# A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling
## 1 Introduction
After that, CWA used the chaotic maps to avoid being stuck into local minima and accelerate convergence speed by assisting two other operators: cross operator and reversal-insertion to improve its exploration capability. Ultimately, CWA used the local search strategy to improve the quality of the best-so-far solution to improve the exploitation capability of CWA. This algorithm was observed using various benchmarks and compared with various optimization algorithms to check its superiority. Further, a new discrete multiobjective approach based on the fireworks algorithm (DMOFWA) has been recently proposed for solving the multi-objective flow shop scheduling problem with sequence-dependent setup times (MOFSP-SDST); this approach was abbreviately called DMOFWA [37]. Inside this approach, two various machine learning techniques have been integrated: The first one called opposition-based learning was used to improve the exploration operator of the standard algorithm to avert entrapment into local minima, and the second one is the clustering analysis and was used to cluster fireworks individuals. To overcome expensive computational costs and local minima problems that might suffer from most of the above-described algorithms, we developed a novel discrete optimization algorithm to tackle the PFSSP in a reasonable time compared to some existing techniques. Recently, a new optimization algorithm, namely generalized optimization algorithm (GNDO), based on the normal distribution theory, has been developed by [PERSON] [38] for tackling the parameter extraction problem of the single diode and double diode photovoltaic models. Due to its high ability to estimate the parameter values that minimize the error rate between the measured I-V curve and the estimated I-V curve, in this paper, we try to observe its performance for tackling the PFSSP. In order to make GNDO applicable to the PFSSP classified as a discrete problem contradicted by the continuous problems tackled using the standard GNDO, the largest ranked value (LRV) rule is used to convert those continuous values into job permutations adequate to solve the PFSSP. Furthermore, this discrete GNDO using the LRV rule is integrated with a local search strategy to avoid being stuck into local minima for reaching better outcomes; this version is named a hybrid GNDO(GNDO). In another attempt to improve the quality of HGNDO, it was integrated with the swap mutation operator applied on the best-so-far solution as another attempt to promote the exploitation capability for reaching better outcomes; this version was abbreviated as HIGNDO. Finally, to improve the quality of the solutions, the local search strategy is improved using the scramble mutation operator and then integrated with HIGNDO to produce a new version named IHGNDO. The proposed algorithms, HGNDO, HIGNDO, and IHGNDO, are verified using 41 well-known instances widely used in the literature and compared with a number of the recent well-established algorithms to verify their efficacy using various performance metrics. The experimental results affirm the superiority of IHGNDO and HIGNDO over the other algorithms in terms of standard deviation, computational cost, and makespan. Generally, our contributions in this work include the following:  
* Develop GNDO using the LRV rule for PFSSP. * Improve GNDO using the swap mutation operator to avoid being stuck into local minima. * Enhance the local search strategy using the scramble mutation operator for accelerating the convergence speed toward the near-optimal solution. * Integrate the improved local search strategy and the standard one with the improved GNDO and GNDO for tackling the PFSSP. * The experimental findings show that IHGNDO and HIGNDO are better in terms of standard deviation and computational cost and final accuracy.