## 2 Materials and Methods  
MCDA assumes preferential independence among criteria and many weighting methods are built on the assumption of the use of an additive value function. Identifying appropriate means of calculating the weightings of criteria involved in a decision, such as sampling from a uniform distribution [57] or the use of rankings [58], is important for differentiating the category of MCDA technique that is suitable for the simulation. As such, we conduct our assessment under the assumption that an additive preference model of the form provided in Equation (1) is being utilized to inform decision making. \[v(x)=\sum_{i=1}^{n}w_{i}v_{i}(x_{i}) \tag{1}\]  
where \(n\) is the total number of criteria being considered, \(w_{i}\) is the weight of the \(i\)-th criteria, \(v_{i}(x_{i})\) is the \(i\)-th value function, and \(x\) is the vector of all criteria values. Within a value function, all weights must add up to 1:  
\[\sum_{i=1}^{n}w_{i}=1 \tag{2}\]  
The use of an additive utility model requires that criteria used in the model are mutually preferentially independent [59]. This means that the weight assigned to any particular criterion is not influenced by the weight assigned to any other attributes. For example, consider choosing between departure times of 6 and 10 a.m. for a flight and their respective costs are $250 and 300. Mutually preferentially independent means that you prefer the cheaper flight to the more expensive one regardless of departure time, and you prefer the later flight to the earlier one regardless of cost. If you prefer the later flight regardless of the ticket price, however, the price dictates your departure preference, then departure time is preferentially independent of cost, but they are not mutually preferentially independent. If the attributes are not mutually preferential independent, there are techniques to combine them using a joint utility function that is beyond the scope of this paper. Additionally, all criteria should be mutually exclusive and collectively exhaustive, that is, they represent the entirety of relevant criteria and each is independent of all others. A taxonomy has been developed to categorize MCDA techniques to help in identifying and conveying the similarities and differences in uses, benefits, and limitations of each of the techniques. We concentrate on two general approaches for assigning weights to preference attributes within the context of a multi attribute utility model: ratio assignment and approximate techniques. Figure 1 provides the two primary classifications of techniques within MCDA, with each followed by four technique classifications. Within each branch of the taxonomy, we have selected the techniques which are most usable in practice. Note that some techniques may be better but are unwieldy. Within each category, the techniques are ordered first by direct and then indirect methods, and then notionally from easiest for the decision maker to implement and use to the more difficult weighting methods, requiring more time and resources to set up the weights for the attributes. [PERSON], [PERSON] [60] suggest that the ratio, swing, tradeoff, and pricing out methods are most commonly used in practice for MCDA. However, more recently, researchers have focused on direct methods for determining weights, including equal and rank-order weighting methods. Weights are often obtained judgmentally with indirect methods [61]; therefore, direct methods that remove some of the subjectivity while determining appropriate weights have become increasingly popular. The difference between ratio assignment and approximate techniques lies in the nature of the questions posed to elicit weights. Ratio assignment techniques assign a score to each attribute based on its absolute importance relative to a standard reference point or relative importance with respect to other attributes. The resulting weights are obtained by taking the ratio of each individual attribute score to the sum of the scores across all attributes.