## 1 Introduction
(2018) minimized the total flow-time of PFSSP by using a Biased Random-Key Genetic Algorithm (BRKGA). Their proposed technique used a shaking strategy to perturb individuals from the elite population, while resetting the remainder of the population. The authors validated their proposed approach by its application to 120 benchmark problems and comparing the results to those obtained by Iterative Greedy Search and Iterated Local Search. The results showed that the proposed approach was more efficient and was able to identify new bounds as well as optimal solutions for the problems. [PERSON] et al. (2018) used the iterated Greedy (IG) search algorithm to solve the distributed permutation flow shop scheduling problems. The authors improved the initialization, construction, and deconstruction procedures, and improved the local search (ILS) to obtain better results. A big advantage of the proposed technique was that it required very little problem-specific knowledge to be implemented. [PERSON] et al. (2018) proposed three constructive heuristics, DLR (inspired by LR heuristic), DNEH (inspired by NEH heuristic), a hybrid DLR-DNEH heuristic, and four metaheuristics--Discrete Artificial Bee Colony (DABC), Scatter Search (SS), ILS, & IG--for minimizing the total flowtime criterion of the distributed permutation flow shop scheduling problem (DPFSP). Both the heuristics and metaheuristic were able to significantly improve upon the existing results in literature. Their results also showed that the trajectory-based metaheuristics were considerably better than the population-based metaheuristics. Though ample work is done in solving PFSSP, the current techniques only provide a single optimal schedule upon execution. Furthermore, real-life operators and schedulers have preferences that cannot be modeled and added to the fitness function of optimization algorithms. The single optimal schedule found using these techniques might theoretically satisfy the objective function. However, it might not be able to satisfy the aforementionedcomplexities. Although multimodal optimization provides multiple similar solutions for the same problem under consideration, it cannot handle complex situations such as machine breakdowns, delayed supply chain, etc., without changing the fitness function and constraints. However, having multiple similar solutions for the same problem could make it possible to provide opportunity for the operators to have the flexibility to handle emergency using their experience, while waiting to solve the underlying problem. Therefore, it is important to obtain multiple optimal schedules that satisfy the objective function, i.e., perform multimodal optimization (MMO). Over the years, many different techniques were developed to accomplish MMO, such as niching techniques [10; 11; 12; 13; 14; 15; 16], differential evolutionary strategies [17; 18; 19; 20; 21; 22; 23; 24], use of GA [25; 26; 27; 28] etc. However, these techniques were only tested on benchmark mathematical problems. Furthermore, the aim of these techniques was to find both global and local optima, which increases their computational complexity. Additionally, it is possible for local optima to have much worse objective value than global optima, which is undesirable in scheduling problems. Therefore, in this study, a new algorithm is proposed for the MMO of PFSSP. This is accomplished by using the k-means clustering algorithm with GA. In each iteration of the proposed algorithm, first, solutions are clustered together, using k-means clustering algorithm. Next, the operators of GA are utilized within each cluster to generate solutions for the next generation. Various techniques used by [PERSON] et al. were also modified and utilized for the MMO of PFSSP and the performance of the algorithms was tested on benchmark PFSSP. The rest of the paper is organized as follows. Section 2 presents information about the modified GA operators used in the proposed algorithm to represent a solution and create new solutions during the iterations of the algorithm.