## 4 The ISFLA for Solving the Simultaneous DCBAP
### Discrete Operators with Self-Adaptive Jump
\[NSE_{i,o}(t)=\sum_{i=1}^{D}[X_{o,k}(t)\approx X_{i,k}(t)] \tag{22}\]  
\[\omega_{i}(t){=}\ \mathrm{D}-NSE_{i,o}(t)-2 \tag{23}\]  
\[BR1_{i}(t){=}\left\{ \begin{array}{cl}\frac{\omega_{i}(t)}{D},&\mathrm{if} \ \ NSE_{i,o}(t)<D-2\\ 0,&\mathrm{if}\ \ NSE_{i,o}(t)\geq D-2\end{array} \right. \tag{24}\]  
In Equation (22), the symbol \"\(\approx\)\" is the comparing operator, and compares two elements with the same position in \(X_{o}(t)\) and \(X_{i}(t)\), and this operator works as follows. \[X_{o,k}(t)\approx X_{i,k}(t){=}\left\{ \begin{array}{cl}1,&\mathrm{if}\ \ X_{o,k}(t)=X_{i,k}(t)\\ 0,&\mathrm{if}\ \ X_{o,k}(t)\
eq X_{i,k}(t)\end{array} \right. \tag{25}\]  
Given \(X_{o}(t)\) = [4,3,2,1,5,6] and \(X_{i}(t)\) = [1,2,3,4,5,6], we can derive \(NSE_{i,o}(t)\) = 2, \(\omega_{i}(t)\) = 2, and the \(BR1_{i}(t)\) = 1/3. Finally, the \(RV_{i}(t)\) and \(D_{i}(t)\) can be derived by Equations (18) and (21), respectively, and Equation (26) can be used to derive the next position of the frog \(i\). \[X_{i}(t+1)=X_{i}(t)D_{i}(t) \tag{26}\]  
In Equation (26), the symbol \"\(6\)\" is as the leaping operator, which is used to determine the next position of a frog. It works on the basis of the following 4 steps:  
1. the operator takes the first non-zero value out of the \(D_{i}(t)\) and replaces the value at the same position in \(X_{i}(t)\),
2. the replaced value takes the position of the non-zero value in \(X_{i}(t)\),
3. repeat the first and second steps until there are no non-zero values in \(D_{i}(t)\),
4. the operator copies the current \(X_{i}(t)\) as \(X_{i}(t+1)\). \[X_{i}(t+1)=X_{i}(t)\ \delta D_{i}(t)=[1,2,3,4,5,6]\ \delta[0,3,0,0,0,0]=[1,3,2,4,5,6]\] (27)  
Take Equation (27) as an example: the operator first takes the non-zero value \"\(3\)\" from \(D_{i}(t)\) and replaces the \"\(2\)\" in \(X_{i}(t)\). Then, the replaced value \"\(2\)\" takes the place of the \"\(3\)\" in \(X_{i}(t)\). Because there are no non-zero elements in \(D_{i}(t)\), the next position of the frog i will be \(X_{i}(t+1)\) = [1,3,2,4,5,6]. This example shows that a non-zero element in \(D_{i}(t)\) can exchange two elements in \(X_{i}(t)\). In terms of distance, the two frogs \(i\) and o are now considered to be closer due to their having four shared position elements, and the frog i possesses the following data: \(NSE_{i,o}(t)\) = 4, \(\omega_{i}(t)\) = 0, \(BR1_{i}(t)\) = 0, and \(RV_{i}(t)\) = [0,0,0,0,0], which will result in \(D_{i}(t)\) = [0,0,0,0,0] being used to stop frog \(i\) jumping directly to the target frog \(o\). In addition to the self-adaptive leap mechanism, the ISFLA also makes it possible to include the following novel features:  
* **Direct-jump prevention:** The direct-jump prevention will stop the generation of binary values of 1 for the \(RV_{i}(t)\) of frog \(i\) in order to prevent it jumping directly to the target frog, as this would waste one local search. This will happen if the frog i is satisfied with the condition \(NSE_{i,o}(t)\geq D-2\). In Equation (22), the term \(-2\) refers to the direct-jump prevention, which prevents frog i from jumping directly to the target frog \(o\). * **Neighborhood jump:** However, if \(BR1_{i}(t)\
eq 0\), there is always a chance for a frog to jump to the target frog directly. Thus, we let the ISFLA count the total number of binary values of 1 generatedso far for the \(RV_{i}(t)\) of frog \(i\); if the threshold \(\omega_{i}\) is reached, instead of allowing a direct jump, the swap(p1, p2) operator, which switches two randomly selected elements in a frog's position vector, will be used to perform a neighborhood search. The neighborhood search avoids wasting local searches for frogs due to the imposition of direct-jump prevention. * **Push jump:** For a frog \(i\) freed from direct-jump prevention but idling at its current position, which happens when \(\sum_{k=1}^{D}RV_{i,k}(t)=0\), the ISFLA will introduce one random binary value 1 into \(RV_{i}(t)\) in order to push jump this frog.