## 2 Multi-Criteria Decision Making
### Combining AHP with SAW  
Addressing the complexity of evaluating olive trees and groves necessitates the adoption of sophisticated methodologies, and among them, the multi-criteria decision-making (MCDM) methods emerge as a potent solution. From the available MCDM methods, we need methods that are easily applied and have valid results. For this purpose, we selected the simple additive weighting (SAW) method and the analytical hierarchy process (AHP) method. Both these multi-criteria evaluation approaches are widely applied in decision-making scenarios and offer systematic and insightful analyses ensuring a comprehensive ranking of alternatives. The versatility extends to various applications, making it a valuable tool for decision-makers dealing with intricate choices. The fundamental premise of SAW lies in systematically weighing criteria importance and their relative significance. The application of the SAW method consists of the following steps:
* Criterion Scoring Scale Creation: The process commences with the establishment of a scoring scale for each criterion source. The final scores, situated in the (0, 1) interval, signify the performance of the criteria source, with 1 denoting optimal and 0 indicating the least favorable outcome. SAW does not have a predefined way of calculating weights. For this purpose, in this step, we used AHP. The application of AHP for the calculation of weights is presented in detail in [21].
* Application of Weights: SAW incorporates a weighted approach, acknowledging the varying importance of each criterion in the decision-making process. The weights, determined by evaluators' perceptions, reflect the relative significance of criteria in the final decision.
* Summation of Values: Employing a summative methodology, SAW calculates a total score for each alternative by adding the products of criteria values and their corresponding weights. This comprehensive score offers a holistic evaluation, considering both the intrinsic merit of each criterion and the assigned importance by evaluators.  
The formula for applying the SAW method used for the implementation needs of the application is as follows:  
\[U\big{(}X_{j}\big{)}=\sum_{i=1}^{6}w_{\mu_{i}}\mu_{ij}+\sum_{i=1}^{7}w_{\alpha _{i}}\alpha_{ij}\]  
where \(\mu_{ij}\) and \(\alpha_{ij}\) are the tangible and intangible criteria for the tree \(X_{j}\).  
The application of AHP is presented in detail in [21] and AHP has been proven to be rather effective in estimating the weights of the criteria. The SAW method's structured approach, on the other hand, ensures a nuanced evaluation, balancing the intricate interplay of criteria. Its application extends beyond olive trees and groves, proving valuable in diverse decision-making scenarios where multiple criteria contribute to the final choice.  
In conclusion, the SAW method stands as a reliable and adaptable tool for multi-criteria decision-making, particularly in the nuanced evaluation of olive trees and groves. Its systematic approach, encompassing criterion scaling, weighted analysis, and summative scoring, contributes to informed and balanced decision processes.  
In the application to be presented, the SAW method will be used to calculate not only the initial value of each olive tree but also the initial value of each olive. These values are further used by the system to select the final set of alternative olive trees and olive groves that are going to be evaluated by the third multi-criteria decision-making theory that is presented in the next section.