# A Group Decision-Making Approach in MCDM: An Application of the Multichoice Best-Worst Method
## 3 Multichoice Best-Worst Method for Group Decision-Making  
Suppose there exist \(n\) criteria, which need to be prioritized. Experts use a linguistic scale to decide the significance of the criterion, such as Equally or Moderately or Extremely Important and their intermediate value see Table 1. Such linguistic terms for pairwise \begin{table}
\begin{tabular}{c c c c} \hline
**Scale** & **Linguistic Term** & **Scale** & **Linguistic Term** \\ \hline
1 & Equally Important (EI) & 2 & Intermediate of EI \& MI (IEM) \\
3 & Moderately Important (MI) & 4 & Intermediate of MI \& I (IMS) \\
5 & Important (I) & 6 & Intermediate of I \& VI (ISV) \\
7 & Very Important (VI) & 8 & Intermediate of VI \& EXI (IVE) \\
9 & Extremely Important (EXI) & & \\ \hline \end{tabular}
\end{table} 
Table 1: Numeric scale and linguistic expressions for pairwise comparison. comparisons are converted for evaluations to numeric values \(o_{ij};i,j=1,2,3,\ldots,n\) to compare the criteria \(C_{1},C_{2},C_{3},\ldots,C_{n}\), where \(o_{ij}\) denotes how much more critical \(C_{i}\) is than \(C_{j}\) using the 1-9 scale shown in Table 1.  
We summarize the five steps involved in MCBWM for group decision making as follows:  
**Step 1. **: Determine the set of decision criteria \(\{C_{1},C_{2},\ldots,C_{n}\}\) and experts \(\{E_{1},E_{2},\ldots,E_{m}\}\). **Step 2. **: In this step, the best criterion (\(C_{B}\)) and the worst criterion (\(C_{W}\)) from the collection of criteria are chosen. For group decision making, it is assumed that all experts use the same best and worst criteria, \(\{E_{1},E_{2},\ldots,E_{m}\}\). **Step 3. **: All experts compare the best criterion (\(C_{B}\)) to all other criteria in pairs, yielding the best-to-others (\(O_{B}^{E_{k}}\)) vector depicted in Equation (7), as follows:  
\[o_{B}^{E_{k}}=(o_{B1}^{E_{k}},o_{B2}^{E_{k}},o_{B3}^{E_{k}},\ldots,o_{Bn}^{E_{ k}})\ \forall\ k=1,2,\ldots m. \tag{7}\]  
where \(o_{Bj}^{E_{k}}\geq 1\), \(j=1,2,\ldots,n\) reflects the level of preference for the best criterion \(C_{B}\) over all other criteria \(C_{j}\), \(j\
eq B\), given by the decision maker \(E_{k}\); \(k=1,2,\ldots,m\). The pairwise comparison is presented in Figure 1.  
**Step 4. ** The others-to-worst (\(O_{W}^{E_{k}}\)) vector representing the pairwise comparison of preferences for each criterion and the least desirable criterion (\(C_{W}\)) for a group of experts is given in Equation (8) as:  
\[O_{W}^{E_{k}}=(o_{1W}^{E_{k}},o_{2W}^{E_{k}},o_{3W}^{E_{k}},\ldots,o_{nW}^{E_{ k}})^{T} \tag{8}\]  
where \(o_{jW}^{E_{k}}\geq 1,j=1,2,\ldots,n\), \(j\
eq W\) shows how much the decision maker \(E_{k}\); \(k=1,2,\ldots,m\) prefers one criterion over another (in this case, the worst criterion \(C_{W}\) over other criteria \(C_{j}\)). **Step 5. **: Combine all experts' pairwise comparison values for the most important criterion \(C_{B}\) over all other criteria \(C_{j},j=1,2,\ldots,n\) when making decisions in groups using a multichoice approach, i.e., MCBWM. Then, the best-to-others (\(O_{B}\)) vector consists of experts' comparison values as choices, as shown in Equation (9) as:  
\[O_{B}=\left(o_{B1},o_{B2},o_{B3},\ldots,o_{Bn}\right) \tag{9}\]  
where \(o_{Bj}=\{o_{Bj}^{E_{1}},o_{Bj}^{E_{2}},o_{Bj}^{E_{3}},\ldots,o_{Bj}^{E_{m}}\}\). Furthermore, \(o_{Bj}=\{o_{Bj}^{1},o_{Bj}^{2},\ldots,o_{Bj}^{1}\}\ \forall\ l=1,\ 2,\ldots,9\) consists of all of the many choices for the comparison of the best criterion \(C_{B}\) with the other criteria \(C_{j}\), such that no two pairwise comparisons provided by the experts in the choice set are same. Similarly, in Equation (10), the preferences for all criteria and the worst criterion are compared pairwise, and the results are presented in the others-to-worst (\(O_{W}\)) vector as follows:  
Figure 1: Pairwise comparison. \[\mathcal{O}_{W}=\left(o_{1W},o_{2W},o_{3W},\ldots,o_{nW}\right)^{T} \tag{10}\]  
where \(o_{jW}=\{o_{jW}^{E_{1}},o_{jW}^{E_{2}},o_{jW}^{E_{3}},\ldots,o_{jW}^{E_{m}}\}\).