#### 3.1. Model Basis

The algorithm presented here assumes that a set of companies and its characteristics (size, industry, age, etc.) are available. The objective is to define an algorithm that maximizes the position of a company compared with its competitors in terms of ITIL implementation. The selection of processes to implement depends on the strategic needs of the company and the degree of implementation of every ITIL process compared to the competitors. The algorithm selects the process that maximizes the benefits of ITIL implementation in front of its competitors.

Let

Ω_{e} be the set of available companies from which characteristics are known and stored. Usually this set of data is taken from a great poll:

Let

E be any company included in the dataset:

Let

Ω_{ne} be the complementary set of

Ω_{e}, that is the set constituted by companies that have not been included in the data set:

It will be noted by

E_{sel} any company included in this last data set:

Let us denote by

Ω_{p} the set of the

p_{ITIL} processes that can be implemented out of the available data:

Let

p_{ITIL} be the number of ITIL processes that can be implemented, so:

Let p_{i} any of the p_{ITIL} potential processes that can be implemented and d_{i} the assessment of p_{i} process in a particular company, E:

The possible value of

d_{i} are:

where

C_{m}(

P_{i}) is a Boolean function that indicates if the

p_{i} process satisfies an implementation degree of

c_{m}. The possible values for the degrees are in the interval [1..

M], so 1 represents the worst option and

M represents the best option. That is:

For example, the c_{1} condition could mean that the process is not implemented, neither will it be in the long term; in that case C_{1}(p_{i}) = True would indicate that p_{i} is not implemented, neither will it be implemented in the long term in company E; in such case d_{i}(E) = 1.

It should be noticed that {1...M} represents the values given to the different degrees of implementation that a p_{i} process may have in company E.

Let

Ω_{v} be the set of the

v parameters that define the characteristics of every company, which are identified by {

v_{1},

v_{2}, …,

v_{v}}. Such parameters could be the type of activity, the age, the number of employees, the geographical area where it operates, etc.,

Following this, let us define the domain of each parameter that belongs to

Ω_{v}:

where

n_{k} represents the number of possible values for the parameter

v_{k}.Next, let us define every

v functions {

V_{1},

V_{2}, …,

V_{v}} which assign a value

v_{ij} to the parameter

v_{i} for the company

E:

In such case, the

v vector of parameters which define the characteristics of a company are given by:

Next,

s_{i}_{1j} is defined as the average of the assessment of the

p_{i} process for the companies that satisfy

v_{1} =

v_{1k}:

In a general way, the average of the assessment of the

p_{i} process for all companies which satisfy

v_{j} =

v_{jk} can be denoted by

s_{ijk}:

Or, expressed in a more formal way:

This means that the average of the assessments

d_{i} for the

p_{i} process for every company

e_{e} whose parameter

V_{j} satisfies

V_{j}(

e_{e}) =

v_{jk}. The ordinality of the set of companies that satisfy such condition is given by:

At this point, it is possible to show the algorithm that allows any company to decide which process should be first implemented.

Let us define

E_{sel} as any company not included in the set

${\Omega}_{e}$:

#### 3.2. 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}:

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:

This indicator can be particularized for a particular process

p_{i} and for all companies with the same characteristics

V(

E_{sel}):

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_{ijt}j address that the p_{i} process is not implemented in the companies with the same characteristics ${v}_{j{t}_{j}}$, 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.

Thus, we can formalize

r_{i} as the relative positioning indicator and

R_{i} as the function to evaluate

r_{i}:

Thus, R_{i} represents the function that evaluates the relative positioning r_{i} if a company e_{x} for the process p_{i} compared to similar companies.

Once this indicator is defined, it is possible to define the selection criteria for the first process to be implemented:

This expression means that the best process and the one that should be first implemented, that is, the one that will most improve the rank of the company (in terms of ITIL implementation) compared to the competitors, is the one with the lowest value for r_{p}, that is, the one that has the lowest relative positioning.

The criteria shown previously needs a small correction: Due to the fact that relative positioning

r has taken into account the situation of the process in the company itself, every process that is already implemented should be excluded; that is, the processes with

D_{i} = 0. This obliges the exclusion of every process that has been already implemented and review only processes with

D_{i} ≠ 0.

Thus, if we select iteratively the P_{opt} process (of course, on every iteration this process will be different as P_{opt} process has already been implemented in the previous iteration and so, dismissed as selectable), we get a sequence of processes to implement that optimizes the degree of implementation compared to the competitors, for the criteria defined (age, size, etc.).