## 4 Proposed Hybrid PSO (PHPSO) for NWFSSP
### Solution Representation  
Solution representation is one of the most important issues in designing a PSO algorithm. To represent the solution, a job-permutation-based encoding scheme [2] has been used very often by earlier researchers for solving NWFSSP. However, as the position of particles in PSO is a continuous character, a standard encoding scheme of PSO cannot be adopted directly for solving NWFSSP. PSO canbe effectively applied by considering dimension size as \"n\" for representing \"n\" jobs, and related particle information is represented as \(X_{i}=\{x_{1},\,x_{2},\,x_{3},\,\ldots,\,x_{n}\}\). As permutations of jobs cannot be presented with the particle alone, it is necessary to find suitable mapping between the job sequence and the position of particles in PSO. So, in this paper, the Ranked-Order-Value (ROV) rule based on the random key value [35; 36] is used to determine the permutation implied by the position values \(x_{ij}\) of particle \(X_{i}\). The ROV rule converts the continuous position values of particle to a discrete job permutation. This enables one to convert the continuous nature of PSO algorithms to apply to the determination of the discrete nature of problems, such as sequencing, which in turn evaluates the performance of a particle. Moreover, permutations of jobs are constructed by considering a job index, which is the rank of each position value of a particle. The ROV rule used in PHPSO handles the particle with the smallest position value first and assigns a rank value i.e., 1, and that which is observed as the smallest is assigned to that position of the particle. In the case of two or more particles with same position values, the position with the smallest dimension number is given priority and assigned a rank value first. The remaining position values are incremented by 1 and subsequently assigned the next rank values as per the dimension number. Then, the second smallest position value will be handled in the same manner. Thus, the position information of a particle is converted to corresponding job permutation \(\sigma_{ij}=[j_{1},j_{2},j_{3},\,\ldots,j_{n}]\). To demonstrate the scheme of the ROV rule, we provide a simple example in Table 1.  
Let us consider that the random position values of particle (for n = 5) observed initially are \(X_{i}=\{5.45,\,4.22,\,4.37,\,5.47,\,4.37\}\). As \(x_{1,2}=4.22\) has the smallest position value of the particle, \(x_{1,2}\) is prioritized first by being assigned the rank value of 1. Next, two particles \(x_{1,3}\) and \(x_{1,5}\) have equal position value i.e., 4.37. Yet, index of \(x_{1,3}\) is smaller as compared to \(x_{1,5}\). So, \(x_{1,3}\) is assigned the next rank value of 2 and \(x_{1,5}\) is incremented to rank value 3. Finally, rank values 4 and 5 are respectively assigned to \(x_{1,1}\) and \(x_{1,4}\). Thus, the job permutation \(\sigma_{ij}=[4,\,1,\,2,\,5,\,3]\) is obtained considering the position information value of each particle and the corresponding rank assignment based on the ROV rule. In the Proposed Hybrid PSO, job permutation based local search approaches are applied rather than a direct consideration of the particle's position information. So, it is necessary to convert the particle's position information to a corresponding job permutation as per the ROV rule when a local search is completed. Because of the simple mechanism of the ROV rule, adjustment for a new particle position is very easy. Local search methods using the position information are handled in the same way as the process adopted for job permutation. For example, in Table 2, if the SWAP operator [2] is used as a local search operator for job permutation; the swapping of job 2 and job 4 corresponds to the swap of position values 4.22 and 4.37.