**3. Methodology**

Simple Additive Bayesian Allocation Network Process (SABANP) utilized the components of the Simple Additive Weighing (SAW) method as the input variables to the Bayesian Belief Network. SAW is an MCDM. To better understand SABANP, it is necessary to provide details of what SAW is and how it is used in the model. The SAW model, which is also known as the WSM or the "weighted average," is a common approach used for multicriteria analysis [17].

One must first calculate the normalized decision matrix for the benefit criteria (higher is better), where *nij* is the normalized score of the *i th* alternative with respect to *j th* criterion, and *rij* are the values in the decision matrix provided by the experts [17, 18]:

$$m\_{\vec{\eta}} = r\_{\vec{\eta}} /\_{r\_{\vec{\eta}}}, i = 1, 2, 3, \dots \\ m, j = 1, 2, 3 \dots, n \tag{2}$$

*rmax ij* is the maximum value of the *i th* alternative with respect to each *j th* criterion in the decision matrix.

For the non-benefit criteria (lower is better), *rmin ij* is the minimum value of the *i th* alternative with respect to each *j th* criterion in the decision matrix [6]:

$$n\_{\vec{\eta}} = r\_{\vec{\eta}}^{\min} \rangle\_{r\_{\vec{\eta}}}, i = \mathbf{1}, 2, 3, \dots \\ m, j = \mathbf{1}, 2, 3 \dots, n \tag{3}$$

The normalized Matrix for IBS is found in **Table 3**.

The best alternative is the one that maximizes *Ai* in Eq. (4) below. The weights (*wj*) are the weighted criteria values, and they sum to 1 as shown in **Table 2**.


*The time for the system to reach obsolescence was modeled with a 95% confidence interval, i.e., the software and hardware would be obsolete within two years.*

#### **Table 3.** *SAW normalized matrix.*

*The Application of Simple Additive Bayesian Allocation Network Process in System… DOI: http://dx.doi.org/10.5772/intechopen.98530*

$$A\_i = \sum\_{j=1}^{n} w\_j n\_{ij}, i = 1, 2, 3 \dots, m \tag{4}$$

The SAW model is governed by additive utility theory [19]. As shown in the equation above, each alternative aggregate value is equivalent to the summation of its multiplication. In a one-dimensional case in which the units are similar—for example, seconds, feet, and dollars—the WSM is easy to use [17, 20]. The approach becomes difficult when applied to decision-making problems that are multidimensional [21]. The weights were assessed during the data collection using the direct weighting method. The direct weighting method allows the decision maker to rank the criteria and provide subjective values to the criteria weights based on the defined rank. However, the weights were not needed in the SABANP model to calculate the best alternative. Eqs. (2) and (3) were used since SABANP requires only the normalization of the experts' inputs. The normalized scores ranging from 0 to 1, as shown in **Table 3**, were transposed into the SABANP model for the analysis.

#### **3.1 Simple additive Bayesian allocation network process (SABANP)**

The SABANP process begins by populating the survey's raw data into the decision matrix. As shown in **Table 4**, the score of criterion C*<sup>j</sup>* with regard to alternative A*<sup>i</sup>* is *rij* and the weight of the Cj *is Wj*. The weights are not required for the analysis. The following steps are required to conduct SABANP analysis:


The true function is given as *T nij* � �, where *T nij* � � <sup>¼</sup> *nij*; (5)

And the false function is given as *F nij* � �, where *F nij* � � <sup>¼</sup> <sup>1</sup> � *T nij* � � <sup>¼</sup> *<sup>n</sup>*<sup>∗</sup> *ij*


**Table 4.** *Raw data decision matrix.*


**Table 5.** *Data decision matrix.*

> 3. Initial probabilities, *nij* and *n*<sup>∗</sup> *ij* are assigned to each variable set {Xij} in the joint distribution function (Eq. (6)) as shown in **Table 5**.

The joint probability distribution (JPD) of the variables set {X11, X21, … , Xmn} is given as follows:

$$P[X\_{11}, X\_{21}, \dots, X\_{mn}] = \prod\_{i=1}^{m} \prod\_{j=1}^{n} P[X\_{ji}|parent(X\_{ji})], i = 1, 2, 3 \dots, m; j = 1, 2, 3 \dots, n \tag{6}$$

In modeling the system, the IBS system data collected from experts with respect to the systems alternatives and criteria were analyzed using the SABANP. NETICA™ software was used to develop the model. NETICA™ is a powerful, easyto-use, complete program for developing belief networks and influence diagrams. It provides an interface for drawing networks, and creating relationships between variables which can be probabilities, equations, or data files.

The systems were modeled using RAM, P&F, PR, CM, DR&TD availability, OA&S, TRL, PT, and O&SR as inherent factors that affect the system's design with effects on the costs and time to obsolescence. The NETICAL™ software model shows the captured image when the model was simulated twice or when N = 2 as displayed in **Figure 1**, where N represents the number of times the model and simulation were run. A graphical representation of it is shown in **Figure 2** for the IBS 1 on DDG-51. **Figures 3** and **4** show the graphical representations of the SABANP models of the IBSs (2 and 3) on CG-47 and CVN-68, respectively.

The research question was centered on the following: Does using the newly derived methodology (SABANP) to evaluate multiple obsolescence characteristics in a System enable one to predict which system is less susceptible to obsolescence?

The null hypothesis is that SABANP cannot predict which System is less susceptible to obsolescence, and the alternative hypothesis is that SABANP can predict which system is less susceptible to obsolescence. Additionally, statistical analysis was not conducted. Rather, the model was ran one hundred times (100x,) and the results were aggregated for accepting or rejecting the null hypothesis. A sensitivity analysis was also performed on the results.

Three questions were used within the survey to validate the SME inputs. These questions were used to cross-examine the survey data received from the experts. The data were analyzed to check for inconsistencies. Individual responses to system rankings and criteria weights were plotted to ensure that there were no outliers, and that the data are attributed to a credible sample of expert practitioners.

*The Application of Simple Additive Bayesian Allocation Network Process in System… DOI: http://dx.doi.org/10.5772/intechopen.98530*
