In practice, the judgments of decision-makers are often uncertain and thus cannot be represented by accurate values. In this study, the opinions of decision-makers are collected based on grey linguistic variables and the data retains the grey nature throughout all the decision-making process. A grey best-worst method (GBWM) is developed for multiple experts multiple criteria decision-making problems that can employ grey linguistic variables as input data to cover uncertainty. An example is solved by the GBWM and then a sensitivity analysis is done to show the robustness of the method. Comparative analyses verify the validity and advantages of the GBWM.

Nowadays, organizations need to make decisions for different matters. Employing a suitable approach to make a correct decision is an ongoing concern of organizations. There are many types of methods to solve multiple criteria decision-making (MCDM) problems, which can be categorized into pairwise comparison-based methods (Doumpos and Zopounidis,

In real-world situations, the input data of decision-making problems include uncertainty and/or incompleteness (Zavadskas

The main objective of the current study is to present the grey BWM (GBWM). Specifically, the contributions of this study can be summarized as follows:

We introduce the GBWM and the result calculated by this method is more reliable than the fuzzy BWM because the GBWM has a smaller inconsistency ratio compared with the fuzzy BWM.

The GBWM uses a linear model that can present the global-optimum weights for MCDM problems, while the existing fuzzy BWM employs a non-linear model with local-optimum weights.

The current study is organized as follows: in Section

AHP is one of the most well-known methods for MCDM. However, the BWM deduces more consistent weights based on the less comparisons compared with the AHP (Rezaei,

Model (

Solving Model (

In this method, the consistency ratio, calculated by Eq. (

Consistency Index (Rezaei,

The maximal preference degree of the best over the worst | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Consistency Index | 0.00 | 0.44 | 1.00 | 1.63 | 2.30 | 3.00 | 3.73 | 4.47 | 5.23 |

Considering the advantages, the BWM has gained ever-increasing investigation in recent years (Mi

In this section, we are going to examine applications of BWM that have been conducted in different fields. Rezaei

Scholars have applied the original BWM into different fields. Gupta and Barua (

For some situations, the uncertain information of experts is easy to access and the uncertain BWMs have attracted the attention of scholars (Mi

Mou

Since there is no research item on the combination of the grey system theory and BWM, the current study attempts to present the GBWM. As mentioned, the grey theory has features such as reality and vague aspects, and can be employed with an incomplete data, therefore, it can be a useful approach to solve decision-making problems (Zavadskas

The grey system theory, proposed in 1980s by Julong (

The comparison between grey system theory and fuzzy set theory.

Uncertainty research | Grey system | Fuzzy math |

Research object | Poor information | Cognitive |

Basic set | Grey number set | Fuzzy set |

Describe method | Possibility function | Membership function |

Procedure | Sequence operator | Cut set |

Data requirement | Any distribution | Known membership |

Emphasis | Intension | Extension |

Objective | Law of reality | Cognitive expression |

Characteristic | Small data | Depend on experience |

In this section, the related preliminaries of grey numbers are reviewed.

A grey number (Liu

To denote the central point of grey numbers, the “Kernel” of grey numbers (Guo

For two grey numbers

The length of

To compare the grey numbers, the greyness degree of

The grey possibility degree for numbers

Different methods have been presented for solving GLP models. Huang

The current study employs the positioned programming method to solve GLP problems in the form of Model (

To solve the GLP model, it ought to be whitened first.

If the values of

After the whitening stage, we are left with Model (

If

This theorem has been proven in Liu

In this section, the GBWM is presented. The eight steps of this method are outlined in detail as follows:

Linguistic variables of decision-makers.

Linguistic variable | Value |

Equally Important (EI) | |

Weakly Important (WI) | |

Fairly Important (FI) | |

Very Important (VI) | |

Absolutely Important (AI) |

In Eq. (

According to the features of absolute value, Model (

To cross multiply the constraints of Model (

Due to the multiplication of variables

In Eq. (

In this section, the nonlinear Model (

The assumptions mentioned in Eq. (

The value of

Based on Eqs. (

To find the optimal grey weights, the grey linear Model (

Grey linguistic variables for determining the significance of each expert.

Very low | Low | Medium low | Medium | Medium-high | High | Very high |

[ |
[ |
[ |
[ |
[ |
[ |
[ |

Ultimately, we have the following results:

where

Consistency Index for linguistic grey numbers.

Linguistic terms | Equally Important (EI) | Weakly Important (WI) | Fairly Important (FI) | Very Important (VI) | Absolutely Important (AI) |

CI-GBWM | 0.00 | 0.20 | 0.71 | 1.31 | 1.96 |

By the sum of the horizontal components of the matrix

In this section, we implement the GBWM to solve an MCDM problem with multiple experts and analyses the important parameters in the GBWM. Then, the comparative analysis with respect to the GBWM and the fuzzy BWM is performed to verify the validity in ranking results and the advantages in keeping high reliability.

In this section, the collected data for an MCDM problem about purchasing a car is described. Different criteria may be considered for purchasing a car. Accordingly, three experts are consulted for a group decision making. The question is which criterion is the most important and how can we find optimal weights for the criteria. The solution is using GBWM for group decision making and under uncertainty conditions.

The degree of preferences of the best criterion to the other criteria.

Experts | Best-to-others | Price | Quality | Comfort | Safety | Style |

1 | Best criterion: price | EI | WI | VI | VI | AI |

2 | Best criterion: price | EI | FI | AI | AI | VI |

3 | Best criterion: quality | WI | EI | AI | VI | AI |

The degree of preference of each criterion to the worst criterion.

Others-to-worst | Expert | ||

1 | 2 | 3 | |

Worst criterion: Style | Worst criterion: Safety | Worst criterion: Comfort | |

Price | AI | AI | VI |

Quality | VI | AI | AI |

Comfort | FI | WI | EI |

Safety | FI | EI | FI |

Style | EI | FI | WI |

After solving Model (

The weight of each criterion based on the opinions of decision-makers.

Variable | P1 (Medium) | P2 (Medium-low) | P3 (Low) | |||

Upper | Lower | Upper | Lower | Upper | Lower | |

0.3107275 | 0.2516854 | 0.4569733 | 0.3334929 | 0.4000000 | 0.2647273 | |

0.4660912 | 0.2726592 | 0.3184965 | 0.1773898 | 0.3733333 | 0.2443636 | |

0.1035758 | 0.0838951 | 0.1364985 | 0.0985499 | 0.1600000 | 0.1047273 | |

0.1864365 | 0.1078652 | 0.0969337 | 0.0638603 | 0.1600000 | 0.1047273 | |

0.1331689 | 0.0838951 | 0.1910979 | 0.1267070 | 0.1066667 | 0.0814545 |

The weights obtained based on the opinions of decision-makers.

The opinions of the experts regarding each criterion have been compared in Fig.

The aggregated weights based on the opinions of decision-makers.

Variable | Lower | Upper |

0.282710725 | 0.39256978 | |

0.22503049 | 0.37427681 | |

0.09769076 | 0.13611538 | |

0.08939217 | 0.14066120 | |

0.09500056 | 0.13694033 |

Final normalized weights.

Variable | Lower bound | Upper bound |

0.286965 | 0.3984682 | |

0.228412 | 0.3799003 | |

0.099159 | 0.1381605 | |

0.090735 | 0.1427746 | |

0.096428 | 0.1389979 |

Comparisons of the final obtained weights of criteria.

Another method which yields a relatively similar result is the formation of a matrix of grey possibility degree which is concluded as Eq. (

Based on the sum of the horizontal components of the matrix in Eq. (

In this section, we aim to analyse the sensitivity of the example solved in the previous section. A primary reason for undertaking sensitivity analysis is to investigate the changes of output parameters resulting from changes in the input data. Since the current study makes use of the positioned programming approach, the parameter

Sensitivity analysis of parameters

As is depicted in Fig.

Sensitivity analysis of parameters

Sensitivity analysis of parameters

It can be seen from Fig.

According to Fig.

In this section, we solve the numerical example by the FBWM (Guo and Zhao,

The deduced weights of criteria and consistency index by the FBWM and GBWM for expert 1.

FBWM (Guo and Zhao, |
GBWM (This paper) CR = |
||||

Weights | Fuzzy weights | Defuzzified | Rank | Grey weights | Rank |

W1 | 0.3731 | 1 | 1 | ||

W2 | 0.2634 | 2 | 2 | ||

W3 | 0.1377 | 3 | 3 | ||

W4 | 0.1377 | 3 | 3 | ||

W5 | 0.0880 | 4 | 4 |

The deduced weights of criteria and consistency index by the FBWM and GBWM for expert 2.

FBWM (Guo and Zhao, |
GBWM (This paper) CR = |
||||

Weights | Fuzzy weights | Defuzzified | Rank | Grey weights | Rank |

W1 | 0.3663 | 1 | 1 | ||

W2 | 0.2866 | 2 | 2 | ||

W3 | 0.1101 | 4 | 4 | ||

W4 | 0.0784 | 5 | 5 | ||

W5 | 0.1586 | 3 | 3 |

The deduced weights of criteria and consistency index by the FBWM and GBWM for expert 3.

FBWM (Guo and Zhao, |
GBWM (This paper) CR = |
||||

Weights | FBWM weights | Defuzzified | Rank | GBWM weights | Rank |

W1 | 0.2731 | 2 | 2 | ||

W2 | 0.3884 | 1 | 1 | ||

W3 | 0.0905 | 5 | 5 | ||

W4 | 0.1438 | 3 | 3 | ||

W5 | 0.1042 | 4 | 4 |

Based on the information given in Tables

The ranking result computed by the GBWM is valid considering the same ranking of alternatives for different input values.

The result calculated by the GBWM is more reliable than the one calculated by the FBWM because the GBWM has a smaller inconsistency ratio compared with that of the FBWM. Fig.

Comparisons of consistency ratio between the GBWM and FBWM.

After defuzzifications of triangular fuzzy weights of criteria, the defuzzied weights belong to the grey weights deduced by the GBWM. The GBWM narrows the feasible space for potential weights of criteria by the positioned programming to obtain the reliability of weights. It can be shown in Fig.

Comparison between the GBWM and FBWM for expert one and criterion Price.

It is interesting to mention that the linguistic variables for both GBWM and FBWM are same, but GBWM employs a grey linear model as a core model and that is why the results are more reliable than of FBWM method. On the other hand, employing grey system theory can contribute to decrease the volume of calculations. Therefore, it is clear why current research suggests using GBMW method. Moreover, GBWM method can provide a crisp solution for decision-maker if there is a need. Decision-makers should just provide suitable values for the parameters in a grey linear model and this is another advantage of GBMW method. In conclusion, the GBWM is valid in deducing the weights of criteria and advantageous in keeping reliable results and requiring less computational complexity.

BWM has shown an acceptable performance, but the fact that it does not consider the uncertainties of the decision-making environment may reduce the performance of the BWM in real-world. In reality, most information is not clear and ambiguous. The grey system theory is a suitable approach for taking into account the uncertainties of the decision-making environment. In this regard, the current study presented the GBWM. The result calculated by the GBWM is more reliable than that calculated by the FBWM because the GBWM has a smaller inconsistency ratio compared with FBWM. Moreover, the GBWM uses a linear model that is able to present global-optimum weights for MCDM problems. It is interesting to mention that GBWM method can provide a crisp solution based on the experts’ needs. However, some limitations include determining suitable parameters for using on position programming method during solving grey linear programming, which it could be examined by scholars in future. Also, future studies may use grey-fuzzy hybrid approaches for the BWM and compare the results with those derived by the GBWM and FBWM methods. Furthermore, scholars can use different methods for linearization of the core model of the BWM and it may provide a better solution for MCDM problems.

The work presented in this paper corresponds to the doctoral dissertation of the first author at Southeast University, China.