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Preventive Maintenance (PM) is a periodic maintenance strategy that has great results for devices in extending their lives, increasing productivity, and, most importantly, helping to avoid unexpected breakdowns and their costly consequences. Preventive maintenance scheduling (PMS) is determining the time for carrying out PM, and it represents a sensitive issue in terms of impact on production if the time for the PM process is not optimally distributed. This study employs hybrid heuristic methods, integrating Genetic Algorithm (GA) and Tabu Search (TS), to address the PMS problem. Notably, the search for an optimal solution remained elusive with GA alone until the inclusion of TS. The resultant optimal solution is achieved swiftly, surpassing the time benchmarks set by conventional methods like integer programming and nonlinear integer programming. A comparison with a published article that used metaheuristics was also applied in order to evaluate the effectiveness of the proposed hybrid approach in terms of solution quality and convergence speed. Moreover, sensitivity analysis underscores the robustness and efficacy of the hybrid approach, consistently yielding optimal solutions across diverse scenarios. The schedule created exceeds standards set by waterworks experts, yielding significant water and electricity surpluses—16.6% and 12.1%, respectively—while simultaneously matching or surpassing total production levels. This method can be used for power plants in private or public sectors to generate an optimal PMS, save money, and avoid water or electricity cuts. In summary, this hybrid approach offers an efficient and effective solution for optimizing PMS, presenting opportunities for enhancement across various industries. [ABSTRACT FROM AUTHOR]

A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if g c d (B , A / B) = 1 (resp. g c d u (B , A / B) = 1) , where g c d u (B , A / B) denotes the greatest common unitary divisor of B and A / B . We denote by σ * * (A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F 2 if the sum of all bi-unitary monic divisors of σ * * (A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F 2 . We prove the nonexistence of odd bi-unitary superperfect polynomials over F 2 . [ABSTRACT FROM AUTHOR]

Preventive maintenance (PM) is a maintenance program with activities created at a determined interval or according to certain principles, designed to reduce the likelihood of failure or deterioration of item performance. This aims to improve overall reliability and system availability. In this research, a preventive maintenance schedule (PMS) was designed for electricity and desalination of water in power plants, subject to meeting relevant constraints. The proposed methodology is used to generate a PMS for the boilers, turbines, and distillers. A nonlinear integer programming (NLIP) model was employed to address this problem. The results of the proposed method were compared with the PMS for a power station in Kuwait. The results were better in terms of the volume of production and in terms of the gap between the available production and demand in order to continue providing consumers with electricity and water without a shortage in the event of a breakdown in equipment. It produces an improvement of 12.12% and 16.58% respectively, for water and electricity. Furthermore, the sensitivity and robustness of the proposed method were analysed by increasing the maintenance duration for some equipment, increasing the demand, and adding various additional conditions. In addition, a comparison of additional conditions with a binary problem method in terms of computer time for the search for an optimal solution was carried out, where the model provided an optimal solution in a reasonable time. Among the most important benefits that the user can obtain for this technique are extending the life of the equipment, increasing efficiency, and reducing expenses. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to use the notion of M -sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type E 6 (K) for fields of characteristic two, Beitrag Zur Algebra und Geometry58 (2017) 529–534.] and using Levi components and unipotent radical subgroups of E 6 (K) to give an elementary and self-contained construction of the stabilizer of two dimensional vector space of 27-dimensional module of type E 6 over a field of characteristic two. This stabilizer is in fact the maximal parabolic subgroup P 2 of E 6 or a Borel subgroup. This construction is elementary on the account that we use not more than little naive linear algebra notions. For more information one can see [M. E. Aschbacher, The 27-dimensional module for E 6 , 1, Invent. Math.89 (1987) 159–195; M. E. Aschbacher, The 27-dimensional module for E 6 , II, J. London Math. Soc.37 (1988) 275–293; M. Bani-Ata, On Lie algebras of type F 4 and D 4 over finite fields of characteristic two, Preprint; B. Cooperstein, Subgroups of the group E 6 (q) which are generated by root-subgroups, J. Algebra46 (1977) 355–388.]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study certain geometric properties of generalized quadrangle (Ω , 𝔏) of type O 6 − (2). We define certain root elements which generate a Lie algebra of type E 6 (K) for fields K of characteristic two. The construction will be mainly based on the geometric properties of the generalized quadrangle (Ω , 𝔏). In fact, we will explicity construct a Chevalley base of this Lie algebra. [ABSTRACT FROM AUTHOR]