# Mathematical Apparatus of Optimal Decision-Making Based on Vector Optimization
## 4 Research, Analysis, and Formulation of the Problem of Decision-Making with Uncertain Data
### The Analysis of Modern Methods of Decision Making to the Experimental Data  
At present, the problems of Equations (37) and (38) are solved by a number of \"simple\" methods based on special criteria, such as Wald, Savage, Hurwitz, and Bayes-Laplace criteria, which provide the basis for decision making.  
The Wald criterion of maximizing the minimal component helps make the optimal decision that ensures the maximal gain among minimal ones, \(\underset{k=\overline{1,K}}{\max}\ \underset{i=\overline{1,M}}{\min}\ f_{i}^{k}\).  
The [PERSON] minimal risk criterion chooses the optimal strategy so that the value of the risk \(r_{\hat{i}}^{k}\) is minimal among maximal values of risks over the columns, \(\underset{i=\overline{1,K}}{\min}\ \underset{k=\overline{1,K}}{\max}\ f_{i}^{k}\). The value of the risk \(r_{\hat{i}}^{k}\) is chosen from the minimal difference between the decision that yields maximal profit \(\underset{i=\overline{1,M}}{\max}\ f_{i}^{k}\), \(k=\overline{1,K}\), and the current value \(f_{i}^{k}\), \(r_{\hat{i}}^{k}=(\underset{i=\overline{1,M}}{\max}\ f_{i}^{k})-f_{i}^{k}\), with their set being the matrix of risks \(R\ =\ \left\|r_{\hat{i}}^{k}\right\|_{\hat{i}=\overline{1,K}}^{k=\overline{1,K}}\).  
The Hurwitz criterion helps choose the strategy that lies somewhere between absolutely pessimistic and optimistic (i.e., the most considerable risk):  
\[\underset{k=\overline{1,K}}{\max}\ \left(\alpha\underset{i=\overline{1,M}}{\min} \ f_{i}^{k}+(1-\alpha)\underset{i=\overline{1,M}}{\max}\ f_{i}^{k}\right),\]  
where \(\alpha\) is the pessimistic coefficient chosen in the interval \(0\leq\alpha\leq 1\).  
The Bayes-Laplace criterion takes into account each possible consequence of all decision options, given their probabilities \(\underset{i=\overline{1,K}}{\max}\ \sum\limits_{i=1}^{K}f_{i}^{k}p_{i}\).  
All these and other methods are widely described in publications on decision making [11]. All have certain drawbacks. For instance, if we analyze the Wald maximin criterion, we can see that by the problem's hypothesis all criteria are in different units. Hence, the first step, which is to choose the minimal component \(f_{k}^{\min}=\underset{i=\overline{1,M}}{\min}\ f_{i}^{k}\), is quite reasonable. However, all \(f_{k}^{\min}\), \(k=\overline{1,K}\), are measured in different units, therefore the second step, which is to maximize the minimal component \(\underset{k=\overline{1,K}}{\max}\ f_{k}^{\min}\), is pointless. Although it brings us slightly closer to a solution, the criteria measurement scale fails to solve the problem since the chosen criteria scales are judgmental.  
We believe that to solve the problem of Equations (37) and (38), we need to form a measure that would allow the evaluation of any decision to be made, including the optimal one. In other words, we need to construct an axiom that shows, based on the set of \(K\) criteria, what makes one alternative better than the other. In turn, the axiom can help derive a principle that determines whether the chosen alternative is optimal. The optimality principle should become the basis for the constructive methods of choosing optimal decisions. We propose such an approach for the vector mathematical programming problem that is essentially close to the decision-making problem of Equations (37) and (38).