## 3 Results
### Limitations of This Approach
#### 3.2.4 Rank Reciprocal Weighting (RR) Technique  
The rank reciprocal method is similar to the ROC and RS methods. It involves use of the reciprocal of ranks, divided by the sum of the reciprocals [61].  
The Rank Summed Weighting technique is comprised of the following two steps: (1) rank order attributes and establish rank indices, and (2) calculate the rank reciprocal weight for each attribute.  
RR Step 1: Rank Order Attributes and Establish Rank Indices  
Consider a finite set of N attributes or criteria deemed relevant by an individual or group of users to a particular decision problem. This first step asks users to agree on a rank ordering of these attributes according to their relative contribution to the expert's overall preference within an additive utility (or value) function framework. A number of approaches exist to assist in holistic ranking, the most popular being pairwise ranking [81]. The resultant ranking is from most important to least important, where the index \(i=1\) is assigned to the most important attribute, and the index \(i=N\) is assigned to the least important attribute. For our car purchase example, we maintain the same ordinal ranking as shown in Table 12 for the previous method.  
RR Step 2: Calculate the Rank Summed Weight for Each Attribute  
The Rank Reciprocal Weighting technique assigns to each of the \(N\) rank ordered attributes a weight \(w_{i}\) according to Equation (8).  
\[wi=(1/i)/\left(\sum_{k=1}^{N}1/k\right) \tag{8}\]  
where, again, the attributes are ordered from most important (\(i=1\)) to least important (\(i=N\)). In the car buying example above, the weights assigned to each attribute can be calculated using the RR technique as shown in Table 15. Again, ordinality of criteria preference remains when compared with previous methods; however, the spread of weights changes due to the predetermined rank reciprocal formula.  
Strengths of This Approach  
Similarly to the rank ordered centroid and rank summed weighting approaches, the rank reciprocal technique provides a mechanism for calculating weights using only an ordinal ranking of relevant attributes. This technique is easily implemented using spreadsheet tools or calculated using a calculator.  
Limitations of This Approach  
As with the previous two methods, rank reciprocal weighting is best used when only an ordering of attributes is possible. When more specific weighting is possible, use of a ratio assignment technique is advised.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Abbreviation** & **Criteria** & **Formula** & **Weight** \\ \hline (P) & \multirow{2}{*}{Purchase Price} & \(w_{1}=1/(i\times\sum_{k=1}^{N}1/k)=\) & \multirow{2}{*}{=0.438} \\ (R) & & \(1/(1\times(1/1)+(1/2)+(1/3)+(1/4)+(1/5))) & \\ \multirow{2}{*}{(S)} & \multirow{2}{*}{Safety} & \(w_{2}=1/(i\times\sum_{k=1}^{N}1/k)=\) & \multirow{2}{*}{=0.218} \\  & & \(1/(2\times(1/1)+(1/2)+(1/3)+(1/4)+(1/5))) & \\ \multirow{2}{*}{(A)} & \multirow{2}{*}{Attractiveness} & \(w_{3}=1/(i\times\sum_{k=1}^{N}1/k)=\) & \multirow{2}{*}{=0.146} \\  & & \(1/(3\times(1/1)+(1/2)+(1/3)+(1/4)+(1/5))) & \\ \multirow{2}{*}{(G)} & \multirow{2}{*}{Gas Mileage} & \(w_{5}=1/(i\times\sum_{k=1}^{N}1/k)=\) & \multirow{2}{*}{=0.109} \\  & & \(1/(5\times(1/1)+(1/2)+(1/3)+(1/4)+(1/5))) & \\ \multirow{2}{*}{(S)} & \multirow{2}{*}{Same} & \(w_{1}+w_{2}+w_{3}+w_{4}+w_{5}\) & \multirow{2}{*}{\(\sim\)1.00} \\ \hline \hline \end{tabular}
\end{table}
Table 15: Weight using RR technique.