Decision-making in fuzzy environments is a mature field of research. Over the last decades, a plethora of methods, based on commonly exploited trapezoidal fuzzy numbers (TrFNs) and triangular fuzzy numbers (TFNs) has been conceived to solve fuzzy multi-attribute decision-making (FMADM) problems (Zavadskas et al., 2020; Mohammadian et al., 2021; Keshavarz-Ghorabaee et al., 2022; Damjanović et al., 2024; Erdmann et al., 2024; Zhang et al., 2025). As is known, many researchers and practitioners use the so-called defuzzification to a point approximation method in order to compare and rank fuzzy numbers (FNs) in the early or later stage of the decision-making process. By defuzzification to a point it is meant the approximation of a given FN by a single deterministic real number (Chen and Hwang, 1992). However, this approximation exhibits serious shortcomings. Most notably, it often leads to the inevitable loss of information existing in the original data, which may seriously affect the reliability and validity of the results obtained by applying FMADM methods (see, e.g. Dymova et al., 2015; Wang, 2015; Sotoudeh-Anvari, 2020; Yatsalo and Martinez, 2018; Çelikbilek, 2019; Sharaf, 2020; Wang et al., 2020). Therefore, it would be of great value to make use of some approximation method, which could lead to more accurate results.

Shadowed sets (SHSs), first introduced by W. Pedrycz in 1998 (see also Pedrycz, 2005; 2009), were proposed as a knowledge-granulation mechanism intended both to capture vagueness in representation and to transform fuzzy sets (FSs) into simpler, more intuitive and more convenient practical use. Several works have studied the SHSs (Cai et al., 2017; Kong and Chen, 2017; Yao et al., 2017; Ibrahim et al., 2020; Gao et al., 2020; Patel and Shah, 2021; Wu et al., 2021; Yang and Yao, 2021; Boffa et al., 2022).

Recently, El-Hawy et al. (2015), Wang et al. (2018) and Landowski (2020) established formal definitions of shadowed numbers (SHNs), shadowed sets of the real line R and their algebraic operations. Any FN can be transformed into a corresponding three-region SHN. Later, Ibrahim et al. (2020, 2021) introduced models with as many as five-region SHSs, or more broadly, n-region SHSs.

While n-region SHSs provide for n > 3, a more precise representation of uncertainty than three-region SHSs, they increase computational complexity and challenges in optimizing multiple thresholds. These factors tend to lead to practical difficulties in interpreting and processing of FSs for at least three reasons. Firstly, processing large amounts of distinct grades poses a significant challenge (Pedrycz, 1998). Secondly, an increased number of thresholds is associated with a more complex approximation of FNs. Finally, additional thresholds increase the risk of misclassification due to the difficulty in distinguishing between slightly different membership grades (William-West and Ibrahim, 2023a; Zhang et al., 2024). For higher efficiency and practicality, this work is confined to three-region SHSs induced from the existing FSs.

In the literature on SHSs, trisecting a FS A ˜ can be achieved by applying the principle of uncertainty relocation or by determining optimal partition thresholds α and β ∈ [ 0 , 1 ] such that α < β (possibly with β = 1 − α). Using the membership function μ A ˜ of the fuzzy set A ˜, the membership grades μ A ˜ ( x ) higher than or equal to β are elevated to 1 and those lower than or equal to α are reduced to 0, while those in between are set to the entire open unit interval (0, 1). For a review and a comparison of different optimized threshold-based methods for constructing SSs from FSs, the reader is referred to William-West and Ibrahim (2023b). While many trisecting methods rely on complex optimization approaches, Grzegorzewski (2013) proposed an alternative framework based on successive approximation intervals. The current research advocates the use of the Grzegorzewski’s approach because, unlike others, it provides analytic solutions to directly calculate the four key points of a three-region SHS.

Compared to Pedrycz’s method, this approach offers at least three distinct advantages. Firstly, it provides direct formulas for SHS parameters applicable to any type of FN. Secondly, it offers a superior representation of uncertainty by preserving the expected interval and width of FN. Finally, through the inverse transform, it allows for the reconstruction of FNs from SHSs – a feature not supported by Pedrycz’s method (Grzegorzewski, 2013).

While FMADM has been extensively studied, its application within the framework of SHS theory has received considerably less interest. Since the inception of the SHS theory in 1998, and over the course of nearly a quarter century, few researchers have proposed and developed FMADM methods based on this mathematical framework. The existing FMADM methods involving SHNs are summarized as follows.

El-Hawy et al. (2015) introduced an enhanced version of shadowed fuzzy numbers (SHFNs), extending the concept to account for multiple types of uncertainty. They also proposed a novel method for ranking SHFNs. Wang et al. (2018) defined Pythagorean shadowed set (PSHS) and presented a data-driven approach for constructing PSHSs models for linguistic terms. Moreover, they have developed a score function for Pythagorean shadowed numbers (PSHNs) and introduced a new FMADM method based on PSHNs. Subsequently, Wang et al. (2020) proposed a q-rung orthopair shadowed set (q-ROSHS) and extended the VIKOR method to solve q-ROSHS FMADM problems involving q-ROSHS. On the other hand, He et al. (2021) modelled linguistics using SHSs and developed a SHS-based clustering model. They also introduced a centroid-based generalized Minkowski distance measure for SHSs and extended the TODIM method to accommodate the SS framework.

To fill the research gap on SHN FMADM, this paper proposes a new FMADM method within the SHN framework called the SHAdowed N Umber based Ranking and S election A pproach (SHANURSA). This approach is designed to ensure sound and defensible selection decisions in a fuzzy environment. Notably, the SHANURSA method is grounded on transforming the most commonly used FNs into three-region SHNs via Grzegorzewski’s (2013) approach, which then serve as the foundational elements of the method. Subsequently, the well-known Minkowski distance metric (MDM) is extended to the weighted MDM tailored for three-region SHNs. A new closeness coefficient is then introduced and defined to rank the available alternatives. lastly, the decision is made by selecting the alternative that maximizes the closeness coefficient (see details in Section 3).

The novelty of this research lies in the simultaneous use of Grzegorzewski’s approach, the weighted MDM for SHNs, in addition to a new and an unusual closeness coefficient. The results obtained in the three illustrative examples presented in Section 4 clearly demonstrate the stability and robustness of the SHANURSA method. Additionally, the proposed method stands out as an effective tool for solving FMCDM problems in the setting of SHS theory without the typical transformation of the FMCDM decision matrix into a beneficial decision matrix beforehand, a step commonly seen in the FMCDM literature.

The remainder of the paper is as follows. We first give clear definitions and notations to key terms used herein. Then, we present a comprehensive, detailed description of the SHANURSA method. Next, we demonstrate the effectiveness of the SHANURSA method using three illustrative examples from literature. Lastly, we summarize key findings and provide directions for future research.

This section briefly reviews several fundamental theoretical concepts, including the definitions of FSs, FNs, nonnegative fuzzy numbers (NNFNs), trapezoidal fuzzy numbers (TrFNs) and triangular fuzzy numbers (TFNs), as well as three-region SHSs and three-region SHNs. Additionally, we introduce a weighted MDM designed for three-region SHNs. Other related terms essential to understanding the proposed method are also defined.

A wide spectrum of methods to solve FMADM problems using the most commonly exploited TrFNs/TFNs has been developed over the years. What is at issue in current research is solving the FMADM problem of choosing a ‘best’ alternative from among a finite, given and fixed choice set (i.e. the predetermined set of feasible alternatives). As is known, to solve the above multi-attribute choice problem, most of the methods proposed in the FMADM literature adopt the ‘rank then select’ view. In fact, the ranking-based view states that at first a ranking of the alternatives from the most preferred to the least preferred (possibly with ties) is obtained; then, an alternative ranked the highest is chosen. Therefore, in the remainder of this work, we concentrate on ranking m ( ⩾ 2 ) discrete multi-attribute alternatives ( A 1 , A 2 , … , A m − 1 , and A m ) characterized by fuzzy attribute values (performance-values)—measured in the same unit or transformed to a common scale—with respect to n ( ⩾ 2) attributes C = ( C 1 , C 2 , … , C n − 1 , and C n ). Thus, the set of attributes is divided into a subset C C of cost attributes and a subset C B of benefit attributes. Furthermore, we assume the representability of the fuzzy attribute values a ˜ i j ’s of every alternative A i with respect to the attribute C j by nonnegative TrFNs/TFNs. Moreover, we need to mention that the types of weights, which are acceptable to reflect the degrees of importance that are associated with the selection attributes, could be either crisp or nonnegative TrFNs/TFNs.

As the SHANURSA method rests on ideal and anti-ideal concepts, the j-th positive ideal SHN, which is written S N j + , will be defined by: (16) S N j + = S N max j , if C j ∈ C B , S N min j , if C j ∈ C C . Likewise, the j-th negative ideal SHN, which is written S N j − , will be given by: (17) S N j − = S N min j , if C j ∈ C B , S N max j , if C j ∈ C C . Therefore, to define the positive ideal alternative (PIA) A + and the negative ideal alternative (NIA) A − , we have to represent them by SHNVs. Therefore, A + (n-dimensional reference vector of desirable SHNs) and A − (n-dimensional reference vector of undesirable SHNs) will be described respectively by the two n-dimensional SHNVs: (18) ∙ S N + ˜ = ( S N 1 + , S N 2 + , … , S N n + ) , (19) ∙ S N − ˜ = ( S N 1 − , S N 2 − , … , S N n − ) . Following, we define the proximity value P ( A i ) of the alternative A i to A + as follows: (20) P ( A i ) = d S H N V ( S N i ˜ , S N + ˜ ) = ∑ j = 1 n d S H N ( S N i j , S N j + ) , i = 1 , 2 , … , m . Similarly, the remoteness value R ( A i ) of A i from A − can be defined as follows: (21) R ( A i ) = d S H N V ( S N i ˜ , S N − ˜ ) = ∑ j = 1 n d S H N ( S N i j , S N j − ) , i = 1 , 2 , … , m .

To rank the alternatives, a new closeness coefficient relative to the ideal alternative is defined. This ranking index incorporates the relative importance of the separation measures from both PIA and NIA. In doing so, it addresses a key limitation of the conventional TOPSIS index. Specifically, the traditional TOPSIS index implicitly assumes that both distances are equally important (Kuo, 2017). In some situations, decision-makers may have distinct preferences regarding proximity to PIA versus avoidance of NIA. This necessity has prompted research efforts. For example, Doukas et al. (2010) developed a formula for separating weights to rank alternatives. A ranking formula was proposed by Kuo (2017) based on the method of Doukas et al. (2010), which normalizes the distance measures of each alternative relative to the positive and the negative ideal alternatives. Nevertheless, this index exhibits a notable drawback as the number of alternatives increases, the value of the ranking index decreases, thereby reducing its interpretability (Sadabadi et al., 2020). To address these drawbacks, Sadabadi et al. (2020) proposed a novel ranking index by modifying the normalization process of the distance measures.

Obviously, the previous research focuses on the simple additive weighting for constructing the closeness coefficient. However, the SAW is a fully compensatory method. To mitigate the compensation effect, this study proposes a different closeness coefficient based on the Weighted Product Model (WPM), which is less compensatory and enhances the robustness of the ranking results. The formula of the proposed closeness coefficient is as follows: (22) C ( A i ) = ( R ( A i ) max k R ( A i ) ) λ × ( min k P ( A i ) P ( A i ) ) 1 − λ , i = 1 , 2 , … , m , 0 < λ < 1 . The parameter λ denotes the weight representing the importance the decision maker assigns to the distance from NIA. Additionally, 1 − λ is the weight representing the importance assigned to the distance to PIA. In cases where the decision-makers do not show a preference for either the positive or negative ideal alternative ( λ = 0.5), the closeness coefficient becomes: (23) C ( A i ) = R ( A i ) max k R ( A i ) × min k P ( A i ) P ( A i ) , i = 1 , 2 , … , m . This corresponds to an equal weighting of the distances to both PIA and NIA.

In this section, we demonstrate the effectiveness of the SHANURSA method using the following three different illustrative examples:

The results are examined from the following three distinct perspectives. (1)

A sensitivity analysis was performed to determine how changes in the parameters k, k ′ , p and θ impacted the final rankings. The parameters were explored over predefined domains: k and k ′ were selected from the set { 0.5 , 0.6 , 0.7 , 0.8 , and 0.9 }, p is varied within { 1 , 2 } and θ takes values of 2 3 , 0.7 , 0.8 , and 0.9. The reliability of the SHANURSA method was assessed through the systematic combination of these values which yielded a total of 200 experimental scenarios denoted by S 1 , S 2 , … , S 200 .