh and Wankhede [117]), among others. The following are the steps in applying the CODAS method to solve MADM problems.

Step 1. Construct the decision matrix \(A\), wherein the performance value of the \(i\) th \(\left(i\in \left\{\mathrm{1,2},\dots , m\right\}\right)\) alternative with respect to the \(j\) th \(\left(j\in \left\{\mathrm{1,2},\dots , n\right\}\right)\) criterion is denoted as \({a}_{ij}\ge 0\).

$$A={\left({a}_{ij}\right)}_{m\times n}=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2n}\\ \vdots & \vdots & \vdots & \vdots \\ {a}_{m1}& {a}_{m2}& \dots & {a}_{mn}\end{array}\right)$$
(25)

Step 2. Generate the normalized decision matrix, denoted as \(N={\left({n}_{ij}\right)}_{m\times n}\), where \({n}_{ij}\) represents the normalized performance values obtained from linear normalization, and \(P\) and \(C\) indicate the sets of benefit and cost criteria, respectively.

$${n}_{ij}=\left\{\begin{array}{c}\frac{{a}_{ij}}{\underset{i}{\mathrm{max}}\;{a}_{ij}}\; if\; j\in P\\ \frac{\underset{i}{\mathrm{min}}\;{a}_{ij}}{{a}_{ij}}\; if\; j\in C \end{array}\; \forall i,j\right.$$
(26)

Step 3. Obtain the weighted normalized decision matrix, denoted as \(B={\left({b}_{ij}\right)}_{m\times n}\),

$${b}_{ij}={w}_{j}{n}_{ij} \forall i,j$$
(27)

wherein \({w}_{j}\) indicates the weight of the \(j\) th criterion, \(\sum_{j=1}^{n}{w}_{j}=1\), and \(0<{w}_{j}<1\). The determination of \({w}_{j}\) can be done using any priority weight generation approach, such as the CRITIC method.

Step 4. Calculate the negative ideal solution vector \({b}_{*}\), where

$${b}_{*}={\left({b}_{j*}\right)}_{1\times n}$$
(28)

and

$${b}_{j*}=\underset{i}{\mathrm{min}}\;{b}_{ij}$$
(29)

Step 5. Generate the Euclidean and Taxicab distances, denoted as \({d}_{i}\) and \({t}_{i}\), respectively, using the following formulations.

$${d}_{i}=\sqrt{{\sum }_{j=1}^{n}{\left({b}_{ij}-{b}_{j*}\right)}^{2}},\forall i=1,\dots m$$
(30)
$$t_{i} = \sum\limits_{{j = 1}}^{n} {\left| {b_{{ij}} - b_{{j*}} } \right|} ,\forall i = 1, \ldots m$$
(31)

Step 6. Construct the relative assessment matrix, denoted as \(H={\left({h}_{ik}\right)}_{m\times m}\),

$${h}_{ik}=\left({d}_{i}-{d}_{k}\right)+\beta \left[\left({d}_{i}-{d}_{k}\right)\times \left({t}_{i}-{t}_{k}\right)\right]$$
(32)

where \(k\in \left\{\mathrm{1,2},\dots ,m\right\}\). The threshold function \(\beta\) can be set by the decision-maker, given a predetermined parameter ρ, and is defined as

$$\beta =\left\{\begin{array}{c}1\; if\; {d}_{i}-{d}_{k}\ge \rho \\\; 0\; if\; {d}_{i}-{d}_{k}<\rho \end{array}\right.$$
(33)

Step 7. Evaluate the assessment score of each alternative, denoted as \({S}_{i}\), which is obtained as follows:

$${S}_{i}=\sum_{k=1}^{m}{h}_{ik} \forall i$$
(34)

wherein the higher the value of \({S}_{i}\), the more preferable the alternative is.

4 The Proposed Procedure for the Integrated Fermatean Fuzzy CRITIC-CODAS Approach

The process of the FF CRITIC-CODAS approach for evaluating digital technologies to address the barriers to circularity in the public sector supply chains is presented in Fig. 2. The proposed methodological framework consists of three phases. Phase I defines the list of barriers most relevant in public sector supply chains, while Phase II determines the importance weights of the identified barriers using the FF CRITIC approach. On the other hand, Phase III evaluates the predetermined digital technologies via the FF CODAS method. The proposed approach is applied as follows.

Fig. 2
figure 2

The methodological framework for the proposed procedure

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4.1 Phase I: Identifying Relevant Circular Supply Chain Barriers in Public Sector Supply Chains

Step 1. Determine the list of circular supply chain barriers. To ensure a comprehensive cutting-edge list derived from a rigorous systematic literature review, the list of barriers to implementing a circular supply chain provided by Aro et al. [5] is adopted in this work. For tractability, see Appendix Table 14 for the list of barriers along with their brief descriptions. With this list, a questionnaire was generated to gather the preliminary assessments. In this study, a group of eight decision-makers was selected to provide their judgments on the relevance of each barrier in public sector supply chains using a predefined scale presented in Table 2. The expert decision-makers were chosen in such a way that satisfies the following qualifications: (1) managerial or administrative experience in a public sector, (2) a minimum of ten years of professional experience in a public organization, (3) sufficient knowledge of the supply chains of the public sector, and (4) background knowledge on the principles of circular economy. In this study, eight experts were asked to elicit judgment to determine barriers relevant to public sector supply chains.

Table 2 Linguistic evaluation scale
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Step 2. Obtain the FF initial evaluation matrices for circular supply chain barriers. Initial evaluation matrices \(R^{k} = \left( {r_{j}^{k} } \right)_{n \times 1}\) were constructed, where \(r_{j}^{k}\) represents the relevance degree of the \(j\) th \(\left( {j = 1,2, \ldots ,n} \right)\) circular supply chain barrier in the context of public sector supply chains elicited by the \(k\) th \(\left( {k = 1,2, \ldots ,K} \right)\) decision-maker in linguistic terms. Using the linguistic scale shown in Table 2, these matrices were transformed into FF evaluation matrices, \(R^{k} = \left( {\left( {r_{{j_{\mu } }}^{k} ,r_{{j_{\nu } }}^{k} } \right)} \right)_{n \times 1}\).

Step 3. Construct the aggregate FF evaluation matrix for circular supply chain barriers. Using the \(FFWA\) operator, the aggregate FF evaluation matrix \(\overline{R} = \left( {\overline{r}_{j} } \right)_{n \times 1}\), \(\overline{r}_{j} = \left( {\overline{r}_{{j_{\mu } }} ,\overline{r}_{{j_{\nu } }} } \right)\), is constructed by,

$$\mathop R\limits^{ - } = FFWA\left( {R^{1} ,R^{2} , \ldots ,R^{K} } \right) = \left( {\left( {\mathop \sum \limits_{k = 1}^{K} w_{k} r_{{j_{\mu } }}^{k} ,\mathop \sum \limits_{k = 1}^{K} w_{k} r_{{j_{\nu } }}^{k} } \right)} \right)_{n \times 1}$$
(35)

where \(w_{k} = \frac{1}{K}\), implying that decision-makers are considered to have similar knowledge and experience concerning circular economy and public sector supply chains. Afterward, using the score function offered by Mishra and Rani [80], \(S\left( {r_{j} } \right)\) was obtained using,

$$S\left( {\overline{r}_{j} } \right) = \frac{1}{2}\left[ {\left( {\left( {\overline{r}_{{j_{\mu } }} } \right)^{3} - \left( {\overline{r}_{{j_{\nu } }} } \right)^{3} - ln\left( {1 + \left( {\overline{r}_{{j_{\pi } }} } \right)^{3} } \right)} \right) + 1} \right]\;\;\;\;\;\forall j$$
(36)

Step 4. Obtain the threshold value \(\theta\) from \(\overline{R}\) using Eq. (37),

$$\theta = \left( {1/n} \right){ }\left( {\overline{r}_{1} \oplus \overline{r}_{2} \oplus \ldots \oplus \overline{r}_{n} } \right)$$
(37)

wherein Eq. (9) was utilized for the addition operation \(\oplus\) of the FFS, and Eq. (11) was used to multiply the FF with a scalar \(\left( {1/n} \right)\) (i.e., \(\left( {1/n} \right) \in {\mathbb{R}}\)). If \(S\left( {r_{j} } \right) \ge\)\(S\left(\theta \right)\), then a barrier is considered relevant to the supply chains in the public sector. The computations yield \(\theta =\left(\mathrm{0.761,0}\right)\) and \(S\left(\theta \right)=0.4983\). Table 3 presents the list of relevant circular supply chain barriers. It shows 21 barriers relevant to public sector supply chains.

Table 3 List of circular supply chain barriers relevant to the public sector
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4.2 Phase II: Obtaining Priority Weights of Circular Supply Chain Barriers Using FF CRITIC

Step 5. Obtain the list of significant digital technologies. Mahroof et al. [72] offer a list of enablers of a circular economy (see Table 4). While it may be arguable that a systematic literature review is needed to develop the list of digital technologies, we contend that the list of Mahroof et al. [72] suffices the needed list based on the following justifications. First, the list of digital technologies provided by Mahroof et al. [72] contains the commonly available digital technologies deemed relevant in achieving a circular economy. Secondly, since a circular supply chain is a conceptual framework within the notion of circular economy, we can draw an extension in such a way that the digital technologies for a circular economy are likewise relevant to a circular supply chain. Thus, the list in Table 4 can potentially be considered sufficient to overcome the circularity barriers to public sector supply chains. The same eight decision-makers were asked to elicit judgment on the degree of importance of digital technologies in overcoming the barriers to supply chains in the public sector (see Table 3).

Table 4 List of digital technologies
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Step 6. Obtain the FF initial decision matrices. Initial decision matrices \({A}^{k}={\left({a}_{ij}^{k}\right)}_{m\times n}\), where \({a}_{ij}^{k}\) represent the degree of importance of an \(i\) th digital technology in overcoming a \(j\) th barrier elicited by the \(k\) th decision-maker \(\left(k=\mathrm{1,2},\dots ,K\right)\) in linguistic terms, were constructed. These matrices were transformed into FF decision matrices \(\tilde{A}^{k} = \left( {\left( {\mathop {\tilde{a}_{\mu }^{k} }\nolimits_{ij} ,\mathop {\tilde{a}_{\nu }^{k} }\nolimits_{ij} } \right)} \right)_{m \times n}\) using the linguistic scale shown in Table 2.

Step 7. Construct the aggregate FF decision matrix \(\overline{A }={\left(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\right)}_{m\times n}\). The aggregate FF decision matrix \(\overline{A }\) is constructed, where \(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\) is obtained using the \(FFWA\) operator in Eq. (15) such that

$$\left( {\mathop {\overline{a}_{\mu } }\nolimits_{ij} ,\mathop {\overline{a}_{\nu } }\nolimits_{ij} } \right) = \left( {\mathop \sum \limits_{k = 1}^{K} w_{k} \mathop {\tilde{a}_{\mu }^{k} }\nolimits_{ij} ,\mathop \sum \limits_{k = 1}^{K} w_{k} \mathop {\tilde{a}_{\nu }^{k} }\nolimits_{ij} } \right)_{m \times n}$$
(38)

where \({w}_{k}=\frac{1}{K}\), given that decision-makers are believed to be equally familiar with the concept of circular economy and public sector supply chain regarding their knowledge and experience. The resulting matrix is shown in Table 5.

Table 5 Aggregate FF decision matrix
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Step 8. Construct the normalized decision matrix. The score function \(S\left({\overline{a} }_{ij}\right)\) (\(\forall i,j\)) was obtained using the score function of Mishra and Rani [80]. Thus,

$$S\left({\overline{a} }_{ij}\right)=\frac{1}{2}\left[\left({\left({{\overline{a} }_{\mu }}_{ij}\right)}^{3}-{\left({{\overline{a} }_{\nu }}_{ij}\right)}^{3}-ln\left(1+{\left({{\overline{a} }_{\pi }}_{ij}\right)}^{3}\right)\right)+1\right] \forall i,j$$
(39)

After converting the aggregate evaluation scores in FFS to crisp values, the resulting matrix was used to create the normalized decision matrix, as shown in Table 6. Subsequently, Eq. (21) was utilized to obtain the standard deviation \({\sigma }_{j}\).

Table 6 Normalized decision matrix
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Step 9. Construct the symmetric matrix. Denoted as \(R\) by utilizing Eq. (22), the symmetric matrix is displayed in Table 7.

Table 7 Symmetric matrix
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Step 10. Calculate the priority weights \({w}_{j}\) for the circular supply chain barriers by computing the information value \({z}_{j}\) with Eq. (23). Then, using Eq. (24), the priority weights \({w}_{j}\) were derived and are illustrated in Table 8.

Table 8 Priority weights of the barriers of circular supply chain
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4.3 Phase 3: Evaluating the Digital Technologies via FF CODAS

Step 11. Obtain the negative ideal solution in FF, \({b}_{*}={\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)}_{1\times n},\) from the aggregate matrix \(\overline{A }={\left(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\right)}_{m\times n}\) where \(\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\) is obtained through \(\underset{i}{\mathrm{min}}\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\). According to Theorem 1, discussed in Sect. 3.1., \(\underset{i}{\mathrm{min}}\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\approx \underset{i}{\mathrm{min}}S\left({\overline{a} }_{ij}\right)\) where \(S\left({\overline{a} }_{ij}\right)\) obtained in Step 8 using the score function [80]. The resulting matrix is presented in Table 9.

Table 9 The negative ideal solution
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Step 12. Determine the Euclidean and Taxicab distances. The Euclidean distance \({d}_{i}\) between the action \(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\) and the ideal negative solution \(\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\) is calculated using,

$${d}_{i}\left(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right),\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\right)=\sum_{j=1}^{n}{w}_{j}\times d\left(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right),\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\right) \forall i$$
(40)

where \({w}_{j}\) represents the priority weight of each barrier obtained in Step 6 in Phase I. The measure \(d\left(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right),\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\right)\) was obtained using the Euclidean distance formula of Senapati and Yager [105],

$$d\left(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right),\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\right)=\sqrt{\frac{1}{2}\left[{\left({\left({{\overline{a} }_{\mu }}_{ij}\right)}^{3}-{\left({{b}_{\mu }}_{j*}\right)}^{3}\right)}^{2}+{\left({\left({{\overline{a} }_{\nu }}_{ij}\right)}^{3}-{\left({{b}_{\nu }}_{j*}\right)}^{3}\right)}^{2}+{\left({\left({{\overline{a} }_{\pi }}_{ij}\right)}^{3}-{\left({{b}_{\pi }}_{j*}\right)}^{3}\right)}^{2}\right]}$$
(41)

This formulation is analogous to how Senapati and Yager [105] framed the Fermatean fuzzy set extension of the TOPSIS method. On the other hand, the Taxicab distance \({t}_{i}\) between the action \(\left({{\overline{a} }_{\mu }}_{ij},{{\overline{a} }_{\nu }}_{ij}\right)\) and the ideal negative solution \(\left({{b}_{\mu }}_{j*},{{b}_{\nu }}_{j*}\right)\) is calculated using the subtraction operation defined in Eq. (13) such that,

$${t}_{i}=\left(\sum_{j=1}^{n}\left|\sqrt[3]{\frac{{\left({{\overline{a} }_{\mu }}_{ij}\right)}^{3}-{\left({{b}_{\mu }}_{j*}\right)}^{3}}{1-{\left({{b}_{\mu }}_{j*}\right)}^{3}}}\right|,\sum_{j=1}^{n}\left| \frac{{\left({{\overline{a} }_{\nu }}_{ij}\right)}^{3}}{{\left({{b}_{\nu }}_{j*}\right)}^{3}}\right|\right) \forall i$$
(42)

Subsequently, the taxicab distance \({t}_{i}\) is transformed to their corresponding crisp values by employing Eq. (16). Table 10 presents the Euclidean and Taxicab distances.

Table 10 Euclidean \({d}_{i}\) and Taxicab distances \({t}_{i}\)
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Step 13. Construct the relative assessment matrix, denoted as \(H={\left({h}_{ik}\right)}_{m\times m}\) by performing Step 6, outlined in Sect. 3.2, wherein the resulting matrix is illustrated in Table 11. A value of \(\rho =0.02\) is set as the threshold parameter.

Table 11 Relative assessment matrix
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Step 14. Determine the rank of each digital technology. Calculate the assessment score (\({S}_{i})\) of each alternative using Eq. (34), wherein the alternative with the highest \({S}_{i}\) value holds the highest rank. Table 12 features the ranking of the digital technologies according to their relevance in addressing circular supply chain barriers in the context of supply chains in the public sector.

Table 12 Ranking of digital technologies according to their relevance in overcoming barriers
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5 Post hoc Analysis

5.1 Sensitivity Analysis

The purpose of the sensitivity analysis is to test the sensitivity of the proposed procedure to possible changes in the evaluations of decision-makers. To this end, a stochastic simulation is performed with the following steps:

Step 1. Approximate the distribution of the evaluations on the \(i\) th digital technology and \(j\) th circular supply chain barrier. The evaluations are derived from the initial decision matrices \({A}^{k}\). Here, instead of \({a}_{ij}^{k}\) we used the evaluation scores described in Table 2 to represent the linguistic terms. This step is performed by first obtaining the standard deviation \({\sigma }_{ij}\) of all the evaluations corresponding to digital technology \(i\) and barrier \(j\). Assume that each \({a}_{ij}^{k}\) (\(\forall i,j,k\)) represents the aggregate knowledge and experience on the evaluation of \(i\) on \(j\). Lastly, assume that all evaluations are normally distributed within the decision-makers’ value judgments, with a mean equal to \({a}_{ij}^{k}\). Given the mean \({a}_{ij}^{k}\) and the standard deviation \({\sigma }_{ij}\), the distribution of the decisions on digital technology \(i\) and barrier \(j\) could then be obtained.

Step 2. Randomize the evaluations on digital technology \(i\) and barrier \(j\). Define \({p}_{ij}\in \left[\mathrm{0,1}\right]\) as the probability corresponding to an evaluation on \(i\) and \(j\). Also, a randomizer operator \(RAND\left(*\right)\) is defined, which generates a random value of *. The inverse of the normal cumulative distribution for the specified mean and standard deviation, a random probability \(RAND({p}_{ij})\), could then be obtained. The value of the inverse of the normal cumulative distribution is the randomized evaluation of decision-maker \(k\) on digital technology \(i\) and barrier \(j\), which is represented as \({a}_{ij}^{*,k}\)

Step 3. Perform the ranking methodology using \(a_{ij}^{k} \leftarrow \left| {a_{ij}^{*,k} } \right|\). In this step, \({a}_{ij}^{*,k}\) is rounded up, as indicated by the use of the ceiling function, so \(\lceil{a}_{ij}^{*,k}\rceil\) is the smallest integer greater than or equal to \({a}_{ij}^{*,k}\). This approximation is performed to mimic the decision-maker’s original decision-making task method by eliciting the evaluation scores in Table 2. The approximation also guarantees that \(1\le \lceil{a}_{ij}^{*,k}\rceil\le 7\). Also, the absolute value \(\left|\lceil{a}_{ij}^{*,k}\rceil\right|\) is obtained to guarantee that \({a}_{ij}^{k}>0\) since \({a}_{ij}^{k}\in \left\{\mathrm{1,2},\dots ,7\right\}\).

Step 4. Perform Step 1 to Step 3 multiple times. In this analysis, Step 1 to Step 3 is performed 1000 times while recording the results of each iteration.

5.2 Comparative Analysis

Two comparative analysis is performed in this study. The first comparative analysis is performed by comparing the results of the FF CRITIC-CODAS and CRITIC-CODAS. The purpose of this analysis is to show the difference in the results when epistemic uncertainty is considered and when it is not. The second comparative analysis is performed to compare the performance of the proposed procedure with other established MADM methods. In this analysis, the proposed integrated FF CRITIC-CODAS procedure is compared with CRITIC-EDAS and CRITIC-WASPAS. The specific procedure undertaken to perform the comparison is as follows:

Step 1. Obtain the decision matrix. The crisp decision matrix used for the computational procedures of both EDAS and WASPAS was obtained using the score function in Eq. (16), which provides the crisp equivalent of the FF evaluations of the experts.

Step 2. Obtain the priority weights of circular supply chain barriers. The barrier (or criteria) weights used to perform both EDAS and WASPAS are obtained using the CRITIC method, specifically those outlined in Sect. 3.2.

Step 3. Perform the sensitivity analysis. The sensitivity of the CRITIC-EDAS and CRITIC-WASPAS methods is assessed by performing Step 1 to Step 4 of the sensitivity analysis procedures detailed previously in Sect. 4.3 and then solving for the results of the CRITIC-EDAS and CRITIC-WASPAS.

Step 4. Normalize the performance score. To make the comparison of the three methods possible, the performance assessment metrics utilized by each method are normalized relative to the minimum and maximum performance of the digital technologies for all iterations of Step 1 to Step 4 of the sensitivity analysis. The performance of each \(i\) is measured using the assessment score \({S}_{i}\) in Eq. (34) in the CODAS method. On the other hand, the EDAS method makes use of the appraisal score \(A{S}_{i}\) (see Keshavarz Ghorabaee et al. [56]), while the WASPAS method makes use of the total relative importance score \({Q}_{i}\) (see Chakraborty and Zavadskas [19]). The metrics mentioned for each method are used to rank the digital technologies. To perform this step, the following mathematical procedures are implemented:

Suppose \({S}_{i}^{t}\), \(A{S}_{i}^{t}\), and \({Q}_{i}^{t}\) are the assessment score, appraisal score, and importance score of digital technology \(i\) at the \(t\) th iteration of Step 1 to Step 4 of the sensitivity analysis procedures, respectively. Define \(M{D}_{i}^{t}\in \left\{{S}_{i}^{t}, A{S}_{i}^{t}, {Q}_{i}^{t}\right\}\) and the normalizing operator \(NORM\left(*\right)\). The \(NORM\left(*\right)\) operator performs the following operation:

$$NORM\left(M{D}_{i}^{t}\right)=\frac{M{D}_{i}^{t}-\underset{{i}^{*}}{\mathrm{min}}M{D}_{{i}^{*}}^{t}}{\underset{{i}^{*}}{\mathrm{max}}M{D}_{{i}^{*}}^{t}-\underset{{i}^{*}}{\mathrm{min}}M{D}_{{i}^{*}}^{t}},\forall i,t$$
(43)

5.3 Results of the Sensitivity and Comparative Analyses

Figure 3 shows the sensitivity of the proposed FF CRITIC-CODAS compared to CRITIC-WASPAS and CRITIC-EDAS. The difference between the three methods is clearly minimal, which indicates that the proposed FF CRITIC-CODAS priority evaluation capability is at par with CRITIC-WASPAS and CRITIC-EDAS but with the added benefit of incorporating the inherent epistemic uncertainties in decision-making. It also shows that the results of the proposed FF CRITIC-CODAS, just the other methods that it is compared with, are moderately sensitive to changes in the evaluations of the decision-makers, which is apparent in the intersections of the range of the box plots corresponding to the digital technologies, specifically the range of the quartiles, where the results concentrate. Interestingly, this observed sensitivity yields a relatively substantial difference if the median of the stochastic simulation is considered for the priority evaluation. Figure 4 shows the difference between the FF CRITIC-CODAS, the FF CRITIC-CODAS while considering the median of the 1000 iterations of the stochastic simulation, and the CRITIC-CODAS.

Fig. 3
figure 3

Comparison of FF CRITIC-CODAS, CRITIC-WASPAS, and CRITIC-WASPAS

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Fig. 4
figure 4

Comparison of FF CRITIC-CODAS, the FF CRITIC-CODAS with stochastic simulation, and CRITIC-CODAS (non-fuzzy variant)

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There are digital technologies with a similar ranking in FFS and crisp set formulations. Specifically, digital twin (DT2), sentiment analysis (DT4), and RFID (DT8) rank sixth, seventh, and eighth, respectively, in both variants of FFS and crisp sets. Meanwhile, other digital technologies have different ranking results when FFS and crisp sets are adopted. Unlike crisp values, this difference is brought about by the FFS considering the degree of uncertainty (i.e., membership and non-membership degrees) of a particular judgment elicited by a decision-maker. Thus, trivial changes are likewise evident in the ranking, as seen in Fig. 4. For instance, ripple effect modelling (DT6) ranks first in FFS while it ranks third in the crisp set. Similarly, simulation (DT5) downgrades from ranking second in FFS to fifth in crisp sets. On the other hand, AI (DT1) ranks third in FFS but first with crisp sets. HR analytics (DT3) ranks fourth in FFS and second in the crisp sets. Meanwhile, blockchain technology (DT8) ranks fifth in FFS and fourth in non-fuzzy sets.

Crisp scores fail to consider the vagueness and uncertainty in the judgments of the decision-makers. Thus, in view of the hesitancy and uncertainty of decision-makers’ judgments, the use of FFS represents the evaluations of digital technologies better. Moreover, considering the practicality of using FF values to represent real-life scenarios wherein uncertain evaluations are more likely to happen, it is more suitable to cope with the decision-makers' hesitancy in eliciting judgments. Contextually, in this work, these sources of hesitancy and uncertainty came from the notion that the concept of supply chain circularity is not yet widely applied in different sectors, particularly the public sector. Thus, utilizing FF values in the proposed CRITIC-CODAS approach to evaluate the relevance of digital technologies in addressing barriers is a more plausible representation.

From Fig. 4, it could also be observed that the results of the CRITIC-CODAS (non-fuzzy set) are relatively closer to the results of the FF CRITIC-CODAS while considering the median of the stochastic simulation that was performed for the sensitivity analysis as the basis for the priority ranking. This is made more apparent through the correlation matrix in Table 13, where the two results have a correlation coefficient of 0.92857. This result is found to be significant, especially when a decision-maker is not very comfortable with a huge difference between the priority ranking results of the non-fuzzy set- (CRITIC-CODAS) and fuzzy set-based (FF CRITIC-CODAS) methods. The decision-maker could then adopt the post-analysis performed here—a stochastic simulation—to add an extra layer of uncertainty consideration. In this particular case, the result of the FF CRITIC-CODAS with stochastic simulation is relatively closer to the result of the CRITIC-CODAS (non-fuzzy set), in which a decision-maker (i.e., manager) might find to be a more comfortable basis for making a decision.

Table 13 Pearson correlation of the results of FF CRITIC-CODAS, the FF CRITIC-CODAS with stochastic simulation, and CRITIC-CODAS (non-fuzzy variant)
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6 Results and Discussion

The visual summary of the weights of the barriers, as shown in Table 8, is presented in Fig. 5. The barriers related to operational risk (B9), restrictive collection regulations (B19), lack of expertise (B24), lack of information sharing (B25), poor demand for environmentally friendly technologies (B36), and immature recycling technology (B40) emerge with the largest weights. Thus, these barriers have the highest influence (cumulative percentage \(\approx 40\%\)) on the ranking.

Fig. 5
figure 5

Priority weights of the barriers

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In this study, we provide a unique analysis of how digital technologies influence the barriers to circularity in public sector supply chains using the proposed FF CRITIC-CODAS approach. Based on the findings, ripple effect modeling (DT6), simulation (DT5), and Artificial Intelligence (AI) (DT1) are the top-priority digital technologies that have significant relevance to the barriers to the circular supply chain in the public sector. The preliminary analysis ensures that the gathered data is sufficient for use as inputs into the proposed method. With the proposed method, AI ranks third according to its relevance in addressing the barriers to circular supply chains. Presented in Fig. 6, ripple effect modeling (DT6) highly affects the barriers to organizational culture and management (B2), lack of effective planning and management for circularity (B3), operational risk (B9), lack of vision (B16), lack of theoretical information (B23), lack of expertise (B24), and size and complexity of supply chains make them difficult to align (B49). For instance, ripple effect modeling (DT6) serves a significant role as a supportive technology for some institutions. It can predict and simulate other supply chain elements that can help create a long-term vision of the circularity of the supply chain.

Fig. 6
figure 6

Isolated heat map of high-relevance digital technologies

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On the other hand, ripple effect modeling is advantageous in terms of product and process design. It can be used to simulate the life cycle assessment of a particular product, which is crucial in holistically understanding the environmental footprint of the product in all life cycle stages. Another digital technology that overcomes the barriers to a circular supply chain is simulation (DT5) which affects the following barriers: lack of performance evaluation system (B1), lack of effective planning and management for circularity concepts (B3), operational risk (B9), lack of vision (B16), lack of theoretical information (B23), lack of expertise, technology and information (B24), and size and complexity of supply chains make them difficult to align (B49). For instance, various simulation platforms can address the lack of effective planning and management for circularity concepts (B3) by simulating all the necessary elements of circular supply chains. Baruffaldi et al. [11] proposed a decision-support tool that utilizes simulation to support logistics managers with warehouse customization. A further example to address the lack of effective planning and management (B3), Kogler and Rauch [67] propose a contingency planning toolbox involving an event simulation model setup to examine outcomes of decisions prior to real, costly, and more permanent changes. Artificial Intelligence is another digital technology that addresses the barriers to circular supply chains. Implementing AI in the public sector requires a strategic course of action to take and ultimately create value. Based on the results, AI affects lack of performance evaluation system (B1), lack of trained intermediate staff, operational risk (B9), lack of circular economy awareness (B22), lack of theoretical information (B23), lack of expertise, technology, and information (B24), lack of sharing information and communication (B25), lack of technology transfers (B37), immature recycling technology (B40), and size and complexity of supply chains make them difficult to align (B49). Digitization is significantly changing the working environment, especially in the public sector, showing that the relevance of AI is evident. However, the public sector should invest highly in addressing these barriers via digital technologies.

7 Managerial Implications

Some possible guidelines for managers to consider implementing digital technologies for a circular public sector supply chain can be outlined. We start with the insights that the priority barriers offer to the circularity agenda in the public sector. Implementing circularity in public sector supply chains is highly impeded by those barriers associated with regulations that restrict the collection, poor demand or acceptance for environmentally superior technologies, lack of expertise, technology, and information, operational risk, immature recycling technologies, and information sharing and communication. The bureaucracy in the governance of the public sector makes the adoption of circular supply chain practices more challenging. For instance, in the Philippines, electronics and other equipment purchased by the government for its operations are prohibited for sale by existing laws unless stringent protocols are observed. Specific measures that would relax these bureaucracies would help promote the reverse logistics of used assets for end-of-life recovery. Aside from relaxing the purchase of recyclable assets, measures that would improve the expertise of the sector, technologies for recycling, and information sharing within the supply chain must be a priority to support the circularity initiatives. Currently, public sectors view a supply chain merely as a procurement tool [9], and in consequence, developing expertise in managing supply chains, acquiring technologies that support end-of-life recovery options, and sharing their interests with supply chain partners are mostly not within their organizational agenda. Thus, changing the perspectives of the public sector on how supply chains must work is a necessary condition in any circularity initiative. Due to the role the public sector plays in the economy, certain measures imposed on public sector supply chains may influence consumption behavior for recycled or remanufactured products. For instance, purchasing decisions that integrate design for remanufacturability or recyclability may further promote circularity in the supply chain.

The succeeding insights are focused on top-priority digital technologies. The first insight is into implementing simulation and ripple effect technology in the public sector. The utility of these technologies in facilitating circularity is their ability to provide a priori knowledge on the effects of implementing the circularity concept in the public sector. Such knowledge is essential for decision-makers to appreciate the circular economy and be motivated to institutionalize measures that facilitate its integration into public institutions. Moreover, when ripple effect analysis is adopted to promote circular public sector supply chains, it provides managers and administrators insights on the (1) event a particular failure can be triggered by an array of factors as well as the consequent set of failures that may occur, (2) specific structures in the public sector supply chains which are most sensitive to the ripple effects, and (3) the most efficient means when such scenarios take place. These insights provide managers and decision-makers with crucial information on the design of their networks and the necessary investments to implement an efficient circular supply chain in the public sector.

The approach Kamble et al. [49] discussed may be adopted to implement simulation and ripple effect modelling in the public sector. Their work emphasizes the importance of the Internet of Things to make simulations possible for tracking material flows and the ripple effects of policies for implementing such flows. However, such a technology-intensive measure may require high expertise and hefty investments. Tracing material flows in the system may require the analysis of complex networks and optimizing combinatorial problems, which is challenging to implement numerically. However, AI technology can be adapted to this end. AI can determine useful patterns that are beyond the limited human cognitive ability. AI can be trained to analyze material flows in the system to produce valuable designs for a circular public sector supply chain. Simulation analysis, being the second most relevant digital technology in overcoming barriers to circularity, can aid the ripple effect modeling, introducing the possibility of changing parameters dynamically during experiments and seeing how these changes affect the performance of transactions and operations in real time.

8 Conclusion and Future Work

Despite the prevalence of determining the barriers to circularity in supply chains, limited attention has been provided to understanding the characterization of these barriers in the context of public sector supply chains, which remains a significant gap in the literature. Furthermore, the role of digital technologies in addressing the barriers to circularity implementation in public sector supply chains remains unexplored. Thus, this work identifies, explores, and provides possible pathways to address the circular supply chain barriers by utilizing digital technologies in the public sector to advance the circularity agenda. In this work, three phases of analytical procedures were designed: (1) determining the list of barriers to implementing circular supply chain in the public sector, (2) obtaining the priority weights of these barriers, and (3) ranking the digital technologies based on their relevance in addressing the identified barriers in public sector supply chains. The proposed method involves integrating the CRITIC and the CODAS methods under the FFS environment to account for uncertainties, especially in performing analysis based on imprecise data.

Findings show that the transition to a circular economy-based supply chain in public sector organizations is associated with a myriad of barriers. Some highly crucial barriers include restrictive collection regulations, poor demand for environmentally benign technologies, lack of expertise, operational risk, immature recycling technology, and lack of information sharing. These barriers are considered more relevant to supply chains in the public sector than those in the private sector counterpart, where public sector decisions are not based on profit maximization. Findings also reveal that the top three most significant digital technologies in addressing barriers in the circular public sector supply chains are ripple effect modelling, simulation, and AI, wherein these digital technologies serve as supportive tools for the management of implementing supply chain circularity initiatives. Ripple effect modeling and simulation offer decision-makers prior knowledge of the systemic impact of a given circularity initiative in its conceptual design stage, providing critical inputs to the design and control measures before the deployment of such an initiative. The efficiency and flexibility that these digital technologies offer could become leverage to decision-makers in modeling different scenarios in its implementation stage. For instance, determining the location and capacity of warehouses that store electronic wastes for further processing back to the upstream supply chain, given certain regulatory requirements of the public sector, can benefit from the use of ripple effect modelling and simulation. On the other hand, AI augments decision-making under complex scenarios where decisions come from dynamic changes in parameters and variables. As a case in point, routing of reverse logistics can be a complex process under dynamic environments, with more uncertainty in decision parameters prevalent in the public sector, such as the arrival times of recyclable wastes, frequent changes in suppliers or contractors, and frequent downtimes of necessary infrastructures resulting from poor maintenance programs, among others.

The findings of the performed analysis can aid policymakers in circularity initiatives implementation in public sector supply chains. Future endeavors related to this work may include an empirical assessment of the effectiveness of the proposed measures through an actual case study. Additionally, the FF CRITIC-CODAS method shares a process-oriented nature with other integrated MADM techniques (e.g., [2, 8, 30]). The discussion obtained using this approach to analyze circularity barriers of the supply chains in the public sector contributes significantly to the knowledge gained through its application. The complexity of the computational steps associated with the proposed approach may be overcome by creating user-oriented software with the required steps in modular format, as Pamucar et al. [88] suggested. Alternative MADM techniques, such as the entropy weight method, analytic hierarchy process, and stepwise weight assessment ratio analysis, could be employed to produce priority weights for circular supply chain barriers as a future research endeavor. Furthermore, extensions of emerging MADM approaches for the sorting process (e.g., EDAS, MARCOS, and RAFSI) may be considered instead of CODAS. An extension of the FFS, the interval-valued FFS, was shown to be more robust than the FSS [96], and future work may explore such an application within the CRITIC and CODAS integration. Finally, other fuzzy environments can be incorporated to assist with making decisions for the problem domain using fuzzy logic, e.g., Picture fuzzy sets (e.g., [99]), spherical fuzzy sets, neutrosophic sets, the emerging \(q\)-rung orthopair fuzzy set as demonstrated by recent MADM studies (e.g., [31]), rough numbers as shown in Gokasar et al. [40], or D numbers such as in Pamucar et al. [89], among others.