## 4 The Algorithm: Space of Convergence Vectors (SCV)
### The Proposed Algorithm
#### 4.2.4 The Mathematical Model of the SCV Algorithm  
For defining the mathematical model that describes the swarm behavior in the SCV algorithm, an important issue is the specificity of the problem under consideration--the design of a new technical system, where people have active and significant interventions in the creation/conceptualization process at every iteration of the algorithm. Thus, we propose an augmented approach, where the computer and humans iteratively collaborate to design the system.  
Hence, a series of aspects that differentiate this problem concerning other optimal design problems using PSO algorithms must be highlighted. The first aspect is about the fact that when solutions associated with each particle are defined and when comparative analysis of various solutions belonging to various particles is done, the premise is that humans are integrated part of the algorithm (e.g., experience, intellect), even if he/she is in the outer space of the mathematical formalism.  
The second aspect concerns that, at every iteration, humans are inspired and influenced by all previous solutions belonging to all particles; in other words, the proposed solution assigned to a certain particle, at a certain iteration, accumulates experiences and knowledge from all previous iterations.  
The third aspect is referring to the fact that, during the formulation of a solution, new connections occur in the human brain, and unexpected creative ideas could occur, thus, randomizing the leading direction in formulating a solution.  
Considering the elements highlighted before, an enhanced relation for describing the velocity of the particle is here considered, in the sense of adding a supplementary term [62]:  
\[v_{i}(t+1)=\omega\cdot v_{i}(t)+c_{1}\cdot r_{1}(t)\cdot[y_{i,t}-x_{i}(t)]+c_{2 }\cdot r_{2}(t)\cdot[z_{t}-x_{i}(t)]+c_{3}\cdot r_{3}(t)\cdot[\mu_{t}-x_{i}(t)] \tag{7}\]  
where \(c_{3}\) is the passive congregation coefficient (e.g., \(\omega=1\), \(c_{1}=0.5\), \(c_{2}=0.5\), \(c_{3}=0.5\) according to [62], \(r_{3}\) is a random real number in the interval [0,1], and \(\underline{u}_{t}\) is the position at iteration \(t\) of a randomly selected particle in the swarm.  
In the problem under consideration, inertia \(\omega\), accelerations \(c_{1}\), \(c_{2}\), \(c_{3}\), as well as redirection of particles (\(r_{1}\), \(r_{2}\), \(r_{3}\)) are actions happening in the mental space of engineers and not in the computer space of mathematical formalism. Therefore, all these factors are considered here of value 1. With these considerations, the mathematical model that describes particles' behavior in the SCV model is proposed in Equations (8) and (9). Equation (8) defines particle's position, and Equation (9) defines particle's velocity:  
\[\left\{\begin{array}{l}1_{i}\{j_{1}(t+1)\}\\ 2_{i}\{j_{2}(t+1)\}\\ 3_{i}\{j_{3}(t+1)\}\\ \ldots\\ n_{i}\{j_{n}(t+1)\}  
Figure 1 describe the GIVs in a subset, in the order of their ranks (that is, the GIV with the highest rank in the subset is \(p_{1}\)). More insights on how these operations are applied are given in the case study from Section 5. The convergence of the algorithm is verified by comparing the evolution of the utility function \(T\) (Equation (4)) from one iteration to the next one, as well as by comparing the evolution of the result of subtracting the target value and the current value for each performance indicator (see Equation (5)).