# A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling
## 1 Introduction  
The permutation flow shop scheduling problem (PFSSP) is a critical problem that needs to be solved accurately and effectively to minimize the makespan criteria. The solution to this problem involves finding the near-optimal permutation of n jobs to be processed in a set of m machines sequentially that will minimize the makespan required, even completing the last job in the last machine [1]. This problem has significant utilization in several fields, especially in industries such as computing designs, procurement, and information processing. According to its significant effectiveness and its nature, which is normally classified as nondeterministic polynomial time (NP)-hard [1, 2, 3, 4, 5, 6], several techniques of exact, heuristic, and meta-heuristic properties have been extensively employed for solving this problem. Some of them will be surveyed in the rest of this section. Exact methods such as linear programming [7] and branch and bound [8] could fulfill the optimal value for the small-scale problem, but for medium-scale and large-scale problems, their performance degrades significantly, in addition to increasing exponentially thecomputational cost. Therefore, the heuristics algorithms have been designed to overcome this expensive computational cost and high dimensionality. Involving the heuristic algorithms, the Nawaz-Enscore-Ham (NEH) algorithm employed by [PERSON] et al. [9] for solving PFSSP could be the most effective heuristic algorithm, and their results are comparable with the meta-heuristic algorithms [10; 11; 12; 13], which are being used for solving several optimization problems in a reasonable time. Broadly speaking, the image segmentation problem is an indispensable process in image processing fields, so several image segmentation methods have been suggested such as clustering, fractal-wavelet techniques [14; 15; 16; 17; 18; 19; 20], region growing, and thresholding; Among those techniques, the threshold-based segmentation technique is the most effective due to the metaheuristic algorithms which could segment the images based on this technique with high accuracy [21]. The particle swarm optimization (PSO)-based memetic algorithm (MA) [22], namely PSOMA, has been proposed for tackling the PFSSP as an attempt to find the near-optimal job permutation that minimizes the maximum completion time. In detail, to adapt the PSOMA for solving the PFSSP, the authors used a ranked order rule to convert the continuous values produced by the standard algorithms into discrete ones. In addition, to improve the quality and diversity of the initialized solutions, the NEH algorithm has been used. Furthermore, to balance between the exploration and exploitation operators, a local search operator has been used to be applied on some solutions selected using the roulette wheel mechanism with a specific probability. Ultimately, to avoid being stuck into local minima, PSOMA used the simulated annealing with multiple neighborhood search strategies. It is worth mentioning that the local search has been used with the PSO for tackling several optimization problems, and this confirms that the local search has a significant influence on the performance after integration; some of those works are comprehensive learning PSO with a local search for multimodal functions [23], PSO with local search [24], and many others [2; 25; 26; 27; 28; 29]. The cuckoo search-based memetic algorithm (HCS) [30] has been adapted using the largest ranked values rule for tackling the PFSSP. Besides, HCS used the NEH algorithm to initialize the population to fulfill better quality and diversity. Furthermore, this algorithm used a fast local search to accelerate the convergence speed in an attempt to improve its exploitation algorithm. This algorithm was compared with a number of optimization algorithms: hybrid genetic algorithm (HGA), particle swarm optimization with variable neighborhood search, and the differential-evolution-based hybrid algorithm (HDE) on four benchmark instances to see its efficacy.