**6. Example of application**

To illustrate finding a satisficing solution, the following example of a bicriteria problem [24] is presented.

$$\begin{aligned} \max \left\{ \left( f\_1(\mathbf{x}) = \mathbf{1} \mathbf{0} \cdot \mathbf{x}\_1, \ f\_2(\mathbf{x}) = \mathbf{x}\_1 + \mathbf{5} \cdot \mathbf{x}\_2 \right) \right\} \\ \mathbf{10} \cdot \mathbf{x}\_1 &\ge \mathbf{50} \\ \mathbf{x}\_1 &\le \mathbf{8} \\ \mathbf{x}\_1 + \mathbf{x}\_2 &\le \mathbf{14} \\ \mathbf{x}\_1 \ge \mathbf{0}, \mathbf{x}\_2 &\ge \mathbf{0} \end{aligned} \tag{7}$$

The first step of multi-criteria analysis is the single-criteria optimization of each evaluation function is a pay-off matrix containing the values of all functions obtained when solving two single-criteria problems. This matrix allows us to estimate the extent of change of each evaluation function on the possible set, and also provides some information about the conflicting nature of the evaluation function. The objective matrix generates a utopia vector representing the best value of each of the separate criteria (**Table 1**).

The multi-criteria analysis is shown in **Table 2**.

At the beginning of the analysis, the decision maker defines his preference as an aspiration point equal to the utopia vector. The resulting value of the function s is negative. The aspiration point is not achievable. The decision maker's requirements are too high. The obtained solution prefers the first function. To improve the solution for the second function in the next iteration, the decision maker explicitly reduces his requirements for the first function and reduces the requirements for the second function. The value of function s is still negative. The aspiration point is not achieved. The decision maker's requirements are too high. The result is that the solution for the

### *Data and Decision Sciences – Recent Advances and Applications*


#### **Table 1.**

*Pay-off matrix with utopia vector.*


#### **Table 2.**

*Interactive analysis of finding a satisfactory solution.*

first function deteriorates and the solution for the second function improves. In the third iteration, the decision maker reduces the requirements for both functions. The value of function s is still negative. The aspiration point is not achieved. The decision

#### *Satisficing Decision-Making DOI: http://dx.doi.org/10.5772/intechopen.107428*

maker's requirements are too high. The solution continues to deteriorate for the first function and improves for the second function. In the fourth iteration, the decision maker continues to reduce the requirements for both functions. The value of function s is still negative. The aspiration point is not achieved. The decision maker's requirements are still too high. The solution continues to deteriorate for the first function and improves for the second function. In the fifth iteration, the decision maker continues to reduce the requirements for both functions. The value of function s is still negative. The aspiration point is not achieved. The decision maker's requirements are too high. The solution continues to deteriorate for the first function and improves for the second function. In the sixth iteration, the decision maker continues to reduce the requirements for both functions. The value of function s is now positive. The aspiration point is exceeded. The decision maker's requirements are too small. The solution continues to deteriorate for the first function and improves for the second function. In the seventh iteration, the decision maker increases the requirements for both functions. The value of function s becomes negative. The aspiration point is not achieved. The decision maker's requirements are too high. The solution improves for the first function and deteriorates for the second function. For the seventh iteration, the corresponding decisions are as follows: *<sup>x</sup>*^<sup>7</sup> <sup>¼</sup> ð Þ 10, 00, 4, 00 . The analysis shows that the solution depends heavily on the first function and affects the solution more.

The final choice of a particular solution depends on the preferences of the decision maker. The example shows that the method allows the decision maker to explore decision choices during interactive analysis and search for a satisfactory solution.
