## 3 Results
### Ratio Assignment Techniques
#### 3.1.1 Direct Assignment Technique (DAT)  
The Direct Assignment Technique (DAT) asks users to assign weights or scores directly to preference attributes. For example, the user may need to divide a fixed pot of points (e.g., 100) among the attributes. Alternatively, users may also be asked to score each attribute over some finite scale (e.g., 0 to 100) and the resulting weights are then calculated by taking the ratio of individual scores to the total score among all attributes. The Direct Assignment Technique is comprised of the following two steps: (1) assign points to each attribute, and (2) normalize the points such that the total is equal to one. DAT Step 1: Assign Points to Each Attribute  
One of two approaches can be adopted for completing this step. The first approach considers a fixed pot of points and asks users to divide the pot among the attributes where attributes of greater importance receive higher scores than those of lesser importance. For example, if the total pot consists of 100 points, users would assign a portion of this total among the set of attributes. The second approach considers a finite range of potential scores and asks user to assign a point value to each attribute according to its importance, where higher importance attributes receive more points than those of lesser importance. For example, if a range of scores ranging from 0 to 100 is considered, users would choose a point value between these limits to establish the relative importance among attributes. As previously mentioned, it is important that an objective is established and that the ranges (swing) for each attribute are defined; therefore, a common example will be used throughout this paper. We will consider the purchase of a car for a small family, early in their careers, with one small child and a short commute. One could see that the relative weighting of attributes might change if the problem definition changed, e.g., if the decision maker had a long commute or large family. The attributes will use notional ranges for the remainder of this paper. Again, the relative weight that a decision maker would apply may be impacted by the range. A narrow purchase price range of $20,000 to $20,200 would have less importance than that of a larger range. The criteria used for the analysis of this choice are Purchase Price, Attractiveness, Reliability, Gas Mileage, and Safety Rating. Assume a fixed pot of 1000 points to be divided among the five attributes. These criteria, their abbreviations, their least and most preferred value, and scores are shown in Table 1.  
Using the point scores assigned to each of the attributes in the previous step, the second step of the Direct Assignment Technique is to calculate attribute weights. This is done by normalizing each attribute score against the total score among all attributes as shown in Equation (3). \[w_{i}=\text{S}_{i}/\sum_{j}S_{j} \tag{3}\]  
In the car buying example, based on a fixed budget of points, the weights for each attribute can be readily calculated using Equation (3). This result is shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Abbreviation** & **Criteria** & **Least Preferred** & **Most Preferred** & **Score** \\ \hline (P) & Purchase Price & \$30,000 & \$15,000 & 400 points \\ (R) & Reliability (Initial Owner complaints) & 150 & 10 & 300 points \\ (S) & Safety & 3 star & 5 star & 150 points \\ (A) & Attractiveness (qualitative) & Low & High & 100 points \\ (G) & Gas Mileage & 20 mpg & 30 mpg & 50 points \\ (P) & Purchase Price & \$30,000 & \$15,000 & 400 points \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Criteria for decision problem. Strengths of This Approach  
This approach is the most straightforward of the techniques presented in this paper for eliciting attribute weights in that it does not require the user to formally establish a rank order of attributes a priori. The number of questions needed to assign weights using the direct assignment technique is equal to the number of preference attributes.