## 4 Methodology
Notwithstanding, we shortly describe the widest utilised initialiser in several (almost all) heuristic-based applications, which employ either one or multiple search agents, as follows. **Random Sample (RX):** An initial solution \(\vec{x}_{n}(t)\) is obtained through a random sample operation such as \(\vec{x}_{n}(t)=\vec{r}\), where \(\vec{r}\in\mathfrak{S}\) is a vector of i.i.d. random numbers with uniform distribution between the boundaries of the feasible domain. If it is a continuous feasible domain between \(-1\) and \(1\), \(\vec{x}=[-1,1]^{D}\subset\mathfrak{S}=\mathbb{R}^{D}\), then \(\vec{r}\
i r_{i}\sim\mathcal{U}(-1,1)\). If it is adiscrete feasible domain between \(1\) and \(m\left(\gg 1\right)\), \(\mathfrak{X}=\{\forall\,y_{i}\in\vec{y},\vec{y}\in\mathfrak{S}|1\leq y_{i}\leq m \}\subset\mathfrak{S}=\mathbb{Z}^{D}\), hence \(\vec{r}\
i r_{i}\sim\mathcal{U}\{1,m\}\). Other feasible domain can also be considered, for example, the mixed continuous-discrete ones. Notice that this initialiser, as well as others, can be implemented for either a single-agent or population-based search procedure. ### Perturbators

In this section, we describe several perturbators found in the selected metaheuristics described at the beginning of Appendix A. Somewhat similar to initialisers, many perturbators can be generalised as population-based procedures, but one may encounter some interesting exceptions. One of these, closely related to this work, are the single-agent perturbators with variable length or dimension. These are found with ease in several 'classic' heuristics dealing with combinatorial problems, where the candidate solution is represented by a set of elements (say, item, values, weights, and so forth) which can vary in length while solving the problem. Therefore, we grouped the extracted perturbators into two main groups: single-agent variable-dimension and population-based fixed-dimension perturbators. Consider that these perturbators can be implemented in both integer and continuous domains. Naturally, for the latter, one must take some additional precautions into account. Other groups from the possible combinations of these characteristics would be worth investigating, but they are out of our scope. #### a.2.1 Single-Agent Variable-Dimension Perturbators

For these simple heuristics, also named as _rules_ or _actions_ in the literature, we consider an arbitrary problem domain \(\mathfrak{S}\). Nevertheless, bear in mind that we use these heuristics in the context of a combinatorial domain such as the heuristic space, \(\mathfrak{S}=\mathfrak{H}\). Recall that \(\vec{x}(t)\) and \(\vec{y}\) correspond to the current and candidate solutions, respectively, and \(\varpi\) is the cardinality of \(\vec{x}(t)\), \(\varpi=\#\vec{x}\). This cardinality or dimensionality is finite and delimited by a lower and upper value such as \(\varpi_{l}\leq\varpi\leq\varpi_{u}\). In the following lines, we briefly describe the actions regarded for this work. **Add:** It inserts randomly chosen element \(y_{i}\in\mathfrak{S}\) at a random l.h.s. position \(i\sim\mathcal{U}\{1,\varpi+1\}\) into \(\vec{x}(t)\). Thus, the candidate sequence will be \(\vec{y}=(x_{1},\ldots,x_{i-1},y_{i},x_{i+1},\ldots,x_{\varpi})^{\intercal}\) with \(\#\vec{y}=\varpi+1\). **Add Many:** It performs the **Add** operation \(\vartheta\) times since this value is chosen at random between \(1\) and the remainder of \(\varpi\) to \(\varpi_{u}\), \(\theta\sim\mathcal{U}\{1,\varpi_{u}-\varpi\}\), where \(\varpi_{u}\) is the maximal cardinality. After doing so, the cardinality of the candidate solution will be \(\#\vec{y}=\varpi+\vartheta\). **Remove:** It discards an element \(x_{i}\) from a random position \(i\sim\mathcal{U}\{1,\varpi\}\) of \(\vec{x}(t)\). Thus, the candidate sequence will be \(\vec{y}=(x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{\varpi})^{\intercal}\) with \(\#\vec{y}=\varpi-1\).