## 3 Results
### Approximate Techniques
#### 3.2.3 Rank Summed Weighting (RS) Technique  
To approximate attribute weights, the Rank Summed Weighting technique uses information on the rank order of attributes on the basis of importance combined with the weighting of each attribute in relation to its rank order [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 summed weight for each attribute.  
RS 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  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Abbreviation** & **Criteria** & **Formula** & **Weight** \\ \hline (P) & Purchase Price & \(w_{1}=1/5\) (\(1+1/2+1/3+1/4+1/5\)) & =0.457 \\ (R) & Reliability & \(w_{2}=1/5\) (\(1/2+1/3+1/4+1/5\)) & =0.257 \\ (S) & Safety & \(w_{3}=1/5\) (\(1/3+1/4+1/5\)) & =0.157 \\ (A) & Attractiveness & \(w_{4}=1/5\) (\(1/4+1/5\)) & =0.090 \\ (G) & Gas Mileage & \(w_{5}=1/5\) (\(1/5\)) & =0.040 \\  & Sum & \(w_{1}+w_{2}+w_{3}+w_{4}+w_{5}\) & \(\sim\)1.00 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Weight using ROC technique.  
important attribute. For our car purchase example, we maintain the same ordinal ranking as shown in Table 12 for the previous method.  
RS Step 2: Calculate the Rank Summed Weight for Each Attribute  
The Rank Summed Weighting technique assigns, to each of \(N\) rank ordered attributes, a weight \(wi\) according to Equation (6).  
\[wi=(N-i+1)/\left(\sum_{k=1}^{N}N-i+1\right)=2(N-i+1)/N(N+1) \tag{6}\]  
with the attributes ordered from most important (\(i=1\)) to least important (\(i=N\)).  
The rank exponent weighting technique is a generalization of the rank sum weighting technique, as shown in Equation (7).  
\[wi=(N-i+1)^{p}/\left(\sum_{k=1}^{N}N-i+1\right)^{p} \tag{7}\]  
In this case, a \(p\) of 0 results in equal weights, \(p=1\) is the rank sum, and increasing \(p\) values further disperses the weight distribution among attributes.  
In the car buying example above, the weights assigned to each attribute can be calculated using the Rank Summed Weighting technique as shown in Table 14. Once again, ordinality of criteria preference remains when compared with previous methods; however, the spread of weights changes due to the predetermined rank summed formula.  
Strengths of This Approach  
The Rank Summed Weighting technique provides a means for coming up with meaningful weights based solely on ordinal rankings of attributes based on importance. This is particularly helpful since, in situations consisting of many users with diverse opinions, rank orderings of attributes may be the only aspect of preference that can achieve consensus agreement. Calculating weights using the Rank Summed Weighting technique can be easily implemented using standard spreadsheet tools or calculated using a calculator.