# Hyper-Heuristic Based on ACO and Local Search for Dynamic Optimization Problems +
## 3 HULK: Hyper-Heuristic Based on ACO and Local Search Operators
### Choosing and Applying Local Search Operator
#### 3.2.2 Unstringing and Stringing Local Search Operator
Figure 3: (**a**) Unstringing Type I, (**b**) unstringing II, (**c**) unstringing Type III, and (**d**) unstringing Type IV. and \((V_{k-1},V_{k})\) and the insertion of arcs \((V_{i+1},V_{j})\), \((V_{i-1},V_{k})\), and \((V_{j-1},V_{k-1})\). As before, the sub tours \((V_{j-1},\dots,V_{k})\) and \((V_{i-1},\dots,V_{j})\) are reversed. * Unstringing Type IV: consider \(V_{j}\) belonging to the neighborhood of \(V_{i+1}\), \(V_{k}\) belonging to the neighborhood of \(V_{i-1}\) with \(V_{k}\) being part of the sub tour \((V_{l+1},\dots,V_{i-2})\), and \(V_{l}\) belonging to the neighborhood of \(V_{j-1}\) with \(V_{l}\) being part of the sub tour \((V_{j+1},\dots,V_{k-1})\). The removal of node \(V_{i}\) results in the deletion of arcs \((V_{i-1},V_{i})\), \((V_{i},V_{i+1})\), \((V_{j-1},V_{j})\), \((V_{l-1},V_{l})\), and \((V_{k},V_{k+1})\) and the insertion of arcs \((V_{k},V_{i-1})\), \((V_{j-1},V_{l})\), \((V_{k+1},V_{l-1})\), and \((V_{j},V_{i+1})\). As before, the sub tours \((V_{k+1},\dots,V_{l-1})\) and \((V_{j},\dots,V_{l-1})\) are reversed. The stringing procedure can be seen as the reverse of the unstringing procedure and consists of four types of insertions as follows:  
* Stringing Type I: Assuming \(V_{k}\
eq V_{i}\) and \(V_{k}\
eq V_{j}\). The insertion of \(V_{x}\) results in the deletion of arcs \((V_{i},V_{i+1})\), \((V_{j},V_{j-1})\) and \((V_{k},V_{k+1})\) and the insertion of arcs \((V_{i},V_{x})\), \((V_{x},V_{j})\), \((V_{i+1},V_{k})\) and \((V_{j+1},V_{k+1})\). Additionally, the sub tours \((V_{i+1},\dots,V_{j})\) and \((V_{j+1},\dots,V_{k})\) are reversed. * Stringing Type II: assuming \(V_{k}\
eq V_{j}\), \(V_{k}\
eq V_{j+1}\), \(V_{l}\
eq V_{i}\), and \(V_{l}\
eq V_{i+1}\). The insertion of \(V_{x}\) results in the deletion of arcs \((V_{i},V_{i+1})\), \((V_{l-1},V_{l})\), \((V_{j},V_{j+1})\) and \((V_{k-1},V_{k})\) and the insertion of arcs \((V_{i},V_{x})\), \((V_{x},V_{j})\), \((V_{l},V_{j+1})\), \((V_{k-1},V_{l-1})\) and \((V_{i+1},V_{k})\). As before, the sub tours \((V_{i+1},\dots,V_{l-1})\) and \((V_{l},\dots,V_{j})\) are reversed. * Stringing Type III: this stringing type can be seen as the inverse of Stringing Type I. Notice that when node \(V_{x}\) is inserted between \(V_{i}\) and \(V_{j}\), the sub tour of nodes is rearranged in such a way that almost the entire sequence is reversed. The objective is to explore other promising regions of the search space. As in Stringing Type I, assume that \(V_{k}\
eq V_{i}\) and \(V_{k}\
eq V_{j}\). The insertion of \(V_{x}\) results in the deletion of arcs \((V_{i-1},V_{i})\), \((V_{j-1},V_{j})\) and \((V_{k-1},V_{k})\) and the insertion of arcs \((V_{i},V_{x})\), \((V_{x},V_{j})\), \((V_{k},V_{j-1})\) and \((V_{k-1},V_{i-1})\). As before, the sub tours \((V_{i},\dots,V_{j-1})\) and \((V_{k},\dots,V_{i-1})\) are reversed. * Stringing Type IV: similarly, this type of insertion can be seen as the reverse of Stringing Type II. As in Stringing Type II, assume that \(V_{k}\
eq V_{j}\), \(V_{k}\
eq V_{j+1}\), \(V_{l}\
eq V_{i}\) and \(V_{l}\
eq V_{i+1}\). The insertion of \(V_{x}\) results in the deletion of arcs \((V_{i-1},V_{i})\), \((V_{l},V_{l+1})\), \((V_{j-1},V_{j})\) and \((V_{k},V_{k+1})\) and the insertion of arcs \((V_{i},V_{x})\), \((V_{x},V_{j})\), \((V_{i-1},V_{l})\), \((V_{l+1},V_{k+1})\) and \((V_{k},V_{j-1})\). As above, the sub tours \((V_{i},\dots,V_{l})\) and \((V_{l+1},\dots,V_{j-1})\) are reversed.