The best-worst method (BWM) is a multi-criteria decision-making method which works based on a pairwise comparison system. Using such a systematic pairwise comparison enhances consistency and reliability of results. The BWM results in single solution when there are two or three criteria, and for problems with fully-consistent systems, with any number of criteria. To obtain the weights of criteria for not fully-consistent comparison systems with more than three criteria, there may be a multiple optimal solution. Although multiple optimality may be desirable in some cases, in other cases, decision-makers prefer to have a unique optimal solution. This study proposes new models which result in a unique solution. The proposed models have less constraints in comparison with the previous models.

In recent years, multi-criteria decision-making (MCDM) methods have been developed by many researchers and applied to many real-world problems (Keshavarz-Ghorabaee

In the past decades, several MADM methods have been proposed and the most popular methods are Best-Worst Method (BWM) (Rezaei,

In decision-making processes, many elements should be considered simultaneously. Therefore, it is very important to have an appropriate approach to obtain unambiguous results. There are many examples in which pairwise comparison (PC) method can be used to draw the final result in a relatively easy way (Koczkodaj,

BWM (Rezaei,

Recently, the use of some PC-based methods has gained more interest, because it is possible to compute the consistency ratio, which boosts the reliability of results. However, researchers have faced difficulties in using PCMs due to rank reversal and great number of comparisons, as well as the inappropriate consistency ratio in big PCMs. The BWM was proposed by Rezaei (

So far, some researchers attempted to develop BWM in fuzzy environment. Guo and Zhao (

Some researchers attempted to develop BWM base on rough numbers and some researchers applied rough BWM in their studies. Zavadskas

Several studies combined BWM and other MADM methods. Tian

Keršuliene

Apart from the aforementioned studies, other methods are proposed to determine the weight of the attributes. Zavadskas and Podvezko (

Kocak

The BWM provides an appropriate system to reflect DM preferences in final weights. However, for not fully-consistent comparison systems with more than three criteria, there may be multiple optimal solutions. Although multiple optimality may be desirable in some cases, in other cases, decision-makers prefer to have a unique solution. This research proposes new models which result in a unique solution. As the proposed model has less constraints in comparison with the previous models, we can mention that it involves less calculations. In Sections

In this section, steps of the BWM to derive the weights of criteria are described. Readers can refer to Rezaei (

Determining a set of decision criteria;

Determining the best (e.g. most desirable, most important) and the worst (e.g. least desirable, least important) criteria;

Determining the preferences of the best criteria over all the other criteria; the Best-to-Others vector would be:

Determining the preferences of all the criteria over the worst criteria; the Others-to-Worst vector would be:

Solving the model and obtaining the optimal criteria weight; according to Eq. (

The above model can be transformed to the following model Eq. (

One of the crucial features of the BWM is its ability to determine the consistency ratio. In the real world, it is not often possible to have full consistency and subjective evaluation with a high level of accuracy (Fülp

Consistency table.

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 |

In the case of multi-optimality, ranges of weights can be determined. Solving Eqs. (

The linear model of the BWM is proposed by Rezaei (

The above model can be transformed to the following linear model (Eq. (

Eq. (

As previously mentioned, the BWM model for not fully-consistent comparison systems with more than three criteria may result in multi-optimality solutions. It means that solving the problem results in different sets of weights for the criteria. This feature of the BWM model can be desirable in some cases. For instance, when debating is important in the decision-making process, DMs may prefer to have more information. Although multiple optimality may be desirable in some cases, in other cases, DMs prefer to have a unique solution.

In this section, we propose two models which guarantee a unique optimal solution. The proposed models use free variables (

We can mention that the number of constraints in the proposed models has significantly decreased.

In the proposed nonlinear model, if

The consistency ratio can be also obtained by Eqs. (

In the proposed linear model, if

The consistency ratio can be also obtained by Eqs. (

When buying a car, a customer considers eight criteria including quality

Best-to-others (BO) and others-to-worst (OW) pairwise comparison vectors.

1 | 1 | 3 | 3 | 2 | 3 | 2 | 6 | |

6 | ||||||||

6 | ||||||||

2 | ||||||||

3 | ||||||||

4 | ||||||||

2 | ||||||||

3 | ||||||||

1 |

According to Table

Solving this example through Eq. (

Optimal interval weights.

In some cases, DMs prefer to have a unique solution. This study proposes two models which result in a unique solution. Table

Comparison of results of different models applied to the same data set.

Criterion | BWM | Linear BWM | Proposed nonlinear model | Proposed linear model |

0.2318 | 0.2308 | 0.2308 | 0.2400 | |

0.1994 | 0.2308 | 0.2308 | 0.2400 | |

0.0882 | 0.0839 | 0.0769 | 0.0800 | |

0.0912 | 0.0839 | 0.0769 | 0.0800 | |

0.1411 | 0.1259 | 0.1538 | 0.1200 | |

0.0882 | 0.0839 | 0.0769 | 0.0800 | |

0.1241 | 0.1259 | 0.1154 | 0.1200 | |

0.0359 | 0.0350 | 0.0385 | 0.0400 |

According to Table

Except in the aforementioned issue, generally, Table

Distances from the centre.

Criterion | Distance | |||

BWM | Linear BWM | Proposed nonlinear model | Proposed linear model | |

0.0064 | 0.0074 | 0.0074 | 0.0018 | |

0.0217 | 0.0097 | 0.0097 | 0.0189 | |

0.0082 | 0.0039 | 0.0031 | 0.0001 | |

0.0025 | 0.0098 | 0.0168 | 0.0137 | |

0.0016 | 0.0168 | 0.0112 | 0.0227 | |

0.0082 | 0.0039 | 0.0031 | 0.0001 | |

0.0117 | 0.0135 | 0.0030 | 0.0076 | |

0.0010 | 0.0019 | 0.0016 | 0.0031 | |

0.0612 | 0.0668 | 0.0559 | 0.0680 |

To compare the proposed models with the existing models, the deviation of priority ratio from initial judgment can be calculated by sum of squared errors (SSE), as defined in Eq. (

SSE values for different models.

BWM | Linear BWM | Proposed nonlinear model | Proposed linear model | |

1.9076 | 2.1459 | 1.2547 | 2 |

This research proposes novel models for BWM to generate a unique solution. For this purpose, having reviewed existing models, novel models are proposed. A numerical example is presented and differences between the previous models and proposed models are discussed. The results show that the previous models and proposed models both provide very close outcomes; however, in cases when there is more than one best (worst) item, we suggest to use the proposed models. The main advantage of applying the proposed models is that we need only

We would like to thank Dr. Jafar Rezaei for his useful comments that improved the manuscript.