**5. Reference point method**

The reference point method combines the simplicity and openness of controlling the interactive analysis process with strict adherence to the principle of efficiency of the generated solutions and complete parameterization of the set of efficient solutions. The reference point method uses aspiration levels as control parameters and always generates efficient solutions.

The preference model used in the reference point method satisfies the following two postulates:


In this model, it is assumed that when solving a decision problem, the decision maker defines aspiration levels as the desired values of individual evaluations. If the values of the evaluations do not achieve the aspiration levels, the decision maker tries to find a better solution. If the values of some evaluations achieve their respective aspiration levels, the decision maker focuses attention on improving the values of those evaluations that have not achieved their aspiration levels. When all evaluations have achieved their aspiration levels, the decision maker is interested in further improving the evaluations if possible.

The reference point method relies on the minimization of a suitably defined achievement scalarizing function that generates a preference relation satisfying postulates P1 and P2. For that reason, it always determines efficient solutions. It is also required that the achievement scalarizing function ensures the completeness of the parameterization of the set of efficient solutions by aspiration levels. This requirement means that for each achievable evaluation vector *y*∈*Y*0, there should be aspiration levels that allow for determining the efficient solution that generates this evaluation vector.

The achievement scalarizing function in the reference point method is as follows [10, 11, 14, 21]:

$$s(\boldsymbol{\mathfrak{y}}, \boldsymbol{\overline{\mathfrak{y}}}) = \min\_{1 \le i \le m} (\boldsymbol{\mathfrak{y}}\_i - \boldsymbol{\overline{\mathfrak{y}}}\_i) + \boldsymbol{e} \cdot \sum\_{i=1}^m (\boldsymbol{\mathfrak{y}}\_i - \boldsymbol{\overline{\mathfrak{y}}}\_i) \tag{6}$$

where

*y* ¼ *y*1, *y*2, … , *ym* � � is an evaluation vector.

*y* ¼ *y*1, *y*2, … , *yk* � � is a vector of aspiration levels.

*ε*—an arbitrarily small regularization parameter.

The maximization of the function *s y*ð Þ , *y* due to *y* ∈*Y*<sup>0</sup> determines the nondominated evaluation vector ^*y* and the generating efficient solution *x*^. The determined efficient solution depends on the values of the aspiration levels *y*. The aspiration levels *yi* , *i* ¼ 1, … , *m* are the parameters that control the interactive analysis process. The parameter *ε* is used to introduce a regularization component to guarantee the efficiency of the solution in case of ambiguity of the minimum of the first component of the function *s y*ð Þ , *y* .

The optimization problem solved by the reference point method does not introduce significant complications into the structure of the original problem. The process of interactive analysis by the reference point method is consistent with the concept of decision support systems. It implements an open process of searching for a satisficing efficient solution on the basis of current preferences determined by aspiration levels. It is easy for the decision maker to understand the expression of current preferences in terms of aspiration levels.

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

In the reference point method, the scalarizing function *s y*ð Þ , *y* is called the achievement function. This name is related to the fact that the values of this function are zero for y <sup>¼</sup> y, positive for y∈<sup>y</sup> <sup>þ</sup> *<sup>D</sup>*<sup>~</sup> , and negative for y <sup>∉</sup> <sup>y</sup> <sup>þ</sup> *<sup>D</sup>*<sup>~</sup> . Therefore, the maximum values of this function can be used not only to calculate efficient outcomes but also to assess the achievability of a given aspiration point y:

