## 4 The Algorithm: Space of Convergence Vectors (SCV)
### Planning the SCV Algorithm  
In the preparatory phase, also called the planning phase, the performance indicators \(i\), \(i\) = 1,..., \(n\) and their related target values \(V_{T}\), \(i\) = 1,..., \(n\) are defined. Further, a normalized-to-1 value weight \(\beta_{i}\), \(i\) = 1,..., \(n\); \(\beta_{1}\) + \(\beta_{2}\) +... + \(\beta_{n}\) = 1, is associated with each performance indicator \(i\), \(i\) = 1,..., \(n\). The utility function \(T\) is defined as follows:  
\[\max T=\left[\sum_{i=1}^{n}\beta_{i}\cdot\left(\frac{V_{Ei}}{V_{Ti}}\right)^{ k(i)}\right]_{|} \begin{array}{c}k(i)=1\ (\max)\\ k(i)=-1\ (\min)\end{array}  \tag{4}\]  
where \(V_{Ei}\), \(i\) = 1,..., \(n\) is the effective value of the performance indicator \(i\), \(i\) = 1,..., \(n\) and \(k(i)\), \(i\) = 1,..., \(n\) is a coefficient that defines the optimization trend of the performance indicator \(i\), \(i\) = 1,..., \(n\). Coefficient \(k(i)\) = 1 if the performance indicator \(i\) must be maximized, \(k(i)\) = \(-\)1 if the performance indicator \(i\) must be minimized. When ideality is achieved, \(T\) = 1 (considering that \(V_{Ei}\), \(i\) = 1,..., \(n\) does not exceed \(V_{Ti}\), \(i\) = 1,..., \(n\)). The convergence of the algorithm is thus described by the following condition:  
\[|V_{Ti}-V_{Ei}|_{t+1}<|V_{Ti}-V_{Ei}|_{t};\ \ |V_{Ti}-V_{Ei}|_{t\to N}\to 0;\ \ i= \overline{1,n} \tag{5}\]  
where \(N\) is the maximum number of planned iterations, and \(t\) is the symbol of the current iteration (\(t\) = 1,..., \(N\)). For each performance indicator \(i\), \(i\) = 1,..., \(n\), the barrier in achieving its target value \(V_{T}\) is formulated. This barrier is expressed in terms of TRIZ-MC language, meaning that a pair of GEPs is associated (please revisit Section 3.2 for the acronym). It is denoted with \(GEP_{B}\) the GEP that must be improved for reaching \(V_{T}\) and with \(GEP_{W}\) the GEP, which is affected because of the action on \(GEP_{B}\). In the TRIZ-MC, at the intersection of the raw \(GEP_{B}\) with the column \(GEP_{W}\) a subset of GIVs is revealed (please consult [32] or other references about TRIZ, including Internet sources for details about the list of 40 GIVs and their locations in the boxes of TRIZ-MC). The elements of this subset are denoted \(GIV_{ir(j)}\), \(i=1,...,n\); \(v(i)=0\),..., \(h_{i}\), where \(h_{i}\) is the size of the subset of GIVs associated with the performance indicator \(i\), \(i=1,...,n\) (\(h_{i}\), \(i=1,...,n\), can take the value \(0\), \(1,2,3\) or \(4\), as they are disclosed by TRIZ-MC). At the end of this process, it is possible that some of the GIVs to be associated with several performance indicators. To each GIV from the subsets, a rank \(b\) is assigned. The rank is the sum of the value weights of the performance indicators to which the respective GIV is associated. For example, considering that the generic inventive vector \(GIV_{e}\) is met in the subsets belonging to the performance indicators \(x\), \(y\) and \(z\), which have the value weights \(\beta_{x}\), \(\beta_{y}\) and \(\beta_{z}\), the rank \(b_{e}\) of \(GIV_{e}\) will be \(=\beta_{x}+\beta_{y}+\beta_{z}\). If it is denoted with \(p(j),j=1,...,40\), the positions of the GIVs in the list of 40 GIVs of TRIZ = MC [32], to each \(GIV_{ir(j)}\), \(i=1,...,n\); \(v(i)=0,...,h_{i}\) from the identified subsets, the corresponding order from the list of 40 GIVs of TRIZ-MC can be assigned. For example, if \(GIV_{e}\) belonging to three subsets that correspond to the performance indicators \(x\), \(y\) and \(z\), is in the position \(p(l)\) (e.g., \(l=21\)) in the list of 40 GIVs of TRIZ-MC, the number \(l\) will be assigned to \(GIV_{e}\) in all the three GIV-subsets of the indicators \(x\), \(y\) and \(z\). The association of GIVs with numbers is necessary when the SCV algorithm is implemented in computer software for a more efficient application. TRIZ method respects both the ideality and convergence paradigms mentioned in Section 2 of the paper [58].