# How to Optimize the Implementation of ITIL through a Process Ordering Algorithm
## 3 Algorithm for Optimal Ordering of ITIL Processes
### Criteria for Establishing the Sequence  
In order to decide which process should be first implemented, it is necessary to define a parameter containing information about the implementation of an ITIL process in other companies with similar characteristics. To define it, and taking in consideration the definitions presented in previous paragraphs, the next parameters are needed; where \(S_{ijk}\) is referred to the set of available set of companies in the database and \(D_{i}\) is specifically referred to the company \(E_{sel}\):  
\[ \begin{array}{c}S_{ijk}=\big{(}M-s_{ijk}\big{)}\forall i\in(1..p_{\text{ITIL }})\wedge j\in(1..v)\wedge k\in\big{(}1..v_{j}\big{)}\\ D_{i}=(M-d_{i})\forall i\in(1..p_{\text{ITIL}})\end{array} . \tag{18}\]  
In this expression, \(M\) represents the maximum value that the implementation of an ITIL process \(p_{i}\) can take. This \(M\) value represents a complete implementation of the \(p_{i}\) process or a very short-term implementation. It can be seen how \(S_{ijk}\) represents, for a specific p\({}_{i}\) process, the distance between the maximum value \(M\) and the average of the values of implementation for that particular process in companies with the same characteristics \(v_{jk}\) as \(E_{sel}\). It can be also seen how \(D_{i}\) represents the distance between the maximum value M from the response of the company \(E_{sel}\). It is necessary to define an indicator or parameter to help us to decide the best process to be implemented next. This indicator should take high values when the relative position (in terms of implementation of a process) is better than the reference companies denoted by \(V(E_{sel})\). This indicator should also take low values when the company is in a worse position than similar companies with characteristics \(V(E_{sel})\). Let \(r\) be the indicator \"relative position\" defined as follows:  
\[r=\frac{S_{a}^{2}S_{\beta}^{2}S_{\gamma}^{2}\ldots S_{\delta}^{2}}{D^{2}}. \tag{19}\]This indicator can be particularized for a particular process \(p_{i}\) and for all companies with the same characteristics \(V(E_{sel})\):  
\[r_{i}=\frac{\prod_{j=1}^{p}S_{ijt_{j}}^{2}}{D_{i}^{2}}\left|\;V_{j}(E_{sel})=v_{ jt_{j}},\right. \tag{20}\]  
where \(r_{i}\) represents the value of the \(r\) indicator for the company \(E_{sel}\) and for the process \(p_{i}\). This expression calculates \(r_{i}\) from the \(S^{2}\) value for each group of companies that satisfy the condition of having the same characteristics as \(E_{sel}\): Low values of \(D_{i}\) indicate that the process \(p_{i}\) is implemented or it will be soon in the company \(E_{sel}\); on the other hand, high values of \(D_{i}\) indicate that process is not implemented, neither will it be in the short term. At the same time, high values for \(S_{ijtj}\) address that the \(p_{i}\) process is not implemented in the companies with the same characteristics \(v_{jt_{j}}\), while low values show this process is implemented in companies with same characteristics, but it is not in \(E_{sel}\). As a consequence, if the \(p_{i}\) process is implemented in similar companies but it is not in the company \(E_{sel}\) (that is, \(E_{sel}\) position is worse than competitors), this leads to a really small value for \(r\), compared to the maximum value it could take; it can be also shown that if a company has the process \(p_{i}\) implemented but similar companies do not (that is, the relative position of \(E_{sel}\) is better than competitors), then \(r\) will take a high value. Thus we can conclude that parameter \(r\) is a measurement of the relative position of a company for a specific process \(p_{i}\) compared to similar companies. Anyway, the value of r goes from 0 to 1.