A complex spherical fuzzy set (CSFS) is a generalization of the spherical fuzzy set (SFS) to express the two-dimensional ambiguous information in which the range of positive, neutral and negative degrees occurs in the complex plane with the unit disk. Considering the vital importance of the concept of CSFSs which is gaining massive attention in the research area of two-dimensional uncertain information, we aim to establish a novel methodology for multi-criteria group decision-making (MCGDM). This methodology allows us to calculate both the weights of the decision-makers (DMs) and the weights of the criteria objectively. For this goal, we first introduce a new entropy measure function that measures the fuzziness degree associated with a CSFS to compute the unknown criteria weights in this methodology. Then, we present an innovative Complex Proportional Assessment (COPRAS) method based on the proposed entropy measure in the complex spherical fuzzy environment. Besides, we solve a strategic supplier selection problem which is very important to maximize the efficiency of the trading companies. Finally, we present some comparative analyses with some existing methods in different set theories, including the entropy measures, to show the feasibility and usefulness of the proposed method in the decision-making process.

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,

Fuzzy set (FS) theory (Zadeh,

Geometric representations of IFS, PyFS, PFS and SFS.

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,

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,

The COPRAS method, introduced by Zavadskas

Some combinations with traditional methods via objective and subjective weighting.

Obj. w. | Some combined versions | Given by | Subj. w. | Some combined versions | Given by |

MEREC | MEREC-ARAS | Rani |
ANP | ANP-TOPSIS | Sakthivel |

MEREC | MEREC-MULTIMOORA | Mishra |
ANP | ANP-DEMATEL | Yang |

MEREC | MEREC-WASPAS | Keshavarz-Ghorabaee ( |
ANP | ANP-COPRAS | Balali |

CRITIC | CRITIC-CoCoSo | Peng |
AHP | AHP-COPRAS | Ecer ( |

CRITIC | CRITIC-WASPAS | Keshavarz-Ghorabaee |
AHP | AHP-TOPSIS | Anser |

BWM | BWM-LBWA-CoCoSo | Torkayesh |
LBWA | BWM-LBWA-CoCoSo | Torkayesh |

BWM | BWM-TOPSIS | Gupta and Barua ( |
LBWA | LBWA-WASPAS | Pamucar |

Entropy | Entropy-COPRAS-MULTIMOORA | Alkan and Albayrak ( |
FUCOM | FUCOM-MABAC | Bozanic |

Entropy | Entropy-WASPAS | Aydoğdu and Gül ( |
FUCOM | FUCOM-MARCOS | Pamucar |

Entropy | Entropy-ARAS | Aydoğdu and Gül ( |
SWARA | SWARA-COPRAS | Rani |

Entropy | Entropy-TOPSIS | Aydoğdu |
SWARA | SWARA-VIKOR | Alimardani |

Literature review for COPRAS method.

Given by | Model | Method | Group | Criteria weights | Application area |

Kumari and Mishra ( |
IFS | COPRAS | X | Obj. (Entropy) | Green supplier selection |

Mishra |
IFS | SWARA-COPRAS | X | Subj. (SWARA) | Select. of an optimal bioenergy production tech. |

Schitea |
IFS | WASPAS-COPRAS-EDAS | X | Subj. | Hydrogen mobility roll-up site selection |

Buyukozkan and Gocer ( |
PyFS | AHP-COPRAS | X | Subj. (AHP) | Digital supply chain partner selection |

Rani |
PyFS | COPRAS | X | Obj. (Entropy) | Pharmacological therapy select. for type-2 diabetes |

Dorfeshan and Mousavi ( |
PyFS | COPRAS-TOPSIS | Marble processing plants project | ||

Chaurasiya and Jain ( |
PyFS | COPRAS | X | Obj. (Entropy) | Multi-criteria healthcare waste treatment problem |

Thaoa and Smarandache ( |
PyFS | COPRAS | X | Obj. (Entropy) | Select. of teaching management system |

Alipour |
PyFS | SWARA-COPRAS | X | Obj. (Entropy) | Fuel cell and hydrogen components supplier select |

X | Subj. (SWARA) | ||||

Lu |
PFS | COPRAS | Obj. (CRITIC) | Green supplier selection | |

Kahraman |
PFS | COPRAS-VIKOR-TOPSIS | X | Subj. (AHP) | A state of the art survey |

Omerali and Kaya ( |
SFS | COPRAS | Subj. | Selection of the augmented reality solution | |

Güner |
SFS | AHP-COPRAS | Subj. (AHP) | Renewable energy selection |

Nowadays, researchers are handling MCDM/MCGDM problems including uncertain two-dimensional data. Especially, the CSFSs have drawn attention to their broader structure when comparing other set theories. Different approaches with several applications in the CSF environment have been presented: Ali

Literature review for the MCDM-MCGDM methods in the CSFSs.

Given by | Model | Method | Group | Criteria weights | Application area |

Ali |
CSFS | TOPSIS | X | Subj. | Select. of organization to extend the income |

Akram |
CSFS | ELECTRE-I | X | Subj. | Select. of location for new branch of a company |

Akram |
CSFS | TOPSIS | Subj. | Select. of best water supply strategy | |

Akram |
CSFS | VIKOR | Subj. | Select. of the advertisement on Facebook | |

Aldemir |
CSFS | TOPSIS based on aggregation op. | Subj. | ||

Zahid |
CSFS | ELECTRE-II | Subj. | Selection of the tech. to treat cad.-contam. water | |

Naeem |
CSFS | Aggregation operators | Obj. | Green supplier selection | |

Aydoğdu |
CSFS | TOPSIS based on entropy | Obj. | Select. of the advertisement on Facebook |

COPRAS method is used for the evaluation of the multi-criteria system of variables for maximizing and minimizing the values. Since this method allows us to compare and also check the final results of measuring easily, it is preferred more over the other existing methods. Also, this method allows being used to implement the comparison and evaluation of variables described hierarchically without requiring such transformation as minimizing the variables. On the other hand, CSFS theory is more powerful with its superior structure to those modern extensions of FS theory which can elaborate the two-dimensional ambiguous information. By considering all positive sides, in this study, we establish a novel method by considering respect to the advantages of CSFSs in describing uncertain information, the useful structure of the COPRAS method in MCGDM problems and the entropy measure which allows for determining the objective weights of the criteria. While the proposed method determines the unknown criteria weights by using the entropy measure, it satisfies that the smaller entropy measure of a criterion among alternatives should be imposed as the bigger weight to that criterion, and otherwise, the smaller weight to that criterion. We can enlist the main objectives of the article as follows:

We establish a novel improved COPRAS method in CSFS. In this method, a new formula is developed to evaluate unknown weight information of both DMs and criteria weights. These weights are calculated by using the entropy measure method to obtain objective weights. For this reason, we propose a new entropy measure function and explain why we need this entropy measure function and what kind of superiority it has over the existing functions.

We solve the problem of “selection of the strategic supplier” by the proposed method as an objective weight of DMs and criteria.

To explicate the adequacy of the proposed strategy and consistency of the result, a comparison analysis and method analysis with the existing method are presented.

The versatility and decision-making skills of our proposed COPRAS method is not only limited to two-dimensional data but also this method exhibits the same accuracy when applied to one-dimensional data inclusive of spherical fuzzy data and picture fuzzy data by taking their phase term equal to zero. Thus, the proposed methodology is a flexible approach that competently manages both traditional and two-dimensional uncertain information with precision.

The proposed COPRAS technique not only deals excellently with CSF information but also can be successfully applied to the complex Pythagorean model and complex intuitionistic model by taking their neutral-membership equal to zero.

The objective weight data of our proposed method is not limited to the COPRAS methodology. Proposed objective criteria weighting schema and objective DMs’ weighting schema can be applied to different CSF-MCGDM methods with the same example if their methods include subjective weighting data.

We compare this method with the CSF-TOPSIS based on entropy method given by Aydoğdu

The rest of the paper is organized as follows. In Section

In this section, we recall some fundamental definitions which will be used in the main sections. Throughout this paper,

Let

The complement of the CSFS

Let

(Akram

Let

Let

A score function

An accuracy function

If

If

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

Let

For a crisp set

For all

It is obvious that

There are four possibilities we have to consider. The first one is

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

Linguistic terms to evaluate the alternatives via criteria (Zahid

Lingusitic terms | CSFNs |

Very good (VG)/Very important (VI) | |

Good (G)/Important (I) | |

Medium good (MG)/Medium important (MI) | |

Medium (M) | |

Medium poor (MP)/Medium unimportant (MUI) | |

Poor (P)/Unimportant(UI) | |

Very poor (VP)/Very unimportant (VUI) |

Then these values establish the complex spherical fuzzy decision matrix (CSFDM)

Flow chart of the CSF-COPRAS technique based on entropy.

The procedure of the new COPRAS method based on entropy consists of the following steps:

The

Suppliers have always been an integral component of a company’s management policy; however, the relationship between companies and their suppliers has traditionally been distant. In today’s global economy of just-in-time (JIT) manufacturing and value-added focus, there is a heightened need to change this adversarial relationship to one of cooperation and seamless integration. JIT requires the vendor to manufacture and deliver to the company the precise quantity and quality of material at the required time. Thus the performance of the supplier becomes a key element in a company’s success or failure. In order to attain the goals of low cost, consistently high quality, flexibility, and quick response, companies have increasingly considered better supplier selection approaches. These approaches require cooperation in sharing costs, benefits, and expertise in attempting to understand one another’s strengths and weaknesses, which in turn leads to single sourcing, supplier, and long-term partnerships. Since the supplier selection process encompasses different functions (such as purchasing, quality, production, etc.) within a company, it is a multi-objective problem, encompassing many tangible and intangible factors in a hierarchical manner. The evaluation of intangible factors requires the assessment of expert judgment, and the hierarchical structure requires decomposition and synthesis of these factors (Bhutta and Huq,

Now, we consider the problem “selection of the strategic supplier selection” given by Igoulalene

Illustrative example (stakeholder preferences given in Igoulalene

G | VG | G | ||

MG | G | MG | ||

VG | G | VG | ||

G | G | G | ||

MG | G | M | ||

M | MG | G | ||

G | MG | MG | ||

MG | M | MG | ||

VG | VG | VG | ||

VG | G | VG | ||

VG | VG | G | ||

VG | VG | G | ||

MG | G | G | ||

M | M | MG | ||

VG | G | G | ||

G | MG | MG | ||

M | MG | MG | ||

MP | M | M | ||

G | G | MG | ||

M | MG | M |

CSFDMsestablished by expert

CSFDMsestablished by expert

CSFDMsestablished by expert

NCSFDM of the expert

NCSFDM of the expert

NCSFDM of the expert

GO matrices.

GO | ||

RIO matrices.

RIO | ||

LIO matrices.

LIO | ||

DGO, DRIO and DLIO matrices.

DGO | |||||

0.1213 | 0.3204 | 0.1112 | 0.2494 | 0.2432 | |

0.2332 | 0.2106 | 0.1858 | 0.1538 | 0.1930 | |

0.1213 | 0.2587 | 0.2210 | 0.1856 | 0.2363 |

DRIO | |||||

0.2766 | 0.4791 | 0.2171 | 0.3612 | 0.3175 | |

0.2171 | 0.4439 | 0.3070 | 0.3146 | 0.2277 | |

0.2766 | 0.3020 | 0.2171 | 0.2171 | 0.2695 |

DLIO | |||||

0.2171 | 0.2774 | 0.3070 | 0.2171 | 0.2695 | |

0.2766 | 0.3497 | 0.2171 | 0.2805 | 0.4746 | |

0.2171 | 0.4727 | 0.3070 | 0.3612 | 0.3175 |

The ACSFDM.

Weights of criteria.

0.2802 | 0.4421 | 0.1876 | 0.3697 | |

0.7198 | 0.5578 | 0.8124 | 0.6303 | |

0.2646 | 0.2051 | 0.2986 | 0.2317 |

The AWCSFDM.