## 1 Introduction

*et al.*, 2018). The MCDM can be classified into multi-attribute decision-making (MADM) and multi-objective decision-making (MODM); the MADM is for evaluation, and the MODM is for design. In the MADM, alternatives are predefined. However, the MODM designs the best alternative by considering various constraints (Hwang and Yoon, 2012).

*et al.*, 2015).

*et al.*2016, 81). Condorcet (1785), Fechner (1860) and Thurstone (1927) also used PC in their studies. However, MADM is one of the most famous fields that has used PC to rank criteria or alternatives (Bozóki

*et al.*, 2016) and Saaty (1977; 1980) made the greatest impact in the popularization of the PC method by proposing Analytic Hierarchy Process (AHP) (Koczkodaj, 1993). The PC methods decompose problems into sub-problems and let experts discriminate between two items at a time. Consequently, problems can be solved easily (Brunelli and Fedrizzi, 2015). Using the PC is useful as it compels the decision maker to think about and analyse the situation more precisely and deeply (Kurttila

*et al.*, 2000). Since the comparison of two items is the simplest type of question for measuring the weights, using the PC has become an interesting topic to researchers (Kim

*et al.*, 2017).

*et al.*, 2016, 2015; Salimi and Rezaei, 2016, 2018; Ren

*et al.*, 2017; Gupta

*et al.*, 2017; Rezaei

*et al.*, 2018). Rezaei (2016) proposed interval analysis for the case of multiple optimal solutions for not-fully consistent comparison systems and when the problem contains more than three criteria. He also proposed a linear model of the BWM. The major difference between the BWM and other PC-based methods is that those methods use the PCM (Eq. (1)) whereas the BWM uses pairwise comparison vectors (PCVs) (Eq. (2)).

##### (1)

\[ A=\left(\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}1\hspace{1em}& {a_{12}}\hspace{1em}& {a_{13}}\hspace{1em}& \cdots \hspace{1em}& {a_{1n}}\hspace{1em}\\ {} \frac{1}{{a_{12}}}\hspace{1em}& 1\hspace{1em}& {a_{23}}\hspace{1em}& \cdots \hspace{1em}& {a_{2n}}\hspace{1em}\\ {} \frac{1}{{a_{13}}}\hspace{1em}& \frac{1}{{a_{23}}}\hspace{1em}& 1\hspace{1em}& \cdots \hspace{1em}& {a_{3n}}\hspace{1em}\\ {} \vdots \hspace{1em}& \vdots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \hspace{1em}\\ {} \frac{1}{{a_{1n}}}\hspace{1em}& \frac{1}{{a_{2n}}}\hspace{1em}& \frac{1}{{a_{3n}}}\hspace{1em}& \cdots \hspace{1em}& 1\hspace{1em}\end{array}\right)\in {\mathbb{R}_{+}^{n\times n}},\]##### (2)

\[\begin{array}{l}\displaystyle \textit{Comparisons between}\hspace{2.5pt}\hspace{2.5pt}{a_{Best}}\hspace{2.5pt}\textit{and other items}:\hspace{2.5pt}A=(\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{a_{1}}\hspace{1em}& {a_{2}}\hspace{1em}& \cdots \hspace{1em}& {a_{n}}\end{array})\in {\mathbb{R}_{+}^{1\times n}},\\ {} \displaystyle \textit{Comparisons between other items and}\hspace{2.5pt}\hspace{2.5pt}{a_{Worst}}:\hspace{2.5pt}{B^{T}}=(\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{b_{1}}\hspace{1em}& {b_{2}}\hspace{1em}& \cdots \hspace{1em}& {b_{n}}\end{array})\in {\mathbb{R}_{+}^{n\times 1}}.\end{array}\]*et al.*(2016) proposed intuitionistic fuzzy multiplicative BWM (IFMBWM) with intuitionistic fuzzy multiplicative preference relations for group decision-making. Hafezalkotob and Hafezalkotob (2017) proposed a method based on group and individual decisions supported on FBWM called GI-FBWM. The study considered a hierarchical decision framework and both democratic and autocratic decision making styles can be considered in the proposed method. Liu

*et al.*(2018a) proposed a two-stage model for supplier selection of green fresh product. In this study, FBWM is applied for subjective criteria weights and Shannon entropy is applied for objective criteria weights. Finally, suppliers are ranked using fuzzy MULTIMOORA method. Aboutorab

*et al.*(2018) combined the concept of Z-numbers with BWM and applied the proposed method in a supplier development problem.

*et al.*(2018) developed a rough SWARA approach and compared the obtained result with rough BWM and rough AHP to determine the weight values. The results showed the correlation of ranks using rough SWARA with the ranks of rough BWM and rough AHP was complete. Stević

*et al.*(2017) proposed an approach based on rough BWM (RBWM) and rough SAW (RSAW) methods and applied the hybrid model to select wagons for internal transport of a logistics company. This study compares the obtained ranks using RBWM-RSAW with the obtained ranks using AHP-TOPSIS, AHP-MABAC, AHP-SAW, BWM-TOPSIS, BWM-MABAC, BWM-SAW, RAHP-RTOPSIS, RAHP-RSAW, RAHP-RMABAC, RBWM-RTOPSIS and RBWM-RMABAC. The results indicate that there is a high correlation between the ranks of the compared models. Stević

*et al.*(2018) proposed an integrated model based on Rough BWM and Rough WASPAS methods. The model was applied to a location selection problem for the construction of roundabout. Through the sensitivity analysis the proposed model was compared with RBWM-RSAW, RBWM-RMABAC, RBWM-RVIKOR, RBWM-RMAIRCA, RBWM-RTOPSIS and RBWM-REDAS. The sensitivity analysis indicated that the model stability was verified. Liu

*et al.*(2018b) combined rough number, the BWM and the VIKOR to solve a Supply chain partner selection problem under cloud computing environment. Pamučar

*et al.*(2018) developed an approach based on interval-valued fuzzy-rough numbers (IVFRN). In this study, BWM and MABAC methods were combined and applied to evaluate firefighting aircraft. Through the sensitivity analysis, the proposed model was compared with the fuzzy and rough extension of the MABAC, COPRAS and VIKOR methods and showed a high degree of stability.

*et al.*(2018) combined the BWM and improved TOPSIS and developed a hybrid multi-criteria group decision-making (MCGDM) model in a green supplier selection problem. You

*et al.*(2016) proposed a hybrid decision framework to solve MCGDM problems based on the BWM and ELECTRE III methods. Yadav

*et al.*(2018) proposed a hybrid BWM-ELECTRE-based framework for effective offshore outsourcing adoption. In this study, having determined the weight values of offshore outsourcing enablers by employing BWM, the automotive case organizations were prioritized using ELECTRE approach. Amoozad Mahdiraji

*et al.*(2018) applied the combination of interval BWM and fuzzy COPRAS to analyse key factors of environmental sustainability in Iranian contemporary architecture.

*et al.*(2010) developed Step-wise weight assessment ratio analysis (SWARA) to determine the weight values of the attributes. Although BWM and SWARA use different mathematical approaches, they are similar in some aspects. Identifying the best and the worst criterion in the BWM method is similar to the first step of the SWARA method. BWM and SWARA methods are more preferable than the AHP method which requires more PCs (Zolfani and Chatterjee, 2019). Zolfani

*et al.*(2018) extended the classic SWARA method to improve the quality of decision making process. In this study, the reliability evaluation of experts’ opinion is incorporated into SWARA method. Zolfani and Chatterjee (2019) compared the results of variability between the criteria priorities for SWARA and BWM in the sustainable housing material selection problem.

*et al.*(2018) proposed a method of weights’ recalculation, and the integration of various estimates into a single one.

*et al.*(2018) developed a quadratic model of Euclidean BWM and minimized the sum of the squared differences instead of maximum number of differences in BWM. However, the proposed model needs more computations in comparison with linear BWM.

## 2 Best-Worst Method (BWM)

- Step 1. Determining a set of decision criteria;
- Step 2. Determining the best (e.g. most desirable, most important) and the worst (e.g. least desirable, least important) criteria;
- Step 5. Solving the model and obtaining the optimal criteria weight; according to Eq. (3), we should find a solution where maximum difference between $ \big|\frac{{w_{B}}}{{w_{j}}}-{a_{Bj}}\big|$ and $ \big|\frac{{w_{j}}}{{w_{W}}}-{a_{jW}}\big|$ for all j is minimized.

##### (3)

\[\begin{array}{l}\displaystyle \min \underset{j}{\max }\left\{\left|\frac{{w_{B}}}{{w_{j}}}-{a_{Bj}}\right|,\left|\frac{{w_{j}}}{{w_{W}}}-{a_{jW}}\right|\right\}\\ {} \displaystyle \hspace{2em}\text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\hspace{2em}\sum \limits_{j}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}\hspace{2em}{w_{j}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]##### (4)

\[\begin{array}{l}\displaystyle \min \xi \\ {} \displaystyle \hspace{1em}\text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\left|\frac{{w_{B}}}{{w_{j}}}-{a_{Bj}}\right|\leqslant \xi ,\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\left|\frac{{w_{j}}}{{w_{W}}}-{a_{jW}}\right|\leqslant \xi ,\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\sum \limits_{j}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}{w_{j}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]*et al.*, 2010). The existence of consistency alone can not show the level of expertise. However, the existence of inconsistency means lack of expertise or necessary information (Brunelli and Fedrizzi, 2015; Forman and Selly, 2000). The consistency ratio is calculated by Eq. (5) and the consistency index is presented in Table 1.

## 3 Interval Weights

##### (6)

\[\begin{array}{l}\displaystyle \min {w_{j}}\\ {} \displaystyle \text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\left|\frac{{w_{B}}}{{w_{j}}}-{a_{Bj}}\right|\leqslant \xi ,\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\left|\frac{{w_{j}}}{{w_{W}}}-{a_{jW}}\right|\leqslant \xi ,\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\sum \limits_{j}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}{w_{j}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j,\end{array}\]##### (7)

\[\begin{array}{l}\displaystyle \max {w_{j}}\\ {} \displaystyle \text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\left|\frac{{w_{B}}}{{w_{j}}}-{a_{Bj}}\right|\leqslant \xi ,\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\left|\frac{{w_{j}}}{{w_{W}}}-{a_{jW}}\right|\leqslant \xi ,\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\sum \limits_{j}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}{w_{j}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]## 4 Linear Model

##### (8)

\[\begin{array}{l}\displaystyle \min \underset{j}{\max }\left\{\left|{w_{B}}-{a_{Bj}}{w_{j}}\right|,\left|{w_{j}}-{a_{jW}}{w_{W}}\right|\right\}\\ {} \displaystyle \hspace{2em}\text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\hspace{2em}\sum \limits_{j}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}\hspace{2em}{w_{j}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]##### (9)

\[\begin{array}{l}\displaystyle \min {\xi ^{L}}\\ {} \displaystyle \text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\left|{w_{B}}-{a_{Bj}}{w_{j}}\right|\leqslant {\xi ^{L}},\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\left|{w_{j}}-{a_{jW}}{w_{W}}\right|\leqslant {\xi ^{L}},\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}{\sum \limits_{j}^{}}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}{w_{j}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]## 5 Proposed Models

### 5.1 Proposing a Nonlinear Model

##### (10)

\[\begin{array}{l}\displaystyle \min z=\sum \limits_{j}({y_{j}^{+}}+{y_{j}^{-}})+\sum \limits_{j}({z_{j}^{+}}+{z_{j}^{-}})\\ {} \displaystyle \text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\hspace{2em}\frac{{w_{B}}}{{w_{j}}}-{a_{Bj}}={y_{j}^{+}}-{y_{j}^{-}},\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\hspace{2em}\frac{{w_{j}}}{{w_{W}}}-{a_{jW}}={z_{j}^{+}}-{z_{j}^{-}},\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\hspace{2em}\sum \limits_{j}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}\hspace{2em}{w_{j}},{y_{j}^{+}},{y_{j}^{-}},{z_{j}^{+}},{z_{j}^{-}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]### 5.2 Proposing a Linear Model

##### (13)

\[\begin{array}{l}\displaystyle \min z=\sum \limits_{j}({y_{j}^{+}}+{y_{j}^{-}})+\sum \limits_{j}({z_{j}^{+}}+{z_{j}^{-}})\\ {} \displaystyle \text{s.t.}\hspace{2.5pt}\\ {} \displaystyle \hspace{2em}\hspace{2em}{w_{B}}-{a_{Bj}}{w_{j}}={y_{j}^{+}}-{y_{j}^{-}},\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\hspace{2em}{w_{j}}-{a_{jW}}{w_{W}}={z_{j}^{+}}-{z_{j}^{-}},\hspace{1em}\text{for all}\hspace{5pt}j,\\ {} \displaystyle \hspace{2em}\hspace{2em}{\sum \limits_{j}^{}}{w_{j}}=1,\\ {} \displaystyle \hspace{2em}\hspace{2em}{w_{j}},{y_{j}^{+}},{y_{j}^{-}},{z_{j}^{+}},{z_{j}^{-}}\geqslant 0,\hspace{1em}\text{for all}\hspace{5pt}j.\end{array}\]## 6 Illustrative Example

##### Table 2

BO | $ {C_{1}}$ | $ {C_{2}}$ | $ {C_{3}}$ | $ {C_{4}}$ | $ {C_{5}}$ | $ {C_{6}}$ | $ {C_{7}}$ | $ {C_{8}}$ |

Best criterion: $ {C_{1}}$ | 1 | 1 | 3 | 3 | 2 | 3 | 2 | 6 |

OW | Worst criterion: $ {C_{8}}$ | |||||||

$ {C_{1}}$ | 6 | |||||||

$ {C_{2}}$ | 6 | |||||||

$ {C_{3}}$ | 2 | |||||||

$ {C_{4}}$ | 3 | |||||||

$ {C_{5}}$ | 4 | |||||||

$ {C_{6}}$ | 2 | |||||||

$ {C_{7}}$ | 3 | |||||||

$ {C_{8}}$ | 1 |

##### Table 3

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

$ {C_{1}}$ | 0.2318 | 0.2308 | 0.2308 | 0.2400 |

$ {C_{2}}$ | 0.1994 | 0.2308 | 0.2308 | 0.2400 |

$ {C_{3}}$ | 0.0882 | 0.0839 | 0.0769 | 0.0800 |

$ {C_{4}}$ | 0.0912 | 0.0839 | 0.0769 | 0.0800 |

$ {C_{5}}$ | 0.1411 | 0.1259 | 0.1538 | 0.1200 |

$ {C_{6}}$ | 0.0882 | 0.0839 | 0.0769 | 0.0800 |

$ {C_{7}}$ | 0.1241 | 0.1259 | 0.1154 | 0.1200 |

$ {C_{8}}$ | 0.0359 | 0.0350 | 0.0385 | 0.0400 |

##### Table 4

Criterion | Distance | |||

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

$ {C_{1}}$ | 0.0064 | 0.0074 | 0.0074 | 0.0018 |

$ {C_{2}}$ | 0.0217 | 0.0097 | 0.0097 | 0.0189 |

$ {C_{3}}$ | 0.0082 | 0.0039 | 0.0031 | 0.0001 |

$ {C_{4}}$ | 0.0025 | 0.0098 | 0.0168 | 0.0137 |

$ {C_{5}}$ | 0.0016 | 0.0168 | 0.0112 | 0.0227 |

$ {C_{6}}$ | 0.0082 | 0.0039 | 0.0031 | 0.0001 |

$ {C_{7}}$ | 0.0117 | 0.0135 | 0.0030 | 0.0076 |

$ {C_{8}}$ | 0.0010 | 0.0019 | 0.0016 | 0.0031 |

Sum | 0.0612 | 0.0668 | 0.0559 | 0.0680 |

## 7 Conclusion

##### Table 5

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

$ SSE$ | 1.9076 | 2.1459 | 1.2547 | 2 |