**3. Multicriteria decision analysis methods**

However, the MCDA approach has certain drawbacks linked to the subjective nature of the preferences granted by decision-makers. Specially, the criteria weighting profile can be considered as the uncertain part as it relies heavily on subjective measurements and has a great influence on the final result.

*Optimization Multicriteria Scheduling Criteria through Analytical Hierarchy Process… DOI: http://dx.doi.org/10.5772/intechopen.96557*

Linkov offers traditional risk analysis and Monte Carlo simulation taking into account the uncertainties underlying the point estimates, to which all considerations and calculations are reduced [16].

Historically Multiple Criteria Evaluation methods were developed to select the best alternative from a set of competing options [17].

#### **3.1 Analytical hierarchy process methods**

The Analytical Hierarchy Process (AHP) is a method that was invented by Professor Thomas Saaty. It provides a decision-making structure that takes into account factors weighed in as a group [18].

The analytical hierarchy process for decision making is a relative measurement theory based on pairwise comparisons. Pairwise comparison matrices are formed either by providing judgments to estimate dominance using absolute numbers from the 1 to 9 fundamental scales of the AHP, or by directly constructing pairwise dominance ratios using real measurements.

The process of synthesis of weighting and addition applied in the hierarchical structure of the AHP combines multidimensional measurement scales into a single "one-dimensional" priority scale [19].

The steps of the AHP method are as follows:


$$\mathbf{r}(\mathbf{Ea}, \mathbf{Eb}) = \frac{\mathbf{1}}{\mathbf{Pc}(\mathbf{Eb}, \mathbf{Ea})} \tag{1}$$


**Figure 1.** *Construction of the hierarchy.*

*Engineering Problems - Uncertainties, Constraints and Optimization Techniques*

$$IC = \frac{\lambda m \infty - n}{n - 1} \tag{2}$$

Where: λ max is the primary maximal value running in matrix of comparisons by pairs; and n: is a large number of comparative elements. Then; the ratio of coherence (RC) defines by:

$$RC = 100.\frac{IC}{ACI} \tag{3}$$

Where: ACI is the means coherence indicator of obtained generating aleatory matrix of judgment equalizes height. The means of indicator coherence is identified in the following **Table 2**.

A value of RC inferior to 10% is generally acceptable; otherwise, comparisons by pairs must be examined again to reduce the incoherence.

• To settle the relative performance of each action:

$$P\_k(e\_1^k) = \sum\_{J=1}^{nk-1} P\_{K-1} \quad (e\_i^{k-1} \ ) P\_k \frac{e\_i^k}{e\_i^{k-1}} \tag{4}$$

With: *Pk e<sup>k</sup>* 1 � � <sup>¼</sup> 1, and: nk � 1 are a large number of elements of the hierarchic level k-1, Pk ek i � � is the terms priority to the element e<sup>i</sup> to the hierarchic level k [20].


**Table 1.**

*Saaty scales.*


**Table 2.** *Means coherence indicator.* *Optimization Multicriteria Scheduling Criteria through Analytical Hierarchy Process… DOI: http://dx.doi.org/10.5772/intechopen.96557*
