**Figure 1.**

*System design characteristics of the SABANP model (N = 2) with netica software.*

Each criterion was modeled on two functions. The function was based on a true or false table. To construct the table, the true table was given an arbitrary phrase like high, provided, available, supported, highly required, time to obsolescence and best. For example, for RAM, the true table states the following: what is the probability that the system has a high RAM? For P&F, the true table states the following: what is the probability that the system's P&F is high? This logic of the true table for these examples is applied to the remaining criteria.

The same logic was applied to the false table. To construct the false table, an arbitrary phrase like low, not provided, not available, unsupported, not required, no time and worst was used. For example, for PT, the false table states the following: what is the probability that Personnel Training is not required for the system? In addition, for time, what is the probability that the system will not be obsolete in two (2) years? **Table 6** represents the true and false table probabilities. The true table probabilities are the normalized values of the expert judgments that are displayed in **Table 3**. The false table is the difference between the probability of each event as described in Eq. (5). The criteria are modeled as events. For example, when one event's true table value is 0.9, the false table value is 0.1. The JPD of the model is given as Eq. (6) and modeled using NETICA™ respectively for IBS 1, IBS 2 and IBS 3 system. Each system is modeled based on the probabilities of each criterion (RAM; P&F; PR; CM; DR&TD availability; OA&S; TRL; PT; and O&SR) given Cost and Time to obsolescence.

**Figure 2.** *The IBS for DDG-51 and its equivalent SABANP model.*

**Figure 3.** *The IBS for CG-47 and its equivalent SABANP model.*

## **3.2 Multi criteria decision making (MCDM)**

What is MCDM? MCDM methods provide a way to combine qualitative data (such as expert opinions) and quantitative data in order to analyze various

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

**Figure 4.** *The IBS for CVN-68 and its equivalent SABANP model.*


#### **Table 6.** *True and false table.*

alternatives [17]. When nonlinear factors are present, MCDM techniques are beneficial for discriminating among alternatives. Nonlinear factors are cases where the units of measurement (e.g., feet, seconds, Fahrenheit, and miles/hr.) are not the same. In the case of linear factors, the units of measurement are the same across the attributes or criteria. MCDM techniques can be categorized as fuzzy, stochastic or deterministic [20]. Additionally, a popular MCDM was examined for validation and or comparison to the SABANP result. The MCDM is TOPSIS. TOPSIS was chosen because of its popular usage in the literature and because it is capable of providing a deterministic data approach that accounts for the expert judgment participation, the dimensional criteria space, the methodical representation, and the explanation.

Four steps are required in any decision-making approach that relies on numerical analyses of alternatives to assess system factors' nonlinearity when selecting an MCDM methodology:


There are several methods used to determine criteria weights, such as weightassessment models [22, 23]. Certain authors have stated that standards are not available for defining which technique yields the most accurate criteria weight because whether the technique is biased is uncertain [24]. Others have suggested pairwise assessment matrices to calculate the significance or weights of criteria [18, 25]. A weighting method can be categorized as subjective or objective, algebraic or statistical, decomposed or holistic, and indirect or direct [17, 26]. In this study, the direct weighting method is selected because this method allows the decision maker to rank the criteria and provide subjective values to the criteria weight based on the defined rank. The direct weighting method was utilized in the analysis of TOPSIS, however, SABANP requires no weight inputs. Often weights are difficult to quantify when there are many experts. The benefit of having no weights is that it simplifies the decision matrix and provides for optimal decision making.

### *3.2.1 TOPSIS analysis*

Established by Yoon [27] and Hwang & Yoon [28], the basic principle underlying TOPSIS is that the selected alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution [29]. **Figure 5**, which is adapted from Adetunji's [6] graphic, depicts an assumption

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

for two criteria, where (*A*�) is the negative ideal solution and (*A*þ) is the positive ideal solution [17, 28]. The negative ideal solution is made up of the worst performance value of all the alternatives. The positive ideal solution is made up of the best performance value of all the alternatives.

Justifying the selection of A1 is difficult [17, 29] as shown in the visual example in **Figure 5**; therefore, the proximity (relative closeness) to each of these performance poles (*A*�) and (*A*þ) is measured in the Euclidean sense [17, 28, 30], for which the square root of the sum of the squared distances along each axis is in the 'attribute space' [30].

The following steps are required to conduct a TOPSIS analysis:

1.The normalized decision matrix is first determined using Eq. (7). The normalized score (*rij*) is calculated to transform the various attribute dimensions *Xij* from the raw data into non-dimensional attributes, thus allowing for comparisons among the attributes [17, 28, 29]:

$$r\_{\vec{\eta}} = {}^{X\_{\vec{\eta}}} \sqrt{\sum\_{i=1}^{n} {}^{X\_{\vec{\eta}}}}, \\
\text{where } i = 1, 2, 3, 4 \dots, m; j = 1, 2, 3, 4 \dots, n \tag{7}$$

**Table 7** shows the normalized decision matrix.

2.Calculate the weighted normalized values (*vij*) by multiplying *rij* by the criterion weights (*w <sup>j</sup>*) [17, 28, 29] (see **Table 8** and Eq. (8)):

$$\boldsymbol{\sigma}\_{\vec{\eta}} = \boldsymbol{w}\_{\vec{f}} \boldsymbol{r}\_{\vec{\eta}}, where \, j = 1, 2, 3, 4 \ldots, n; i = 1, 2, 3, 4 \ldots, m \tag{8}$$

The weighted normalized values are shown in the weight normalized matrix in **Table 8**.

3.Evaluate the positive (*A*<sup>+</sup> ) and negative (*A*�) ideal solution using Eq. (9) [17, 28, 29]:


**Table 7.**

*TOPSIS normalized decision matrix for IBS.*


#### **Table 8.**

*TOPSIS weighted normalized decision matrix for IBS.*

$$\begin{split} A^{+} &= \left\{ \left( \begin{array}{c} \max \\ i \end{array} v\_{\vec{\eta}} \mid j \in I \right), \left( \begin{array}{c} \min \\ v\_{\vec{\eta}} \mid j \in I^{+} \end{array} \right) \mid where \ i = 1, 2, 3, 4 \dots, m \right\} \\ &= \left\{ v\_{1}^{+}, v\_{2}^{+}, v\_{3}^{+}, \dots, v\_{j}^{+}, \dots, v\_{n}^{+} \right\}, \\ A^{-} &= \left\{ \left( \begin{array}{c} \min \\ v\_{\vec{\eta}} \mid j \in I \end{array} \right), \left( \begin{array}{c} \max \\ v\_{\vec{\eta}} \mid j \in I^{-} \end{array} \right) \mid where \ i = 1, 2, 3, 4 \dots, m. \right\} \\ &= \left\{ v\_{1}^{-}, v\_{2}^{-}, v\_{3}^{-}, \dots, v\_{j}^{-}, \dots, v\_{n}^{-} \right\}, \\\\ \end{split}$$

$$\begin{split} \mathcal{J}^{+} = \{ \mathbf{j} = 1, 2, 3, 4 \dots, n \} \text{ yields to the benefit criteria} \}. \end{split}$$

$$f^{-} = \{ \begin{array}{c} j = 1, 2, 3, 4 \dots, n \} \text{ yields to the benefit criteria} \}. \end{split}$$

The *A*<sup>+</sup> and *A*� solutions are shown in **Tables 9** and **10** respectively.


**Table 9.** *TOPSIS positive ideal solutions for IBS.* *The Application of Simple Additive Bayesian Allocation Network Process in System… DOI: http://dx.doi.org/10.5772/intechopen.98530*


### **Table 10.**

*TOPSIS negative ideal solutions for IBS.*

4.Calculate each alternative separation from the positive (*si +* ) and negative (*si* �) ideal solutions (use the *n*-dimensional Euclidean distance) using Eq. (10) [17, 28, 29]:

$$S\_i^+ = \sqrt{\sum\_{j=1}^n (v\_{ji} \cdot v^+)^2 \, where \, i = 1, 2, 3, 4 \dots m, \tag{10}$$

$$\mathcal{S}\_i^- = \sqrt{\sum\_{j=1}^n \left(v\_{ij} \cdot v^-\right)^2} \text{ where } i = 1, 2, 3, 4 \dots m$$

**Tables 11** and **12** show the separation measures between the positive and negative ideal solutions.

5.Calculate the relative closeness of each alternative to the ideal solution (*ci\**) [17, 28, 29] using Eq. (11):


**Table 11.**

*TOPSIS separation Mmasures to positive ideal solutions for IBS.*


**Table 12.**

*TOPSIS separation measures to negative ideal solutions for IBS.*


**Table 13.** *Experts' demographics.*

$$\mathcal{c}\_{i}^{\*} = \mathbb{S}\_{i}^{-} \langle \mathbb{S}\_{i}^{-} + \mathbb{S}\_{i}^{+} \rangle, \mathbf{0} < c\_{i}^{\*} < \mathbf{1}, i = \mathbf{1}, 2, 3, 4 \dots, m \tag{11}$$

*Ai equals A*<sup>þ</sup> ð Þ *if c* <sup>∗</sup> *<sup>i</sup> equals* 1; *Ai equals A*� ð Þ *if c* <sup>∗</sup> *<sup>i</sup> equals* 0

6. Sort the order of preference alternatives by the descending order of *ci\**: The nearer *ci* <sup>∗</sup> is to one indicates a higher importance of the alternative [17, 28, 29].

### **3.3 Delivery mechanism**

Before administering the research survey, each research participant was provided an information sheet and consent form to complete and instructions on how to complete the survey. Once retrieved, the survey files were password protected and saved. The experts' participation in the study was voluntary and anonymous. The breakdown of the experts' demographics is shown in **Table 13**.
