In our world which is becoming a more global marketplace, the global environment is forcing companies to take almost everything into consideration at the same time, remain competitive and respond to rapidly changing markets. In this aspect, supply chain management and strategic sourcing have been one of the fastest-growing and most important areas of management in companies. Since technological complexity has affected the logistics and supply chains directly, the supply chain management has to adapt to these complex and dynamic factors. So, in this trading world, the search for new and strategic suppliers is a continuous priority for companies in order to upgrade the variety and typology of their product range. Hence, supplier selection represents one of the most important functions to be performed by the purchasing department that determines the long-term viability of a company. Strategic supplier selection is a multi-criteria problem that includes both qualitative and quantitative criteria. In order to select the best suppliers, it is necessary to make a tradeoff between tangible and intangible criteria, some of which may conflict. In this case, we are required to handle a decision-making problem.

Decision-making is the process of identifying different and possible alternatives that can solve a problem and choosing the one that will best meet the expectations among these alternatives. Since complexity prolongs the decision-making process, as it requires the evaluation of many alternatives according to many criteria in the process, many studies and decision-making methods have been developed in the literature to work with complex data and make an appropriate choice (Chen, 1988; Maji et al., 2001). Multi-criteria decision-making (MCDM) is one of the decision-making methods based on an expert’s opinion. If more than one expert is attending, this method is called MCGDM. In literature, there are many techniques that have been developed to solve MCDM and MCGDM problems such as the Analytic Hierarchy Process (AHP) (Saaty, 1980), Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) (Hwang and Yoon, 1981), COPRAS (Zavadskas et al., 1994), VIsekriterijumsko optimizacija Kompromisno Rangiranje (VIKOR) (Opricovic, 1998), Multi-Objective Optimization by Ratio Analysis (MOORA) (Brauers and Zavadskas, 2006) and so on.

Fuzzy set (FS) theory (Zadeh, 1965) is an effective tool for solving decision-making problems in this increasingly complex world, as FS is a way of thinking used to describe the imprecise. However, since the evaluation is only made on the degree of membership in FS theory, this is also insufficient to solve complex problems. For this reason, many generalizations of FS theory have been made in the literature. The intuitionistic fuzzy set (IFS) (Atanassov, 2003) which is one of these generalizations, refers to a set whose sum of positive-membership degree and negative-membership degree is less than or equal to 1. After, the IFS theory was extended to Pythagorean fuzzy set (PyFS) (Yager, 2013) theory by considering the sum of the squares of its positive-membership degree and negative-membership degree is less than or equal to 1. The other extension of IFS is Picture Fuzzy Set (PFS) (Cuong, 2013) which has positive-membership, neutral-membership and negative-membership degrees and the sum of these degrees is less than or equal to 1. PFS is able to deal with problems that have more answers. Then the theory of SFS has been developed by Mahmood et al. (2018) to encounter situations that PFS cannot meet. In the SFS, the sum of the squares of positive-membership, neutral-membership and negative-membership degrees is less than or equal to 1. Many authors have studied on these sets (Aydoğdu and Gül, 2020; Güner and Aygün, 2020, 2022). Geometric representations of the theories IFS, PyFS, PFS and SFS are shown in Fig. 1. The mentioned set theories are highly proficient and skilled to carry ambiguous information but their capabilities are limited to handle one-dimensional data. Many MCDM problems comprise two-dimensional data but the existing MCDM strategies are incompetent to handle the two-dimensional information. To handle such phenomena, complex generalizations of FSs mentioned above have been studied by Ramot et al. (2002, 2003), Alkouri and Salleh (2013), Ullah et al. (2020), Akram et al. (2021c) and these sets have been applied to many decision-making problems. Azam et al. (2022) gave an example of evaluating the enterprise’s information security management issue in a particular organization on complex intuitionistic fuzzy sets (CIFSs). Akram et al. (2020) made an example of selecting the best capable ERP systems as candidates after collecting information about ERP vendors and systems from all aspects of the complex picture fuzzy set (CPFS) environment. Recently, Akram et al. (2021c) introduced the theory of CSFSs to handle the two-dimensional data where we consider the degrees of positive-membership, neutral-membership, negative-membership and refusal that lie inside a complex unit circle. According to this theory, the sum of squares of their amplitude (and phase terms) can not exceed 1. In this way, lots of decision-making problems, consisting of the mentioned data, became solvable by using the developed MCGDM methods.

One of the most critical steps in MCDM/MCGDM techniques is to determine the weights of the criteria because the weights directly affect the ranking of the alternatives. For this reason, many methods have been developed to calculate criterion weights. Some of these are subjective and some are weighting methods based on an objective point of view. Methods such as AHP (Saaty, 1980), Analytic Network Process (ANP) (Saaty, 1996), Step-Wise Weight Assessment Ratio Analysis (SWARA) (Keršuliene et al., 2010), Full Consistency Method (FUCOM) (Pamucar et al., 2018) and Level Based Weight Assessment (LBWA) (Žižović and Pamucar, 2019) are among the subjective weighting methods in which the preferences of DMs are taken into account. In some objective weighting methods such as Entropy (De Luca and Termini, 1972), CRiteria Importance Through Intercriteria Correlation (CRITIC) by Diakoulaki et al. (1995), Best Worst Method (BWM) (Rezaei, 2015) and Method based on the Removal Effects of Criteria (MEREC) by Keshavarz-Ghorabaee et al. (2021), the mathematical model is solved without considering the ideas of the DMs.

Entropy is the random measurement of the uncertainty in a process or the amount of information produced. It is also relevant to questions about how to measure the uncertainty of the entropy fuzzy environment. Many authors (De Luca and Termini, 1972, 1977; Xuecheng, 1992; Fan and Xie, 1999) introduced the axiom construction of FS entropy. Hung and Yang (2006) extended these ideas to construct the concept of the fuzzy entropy of IFSs. Thaoa and Smarandache (2019) extended the fuzzy entropy of Hung and Yang (2006) to PFSs. Many authors (Thaoa and Smarandache, 2019; Rani et al., 2020b; Alipour et al., 2021; Gül and Aydoğdu, 2021; Chaurasiya and Jain, 2022) gave the entropy measure on PFS to solve many decision-making problems. Aydoğdu and Gül (2020) proposed a novel entropy measure for SFSs and applied this entropy to solve the MCGDM problems. Also, Naeem et al. (2022) and Aydoğdu et al. (2023) defined the new entropy measure functions to calculate the weights of criteria objectively. In Table 1, one can find some remarkable studies that are combined with the traditional methods and the mentioned objective and subjective weighting approaches.

In this section, we recall some fundamental definitions which will be used in the main sections. Throughout this paper, X will denote the set of the universe. Definition 1

Let f , g , h : X → [ 0 , 1 ], α , β , γ : X → [ 0 , 2 π ] and i = − 1 . A CSFS over X is of the form C = { ( x , f ( x ) e i α ( x ) , g ( x ) e i β ( x ) , h ( x ) e i γ ( x ) ) | x ∈ X } if the conditions f 2 ( x ) + g 2 ( x ) + h 2 ( x ) ⩽ 1 and ( α ( x ) 2 π ) 2 + ( β ( x ) 2 π ) 2 + ( γ ( x ) 2 π ) 2 ⩽ 1 are satisfied for all x ∈ X. The functions f ( x ) = f ( x ) e i α ( x ) , g ( x ) = g ( x ) e i β ( x ) and h ( x ) = g ( x ) e i γ ( x ) denote the positive-membership, neutral-membership and negative-membership of x to C which are restricted to the unit circle and consists of two terms such as amplitude term and phase term. The refusal function is given by t ( x ) = 1 − f 2 ( x ) − g 2 ( x ) − h 2 ( x ) e i 2 π 1 − ( α ( x ) 2 π ) 2 − ( β ( x ) 2 π ) 2 − ( γ ( x ) 2 π ) 2 for all x ∈ X. We denote the collection of CSFSs over X with CSFS(X). The triplet C = ( f , g , h ) is called a complex spherical fuzzy number (CSFN) where f = f e i α , g = g e i β , h = h e i γ for f , g , h ∈ [ 0 , 1 ], α , β , γ ∈ [ 0 , 2 π ] satisfying f 2 + g 2 + h 2 ⩽ 1 and ( α 2 π ) 2 + ( β 2 π ) 2 + ( γ 2 π ) 2 ⩽ 1.

In this section, we give a novel entropy to measure the fuzziness of CSFSs in the process of decision-making. Definition 8.

Let C, C 1 and C 2 be CSFSs on X. A mapping E : CSFS ( X ) → [ 0 , 1 ] is said to be an entropy measure function on CSFS if E satisfies all of the conditions: 1.

E ( C ) = 0 if C is acrisp set.

In this section, we establish the COPRAS method to solve MCGDM problems in the complex spherical fuzzy environment when the information of both weights of DMs and criteria are completely unknown. With this aim, we calculate the weights of DMs based on Euclidean distance and the weights of criteria based on the proposed new entropy measure.

Let A = { A 1 , A 2 , … , A k } be the set of k alternatives, C = { C 1 , C 2 , … , C m } be the set of m criteria and E = { E 1 , E 2 , … , E n } be the set of n experts (DMs) hired for decision-making. Each expert E r evaluates the alternatives A p with respect to C q by considering the influence of C q on the alternatives A p and by using the linguistic table given in Table 4.