## 2 Multi-Criteria Decision Making
### VIKOR (VlseKriterijumska Optimizacija I Kompronisno Resenje): A Comprehensive Analysis  
The VIKOR method, introduced to the academic realm in 2004 by [PERSON] and [PERSON] [28], stands as a pivotal approach in multi-criteria decision-making. Its inception aimed to tackle the intricate challenges of evaluation problems by presenting solutions that strike a balance between maximizing returns and minimizing risks. Over the years, VIKOR has become a cornerstone in decision-making processes, offering a nuanced perspective for assessing and selecting alternatives. At its core, the VIKOR method operates on the principles of maximum performance and minimal risk. By normalizing data, calculating overall scores, and employing VIKOR indices, the method ensures a judicious equilibrium between achieving the highest possible return and mitigating associated risks. The VIKOR method has the following methodology. 1. Data Normalization: The initial step involves standardizing the input data, ensuring uniformity and comparability across diverse criteria. 2. Calculation of Overall Scores: VIKOR computes comprehensive scores for each alternative, synthesizing the normalized data into a coherent representation of their overall performance. 3. VIKOR Indices: Leveraging specific indices, the method gauges the compromise solution for each alternative, offering a nuanced understanding of their relative standings. 4. Final Selection Determination: The ultimate aim is to pinpoint the alternative that strikes the most balanced compromise between maximizing returns and minimizing risks. Subsequently, the algorithm proceeds to calculate the values \(\mu_{ij}\) and \(\alpha_{ij}\) for the tangible and intangible criteria and \(w_{\mu_{i}},\;\;w_{a_{i}}\) represent the weights of the criteria and signify their relative importance. Following this, the values \(Q\), \(R\), and \(S\) are calculated for the given values. S is the Utility Measure and represents the distance from an ideal solution. The lower the S value, the closer the alternative is to the ideal solution in an overall sense. R is the Regret Measure and represents the maximum normalized gap (or worst deviation) among all criteria for an alternative. Q is the VIKOR Index, which means the Compromise Measure and it is used as the final ranking score that balances both S and R using a compromise approach. The alternatives are then ranked based on the values of Q in ascending order using the values of S and R.  
The analysis of the method's stability relies on a combination of the results from three tables. Stability is confirmed when the best element in the Q list is also the best element in either the S or R list. More specifically, the application of compromise ranking algorithm VIKOR [29] is based on the values of the criteria \(\mu_{1}\ldots\mu_{6}\) and \(a_{1}\ldots a_{7}\) and is briefly reviewed as follows:  
Estimating the best \(\mu_{1}^{+}\ldots\mu_{6}^{+},\alpha_{1}^{+}\ldots\alpha_{7}^{+}\) and worst \(\mu_{1}^{-}\ldots\mu_{6}^{-},\alpha_{1}^{-}\ldots\alpha_{7}^{-}\) values of all criteria taking into account all values of \(\mu_{1}\ldots\mu_{6}\) and \(a_{1}\ldots a_{7}\)  
\[\mu_{j}^{+}=\underset{1\leq i\leq 6}{max}(\mu_{ij})\alpha_{j}^{+}=\underset{1 \leq k\leq 7}{max}\Big{(}a_{kj}\Big{)}\mu_{j}^{-}=\underset{1\leq i\leq 6}{min} (\mu_{ij})\alpha_{j}^{-}=\underset{1\leq k\leq 7}{min}\Big{(}a_{kj}\Big{)}\]  
Computing the values of \(S_{i}\) and computing the values of \(S_{i}\) and \(R_{i}\)  
Where \(w_{\mu}\) and \(w_{a}\) are the weights of criteria \(\mu\) and \(\alpha\), representing the decision maker's relative preference for the importance of the corresponding criteria. Computing the values of \(S^{*}\), \(S^{-}\) and \(R^{*}\), \(R^{-}\)  
\[S^{*}=minS_{i},\;S^{-}=maxS_{i}R^{*}=minR_{i},\;R^{-}=maxR_{i}\]  
Determining the value of \(Q_{i}\) for the alternative trees i and ranking the alternatives by the values of \(Q_{i}\).